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(download) - view tree Revision 1.26, Thu Aug 15 00:54:45 2002 UTC (10 years, 9 months ago) by bcfuchs Branch: MAIN Changes since 1.25: +4 -12 lines hy&longs;\. </s><s>[:rom:num:lc:cap] remove bound |
<?xml version="1.0"?> <!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info> <author>Fabri, Honoré</author> <title>Tractatus physicus de motu locali</title> <date>1646</date> <place>Lyon</place> <translator></translator> <lang>la</lang> <cvs_file>fabri_tract_01_la_1646</cvs_file> <cvs_version></cvs_version> <locator>0000000026.xml</locator> </info> <text> <front> </front> <body> <chap> <pb/><p type="main"> <s><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 type="main"> <s><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 type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Ex prælectionibus<emph.end type="italics"/> R. P. HONORATI FABR<gap/>, <lb/><emph type="italics"/>Societatis<emph.end type="italics"/> IESV.<emph.end type="center"/></s></p><figure></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>LVGDVNI,<emph.end type="italics"/><lb/>Apud IOANNEM CHAMPION, <lb/>in foro Cambij.<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>M. </s> <s>D C. XLVI.<emph.end type="italics"/><lb/>Cum Priuilegio Regis, & Approbatione Doctorum.<emph.end type="center"/></s></p><pb/><figure></figure><p type="main"> <s><emph type="center"/>AMPLISSIMO, <lb/>NOBILISSIMOQVE DOMINO,<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/>PETRVS MOVSNERIVS,<emph.end type="center"/></s></p><p type="main"> <s><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: non muta&longs;ti merita, non mu­<lb/>tabo officia: multos non expo&longs;cam Patronos, qui <lb/>iam omnium optimum, & meriti&longs;simum habeo; 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: Nec leuem toti rei li­<lb/>terariæ iniuriam inferrem, &longs;i alium illi, quàm li-<emph.end type="italics"/><pb/><emph type="italics"/>teratum Mecænatem accer&longs;erem: & 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: Tu prudens Intelli­<lb/>gentia, huic orbi &longs;emper a&longs;si&longs;tis; 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"/></s></p><pb/><figure></figure><p type="main"> <s><emph type="center"/>PRÆFATIO.<emph.end type="center"/></s></p><p type="main"> <s>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: Multi &longs;anè hactenus in hac mate­<lb/>ria feliciter de&longs;udarunt; & quidem præ cæteris magnus <lb/>ille Galileus, qui mirificâ, & ferè diuinâ ingenijacie, <lb/>motum localem eò perduxit, quò mortalium nemo per­<lb/>duxerat; quia tamen multa omi&longs;it, quæ ad motum &longs;pe­<lb/>ctant, vt nemo ne&longs;cit; 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; vt Phy­<lb/>&longs;icæ con&longs;ulamus, aliam inimus viam: Geometriam qui­<lb/>dem adhibemus, ad explicandas, exponenda&longs;que præ­<lb/>dictas illas proportiones, quæ motibus in&longs;unt; &longs;ed effe­<lb/>ctus illos prædictis proportionibus affixos ad principia <lb/>Phy&longs;ica reducimus; id e&longs;t, cùm &longs;upponamus quòd &longs;int, <lb/>propter quid &longs;int demon&longs;tramus: in votis erat motus <lb/>omnes vno volumine complecti; id e&longs;t effectus omnes <lb/>cuiu&longs;uis potentiæ motricis; tres enim agno&longs;cimus hu­<lb/>iu&longs;modi potentias: primam naturalem voco, quæ e&longs;t <lb/>grauium: alteram animalem, quæ e&longs;t animantium: ter­<lb/>tiam mediam, quæ ten&longs;orum e&longs;t vel compre&longs;&longs;orum: In <lb/>hoc tractatu tùm à motu progre&longs;&longs;iuo animantium, tùm <lb/>ab alijs motibus, qui in animato corpore, neruorum & <pb/>mu&longs;culorum opera fiunt, penitus ab&longs;tinemus; 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; 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>Fortè aliquis poten­<lb/>tias machanicas de&longs;ideraret, lineas, motus, & cæle&longs;tes <lb/>&longs;piras; &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>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>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: omitto etiam quæ in me quidam iniquè <lb/>certè rerum æ&longs;timatores iactarunt: reponere po&longs;&longs;em cum <lb/>fænore; &longs;ed nos talem con&longs;uetudinem non habemus; 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/><p type="main"> <s>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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>SYNOPSIS LIBRORVM<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>huius tractatus.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="table"> <s>TABELLE WAR HIER</s></p><pb/><figure></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>SYNOPSIS AMPLIOR.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De Impetu.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. IMPETVS e&longs;t qualitas exigens motum &longs;ui &longs;ubiecti: <lb/>datur impetus; quia non pote&longs;t e&longs;&longs;e alia cau&longs;a exi­<lb/>gitiua motus: adde quòd, potentia motrix e&longs;t acti­<lb/>ua; igitur aliquid producit, &longs;ed non aliud quàm <lb/>impetum, vt con&longs;tat ex dictis de motu: 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: non e&longs;t modus, <lb/>quia di&longs;tinguitur ab effectu &longs;uo formali &longs;ecundario: impetus non <lb/>producitur in eo mobili, quod moueri non pote&longs;t à potentia mo­<lb/>trice applicata: & produci tantùm pote&longs;t, vel in omni parte, vel <lb/>in nulla; alioquin e&longs;&longs;et fru&longs;trà; & gratis ponitur ne&longs;cio quis impe­<lb/>tus inefficax. </s></p><p type="main"> <s>2. Primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, ne &longs;imulim­<lb/>petus &longs;it in duobus locis. </s> <s>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: agit tan­<lb/>tùm ad extra, vt tollat impedimentum: hinc, cùm pro diuer&longs;a <lb/>applicatione &longs;it diuer&longs;um impedimentum, modò plùs, modò minùs <lb/>agit; maximè verò, cum maximum e&longs;t impedimentum: 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 type="main"> <s>3. Impetus inten&longs;us producere pote&longs;t remi&longs;&longs;um, minotis mobi­<lb/>lis in maiore; & 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/>ta applicatio, cum maximo impedimento, quod reuerâ tunc e&longs;t, <lb/>cùm linea directionis connectit centra grauitatis vtriu&longs;que. </s> <s>Datur <lb/>impetus alio impetu perfectior, & imperfectior, &longs;ine quo non po­<lb/>te&longs;t explicari natura vectis: 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 type="main"> <s>4. Propagatur impetus vniformiter tantùm, cùm omnes partes <lb/>corporis mouentur moctu recto æquali: ibi enim e&longs;t æqualis cau&longs;a, <lb/>vbi e&longs;t æqualis effectus: in motu circulari applicata potentia cen­<lb/>tro vectis, producitur æqualis perfectionis versùs circunferentiam, <lb/>& inæqualis numerus; applicata verò potentia circunferentiæ, pro­<lb/>ducitur æqualis numerus, &longs;ed inæqualis perfectionis versùs cen­<lb/>trum; quia potentia non pote&longs;t producere immediatè perfectiorem, <lb/>& imperfectiorem in infinitum: eadem potentia nece&longs;&longs;aria æquali­<lb/>bus temporibus, & ii&longs;dem cireun&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 type="main"> <s>5. Impetus innatus e&longs;t tantùm determinatus ad lineam perpen­<lb/>dicularem deor&longs;um; alioquin &longs;i ad aliam determinari po&longs;&longs;et, primo <lb/>e&longs;&longs;et æqualis motus per inclinatam, & perpendicularem; corpus <lb/>graue mi&longs;&longs;um per lineam inclinatam ab eo non declinaret; imò im­<lb/>petus &longs;emel productus (&longs;i liberum e&longs;&longs;et medium) non de&longs;trueretur: <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 type="main"> <s>6. Impetus indifferens determinatur ad lineam multis modis: <lb/>primò, à potentia motrice: &longs;ecundò, ab impetu: tertiò, ab alio impe­<lb/>tu concurrente; quartò, ab obice occurrente: 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>Impetus e&longs;t capax inten&longs;ionis; <lb/>quia aliquando de&longs;truitur ex parte: eius exten&longs;io commen&longs;uratur <lb/>exten&longs;ioni mobilis; 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 type="main"> <s>7. Impetus non e&longs;t contrarius alteri ratione entitatis; quia qui­<lb/>libet cum quolibet in eodem &longs;ubiecto coëxi&longs;tere pote&longs;t: pugnat <lb/>tamen vnus cum alio ratione determinationis: hinc vnus impetus <lb/>pugnat cum alio ratione lineæ motus: hinc vnus videtur de&longs;trui ab <pb/>alio; quanquam impetus tantùm de&longs;truitur, cùm e&longs;t fru&longs;trà: 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 type="main"> <s>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; <lb/>&longs;i ad eandem, nihil impetus de&longs;truitur, &longs;ed e&longs;t duplò velocior mo­<lb/>tus; &longs;i ad diuer&longs;as, vel &longs;unt oppo&longs;itæ ex diametro, vel concurrentes <lb/>faciunt angulum; &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>Impetus innatus nunquam de&longs;truitur: 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>Porrò <lb/>nunquam de&longs;truitur; 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>Sed de grauitate aliàs. </s></p><figure></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu naturali deor&longs;um.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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>Eius finis <lb/>e&longs;t globi terre&longs;tris compactio, &c. </s> <s>E&longs;t autem motus naturalis ab <lb/>impetu: primò, quia eius acceleratio &longs;ine impetu explicari non po­<lb/>te&longs;t: &longs;ecundò, quia, cùm graue deor&longs;um cadens imprimat impetum <lb/>in corpore occcurrente, certè debet habere impetum: nec alio ar­<lb/>gumento mihi probabis, Solem e&longs;&longs;e lucidum, ignem calidum. </s></p><p type="main"> <s>2. Motus hic e&longs;t naturaliter acceleratus, &longs;cilicet, ab intrin&longs;eco; <lb/>patet experientiâ. </s> <s>Ratio e&longs;t: 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;atia; certè &longs;ecundo in&longs;tanti producit <lb/>nouum impetum: idem dica de tertio, quarto, &c. </s> <s>igitur cre&longs;cit <lb/>cau&longs;a motus; igitur & motus: quæ ratio clari&longs;&longs;ima e&longs;t: hinc æquali­<lb/>bus remporibns æqualia acquiruntur velocitatis momenta; quia <lb/>cau&longs;a nece&longs;&longs;aria æqualibus temporibus, æqualem effectum produ­<lb/>cit: quid clarius? </s></p><p type="main"> <s>3. Hinc non pote&longs;t cre&longs;cere hic impetus &longs;ecundùm porportio-<pb/>nem duplicatam temporum, cùm cre&longs;cat &longs;ecundùm proportionem <lb/>temporum, etïam ex mente Galilei: cre&longs;eit autem velocitas, vt im­<lb/>petus; effectus, &longs;cilicet, vt cau&longs;a: idem dico de motu, ratione velo­<lb/>citatis; 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 cou&longs;ideretur crementum in in&longs;tan­<lb/>&longs;tantibus, 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 type="main"> <s>4. Hoc autem facilè pote&longs;t <expan abbr="demõ&longs;trari">demon&longs;trari</expan>: 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: 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 type="main"> <s>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 tempotis <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>quam Galileus excogitauit, vt &longs;ine &longs;crupulo hæc a&longs;­<lb/>&longs;umi po&longs;&longs;it: hinc &longs;patia &longs;unt ferè vt temporum quadrata: dixi, ferè: <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. </s></p><p type="main"> <s>6. Hinc ratio euidens maioris ictus inflicti à corpore graui, <lb/>cùm ex maiori altitudine cadit. </s> <s>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>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; quia motus fit &longs;emper velocior: cùm graue <lb/>de&longs;cendit in medio, quod re&longs;i&longs;tit, minùs aecuratè &longs;eruantur prædi­<lb/>ctæ proportiones, quæ in vacuo modico accurati&longs;&longs;imè &longs;eruaren­<lb/>tur. </s></p><p type="main"> <s>7. Re&longs;i&longs;tentia medij non e&longs;t propter vllam formam improportio­<lb/>nat am, qua&longs;i verò impetus &longs;it forma improportionata aëri: &longs;ed in <pb/>duobus præ&longs;ertim con&longs;i&longs;tit; primò, eò quòd medium detrahat ali­<lb/>quid grauitationis corporis grauis; &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; 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>de quo <lb/>aliàs. </s></p><p type="main"> <s>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>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>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 type="main"> <s>9. Figura corporis grauis deor&longs;um cadentis motum vel retardat <lb/>vel accelerat; retardat quidem, &longs;i plures partes medij amouendæ <lb/>&longs;unt vel pauciores velociori motu; accelerat è contratio: 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>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 type="main"> <s>10. Duo globi, vel cubi eiu&longs;dem materiæ æquè velociter de&longs;­<lb/>cendunt: 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; certè ratio compo&longs;ita vtriu&longs;que erit eadem in vtroque cu­<lb/>bo: igitur æqualiter de&longs;cendet vterque. </s></p><p type="main"> <s>11. Si tamen &longs;int diuer&longs;æ materiæ, hand 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>Corpu&longs;cula etiam ex graui&longs;&longs;ima ma­<lb/>teria tardi&longs;&longs;imè de&longs;cendunt: tum, quia à filamentis illis, quibus par­<lb/>tes aëris implicantur, facilè detinentur; analogiam habes in lapil­<lb/>lo, qui ab araneæ tela intercipitur: tum, quia, cùm lati&longs;&longs;imam ali­<lb/>quando habeant &longs;uperficiem pro modica mole, minimam habent <pb/><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 type="main"> <s>12. Singulis in&longs;tantibus motus naturaliter accelerati cre&longs;cit <lb/>re&longs;i&longs;tentia; quia, cùm motus cre&longs;cat, æqualibus temporibus, plures <lb/>partes medij occurrunt; cre&longs;cunt tamen vires in eadem proportio­<lb/>ne, &longs;cilicet, impetus: igitur non mutatur progre&longs;&longs;io motus. </s> <s>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></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu violento &longs;ur&longs;um.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. MOtus violentus &longs;ur&longs;um vulgò dicitur e&longs;&longs;e à principio ex­<lb/>trin&longs;eco. </s> <s>Triplici modo accidere pote&longs;t: primò, &longs;i reuerà <lb/>imprimatur impetus ab extrin&longs;eco, vt, cùm mitto lapidem &longs;ur&longs;um: <lb/>&longs;ecundò, &longs;i corpus deor&longs;um cadens deinde reflectatur &longs;ur&longs;um; 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: tertiò, &longs;i terra vtrinque e&longs;&longs;et peruia; nam lapis haud <lb/>dubiè non &longs;i&longs;teret in centro, &longs;altem po&longs;t primum de&longs;cen&longs;um; igitur <lb/>a&longs;cenderet per eandem lineam; 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 type="main"> <s>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;centu <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: 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>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 type="main"> <s>3. Hinc nece&longs;&longs;ariò e&longs;t retardatus: igitur de&longs;truitur impetus, non <lb/>quidem ab ip&longs;a medij re&longs;i&longs;tentia; quippe idem medium non magis <lb/>re&longs;i&longs;tit motui &longs;ur&longs;um, quàm motui deor&longs;um, vt patet: igitur de&longs;trui­<lb/>tur ille impetus motus violenti ab impetu innato aliquo modo; non <lb/>quidem vt à contrario ratione entitatis, &longs;ed ratione determinatio­<lb/>nis: cùm enim impetus innatus exigat motum deor&longs;um, & alius &longs;ur­<lb/>&longs;um: 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/><p type="main"> <s>4. Hinc &longs;ingulis temporibus æqualibus de&longs;truitur gradus impe­<lb/>tui innato; e&longs;t enim eadem ratio pro omnibus: igitur temporibus <lb/>æqualibus de&longs;truitur æqualis impetus: igitur amittit ille motus <lb/>æqualia velocitatis momenta: igitur e&longs;t naturaliter retardatus: 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 type="main"> <s>5. Hinc impetus imperfectior initio de&longs;truitur: quia, cùm motus <lb/>ille &longs;it velocior initio, in&longs;tantia &longs;unt minora: atqui minori tempore <lb/>minùs retardatur: igitur inperfectior impetus de&longs;truitur; 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 type="main"> <s>6. Hinc vltimus gradus impetus violentie&longs;t omnium perfecti&longs;­<lb/>&longs;imus, vt con&longs;tat. </s> <s>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: vbi enim ventum e&longs;&longs;et ad in&longs;tans æqualitatis, neutrum præ­<lb/>ualere po&longs;&longs;et: igitur in&longs;tanti &longs;equenti e&longs;&longs;et quies: cùm tamen &longs;int <lb/>diuer&longs;æ perfectionis, perfectior præualet: vter autem &longs;it perfectior, <lb/>dicemus infrà. </s></p><p type="main"> <s>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: &longs;i verò imprimitur ab alia potentia motrice, <lb/>tunc etiam imperfectior e&longs;t impetu innato; nam inæqualis e&longs;t; alio­<lb/>quin, &longs;i e&longs;&longs;et æqualis, &longs;imul e&longs;&longs;ent in eodem &longs;ubiecto duo gradus <lb/>homogenei: præ&longs;tat autem e&longs;&longs;e imperfectiorem, quàm perfectio­<lb/>rem, vt plura impetus puncta à potentia imprimantur; 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 type="main"> <s>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; 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><lb/>Si motus violentus e&longs;&longs;et æquabilis, percurreret proiectum &longs;patium <lb/>ferè duplum eo tempore, quo retardato percurrit &longs;ubduplum: hinc <lb/>&longs;onus tam citò auditur; quia propagatur cum particulis aëris æqua­<lb/>bili ferè motu: e&longs;&longs;e autem &longs;patium ferè duplum, probatur ex eo, <pb/>quòd &longs;patium motu æquabili decur&longs;um re&longs;pondet rectangulo; 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 type="main"> <s>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; tantumdem enim <lb/>impetus in de&longs;cen&longs;u acquiritur, quantùm in a&longs;cen&longs;u deperditur. </s> <s><lb/>Impetus primo in&longs;tanti, quo e&longs;t, agit, &longs;i e&longs;t aliquod impedimen­<lb/>tum; e&longs;t enim cau&longs;a nece&longs;&longs;aria: primo in&longs;tanti motus aliquid im­<lb/>petus de&longs;truitur: &longs;iue præce&longs;&longs;erit motus violentus, &longs;iue non præce&longs;­<lb/>&longs;erit, corpus graue æquali motu deor&longs;um cadit: re&longs;i&longs;tentia aëris e&longs;t <lb/>quidem maior initio; &longs;ed etiam &longs;unt maiores vires. </s></p><figure></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu in planis inclinatis.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. PLanum inclinatum e&longs;t &longs;ur&longs;um, vel deor&longs;um: 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: impeditur autem motus naturalis in plano <lb/>prædicto, quia impeditur eius linea: ideò e&longs;t tardior hic motus in <lb/>plano inclinato, quàm in perpendiculari: in ea porrò porportione <lb/>e&longs;t tardior, in qua perpendiculum e&longs;t minus linea inclinata, eiu&longs;dem <lb/>&longs;cilicet, altitudinis; 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 type="main"> <s>2. Hinc motus &longs;unt vt lineæ permutando: hinc mobile de&longs;cendit <lb/>per &longs;e in prædicto plano: licet enim motus impediatur, non tamen <lb/>tous, impetus, qui acquiritur in eodem plano e&longs;t imperfectior ac­<lb/>qui&longs;ito in perpendiculari in eadem proportione; nam impetus &longs;unt <lb/>vt motus: 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 type="main"> <s>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: motus in plano inclinato deor&longs;um e&longs;t acceleratus <lb/>iuxta eandem proportionem, iuxta quam acceleratur in perpendi-<pb/>culo: tempora, quibus pereurruntur 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 type="main"> <s>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>Hinc habes <lb/>apud Galileum, per omnes chordas circuli erecti de&longs;cen&longs;um fieri <lb/>æqualibus temporibus. </s> <s>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 type="main"> <s>5. Porrò minùs grauitat in ip&longs;um planum inclinatum corpus gra­<lb/>ue, quàm in planum horizontale: 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>Proiicitur mobile <lb/>faciliùs per inclinatum planum &longs;ur&longs;um, quàm per ip&longs;am perpendi­<lb/>cularem: patet experientia: euius 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 type="main"> <s>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: 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>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 type="main"> <s>7. Hinc minùs de&longs;truitur impetus in plano inclinato &longs;ur&longs;um, <lb/>quàm in perpendiculo; quia diutiùs durat: 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: hinc accedit propriùs hic motus ad æquabilem: 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: 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;ecantemé&longs;tque eadem proportio motuum. </s></p><p type="main"> <s>8. Corpus graue in &longs;uperficie quadrantis caua, deor&longs;um cadit <lb/>motu naturaliter accelerato; quia &longs;ingulis in&longs;tantibus accedit nouus <pb/>impetus; non tamen æqualibus temperibus, acquiruntur æqualia <lb/>velocitatis momenta; quia in &longs;ingulis punctis quadrantis, e&longs;t diuer­<lb/>&longs;a tangens; 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 type="main"> <s>9. De&longs;cendie etiam in &longs;uper&longs;icie conuexa globi erecti motu ac­<lb/>celerato; initio quidem, in minore proportione; &longs;ub finem, in maio­<lb/>re; vnde e&longs;t inuer&longs;a prioris: 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: &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;trueretut impetus im­<lb/>pre&longs;&longs;us, vt con&longs;tat; 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></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu mixto ex rectis.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. DAri motum mixtum ille non dubitat, qui di&longs;cum proiicit. </s> <s><lb/>Mixtus ex duobus rectis æquabilibus e&longs;t rectus, e&longs;t que <lb/>diagonalis vtriu&longs;que: hinc de&longs;truitur aliquid impetus, iuxta pro­<lb/>portionem differentiæ diagonalis, & vtriu&longs;que lateris &longs;imul &longs;ump­<lb/>pti; quia, &longs;eilicet, e&longs;t fru&longs;trà: quò maior e&longs;t angulus, quem faciunt li­<lb/>neæ determinationum, minor e&longs;t diagonalis; igitur plùs impetus <lb/>de&longs;truitur, donectandem concurrant in oppo&longs;itas lineas, tunc enim <lb/>totius impetus de&longs;truitur. </s></p><p type="main"> <s>2. <expan abbr="Qoũ">Qoum</expan> minor e&longs;t, vel acutior prædictus angulus, minùs impetus <lb/>de&longs;truitur; quia diagonalis maior e&longs;t; donec tandem conueniant in <lb/>eandem lineam, tunc enim nihil de&longs;truitur: datur de facto hic mo­<lb/>tus in rerum natura; talis e&longs;t motus nauis à duobus ventis im pre&longs;­<lb/>&longs;us; vel eiu&longs;dem partis aëris; imò & ip&longs;ius venti: motus mixtus ex <lb/>duobus retardatis iuxta eandem progre&longs;&longs;ionem e&longs;t rectus; quia fit <lb/>per hypothenu&longs;im triangulorum proporionalium: idem dico de <lb/>duobus acceleratis. </s></p><p type="main"> <s>3. Si mixtus &longs;it ex æquali, & accelerato, velex 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: dum proiicitur corpus graue per horizon-<pb/>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 type="main"> <s>4. Non tamen impetus acqui&longs;itus in eo motu e&longs;t eiu&longs;dem perfe­<lb/>ctionis cum illo, qui acquiteretur in perpendiculari eiu&longs;dem longi­<lb/>tudinis; &longs;ed tantùm eiu&longs;dem altitudinis: nam perinde cre&longs;cit ille <lb/>impetus, atque cre&longs;ceret in diuer&longs;is planis inclinaris: impetus verò <lb/>violentus in hoc motu retardatur; tùm, quia, &longs;i maneret idem, maior <lb/>e&longs;&longs;er 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 type="main"> <s>5. Adde, quòd e&longs;t duplex determinatio: igitur aliquid de&longs;trui de­<lb/>bet, non acqui&longs;iti; igitur impre&longs;&longs;i: de&longs;trui autem non dicitur acqui­<lb/>fitus, quòd, fcilicet, plùs de nouo accedat, quàm percat; e&longs;t enim ac­<lb/>celeratus: adde, quòd non infligitur tantus ictus &longs;ub finem; igitur <lb/>de&longs;truitur aliquid impetus, non acqui&longs;iti, eo modo, quo diximus; <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 type="main"> <s>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: tardiùs cadit mobile ita proiectum in planùm horizontale &longs;ub­<lb/>iectum, quàm cum &longs;ua &longs;ponte, ex eadem altitudine de&longs;cendit: cuius <lb/>rei clari&longs;&longs;ima e&longs;t experientia: ratio e&longs;t; quia impetus acqui&longs;itus in <lb/>hoc iactu non e&longs;t eiu&longs;dem perfectionis, cùm acqui&longs;ito in perpendi­<lb/>cnlo: cùm proiicitur mobile per inclinatam &longs;ur&longs;um, mouetur motu <lb/>mixto ex naturali æquabili, & violento retardato: patet prima pars; <lb/>quia acceleratur tantùm naturalis deor&longs;um, &longs;altem in inclinata: &longs;e­<lb/>cunda pars etiam patet; quia &longs;ub finem minor e&longs;t ictus. </s></p><p type="main"> <s>7. Hinc linea motus e&longs;t curua: iuxta diuer&longs;am progre&longs;&longs;ionem de­<lb/>&longs;truit ur hic impetus impre&longs;&longs;us: tùm pro diuer&longs;a inclinatione plani, <lb/>cuius etiam hîc habetur ratio; nam &longs;ingulis in&longs;tantibus mutatur: <lb/>tùm, quia modò plùs impetus e&longs;t fru&longs;trà, modò minùs; plùs <lb/>certè, cùm linea determinationis impetus impre&longs;&longs;i facit obtu­<lb/>&longs;iorem: atqui initio e&longs;t obtu&longs;ior; &longs;ub finem verò a&longs;cen&longs;us acu­<lb/>tior. </s></p><p type="main"> <s>8. A&longs;cen&longs;us proiecti per inclinatam diutiùs durat, quàm de&longs;­<lb/>cen&longs;us, ratione ciu&longs;dem plani horizontalis; quia, &longs;cilicer, a&longs;­<lb/>cen&longs;us longior e&longs;t, quàm de&longs;cen&longs;us: e&longs;t autem longior; 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/>porrò in de&longs;cen&longs;u e&longs;t motus mixtus ex accelerato naturali, <lb/>& retardato violento, vt con&longs;tat ex dictis: 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>Ratio e&longs;t: quia per verti­<lb/>calem &longs;ur&longs;um, nihil acquiritur in plano horizontali, ex quo fit ia­<lb/>ctus; nihil etiam per ip&longs;am horizontalem; igitur plùs acquiritur per <lb/>illam, quæ maximè ab vtraque &longs;imul recedit. </s></p><p type="main"> <s>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. </s> <s>45. <lb/>di&longs;tantes. </s> <s>Cùm emitcitur mobile per inclinatm 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; ille autem primus accelera­<lb/>tur per acce&longs;&longs;ionem impetus perfectionis quàm in iactu per ho­<lb/>rizontalem; &longs;ed imperfectionis, quàm in perpendiculo: 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 type="main"> <s>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: mouetur &longs;ur&longs;um per our­<lb/>uam, &longs;empérque capiti iaculatoris imminet; quippe tantùm acqui­<lb/>rit in horizontali, quantùm nauis: in de&longs;cen&longs;u verò e&longs;t motus mix­<lb/>ex horizontali retardato, & naturali accelerato: 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 type="main"> <s>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 type="main"> <s>12. Cùm è naui mobili emittitur &longs;agitta per horizontalem, quæ fa­<lb/>cit anglum rectum cum linea directionis nauis, fertur qua&longs;i per dia­<lb/>gonalem vtriu&longs;que, &longs;altem per aliquod &longs;patium: cùm verò emitti-<pb/>tur per horizontalem, quæ conueniat cum eadem linea directionis, <lb/>&longs;actus e&longs;t longior toto illo &longs;patio, quod nauis decurrit, dum iactus <lb/>durat; 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; imò &longs;agitta po&longs;&longs;et retorqueri in iaculatorem: <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></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu reflexo.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. MOtus reflexi vera cau&longs;a e&longs;t impetus prior, ad nouam li­<lb/>neam determinatus ab occurrente obice; planum refle­<lb/>ctens e&longs;t cau&longs;a nouæ determinationis &longs;uo modo; cau&longs;am enim di­<lb/>co eam, ex qua aliquid &longs;equitur: 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; quæ omnia <lb/>patent ex terminis: hinc nullus impetus producitur à plano refle­<lb/>ctente; quippe prior pote&longs;t determinariad nouam lineam: adde, <lb/>quòd planum, quod caret impetu, impetum producere non pote&longs;t. </s></p><p type="main"> <s>2. Imò nihil impetus de&longs;truirur in reflexione pura per &longs;e; quia ni­<lb/>hil impetus e&longs;t fru&longs;trà per &longs;e in pura reflexione; 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: hinc &longs;uppo&longs;ito eodem iactu, perpendi­<lb/>cularis reflexa e&longs;t omnium reflexarum minima; quia per eam li­<lb/>neam maximus ictus infligitur; 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 type="main"> <s>3. Determinatio noua dupla e&longs;t prioris, po&longs;ita linea incidentiæ <lb/>perpendiculari, & po&longs;ito etiam plano reflectente immobili; quia <lb/>alioquin anguli reflexionis non e&longs;&longs;ent æquales angulis incidentiæ: <lb/>&longs;i globus reflectens &longs;it æqualisimpacto, 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: igitur, &longs;i æqualis re&longs;i&longs;tit, & cedit, certè æqualiter ce­<lb/>dit, & re&longs;i&longs;tit: hinc noua determinatio æqualis e&longs;t priori: hinc glo­<lb/>bus impact is &longs;i&longs;tit immobilis; quia ex duabus determinationibus <lb/>oppo&longs;itis neutra præualet. </s></p><pb/><p type="main"> <s>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: hinc, cùm à tota ce&longs;&longs;ione ad æqua­<lb/>litatem prædictam acquiratur tantùm noua determinato æqualis <lb/>priori; igitur ab eadem æqualitate ad nullam ce&longs;&longs;ionem tantun­<lb/>dem acquiritur; igitur dupla prioris, vt iam &longs;uprà dictum e&longs;t; 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 type="main"> <s>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; quia &longs;unt hæ, vt ictus, per vtran­<lb/>que lineam; ictus verò vt grauitationes in horizontale planum, & <lb/>in planum inclinatum, &longs;ub angulo complementi anguli incidentiæ: <lb/>hinc noua determinatio per lineam obliquam, e&longs;t vt dupla &longs;inus re­<lb/>cti anguli incidentiæ, ad &longs;inum totum: 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 type="main"> <s>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. </s> <s>17. problem. </s> <s>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<gap/>&longs;ed minimè demon&longs;tratur ex certis principiis &longs;ine petitio­<lb/>ne principij. </s> <s>In puncto reflexionis, po&longs;ita hypothe&longs;i plani immo­<lb/>bilis reflectentis, nulla datur quies; 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 type="main"> <s>7. Omnes lineæ reflexæ per &longs;e &longs;unt æqualis longitudinis, & ab <lb/>eodem puncto contactus, ad communem peripheriam terminan­<lb/>tur: &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; 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 v<gap/><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 type="main"> <s>8. Si globus impactus &longs;it minor corpore reflectente, reflectitur <lb/>eriam per ip&longs;am perpendicularem, & determinatio noua e&longs;t dupla­<lb/>prioris, minùs ratione globorum v. </s> <s>g. </s> <s>&longs;i globus impactus &longs;it &longs;ubdu-<pb/>plus, determinatio noua e&longs;t dupla prioris, minùs vna quarta, <lb/>&c. </s> <s>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 type="main"> <s>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: 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>Porrò in hac <lb/>hypothe&longs;i vltima, determinatio noua per ip&longs;am perpendicularem <lb/>e&longs;t minor priore: hinc non modò nulla fit reflexio in perpendicula­<lb/>ri, &longs;ed linea directa vlteriùs propagatur; quia prior determinatio <lb/>præualet. </s></p><p type="main"> <s>10. Detrahitur priori portio æqualis rationi globorum; v. </s> <s>g. </s> <s>glo­<lb/>bus reflectens e&longs;t &longs;ubduplus impacto de trahitur priori determina­<lb/>tioni vna &longs;ecunda; e&longs;t &longs;ubquadruplus, vna quarta; atque ita dein­<lb/>ceps: ratio patet ex dictis: 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 type="main"> <s>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; quia æqualis in æqualis æqualem impetum imprimit: non <lb/>e&longs;t tamen motus reflexus; quia totus prior impetus de&longs;truitur, vt <lb/>patet ex dictis: &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. </s> <s><lb/>g. </s> <s>æqualem motum imprimere globo B, &longs;iue hic moueatur, &longs;iue <lb/>quie&longs;cat. </s></p><p type="main"> <s>12. Si verò linea incidentiæ &longs;it obliqua, vterque globus reflecte­<lb/>tur pror&longs;us vt à plano immobili: hinc reflexio &longs;it ad angulos æqua­<lb/>les, & lineæ omnes reflexionis &longs;unt æquales: ratio e&longs;t; 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, resetiam determinari pote&longs;t ex <lb/>præmi&longs;&longs;is. </s></p><pb/><p type="main"> <s>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: maior verò, &longs;i duplus e&longs;t alterius, &longs;i&longs;tit immobilis in puncto <lb/>contactus; &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>Pote&longs;t impetus e&longs;&longs;e æqualis alteri, & præualere; pote&longs;t <lb/>æqualem impetnm producere hoc in&longs;tanti, & &longs;tatim in&longs;tanti, quod <lb/>&longs;equitur, totus de&longs;trui. </s></p><p type="main"> <s>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: tùm <lb/>deinde, licèt nihil pror&longs;us addatur, versùs occa&longs;um; quod accidit, <lb/>cum globus vtroque motu, centri, &longs;cilicet, & orbis, mouetur, &longs;ed <lb/>contrario; primùm enim motus centri plæualet, &longs;ed facilè cedit <lb/>propter attritum maiorem partium. </s> <s>Nullus datur propriè motus <lb/>refractus: licèt enim incuruetur linea motus, dum per aquam &longs;u­<lb/>bit mobile; hæc tamen e&longs;t reflexionis &longs;pecies. </s></p><p type="main"> <s>15. Globus reflectens, qui ab ictu alterius mouetur, non mouetur <lb/>in&longs;tanti contactus; quia impetus primo in&longs;tanti, quo e&longs;t, non mo­<lb/>uetur; producitur enim impetus primo in&longs;tanti contactus: &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; quia veri&longs;&longs;ima cau&longs;a reflexionis <lb/>con&longs;i&longs;tit in noua determinatione: per reflexionem po&longs;&longs;unt colligi <lb/>plures partes aëris &longs;onori ad Echometriam: &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></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu circulari.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. DAri motum circularem, probatur infinitis ferè experimen­<lb/>tis: cuius ratio à priori e&longs;t, quòd po&longs;&longs;int extremitates eiu&longs;­<lb/>dem cylindri in partes oppo&longs;itas pelli; vnde &longs;equitur nece&longs;&longs;ariò <lb/>motus circularis; quem ij negare coguntur, qui ex punctis mathe­<lb/>maticis quantitatem componunt. </s> <s>Motus circularis in &longs;ublunaribus <lb/>oritur ex recto impedito; quia, &longs;cilicet, determinatur tantùm im­<lb/>petus ad lineam rectam: hinc quidam motus circularis e&longs;t merè <lb/>per accidens, vt cùm retinetur extremitas funependuli, &longs;eu <pb/>fundæ, quæ &longs;i demittatur, &longs;equitur motus rectus: 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 type="main"> <s>2. Quippe tale e&longs;t naturæ in&longs;titutum, vt eo motu corpora mo­<lb/>ueantur, quo faciliùs moueri po&longs;&longs;unt: 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: æqualitempore: 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 type="main"> <s>3. Hinc, &longs;i tollatur impedimentum, &longs;tatim per tangentem or­<lb/>bis fit motus, vt patet in funda: inæqualiter partes radij prædicti <lb/>orbis mouentur, iuxta proportionem di&longs;tantiæ maioris, & minoris <lb/>à centro: hinc propagatio impetus inæqualis, de qua iam &longs;uprà, <lb/>&longs;ingulis in&longs;tantibus & punctis e&longs;t noua determinatio; quia, &longs;cilicet, <lb/>&longs;ingulis punctis &longs;ua tangens re&longs;pondet: 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 type="main"> <s>4 Dari impetum in motu circulari certi&longs;&longs;imum e&longs;t: punctum phy­<lb/>&longs;icum e&longs;t capax huius motus; cuius finis multiplex e&longs;t; 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: 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 moru circulari carere non pote&longs;t. </s></p><p type="main"> <s>5. Motus circularis rotæ in plano verticali e&longs;t æquabilis per &longs;e; <lb/>quia nihil e&longs;t, quod impetum &longs;emel impre&longs;&longs;um de&longs;truat: licèt enim <lb/>&longs;ingulis in&longs;tantibus &longs;it noua determinatio, nullus tamen impetus <lb/>e&longs;t fru&longs;trà; quippe illud &longs;patium acquiritur in linea curua, quod in <lb/>recta, &longs;i nullum e&longs;&longs;et impedimentum, percurreret: quemadmodum <lb/>enim in reflexione, quæ fit à plano immobili, nullus de&longs;truitur im­<lb/>petus; ita nullus hîc de&longs;truitur; 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/><p type="main"> <s>6. Hinc per &longs;e motus circularis integri orbis e&longs;t perpetuus; de­<lb/>&longs;truitur tamen per accidens, &longs;cilicet, propter attritum axis: hinc <lb/>tam diu durat hic motus: 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; nec <lb/>vnquam ce&longs;&longs;aret hic motus &longs;ine prædicto attritu, & partium a&longs;peri­<lb/>tate: nec quidquam ob&longs;tat, quòd aliquæ partes rotæ, quæ in circu­<lb/>lo verticali voluitur, a&longs;cendant; 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>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 type="main"> <s>7. Cùm mouetur æquali ni&longs;u acus circa immobile centrum, tùm <lb/>in plano <expan abbr="horizõrali">horizorrali</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; quia per &longs;e <lb/>mouetur motu æquabili: per accidens tamen &longs;ecus accidit; quippe <lb/>maior e&longs;t maioris attritus: dixi, cùm mouetur æquali ni&longs;u; nam &longs;æpè <lb/>contingit, maiore ni&longs;u potentiam motricem agere circa maiorem; <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 type="main"> <s>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, co tempore, quo maior decurret vnicam: 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; 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 type="main"> <s>9. Rota in plano verticali faciliùs mouetur, quàm in horizonta­<lb/>li; quia in illo mouetur per minimam impetus, vel potentiæ acce&longs;­<lb/>&longs;ionem; &longs;ecùs in i&longs;to; quippe per minimam acce&longs;&longs;ionem tollitur <lb/>æquilibrium; imò moueri pote&longs;t in plano verticali, licèt nullus im­<lb/>primatur impetus rotæ, v. </s> <s>g. </s> <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 type="main"> <s>10. Si cylindrus in plano horizontali læuigato in alteaa extremi­<lb/>tate per tangentem impellatur, mouebitur motu circulati, &longs;cilicet, <pb/>faciliori, cirra 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 type="main"> <s>11. Cùm rota mouetur in verticali, atque præponderat alter &longs;emi­<lb/>circulus, haud dubiè hic præponderans producit impetum in alio <lb/>&longs;emicirculo: hinc fortè e&longs;t, quòd mirere, impetus determinatus <lb/>deor&longs;um producit alium &longs;ur&longs;um: hinc impetus vnius partis mobi­<lb/>lis pote&longs;t producere&longs;imilem in alia parte continua; quod tantùm in <lb/>hoc ca&longs;u locum habet: 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></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu funependuli.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. FVnependulum de&longs;cendit per arcum motu naturaliter acce­<lb/>lerato: experientia clari&longs;&longs;ima e&longs;t: cùm enim ex maiori &longs;ubli­<lb/>mitate de&longs;cendit, maiorem ictum infligit. </s> <s>Ratio à priori e&longs;t quia <lb/>priori impetui acqui&longs;ito nouus accedit: non acceleratur in eadem <lb/>proportione, in qua &longs;uprà dictum e&longs;t accelerari in linea recta; quia <lb/>in hac acceleratur vniformiter, id e&longs;t, æqualibus temporibus, <lb/>æqualia acquiruntur velocitatis momenta; quia vel e&longs;t &longs;emper ea­<lb/>dem inclinatio plani, vel idem perpendiculum: at verò in fune­<lb/>pendulo in &longs;ingulis punctis e&longs;t noua tangens; igitur noua inclina­<lb/>tio plani; igitur noua ratio motus. </s></p><p type="main"> <s>2. Initio acceleratur motus per maiora crementa, &longs;ub finem per mi­<lb/>nora; v.g. </s> <s>&longs;i dato tempore acqui&longs;iuit vnum gradum impetus initio, <lb/>æquali deinde tempore acquiret minùs: ratio clara e&longs;t: quia, vt ac­<lb/>quireret æqualem, deberet e&longs;&longs;e eadem pl<gap/>ni inclinatio; &longs;ed &longs;empes <lb/>cre&longs;cit Inclinatio; igitur &longs;emper imminuitur impetus æquali <expan abbr="t&etilde;pore">tempore</expan> <lb/>acqui&longs;itus: 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 type="main"> <s>3. Hinc breuiore tempore de&longs;cendit per radium perpendicula­<lb/>rem, quàm per quadrantis arcum eiu&longs;dem radij; tùm quia breuior <lb/>e&longs;t linea; tùm, quia in perpendiculari acceleratur motus per maiora <lb/>crementa. </s> <s>Vibrario maior eiu&longs;dem funependuli æquali ferè tem-<pb/>pore cum minore perficitur: ratio <gap/> 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: dixi <lb/>ferè: 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">analyticcam</expan> remittimus. </s></p><p type="main"> <s>4. Non a&longs;cendit funependulum ad eam altitudinem, ex qua priùs <lb/>de&longs;cenderat: clara e&longs;t experientia: neque ratio tantùm petitur ab <lb/>aëris re&longs;i&longs;tentia; tam enim re&longs;i&longs;tit de&longs;cen&longs;ui, quàm a&longs;cen&longs;ui; &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: quippe impetus <lb/>innatus ad totum de&longs;cen&longs;um, &longs;ed nullo modo ad a&longs;cen&longs;um con­<lb/>currit: hinc in maiori vibratione imminuitur motus, & &longs;patium in <lb/>maiori proportione, quàm in minori; 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 type="main"> <s>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: <lb/>hinc etiam in funependulo maiori dintiùs durant huiu&longs;modi vi­<lb/>brationes, idque in arcubus paulò maioribus; quia &longs;ubten&longs;æ his <lb/>arcubus &longs;unt inclinatiores: hinc refutabis eos, qui dicunt, vibra­<lb/>tiones funependuli in vacuo fore perpetuas: 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 type="main"> <s>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; quippe e&longs;t omnium imperfecti&longs;&longs;imum; 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: &longs;i &longs;int funependula inæqualia, vibrationes non &longs;unt æquè diu­<lb/>turnæ: ratio e&longs;t: quia, &longs;i a&longs;&longs;umamtur, v.g.duo quadrantes inæquales, <lb/>&longs;unt eju&longs;dem inclinationis; igitur minor citiùs percurritur. </s></p><p type="main"> <s>7. Porrò tempora vibrationum &longs;unt in ratione &longs;ubduplicata ar­<lb/>cuum &longs;imilium, vel chordarum &longs;imilium, vel radiorum; id e&longs;t, vt <lb/>radices &longs;patiorum &longs;imilium: verbi gratia, &longs;it quadruplus alterius, <lb/>tempus vibrationis maioris e&longs;t duplum temporis vibrationis mino­<lb/>ris; 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/><p type="main"> <s>8. In maiori quadrante, circa &longs;upremam extremitatem, e&longs;t minor <lb/>inclinatio, quàm in minore; hic enim &longs;tatim detorquetur à perpen­<lb/>diculo, cum quo facit angulum maiorem: at verò circa infirmam <lb/>extremitatem, e&longs;t maior inclinario in maiore, quàm in minore: 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 type="main"> <s>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: hinc &longs;unt ijdem ictus: <lb/>numerus, vibrationum non e&longs;t infinitus, licèt in vacuo vibraretur <lb/>funependulum; quia, cùm &longs;ingulæ imminuantur, & infinitis pun­<lb/>ctis non con&longs;tent; tandem ad vltimam peruenitur: 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 type="main"> <s>10. Funependulum in fine a&longs;cen&longs;us non quie&longs;cit vno in&longs;tanti; <lb/>quia impetui innato <expan abbr="nũquam">nunquam</expan> redditur æqualis acqui&longs;itus; po&longs;ita ta­<lb/>men illa æqualirate, in&longs;tanti &longs;equenti e&longs;&longs;et quies: funependulum <lb/>grauius citiùs de&longs;cendit; e&longs;t enim eadem ratio, quæ fuit pro mo­<lb/>tu naturali; 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 type="main"> <s>11. Ratio facilis e&longs;t; quia partes &longs;olidæ, quæ accedunt propiùs <lb/>ad extremitatem immobilem, accelerant motum aliarum, quæ <lb/>ad mobilem extremitatem accedunt; faciunt enim arcum mino­<lb/>rem: hinc a&longs;cen&longs;us non peruenit ad tantam &longs;ublimitatem; quia, vt <lb/>prædictæ partes accelerant motum aliarum in de&longs;cen&longs;u, ita retar­<lb/>dant in de&longs;cen&longs;u: hinc citiùs quie&longs;cit hoc penduli genus, quàm <lb/>aliud: 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. </s> <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></figure><p type="main"> <s><emph type="center"/><emph type="italics"/>De motu mixto ex circulari.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>1. ROta, quæ mouetur in &longs;uperficie plana, mouetur motu mixto <lb/>ex recto centri, & circulari orbis: axis tantùm rotæ mouetur <lb/>motu recto: punctum contactus rotæ mouetur motu tardi&longs;&longs;imo, <pb/>quando motus centri, & &longs;uprema rotæ pars in eandem partem &longs;e­<lb/>runtur; punctum verò oppo&longs;itum veloci&longs;&longs;imo, quia in motu huius <lb/>rotus motus orbis additur motui centri; 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 type="main"> <s>2. Hinc omnia puncta eiu&longs;dem circuli rotæ mobilis in plano <lb/>hoc motu mixto mouentur in æquali motu: hoc etiam motu mo­<lb/>uetur globus de&longs;cendens in plano inclinato, in quo reuerâ motu <lb/>hæc habes: 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 type="main"> <s>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>Ca­<lb/>lamus volatilis, cuius mi&longs;&longs;io frequens, & repercu&longs;&longs;io, ludi non in­<lb/>grati copiam facit: mouetur motu mixto ex recto, & circulari: in <lb/>hoc porrò motu præit calami caput, & &longs;equuntur pennæ; 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 type="main"> <s>4. Hinc, &longs;i quando accidat, penas educi ex theca in libero medio; <lb/>&longs;tatim theca velociori motu mouetur, cùm tamen pennæ ip&longs;æ &longs;i­<lb/>&longs;tant: 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;tente, in alteram <lb/>partem qua&longs;i reflectentur, &longs;int nece&longs;&longs;aria conditio huius traductio­<lb/>nis. </s></p><p type="main"> <s>5. Hinc motu recto prædictum iaculum in vacuo tantùm mo­<lb/>ueretur, vt patet: hinc: cùm pennæ &longs;unt explicatiores, tardiùs; cùm <lb/>verò contractiores, velociùs mouetur, etiam motu orbis; cui non <lb/>minùs aër re&longs;i&longs;tit, in pennis, &longs;cilicet, quàm motui axis: hinc, &longs;i theca <lb/>&longs;it grauior, velociùs; &longs;i leuior, tardiùs iaculum fertur; etiam tenera <lb/>plumarum lanugo tarditatem conciliat: 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 type="main"> <s>6. Sunt infiniti propemodum motus mixti; v. </s> <s>g. </s> <s>cylindri ab alte­<lb/>ra extremitate rotata emi&longs;&longs;i; longioris ha&longs;tæ, quæ &longs;ur&longs;um facta cir­<lb/>cuitione emittitur; brachij, gladij, &c. </s> <s>&longs;ed poti&longs;&longs;imùm turbinis, qui <pb/>vel &longs;eutica, vel funiculo in torto circumagitur, in quo clari&longs;&longs;i­<lb/>mè apparet motus centri, & orbis: ratio motus orbis e&longs;t impe­<lb/>tus impre&longs;&longs;us vtrique extremitati diametri va&longs;is in partes contra­<lb/>rias; ratio verò motus centri e&longs;t, quia adducitur funiculo vel ex­<lb/>ploditur, &longs;eu expellitur &longs;cutica: huius motus phænomena &longs;unt ferè <lb/>infinita: &longs;ingula ex no&longs;tris principiis facilè explicantur. </s></p><figure></figure><p type="main"> <s><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 type="main"> <s>1. CVm &longs;u&longs;tinetur manus, &longs;eu brachium, in &longs;itu horizontali im­<lb/>mobile, producitur nece&longs;&longs;aliò impetus æqualis impetui gra­<lb/>uitationis; alioquin, &longs;i maior e&longs;&longs;et, &longs;ur&longs;um ferretur brachium; &longs;i verò <lb/>minor, deor&longs;um: quia præualeret grauitatio, porrò hic impetus pro­<lb/>ducitur tantùm à potentia motrice animantis, in &longs;ingulari organo; <lb/>non verò in aliis partibus, etiam animatis, ni&longs;i quando mouentur; <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>Si anima immediatè in toto <lb/>corpore po&longs;&longs;et producere impetum, homo facilè volare po&longs;&longs;et. </s></p><p type="main"> <s>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; &longs;ecùs verò, &longs;i attolla­<lb/>tur: &longs;ic per quamlibet lineam corpus retineri pote&longs;t &longs;ine impetu in <lb/>eo corpore producto per &longs;e: 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; &longs;ed per acci­<lb/>dens, propter mollitiem, & ten&longs;ionem partium: cùm verò defertur <lb/>aliquid coniunctum, producitur haud dubiè æqualis impetus; hinc <lb/>&longs;eparari non pote&longs;t; quia æqualis e&longs;t motus latoris, & delati: exem­<lb/>plum habes in naui. </s></p><p type="main"> <s>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>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: 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 type="main"> <s>4. Cùm trahitur aliquod corpus impetus impre&longs;&longs;us in vna parte <lb/>non producit impetum in alia, alioquim daretur proce&longs;&longs;us in in&longs;i­<lb/>nitum; &longs;i chorda vtrinque trahatur, rumpetur in medio: &longs;i affixa <lb/>extremitati immobili, trahatur à potentia applicata alteri extremi-<pb/>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>Ratio communis i&longs;torum omnium e&longs;t: 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 type="main"> <s>5. Vt quodlibet pondus faciliùs trahatur, &longs;inguli equi trahere <lb/>debent fune communi, potiùs quàm bigati; quia tunc nihil ferè pe­<lb/>rit impetus: 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;im trahere? </s> <s>o&longs;&longs;ent, quod fal&longs;um e&longs;t: ideò currus <lb/>paulò po&longs;t initium motus faciliùs mouetur; 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 type="main"> <s>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/>norum ten&longs;ione, &c. </s> <s>dum trahitur vnco annullus 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 type="main"> <s>7. Cùm trahitur cylindrus vtrinque æqualiter, qui neque flecti, <lb/>neque tendi pote&longs;t, nullum impetum accipit; imò in tractione nul­<lb/>lus impetus e&longs;t inutilis: brachium infligit maiorem ictum, cùm ma­<lb/>iorem <expan abbr="arcũ">arcum</expan> de&longs;cribit &longs;uo motu; quia, &longs;cilicet, mouetur motu natu­<lb/>raliter accelerato: hinc auer&longs;a manu validior impingitur colaphus, <lb/>quàm aduer&longs;a; quia illa maiorem arcum de&longs;cribit: 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 type="main"> <s>8. Maiore fu&longs;te maior ictus infligitur; quia potentia toto ni&longs;u <lb/>agens, diutiùs manet applicata maiori, quàm minori; &longs;untque ictus <lb/>in ratione &longs;ubduplicata vtriu&longs;que fu&longs;tis; v. </s> <s>g. </s> <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: idem dicatur de mi&longs;&longs;o lapide: principium huius <lb/>veritatis pender<gap/> iis, quæ diximus lib. 2. de motu naturali­<lb/>ter accelerate, luxta progre&longs;&longs;ionem numerorum imparium, <lb/>1. 3. 5. &c. </s></p><p type="main"> <s>9. Fu&longs;tis circa centrum immobile vibratus, maximum ictum in-<pb/>fligit, non quidem in centro grauitatis, id e&longs;t, in medio, &longs;i &longs;it cy­<lb/>lindrus, vel parallelipedum; nec in extremitate mobili; &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; quia tunc totus impetus agit, <lb/>quantùm pote&longs;t; 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 type="main"> <s>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: quia non <lb/>tantùm agit impetus extrin&longs;ecùs adueniens; 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 type="main"> <s>11. Rotato flagello ideò maxima vis ine&longs;t, quia diutiùs potentia <lb/>manet applicata: hinc vides hoc principium e&longs;&longs;e vniuer&longs;ali&longs;&longs;imum, <lb/>quod iactis, pul&longs;is, & impactis competit; de malleorum ictu idem <lb/>pror&longs;us dicendum e&longs;t, quod de fu&longs;te; &longs;i autem mallei cadant <lb/>ex eadem altitudine, motu naturali accelerato, ictus &longs;unt vt <lb/>mallei, quia duplus malleus, v. </s> <s>g. </s> <s>duplum impetum acquirit: nam <lb/>&longs;ingulæ partes &longs;eor&longs;im æqualem impetum acquirunt. </s></p><p type="main"> <s>12. Si verò ex diuer&longs;a altitudine cadant, vel &longs;unt æquales, vel <lb/>inæquales: &longs;i primum, ictus &longs;unt vt tempora, quibus cadunt: &longs;i <lb/>&longs;ecundum, ictus &longs;unt in ratione compo&longs;ita temporum, & mal­<lb/>leorum: &longs;i &longs;unt infinitæ, partes actu, nulla e&longs;t proportio percu&longs;&longs;ionis <lb/>granuli cadentis, & rupis ingentis grauitantis; &longs;ed hoc vltimum fal­<lb/>&longs;um e&longs;&longs;e con&longs;tat; 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 type="main"> <s>13. Leui&longs;&longs;imi lapides vix emittuntur ad modicam di&longs;tantiam; <lb/>quia &longs;tatim &longs;eparantur à potentia: 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; quia tunc totus impetus agit, totus enim impe­<lb/>ditur: 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. <pb/><figure id="fig1"></figure><pb/><figure id="fig2"></figure><pb/><figure id="fig3"></figure><pb/><figure id="fig4"></figure><pb/><figure id="fig5"></figure><pb/><figure id="fig6"></figure><pb/><figure id="fig7"></figure><pb/><figure id="fig8"></figure><pb/><figure id="fig9"></figure><pb/><figure id="fig10"></figure><pb/><figure id="fig11"></figure></s></p><pb pagenum="1"/><figure></figure><p type="main"> <s><emph type="center"/>LIBER PRIMVS,<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>DE IMPETV.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; cum enim impetus &longs;it cau&longs;a <lb/>immediata motus, vt fusè demon&longs;tra­<lb/>bimus infrà; & 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>DEFINITIO I.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Definitio II.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus velox e&longs;t quo percurritur maius &longs;patium æquali tempore, vet <lb/>æquale &longs;patium minori tempore; contrà verò motus tardus.<emph.end type="italics"/></s></p><pb pagenum="2"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Definitio III.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s>Dico e&longs;&longs;e qualitatem &longs;iue di&longs;tincta &longs;it, &longs;iue non di&longs;tincta; quod hîc <lb/>certè non di&longs;cutio; nec enim affirmo in hac definitione dari impetum; <lb/>&longs;ed definio tantùm quid &longs;it impetus; qui reuera aliud non e&longs;t, &longs;i e&longs;t: <lb/>quippe id tantùm concipio, cum impetum appello; &longs;iue &longs;it, &longs;iue non &longs;it, <lb/>ne quis fortè initio &longs;tatim mihi litem intendat; quemadmodum definit <lb/>circulum Geometra; licèt non a&longs;&longs;erat dari perfectum circulum; ita Phy­<lb/>&longs;icus definit impetum, quamuis non affirmet dari impetum; quod tamen <lb/>in &longs;exto Theoremate demon&longs;trabimus; 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 in&longs;rà in Theorematis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is I.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Datur motus localis<emph.end type="italics"/>; quis enim non videt volantem auem, natantem <lb/>pi&longs;cem; currentem equum, rotatum globum; denique vnum corpus mi­<lb/>grans è loco in locam? </s> <s>&longs;ed hoc e&longs;t moueri per Def. </s> <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 type="main"> <s>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>Quis enim oculos in Solem intendens, primo intuitu Solem &longs;ta­<lb/>re non iudicet? </s> <s>cum tamen deinde pernici&longs;&longs;imo cur&longs;u rotari demon&longs;tre­<lb/>mus; adde alias oculorum fallacias circa motum; &longs;ic rotata &longs;cintilla, vel <lb/>carbo accen&longs;us immotum orbem de&longs;cribere videtur; &longs;ic nota inu&longs;ta <lb/>trocho, dum celerrimè rotatur, orbem etiam immobilem de&longs;cribere iu­<lb/>dicatur; &longs;ic &longs;tella cadens, vel exhalatio continenti &longs;ucce&longs;&longs;ione accen&longs;a <lb/>moueri videtur; licet minimè moucatur; idem dicendum de puluere <lb/>tormentario, vel alia qualibet materia; quæ continuata con&longs;ecutione <lb/>accenditur; immò trochus ip&longs;e in orbem celerrimè agitatus, quie&longs;cere <lb/>videtur; &longs;ic qui vertigine laborant, ea moueri exi&longs;timant, quæ quie&longs;cunt; <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; <lb/>&longs;ic corum quæ motu æquali feruntur, remotiora tardiùs moueri viden­<lb/>tur; immò &longs;i per eandem lincam 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 co quòd oculis aliquid moucri videatur. </s></p><p type="main"> <s>Re&longs;pondeo equidem &longs;ateri me, vi&longs;um ip&longs;um plurimis &longs;ube&longs;&longs;e fraudi­<lb/>bus; attamen &longs;i rectè oculus admoueatur, iu&longs;ta di&longs;tantià, nec vllum &longs;it <lb/>impedimentum exterius nec interius; 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"/>te&longs;t? </s> <s>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; &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 conuc­<lb/>nire non po&longs;&longs;um. </s></p><p type="main"> <s>Porrò quod &longs;pectat ad fallacias illas quæ &longs;upra adductæ &longs;unt; certum <lb/>e&longs;t vel obiectum e&longs;&longs;e remotius, quam par &longs;it; vel moueri celeriùs, vel <lb/>e&longs;&longs;e aliquod impedimentum interius; 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 type="main"> <s>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 type="main"> <s>Prima duci pote&longs;t à fine motus; 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; &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; 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><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; 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 type="main"> <s>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 type="main"> <s>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 type="main"> <s>Quarta petitur à termino motus; cum enim globus proiectus &longs;it in <lb/>nouo loco in quo ante non erat; 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 type="main"> <s>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 type="main"> <s>Sexta ab ip&longs;a Mechanica, quæ organa motui mini&longs;trat: quis enim ne­<lb/>garet maius momentum e&longs;&longs;e cum maiori di&longs;tantiâ coniunctum; &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>Igitur certum e&longs;t <lb/>dari motum localem. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is II.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Datur quies, id e&longs;t priuatio motus.<emph.end type="italics"/> Hæc hypothe&longs;is etiam ceita e&longs;t, <pb pagenum="4"/>Quis enim neget &longs;edentem humi, vel decumbentem in lecto quie&longs;ceret <lb/>con&longs;ule &longs;en&longs;us rectè applicatos; 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>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>Secundò maximum incommodum e&longs;&longs;et, &longs;i res &longs;e­<lb/>mel mota perpetuò moueretur. </s> <s>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 type="main"> <s>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 type="main"> <s>Re&longs;pondeo primò illam Copernici &longs;ententiam e&longs;le 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 type="main"> <s>Dices iuxta hypothe&longs;im Heraclidis Pontici, terra ip&longs;a, Sol etiam, & <lb/>&longs;tellæ mouentur. </s> <s>Re&longs;pondeo primò hypothe&longs;un illam e&longs;&longs;e fal&longs;am, vt &longs;uo <lb/>loco videbimus; &longs;ecundò etiam data illa hypothe&longs;i po&longs;&longs;et dari quies; &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;un, 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 type="main"> <s>Quod verò &longs;pectat ad fallacias oculi circa quietem; codem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is III.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Aliquid mouetur quod incœpit moueri.<emph.end type="italics"/></s><s> Video lapidem quie&longs;centem, <lb/>qui deinde proiectus mouetur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is IV.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Aliquid mouetur quod tandem de&longs;init moueri, vel incipit quie&longs;cere.<emph.end type="italics"/></s><s> Vi­<lb/>deo rotatam pilam, quæ tandem quie&longs;cit, cadentem lapidem, qui tan­<lb/>dem &longs;i&longs;tit, &c. </s> <s>igitur certa e&longs;t hæc hypothe&longs;is. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is V.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Idem mouetur modò tardiùs, modò velociùs.<emph.end type="italics"/></s><s> Video rotatum globum, <lb/>qui &longs;en&longs;im quie&longs;cit: &longs;entio ab codem globo modò maiorem, modò mi­<lb/>norem ictum infligi, &c. </s> <s>igitur e&longs;t certa hypothe&longs;is. </s></p><pb pagenum="5"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is VI.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Corpus proiectum etiam à potentiâ motrice &longs;eiunctum adhuc moue tur.<emph.end type="italics"/><lb/>Oculos omnium te&longs;tes appello. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is VII.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Corpus proiectum, & in aliud impactum illud ip&longs;um impellit, & mouet.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is VIII.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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> 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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma I.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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> Hoc ip&longs;um iam ptæ­<lb/>mi&longs;unus Logicæ no&longs;træ demon&longs;tratiuæ, complectiturque prima illa <lb/>principia Metaphy&longs;icæ. </s></p><p type="main"> <s>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 type="main"> <s>2. <emph type="italics"/>Quodlibet e&longs;t, vel non est.<emph.end type="italics"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma II.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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> Nihil enim a &longs;e ip&longs;o &longs;eparari po&longs;t; 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>quia &longs;cilicet vnus <lb/>non e&longs;t alius inquies; & quare vnus non e&longs;t alius? </s> <s>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>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 type="main"> <s>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/>temus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma III.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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> 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; illud porrò argumentum <lb/>duci pote&longs;t vel ab experimento certo; &longs;ic probo exi&longs;tere rem aliquam, <lb/>quam video; vel ab aliqua ratione; &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; vel co 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/><gap/>ium ad aliquem finem à natura in&longs;titutum, quo natura ip&longs;a &longs;ine ab&longs;ur-<pb pagenum="6"/>do, vel graui&longs;&longs;imo incommodo carere non pote&longs;t, probo illud ip&longs;um <lb/>e&longs;&longs;e; vel demùm ex aliqua reuelatione certa in rebus fidei; igitur hoc <lb/>Axioma certum e&longs;t phy&longs;icè; quod ni&longs;i recipiatur à Philo&longs;ophis; 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><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/>reuoea principium illud commune. </s></p><p type="main"> <s>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 type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma IV.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s>Hoc Axioma explicatione modicâ indiget: Determinatum illud <lb/>apello, quod illud ip&longs;um e&longs;t, quod e&longs;t, & nihil aliud; quod e&longs;t hoc, id <lb/>e&longs;t ab omni alio di&longs;tinctum; atqui quidquid productum e&longs;t, &longs;ingulare <lb/>e&longs;t, id e&longs;t, e&longs;t hoc; &longs;i enim producitur, alicubi producitur, & ali­<lb/>quando, ergo dici pote&longs;t, e&longs;t hîc, e&longs;t nune; igitur determinatum e&longs;t. </s> <s><lb/>Aliquis fortè &longs;tatim opponet mihi partes indeterminatas quantitatis: &longs;ed <lb/>pro&longs;ectò nulla pars actu e&longs;t quæ non &longs;it hæc, & non alia; 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. </s> <s>Dices aliquid po&longs;&longs;e e&longs;&longs;e nullibi; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma V.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quod vnum e&longs;t, determinatum e&longs;t.<emph.end type="italics"/></s><s> Quia quod vnum e&longs;t, e&longs;t hoc, & <lb/>nihil aliud; nihil enim aliud e&longs;t vnum, ni&longs;i indiui &longs;um in &longs;e, & diui­<lb/>&longs;um à quolibet alio: quipnè 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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma VI.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quidquid e&longs;t, fru&longs;trà non e&longs;t.<emph.end type="italics"/></s><s> 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; finem autem rei ex ip&longs;o v&longs;u cogno­<lb/>&longs;cimus; v&longs;um verò ip&longs;o ferè &longs;en&longs;u: quod vt breui inductione confirme­<lb/>mus, quidquid exi&longs;tit vel e&longs;t &longs;ub&longs;tantia, vel accidens; &longs;i &longs;ub&longs;tantia, vel <lb/>incorporea, vel corporea; &longs;i incorporea, vel e&longs;t Deus, vel Angelus, vel <pb pagenum="7"/>Animarationalis; atqui nihil horum fru&longs;trà e&longs;t, vt con&longs;tat; &longs;i corporea, <lb/>vel e&longs;t corpus, vel forma; &longs;i corpus, vel elementum, vel mixtum; <lb/>vtrumque &longs;uum finem habet, & con&longs;tantem v&longs;um; &longs;i forma quamdiu <lb/>e&longs;t principium actionum compo&longs;iti fru&longs;trà non e&longs;t; quippe ad cum finem <lb/>e&longs;t in&longs;tituta; 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; 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; at verò <lb/>anima rationalis, quæ aliquas actiones inorganicas habet, fru&longs;trà non <lb/>e&longs;t etiam &longs;eparata, igitur immortalis e&longs;t: vtramque rationem &longs;uo loco fu­<lb/>sè demon&longs;trabimus; &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; quem &longs;cilicet præ&longs;tat in &longs;uo &longs;ubiecto, cui certè &longs;i ni­<lb/>hil præ&longs;taret, in co fru&longs;trà e&longs;&longs;et; &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>Igitur tunc effet fru&longs;trà accidens, cum &longs;uo illo effectu careret; <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; &longs;ed non e&longs;t huius loci: cùm verò audis fi­<lb/>nem: 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 type="main"> <s>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 type="main"> <s>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>tot fluitantes atomi? </s> <s>tot in&longs;ecta? </s> <s>& vermiculi: Re&longs;pondeo <lb/>quamlibet &longs;tellam, quodlibet in&longs;ectum, &longs;eu vermiculum &longs;uis pollere pro­<lb/>prietatibus; igitur fru&longs;trà non e&longs;<gap/> & quodlibet punctum, quamlibet ato­<lb/>mum, & quamlibet guttulam aquæ e&longs;&longs;e partem huius vniuer&longs;itatis: quod <lb/>enim dices de vna, dicam de omnibus; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma VII.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Tunc ponenda e&longs;t forma distincta &longs;ub&longs;t antialis 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 type="main"> <s>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; atqui formam &longs;ub&longs;tantialem &longs;en&longs;u <lb/>non percipis immediatè; igitur aliquem eius effectum &longs;en&longs;ibilem vel me­<lb/>diatè, vel immediatè; qui certè &longs;i tribui po&longs;&longs;it materiæ, haud dubiè per il­<lb/>lum &longs;ormam non probabis, ni&longs;i formæ ip&longs;ius e&longs;&longs;e antè demon&longs;tres; &longs;i ve­<lb/>to e&longs;t forma accidentalis, quam &longs;en&longs;u percipis; certè id tantùm accidit ex <pb pagenum="8"/>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 type="main"> <s>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; 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; quæ <lb/>omnia fal&longs;i&longs;&longs;una 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 type="main"> <s>Igitur nunquam claudicat hic equus O<gap/>mi, vt vulgò dicitur, &longs;i hoc <lb/>fræno regatur, & præ&longs;cripto ambulet pa&longs;&longs;u. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s>Primum certum e&longs;t etiam certitudine metaphy&longs;ica, &longs;eu geometrica. </s> <s><lb/>Secundum, Quartum, & Quintum per Primum demon&longs;trari po&longs;&longs;unt. </s> <s><lb/>Tertium e&longs;t veluti communis po&longs;itio, &longs;eu commune po&longs;tulatum, in quo <lb/>docti omnes conucniunt; quippe nihil &longs;ine ratione dici debet à philo&longs;o­<lb/>pho; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma VIII.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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 e&longs;t quid­<lb/>quid incipit e&longs;&longs;e ab alio e&longs;t; quippe à &longs;e e&longs;&longs;e non pote&longs;t; nihil enim à &longs;e <lb/>ip&longs;o dependere pote&longs;t &longs;eu produci; 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: præterea quidquid produci­<lb/>tur aliquando producitur, & alicubi, vt certi&longs;&longs;imum e&longs;t; &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; Igitur &longs;i aliquid &longs;e producit; cur hîc potiùs quam illîc? </s> <s>cur <lb/>nunc potius quam antè? </s> <s>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>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; &longs;ufficit modò habere phy&longs;icam, quod ex om­<lb/>nibus hypothe&longs;ibus demon&longs;tratur; &longs;i enim aliquid de nouo produci­<lb/>tur, quod certum e&longs;t, abalio produci video: calor ab igne mediatè <lb/>vel immediatè, impetus à potentia motrice, vel ab alio impetu: cuncta <lb/>hæc &longs;i reueraproducuntur de quo alibi, ab alio produci con&longs;tat; in Me­<lb/>taphy&longs;ica hoc ip&longs;um geometricè demon&longs;trabimus; cum enim agere &longs;up­<lb/>ponat e&longs;&longs;e; quippe omnis actio alicuius agentis e&longs;t; & 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma IX.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cau&longs;a debet exi&longs;tere vt immediatè agat.<emph.end type="italics"/></s><s> Hoc certum e&longs;t; quia agere <lb/>&longs;upponit e&longs;&longs;e; 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. </s> <s>Iam vero &longs;ufficiat certum e&longs;&longs;e phi&longs;icè, vt con&longs;tat ex omnibus <lb/>hypoth. </s> <s>phy&longs;icis; nihil enim videmus agere, ni&longs;i quod e&longs;t; &longs;i enim age­<lb/>ret quod non e&longs;t; cur potius hîc, & nunc quam alibi, & aliàs? </s> <s>cur in <lb/>hoc &longs;ubiecto potius quàm in alio? </s></p><p type="main"> <s>Dices, finis qui non e&longs;t influit; igitur agit; Re&longs;pondeo finem non <lb/>agere, nec influere ni&longs;i obiectiuè; 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; 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 type="main"> <s>Dices, cau&longs;a principalis pulli exclu&longs;i pote&longs;t non e&longs;&longs;e; hæc omnia di­<lb/>&longs;cutiemus &longs;uo loco cum de generatione animalium; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma X.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cau&longs;a debet e&longs;&longs;e applicata vt immediatè agat.<emph.end type="italics"/></s><s> 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>quidquid &longs;it, certum e&longs;t phy&longs;icè; nec enim ignis, qui e&longs;t <lb/>Romæ, calefacit Lugduni. </s></p><p type="main"> <s>Dices dari fortè actionem in di&longs;tans; 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; 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 type="main"> <s>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>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 type="main"> <s>Dices etiam partes interiores cau&longs;æ v. </s> <s>g. </s> <s>Solis agunt, &longs;ed non agunt <lb/>per totum medium; alioquin agerent in alias partes Solis, à quibus <lb/>obteguntur. </s> <s>Re&longs;pondeo, diffu&longs;ionem vel propagationem actioms in­<lb/>choari tantum ab ipsâ &longs;uperficie Solis; quippe omnes partes agunt <lb/>actione communi, de quo infrà; atqui actio communis à communi m<gap/>­<lb/>dio incipit. </s></p><p type="main"> <s>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 type="main"> <s>Re&longs;pondeo, ignem quidem accendi in data di&longs;tantia; at non &longs;ine <pb pagenum="10"/>aliqua applicatione, &longs;altem virtutis, in quo non e&longs;t difficultas; 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 magnte. </s> <s>e </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma XI.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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> 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. </s> <s>g. </s> <s>calor e&longs;t cau&longs;a &longs;ufficiens caloris, vt con&longs;tat in aqua calida; <lb/>igitur &longs;i calor e&longs;t applicatus &longs;ubiecto, in quo producitur calor non &longs;upe­<lb/>rans vires caloris applicati; dicendum e&longs;t calorem illum ab hoc produ­<lb/>ci; cum calor &longs;it cau&longs;a nece&longs;&longs;aria; igitur &longs;i &longs;it applicatus &longs;ubjecto apto, <lb/>nece&longs;&longs;ariò agit; 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 type="main"> <s>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; licet aliud &longs;it <lb/>coniunctum, ex quo &longs;eor&longs;im &longs;umpto applicato non &longs;equitur effectus<emph.end type="italics"/>; v. </s> <s>g. </s> <s>ex <lb/>applicatione aquæ calidæ &longs;equitur productio caloris; ex applicatione &longs;o­<lb/>lius aquæ non &longs;equitur; 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 type="main"> <s>Dices non e&longs;&longs;e certum calorem produci; Re&longs;pondeo, negando; &longs;ed, <lb/>quidquid &longs;it, loquor tantùm hypotheticè; dixi enim &longs;i producatur, à <lb/>calore aquæ inhærente producitur. </s></p><p type="main"> <s>Dices produci po&longs;&longs;e ab aliqua cau&longs;a ignota po&longs;ita dumtaxat tali, vel <lb/>tali conditione. </s> <s>Re&longs;pondeo, hoc reuera geometricè non probari, &longs;ed <lb/>tantùm phy&longs;icè; quidquid &longs;it, voco cau&longs;am id, ex cuius applicatione <lb/>&longs;equitur &longs;emper effectus, & nunquam aliàs; nam phy&longs;icè loquendo, &longs;iue <lb/>&longs;it alia cau&longs;a, &longs;iue non, codem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma XII.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ca<gap/> nece&longs;&longs;aria &longs;ubiecto apto applicata, & non impedita &longs;emper agit, & <lb/>quantum pote&longs;t.<emph.end type="italics"/></s><s> Hoc Axioma duas partes habet; prima certa e&longs;t per hy­<lb/>poth. </s> <s>8. & per definitionem cau&longs;æ nece&longs;&longs;ariæ, quæ in hoc differt à libe­<lb/>tâ: Secunda pars probatur; 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><lb/>nam &longs;i vnam partem effectus omittat; cur vnam potiùs quam aliam? </s> <s><lb/>cur non duas? </s> <s>cur non omnes? </s> <s>denique video cau&longs;am eandem eidem <lb/>&longs;ubiecto codem modo applicatam, eundem &longs;emper effectum producere <lb/>per Hyp. </s> <s>8. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma XIII<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Exten&longs;io cau&longs;a non intendit effectum ad intra.<emph.end type="italics"/></s><s> Quælibet pars maioris <lb/>ignis non habet calorem inten&longs;iorem, quàm quælibet pars minoris; idem <pb pagenum="11"/>dico de grauitate plumbi, &c. </s> <s>nec enim libra plumbi coniuncta cum <lb/>alia habet diuer&longs;am grauitatem ab eâ, quam habet &longs;eparata. </s></p><p type="main"> <s>Dixi ad intra; quia ad extra multum iuuat exten&longs;io; &longs;ic maior ignis <lb/>longiùs diffundit &longs;uum calorem; corpus grauiùs cadens majorem ictum <lb/>infligit; Ad hoc Axioma reuocatur i&longs;tud. </s></p><p type="main"> <s>1. <emph type="italics"/>Omnes partes eiu&longs;dem cau&longs;æ agunt ad extra actions 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; ad quam diffundit coniunctum cum aliis; idem dico de igne maiori, <lb/>& minori; de quibus omnibus &longs;uo loco. </s> <s>Huc etiam reuoca dicta illa <lb/>communia. </s></p><p type="main"> <s>2. <emph type="italics"/>Plures partes cau&longs;a plures partes effectus producunt, & vici&longs;&longs;im.<emph.end type="italics"/></s></p><p type="main"> <s>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 type="main"> <s>4. <emph type="italics"/>Perfectior effectus, vel imperfectior arguit cau&longs;am perfectiorem, vel im­<lb/>perfectiorem, &longs;uppo&longs;itâ eâdem applicatione; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma XIV.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quidquid de&longs;truitur non e&longs;t à &longs;e.<emph.end type="italics"/></s><s> Hoc Axioma geometricum e&longs;t; Quod <lb/>enim e&longs;t à &longs;e, nece&longs;&longs;ariò e&longs;t; cùm à libertate &longs;eu voluntate alterius non <lb/>pendeat; cum enim primo in&longs;tanti quo res e&longs;t, non &longs;it à &longs;e per Axiom. </s> <s>8. <lb/>de &longs;ecundo idem dici debet, quod de primo, vt patet: 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; &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 type="main"> <s>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>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>quæ omnia fusè <lb/>in Metaphy&longs;ica demon&longs;trabimus; 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; &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 type="main"> <s>Huc reuoca Axiomata &longs;equentia, quæ ex hoc vno deducuntur. </s></p><p type="main"> <s>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 type="main"> <s>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> Hoc etiam ex hypothe&longs;ibus &longs;equitur; cum <lb/>enim de&longs;trui &longs;it idem ac de&longs;inere con&longs;eruari; 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 type="main"> <s>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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma XV.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Contraria pugnant pro rata.<emph.end type="italics"/></s><s> Nec enim alia regula e&longs;&longs;e pote&longs;t; &longs;ic minot <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><lb/>Ratio e&longs;t, quia plùs vel minùs contrarij de&longs;truere, multam habet ex­<lb/>ten&longs;ionem. </s> <s>v.g. </s> <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>Si infra, cur potius 4. quam 3. <lb/>2. &c. </s> <s>Igitur cum plures &longs;int termini tùm infra, tùm &longs;upra 5. cur potius <lb/>vnus quàm alius? </s> <s>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>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: 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>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;alrem ex parte. </s></p><p type="main"> <s>Ob&longs;eruabis hæc Axiomata &longs;altem maiori ex parte e&longs;&longs;e metaph. </s> <s>quæ <lb/>nos fusè in Theorematis metaph. </s> <s>explicabimus, & demon&longs;trabimus; &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 type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Postulatum,<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus e&longs;t aliquid realiter di&longs;tinctum à mobili.<emph.end type="italics"/></s><s> Demon&longs;tratur; Motus <lb/>e&longs;t in mobili, in quo antè non erat per hypoth. </s> <s>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; igi­<lb/>tur à motu &longs;eparatum; igitur realiter di&longs;tinctum per Ax. 2. præterea <lb/>moueri, & non moueri &longs;unt prædicata contradictoria, vt con&longs;tat; igi­<lb/>tur cidem &longs;imul ine&longs;&longs;e non po&longs;&longs;unt per Ax. 1. igitur cum co non &longs;unt <lb/>idem; alioquin &longs;imul e&longs;&longs;ent; igitur alterum illorum e&longs;t di&longs;tinctum à <lb/>mobili; non quies, vt con&longs;tat, quæ e&longs;t tantùm negatio motus, &longs;eu per­<lb/>&longs;euerantia in codem loco; igitur nullam dicit mutationem; at verò <lb/>motus mutationem dicit, per Def. </s> <s>1. hoc Theorema fusè demon&longs;trabo <lb/>in Metaph. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus non pote&longs;t dici propriè preductus immediatè, vel effectus immedia­<lb/>tus cau&longs;æ efficientis.<emph.end type="italics"/></s><s> Demon&longs;t. </s> <s>Motus e&longs;t mutatio, &longs;eu tian&longs;itus ex loco <lb/>in locum per Def. </s> <s>1. &longs;ed mutatio propriè non producitur; qu<gap/>ppè pro­<lb/>ductio tantùm terminatur ad ens; nihil enim ni&longs;i ens produci pote&longs;t; <pb pagenum="13"/>atqui nulla mutatio dicit tantùm ens; præ&longs;ertim hæc, quæ tantùm dicit <lb/>terminum à quo, ide&longs;t locum relictum; & terminum ad quem, id e&longs;t lo­<lb/>cum immediatum acqui&longs;itum; nam &longs;eparato quocunque alio ab ip&longs;o <lb/>mobili; 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; illud autem e&longs;t motus, qui certè non <lb/>dicit tantùm locum relictum &longs;ine acqui&longs;ito; alioqui &longs;i mobile de&longs;true­<lb/>retur, diceretur moueri; nec etiam locum acqui&longs;itum &longs;ine priori relicto: <lb/>alioqui &longs;i mobile primò produceretur, diceretur moueri localiter; igitur <lb/>motus neutrum dicit &longs;eor&longs;im; &longs;i primum, diceretur de&longs;tructus; &longs;i &longs;ecun­<lb/>dum, diceretur aliquo modo productus, vel potiùs acqui&longs;itus; at vtrum­<lb/>que coniunctim, &longs;imulque e&longs;&longs;entialiter dicit motus; nec enim conci­<lb/>pio aliud, dum concipio motum: porrò vtrumque &longs;imul &longs;umptum indi­<lb/>ui&longs;ibiliter non pote&longs;t dici, vel de&longs;tructum propriè, vel productum; Di<gap/><lb/>xi propriè; nam impropriè dici pote&longs;t motus productus. </s></p><p type="main"> <s>Dices Motus e&longs;t ens, non à &longs;e; igitur ab alio; igitur motus e&longs;t pro­<lb/>ductus. </s> <s>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; <lb/>quòd certè non pote&longs;t dici propriè productum, &longs;ed re&longs;ultans, vt relatio; <lb/>nam producatur, &longs;i fieri pote&longs;t; certè e&longs;t aliquid, quod tam facilè de­<lb/>&longs;trui pote&longs;t, quam produci; 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 type="main"> <s>Dices potentia motrix e&longs;t actiua; igitur agit; igitur producit, &longs;ed ni­<lb/>hil ni&longs;i motum. </s> <s>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 type="main"> <s>Nec e&longs;t quod aliqui ita mirentur hæc à me dici; cum certum &longs;it effe­<lb/>ctus &longs;ormales &longs;ecundarios principum ferè qualitatum tales e&longs;&longs;e, vt mini­<lb/>mè producantur; &longs;ed qua&longs;i re&longs;ultent ab exigentia; v. </s> <s>g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus e&longs;t ab alio di&longs;tincto in aliquo genere cau&longs;æ.<emph.end type="italics"/></s><s> 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis motum localem e&longs;&longs;e duplicis generis; 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; cum etiam &longs;ine motu defatigetur, vt cum <lb/>quisalium pellit, à quo pellitur æquali ni&longs;u; patet etiam in manu &longs;u­<lb/><gap/>ine nte aliquod pondus, quæ non mouetur; licet reuerâ etiam &longs;ummo <pb pagenum="14"/>conatu agat: immò &longs;i potentia motrix produceret motum primum, non <lb/>impetum in corpore proiecto; nulla deinde e&longs;&longs;et cau&longs;a applicata ad pro­<lb/>ducendum impetum: Itaque hic motus primi generis, &longs;i comparetur <lb/>cum potentia motrice, e&longs;t verè in&longs;luxus, vel actio; &longs;i cum termino, e&longs;t <lb/>eius fieri, &longs;eu dependentia; &longs;i cum &longs;ubiecto, &longs;eu mobili e&longs;t pa&longs;&longs;io; 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>Motus &longs;ecundi generis e&longs;t mutatio, &longs;eu tran&longs;itus ex <lb/>vno loco in alium; hoc e&longs;t finis, vel effectus formalis &longs;ecundarius, <lb/>quem exigit impetus; & 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cau&longs;ailla 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> Sic calor exigit rarefactionem, vel <lb/>re&longs;olutiouem, impetus motum; cum enim non &longs;it cau&longs;a efficiens per Th. </s> <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; vt patet ex ip&longs;a cau&longs;arum <lb/>enumeratione; non e&longs;t materialis, quia non recipit motum, ni&longs;i ab alio; <lb/>nec finalis, quæ &longs;upponit alias; cum ip&longs;a non &longs;it dum ponitur <lb/>effectus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>haud dubiè &longs;ub&longs;tantia lapidis non e&longs;t cau&longs;a <lb/>huius motus; 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; igitur &longs;emper cau&longs;aret per Ax.12. præ­<lb/>terea potentia motrix proiicientis verè agit, cum etiam defatigetur; igi­<lb/>turaliquid producit, non motum immediatè, qui produci non pote&longs;t pro <lb/>prièper Th. 2. Adde quod motus &longs;ecundi generis habet tantùm caulam <lb/>immediatam exigentem, &longs;ed potentia motrix non exigit; quia primò <lb/>non defatigaretur exigendo; &longs;ecundò quia lapis &longs;eparatus à manu etiam <lb/>mouetur, &longs;ed non ad exigentiam potentiæ motricis, vt patet; quia &longs;tatim <lb/>po&longs;t &longs;eparationem pote&longs;t illa potentia de&longs;trui, licèt lapis longo pò&longs;t <lb/>temporemoueatur; &longs;ed quod non e&longs;t, nihil exigit. </s></p><p type="main"> <s>Aliquis fortè diceret potentiam motriœm exigere primam partem <lb/>motus, quæ deinde &longs;ecundam exigit, & &longs;ecunda tertiam, tertia quar­<lb/>tam, &c. </s> <s>Sed contra; quæro quid &longs;it prima illa pars motus; 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; 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"/><p type="main"> <s>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>Re&longs;p. negando, tùm quia omnis potentia motrix <lb/>agit; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Datur impetus.<emph.end type="italics"/></s><s> 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 type="main"> <s>Demon&longs;tro primò dari impetum: 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; igitur habet cau&longs;am, &longs;ed non aliam, quam impetum, quod pro­<lb/>bo: 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; 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 type="main"> <s>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 type="main"> <s>Secundò probatur; potentia motrix e&longs;t actiua, quia defatigatur, quis <lb/>hoc neget? </s> <s>igitur aliquid producit; non motum, qui propriè non pro­<lb/>ducitur per Th.2. igitur aliquid aliud; voco impetum; adde quod etiam <lb/>&longs;ine motu agit, & defatigatur vt iam dictum e&longs;t; 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 type="main"> <s>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><lb/>atqui potentia motrix non e&longs;t can&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. Non e&longs;t etiam cau&longs;a &longs;ub&longs;tantia <gap/>pilæ <lb/>mobilis per Th.5.5. nec priores pattes motus per re&longs;p. </s> <s>ad primam in­<lb/>&longs;tantiam Th 5. igitur aliquid aliud; voco impetum. </s></p><p type="main"> <s>Quartò probatur; 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>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; &longs;ed decre&longs;cit &longs;ub&longs;tantia mobilis, nec potentia mo-<pb pagenum="16"/>trix, vel corpus prius impactum; ergo e&longs;t alia cau&longs;a præ&longs;ens, quæ mi­<lb/>nuitur; voco impetum. </s></p><p type="main"> <s>Quintò corpus graue deor&longs;um cadens accelerat &longs;uum motum, vt patet <lb/>experientia; quæ maximè clara e&longs;t in funependulis, de qua in &longs;equen­<lb/>tibus libris; igitur debet e&longs;&longs;e cau&longs;a huius motus velocioris; 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 inttin&longs;ecum, <lb/>voco impetum. </s> <s>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 type="main"> <s>Diceret fortè alius hæc omnia e&longs;&longs;e dubia; nam fieri pote&longs;t vt Deus <lb/>tantùm moueat; quod &longs;ine impetu fieri po&longs;&longs;e certum e&longs;t; Re&longs;p. equi­<lb/>dem per miraculum hoc fieri po&longs;&longs;e; &longs;ed quemadmadum certum e&longs;t phy­<lb/>ficè 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>adde quod Deus, vt auctor naturæ <lb/>e&longs;t, agit tantùm; 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 type="main"> <s>Dicerent alij hoc totum prouenire à corpu&longs;culis; vel atomis, vel fila­<lb/>mentis &longs;ine vlla actione; equidem non reiicio corpu&longs;cula, & perennia <lb/>corporum effluuia: 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><lb/>immò maximam &longs;uarum partium partem; igitur cum deinde proiicitur <lb/>idem globus, illæ partes mouentur; dari igitur debet cau&longs;a huius motus <lb/>per Ax.8, igitur impetus: nec dicas moueri illas partes à corpu&longs;culis; quia <lb/>antè erant eadem, immò plura corpu&longs;cula; & tamen non mouebant: 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>vel enim mutantur, vel non mutantur; &longs;ecundum dici <lb/>non pote&longs;t; quia vt excitentur, ex non excitatis mutari debent; igitur <lb/>per aliquid: 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 moucbuntur, & excitabuntur per Ax.12. quia &longs;unt applicatæ <lb/>cau&longs;æ nece&longs;&longs;ariæ. </s></p><p type="main"> <s>Dico &longs;ecundò minimum ex his corpu&longs;culis non &longs;emper moueri; po­<lb/>te&longs;t enim &longs;i&longs;tere; quis hoc neget? </s> <s>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 type="main"> <s>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 type="main"> <s>Diceret aliquis lapidem impul&longs;um ab aëre deinde propelli; &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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus est aliquid distinctum à &longs;ubstantiâ mobilis.<emph.end type="italics"/></s><s> Demon&longs;tratur. </s> <s><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. </s> <s>3. & Th. 6. de codem contradictoria dici <lb/>non po&longs;&longs;unt per Ax. 1. n. </s> <s>3. Igitur impetus non e&longs;t idem cum &longs;ub&longs;tantià <lb/>mobilis; igitur di&longs;tinctus; dcinde &longs;eparari pote&longs;t à &longs;ub&longs;tantia mobilis <lb/>per Hypoth. </s> <s>4. igitur e&longs;t di&longs;tinctus per Ax. 2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus est accidens<emph.end type="italics"/>; 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 moucri, vt con&longs;tat ex po&longs;t. </s> <s>& ex Hypoth. </s> <s>3. & 4. igitur di&longs;tingui­<lb/>tur à motu; igitur & ab impetu per Ax. 2. igitur impetus non e&longs;t &longs;ub­<lb/>&longs;tantia; igitur accidens. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus non e&longs;t modus.<emph.end type="italics"/></s><s> Modus duplicis generis e&longs;&longs;e pote&longs;t: 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>Modus &longs;ecundi generis non <lb/>e&longs;t entitas quidem di&longs;tincta; e&longs;t tamen &longs;tatus quidam corporis; &longs;ic &longs;e&longs;&longs;i<gap/><lb/>e&longs;t modus, conden&longs;atio, compre&longs;&longs;io, &c. </s> <s>His po&longs;itis Impetus non e&longs;t mo­<lb/>dus primi generis; nihil enim probat impetum e&longs;&longs;e modum, quod etiam <lb/>non probet calorem, & luccm e&longs;&longs;e modos; dicere autem omnia acci­<lb/>dentia e&longs;&longs;e modos non debemus, de quo &longs;uo loco; 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; v. </s> <s>g. </s> <s>actio non pote&longs;t concipi ni&longs;i &longs;it <lb/>alicuius actio; nec fieri &longs;ine facto; nec via &longs;ine termino; nec dependen­<lb/>tia &longs;ine dependente; at verò po&longs;&longs;um concipere calorem, & impetum <lb/>&longs;ine alio, quod &longs;it actu; 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><lb/>quod exigunt, hoc e&longs;t &longs;ine &longs;uo &longs;ine; igitur &longs;ine &longs;ubiecto; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus e&longs;t qualitas Phy&longs;ica.<emph.end type="italics"/></s><s> Sequitur ex dictis; cum nec &longs;it motus. </s> <s><lb/>nec &longs;ub&longs;tantia, nec modus, nec quidquam negatiuum, alioquin exige­<lb/>ret; igitur e&longs;t aliud accidens; vocetur qualitas. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus est qualitas Phy&longs;ica.<emph.end type="italics"/></s><s> 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. </s> <s>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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus est qualitas permanens.<emph.end type="italics"/></s><s> Quia lapis proiectus etiam &longs;eparatus <lb/>mouetur aliquandiu per Hyp. </s> <s>6. igitur durat eius cau&longs;a, &longs;cilicet impe­<lb/>tus; igitur e&longs;t qualitas permanens. </s></p><p type="main"> <s>Diceret fortè aliquis lapidem proiectum pelli ab aëre à tergo in&longs;tan­<lb/>te, vt voluit Ari&longs;toteles pluribus in locis; &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; Primò quia non <lb/>modò non iuuat aër; &longs;ed ctiam impedit motum proiecti, quod de omni <lb/>medio nece&longs;&longs;ariò dicendum e&longs;t, vt patet experientiâ; vnde quo cra&longs;&longs;ius, <lb/>&longs;eu den&longs;ius e&longs;t <expan abbr="mediũ">medium</expan>, motum potentiùs ratardat, vt videmus in proiectis <lb/>per aquam; rationem à priori afferemus infrà, cum de re&longs;i&longs;tentia medij: <lb/>Secundò, quis dicat pilam rotatam in &longs;olo moueri aëris appul&longs;u? </s> <s>cum <lb/>alia corpora, quæ pila rotata præterlambendo qua&longs;i allambit, nullo mo­<lb/>do moueantur; præ&longs;ertim granula pulueris. </s> <s>Tertiò, an fortè aër id præ­<lb/>&longs;tare pote&longs;t &longs;ine vi impre&longs;&longs;a; igitur non minus ip&longs;i pilæ proiectæ, quam <lb/>aëri ambienti imprimi poterit: Quartò, nullus aër à tergo pellitur; &longs;ed <lb/>potius ip&longs;a pila aducr&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>Quintò denique, cum diu moueatur cadem <lb/>pars aëris, haud dubiè in ca manet vis impre&longs;&longs;a; igitur impetus erit ad­<lb/>huc qualitas permanens. </s></p><p type="main"> <s>Ad id quod obiicitur ex Ari&longs;totele; aliqui putant inclina&longs;&longs;e in cam &longs;en­<lb/>tentiam; cùm tam en no&longs;tram tencant illu&longs;tres Peripatetici, quorum no­<lb/>minibus parco, ne tot citationes paginas impleant; vide apud Conim­<lb/>bric. </s> <s>l. </s> <s>7. Phy&longs;. cap. 2. Aliqui excu&longs;ant ip&longs;um Ari&longs;torelem, putantque <lb/>non e&longs;&longs;e locutum ex propriâ &longs;ententiâ: Alij dicunt Ari&longs;totelem quidem <lb/>tribui&longs;&longs;e aliquam vim extrin&longs;ecam aëri; non tamen nega&longs;&longs;e intrin&longs;ecam <lb/>impetus; quidquid &longs;it, ip&longs;a verba Ari&longs;totclis 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus non producit motum.<emph.end type="italics"/></s><s> Probatur, quia motus non dicitur pro­<lb/>ductus per Th. 2. Adde &longs;i vis rationem metaphy&longs;icam; quia nihil cogit <lb/>dicere accidens aliquod, ex iis &longs;cilicet, quæ &longs;en&longs;u percipimus, agere ad <lb/>intra; quod videtur e&longs;&longs;e proprium &longs;ub&longs;tantiæ, &longs;altem naturaliter; vt <lb/>demon&longs;trabimus in Metaph. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus exigit motum, id est fluxum mobilis in loco<emph.end type="italics"/>; 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; quæ vel e&longs;t localis, hoc e&longs;t fluxus quidam: <pb pagenum="19"/>per &longs;patium loci; vel alteratiua, vt vulgò vocatur; quà &longs;cilicet vel re­<lb/>&longs;oluuntur partes, vel rare&longs;iunt, vel lique&longs;cunt, vel concre&longs;cunt &c. </s> <s>vel <lb/>demùm mutant &longs;en&longs;ibilem &longs;tatum; 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>&c. </s> <s>de quibus aliàs. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus e&longs;t effectus formalis &longs;ecundarius impetus.<emph.end type="italics"/></s><s> Cum enim &longs;it cau&longs;a <lb/>exigens per Th. 121. Voco effectum formalem &longs;ecundarium, quem in <lb/>mobili exigit impetus; quippe, vt iam dictum e&longs;t, cau&longs;a exigens redu­<lb/>citur ad formalem; nec enim cau&longs;at aliquid producendo, quod &longs;pectat ad <lb/>efficientem; nec mouendo, quod &longs;pectat ad finalem; nec determinando, <lb/>quod &longs;pectat ad obiectiuam; nec recipiendo, quod &longs;pectat ad materia­<lb/>lem; nec dirigendo, quod &longs;pectat ad idæalem, vel exemplarem; &longs;ed <lb/>exigendo; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus e&longs;t finis intrin&longs;ecus impetus.<emph.end type="italics"/></s><s> Dum finem audis intrin&longs;ecum, <lb/>cogita quæ&longs;o aliquid phy&longs;icum; e&longs;t enim id, propter quod talis, vel ta­<lb/>lis forma in&longs;tituta e&longs;t: quid enim aliud e&longs;&longs;e pote&longs;t; finem enim rerum <lb/>naturalium ex ip&longs;o v&longs;u cogno&longs;cimus; immò idem e&longs;t finis cum ip&longs;o v&longs;u; <lb/>cum igitur impetus illum tantùm v&longs;um habeat, quem in ip&longs;o mobili <lb/>præ&longs;tare cernimus, &longs;cilicet motum; dicendum e&longs;t motum e&longs;&longs;e finem in­<lb/>trin&longs;ecum impetus; 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>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 veiut <lb/>appetentiâ concupi&longs;cit, vel exigit. </s> <s>Dixi mediatè, vel immediatè; num <lb/>reuera datur fortè aliquis impetus, vt dicemus infrà; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ni&longs;i e&longs;&longs;et motus non e&longs;&longs;et impetus.<emph.end type="italics"/></s><s> 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 finc ca­<lb/>reret impetus; igitur non e&longs;&longs;et, vt patet, igitur non e&longs;&longs;et; 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 vtrecti <lb/>di&longs;tinguamus ab acqui&longs;ito quem vocabimus impetum accelerationis) <lb/>qui &longs;ine motu con&longs;eruatur in corpore grauitante; quia ni&longs;i po&longs;&longs;ibilis e&longs;­<lb/>&longs;et motus deor&longs;um nulla e&longs;&longs;et grauitatio; quippe grauitare e&longs;t dcor­<lb/>&longs;um inclinari, motumque inclinationis impediri; hinc dicemus <pb pagenum="20"/>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>immò dicemus infrà primo in&longs;tanti, <lb/>quo e&longs;t impetus, nondum e&longs;&longs;e motum. </s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ni&longs;i e&longs;&longs;et impetus, non e&longs;&longs;et naturaliter motus.<emph.end type="italics"/></s><s> Quia ni&longs;i e&longs;&longs;etcau&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; &longs;iue &longs;it naturalis in grauibus, & leuibus, &longs;iue &longs;evitalis <lb/>in viuentibus; &longs;iue &longs;it media in compre&longs;&longs;is, & dilatatis; &longs;iue alia quæli­<lb/>bet: &longs;ed omnis potentia motrix e&longs;t actiua, quia mouet; ergo agit, &longs;ed <lb/>motum non producit per Th. 2. Igitur impetum, qui deinde exigit mo­<lb/>tum per Th. 14. Dixi naturaliter; quia non e&longs;t dubium, quin Deus &longs;ine <lb/>impetu aliquo modo mouere po&longs;&longs;it; ide&longs;t, facere &longs;ine impetu, vt corpus <lb/>mutet locum: nec dicas Deum non po&longs;&longs;e &longs;upplere vices cau&longs;æ formalis; <lb/>nam concedo id quidem pro effectu formali primario; nec enim Deus <lb/>pote&longs;t facere, vt aliquid &longs;it calidum &longs;ine calore; cum e&longs;&longs;e calidum &longs;it <lb/>idem, ac e&longs;&longs;e habens calorem; id tamen nego pro effectu &longs;ecundario, <lb/>quem &longs;cilicet cau&longs;a formalis exigit: Etenim &longs;icut pote&longs;t &longs;ummo iure non <lb/>&longs;atisfacere exigentiæ; 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>quanquam vt verum fatear non e&longs;&longs;et propriè motus, &longs;ed <lb/>qua&longs;i continuæ reproductionis modus; nam motus dicit aliquam pa&longs;­<lb/>&longs;ionem; &longs;cilicet actum entis in potentiâ, vt aiunt. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s></p><p type="main"> <s><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; quia nullus impetus exigeret; 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;etmotus, vt iam dictum e&longs;t, quia non e&longs;­<lb/>&longs;et pa&longs;&longs;io. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Corpus illud æquali &longs;emper motu ferretur per &longs;e<emph.end type="italics"/>; Quia e&longs;&longs;et &longs;emper ca­<lb/>dem cau&longs;a nece&longs;&longs;aria motus, id e&longs;t, ip&longs;a entitas corporis; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si e&longs;&longs;et aliquod corpus e&longs;&longs;entialiter mobile, impetu non indigeret.<emph.end type="italics"/></s><s> Probatur; <lb/>quia in tantum indiget mobile impetu vt impetus exigat motum; &longs;ed <lb/>corpus illud per &longs;uam e&longs;&longs;entiam exigeret motum; igitur non indigeret <lb/>impetu; po&longs;&longs;et tamen impediri cius motus, vt patet; immò e&longs;&longs;et capax <lb/>recipiendi impetus., &longs;iue quem in ip&longs;o produceret, &longs;iue quem ab a lia <pb pagenum="21"/>cau&longs;a extrin&longs;eca acciperet. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s></p><p type="main"> <s><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> Pro­<lb/>batur; quia, ni&longs;i e&longs;&longs;et motus, non e&longs;&longs;et impetus per Th. 17. igitur &longs;ubic­<lb/>ctum incapax motus e&longs;t incapax impetus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s></p><p type="main"> <s><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> 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s></p><p type="main"> <s><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; quia impetus in eo non e&longs;&longs;et fru&longs;trà; haberet enim <lb/>fuum effectum formalem, & finem intrin&longs;ecum. </s> <s>Probatur 2.pars; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Omne corpus finitum e&longs;t capax motus, & impetus.<emph.end type="italics"/></s><s> Probatur 1. pars; <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>Dices fortè quia affixum e&longs;&longs;et e&longs;&longs;entialiter tali, vel tali lo­<lb/>co, &longs;ed contra; quia de&longs;truantur omnia, præter ip&longs;um corpus; certè <lb/>nulli affixum manet. </s> <s>Dices &longs;patio imaginario; apage i&longs;tas nugas: <lb/>de i&longs;to &longs;patio plura demon&longs;trabimus in Metaphy. </s> <s>Probatur 2. pars; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quod dur at tantùmvno in&longs;tanti, e&longs;t incapax motus, & impetus.<emph.end type="italics"/></s><s> Pro­<lb/>batur, quia non e&longs;t moueri, ni&longs;i relinquat locum, & acquirat alium; &longs;ed <lb/>1. acquirere locum, e&longs;t 1. e&longs;&longs;e in illo loco; & relinquere locum e&longs;t, <lb/>1. non e&longs;&longs;e in co loco; 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; & phy­<lb/>ficè certum e&longs;t ex omni hypothe&longs;i; 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; &longs;ed quod durat <lb/>tantùm in in&longs;tanti, non habet antè, neque po&longs;t; igitur quod durat tan­<lb/>tùm vno in&longs;tanti, moucri non pote&longs;t; igitur e&longs;t incapax motus; igitur <lb/>& impetus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Deus e&longs;t incapax motus, & impetus<emph.end type="italics"/>: Tum quia vbique, e&longs;t igitur <lb/>nouum locum acquirere non pote&longs;t; igitur nec moueri per Deffinitio­<lb/>nem 1. tùm quia æternitas Dei tota &longs;imul e&longs;t; igitur nec fuit antè, ne­<lb/>que po&longs;t in ca; igitur non pote&longs;t dici antè habui&longs;&longs;e locum, quo nunc <lb/>caret: & nunc non habere illum quo caret; tùm quia immutabilitas <pb pagenum="22"/>Dei hoc prohibet; nam moucri, e&longs;t affici intrin&longs;ecè; 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>præterea &longs;i &longs;unt, &longs;unt e&longs;&longs;entia­<lb/>liter immobiles; igitur valet &longs;emper ratio allata; igitur Deus e&longs;t inca­<lb/>pax motus; igitur & impetus. </s></p><p type="main"> <s>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>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; atqui &longs;i e&longs;t immen&longs;us, non pote&longs;t mutari intrin&longs;ecè per motum; <lb/>quia nullum locum de nouo acquireret; &longs;ed de hoc motu aliàs, cum de <lb/>infinito; vel de puncto phy&longs;ico mobili; quidquid &longs;it. </s> <s>valet &longs;altem <lb/>1. ratio pro motu recto, & aliæ duæ pro omni motu. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus ip&longs;e mouerinon pote&longs;t.<emph.end type="italics"/></s><s> Quia cum tantùm dicat mutationem <lb/>loci; certè mutatio non e&longs;t in loco; dicit enim tantùm locum relictum <lb/>eo in&longs;tanti, quo nouus acquiritur. </s> <s>Præterea quod e&longs;t in loco dicit tan­<lb/>rùm ens phy&longs;icum; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Duratio moueri non pote&longs;t.<emph.end type="italics"/></s><s> 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc actio moueri non pote&longs;t<emph.end type="italics"/>; 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 moucri non po&longs;&longs;it, per Th. 29. certè neque <lb/>actio moueri pote&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc in tanta rerum creatarum multitudine &longs;unt tantùm duæ, quæ <lb/>&longs;unt e&longs;&longs;entialiter immobiles; &longs;cilicet motus, & actio; quorum ille cum <lb/>&longs;it mutatio non e&longs;t adæquatè aliquid po&longs;itiuum; &longs;ecus actio. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Hinc &longs;unt tantùm duo adæquatè po&longs;itiua, quæ moucri non po&longs;&longs;uat; <lb/>&longs;cilicet Deus, & actio; Deus, qui &longs;emper e&longs;t; actio, quæ tantùm vno <lb/>in&longs;tanti e&longs;t; Deus vbique e&longs;&longs;entialiter; actio hic tantum e&longs;&longs;entialiter; <lb/>Deus primum ens; actio infinitum ens; e&longs;t enim modus; Deus primum <lb/>mouens; actio ip&longs;e motus; &longs;cilicet primi generis, de quo in &longs;ect. </s> <s>Th.3. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Hinc &longs;i res aliqua creata per actionem tantæ perfectionis, quæ mille <lb/>annis e&longs;&longs;entialiter re&longs;ponderet, con&longs;eruaretur; certè per totum illud <lb/>tempus moucri non po&longs;&longs;et; e&longs;&longs;et enim vnicum in&longs;tans, hoc e&longs;t duratio <pb pagenum="23"/>tota &longs;imul; &longs;ed codem in&longs;tanti in pluribus locis e&longs;&longs;e non pote&longs;t; igitur <lb/>nec moueri; adde quod per cam actionem &longs;um in loco, per quam &longs;um <lb/>in tempore; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scolium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis primò ex dictis præclarum naturæ in&longs;titutum; cum enim <lb/>corpus moueri &longs;emper non debeat, (quippe hoc e&longs;&longs;et maximè incom­<lb/>modum) certè per &longs;uam entitatem moucri non exigit; alioquin &longs;emper <lb/>moueretur; igitur per aliud ab entitate di&longs;tinctum, id e&longs;t per impetum; <lb/>itaque licet per &longs;uam entitatem exigat fluxum in tempore, id e&longs;t con&longs;er­<lb/>nari, & durare; id e&longs;t nouam &longs;emper actionem con&longs;eruatiuam; quia <lb/>maximum eius bonum e&longs;t durare vel exi&longs;tere; Igitur per &longs;e ip&longs;um illud <lb/>exigit; 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 type="main"> <s>Ob&longs;eruabis &longs;ecundò, cum idem corpus aliquando velociùs, tardiùs <lb/>aliquando moueri exigat; &longs;i per &longs;uam entitatem moueri exigeret, eo­<lb/>dem &longs;emper ferretur motu; quia cadem &longs;emper e&longs;&longs;et exigentia; igitur <lb/>debet e&longs;&longs;e aliquid aliud; illud autem e&longs;t impetus, qui aliquando maior <lb/>&longs;eu perfectior, aliquando verò minor e&longs;t; igitur maiorem &longs;eu velcio­<lb/>rem motum aliquando exigit, aliquando minorem, &longs;eu tardiorem; <lb/>cum enim motus &longs;it eius finis intrin&longs;ecus, vt re&longs;olutio e&longs;t finis caloris <lb/>vel rarefactio; 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 type="main"> <s>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 cius effe­<lb/>ctus formalis &longs;ecundarius: nec alia regula, præter eam excogitari pote&longs;t; <lb/>cum enim aliqua forma ad talem, vel talem finem à natura in&longs;tituta e&longs;t; <lb/>certè propter illum finem e&longs;t, igitur in co non e&longs;t, in quo &longs;uum finem <lb/>con&longs;equi non pote&longs;t; alioquin fru&longs;trà e&longs;&longs;et; & contra in eo e&longs;&longs;e pote&longs;t, <lb/>in quo fru&longs;trà non e&longs;t; cum &longs;cilicet in eo &longs;uum finem con&longs;equatur; ad­<lb/>de quod finis ille intrin&longs;ecus phy&longs;icus &longs;cilicet, non moralis, aliquis no­<lb/>uus effectus e&longs;t; 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 e&longs;&longs;ectu; 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 type="main"> <s>Ob&longs;eruabis quartò egregiam rationem; propter quam res eadem in <lb/>pluribus locis naturaliter e&longs;&longs;e non pote&longs;t; quippe cum res fuerit primo <lb/>producta in aliquo loco, illa certè nouum locum acquirere non pote&longs;t <lb/>naturaliter; ni&longs;i per motum, atqui motus dicit nece&longs;&longs;ario priorem lo­<lb/>tum relictum, & nouum acqui&longs;itum; igitur cum tot acquirantur loca <lb/>per motum, quot relinquuntur; &longs;i ante motum vnus tantùm erat eiu&longs;­<lb/>dem rci locus, po&longs;t motum etiam vnus e&longs;t: quod autem producatur tan-<pb pagenum="24"/>tùm res in vno loco patet; vel enim à cau&longs;a prima vel ab aliqua 2. pro­<lb/>ducitur; &longs;i à 2. ergo ab aliqua aplicata; 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; 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; &longs;ingulis terminis &longs;ingulæ <lb/>viæ; immò hoc requiri videtur, &longs;eu &longs;pectare ad huius vniuer&longs;itatis or­<lb/>dinem; quippe &longs;i res eadem in pluribus locis e&longs;&longs;et; cur potius in duo­<lb/>bus quam in tribus? </s> <s>deinde multiplex iure po&longs;&longs;et exi&longs;timari; denique <lb/>quod vnum e&longs;t in entitate creata, &longs;eu dependente ab cadem cau&longs;a, vnum <lb/>e&longs;t etiam in dependentia; quæ e&longs;t actio, per quam dependet; &longs;ed de his <lb/>aliàs. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s></p><p type="main"> <s><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> 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; &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. </s> <s>17. adde quod non pote&longs;t cogno&longs;oi <lb/>impetus, ni&longs;i vel ex motu, vel ex ictu, vel ex contrario ni&longs;u, vel <lb/>impul&longs;u; &longs;ed nihil horum cernitur in rupe quam ferio; Igitur non <lb/>e&longs;t dicendum in ea produci impetum, cuius rationem afferemus infrà; <lb/>nunc&longs;atis e&longs;t Ax. 3. id manife&longs;tè probari; nam qui diceret in rupe im­<lb/>mobili impetum imprimi; certè po&longs;itiuo argumento probare tenere­<lb/>tur, quod tantùm duci pote&longs;t, vel ab experimento; atqui hîc nullum e&longs;t; <lb/>vel à nece&longs;&longs;itate, quæ nulla e&longs;t; vel ex alio quocumque capite, quod <lb/>nullum excogitari pote&longs;t; &longs;ed maiorem lucem huic Th. 3. ex proximè <lb/>&longs;equentibus accer&longs;emus; nec e&longs;t quòd aliqui dicant produci impetum <lb/>inefficacem; qui cum fru&longs;trà &longs;it, &longs;i e&longs;t, ex nullo capite probari pote&longs;t: ad­<lb/>de quòd de&longs;truitur impetus, ne &longs;it fru&longs;trà; Igitur non producitur, ne &longs;it <lb/>fru&longs;trà; nam con&longs;eruatio e&longs;t vera actio, vt dicemus &longs;uo loco; Igitur &longs;i <lb/>hæc non ponitur, ne aliquid &longs;it fru&longs;trà; etiam 1. productio poni non <lb/>debet; vnde commentum illud impetus inefficacis pror&longs;us inefficax e&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ideo potentia motrix non producit impetum in pradicta rupe.<emph.end type="italics"/> v.g. <emph type="italics"/>quia de­<lb/>bilior e&longs;t.<emph.end type="italics"/></s><s> Probatur, & explicatur; quippedebilior potentia minorem ef­<lb/>fectum producit per. </s> <s>Ax. 13. <emph type="italics"/>num.<emph.end type="italics"/> 2. igitur pauciores partes impetus <lb/>æquales vni certæper 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; cum vna pars impetus duobus parti­<lb/>bus &longs;ubiecti ine&longs;&longs;e non po&longs;&longs;it; licet plures vni &longs;imul in e&longs;&longs;e po&longs;&longs;int; <lb/>non e&longs;t mirum &longs;i nullus impetus producatur; 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"/><p type="main"> <s>Ob&longs;eruabis autem nouum quoddam genús re&longs;i&longs;tentiæ; 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; at verò <lb/>omnes &longs;imul, & coniunctim con&longs;ideratæ; 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; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Vel producitur impetus in omnibus &longs;ubiecti partibus vnitis, vel in nulla; <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>Probatur igitur <lb/>1. quia &longs;i non producatur in omnibus partibus, & nulla &longs;eparetur ab <lb/>alijs; certè nulla mouetur, vt certum e&longs;t; igitur nulla habet impetum; <lb/>quia ibi non e&longs;t cau&longs;a formalis, vbi non e&longs;t effectus formalis; alicquin <lb/>e&longs;&longs;et fru&longs;trà, contra Ax. 6.2. Tu dicis produci impetum in aliquot parti­<lb/>hus; hoc dicis, hoc proba? </s> <s>an potes digno&longs;cere impetum ni&longs;i ex motu? </s> <s><lb/>vel con&longs;eruaretur hîc impetus &longs;equentibus in&longs;tantibus, vel &longs;tatim &longs;ecun­<lb/>do in&longs;tanti de&longs;trucretur. </s> <s>Primum dicere ab&longs;urdume&longs;t; quia &longs;i hoc e&longs;&longs;et <lb/>multisictibus repetitis tandem moueretur totum mobile; &longs;i verò de­<lb/>ftrui dicatur. </s> <s>Secundo in&longs;tanti; eadem ratio probat non produci. </s> <s>Pri­<lb/>mo in&longs;tanti, quæ prob<gap/>t de&longs;trui. </s> <s>Secundo nam ideo de&longs;truitur. </s> <s>Secun­<lb/>do quia e&longs;t fru&longs;trà, &longs;ed non minus e&longs;t fru&longs;trà. </s> <s>Primo igitur non produ­<lb/>citur. </s> <s>Primo 4. probatur; quia cum non &longs;ufficiant partes impetus, quas <lb/>dixi produci, vt omnibus partibus &longs;ubiecti di&longs;tribuantur; 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>Igitur certum e&longs;tvel produci in omnibus, vel <lb/>in nulla, ni&longs;i forte aliquæ auolent, &longs;ed tunc &longs;eparantur. </s></p><p type="main"> <s>Obiiciet aliquis 1. e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam applicatam &longs;ubiecto ap­<lb/>to: igitur agit per Ax. 12. Re&longs;pondeo e&longs;&longs;e impeditam; nam re&longs;i&longs;tentia <lb/>&longs;ubiecti &longs;uperat vires potentiæ vt dictum e&longs;t; immò in ip&longs;o motu re­<lb/>torqueo argumentum; licèt enim &longs;it applicata cau&longs;a nec&longs;&longs;aria mouens, <lb/>non tamen mouet. </s></p><p type="main"> <s>Obiiciet 2. Ignis applicatus agit in nonnullas partes fubiecti, licèt <lb/>non agat in omnes; igitur & potentia motrix. </s> <s>Re&longs;pondeo non e&longs;&longs;e pa­<lb/>ritatem; 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 type="main"> <s>Obiiciet. </s> <s>3. &longs;int duo trahentes idem mobile; ita vt &longs;eor&longs;im neuter <lb/>trahere po&longs;&longs;it, coniunctim verò vterque po&longs;&longs;it; certè &longs;i alter non pro­<lb/>ducit impetum &longs;cor&longs;un, nec etiam coniunctim producet; nec enim au­<lb/>gentur eius vires ab altero: 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"/>&longs;eor&longs;im producere pote&longs;t. </s></p><p type="main"> <s>Dices &longs;i vterque coniunctim producit effectum: &longs;int v. </s> <s>g. </s> <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>Re&longs;pondeo &longs;ingulos producere 100. <lb/>actione &longs;cilicet communi indiui&longs;ibiliter; &longs;int enim duo trahentes A. & <lb/>B. A. producit 100. &longs;ed non &longs;olus; B. producit ea&longs;dem 100. &longs;ed non &longs;o­<lb/>lus; &longs;ed explicabimus hunc modum actionis communis in Metaphys. </s> <s><lb/>quod autem agant actione communi patet per Ax. 13. </s></p><p type="main"> <s>Obiicies 4. producitur &longs;onus &longs;i ferias rupem; igitur & impetus; 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; deinde mallej motus impacti in rupem facit &longs;onum; <lb/>quidquid tandem &longs;it &longs;onus, de quo hîc non di&longs;puto: adde qnod in ru­<lb/>pe &longs;unt &longs;emper aliquæ partes trem ulæ, quæ modico tantùm, coque flexi­<lb/>bili nexu cum alijs partibus copu lantur; adde aliquam compre&longs;&longs;ionem, <lb/>ex qua modicæ vibrationes &longs;equuntur. </s></p><p type="main"> <s>Obiicies 5. Quando aliquæ partes auolant ab ictu, haud dubiè auo­<lb/>lant propter impetum impre&longs;&longs;um: 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>Re&longs;pondeo equidem im­<lb/>petum produci in illis partibus antequam auolent; &longs;ed ideo produci vt <lb/>deinde auolent nam tota ratio cur non producatur, e&longs;t ne &longs;it fru&longs;trà; &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 type="main"> <s>Dices; igitur primo in&longs;tanti impetus ille e&longs;t fru&longs;trà; in quo non <lb/>habet &longs;uum effectum; 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 i n&longs;tant i &longs;equenti, de <lb/>cum impetus ponat tantùm motum quo aliàs. </s></p><p type="main"> <s>Dices; &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>Re&longs;pondeo non &longs;tare per cau&longs;am nece&longs;&longs;ariam, quin &longs;em­<lb/>per agat; &longs;ed per &longs;ubiectum, quod &longs;i aptum e&longs;t, & capax e&longs;&longs;ectus; haud <lb/>dubiè co ip&longs;o cau&longs;a nece&longs;&longs;aria applicata in ip&longs;um aget; &longs;i verò ineptum. </s> <s><lb/>haud dubiè non aget; nam ad hoc vt producatur effectus in &longs;ubiecto; <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; igitur cum &longs;it talis ordo à natura in&longs;titutus, ne aliquid &longs;it fru&longs;trà; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s>Ob&longs;eruabis 2. etiam maximam corporis molem à debili potentia mi-<pb pagenum="27"/>nimo etiam ictu moucri; quod etiam ob&longs;eruauit Galileus in &longs;uis dialo­<lb/>gis de motu; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc egregia ratio erui pote&longs;t, cur ingens corporis moles à debili po­<lb/>tenria loco moueri non po&longs;&longs;it; cum enim tot &longs;altem requirantur partes <lb/>impetus, quot &longs;unt partes &longs;ubiecti: quia vel in omnibus, vel in nulla <lb/>producitur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>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 velocioremmotum imprimat; quia &longs;cilicet <lb/>partes impetus producti di&longs;tribuuntur pluribus partibus &longs;ubiecti in ma­<lb/>iori lapide, & paucioribus in minori; igitur &longs;ingulæ partes minoris <lb/>habent plures partes impetus, vt manife&longs;tè coa&longs;tat; ergo ille impetus <lb/>inten&longs;ior e&longs;t; igitur maiorem exigit &longs;eu perfectiorem motum per Ax. </s> <s><lb/>13. num.2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>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; nam ratio velocitatum <lb/>e&longs;t inu<gap/>r&longs;a ponderum v. </s> <s>g. </s> <s>&longs;it pondùs 4. librarum; 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; haud dubiè velocitas mobilis 2. libra­<lb/>rum erit dupla velocitatis mobilis 4. librarum; quia cum &longs;int duplo <lb/>plures partes &longs;ubiecti in hoc mobili quàm in illo (accipio enim vtrum­<lb/>que eiu&longs;dem marcriæ, vt omnes lites fugiam) igitur in minori e&longs;t duplo <lb/>inten&longs;ior impetus: Igitur duplo velocior motus; dixi, &longs;i fiat æquali <lb/>ni&longs;u, & æquali tempore; quia reuerâ non fit in tempore æquali, &longs;ed <lb/>inæquali, &longs;i &longs;upponatur idem arcus brachij v. </s> <s>g. </s> <s>iacientis; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s>Hinc facilè determinari pote&longs;t proportio impetus impre&longs;&longs;i cognitâ <lb/>grauitate mobilium; v. </s> <s>g. </s> <s>&longs;it mobile grauc vt4. & aliud graue vt 2. haud <lb/>dubiè vt moucatur 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;cquitur ex dictis; vt enim tot&longs;int gradus impetus <pb pagenum="28"/>in qualibet parte minoris, quot &longs;unt in qualibet parte minoris; haud <lb/>dubiè impetus maioris habet eandem rationem ad impetum minoris; <lb/>quam habet maius ad minus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s>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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s></p><p type="main"> <s><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> Probatur; 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, codem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s></p><p type="main"> <s><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> Quia non pote&longs;t <lb/>moueri, ni&longs;i acquirat nouum locum, & priorem relinquat; igitur, vel &longs;i­<lb/>mul in vtroque e&longs;t, quod dici non pote&longs;t; vel in relicto antè fuit; igitur <lb/>non e&longs;t primum in&longs;tans, contra &longs;uppo&longs;itionem. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s></p><p type="main"> <s><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> Nam moueatur mobile quodlibet; & 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. </s> <s><lb/>35. quo tamen in&longs;tanti mouetur prædictum mobilc. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Collorarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Hinc etiam aliud paradoxon; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Hinc pote&longs;t aliquid dato in&longs;tanti carere &longs;uo fine; 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 haberemotum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus pars recepta in parte &longs;ubiecti non exigit motum aliarum partium<emph.end type="italics"/><pb pagenum="29"/><emph type="italics"/>eiu&longs;dem &longs;ubiecti, licèt coniunctarum.<emph.end type="italics"/></s><s> Probatur 1. quia alioquin vna pars <lb/>impetus &longs;ufficeret ad mouendam ingentem rupem; quod ab&longs;urdum e&longs;t. </s> <s><lb/>2. &longs;icut vna pars caloris non re&longs;oluit alias partes &longs;ubiecti; ita nec im­<lb/>petus. </s> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc partes impetus non cau&longs;ant motum in &longs;uo &longs;ubiecto actione, vel <lb/>exigentia communi; quia quælibet pars impetus exigit tantùm motum <lb/>&longs;ui &longs;ubiecti; id e&longs;t illius partis, quàm afficit; quod etiam probatur per <lb/>Ax. 13. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>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; quia &longs;cilicet impetus vnius partis non iuuat mo­<lb/>tum alterius: præterca tam facilè 2, partes impetus in 2. partibus &longs;ubie­<lb/>cti receptæ ca&longs;dem mouent, quàm 100. alias 100. dixi per &longs;e; 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus recipitur tantùm in ip&longs;a &longs;ub&longs;tantia &longs;ubiecti naturaliter.<emph.end type="italics"/> v. </s> <s>g. </s> <s>&longs;i <lb/>mobile &longs;it ferrum calidum, recipitur in ip&longs;a &longs;ub&longs;tantia ferri; non verò <lb/>in ip&longs;o calore (ex &longs;uppo&longs;itione quod calor &longs;it accidens, vt aliàs demon­<lb/>&longs;trabimus; nec in alijs accidentibus, &longs;i quæ &longs;unt, in eodem &longs;ubiecto; 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; plures quoque partes impetus <lb/>producendæ e&longs;&longs;ent; igitur maiori potentiâ opus e&longs;&longs;et per Ax. 13. n. </s> <s>4. <lb/>Igitur difficiliùs mouerentur, quod e&longs;t ab&longs;urdum. </s> <s>Diceret fortè ali­<lb/>quis eundem impetum recipi &longs;imul in &longs;ub&longs;tantia & in ip&longs;is accidenti­<lb/>bus; &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: 2. qui hoc diceret, deberet probare; nam codem modo <lb/>mouetur corpus &longs;iue afficiatur pluribus accidentibus, &longs;iue paucioribus; <lb/>igitur non euincit experientia recipiin illis impetum, nec etiam ratio, <lb/>vt dicam paulò po&longs;t. </s> <s>Ratio à priori e&longs;&longs;e pote&longs;t; 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; igitur cum exigat con&longs;erua­<lb/>ri, & exi&longs;tere; co tantùm modo, quo pote&longs;t naturaliter con&longs;eruari & <lb/>exi&longs;tere; certè exigit con&longs;eruari, & ine&longs;&longs;e &longs;ubiecto; 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 cnm co coniu nctum mutare debet. </s></p><p type="main"> <s>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>cum <lb/>hoc ip&longs;um ad omnem vnionem &longs;pectare videatur; Re&longs;pondeo vnam <pb pagenum="30"/>partem plumbi ita coniungi cum alia, vt etiam &longs;eparata naturaliter <lb/>exi&longs;tere po&longs;&longs;it; igitur non e&longs;t par ratio; præterea vna pars plumbi non <lb/>e&longs;t in loco alterius; nec enim inuicem penetrantur cum &longs;it compene­<lb/>tratio accidentium cum &longs;ubiecto; 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; denique non vide­<lb/>rur cur potius produceretur in vna parte, quam in alia; quæ omnia <lb/>iam &longs;uprà Th. 33. demon&longs;trauimus. </s></p><p type="main"> <s>Adde quod &longs;i impetus produceretur in ip&longs;is accidentibus, etiam in <lb/>ip&longs;o impetu prius producto alius impetus produceretur; 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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc manife&longs;tè patet, quid dicendum &longs;it de anima bruti, quæ mouc­<lb/>tur etiam &longs;ine impetu; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Idem quoque de Anima rationali dicendum e&longs;&longs;e videtur; licèt <lb/>enim à corpore &longs;eparata naturaliter exi&longs;tere po&longs;&longs;it; tandiù tamen cum <lb/>corpore manet coniuncta, quandiu agere pote&longs;t in organis corporcis; <lb/>ac proinde exigit con&longs;eruari in corpore ip&longs;o, quandiu &longs;uas operatio­<lb/>nes organicas in co exercere pote&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Hinc patet ratio manife&longs;ta ad quæ&longs;itum illud; quomodo &longs;cilicet po­<lb/>tentia motrix materialis v.g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Scolium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s>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 type="main"> <s>Ob&longs;eruabis tertiò etiam Animam rationalem &longs;eparatam, modò &longs;it <lb/>cum impenetrabilitate coniuncta, capacem e&longs;&longs;e impetus; quem etiam <lb/>à potentia motrice corporea recipere pote&longs;t; idem dictum e&longs;to de An­<lb/>gelo; &longs;ed de vtroque aliàs. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quando corpus pellitur ab alio corpore per impetum impre&longs;&longs;um; band da­<lb/>biè impetus ille impre&longs;&longs;us ab aliqua cau&longs;a efficiente producijur<emph.end type="italics"/>; patet <lb/>per Ax. 8. </s></p><pb pagenum="31"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ille impetus non producitur à &longs;ub&longs;tantia corporis in aliud impacti.<emph.end type="italics"/></s><s> Proba­<lb/>tur; 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><lb/>Dicunt aliqui requiri <expan abbr="motũ">motum</expan> præuium, vt agat; &longs;ed contra; nam motus <lb/>præuius non requiritur vt cau&longs;a, vt patet; quia cau&longs;a vt agat debet exi­<lb/>&longs;tere per Ax. 9. Igitur requiritur, vt conditio, quod dici non pote&longs;t; <lb/>quia primo etiam conditio debet e&longs;&longs;e præ&longs;ens; &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; ni&longs;i vel vt tollat impedimentum, vel vt applicet cau&longs;am &longs;ubie­<lb/>cto apto; præterea motus præuius non e&longs;t; igitur codem modo &longs;e <lb/>habet, ac &longs;i nunquam extiti&longs;&longs;et; & &longs;i co 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 type="main"> <s>Diceret alius impetum e&longs;&longs;e tantùm conditionem, quæ &longs;emper e&longs;t <lb/>de præ&longs;enti: ad hanc in&longs;tantiam non valet &longs;uperior re&longs;pon&longs;io; & certè <lb/>&longs;i eo ip&longs;o in&longs;tanti contactus noua fieret impetus acce&longs;&longs;io; haud dubiè <lb/>maior e&longs;&longs;et ictus; licèt cum codem motu præuio, & tamen idem e&longs;&longs;et <lb/>corpus <expan abbr="impactũ">impactum</expan>, Igitnr 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; dum &longs;cili­<lb/>cet &longs;ubiectum, cui applicatur &longs;it capax motus; 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>1. voco enim illud cau&longs;am, <lb/>ex cuius applicatione &longs;emper &longs;equitur &longs;imilis effectus; alioquin &longs;i hoc <lb/>neges; proba mihi aliter ignem accendi ab alio igne; 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>quo enim medio, vel argu­<lb/>mento id euinces? </s> <s>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>v. </s> <s>g. </s> <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>igitur illius e&longs;t tollere impedimentnm, cuius e&longs;t exigere motum, <lb/>corpus ip&longs;um graue non exigit motum &longs;ur&longs;um, &longs;ed impetus; igitur im­<lb/>petus e&longs;t tollerc impedimentum &longs;ui effectus; igitur producere impetum, <lb/>quo vno tolli tantùm pote&longs;t: 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>quæ ratio? </s> <s>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>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>prætereà e&longs;t-ne effectus ad extra? </s> <s>deinde idem dico de calore; <pb pagenum="32"/>qui reuera habet effectum formalem &longs;ecundarium ad intra, &longs;cilicet rare­<lb/>factionem, quæ e&longs;t mutatio exten&longs;ionis; quemadmodum motus e&longs;t mu­<lb/>tatio loci, vel vbicationis; igitur cum hoc |non ob&longs;tante, calor pro­<lb/>ducat calorem ad extra; cur impetus non producit impetum? </s> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Aliquis impetus non producitur ab alio impetu.<emph.end type="italics"/></s><s> Probatur, quia aliquis <lb/>impetus producitur ad intra à potentia motrice, vt patet. </s> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus producitur &longs;emper ad extra ab alio impetu.<emph.end type="italics"/></s><s> 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc impetus habet duplex munus cau&longs;æ; &longs;cilicet cau&longs;æ exigentis ad intra <lb/>& efficientis ad extra<emph.end type="italics"/>; vtrumque patet ex dictis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus agit tantùm ad extra, vt tollat impedimentum motus<emph.end type="italics"/>; cum enim <lb/>motus &longs;it finis intrin&longs;ecus impetus; certè &longs;i nihil impediret motum, <lb/>haud dubiè gauderet impetus &longs;uo fine; igitur fru&longs;trà quidquam aliud <lb/>de&longs;ideraret; præterea licèt applicetur à tergo aliud mobile; non tamen <lb/>propterea in eo producit, vt con&longs;tat experientiâ; denique cum tan­<lb/>tùm impetum cogno&longs;camus per motum; cum nequidem e&longs;&longs;et impetus, <lb/>&longs;i non e&longs;&longs;et motus, per Th. 17.ce rtè totus e&longs;t impetus propter motum <lb/>qui e&longs;t eius finis; igitur non agit ni&longs;i propter motum: &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc non &longs;imul agit impetus in orbem &longs;ed tantùm per lineam <lb/>&longs;ui motus; cui &longs;i nullum corpus occurrit reuerà non agit,<emph.end type="italics"/> Ratio e&longs;t; quia li­<lb/>cèt aliud corpus mobili admoueatur in alia linea; cum non impediat <lb/>eius motum, vt &longs;uppono; cum agat tantùm impetus ad extra, vt tollat, <pb pagenum="33"/>impedimentum motu &longs;ui &longs;ubiecti, in eo non agit, quod non impedit; & <lb/>cum impediatur tantùm in vna linea, in ca tantùm agit; igitur non <lb/>agit in orbem. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis primò, hanc primam e&longs;&longs;e difficultatem; cum in hoc im­<lb/>petus maximè differat ab alijs qualitalibus &longs;i quæ &longs;unt, quæ agunt in or­<lb/>bem, vt dicemus &longs;uo loco. </s></p><p type="main"> <s>Ob&longs;eruabis &longs;ecundò, hanc etiam e&longs;&longs;e communem illorum &longs;ententiam, <lb/>qui dicunt imptum 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 type="main"> <s>Ob&longs;eruabis tertiò, impetum in vtroque muncre cau&longs;æ &longs;ube&longs;&longs;e tantùm <lb/>vni lineæ; &longs;cilicet exigit motum per vnam lineam; cum per plures &longs;i­<lb/>mul motus e&longs;&longs;e non po&longs;&longs;it; ne idem mobile &longs;imul e&longs;&longs;et in pluribus lo­<lb/>cis; & producit impetum per vnam lincam; cum producat tantùm pro­<lb/>pter motum. </s></p><p type="main"> <s>Ob&longs;eruabis quartò, alias qualitates, &longs;i quæ &longs;unt, non agere ad extra, <lb/>vt tollant impedimentum &longs;ui effectus ad intra; qui &longs;cilicet ab impedi­<lb/>mento extrin&longs;eco impediri non pote&longs;t; vt accidit in ip&longs;o impetu; etenim <lb/>corpus non pote&longs;t moucri 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 amoucri debet, accepto dumtaxat impetu <lb/>ab alio mobili. </s></p><p type="main"> <s>Ob&longs;eruabis quintò nonnullos e&longs;&longs;e, qui volunt motum vnius corporis <lb/>transferri in aliud corpus; &longs;ed mera e&longs;t metaphora; nihil cnim 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;cicilet nouum produci. </s></p><p type="main"> <s>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; quia lumen, & calor &longs;unt veræ qualita­<lb/>tates, non corpu&longs;cula, vt demon&longs;trabimus in 5. tractatu: Adde quod li­<lb/>cet ferrum candens aliud frigidum impellat, etiam veloci&longs;&longs;imè; hoc ip­<lb/>&longs;um æquè frigidum manet; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Omnes partes impetus mobilis agunt ad extra actione communi.<emph.end type="italics"/></s><s> Probatur <lb/>per Ax. 13. n. </s> <s>1. ni&longs;i enim agerent actione communi &longs;ed quælibet &longs;uam <lb/>produceret; cur potius in hac parte &longs;ubiecti, quam in alia, demde ap­<lb/>pìicatur tantùm vna immediatè; Igitur agunt omnes actione commu­<lb/>ni; omnes inquam illæ, quæ impediuntur; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s></p><p type="main"> <s><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> 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>2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Vides primò in hoc ca&longs;u compen&longs;ari inten&longs;ionem ab exten&longs;ione; <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 type="main"> <s>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 type="main"> <s>Tertiò, Hinc corpora illa, quorum partes vel nullo vel modico nexu <lb/>copulantur, minimo ferè impul&longs;u commouentur; &longs;ic aër & aqua mini­<lb/>mo flante vento agitantur, nubes pelluntur; 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 moucri pote&longs;t, quod in aqua gelu concreta minimè accidit. </s></p><p type="main"> <s>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 type="main"> <s>Quintò, Hinc diuino penè con&longs;ilio factum e&longs;t, vt partes terre&longs;tris <lb/>globi arctiore fibula copulentur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus, cuius motus non impeditur, non agit ad extrà.<emph.end type="italics"/></s><s> 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc egregium paradoxon erui pote&longs;t; quod &longs;cilicet cau&longs;a nece&longs;&longs;aria <lb/>etiam immediatè applicata, & non impedita in &longs;ubiecto apto non agit; <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; &longs;i agat in orbem, omnis applicatio &longs;ufficiens <lb/>e&longs;t: &longs;i verò agat tantùm per vnam lineam; certè applicari debet in ca <lb/>linea; alioquin non aget defectu debitæ applicationis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Hinc etiam aliud paradoxon non minus iucundum; cau&longs;a nece&longs;&longs;aria <pb pagenum="35"/>appllcata, & non impedita non agit; at verò agit impedita; &longs;cilicet <lb/>impetus qui tantùm agit, vt tollat impedimentum; igitur, &longs;i non <lb/>impediatur non agit. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quo minùs impeditur impetus, minùs agit ad extra, & contrà; quo plùs <lb/>impeditur, plùs agit.<emph.end type="italics"/></s><s> Cum enim ideò agat ad extra, vt tollat impedi­<lb/>mentum; certè &longs;i nullum e&longs;t, nihil agit, &longs;i minùs, minùs agit; igitur <lb/>agit pro rata, id e&longs;t, pro diucr&longs;a impedimenti ratione. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si linea motus, quam directionis appellant, ducatur per centrum vriu&longs;que <lb/>corporis, maximum est impedimentum,<emph.end type="italics"/> vt patet. </s> <s>&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; certè glo­<lb/>bus B maximum ponit impedimentum, quod ab co poni po&longs;&longs;it; Igitur <lb/>impetus globi A agit quantùm pote&longs;t in globum B; vt &longs;cilicet maxi­<lb/>mum impedimentum remoucat. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si linea motus vel ip&longs;ius parallela cadat perpendiculariter in extremam <lb/>diametrum globi immobilis: haud dubiè nihil impedit<emph.end type="italics"/>; &longs;it enim globus <lb/>mobilis A, Immobilis B, linea directionis &longs;it GA, ip&longs;i parallela FC; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si linea motus &longs;it inter vtramque; est minus impedimentum.<emph.end type="italics"/> &longs;it globus <lb/>immobilis BA; &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; certè minus e&longs;t impedimentum in contactu D, quàm in C; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc producitur in contactu<emph.end type="italics"/> C, <emph type="italics"/>totus impetus; in contactu<emph.end type="italics"/> D, <emph type="italics"/>minùs; <gap/><lb/>contactu<emph.end type="italics"/> E <emph type="italics"/>adhuc minùs; in<emph.end type="italics"/> B <emph type="italics"/>nihil<emph.end type="italics"/>; quia in ca proportione producitui <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; & 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc eadem cau&longs;a nece&longs;&longs;aria etiam immedia&longs;e applicata diuer&longs;um impe<emph.end type="italics"/><pb pagenum="36"/><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc ratio multorum effectuum phy&longs;icorum e. </s> <s>ui potest<emph.end type="italics"/>; cur &longs;cilicet cor­<lb/>pus incidens in aliud perpendiculariter maximum ictum infligat; quia <lb/>&longs;cilicet maximum impetum producit, qui po&longs;&longs;it ab eo produci; 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>Huc etiam reuoca tormenta bellica, quæ <lb/>vel directo, vel obliquo ictu muros verberant; hinc perpendicularis <lb/>forti&longs;&longs;ima e&longs;t; licèt cadem ratio pro motu corporum non valeat, quæ <lb/>valet pro diffu&longs;ione, &longs;eu propagatione qualitatum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s></p><p type="main"> <s>Hinc pote&longs;t determinari quota pars impetus producatur, & quantus <lb/>&longs;it ictus; cognito &longs;cilicet & &longs;uppo&longs;ito co impetus gradu, qui producitur, <lb/>cum totus producitur, vt fit in perpendiculari; quippe tota men&longs;ura <lb/>impetus continetur in arcu CB; quam proportionem nos infrà demon­<lb/>&longs;trabimus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s></p><p type="main"> <s><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; quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria; Igitur idem impetus codem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc impetus remi&longs;&longs;us potest producere inten&longs;um; & hæc e&longs;t altera diffcul­<lb/>tas; cum &longs;cilicet maior globus in minorem impingitur<emph.end type="italics"/>; cum enim omnes <lb/>partes impetus maioris globi agant actione communi per Th. 46. & <lb/>cum agant quantùm maximè po&longs;&longs;unt; in minore globo, tot partes pro­<lb/>ducunt impetus, quot in maiore, vt patet; igitur in minore globo pau­<lb/>cioribus partibus &longs;ubiecti di&longs;tribuuntur plures partes impetus; crgo in <lb/>qualibet parte &longs;ubiecti &longs;unt plures; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quia &longs;cilicet pauciores partes impetus di&longs;tribuun­<lb/>tur pluribus partibus &longs;ubiecti; igitur quælibet &longs;ubiecti pauciores impe­<lb/>tus habet; quæ omnia con&longs;tant ex dictis. </s></p><pb pagenum="37"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis primò, &longs;ingularem impetus proprictatem, quæ alijs qua­<lb/>litatibus minimè competit; nam aliæ qualitates v. </s> <s>g. </s> <s>calor; lumen in <lb/>cadem di&longs;tantia effectum &longs;emper æquè inten&longs;um producunt; &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; <lb/>cuius ratio ex eo capite petitur; quòd impetus agat tantùm ad extra <lb/>propter &longs;uum effectum ad intra, vt &longs;cilicet tollat impedimentum; igi­<lb/>tur in totum, quod impedit, agit; igitur non habet certam, & deter­<lb/>minatam &longs;phæram; cum tantùm agat in obicem, &longs;iue &longs;it maior, &longs;iue <lb/>minor: 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; 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/>maiorc. </s></p><p type="main"> <s>Prætereà, cum dixi omnes partcs mobilis actione communi agere ad <lb/>extra; ita primò intelligi debet, vt omnes illæ partes moueantur: &longs;ecun­<lb/>dò, vt linea motus, &longs;eu directionis per centra grauitatis vtriu&longs;que glo­<lb/>bi v, g. </s> <s>ducatur; alioquin, vel omnes actione communi non agunt, vel <lb/>minus agunt, de quo infrà; &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>in qua <lb/>linea directionis e&longs;t DE. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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 æqualemi. </s> <s>Probatur primò, <lb/>exemplo aliarum qualitatum; &longs;ecundò, quia ideo agit vt tollat impedi­<lb/>mentum, hoc e&longs;t vt corpus illud amoueat loco; igitur æquali motu per <lb/>&longs;e; alioquin ni&longs;i æquali motu amoueret, non tolleret impedimentum, <lb/>vt pater; tertiò &longs;int 30. partes impetus, certèvel producent plures vel <lb/>pauciores, vel totidem, non plures; cur enim potius 31. quam 32. <lb/>nec etiam pauciores; cur enim potius 20. quam 18, &c. </s> <s>Igitur totidem; <lb/>quia cum &longs;int plures numeri plurium partium &longs;upra 30. & pauciorum <lb/>infra vt patet; &longs;itque tantùm vnicus numerus æqualium; 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 potenri&longs;&longs;ima: quartò, quia actus &longs;ecundus, re&longs;pondet actui <lb/>primo, id e&longs;t, effectus productus virtuti cau&longs;æ producentis; 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 co contactu iuxta modum prædictum, ca quo-<pb pagenum="38"/>que lege, vt impetu<gap/> agat quantum pote&longs;t, & omnes partes mobilis <lb/>moueantur æquali motu. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc reijcics 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; quia duo globi æquales in&longs;tanti contactus <lb/>perinde &longs;e habent, atque &longs;i conflarent vnum corpus; &longs;ed &longs;i conflarent <lb/>vnum corpus quilibet &longs;ubduplum impetum haberet; &longs;i verò globus cum <lb/>alio &longs;ubduplo faceret vnum mobilc; haud dubiè minor, id e&longs;t, &longs;ubduplus <lb/>haberet tantùm &longs;ubtriplum impetum; atque ita deinceps; hoc totum <lb/>fal&longs;i&longs;&longs;imum e&longs;t; nam primò &longs;i globus æqualis acciperet tantùm &longs;ubdu­<lb/>plum impetum ab alio, &longs;ubduplo tantùm motu ferretur; 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; 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; nec enim migrat de <lb/>de &longs;ubiecto in &longs;ubiectum, &longs;ed producit &longs;ibi æqualem: 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; &longs;i e&longs;&longs;ent æquales, qui&longs;que &longs;ubduplum: &longs;i alter &longs;ubduplus &longs;ubtri­<lb/>plum, &c. </s> <s>&longs;ed hæc &longs;unt &longs;atis facilia. </s></p><p type="main"> <s>Obijci fortè po&longs;&longs;et ab aliquo primò experientia; videmus enim &longs;æpè <lb/>globum impul&longs;um in ludo Tudiculario moueri tardiùs globo impellen­<lb/>te; re&longs;pondeo id &longs;æpè accidere; tùm quia linea directionis non connec­<lb/>tit centra vtriu&longs;que globi; igitur minor e&longs;t ictus per Th 52. tùm quia <lb/>globus impellens, vel impul&longs;us deficiunt à perfecta &longs;phæra; 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 type="main"> <s>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; igitur illa pars impetus, quæ de&longs;truitur, non <lb/>producit nouum impetum in globo impul&longs;o; Re&longs;pondco de&longs;truiquidem <lb/>aliquid impetus in globo impacto, vt videbimus infrà; cum tamen de­<lb/>&longs;truatur tantùm &longs;equenti po&longs;t ictum in&longs;tanti; 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; adde quod illud ip&longs;um repugnat manife&longs;tæ experientiæ; <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 type="main"> <s>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; modò illi globi ita &longs;tatuantur, vt <lb/>linea directionis per omnium centra rectà ducatur; Re&longs;pondco, vel il-<pb pagenum="39"/>los omnes globos ita e&longs;&longs;e contiguos, vt mutuo contactu &longs;e inuicem tan­<lb/>gant; vel aliquod &longs;patium inter &longs;ingulos intercipi; &longs;i primum, produci­<lb/>tur impetus à potentia motrice in omnibus, &longs;i &longs;ufficiens e&longs;t; non verò <lb/>vnus globus in alio, vt con&longs;tat; &longs;icut duo pondera &longs;imul attollo, quorum <lb/>vnum alteri incumbit: &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: idem dico de B, & C, vnde &longs;i nihil impetus in co primo motu <lb/>periret & linea directionis omnium centra perfectè connecteret; ita vt <lb/>omnium ictus illi omnino &longs;ine vlla deflexione re&longs;ponderent; 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; &longs;ed vix illa omnia de quibus &longs;uprà po&longs;&longs;unt ob&longs;eruari; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s></p><p type="main"> <s><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; id e&longs;t, nullum corpus po&longs;le maiorem veloci­<lb/>tatis gradum alteri corpori imprimere; co &longs;cilicet gradu, quem ip&longs;um <lb/>habet; 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>Quis enim non vidit vel maius &longs;axum in aliud <lb/>etiam tardo motu impactum maiorem motum, & impetum imprimere? </s> <s><lb/>quis non vidit maiores illas onerarias naues etiam pigro, & tardo motu <lb/>labentes maximum impetum minori occurrenti cymbæ ctiam impri­<lb/>mere? </s> <s>Rationem habes in Th. 47. &longs;ed dices; igitur aliquis velocitatis <lb/>gradus nullam habet cau&longs;am; igitur e&longs;t à nihilo, quod dici non pote&longs;t. </s> <s><lb/>Re&longs;pondeo, plures partes impetus non produci in minore globo, quàm <lb/>&longs;int in maiore; igitur nulla pars e&longs;t impetus minoris globi, quæ &longs;ui <lb/>cau&longs;am &longs;ufficientem non habeat; &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; cum &longs;cilicct 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>1. quid clarius. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Globus minor imprimit maiori remi&longs;&longs;iorem impetum & tardiorem motum <lb/>& æqualis, aqitali æqualem<emph.end type="italics"/>; hæc omnia probantur per Th. 60. & præ-, <lb/>cedentia. </s></p><pb pagenum="40"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis primò, vtrumque globum e&longs;&longs;e eiu&longs;dem materiæ; &longs;i enim <lb/>&longs;int diuer&longs;æ materiæ, &longs;ecùs accidit, quàm diximus; &longs;i v. </s> <s>g. </s> <s>æncus mi­<lb/>nor pellatur ab cburneo maiore, maiorem motum hic illi non impri­<lb/>met; licèt enim &longs;it maior exten&longs;io eburnci; e&longs;t tamen minus pondus; <lb/>igitur pauciores partes. </s></p><p type="main"> <s>Secundò, cos globos accipiendos e&longs;&longs;e, quorum partes, vel non auo­<lb/>lent ab ictu, vel non comprimantur; comprimuntur in plumbeis, <lb/>æncis, & 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 type="main"> <s>Tcrtiò reiice commentum illorum, qui dicunt corpus illud e&longs;&longs;e ma­<lb/>joris vclocitatis capax, quod plures habet partes materiæ &longs;ub eadem <lb/>quantitate; nam &longs;uppo&longs;ita cadem re&longs;i&longs;tentiæ ratione, omne corpus e&longs;t <lb/>capax illius vclocitatis, cuius alind e&longs;t capax; cum nullus &longs;it motus, quo <lb/>non po&longs;&longs;it dari velocior, & tardior, vt dicemus infrà; immò &longs;it glo­<lb/>bus plumbeus 12. librarum, &longs;it eburncus eiu&longs;dem diametri 2. librarum, <lb/>v. </s> <s>g. </s> <s>haud dubiè cadem potentia producet inten&longs;iorem impetum in <lb/>cburnco, vt patet experientia, & ratio con&longs;tat ex dictis; qua&longs;i verò &longs;it <lb/>aliqua materiæ inertia, quæ motum re&longs;puat; licèt fortè maior &longs;it pro­<lb/>portio re&longs;i&longs;tentiæ medij comparatz cum globo cburneo, quàm compa­<lb/>ratæ cum plumbeo; &longs;ed de re&longs;i&longs;tentia de percu&longs;&longs;ionc, & de &longs;patio age­<lb/>mus infra. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Omnis globus, qui in alium, qui mouetur impingitur, dum hie mouetur, vt­<lb/>lociùs mouetur eo &c. </s> <s>in quem impingi<gap/><emph.end type="italics"/> patet; alioquin numquam a&longs;&longs;cqui <lb/>po&longs;&longs;et, quod ex ip&longs;is terminis con&longs;tat. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ex hac hyp<gap/>e&longs;i globus impactus producit in alie mouas partes impetus<emph.end type="italics"/>; <lb/>quia impeditur cius motus, igitur vt tollat impedimentum, agit ad <lb/>cxtra per Th. 44. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hie impetus neuus preductus miner e&longs;t eo qus preduceretur in codem glob­<lb/>immobili<emph.end type="italics"/>: ratio c&longs;t; quia &longs;i &longs;i&longs;tcret, maius e&longs;&longs;et impedimentum, quia <lb/>totum motum impediret, cuius tantùm partem impedit, dum mouctur, <lb/>licèt paulò tardius; igitur minus agit ad cxtra per Th. 49. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Mobile adbarens alseri mobili à terge; dum vtr<gap/>que æqu<gap/> velociter <lb/>feratur nullum preducis in ce i<gap/>pesum.<emph.end type="italics"/></s><s> Probatur, quia mobile quod præit, <lb/>non impedit motum &longs;ub&longs;equentis; igitur nullum impetum ab co acci <lb/>pit per Th. 48. </s></p><pb pagenum="41"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc paradoxon egregium &longs;i quod aliud; 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. </s> <s>g. </s> <s>&longs;it globus A quie&longs;cens, cui unprimantur ab alio B 40. gra­<lb/>dus velocitatis: id e&longs;t æqualis impetus impetui percutientis, iam verò <lb/>moueatur A, cum 20. grad. </s> <s>vclocitatis, & B, qui mouetur cum 40. <lb/>impingatur, certè cum impediatur tantùm &longs;ubduplum motus, produce­<lb/>tur tantùm &longs;ubduplum impetus, id c&longs;t 20. qui &longs;i addantur 20. grad. </s> <s>erunt <lb/>40. quæ omnia con&longs;tant per Th.49.48.&c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>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 co, 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Hinc reiicies aliquos, quorum &longs;ententiam habes apud Doctum Mer­<lb/>&longs;emium, <emph type="italics"/>in prop.<emph.end type="italics"/> 20. <emph type="italics"/>phæn. </s> <s>mech. </s> <s>quorum &longs;unt hæc verba; &longs;i malleus p<gap/><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> Supponit primò hæc &longs;ententia mal­<lb/>leum e&longs;&longs;e duplum pilæ percu&longs;&longs;æ. </s> <s>Secundò, mallcum 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>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 quictem propiùs accedit; mi­<lb/>norem verò paulò po&longs;t initium motus, vt con&longs;tat ex dictis, & exip&longs;a ex­<lb/>perientia; potc&longs;t quidem in aliquo puncto &longs;ui motus &longs;ecunda vice per­<lb/>cuti, in quo &longs;ubtriplum tantùm motum imprimet; hoc e&longs;t eo in&longs;tanti­<lb/>quo tantùm ami&longs;it tertiam fui impetus partem; tum deindc in tertia <lb/>percu&longs;&longs;ione pote&longs;t tantùm (1/27) motus partem illi tribuere; co &longs;cilicet in­<lb/>&longs;tanti, quo tantùm ami&longs;it (1/27) &longs;ui impetus partem; &longs;ed in alijs cemporis <lb/>punctis longè alia crit impetus producti ratio; Igitur tota hæc progre&longs;­<lb/>&longs;io gratis omninò fuit excogitata. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s>Hinc ctiam po&longs;t &longs;ecundam percu&longs;&longs;ioncm æquale &longs;patium conficier al­<lb/>teri, quod iam confecit po&longs;t primam æqualibus temporibus; igitur æqua­<lb/>lis e&longs;t velocitas vtriu&longs;que motus; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s>Ex dictis ctiam 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; 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>Igitur velocitates motuum &longs;unt in ratiòne ponderum <lb/>permutando. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si corpus percu&longs;&longs;um &longs;it oblongum, & percu&longs;&longs;io fiat in centro grauitatis eiu&longs;­<lb/>dem corpois; producitur impetus in percu&longs;&longs;io æqualis impetui percutientis<emph.end type="italics"/>; &longs;ed <lb/>opus e&longs;t aliqua figura: Sit corpus AD, parallelipedum; diuidatur æqua­<lb/>liter in E ita vt E &longs;it centrum grauitatis; &longs;i percu&longs;&longs;io &longs;iat in E per lineam <lb/>perpendicularem HE, producetur impetus in corpore AD æqualis im­<lb/>petui corporis percutientis; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s></p><p type="main"> <s><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; &longs;ed hic <lb/>motus e&longs;t facilior quam ille; igitur minus e&longs;t impedimentum; (&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: 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 ficri <lb/>debeat. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ex duobus capitibus minus e&longs;&longs;e pote&longs;t impedimentum; primum e&longs;t, <lb/>quod petitur à puncto contactus, &longs;ecundum à linea incidentiæ; v. </s> <s>g. </s> <s>&longs;i <lb/>accipiatur punctum E, in quo e&longs;t centrum grauitatis corporis AD, & in <lb/>eo &longs;iat percu&longs;&longs;io; maximum e&longs;t impedimentum ratione puncti conta­<lb/>ctus, in quo fit percu&longs;&longs;io; &longs;i verò percu&longs;&longs;io fiat per lincam perpendicu­<lb/>larem HE, maximum e&longs;t impedimentum, ratione lineæ; &longs;i autem ex <lb/>vtroque capite &longs;imul accidat impedimentum, maximum e&longs;t omnium; <lb/>iam verò &longs;i accipiatur punctum E, & linea percu&longs;sionis ME; minor e&longs;t <lb/>percu&longs;sio ratione lineæ non puncti; 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; <lb/>&longs;i demum accipiatur punctum E, & linea ME, minor e&longs;t percu&longs;sio ra-<pb pagenum="43"/>tione lineæ non puncti; accipiatur punctum N linea percu&longs;&longs;ionis MN, <lb/>minor e&longs;t percu&longs;&longs;io ratione puncti non lineæ; &longs;i accipiatur punctum N, <lb/>& linea IN, minor e&longs;t percu&longs;&longs;io ratione vtriu&longs;que: &longs;i demum accipia­<lb/>tur punctum E & linea HE, maior e&longs;t percu&longs;&longs;io ratione vtriu&longs;que; igi­<lb/>tur &longs;unt quatuor coniugationes; &longs;eu quatuor cla&longs;&longs;es diucr&longs;arum percu&longs;­<lb/>&longs;ionum. </s></p><p type="main"> <s>Hinc compen&longs;ari pote&longs;t ratione vnius quod dee&longs;t ratione alterius, <lb/>v. </s> <s>g. </s> <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;<gap/>, & <lb/>demon&longs;trabimus totam i&longs;tam rem, quæ multùm curio&longs;itatis habet, & <lb/>vtilitatis. </s></p><p type="main"> <s>Determinabimus etiam dato puncto percu&longs;&longs;ionis F v.g. </s> <s>cum &longs;equatur <lb/>motus vectis, quodnam &longs;it centrum vectis &longs;eu huius motus. </s></p><p type="main"> <s>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; & data percu&longs;&longs;ione per lineam quantumuis inclinatam, po&longs;&longs;e da­<lb/>ri æqualem per lineam perpendicularem; & 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s></p><p type="main"> <s><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 quæntùm pote&longs;t.<emph.end type="italics"/> v. </s> <s>g. </s> <s>&longs;it parallelipedum EB; quod <lb/>moueatur motu recto parallclo, lineis CD, HG, &c. </s> <s>&longs;itque globus in <lb/>D; 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 parallcli­<lb/>pedi, quàm parallclip edum motum globi impacti per lineam ID; impedit <lb/>inquam ratione oppo&longs;itionis; quia centra grauitatis vtriu&longs;que con­<lb/>currunt in cadem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; vt con&longs;tat ex dictis in &longs;uperioribus Theorematis; <lb/>in qua vero proportionc determinabimus aliàs. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si corpus percutiens non &longs;it par allelipedum, &longs;ed alterius &longs;iguræ v.g.<emph.end type="italics"/> <emph type="italics"/>trigo­<lb/>non,<emph.end type="italics"/> ADE, &longs;itque maioris facilitatis gratia Orthonium; eiu&longs;que motus <lb/>&longs;it parallelus lineis ED, BC: &longs;it autem DA dupla DE; &longs;itque diui&longs;a to­<lb/>ta DA æqualiter in C, in C non erit maximus ictus; quia in C non <pb pagenum="44"/>e&longs;t centrum grauitatis, vt patet; vt autem habeatur centrum impre&longs;&longs;io­<lb/>nis; a&longs;&longs;umatur AN media proportionalis inter totam AD, & &longs;ubdu­<lb/>plum AC; certè cum triangulum ANO &longs;it &longs;ubduplum totius ADE, <lb/>vt con&longs;tat ex Geometria, & æquale trapezo ND EO; erit impetus in <lb/>vtroque æqualis; igitur in N erit centrum impre&longs;&longs;ionis, vel impetus; vt <lb/>autem habeatur centrum percu&longs;&longs;ionis; 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; quippe in I totus impeditur impetus <lb/>grauitatis vtrimque, cum &longs;it in æquilibrio; 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; moucatur 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; igitur punctum A duplò velociùs mouetur quàm B, vt con&longs;tat; igi­<lb/>tur habet duplò maiorem impetum; cum effectum habeat duplò maio­<lb/>rem per Ax. 13. n. </s> <s>4. igitur cum totus motus &longs;egmenti AB &longs;it ad to­<lb/>tum motum &longs;egmenti BC, vt &longs;patia acqui&longs;ita; certè &longs;patia acqui&longs;ita <lb/>&longs;unt vt arcus; igitur & trapezus BAED, continet 3/4 totius CAE, vt <lb/>con&longs;tat; &longs;unt enim &longs;ectores &longs;imilis in ratione duplicata radiorum; igi­<lb/>tur totus motus &longs;egmenti BC &longs;ubquadruplus motus totius CA;, gitur <lb/>& impetus; vt autem habeatur centrum impre&longs;&longs;ionis, vel impetus; &longs;it &longs;e­<lb/>ctor CHI, &longs;ubduplus totius CAE quod quomodo fiat, patet ex Geo­<lb/>metria; 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: vt autem habeatur percu&longs;&longs;ionis, a&longs;­<lb/>&longs;umatur CY dupla YA; Dico punctum Y e&longs;&longs;c 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc etiam&longs;oluctur, quod proponunt aliqui; &longs;eu potiùs quærunt; <lb/>in quà &longs;cilicet parte maiorem ictum infligat en&longs;is; &longs;i enim &longs;it ciu&longs;dem <lb/>cra&longs;&longs;itici 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Huc etiam reuoca clauarum ictus, vel aliorum corporum, quæ ad in <lb/>&longs;tar &longs;eu conorum, &longs;eu py<gap/>idum 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Colligemus etiam quid dicendum &longs;it de malleorum ictu; &longs;it enim <lb/>malleus F æqualis malleo G (in his vna fere manubrij longitudinis ha­<lb/>betur ratio) ducatur arcus NM, itemque OG; ictus mallei G e&longs;t ferè <lb/>&longs;ubduplus alterius, dum vterque malleus &longs;it æqualis; dixi ferè, quia <lb/>motus totius mallei G non e&longs;t omninò &longs;ubduplus motus mallci F, quia <lb/>&longs;cilicet trapezus OD e&longs;t minor &longs;ubduplo alterius NE; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Hîc ob&longs;erua nouum di&longs;crimen, quod intercedit inter impetum, & <lb/>alias qualitates; 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 cumdem &longs;emper effectum pro­<lb/>ducit; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus &longs;imilis, id e&longs;t, ad eamdem lineam determinatus, & æqualis in in­<lb/>ten&longs;ione, non pote&longs;t intendere alium &longs;imilem<emph.end type="italics"/>; Probatur, quia agit tantùm ad <lb/>extra, vt tollat impedimentum per Th. 44. &longs;ed eorum mobilium, quæ <lb/>ver&longs;us eamdem partem pari velocitate mouentur, neutrum impedit al­<lb/>terius motum, vt con&longs;tat; igitur impetus &longs;imilis, &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;erua de impetu &longs;imili id tantùm dici; &longs;imili inquam id e&longs;t non <lb/>modò eiu&longs;dem inten&longs;ionis; &longs;ed etiam eiu&longs;dem lineæ: &longs;i enim alterum <lb/>de&longs;it, haud dubiè &longs;imilis impetus non e&longs;t; &longs;ic impetus quatuor grad. </s> <s>in­<lb/>tendere pote&longs;t impetum duorum graduum; licèt vterque ad eamdem li­<lb/>neam &longs;it determinatus; &longs;i verò ad diuer&longs;as lineas determinentur; etiam <lb/>impetus vt duo pote&longs;t intendere impetum vt quatuor. </s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Hinc &longs;imile pote&longs;t in aliquo ca&longs;u agere in &longs;imile; vnde rectè colligo <lb/>id tantùm dictum e&longs;&longs;e ab Ari&longs;totele de qualitatibus alteratiuis; quid <lb/>verò accidat, cum mobile grauc mobili alteri &longs;uperponitur; dicemus <lb/>infrà<gap/></s></p><pb pagenum="46"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Exten&longs;io impetus respondet extentioni &longs;ui &longs;nbiecti, &longs;cilicet mobilis<emph.end type="italics"/>; cum <lb/>enim extra &longs;ubjectum e&longs;&longs;e non po&longs;&longs;it, cum &longs;it qualitas; certè ibi e&longs;t, vbi <lb/>&longs;ubjectum e&longs;t; nam penetratur accidens cum ip&longs;o &longs;ujecto. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scolium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; nec enim alliud e&longs;t, vnde po&longs;&longs;it <lb/>determinari extentio qualitatum, præter ip&longs;am exten&longs;ionem &longs;ubjecti; <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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Datur impetus altero impetu perfectior &longs;ecundum entitatem<emph.end type="italics"/>; dixi &longs;ecun­<lb/>dum entitatem; quia iam dictum e&longs;t &longs;uprà dari perfectiorem &longs;ecundum <lb/>inten&longs;ionem; huius Theorematis veritas mihi maximè demon&longs;tranda <lb/>e&longs;t, ex quo tàm multa infrà deducemus; &longs;ic autem probamus; 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; 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 fotmali poti&longs;&longs;imum <lb/>dicidebet; præterea-datur aliquod pondus, quod data potentia &longs;ine me­<lb/>chanico organo moucre non pote&longs;t, licèt cum organo facilè moueat; hæc <lb/>hypothe&longs;is certa e&longs;t; 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. </s> <s><lb/>33. cum verò non mouet, non producit tot partes impetus vt con&longs;tat ex <lb/>dictis; igitur producit plures cnm organo in mobili, quàm &longs;ine organo; <lb/>igitur imperfectiores, quod demon&longs;tro: &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; <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; quia pondus quod non pote&longs;t attolli in B, attollitur in <lb/>E, vel in F, vt patet ex dictis; præterea punctum F mouetur tardius, quàm <lb/>B; quia motus &longs;unt vt arcus, atcus vt &longs;emidiametri, hæ demum vt AF, <lb/>ad AB; igitur motus puncti F, e&longs;t tardior, vel imperfectior; 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 entiratis, <lb/>quæ imperfectior e&longs;t; igitur datur impetus altero impetu imperfectior. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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"/>per Axioma illud Metaph. probari, <emph type="italics"/>Data quacumque creatura dari potest <lb/>perfectior, vel imperfectior.<emph.end type="italics"/></s></p><p type="main"> <s>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 type="main"> <s>Quartò, &longs;i daretur punctum impetus in inten&longs;ione: non po&longs;&longs;et dari <lb/>motus tatdior in infinitum &longs;ine diuer&longs;is gradibus perfectionis. </s></p><p type="main"> <s>Quintò, &longs;ine hac diuer&longs;a impetus perfectione nonp o&longs;&longs;et explicari <lb/>productio continua impetus, quæ &longs;it temporibus inæqualibus, neque de­<lb/>&longs;tructio ciu&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 type="main"> <s>Sextò, Denique ratio propo&longs;ita rem i&longs;tam euincit; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s></p><p type="main"> <s><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> Primò, quia cum retardari po&longs;&longs;it <lb/>hic motus, & de&longs;trui &longs;ucce&longs;&longs;inè hic impetus; cumque in&longs;tantia motus <lb/>velocioris &longs;int breuiora; certè initio motus, breuiori &longs;cilicet tempore <lb/>imperfectior impetus de&longs;trui tantùm pote&longs;t; cum enim æqualis æquali­<lb/>bus temporibus; certè inæqualis inæqualibus. </s> <s>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 ctiam in commune e&longs;t calori, lumini, &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc difficiliùs attollitur pertica CA ex puncto C motu circulari, <lb/>quàm ex puncto B motu recto; quia &longs;cilicet, cum motu recto ex puncto B <lb/>attollitur, omnes partes mouentur motu æquali; igitur impetus æqualiter <lb/>omnibus di&longs;tribuitur; igitur modò producantur tot partes impetus, quot <lb/>&longs;unt partes in mobili; haud dubiè attolletur: at verò, cum motu circulari <lb/>ex puncto C attollitur, omnes partcs inæquali motu attolluntur; igitur <lb/>plures &longs;unt nece&longs;&longs;ariæ, vt attollatur motu circulari; igitur difficiliùs iuxta <lb/>experimentum; adde quod cum applicatur potentia in C, punctum A, <lb/>maius momentum habet, de quo aùàs. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Hinc ratio cuidens illius experimenti, quo manife&longs;tè con&longs;tat perti-<pb pagenum="48"/>cam CA, ex A, facilius attolli motu recto, quàm circulari; cum &longs;ci­<lb/>licct cuiu&longs;dam qua&longs;i reflexionis opera eodem tempore vtraque extremi­<lb/>tas æquali motu attollitur. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s></p><p type="main"> <s><arrow.to.target n="note1"></arrow.to.target></s></p><p type="margin"> <s><margin.target id="note1"></margin.target><emph type="italics"/>Fig.<emph.end type="italics"/>7. <lb/><emph type="italics"/>Tab.<emph.end type="italics"/>1.</s></p><p type="main"> <s><emph type="italics"/>Si verò applicetur potentia extra centrum vectis v. </s> <s>g. </s> <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; <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/>turenim ex. </s> <s>gr. </s> <s>vnus tantùm gradus in F, & in E, productus) at verò <lb/>producuntur alij imperfectiones. </s> <s>v.g. </s> <s>in D, præter æquè perfectum pro­<lb/>ducuntur 3. alij adæquantes perfectionem prioris; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quæ non pote&longs;t explicari per impetum remi&longs;&longs;io­<lb/>rem, vel inten&longs;iorem; nam cum &longs;it tantùm impetus in&longs;titutus propter <lb/>motum; certè ille tantùm impetus produci pote&longs;t, ex quo pote&longs;t &longs;equi <lb/>motus; igitur &longs;i tali tantùm motu data pars mobilis moueri pote&longs;t; haud <lb/>dubiètalis tantùm impetus, ex quo &longs;equitur talis motus, in ea produ­<lb/>cetur, & tali modo. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s></p><p type="main"> <s><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; &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 porti mobilis in&longs;unt<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus perfectus producere pote&longs;t imperfectum<emph.end type="italics"/>; patet in vecte; 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 ciu&longs;dem perfectionis cum impetu producto in ip&longs;a ve­<lb/>ctis extremitate, ex qua pendet; cum &longs;it vrriu&longs;que æqualis motus; &longs;ed <lb/>ver&longs;us centrum ciu&longs;dem vectis producitur impetus imperfectior per <lb/>Th.82. igitur imperfectus à perfecto producitur. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s></p><p type="main"> <s><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; dixi <lb/>perfectum ab imperfecto; &longs;cilicet &longs;i con&longs;ideretur perfectio ratione en-<pb pagenum="49"/>titatis; cum reuerâ, vt dictum e&longs;t &longs;uprà, remi&longs;&longs;us producat inten&longs;um, <lb/>quod in vecte clari&longs;&longs;unum e&longs;t; 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: 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; non tamen in <lb/>ratione cau&longs;æ totalis: &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 type="main"> <s>Ob&longs;eruabis &longs;ecundò præclarum naturæ in&longs;titutum, quo factum e&longs;t; <lb/>vt cum vires hominum maiora pondera leuare non po&longs;&longs;int, &longs;i &longs;eor&longs;un <lb/>con&longs;iderentur; cum organis tamen mechanicis conjunctæ nullum pon­<lb/>dus quantumuis immane leuare non po&longs;&longs;int; 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 type="main"> <s>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 type="main"> <s>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 type="main"> <s>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; &longs;i primum, erit primus tantùm gradus maximæ perfectio­<lb/>nis; ita vt perfectiorem producere non po&longs;&longs;it, ad quem e&longs;t determinata <lb/>potentia; imperfectiorem tamen impetu innato, de quo infrà; &longs;i verò <lb/>&longs;ecundum, producet in &longs;ingulis paitibus eumdem gradum perfecti&longs;&longs;i­<lb/>mum cum aliis pluribus, vel paucioribus heterogeneis, & imperfectio­<lb/>ribus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Potentia naturalis grauium producit tantùm vno in&longs;tanti ad intra vnicum <lb/>punctum impetus in quolibet puncto &longs;ubiecti; &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"/>; 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>per Ax. 7. & 3. <lb/>n. </s> <s>1. Præterea non e&longs;t, cur potius produceret 2. quàm 3. 4. &c. </s> <s>atqui <lb/>quod vnum e&longs;t, determinatum e&longs;t per Ax. 5. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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; determinatur autem <pb pagenum="50"/>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Eadem potentia inæqualibus temporibus impetum inæqualem in perfectio­<lb/>ne producit<emph.end type="italics"/>; 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>4. &longs;ed non <lb/>pote&longs;t minùs agere ratione numeri, vt patet; igitur ratione perfectio­<lb/>nis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s></p><p type="main"> <s><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; 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>2. igitur minori <lb/>tempore minùs de&longs;truitur per Ax. </s> <s>13. 4. alioquin totus &longs;imul debe­<lb/>ret de&longs;trui. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc, quò potentia diutiùs manet applicata (putà malleo) percu&longs;&longs;io ma­<lb/>ior e&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Hinc, quò impedimentum diutiùs manet applicatum, illa de&longs;tructio <lb/>e&longs;t maior. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Hinc præclara eruitur ratio, cur maior lapis, quàm minor impactus <lb/>maiorem ictum infligat; licèt tot partes impetus eodem in&longs;tanti produ­<lb/>cantur in vno, quot in alio: 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 90.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s>Probatur; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s></p><p type="main"> <s><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><lb/>accidentis; de qua in Metaphy&longs;icâ; 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; cum ip&longs;is etiam oculis <lb/>qua&longs;i ocrnatur; cum enim percutis corpus oblongum AE, & cadit ictus <lb/>in extremitatem A, corpus ip&longs;um totum &longs;unul 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>In propagatione impetus prima pars<emph.end type="italics"/> A v. </s> <s>g. <emph type="italics"/>non producit partem<emph.end type="italics"/> B, <emph type="italics"/>& <lb/>hæc<emph.end type="italics"/> C; <emph type="italics"/>hæc verò<emph.end type="italics"/> D, <emph type="italics"/>atque ita deinceps<emph.end type="italics"/>; Probatur. </s> <s>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>Secundò, Minimum granum &longs;uperpo&longs;itum rupi, to­<lb/>tam ip&longs;am rupem mouere po&longs;&longs;et. </s> <s>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>Quintò, quia impetus partis B, tàm ageret in A, trahendo, <lb/>quàm in C pellendo; cum impetus vtroque modo propagetur. </s> <s>Sextò, &longs;i <lb/>applicaretur potentia in C, non video, cur impetus partis C, ageret po­<lb/>tius versùs E, quàm versùs A? alioquin cadem 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>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>Octauò, &longs;i <lb/>impetus vnius partis producit in alia; &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 type="main"> <s>Ob&longs;eruabis illud quidem verum e&longs;&longs;e in motu recto, &longs;ecus in circulari; <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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 93.<emph.end type="center"/></s></p><p type="main"> <s><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> Proba­<lb/>tur; &longs;it enim applicata potentia in A, dico &longs;imul produci impetum in <lb/>BCDE; 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; vtrumque e&longs;t ab&longs;ur­<lb/>dum; nam totum AE, &longs;imul mouetur. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 94.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Tribus tantùm modis propagari pote&longs;t impetus ratione inten&longs;ionis.<emph.end type="italics"/></s><s> Primò <lb/>&longs;i æqualiter omnibus partibus &longs;ubjecti di&longs;tribuatur; id e&longs;t vniformiter. </s> <s><lb/>Secundò, &longs;i plùs partibus propioribus, & minùs remotioribus. </s> <s>Tertiò, è <lb/>contra, &longs;i plùs remotioribus, & minùs propioribus; tribus etiam ratione <lb/>perfectionis eo modo, quo diximus de inten&longs;ione; at verò nouem mo­<lb/>dis propagari pote&longs;t ratione vtriu&longs;que; patet ex regula combinationum; <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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 95.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus propagatur vniformiter in mobili, cuius omnes partes mouentur <lb/>æquali motu<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cum duo corpora &longs;e&longs;e mutuò tangunt, impetus in vtroque propagatur<emph.end type="italics"/> &longs;int <lb/>v. </s> <s>g. </s> <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: <lb/>probatur primò, quia &longs;e habent per modum vnius, vt patet ex re&longs;i&longs;ten­<lb/>&longs;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>Tertiò, Ratio à priori e&longs;t; quia idco <lb/>producitur, & propagatur impetus in toto A; 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 moucatur B; <lb/>igitur in vtroque &longs;imul, & æqualiter propagatur impetus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Hinc ratio manife&longs;ta cur maior &longs;it re&longs;i&longs;tentia duorum quàm vnius. </s></p><pb pagenum="53"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Hinc percu&longs;&longs;io vel ictus globi B, cui alter A à tergo immediatè in­<lb/>&longs;t&longs;tit maior e&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s>Hinc pondus alteri &longs;uperpo&longs;itum actione communi cum alio graui­<lb/>tat in &longs;uppo&longs;itam manum. </s> <s>v. </s> <s>g. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s>Hinc potentia applicata in D, minùs impetus &longs;ingulis imprimit. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s>Hinc demum licèt impetus ratione inten&longs;ionis &longs;it æqualis in vtroque <lb/>globo; attamen, &longs;i accipiatur numerus partium vtriu&longs;que impetus, im­<lb/>petus &longs;unt vt globi v. </s> <s>g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 8.<emph.end type="center"/></s></p><p type="main"> <s>Hinc demum colligi pote&longs;t non modò impetum produci in globo B <lb/>v. </s> <s>g. </s> <s>verùm etiam in aëre ambiente, cui &longs;cilicet globus contiguus e&longs;t; <lb/>qui reuera aër facilè amouetur; tùm quia propter raritatem pauci&longs;&longs;imæ <lb/>partes mouendæ &longs;unt; tùm quia facilè diuiduntur, de quibus alias; 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; nec enim <lb/>totus aër agitari debet; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 97.<emph.end type="center"/></s></p><p type="main"> <s><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>Th. 73. &longs;it centrum motus C; haud dubiè plures <lb/>partes impetus producuntur in B, quàm in C, & plures in A, quam in B; <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>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"/>versùs circumferentiam, non perfectior, patet per Th. 8. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 98.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; &longs;int enim punctum B, & pnnctum A: 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; arcus &longs;unt, vt &longs;emidiametri AC, BC; igitur vt <lb/>di&longs;tantiæ quòd erat demon&longs;trandum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Hinc hæc propagatio e&longs;t omninò inuer&longs;a illius, quæ aliis qualitatibus <lb/>competit, vt patet. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s>Hinc etiam manife&longs;ta ratio &longs;equitur illius experimenti, quod propo­<lb/>&longs;uimus corol. </s> <s>2. Th. 80. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s>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; po&longs;ito &longs;cilicet centro motus in altera extremitate, cui applica­<lb/>tur potentia; quippe impetus propagatus in motu circulari e&longs;t &longs;umma <lb/>terminorum; propagatus verò in motu recto e&longs;t vltimus terminorum, <lb/>v.g. </s> <s>&longs;int &longs;ex puncta &longs;ubiecti; in quolibet producatur impetus vt vnum; <lb/>haud dubiè erit motus rectus; vt verò &longs;it motus circularis in primo <lb/>puncto; 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 type="main"> <s>Dixi, &longs;i tantùm habeatur ratio impetus; quia &longs;i addatur ratio graui­<lb/>tationis, &longs;eu momenti; haud dubiè maior erit adhuc difficultas, de <lb/>quo infrà in Schol. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s>Hinc quò longior e&longs;t cylindrus, v. </s> <s>g. </s> <s>cre&longs;cit proportio maioris illius <lb/>facilitatis, vt patet inductione; nam &longs;i &longs;int tantùm 2. puncta, proportio <lb/>erit 3. ad 2.; &longs;it tria 6. ad 3.; &longs;i 4. 10. ad 4. &longs;i 5. 15. ad 5.; &longs;i 6. 21. ad 6. <pb pagenum="55"/>&longs;i 7. 28. ad 7; &longs;i 8. 36. ad 8; &longs;i 9. 45. ad 9; atque ita deinceps; ex quibus primò <lb/>vides cre&longs;cere &longs;emper proportionem. </s> <s>Secundò inter duplam, & triplam <lb/>rationem, &longs;cilicet 6. ad 3. & 15. ad 5. intercedere 2 1/2; inter triplam & <lb/>quadruplam intercedere 3. 1/2; inter quadruplam & quintuplam inter­<lb/>cedere 4 1/2; atque ita deinceps. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s></p><p type="main"> <s>Colligo denique po&longs;&longs;e in motu recto cum maiore ni&longs;u produci inten­<lb/>&longs;iorem impetum in data ratione; &longs;it enim cylindrus AB, qui moueatur <lb/>circa centrum A, percurrátque B, arcum BD; qui accipiatur vt recta, <lb/>quæ à minimis arcubus &longs;en&longs;u di&longs;tingui non pote&longs;t; haud dubiè &longs;i eo <lb/>tempore, vel æquali, quo AB tran&longs;it in AD; 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; quia impetus &longs;unt vt motus; motus verò vt <lb/>&longs;patia, quæ percurruntur æqualibus temporibus; &longs;ed &longs;patium rectanguli <lb/>AD, e&longs;t duplum trianguli ADB; igitur & motus; igitur & impetus; &longs;i <lb/>verò AB tran&longs;eat in EL, ita vt AF, &longs;it dupla AE; impetus erunt <lb/>æquales; quia rectangulum AC, e&longs;t æquale triangulo ABD. </s></p><p type="main"> <s>Dixi arcum BD, accipi vt lineam rectam; 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s>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. </s> <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 type="main"> <s>Tertiò, &longs;i &longs;upponatur fulcitus in B; vt retineatur in æquilibrio, debet <lb/>addi momentum in A; &longs;eu debet retineri ab ip&longs;a potentiâ applicata <lb/>in A. </s></p><p type="main"> <s>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. </s></p><p type="main"> <s>Quintò, &longs;i proprio pondere frangeretur BG, haud dubiè in B frange­<lb/>retur; 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; haud dubiè <lb/>pondus in H, facit momentum &longs;ubduplum eiu&longs;dem in G, vt patet; &longs;unt <lb/>enim vt di&longs;tantiæ; igitur cum &longs;egmentum HG tantùm addat momenti <lb/>&longs;upra H, quantùm detrahit HB; certè momentum totius ponderis BG, <pb pagenum="56"/>e&longs;t tantùm &longs;ubduplum eiu&longs;dem po&longs;iti in G; itaque &longs;it BG, 10. librarum, <lb/>æquiualet 5. libris &longs;tatutis in G, & AB, vni libræ po&longs;itæ in A; &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; igitur vt &longs;tatuatur æquilibrium, <lb/>debent e&longs;&longs;e 24. libræ in A, &longs;eu vires æquiualentes; 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>Primò ex ip&longs;o pondere ab&longs;olutè <lb/>&longs;umpto, quæ communis e&longs;t motui propagationis. </s> <s>Secundò, ex momento <lb/>eiu&longs;dem ponderis; Tertiò, ex tali genere propagationis, de quo &longs;uprà; <lb/>quæ omnia &longs;unt apprimè tenenda, ne quis error &longs;ubrepat. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 99.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cum applicatur potentia circumferentiæ motus circularis; ita propagatur <lb/>impetus, vt plures partes ver&longs;us centrum motus producantur in pondere, quod <lb/>attollitur<emph.end type="italics"/>; &longs;it enim idem cylindrus CA; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Scio quidem hoc ip&longs;um à nemine hactenus, quod &longs;ciam, explicatum <lb/>e&longs;&longs;e; atque fore vt à multis tanquam nouum, & in&longs;olens minùs fortè <lb/>probetur: quamquam illa hypothe&longs;is hoc ip&longs;um euincit, vulgaris certè, <lb/>& nemini qua&longs;i non nota; qua nempè dicimus in omnibus partibus mo­<lb/>bilis, quod actu mouetur, impetum produci; & &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 100.<emph.end type="center"/></s></p><p type="main"> <s><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> Probatur per Th.81. ex motu imper­<lb/>ctiore, 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 101.<emph.end type="center"/></s></p><p type="main"> <s><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"/>: 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 102.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; probatur, quia pondus ver&longs;us centrum mouetur <lb/>minore motu, vt con&longs;tat; igitur ab imperfectiore impetu; &longs;ed non e&longs;t <lb/>imperfectior tantùm ratione numeri, id e&longs;t, pauciorum partium impe­<lb/>tus; quia &longs;i hoc e&longs;&longs;et, &longs;it vectis AC, motus B, e&longs;t &longs;ubduplus motus <lb/>A; igitur &longs;i e&longs;t impetus eiu&longs;dem perfectionis entitatiuæ, vt &longs;ic loquar; <lb/>ita &longs;e habet numerus partium impetus in B, ad numerum partium in A, <lb/>vt motus B, ad motum A; & hic vt arcus BD, ad arcum AE; & hic vt <lb/>BC, ad AC; igitur e&longs;t &longs;ubduplus; igitur æqualis omninò producitur <lb/>impetus ab eadem potentia in vecte AC, &longs;iue applicetur centro C, &longs;iue <lb/>circumferentiæ A; igitur æquè facilè; quod e&longs;t contra experientiam; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 103.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 104.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Decre&longs;cit impetus ver&longs;us centrum iuxta rationem distantiarum<emph.end type="italics"/>; probatur <lb/>quia decre&longs;cit iuxta rationem motuum; & hæc iuxta rationem di&longs;tan­<lb/>tiarum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 105.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Non decre&longs;cit numerus partium impetus à circumferentia ad centrum<emph.end type="italics"/>; <lb/>probatur, quia cum à circumferentia ad centrum ita propagetur impe­<lb/>tus, vt vnicum tantùm punctum producatur in ip&longs;a extremitate mobilis; <lb/>certè non pote&longs;t minùs impetus produci ver&longs;us centrum ratione nume­<lb/>ri; igitur non decre&longs;cit numerus; hinc producitur nece&longs;&longs;ariò imperfe­<lb/>ctior ver&longs;us centrum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 106.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Non producuntur plures partes impetus in vecte ver&longs;us centrum, id est, non <lb/>&longs;unt plures in punclo 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>Secundò, cur <lb/>potiùs in vna proportione, quàm in alia? </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 107.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; <lb/>probatur, quia non producitur numerus minor per Th.105. neque maior <lb/>per Th. 106. igitur æqualis; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis, quod aliquando in mentem venerat; &longs;cilicet, ver&longs;us cen­<lb/>trum produci maiorem numerum in ratione di&longs;tantiarum permutando; <lb/>& imperfectiorem in ratione duplicata earumdem di&longs;tantiarum, etiam <lb/>permutando, v. </s> <s>g. </s> <s>&longs;it idem vectis AC &longs;ectus bifariam in B; in puncto <lb/>B producitur numerus duplus producti in A; at verò perfectio impetus <lb/>in B e&longs;t ad perfectionem impetus in A, vt quadratum BC ad quadra­<lb/>tum AC; vel in ratione &longs;ubquadrupla, licèt tota collectio impetus B <lb/>&longs;it tantùm &longs;ubdupla perfectione collectionis impetus A; &longs;ed hoc profe­<lb/>ctò dici non pote&longs;t; nam &longs;int in A 4. partes impetus; igitur in B erunt <lb/>8. applicetur autem pondus in B. </s> <s>Primò producentur in co partes 8. <lb/>impetus perfectionis &longs;ubquadruplæ; &longs;i comparentur cum partibus A, <lb/>tum producentur 16. quæ æquiualent 4 A; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 108.<emph.end type="center"/></s></p><p type="main"> <s><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>in &longs;ingulis punctis vectis &longs;ingula puncta impetus, &longs;ed diuer&longs;æ <lb/>perfectionis; 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; 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 moucbitur; atque ita deinceps; eum enim ap­<lb/>ponitur pondus in B, producuntur in eo partes impetus omnes eiu&longs;dem <lb/>perfectionis; quæ &longs;cilicet re&longs;pondet B, id e&longs;t, quæ e&longs;t &longs;ubdupla perfectio­<lb/>nis impetus A; igitur plures partes producuntur, quàm &longs;i e&longs;&longs;ent perfe­<lb/>ctionis A; &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><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 109.<emph.end type="center"/></s></p><p type="main"> <s><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; vt impetus puncti B &longs;it &longs;ubduplus in perfectione, puncti R <lb/>fubtriplus: iam verò &longs;it vectis &longs;ubduplus prioris BC, &longs;ectus bifariam in <lb/>Z; &longs;i impetus productus in B, qu&ecedil; 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 æmpore percurrant arcus æquales, &longs;cilicet AV, & BD) certè im-<pb pagenum="59"/>pertus productus in Z e&longs;t æqualis producto in B, cum B pertinet ad ma­<lb/>iorem vectem; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 110.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 111.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;erua tamen quacumque data potentia po&longs;&longs;e dari minorem; quia <lb/>quocumque dato motu, etiam recto, pote&longs;t dari tardior; igitur quocum­<lb/>que impetu imperfectior; igitur quando appellaui potentiam minimam; <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 type="main"> <s>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 ciu&longs;dem perfectionis, ciu&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. </s> <s>v. </s> <s>g. </s> <s>illa, quæ applicatur <lb/>vecti. </s> <s>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; &longs;i 5. ad 3. <lb/>&longs;i 6. ad 3. 1/2; &longs;i 7. ad 4. &longs;i 8. ad 4. 1/2. Vides egregiam progre&longs;&longs;ionem; &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; producatur vnum punctum impetus ciu&longs;dem <lb/>perfectionis cum eo, quod productum e&longs;t in A; 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; perfectiones enim &longs;unt vt lon-<pb pagenum="60"/>gitudines; quæ &longs;i colligantur, habebis characterem totius impetus, 2 1/2: <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; igitur vectis maior maiorem potentiam ad mouendum ip&longs;um ve­<lb/>ctem requirit; non certè in de&longs;cen&longs;u; quippe &longs;uo pondere de&longs;cendit, &longs;ed <lb/>in plano horizontali; ni&longs;i enim potentia po&longs;&longs;it mouere vectem; haud <lb/>dubiè nullum pondus vecte mouebit. </s></p><p type="main"> <s>At verò &longs;i potentia &longs;it tantùm dupla minimæ, quæ datum vectem mo­<lb/>uere po&longs;&longs;it; 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 type="main"> <s>Ob&longs;eruabis demum in mechanicis nullam ferè haberi rationem pon­<lb/>deris ip&longs;ius vectis; parum enim pro nihilo computatur: Ex his tamen <lb/>erui po&longs;&longs;unt veri&longs;&longs;imæ rationes Phy&longs;icæ proportionum vectis AH; &longs;ia­<lb/>que A extremitas, H centrum; &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; certè in B moue­<lb/>bitur pondus K duplum I; quia, cum impetus productus in B, &longs;it &longs;ubdu­<lb/>plus in perfectione illius, qui producitur in A; vt æqualis producatur in <lb/>B, & in A, debent produci in B duplò plures partes impetus; igitur du­<lb/>plò maius pondus mouebit; at verò in C mouebitur pondus L quadru­<lb/>plum I, in D octuplum, atque ita deinceps; donec tandem in G mot ea­<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 type="main"> <s>Ob&longs;eruabis etiam in omni vecte ab&longs;trahendo ab eius pondere, & ap­<lb/>plicata cadem potentia, hoc e&longs;&longs;e commune; vt po&longs;&longs;it quodcumque pon­<lb/>dus attolli, licèt difficiliùs in minore; quia hic non pote&longs;t in tam mul­<lb/>tas partes aliquotas &longs;en&longs;ibiliter diuidi, in medio tamen vecte duplum <lb/>femper pondus mouetur; &longs;iue ip&longs;e vectis &longs;it maior, &longs;iue minor. </s></p><p type="main"> <s>Ob&longs;eruabis deinde, &longs;i centrum vectis non &longs;it in altera extremitate, <lb/>&longs;ed. </s> <s>v.g. </s> <s>in C; haud dubiè producitur in H, & in B impetus æqualis; quia <lb/>æqualiter di&longs;tat vtrumque punctum à centro C; 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 type="main"> <s>Ob&longs;eruabis denique triplicem propagationem impetus e&longs;&longs;e legiti­<lb/>mam. </s> <s>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 heterogencos. </s> <s>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>Tertia e&longs;t in vecte, cum propagatur per partes <lb/>æquales in numero, & inæquales in perfectione. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 112.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus debet determinari ad aliquam lineam motus<emph.end type="italics"/>; 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"/>tum, ni&longs;i per aliquam lineam, vt patet; &longs;ed hoc e&longs;t impetum e&longs;&longs;e de­<lb/>terminatum ad aliquam lineam motus; præterea &longs;i non e&longs;t determina­<lb/>tus ad aliquam lineam; 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>igitur debet determinari. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 113.<emph.end type="center"/></s></p><p type="main"> <s><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; præterea corpus graue <lb/>in diuer&longs;is planis inclinatis de&longs;cendit; igitur per diuer&longs;as lineas; deinde <lb/>pila reflectitur propter impetum priorem, qui tantùm mutat lineam, vt <lb/>dicemus infrà; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 114.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc idem impetus ad plures lineas potest determinari &longs;eor&longs;im<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; vnam verò li­<lb/>neam differre ab alia ratione terminorum v. </s> <s>g. </s> <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; <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 type="main"> <s>Secundò ob&longs;eruabis aliquando videri e&longs;&longs;e vtrumque terminum com­<lb/>munem licèt differant lineæ; &longs;it linea recta BE, habet communes ter­<lb/>minos cum curua BFE, licèt omninò differat ab illa; at profectò licèt <lb/>BE videatur e&longs;&longs;e vnica &longs;implex linea duobus terminis clau&longs;a; con&longs;tat <lb/>ramen ex pluribus aliis continuata, rectáque &longs;erie iunctis; 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 type="main"> <s>Tertiò linea motus non differt ab ip&longs;o motu continuo tractu, &longs;eu <lb/>fluxu qua&longs;i labenti: Porrò vnus motus differt ab alio, vel ratione velo­<lb/>citatis, vel ratione terminorum; &longs;ed hæc parum difficultatis habent. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 115.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus aliquis ad vnam tantùm lineam pote&longs;t e&longs;&longs;e determinatus<emph.end type="italics"/>; v. </s> <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"/><emph type="italics"/>plures determinari po&longs;&longs;e, vt videbimus cum de motureflexo<emph.end type="italics"/>; probatur quia <lb/>motus deor&longs;um e&longs;t finis huius impetus; 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; <lb/>tàm enim omne graue non impeditum tendit deor&longs;um, quàm omnis <lb/>ignis e&longs;t calidus; igitur &longs;i e&longs;t proprietas omnis ignis e&longs;&longs;e calidum, quia <lb/>omni competit; ita omni graui competit tendere infrà leuius, modò <lb/>non impediatur; igitur e&longs;t eius proprietas; igitur ille impetus e&longs;t de­<lb/>terminatus ad lineam quæ tendit deor&longs;um; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 116.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus determinatur aliquando ad lineam motus à potentia motrice<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 117.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; &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; pro cuius diuer&longs;a inclinatione <lb/>diuer&longs;a e&longs;t linea motus deor&longs;um; 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>vel ab appetitu &longs;eu libero, &longs;eu <lb/>&longs;en&longs;itiuo; &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; &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. </s> <s>g. </s> <s>tu&longs;&longs;is, &longs;ingultus, &longs;ternuationis, &c. </s> <s><lb/>de quibus fusè &longs;uo loco. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 118.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus determinatur aliquando ad lineam ab alio impetu producente<emph.end type="italics"/>; <lb/>&longs;ic impetus corporis proiecti determinatur ab impetu vel organi vel <lb/>manus proiicientis; quia nihil e&longs;t aliud à quo determinari po&longs;&longs;it, vt <lb/>patet; adde figuram organi, di&longs;po&longs;itionem &longs;eu &longs;itum mobilis, quod ma­<lb/>nu tenetur; impedimenti etiam habetur ratio v. </s> <s>g. </s> <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: &longs;i enim altera extremi­<lb/>tas adhuc hæreat in manu, dum altera mouetur, vt cum quis baculo <lb/>ferit; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 119.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus determinatus ad vnam lineam pote&longs;t ad aliam in &longs;uo fluxu deter­<lb/>minatu<emph.end type="italics"/>; 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>Probatur etiam ex impetu proiectorum, quæ mutant lineam mo­<lb/>tus manente adhuc priore impetu &longs;altem ex parte. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 120.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Corpus proiectum in aliud ita illud impellit, vt determinet lineam motus <lb/>ratione puncti contactus<emph.end type="italics"/>; Sit enim, ne mnltiplicemus 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; &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: experientia huius rei certa <lb/>e&longs;t, nec ignorant qui in ludo minoris tudiculæ ver&longs;ati &longs;unt; 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; coniungitur autem vtrumque hemi&longs;phærium circulo A, <lb/>vel B, in priore figura, e&longs;tque vtriu&longs;que communis &longs;ectio; 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: præterea cum <lb/>motus orbis globi determinetur à motu centri; cum &longs;cilicet globus in <lb/>globum impingitur; 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 cuincit. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 121.<emph.end type="center"/></s></p><p type="main"> <s><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 eamdem lineam globus impul&longs;us determinabitur,<emph.end type="italics"/> v. </s> <s>g. </s> <s>li­<lb/>cet globus P. ciu&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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 122.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Determinatur impetus corporis proiecti impacti in corpus reflectens ad no­<lb/>uam lineam<emph.end type="italics"/>; patet experientiâ in pilâ reflexâ; reflexionis autem ratio­<lb/>nem afferemus in lib. de motu reflexo. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 123.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Non determinatur tantùm ratione puncti contactus.<emph.end type="italics"/></s><s> 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 124.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 125.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 126.<emph.end type="center"/></s></p><p type="main"> <s><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; & 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 127.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; Patet ex­<lb/>perientia in pilis illis eburneis, quas de&longs;iderat ludus minoris tudiculæ; <lb/>nec e&longs;t vlla ratio, cur determinatio &longs;it maior potiùs, quàm minor, cum <lb/>vtraque pila &longs;it æqualis; &longs;i enim maior e&longs;&longs;et, vel minor; cur potiùs vno <lb/>gradu, quàm duobus? </s> <s>quàm tribus? </s> <s>Præterca, cum re&longs;i&longs;tens, vel im­<lb/>pediens e&longs;t æquale agenti; cer<gap/>e &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. Ita re&longs;i&longs;tens, vel impediens refundit æquale <lb/>impedimentum, quod tantùm &longs;umi pote&longs;t ex æqualitate mobilium; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 128.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc ratio manife&longs;ta illius mirifici effectus, &longs;cilicet quietis pilæ impactæ<emph.end type="italics"/>; <lb/>quippe hæc quie&longs;cet illicò ab ictu; quia &longs;cilicet, cum noua determina­<lb/>tio &longs;it æqualis priori, non e&longs;t vlla ratio, cur alterutra præualeat; 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>de quo infrà. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 129.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quæ<gap/> 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/>connectante centra, quæ &longs;cilicet per punctum contactus à centro impacli globi<emph.end type="italics"/><pb pagenum="65"/><emph type="italics"/>ad centrum alterius ducitur<emph.end type="italics"/>; quippe nihil e&longs;t aliud à quo determinari. </s> <s><lb/>po&longs;&longs;it, vt patet; non determinatur etiam ab alterutra &longs;eor&longs;im, vt con­<lb/>&longs;tat, igitur ab vtraque conjunctim; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 130.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc globus &longs;ic impactus nunquam quie&longs;cit<emph.end type="italics"/>; ratio e&longs;t, quia vtraque linea <lb/>determinationis cum angulum faciat, in communem lineam abit; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 131.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 132.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si globus major in minorem impingatur per lineam directionis, quæ conne­<lb/>ctat centra, &longs;eruat eamdem lineam<emph.end type="italics"/>; patet etiam experientiâ, cuius ratio e&longs;t <lb/>minor re&longs;i&longs;tentia minoris globi; &longs;i verò &longs;it alia linea directionis, omni­<lb/>nò reflectitur &longs;uo modo; id e&longs;t mutat lineam; &longs;ed de his omnibus fusè <lb/>aliàs; hîc tantùm &longs;ufficiat indica&longs;&longs;e; (&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; &longs;i æqualis e&longs;t, globus impactus &longs;i&longs;tit; &longs;i <lb/>maior, reflectitur; &longs;i minor, eamdem lineam, &longs;ed lentiùs pro rata pro­<lb/>&longs;equitur. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 133.<emph.end type="center"/></s></p><p type="main"> <s><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; &longs;it enim <lb/>globus vtrimque gemino malleo percu&longs;&longs;us æquali ictu; haud dubiè &longs;i&longs;tit; <lb/>cur enim potiùs in vnam partem quam in aliam? </s> <s>cum &longs;imul in vtramque <lb/>moueri non po&longs;&longs;it. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 134.<emph.end type="center"/></s></p><p type="main"> <s><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; 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>de quo aliàs. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 135.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Siduo globi projecti &longs;ibi inuicem occurrant in lineæ directionis connectente <lb/>centra, reflectitur vterque æquali motu, quo antè.<emph.end type="italics"/></s><s> Probatur; &longs;unt enim globi <pb pagenum="66"/>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/>camdem rationem; 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; igitur in vtroque globo remanet æqualis im­<lb/>petus priori; igitur æquali motu vterque mouetur, quod erat dem. </s> <s>& hæc <lb/>e&longs;t ratio veri&longs;&longs;ima toties probatæ experientiæ. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 136.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 137.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; &longs;it enim mobile in A. v. </s> <s>g. </s> <s>globus, <lb/>cui &longs;imul imprimatur impetus determinatus ad lineam AD, in plano <lb/>horizontali AF; &longs;i vterque &longs;it æqualis, ad nouam lineam determinabi­<lb/>tur AE; 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; igitur debet ferri in E; igitur per diagonalem AE; clara e&longs;t <lb/>omninò experientia; 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; 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 type="main"> <s>Si verò impetus per AC &longs;it duplus impetus per AD; accipiatur AC <lb/>dupla AD, ducatur DF æqualis & parallela AC; 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 138.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si &longs;it duplex impetus in eodem mobili ad eamdem lineam determinatus, non <lb/>mutabitur linea; &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; deinde produca­<lb/>tur &longs;imul alius impetus æqualis priori in eodem mobili per lineam AB; <lb/>Dico quod eodem tempore percurretur tota AE, dupla &longs;cilicet AB; <lb/>quia &longs;cilicet dupla cau&longs;a non impedita duplum effectum habet per Ax. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 139.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si lineæ duplicis impetus, faciunt angulum acutiorem, longius erit &longs;patium<emph.end type="italics"/><pb pagenum="67"/><emph type="italics"/>acqui&longs;itum<emph.end type="italics"/>: &longs;int duæ lineæ IK IL, mobili &longs;eilicet &longs;tatuto in I; <lb/>haud dubiè noua linea erit IM; & quo angulus KIL, erit acutior (&longs;up­<lb/>po&longs;itis æqualibus &longs;emper lateribus IK IL) Diagonalis IM, erit ma­<lb/>ior; donec tandem IL & IK coeant in eandem lineam; tunc enim li­<lb/>nea erit dupla IK per Th. &longs;uperius: 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 140.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; &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; noua linea erit EG, eo rectè breuior, <lb/>quò angulus e&longs;t obtu&longs;ior, non tamen iuxta rationem angulorum; donec <lb/>tandem de&longs;inat angulus, & ED EF coëant in vnam lineam; 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; quippe quæ cum certis experimentis, & <lb/>clari&longs;&longs;imis principiis con&longs;entiant; &longs;ed de his plura infrà. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 141.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; nec <lb/>enim AE e&longs;t dupla AB, vt con&longs;tat; 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 eamdem 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; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 142.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis vt faciliùs concipias duos impetus ad duas lineas deter­<lb/>minatos; finge tibi nauim à diuer&longs;is ventis impul&longs;am, &longs;eu lapidempro­<lb/>jectum è naui mobili; &longs;ed de his plura in lib.4. cum de motu mixto. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 143.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus &longs;emel productus, quamdiu durat motus, con&longs;eruatur.<emph.end type="italics"/></s><s> 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 144.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus non con&longs;eruatur à cau&longs;a primò productiua.<emph.end type="italics"/></s><s> Probatur; quia proii-<pb pagenum="68"/>ciatur mobile per Po&longs;tulatum, etiam mouetur &longs;eparatum à potentia mo­<lb/>trice per hypoth. </s> <s>6. igitur non con&longs;eruatur à potentia motrice per Ax. </s> <s><lb/>10. igitur nec à causâ primò productiua. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 145.<emph.end type="center"/></s></p><p type="main"> <s><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>1. non à cau&longs;a primò productiua per Th.144.igi­<lb/>tur ab alia, eaque applicata per Ax. </s> <s>10. quæcumque tandem illa &longs;it, ali­<lb/>quando cau&longs;am primam e&longs;&longs;e demon&longs;trabimus; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 146.<emph.end type="center"/></s></p><p type="main"> <s><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> 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; nam reuerâ impetus pro­<lb/>ductus deor&longs;um à corpore graui motu naturaliter accelerato de&longs;truitur, <lb/>vt patet; præterea &longs;i corpus graue con&longs;eruaret impetum primò produ­<lb/>ctum, non produceret nouum contra experientiam; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; 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 type="main"> <s>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; ne &longs;i ab alia con&longs;eruarentur, <lb/>de&longs;truerentur vmquam; cum earum de&longs;tructionem nihil exigeret per <lb/>Ax.14.n.2. & 3. at verò qualitates, quæ contrarias habent: &longs;i quæ &longs;unt, <lb/>à cau&longs;a primò productiua minimè con&longs;eruantur; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 147.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Tamdiu con&longs;eruatur impetus, quamdiu nihil exigit eius destructionem<emph.end type="italics"/>; quia <lb/>de&longs;truitur tantùm ad exigentiam alicuius, quidquid tandem illud &longs;it, de <pb pagenum="69"/>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. </s> <s><lb/>14.num.3. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Inde certa ducitur ratio, cur mobile etiam &longs;eparatum à manu mouea­<lb/>tur; quia &longs;cilicet ip&longs;i adhuc ine&longs;t impetus, qui e&longs;t cau&longs;a motus; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Hinc etiam rationem ducemus æquè præclaram in lib.2. motus natu­<lb/>raliter accelerati. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 148.<emph.end type="center"/></s></p><p type="main"> <s><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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 149.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>In lineis oppo&longs;itis impetus de&longs;truitur ab impetu &longs;uo modo<emph.end type="italics"/>; &longs;it enim globus <lb/>proiectus ver&longs;us au&longs;trum; cui deinde imprimatur nouus impetus ver­<lb/>&longs;us Boream; 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, igitar nihil aliud exigit per Ax. 10. <lb/>hæc porrò exigentia non e&longs;t immediata, &longs;ed mediata, vt dixi. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 150.<emph.end type="center"/></s></p><p type="main"> <s><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; igitur ad exigentiam ali­<lb/>cuius, &longs;ed illud tantùm e&longs;t impetus innatus; nec enim e&longs;t &longs;ub&longs;tantia <lb/>corporis; tùm quia qualitas &longs;ub&longs;tantiæ non opponitur; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 151.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 152.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc &longs;i excipias tantùm impetum natur alem 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"/><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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s>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 type="main"> <s>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 type="main"> <s>Quartò, cum impetus &longs;uo fine caret, fru&longs;trà e&longs;t; 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; 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; igitur minor e&longs;t motus; 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; 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; &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 type="main"> <s>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>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; 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 type="main"> <s>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; 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 type="main"> <s>Sextò, Impetus naturalis innatus nunquam de&longs;truitur; 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"/><p type="main"> <s>Septimò, Impetus &longs;ur&longs;um de&longs;truitur etiam, quia e&longs;t fru&longs;trà; quippe <lb/>naturalis detrahit aliquid &longs;patij pro rata; igitur ne aliquid impetus &longs;it <lb/>fru&longs;trà, de&longs;truitur; idem dico de impetu per inclinatam &longs;ur&longs;um, licèt <lb/>minùs de&longs;truatur quàm in perpendiculari &longs;ur&longs;um; idem de impetu per <lb/>inclinatam deor&longs;um, &longs;ed minùs adhuc, &longs;ed hæc acuratiori medita ioni <lb/>&longs;unt relinquenda; quod reuerâ præ&longs;tabimus in lib.4. de motu mixto; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 153.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; Probatur primò, quia vterque ad omnem <lb/>lineam e&longs;t indifferens per Th.113. igitur vnus non c&longs;t alteri contrarius <lb/>ratione entitatis; cùm vterque &longs;imilem motum, immò eumdem habere <lb/>po&longs;&longs;it, vt patet ex dictis: Igitur ratione tanuùn 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 154.<emph.end type="center"/></s></p><p type="main"> <s><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> Probatur eodem modo; quia determinari pote&longs;t ad omnem li­<lb/>neam, vt patet ex reflexione grauis cadentis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 155.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 156.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus ex naturali acqui&longs;ito pote&longs;t fieri violentus<emph.end type="italics"/>; vt patet in motu re­<lb/>flexo grauium; ratio e&longs;t. </s> <s>quia mutatur linea. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 157.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus ex non contrario eidem fit contrarius<emph.end type="italics"/>; vt patet in eodem ca&longs;u; <lb/>nam impetus naturalis innatus, qui in de&longs;cen&longs;u non erat contrarius <lb/>ac qui&longs;ito, in inotu &longs;ur&longs;um reflexo fit contrarius. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 158.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis cogno&longs;ci tantùm contrarietatem qualitatum ex mutua de­<lb/>&longs;t ructione; cur verò vna qualitas dicatur de&longs;truere aliam, & cur illam <pb pagenum="72"/>de&longs;tructionem exigat; maximum my&longs;terium e&longs;t, quod alibi enucleabi­<lb/>mus; quàm multa enim &longs;uper hac re tacuere Philo&longs;ophi! </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 159.<emph.end type="center"/></s></p><p type="main"> <s><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; <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 type="main"> <s>Iam ferè tumultuatim, &longs;i quæ &longs;unt reliqua, Theoremata congeremus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 160.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 161.<emph.end type="center"/></s></p><p type="main"> <s>Idem impetus pote&longs;t eumdem alium aliquando plùs, aliquando minùs <lb/>intendere. </s> <s>v. </s> <s>g. </s> <s>4. gradus impetus additi aliis 4. per eamdem lineam <lb/>iidem ei&longs;dem, minùs intendunt, vt iam &longs;uprà &longs;atis fusè dictum e&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 162.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 163.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 164.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ideo producitur hic impetus numero potiùs, quàm alius omninò &longs;imilis<emph.end type="italics"/>; quia <lb/>potentia motrix e&longs;t determinata ad tale indiuiduum &longs;iue à &longs;e, &longs;iue ab <lb/>alio; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis breuiter aliqua, quæ fortè in no&longs;tris Theorematis fuere <lb/>omi&longs;&longs;a. </s></p><p type="main"> <s>Primò qualitates, quæ à cau&longs;a primò productiua con&longs;eruantur, ab ea <lb/>intendi non po&longs;&longs;e; quia &longs;ingulis in&longs;tantibus nouum effectum non pro­<lb/>ducit; 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 type="main"> <s>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 type="main"> <s>Tertiò aliqua carere contrario, non tamen con&longs;eruari à cau&longs;a primò <lb/>productiua. </s> <s>v.g. </s> <s>anima bruti, quæ de&longs;truitur ad exigentiam totius natu­<lb/>ræ, nç &longs;it fru&longs;trà. </s></p><pb pagenum="73"/><p type="main"> <s>Quartò, impetum inten&longs;iorem in projectis diutiùs durare; quia cum <lb/>&longs;en&longs;im de&longs;truatur; certè plures partes maiori tempore de&longs;truuntur, quàm <lb/>pauciores. </s></p><p type="main"> <s>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 type="main"> <s>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 type="main"> <s>Septimò, denique totam rem i&longs;tam, quæ pertinet ad impetum paulò <lb/>fu&longs;ius pertractatam in hoc primo libro; quòd &longs;cilicet ab ea reliqua ferè <lb/>omnia pendeant, quæ in hoc tractatu habentur; &longs;ed de his &longs;atis. <lb/><figure id="fig12"></figure></s></p><pb pagenum="74"/><figure></figure><p type="main"> <s><emph type="center"/>LIBER SECVNDVS, <lb/><emph type="italics"/>DE MOTV NATVRALI.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>MOtus localis naturalis latè &longs;umptus e&longs;t, <lb/>qui ab aliqua causâ naturali ponitur; <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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Definitio<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s>Definit aliter hunc motum Galileus; dicit enim eum e&longs;&longs;e, qui æquali­<lb/>bus temporibus æqualia acquirit velocitatis momenta; &longs;ed profectò non <lb/>conuenit hæc definitio omni motui naturaliter accelerato, v. </s> <s>g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Corpus graue cadit deor&longs;um, & cadens ex maiori altitudine maioremictum <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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Omne tempus &longs;en&longs;ibile non e&longs;t; 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>quot in apice acus puncta? </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus additus alteri, & determinatus ad eamdem lineam, facit maiorem <lb/>& inten&longs;iorom impetum<emph.end type="italics"/>; patet, & vici&longs;&longs;im, & detractus alteri minorem <lb/>facit, & vici&longs;&longs;im. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>1. & quâ pro­<lb/>portione illa decre&longs;cit, hic decre&longs;cit, & vici&longs;&longs;im. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Eadem cau&longs;a nece&longs;&longs;aria non impedita &longs;ubjecto ap<gap/> applicata æqualibus <lb/>temporibus æquælem effectum producit, & contrà.<emph.end type="italics"/></s><s> Probatur per Ax.12.l. </s> <s>1. & <lb/>vici&longs;&longs;im æqualis effectus &longs;upponit æqualem cau&longs;am. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s><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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <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. </s></p><p type="main"> <s>Hoc Axioma certi&longs;&longs;imum e&longs;t; 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; quippe vires vt 12. tam facilè mouebunt 12. libras, <lb/>quàm vires vt 4. 4.libras; &longs;ed faciliùs, quàm vires vt 20. 30.libras, & dif­<lb/>ficiliùs quàm vires vt 4. 3. libras; quid clarius? </s> <s>Igitur illa cau&longs;a faciliùs <pb pagenum="76"/>&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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Si quando appellandum erit aliquod Axioma vel Theorema lib. 1.ci­<lb/>tabitur Liber. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Datur motus localis naturalis, i&longs;que ab intrin&longs;eco.<emph.end type="italics"/></s><s> Probatur; corpus gra­<lb/>ue mouetur localiter deor&longs;um per hypoth. </s> <s>hic motus e&longs;t ab intrin&longs;eco, <lb/>quod probatur; 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><lb/>Primò non e&longs;t à cau&longs;a prima, vt aliquis fortè minùs prudenter, & magis <lb/>piè, quàm par &longs;it, diceret; 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; <lb/>&longs;ed hic effectus pote&longs;t habere cau&longs;am &longs;ecundam applicatam, quam a&longs;&longs;i­<lb/>gnabimus infrà; deinde cau&longs;a prima agit tantùm naturaliter iuxta exi­<lb/>gentiam cau&longs;arum &longs;ecundarum; igitur ideo moueret corpus graue deor­<lb/>&longs;um; quia tunc motum corpus graue exigeret; &longs;ed lioc milri &longs;u&longs;ficit, vt <lb/>dicatur hic motus e&longs;&longs;e ab intrin&longs;eco; 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; 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; 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;oruationem; &longs;ic animal exigit vide­<lb/>re, audire, &longs;entire, moueri; igitur habet virtutem intrin&longs;ecam, per quam <lb/>videat, audiat, & moueatur; &longs;ic ignis exigit calefacere, lucere; aër, vel aqua <lb/>frigefacere, quidquid tandem &longs;int i&longs;tæ qualitates, de quibus alibi; &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;cilicer corpus leuius &longs;ub­<lb/>e&longs;t, quàm &longs;it animali progredi, vrere igni, lucere, &c. </s></p><p type="main"> <s>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; 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; & nemo e&longs;&longs;et, qui non <lb/>exi&longs;timaret lucom hanc & calorem ltunc e&longs;&longs;e ab igne; 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"/><p type="main"> <s>Secundò hic motus non e&longs;t ab aëre ambiente; probatur, ruderet aër <lb/>deor&longs;um corpus graue, quia leuior e&longs;t, id e&longs;t ne &longs;uprà &longs;e corpus grauins <lb/>haberet; &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; e&longs;t enim par omni­<lb/>nò ratio: Præterea &longs;i aër trudit deor&longs;inn corpus graue, quia ip&longs;i loco <lb/>cedit; certè ip&longs;e aër mouetur, igitur ab intrin&longs;eco; &longs;i enim vna pars aë­<lb/>ris pellit aliam, & hæc aliam, tandem ad aliquam peruenitur, quæ &longs;e ip­<lb/>&longs;am mouet; igitur motus illius e&longs;t ab intrin&longs;eco; igitur motus natura­<lb/>lis; deinde non modò lapis de&longs;cendit per aëra, &longs;ed per mediam aquam; <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; nam corpus den&longs;um maiore vi pellit, <lb/>quàm rarum, vt con&longs;tat exprientiâ; cum tamen corpus graue per me­<lb/>dium den&longs;ius difficiliùs de cendat; igitur medium ip&longs;um re&longs;i&longs;tit motui, <lb/>quis hoc neget? </s> <s>igitur non e&longs;t cau&longs;a motus, quem impedit. </s></p><p type="main"> <s>Denique &longs;i corpus graue non tendit, fertur que deor&longs;um &longs;uá &longs;ponte, <lb/>&longs;ed ab aëre extru&longs;um; igitur dum vix &longs;u&longs;tineo manu; o. </s> <s>libras ferri, &longs;eu <lb/>plumbi; hæc vis illata manui, quam probè &longs;entio, e&longs;t ab aëre impel­<lb/>lente plumbum, quod e&longs;t ridiculum, cnm 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;unr tendere. </s></p><p type="main"> <s>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; Primò per qualitatem <lb/>quamdam diffu&longs;am, quod dici non pote&longs;t; quia capillus traheretur faci­<lb/>liùs, quàm ingens &longs;axum, quàm ma&longs;&longs;a, &longs;eu lamina; & faciliùs eadem po­<lb/>tentia motrix minus pondus moueret quàm maius, cæteris paribus; præ­<lb/>terea manum meam æqualiter traheret, &longs;iue &longs;it cum aliquo pondere con­<lb/>iuncta, &longs;iuc &longs;it nuda &longs;ine pondere; deinde illa virtus tractrix ita diffun­<lb/>ditur, vt in maiori di&longs;tantia &longs;it infirmior, fortior in minori; alioqui <lb/>diffunderetur in infinitum, quod dici non pote&longs;t; igitur &longs;i idem lapis <lb/>demittatur ex maiore altitudine, tum ex minore; haud dubiè morus ille <lb/>primus initio e&longs;&longs;et tardior i&longs;to contra experientiam; deinde in &longs;pecu al­<lb/>ti&longs;&longs;ima &longs;ubterranea trahi po&longs;&longs;et corpus virdequaque, &longs;icut in magnete; <lb/>quæ omnia intelligi non po&longs;&longs;unt; denique virtutes illas &longs;eu qualitates <lb/>tractrices refellemus &longs;uo loco. </s></p><p type="main"> <s>Secundò, aliqui dicunt hoc totum fieri per vim quamdam &longs;ympathi­<lb/>cam, quod etiam fal&longs;i&longs;&longs;imum e&longs;t; tùm quia hæc &longs;ympathia explicari <lb/>non pote&longs;t; tùm quia vel ter<gap/>ip&longs;a producit aliquid in corpore graui, <lb/>quod in aëre libratur; vel corpus in &longs;e ip&longs;o; &longs;i primum; refellitur ii&longs;­<lb/>dem omninò rationibus, quibus ip&longs;am vim terræ tractricem &longs;uptà expu­<lb/>gnauimus; &longs;i verò &longs;ccundum, hoc ip&longs;um e&longs;t, quod &longs;uprà diximus. </s></p><p type="main"> <s>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; &longs;ed filamen­<lb/>tis quibu&longs;dam, &longs;eu ductili terræ profluuio, quod illius capillitium vo­<lb/>cant; idque tantùm fieri probant ducta ab electro analogiâ, quod pa­<lb/><gap/>am & minutiora corpu&longs;cula hac câdem arte trahit; &longs;ed profectò gra-<pb pagenum="78"/>uiores &longs;unt di&longs;&longs;icultates, quam vt illis fieri &longs;atis queat; 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 type="main"> <s>Secundò, corpus vicinius etiam faciliùs abriperetur. </s></p><p type="main"> <s>Tertiò, numquid flante vento, vel imbre cadente di&longs;&longs;ipantur hæc &longs;i­<lb/>lamenta? </s> <s>quod etiam videmus in electro. </s></p><p type="main"> <s>Quartò, manum meam æquè facilè traheret terra his funiculis &longs;eu <lb/>pondere grauatam, &longs;eu vacuam. </s></p><p type="main"> <s>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 type="main"> <s>Sextò, hæc filamenta, quæ deinde reducuntur, debent habere cau­<lb/>&longs;am huius reductionis non extrin&longs;ecam; igitur intrin&longs;ecam; igitur datur <lb/>motus naturalis. </s></p><p type="main"> <s>Soptimò, hæc filamenta per mediam flammam non traherent, quod <lb/>etiam fieri videmus in electro. </s></p><p type="main"> <s>Quartò, motus naturalis non e&longs;t à virtute quadam pellente, quam <lb/>cælo quidam affingunt; nam vel ab omni parte cæli deor&longs;um trudere­<lb/>tur, vel ab vnâ; &longs;i ab vna; igitur in omni cæli plaga corpus non fertur <lb/>deor&longs;um; &longs;i ab omni, ergo cum pellatur corpus per plures lineas etiam <lb/>oppo&longs;itas moueri non pote&longs;t: 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 type="main"> <s>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; quòd &longs;cilicet propter &longs;uam tenuitatem ab illis fi­<lb/>lamentis non ita intercipi vel implicari po&longs;&longs;int; &longs;ed qua&longs;i pi&longs;ces per fo­<lb/>ramina retium euadant; &longs;ed profectò longè alia ratio e&longs;t, qaàm &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 type="main"> <s>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>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 c<gap/>claui, quàm &longs;ub dio, <lb/>vel aperto cælo. </s> <s>Secundò, in &longs;ubterraneis locis etiam grauia æquè veloci­<lb/>ter de&longs;cendunt; licèr eò lumen non penetret; quod &longs;i aliquis ob&longs;tinatè, <lb/>id a&longs;&longs;ereret; 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>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>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>Quintò maxima huius materiæ copia collecta &longs;eu &longs;peculi opera <pb pagenum="79"/>&longs;eu vitri, maiore vi corpora deor&longs;um truderet; 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>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>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 type="main"> <s>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 type="main"> <s>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;rs 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/>fali&longs;&longs;imum; <emph type="italics"/>Quidquid mouetur; ab alio mouetur<emph.end type="italics"/>; &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; ad quem etiam, vt ip&longs;i exi&longs;timant, forma &longs;ub&longs;tantialis <lb/>concurrit; In hoc quippe conueniunt omnes tùm &longs;ectarum Principes, <lb/>tùm recentiores: 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>25. vbi ait grauibus & leuibus ine&longs;&longs;e principium <lb/>actiuum &longs;uorum motuum; 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 type="main"> <s>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; quia fortè iam <lb/>interiit, nec ab vllo alio extrin&longs;eco, vt con&longs;tat inductione; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus naturalis est aliquid distinctum realiter à mobili:<emph.end type="italics"/> Probatur; <lb/>mobile ip&longs;um aliquando quie&longs;cit per hypoth.4.lib.1. igitur e&longs;t &longs;ine mo­<lb/>tu; igitur &longs;eparatum à motu; 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; nam quotie&longs;cunque duo prædicata, vel <pb pagenum="80"/>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; alioqui &longs;i vtrumque idem e&longs;&longs;e cum vno <lb/>tertio vere dicitur; <emph type="italics"/>mouetur, non monetur,<emph.end type="italics"/> quæ &longs;unt prædicata contradi­<lb/>ctoria; igitur vel moueri, vel non moueri dicit di&longs;tinctum realiter à mo­<lb/>bili; Secundum e&longs;t mera negatio; nam eo ip&longs;o, quod mobile e&longs;t &longs;ine vllo <lb/>addito, non mouetur; igitur &longs;uprà ip&longs;um mobile dicit puram putam ne­<lb/>gationem motus; igitur moueri, dicit aliquid di&longs;tinctum. </s></p><p type="main"> <s>Præterea quotie&longs;cunque prædicatum aliquod tribuitur in propo&longs;i­<lb/>tione affirmatiua falsâ; certè prædicatum illud non ine&longs;t &longs;ubiecto; alio­<lb/>quin e&longs;&longs;et vera, vt patet; igitur di&longs;tinguitur à &longs;ubiecto realiter; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><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>1. maior e&longs;t effectus, igitur maior cau&longs;a, id e&longs;t <lb/>motus; igitur cau&longs;a motus per Ax.2. &longs;ed e&longs;t eadem entitas mobilis, vt <lb/>patet; 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>3. & fune­<lb/>pendulum &longs;uam vibrationem per hypoth. </s> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus naturalis non e&longs;t immediatè ab ip&longs;a grauitate.<emph.end type="italics"/></s><s> 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>Fuit aliquis non infunæ 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc motus naturalis e&longs;t ab impetu.<emph.end type="italics"/></s><s> 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 grauitace 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 pet definitionem <lb/>3. lib. 1. </s></p><pb pagenum="81"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ille impetus ab aliqua cau&longs;a producitur.<emph.end type="italics"/></s><s> Probatur, quia quidquid de no­<lb/>uo e&longs;t, habet cau&longs;am per Ax.8. lib. 1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s></p><p type="main"> <s>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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc produci tantùm pote&longs;t ab ip&longs;a &longs;ubstantia corporis grauis; nam graui­<lb/>tas e&longs;t ip&longs;e impetus innatus, de qua infrà:<emph.end type="italics"/> probatur; quia nihil e&longs;t aliud in­<lb/>trin&longs;ecum, à quo produci po&longs;&longs;it; quòd autem non produc tur ab alio im­<lb/>petu ad intra, patet per Th.41. lib. 1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus productus primo instanti durat proximè &longs;equenti.<emph.end type="italics"/></s><s> Probatur pri­<lb/>mò; quia &longs;emper habet &longs;uum effectum formalem; vel grauitationis, &longs;i <lb/>impeditur; vel motus in medio libero; igitur non e&longs;t fru&longs;trà; igitur <lb/>non de&longs;truitur per Th.162.lib.1. nihil enim exigit de&longs;tructionem; 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. </s> <s>glans plumbea ab ingenti rupe: 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. </s> <s>Th. </s> <s><lb/>152.num.6.lib.1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus ille innatus, qui durat &longs;ecundo instanti, con&longs;eruatur ab aiiqua cau­<lb/>&longs;a<emph.end type="italics"/>; e&longs;t certum per Ax. 14.lib.1.num.1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Non con&longs;eruatur à cau&longs;a primò productiua.<emph.end type="italics"/></s><s> 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; nam <lb/>cau&longs;a con&longs;eruans verè influit; igitur &longs;i e&longs;t can&longs;a nece&longs;&longs;aria primo, & &longs;e­<lb/>cundo in&longs;tanti æquali ni&longs;u influit; influit enim quantum pote&longs;t per Ax. </s> <s><lb/>12.lib.1.quòd autem impetus intendatur, demon&longs;trabimus infrà; con&longs;ule <lb/>Schol.Th.146.lib.1.in quo habes rationem præclari natura in&longs;tituti; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s></p><p type="main"> <s><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>&c.<emph.end type="italics"/></s><s> Pro­<lb/>batur primò; 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; 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>Probatur &longs;ecundò; cre&longs;cit motus grauium in <lb/>libero medio per hypoth. </s> <s>1.2.3. igitur cre&longs;cit impetus; quia cum motus <lb/>naturalis &longs;it ab impetu per Th.5. quâ proportionc cre&longs;cit effectus, &longs;cilicet <lb/>&longs;ormalis, & exigentiæ; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Hinc reiicies quorumdam placitum, qui volunt cau&longs;am velociotis <lb/>motus e&longs;&longs;e grauitatem ip&longs;am, quatenus connotat motum præuium; quia <lb/>&longs;cilicet grauitas non producit illum maiorem impetum ad extra, vt con­<lb/>&longs;tat; nec &longs;ub&longs;tantia ip&longs;ius corporis grauis per Th.40.lib.1.igitur ip&longs;e im­<lb/>petus: 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. </s> <s><lb/>40.lib.1. vbi hæc fusè di&longs;cu&longs;&longs;imus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quia &longs;cilicet nec con­<lb/><gap/>antur à cau&longs;a primo productiua per Th.144.libri: nec aliquid exigit <lb/>de&longs;tructionem; non contrarius impetus, quia nullus e&longs;t applicatus, vt <lb/>con&longs;tat; non re&longs;i&longs;tentia medij, quæ quidem alicuius momenti e&longs;t; &longs;ed <lb/>non tanti, vt impedire po&longs;&longs;it motum omninò, vt con&longs;tat; nam &longs;uppono <lb/>liberum medium, igitur nec de&longs;truere impetum; cum tamdiu duret cau­<lb/>&longs;a quamdiu durat effectus, vt patet; igitur nihil e&longs;t quod exigat impe­<lb/>ms huius de&longs;tructionem; igitur non de&longs;truitur per Ax. 14. lib.1. <lb/><expan abbr="qūanta">qunanta</expan> verò &longs;it, & quid &longs;it cuiu&longs;libet medij re&longs;i&longs;tentia, dicemus <lb/>infrà. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s></p><p type="main"> <s><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> Probatur primò, non cre&longs;cit <lb/>corporis grauis &longs;eu grauitas, &longs;cu grauitatio, vt con&longs;tat experientiâ; igitur <lb/>non cre&longs;cit impectus; 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 cuidens; iam demon&longs;tratur <lb/>propter quid &longs;it; impetus &longs;ecundo in&longs;tanti productus e&longs;&longs;et fru&longs;trà; careret <pb pagenum="83"/>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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;erua quæ&longs;o, quod iam &longs;uprà indicattun e&longs;t, e&longs;&longs;e tres veluti &longs;pecies <lb/>impetus. </s> <s>Prima e&longs;t impetus naturalis innati. </s> <s>Secunda naturalis acqui&longs;iti. </s> <s><lb/>Tertia violenti; innatus e&longs;t qui vel à generante &longs;imul cum corpore <lb/>graui productus e&longs;t; qui&longs;quis tandem &longs;it generans, de quo aliàs; 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; 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; certè de­<lb/>bet e&longs;&longs;e in co aliquid per quod tùm cogno&longs;ci po&longs;&longs;it eins pondus, tùm in­<lb/>cipiat moueri deor&longs;um; quippe maximè corpora ex pondere cogno&longs;ci­<lb/>mus, vnumque ab alio di&longs;tinguimus; igitur debet e&longs;&longs;e aliquid, quod &longs;en­<lb/>&longs;um afficiat, vt cogno&longs;ci po&longs;&longs;it; atqui illud ip&longs;um non e&longs;t &longs;ub&longs;tantia cor­<lb/>poris; nam corpus graue meæ manui &longs;u&longs;tinenti impetum imptimit; <lb/>immò vim alterius impetus infringit; 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; 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: 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; &longs;ecundum verò &longs;i caret pri­<lb/>mo; 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 type="main"> <s>Ex his verò concludendum &longs;upere&longs;t impetum innatum e&longs;&longs;e proprie­<lb/>tatem quarto modo, vt vulgò aiunt, corporis grauis; ac proinde ab illo <lb/>in&longs;eparabilem; quid verò fiat de illo, cum corpus graue fit leuc; &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 corport <lb/>graui nunquam &longs;eparari, quandiu remanet graue. </s></p><p type="main"> <s>Impetus naturalis acqui&longs;itus producitur ab codem principio intrin­<lb/>&longs;eco; hinc dicitur naturalis: dicitur verò acqui&longs;itus, quia non e&longs;t inna­<lb/>tus; &longs;ed &longs;eparatur à corpore graui; quod &longs;emper co caret, quandiu <lb/>quie&longs;cit: &longs;ed innato tantùm accedit ad motus accelerationem, & ad alia <lb/>quamplurima, quæ ex ea &longs;equuntur; putà maiorem percu&longs;&longs;ionem, re&longs;i­<lb/>&longs;tentiam, vim, & ad tollendum totius naturæ languiorem; quo certè af­<lb/>ficeretur, &longs;i corpus grauc tardi&longs;&longs;imo motu deor&longs;um ferretur, de quo in­<lb/>frà; Porrò impetus acqui&longs;itus in multis differt ab innato; primò quia <pb pagenum="84"/>de&longs;truitur à corpore re&longs;i&longs;tente co modo, quo diximus, & dicemus infrà. </s> <s><lb/>Secundò, quia determinari pote&longs;t ad omnem lincam. </s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s></p><p type="main"> <s><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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc motus naturalis deor&longs;um acceleratur<emph.end type="italics"/>; 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; &longs;ie cnim <lb/>hypothe&longs;is in Theorema conuerti pote&longs;t, vt fæpè monuimus in metho­<lb/>do; igitur probatur hoc Theorema facilè; cre&longs;cit impetus in corpore gra­<lb/>ui, quod tendit deor&longs;um in libero medio per T. 15. igitar cre&longs;cit cau&longs;a <lb/>motus; nam impetus e&longs;t can&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: nec e&longs;t quod aliquis exi&longs;timet hic à me committi <lb/>vitio&longs;um argumentationis circulum; quippe probaui &longs;uprà cre&longs;cere im­<lb/>petum, quia cre&longs;cit motus; iam verò probo cre&longs;cere motum, quia cre&longs;­<lb/>cit impetus; nam primò probaui produci nouum impetum in Th.12. co <lb/>quod &longs;ecundo in&longs;tanti. </s> <s>v.g. </s> <s>&longs;it cadem 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; &longs;ecundò verò probaui ex hypothe&longs;i certa; 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; igitur nullus committitur circulus; nam <lb/>diuer&longs;a e&longs;t omninò cognitio. </s> <s>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>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>v. </s> <s>g. </s> <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>at verò demon&longs;trabo propter quid hæc &longs;int, &longs;ed de his <lb/>&longs;atis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis etiam aliud naturæ in&longs;titutum, quo &longs;cilicet factum e&longs;t, vt <pb pagenum="85"/>corpora grauia motu naturali accelerato deor&longs;um ferantur; &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; 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. </s> <s>quem acqui&longs;iuit in fpatio 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Æqualibus temporibus æqualis impetus producitur, &longs;i &longs;it eadem applica­<lb/>tio, idemque impedimentum<emph.end type="italics"/>; probatur, quia cau&longs;a huius impetus e&longs;t ne­<lb/>ce&longs;&longs;aria; &longs;ed cadem cau&longs;a nece&longs;&longs;aria æqualibus temporibus æqualem <lb/>impetum producit per Ax.3. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s></p><p type="main"> <s><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, ctiam cre&longs;cit effectus per Ax.2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc æqualibus temporibus in de&longs;cen&longs;u corpus graue acquirit aqualia ve­<lb/>locitatis, vel acce&longs;erationis momenta<emph.end type="italics"/>; hoc ip&longs;um e&longs;t quod definitionis lo­<lb/>co Galileus in dialogo tertio de motu naturali a&longs;&longs;umit; 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. </s> <s><lb/>18. &longs;ed temporibus æqualibus acquiruntur æquales impetus gradus per <lb/>Th.17. igitur æqualia velocitatis momenta, vel incrementa. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Spatia que per curruntur motu aquabili æqualibus temporibus &longs;unt æqualia<emph.end type="italics"/>; <lb/>Probatur per Def.2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Duo motus æquabiles, qui durant æqualibus temporibus, &longs;unt vt &longs;patia<emph.end type="italics"/>; <lb/>patet; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>1. Igitur eò veiocior, quò minori tem­<lb/>pore. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s></p><p type="main"> <s><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"/><emph type="italics"/>est maius alio<emph.end type="italics"/>; patet, quia æqualia &longs;unt æqualibus temporibus per Th. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s></p><p type="main"> <s><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>2. l. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Tempus, quo conficitur &longs;patium æquale &longs;ed uelociore motu, est minus eo <lb/>quo conficitur tardiore<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s></p><p type="main"> <s><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> Probatur per Th. </s> <s><lb/>24. & 23. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si idem mobile temporibus æqualibus pereurrat duo &longs;patia motu æquabili, <lb/>&longs;ed inæquali velocitate; &longs;patia erunt vt velocitates, & hæ vt illa; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si percurrantnr 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si duo mobilia mouentur motu æquabili, &longs;ed inæquali velocitate, & <gap/><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 crunt 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 crit 6/2 <lb/>&longs;eu 3. vt con&longs;tat ex ip&longs;is elementis. </s></p><pb pagenum="87"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si duo mobilia ferantur motu æquabili per diuer&longs;a &longs;patia, & diuer&longs;a velo­<lb/>eitate, tempora erunt in ratione compo&longs;ita ex ratione &longs;paliorum & ratioue <lb/>velocitatum permutata<emph.end type="italics"/>; probatur eodem modo quo &longs;uperius Th. 30. &longs;it <lb/>ratio &longs;patiorum 4/1, velocitatum 4/2; permutetur hæc 1/4; componetur ex <lb/>vtraque 4/1, ide&longs;t 1/2, quæ e&longs;t ratio temporum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si duo mobilia æquabili motu ferantur per diuer&longs;a &longs;patia, & inæqualibus <lb/>temporibus; 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 crit 2/2, ide&longs;t 4. <lb/>quæ e&longs;t ratio velocitatum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s></p><p type="main"> <s><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 cadem cau&longs;a, igitur æqualem effectum producit per <lb/>Ax.12. l.1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s></p><p type="main"> <s><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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc &longs;ingalis in&longs;tantibus æqualiter cre&longs;cit & intenditur impetus<emph.end type="italics"/> per Th. </s> <s><lb/>34. igitur æqualiter etiam &longs;ingulis in&longs;tantibus cre&longs;cit velocitas motus <lb/>per Ax.2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; 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; nam dato quocun­<lb/>que tempore finito pote&longs;t dari maius & minus, quod certum e&longs;t; igitur <lb/>totum illud tempus, quo producitur primus impetus acqui&longs;itus, vo-<pb pagenum="88"/>co in&longs;tans primum motus; cui æqualia deinde &longs;uccedunt tem­<lb/>pora. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc cre&longs;cit impetus iuxta progre&longs;&longs;ionem arithmeticam; cum &longs;ingula in­<lb/>&longs;tantia æqualem impetum addant<emph.end type="italics"/>; &longs;i primo in&longs;tanti &longs;it vnus gradus, crunt <lb/>duo; productus &longs;cilicet alteri additus qui con&longs;eruatur, tertio erunt;. </s> <s><lb/>quarto 4. quinto 5. &c. </s> <s>igitur cre&longs;cit &longs;ecundum progre&longs;&longs;ionem arith­<lb/>meticam. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>2. l. </s> <s>1. idem dico de tertio, quar­<lb/>to, &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s></p><p type="main"> <s><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> 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc &longs;patium qucd acquiritur &longs;ecundo in&longs;tanti e&longs;t duplum illius, qaod ac­<lb/>quiritur primo.<emph.end type="italics"/></s><s> Probatur, quia velocitas e&longs;t dupla per Th 38. igitur &longs;pa­<lb/>tium duplum, & triplum tertio, quadruplum quarto, &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc quodlibet &longs;patium cre&longs;cit æqualiter &longs;ingulis in&longs;tantibus æqualibus<emph.end type="italics"/>; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc &longs;patia cre&longs;cunt &longs;ingulis in&longs;tantibus æqualibus &longs;ecundùm pregre&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 crit <lb/>2. tertio 3. quarto 4. &c. </s> <s>hinc &longs;patia acqui&longs;ita &longs;ingulis in&longs;tantibus &longs;unt <lb/>vt &longs;eries numerorm, qui componunt progre&longs;&longs;ionem &longs;implicem, &longs;cilicet <lb/>1.2.3.4.5.6. &c. </s> <s>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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quod certè verum e&longs;t, dum intelligatur legi­<lb/>timus horum verborum &longs;en&longs;us; quidquid reclamet Saluiatus apud <lb/>Galil. </s> <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><lb/>quippe &longs;i comparetur velocitas vnius in&longs;tantis cum velocitate alterius; <lb/>haud dubiè erit eadem vtriu&longs;que ratio, quæ &longs;patiorum; &longs;i enim vno in­<lb/>&longs;tanti percurritur vnum &longs;patium cum vno velocitatis gradu; 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; quia <lb/>motus qui fit in in&longs;tanti debet con&longs;iderari vt æquabilis; 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; nec demum ob&longs;tat quod dicit, dari <lb/>non po&longs;&longs;e motum in&longs;tantaneum, quod multi haud dubiè negabunt; 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; quia quacunque duratione <lb/>data pote&longs;t dari minor; igitur quocunque dato motu pote&longs;t dari minor. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; &longs;i enim accipiantur partes temporis maiores, quæ <lb/>conflentur ex multis in&longs;tantibus; 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; in prima ac­<lb/>quirentur 3.&longs;patia, in &longs;ecunda 7.vt patet: &longs;ed quia velocitas primæ par­<lb/>tis temporis non e&longs;t æquabilis, nec etiam velocitas &longs;ecundæ; addantur <lb/>velocitates primi & &longs;ecundi in&longs;tantis, itemque &longs;eor&longs;im velocitates tertij, <lb/>& quarti; certè ratio collectorum crit 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 type="main"> <s>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 type="main"> <s>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; nec etiam &longs;patium quod ac­<lb/>quiritur primo in&longs;tanti; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Collectio &longs;patiorum e&longs;t &longs;umma terminorum huius progre&longs;&longs;ionis arithmeticæ<emph.end type="italics"/>; <lb/>Gùm enim ratio &longs;patiorum &longs;it vt ratio velocitatum; 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>certè collectio &longs;patiorum e&longs;t &longs;umma ter­<lb/>minorum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quippe hæc omnia &longs;unt in eadem ratio­<lb/>ne; quæ omnia con&longs;tant ex regulis arithmeticis præter alia multa data, <lb/>quæ lubens omitto; 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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quippe habebis in duplo quotientis numerum termino­<lb/>rum v.g. </s> <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; &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 Arithinetici; 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. </s> <s>&longs;it &longs;umma 21. duplicetur, erit 42. rad. </s> <s>9. 6. dat numerum <lb/>terminorum; &longs;it &longs;umma 36. duplicetur, erit 72.rad.9.8. dabit numerum <lb/>terminorum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s></p><p type="main"> <s><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"/><emph type="italics"/>tertij ad &longs;ecundum quàm quarti ad tertium, atque ita deinceps<emph.end type="italics"/>; &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; &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 percurtitur 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s></p><p type="main"> <s><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; &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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si componatur æquabilis motus ex &longs;ubdupla velocitate maxima, & mini­<lb/>ma, æquali tempore, idem &longs;patium percurretur hoc motu naturaliter aceclera­<lb/>to<emph.end type="italics"/>; &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; 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; illudque ducere in nu­<lb/>merum terminorum per regulam arithmeticam; atqui eadom 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s></p><p type="main"> <s><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 arithinetica, &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>vt con&longs;tat ex dictis; 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; <lb/>primæ parti tria ex prædictis &longs;patiis re&longs;pondeant; 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; primæ parti re&longs;pon-<pb pagenum="92"/>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;empet <lb/>æquale, &longs;iue a&longs;&longs;umantur partes maiores temporis, &longs;iuc minores, v. </s> <s>g. </s> <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: 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; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s></p><p type="main"> <s><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;uppofita &longs;emper eadem proare&longs;&longs;ione arithmetica in ip&longs;o incremento<emph.end type="italics"/>; <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; in &longs;ecunda <lb/>percurretur IO, id e&longs;t, duplum HI; at verò diuidatur pars temporis <lb/>AG in duas æquales AF, FG, & con&longs;equenter totum tempus AH in 4. <lb/>æquales; haud dubiè in prima AF percurretur NP &longs;ubtripla HI, & in <lb/>&longs;ecunda FG percurretur PK dupla NP; igitur in 4. partibus temporis <lb/>AH percurretur &longs;patium decuplum PN, &longs;ed HO e&longs;t tantùm nouecupla <lb/>NP; 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 type="main"> <s>Similiter AF diuidatur bifariam in E. & tota AH in 8. æquales AE; <lb/>certè primis 4.percurretur idem &longs;patium ML æquale NK & HI; igitur <lb/>in prima AE percurretur MR. cuius ML &longs;it decupla; 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; 6. æqualium RM; &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 type="main"> <s>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; & &longs;i accipias AG partes temporis AG. GH. <lb/>habebitur HO; &longs;i verò 4.æquales AF, cre&longs;cet &longs;patium &longs;eu &longs;umma 1/9 HO; <lb/>&longs;i autem 8. æquales AE cre&longs;cet 1/5 HO; &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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s></p><p type="main"> <s><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/>(<gap/>0/25), &longs;i 12. (57/21), &longs;i 14. (77/28), atque ita deinceps. </s></p><p type="main"> <s>Ex quo ob&longs;erua mirabilem con&longs;equutionem; 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; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 50.<emph.end type="center"/></s></p><p type="main"> <s><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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s></p><p type="main"> <s><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 &longs;undamini &longs;uperædificatur ingens &longs;æpè ædificij <lb/>moles. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s></p><p type="main"> <s><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 &longs;acta <lb/>acce&longs;&longs;ione maximus ferè euadit. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc ratio, cur tanta &longs;it differentia impetus grauit ationis, & percu&longs;&longs;ionis <lb/>ab eodem mobili<emph.end type="italics"/>; quia &longs;cilicet quantumuis tempore breui&longs;&longs;imo mouea­<lb/>tur, plurimis tamen cius 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>v. </s> <s>g. </s> <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; igitur effectus etiam millecuplus; quæ omnia con&longs;tant <lb/>ex dictis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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; 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"/>plaillius, 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 type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; &longs;i enim cadunt æqualibus temporibus, percu&longs;&longs;io­<lb/>nes erunt vt corpora &longs;eu grauitates, vt patet v.g. </s> <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; &longs;i verò tempora motus &longs;unt inæ­<lb/>qualia, & grauitates æquales, percu&longs;&longs;iones erunt vt tempora; &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. </s> <s>g. </s> <s>&longs;it corpus duarum librarum, <lb/>& alterum trium librarum; primum moueatur per 5. in&longs;tantia, & &longs;ecun­<lb/>dum 2.per 5. ratio grauitatum e&longs;t 3/2; ratio temporum e&longs;t 7/5; compo&longs;ita <lb/>ex vtraque erit (21/10); & hæc e&longs;t ratio percu&longs;&longs;ionum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc pote&longs;t &longs;ciri ratio percu&longs;&longs;ionis. </s> <s>& 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"/>; cum enim <lb/>&longs;ingulis in&longs;tantibus æqualis impetus accedat, vt &longs;æpè dictum e&longs;t; certè <lb/>erit percu&longs;&longs;io ad grauitationem, vt numerus in&longs;tantium motus ad vnita­<lb/>tem, v.g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; v.g. </s> <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. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc primo in&longs;tanti grauit ationis 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/>elara ex dictis. </s></p><pb pagenum="95"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; 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 type="main"> <s>Secundò non pote&longs;t ad amu&longs;&longs;im tempus cum tempore componi ad <lb/>æqualitatem, vel aliam datam rationem; licèt enim vnum tempus &longs;en&longs;i­<lb/>bile haberet mille in&longs;tantia &longs;upra aliud; 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 type="main"> <s>Tertiò, idem dico de percu&longs;&longs;ionum ratione; 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; quippe non pote&longs;t di&longs;tingui <lb/>ab alia ni&longs;i vel ex &longs;patio; atqui di&longs;cerni non pote&longs;t, an vnum &longs;patium <lb/>&longs;uperet aliud mille punctis; vel ex &longs;ono; atqui &longs;onus pote&longs;t diuidi in in­<lb/>finitos ferè gradus &longs;en&longs;u minimè perceptibiles; 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; nec dicas aliquot in&longs;tantia parùm di&longs;cti­<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; <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; at certè neuter &longs;en&longs;u percipi pote&longs;t; ne­<lb/>que tanti e&longs;t hoc &longs;cire. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc &longs;i corpus graue de&longs;cenderet motu æquabili eoque æquali motui primi <lb/>m&longs;tantis; certè vix modicum &longs;patium post multos annos decurreret<emph.end type="italics"/>; &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; in &longs;ecundo minuto &longs;upponamus e&longs;&longs;e <lb/>mille in&longs;tantia, quamuis infinita penè contineat; &longs;itque in primo in­<lb/>&longs;tanti motus vnus gradus impetus; &longs;ic enim vocetur illa pars impetus, que <lb/>producitur primo in&longs;tanti; certè po&longs;t mille in&longs;tantia motus, erunt mille <lb/>gradus impetus; iam vcrò &longs;i accipiatur &longs;ubduplum maximæ & minimæ <lb/>velocitatis; id e&longs;t vnius gradus, & mille graduum, &longs;cilicet 500. 1/2 tri­<lb/>buaturque motui æquabili; 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; &longs;i verò accipiantur plura <lb/>in&longs;tantia, v.g. </s> <s>1000000.in&longs;tantia, percurrentur 12. pedes 500000. &longs;e­<lb/>cundis minutis; &longs;i verò 1000000000000. percurremur 500000000000. <lb/>&longs;ecundis, id e&longs;t 8333333333. minutis, id e&longs;t 138888888. horis <pb pagenum="96"/>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s></p><p type="main"> <s><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 panlò minùs<emph.end type="italics"/>; quia conficit <lb/>idem motu æquabili; cuius velocitas e&longs;t &longs;ubdupla maximæ & minimæ; <lb/>&longs;ed minima velocitas primi in&longs;tantis pro nihilo reputatur; igitur acci­<lb/>piatur tantùm &longs;ubduplum maximæ, igitur cum velocitate æquali maxi­<lb/>mæ, eodem tempore duplum &longs;patium percurretur; igitur in vno minuto <lb/>&longs;ecundo, v.g. </s> <s>24. pedes; igitur in vno minuto primo codem motu æqua­<lb/>bili 1440. pedes percurrentur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus naturaliter acceleratus non propagatur per omnes tarditatis gra­<lb/>dus<emph.end type="italics"/>; 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; prætereà licèt e&longs;&longs;ent infinita in&longs;tantia, non fieret adhuc per omnes <lb/>tarditatis gradus hæc propagatio; quia daretur aliquis gradus tarditatis, <lb/>quem non comprehenderet hæc graduum &longs;eries; nam incipit moucri <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. </s> <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 type="main"> <s>Hinc duplici nomine reiice Galilæum qui hoc a&longs;&longs;erit. </s> <s>Primò, quia <lb/>fru&longs;trà ponit infinita in&longs;tantia &longs;ine nece&longs;&longs;itate; &longs;ecundò, quia ratio, quam <lb/>habet, non conuincit; vocat enim quietem tarditatem infinitam; 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>Secundò, quia tàm ex quiete &longs;equi po­<lb/>te&longs;t immediatè velox motus, quàm tardus, vt patet in proiectis. </s> <s>Tertiò, <lb/>quia motus incipit; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Ex dictis facilè reiicies primò &longs;ententiam illorum, qui negant mo-<pb pagenum="97"/><gap/>m 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Secundò reiicies illos, qui volunt accelerationem motus e&longs;&longs;e, vel à vi <lb/>magnetica, quâ terra trahit ad &longs;e omnia grauia; vel ab alia vi occulta, <lb/>quâ cœlum pellit deor&longs;um; vel à cœle&longs;ti illa, imò potiùs fabulosâ mate­<lb/>riâ; vel demum ab ip&longs;a vi fympathicâ, 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, nefru&longs;trà tempus <lb/>in iis iterum refellendis teramus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Tertiò reiicies, qui volunt motum accelerari ex aëris à tergo impel­<lb/>lentis appul&longs;u, quod ridiculum e&longs;t: licèt enim Ari&longs;toscles videatur illud <lb/>&longs;en&longs;i&longs;&longs;e de projectis, quod examinabimus &longs;uo loco; nunquam tamen hoc <lb/>dixit de motu naturali; 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; hinc <lb/>dictum illud vulgare, <emph type="italics"/>vire&longs;que acquirit eundo<emph.end type="italics"/>; 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<gap/> illud <lb/>autem e&longs;&longs;e impetum con&longs;tat. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s>Quartò ex his &longs;ententia Ari&longs;totelica de motu accelerato optunè vin­<lb/>dicatur; 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, cer<gap/>imi&longs;que principiis <lb/>demon&longs;tratum fuit. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s>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>& &longs;patia &longs;int vt <lb/>quadrata temporum v. </s> <s>g. </s> <s>&longs;i primo in&longs;tanti acquiritur 1.&longs;patium: &longs;ecundo <lb/>acquiruntur 3. tertio 5. quarto 7. &c. </s> <s>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 falfa e&longs;&longs;e con&longs;tat; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 7.<emph.end type="center"/></s></p><p type="main"> <s>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>quod etiam ex eadem ratione facilè confutatur: 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; &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 cius pleni&longs;&longs;imam <lb/>confirmationem pertineat. <lb/><gap desc="hr tag"/></s></p><p type="main"> <s><emph type="center"/>DISSERTATIO<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>De Motu naturaliter accelerato.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>DVæ &longs;unt poti&longs;&longs;imùm in hac materia celebres &longs;ententiæ; Prima e&longs;t <lb/>Galilei, & ferè omnium recentiorum, qui po&longs;t Galileum de motu <lb/>&longs;crip&longs;erunt; inter quos, ne omittam Genuen&longs;em Patricium, Balianum<gap/><lb/>Doctus Mer&longs;ennus, & eruditus Ga&longs;&longs;endus primum locum obtinent; <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 type="main"> <s>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 laud<gap/><lb/>appellat idem Mer&longs;ennus) tribuit. </s> <s>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 type="main"> <s>Sunt tres aliæ &longs;ententiæ, quæ ab eodem Mer&longs;enno referuntur; prima <lb/>e&longs;t quæ progre&longs;&longs;ionem &longs;patiorum eamdem 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 co punctò, in quo incipit motus. </s> <s>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 type="main"> <s>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>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"/>corpus graue &longs;uo motu percurrit; & &longs;ecundo tempore æquali BC, quæ <lb/>tripla e&longs;t AB, tertio CD quintupla quarto DE &longs;eptupla, quinto EF <lb/>nouecupla; vides primò &longs;eriem numerorum imparium 1. 3. 5. 7. 9.atque <lb/>ita deinceps. </s> <s>Secundò vides &longs;patia e&longs;&longs;e in ratione duplicata temporum, <lb/>hoc e&longs;t vt temporum quadrata. </s> <s>v.g. </s> <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. </s></p><p type="main"> <s>Hæc &longs;ententia ingeniosè à Galileo excogitata ex duplici capite à &longs;uis <lb/>auctoribus confirmatur; primò experientiâ, &longs;ecundò ratione. </s> <s>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>v. </s> <s>g. </s> <s>in linea AF; 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 type="main"> <s>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>&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 type="main"> <s>Tertium experimentum petitur ex funependulis; 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; eleuetur vter­<lb/>que funis, cui pondus æquale &longs;it appen&longs;um v&longs;que ad horizontalem <lb/>CDE & alterum ex D; 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>v. </s> <s>g. </s> <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; &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 type="main"> <s>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; item tempus, quo de&longs;cendit <pb pagenum="100"/>per arcum DM &longs;ubquadruplum DF.ad tempus, quo de&longs;cendit per arcum <lb/>EK &longs;ubquadruplum EG; 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; 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; igitur in his rectis de&longs;cendunt grauia iuxta progre&longs;&longs;ionem præ­<lb/>dictam; 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; <lb/>&longs;ed tempus, quo percurritur EG e&longs;t duplum illius, quo percurritur DF; <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 type="main"> <s>Quod autem tempus, quo percurritur EG &longs;it duplum illius, quo per­<lb/>curritur DF, patet experientiâ; nam &longs;i numerentur ducentæ vibrationes <lb/>funependuli CD; eodem tempore numerabuntur centum vibrationes <lb/>maioris CE; igitur vibrationum minoris numerus e&longs;t duplus numeri vi­<lb/>brationum maioris, dum &longs;imul vibrantur; igitur eo tempore, quo fiunt <lb/>100.maioris, fient 200. minoris; nam ommes vibrationes eiu&longs;dem fune­<lb/>penduli &longs;unt æquò diuturnæ, licèt fiant per arcus inæquales ciu&longs;dem. </s> <s><lb/>quadrantis, vt &longs;æpè ob&longs;eruatum e&longs;t. </s> <s>In his tribus poti&longs;&longs;imum experimen­<lb/>tis fundatur hæc hypothe&longs;is Galilei, quæ nec clariùs meo. </s> <s>iudicio, nec <lb/>&longs;inceriùs exponi po&longs;&longs;unt. </s></p><p type="main"> <s>Antequam rationes, quæ pro hac &longs;ententia facere videntur, propona­<lb/>mus, refellamu&longs;que; oftendo primò quomodo cum his experimentis <lb/>&longs;tare po&longs;&longs;it no&longs;tra hypothe&longs;is; igitur ex iis hypothe&longs;is Galilei rectè de­<lb/>duci non pote&longs;t: quippe hæc e&longs;t certi&longs;&longs;ima regula, quam nemo Philo&longs;o­<lb/>phus negare au&longs;it: Quotie&longs;cumque aliquod experimentum tale e&longs;t, vt <lb/>cum co &longs;tare po&longs;&longs;int contrariæ hypothe&longs;es; 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 type="main"> <s>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; quin vel aliqua puncta in &longs;patiis, vel in&longs;tantia in <lb/>temporibus de&longs;int, vel &longs;uper&longs;int; &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>cum tamen experimentum omne phy&longs;icum &longs;en&longs;ui <lb/>&longs;ube&longs;&longs;e po&longs;&longs;it; 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"/>facilè accipi pote&longs;t, cum nullum di&longs;erimen &longs;en&longs;ibile e&longs;t. </s></p><p type="main"> <s>Adde quod non de&longs;unt viri graui&longs;&longs;uni qui dicant &longs;e vix ob&longs;eruare po­<lb/>tui&longs;&longs;e hanc &longs;patiorum progre&longs;&longs;ionem; plures appellare po&longs;&longs;em; vnus <lb/>Ga&longs;&longs;endus e&longs;t in&longs;tar omnium; 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; &longs;ed ego omitto, quia etiam &longs;ua incer­<lb/>titudine laborant; 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: Equidem &longs;en&longs;u percipio practicam hanc e&longs;&longs;e maiorem pede; <lb/>at tot lineis vel <expan abbr="pũctis">punctis</expan> &longs;uperare ne Argus quidem certò, ac di&longs;tinctè cer­<lb/>neret: Sed efficaciter, meo iudicio, hanc Galilei hypothe&longs;im refello; &longs;int <lb/> 2.partes temporis æquales AE, EF, eæque &longs;en&longs;ibiles; nec enim aliæ a&longs;­<lb/>&longs;iuni po&longs;&longs;unt; &longs;intque minintæ omnium &longs;en&longs;ibilium; haud dubiè con&longs;tant <lb/>&longs;ingulæ infinitis ferè aliis in&longs;en&longs;ibilibus, vt patet; igitur &longs;ic ratiocinatur <lb/>Galileus; in prima parte temporis AE corpus graue percurrit &longs;patium <lb/>GH, & in &longs;ecunda æquali EF percurrit &longs;patium HL triplum prioris; <lb/>igitur &longs;patia &longs;unt vt quadrata temporum, rectè; &longs;ed antequam vlterius <lb/>progrediar; Quæro vel à Galileo, vel à quolibet alro, vtrum &longs;patium <lb/>HL &longs;it omnino triplum? </s> <s>& &longs;i aliquis contenderet dec&longs;&longs;e (1/1000000) GH <lb/>vtrum experimento præ&longs;enti conuinci po&longs;&longs;it? </s> <s>nemo, vt puto, id a&longs;&longs;erere <lb/>au&longs;it; 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; &longs;i prima parte temporis AE percurratur &longs;pa­<lb/>tium GH, &longs;ecunda EF. percurretur tantùm HK duplum GH; igitur <lb/>minus e&longs;t hoc &longs;patium vero &longs;patio 1/4. &longs;cilicet tota KL; 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; at po&longs;ito quod &longs;int <lb/>plures termini, vt reuerâ &longs;unt; nam in totidem terminis fit progre&longs;&longs;io, in <lb/>quibus fit augmentum impetus, vel accelorationis acce&longs;&longs;io; atqui hæc <lb/>fit in &longs;ingulis in&longs;tantibus, licèt finitis, igitur & progre&longs;&longs;ro; Quare duæ <lb/>partes temporis AE, EF diuidantur in 4. æquales AD; certè in duabus <lb/>primis percurretur &longs;patium. </s> <s>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>&longs;ecunda 2. igitur QV continet tria &longs;patia; tertia verò 3. quarta <lb/>4.ergo hæ duæ vltimæ 7. &longs;ed QM e&longs;t dupla QV; igitur continet 6. igi­<lb/>tur MK e&longs;t 1/3 VQ, vel KL; igitur KM e&longs;t (1/12) GL; 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; haud dubiè 4. primis percurretur &longs;patium XT æqualc <lb/>GH, quod debet diuidi in 10. &longs;patia; nam 4. terminis, &longs;eu temporibus <lb/>re&longs;pondent &longs;paria 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 type="main"> <s>Prærerea diuidatur tempus AF in 16. partes æquales AB; haud dubiè <pb pagenum="102"/>8 primis acquiritur &longs;patium YS æquale GH; quod debet diuidi in &longs;pa­<lb/>tiola 36, quæ re&longs;poudent 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 type="main"> <s>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 type="main"> <s>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>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>igitur etiam &longs;uppo&longs;ita progre&longs;&longs;ione arith­<lb/>metica, quæ fiat in finitis in&longs;tantibus; &longs;i ob&longs;eruetur acurati&longs;&longs;imè &longs;patium, <lb/>quod percurritur in vna parte temporis &longs;en&longs;ibili v. </s> <s>g. </s> <s>&longs;patium GH in <lb/>parte temporis AE; &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 co tan­<lb/>tùm euinci pote&longs;t. </s></p><p type="main"> <s>Hinc obiter ob&longs;erua progre&longs;&longs;ionem differentiarum; quippe &longs;i &longs;int <lb/>tantùm 2. partes temporis, differentia e&longs;t 1/4; &longs;i 4.1/6 &longs;i 8. (1/10); &longs;i 16.(1/18); &longs;i 32. <lb/>(1/34); &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 type="main"> <s>Eodem modo &longs;oluendum e&longs;t &longs;ecundum experimentum rotati globi in <lb/>plano decliui; præ&longs;ertim cum globus ab incur&longs;u a&longs;periorum partium <lb/>tùm globi, tùm plani &longs;altuatim de&longs;cendat; 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 type="main"> <s>Quod &longs;pectat ad tertium experimentum; 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>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; quippe vibratio maior plùs temporis; minor ve­<lb/>rò minùs in &longs;uo de&longs;cen&longs;u ponit; dimittantur enim duo funependula æ­<lb/>qualia; alterum quidem ex altitudine 90.graduum, alterum ex altitudine <pb pagenum="103"/>10. vel 15.graduum; ita vt &longs;imul vibrationes &longs;uas incipiant; numerentur <lb/>vibrationes vtriu&longs;que, vbi 100. è minoribus numeratç fuerint, numera­<lb/>buntur circiter 97. è maioribus, quod &longs;æpiùs ob&longs;eruaui te&longs;tibus etiam <lb/>adhibitis; hoc ip&longs;um etiam ob&longs;eruarunt alij; 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: rationem huius effectus in libro de <lb/>funependulis explicabimus; immò &longs;i omnes vibratìones maiores primæ <lb/>vibrationi 90. grad. </s> <s>e&longs;&longs;ent æquales, & aliæ minores alterius funependu­<lb/>li &longs;en&longs;un, vt &longs;it, minuerentur; vix 90. maiores numerare po&longs;&longs;es, iam enu­<lb/>meratis 100. ex minoribus; &longs;ed de his omnibus &longs;uo loco; 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 type="main"> <s>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. </s> <s>funependulum longitudinis 4. pedum facere vnam vibratio­<lb/>nem eo tempore, quo funependulum longitudinis vnius pedis facit duas; <lb/>quod primò in multis vibrationibus non tàm accuratè ob&longs;eruatur; <expan abbr="&longs;ecū-dò">&longs;ecun­<lb/>dò</expan> licèt ob&longs;eruaretur &longs;en&longs;ibiliter, id emre&longs;ponderi debet, quod &longs;uprà in <lb/>&longs;ingulis vibrationibus e&longs;&longs;e tantùm di&longs;crimen; uod 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 type="main"> <s>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; cùm <lb/>tamen diuer&longs;a &longs;it inclinatio minoris, & maioris quadrantis: quippe <lb/>principium maioris accedit propiùs ad perpendicularem; facit enim <lb/>angulum contingentiæ minorem; alia verò extremitas accedit propiùs <lb/>ad horizontalem propter rationem prædictam; 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 type="main"> <s>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 fai&longs;um e&longs;t; quia in eodem plano inclinato &longs;upponitur eadem <lb/>inclinatio; &longs;ecus in quadrante, cuius &longs;ingula puncta nouam faciunt in­<lb/>clinationem: 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 type="main"> <s>Præterea, &longs;it ita vt &longs;upponitur; 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. </s> <s>demi&longs;&longs;i; quæ vix e&longs;&longs;e <lb/>po&longs;&longs;unt 1800; &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 nnmerantur; annuo quidem, &longs;i res tantùm &longs;en&longs;ibiliter con&longs;ide<gap/><lb/>retur; &longs;in verò &longs;ecùs, id pernego; &longs;ed dico dee&longs;&longs;e v. </s> <s>g. </s> <s>1000000. puncta <lb/>&longs;patij, quæ di&longs;cerni non po&longs;&longs;unt, ita vt primæ vibrationi 1000. puir­<lb/>cta &longs;ecundæ, 2000. tertiæ 3000. &c. </s> <s>vltimæ verò, &longs;eu mille&longs;im<gap/><pb pagenum="104"/>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>Igitur &longs;uppo&longs;itâ progre&longs;&longs;ione &longs;pa­<lb/>tiorum arithinetica 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; a&longs;&longs;umptis &longs;cilicet partibus temporis &longs;en&longs;ibilibus, vt differentia <lb/>di&longs;cernit non po&longs;&longs;it; immò nec duplum diffetentiæ, nec centuplum, nec <lb/>millecuplum; &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 type="main"> <s>Prima ratio, quam affert Galileus e&longs;t; quia cum natura in &longs;uis opera­<lb/>tionibus adhibeat &longs;implici&longs;&longs;ima media; & 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; 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 type="main"> <s>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>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; &longs;ed nulla e&longs;t ratio, cur Galileus adhibeat <lb/>&longs;uam, vti videbimus; nos verò ratione demon&longs;tratiuâ probamus no&longs;tram; <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 type="main"> <s>Secunda ratio e&longs;t; 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. </s> <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. </s></p><p type="main"> <s>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: &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;patio<gap/>m 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; ducatur in 6.id e&longs;t in numerum termi­<lb/>norum, vel in&longs;tantium; &longs;umma erit 21. igitur quod tribuit Galilcus &longs;uæ <lb/>progre&longs;&longs;ioni, etiam no&longs;træ competit. </s></p><p type="main"> <s>Tertia ratio petitur ex mathe&longs;i &longs;it enim linea AE diui&longs;a in qautuor <lb/>partes æquales, quæ nobis repre&longs;entent 4. partes temporis æquales; <lb/>haud dubiè, cùm acquirantur temporibus æqualibus æqualia velocitatis <lb/>momenta; haud dubiè, inquam, his 4. temporibus AB, BC, CD, DE, ac-<pb pagenum="105"/>quirentur æquales velocitatis gradus; &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; 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; denique in fine quartæ DE quadruplam EF; quip­<lb/>pe cum in parte BC remaneat tota velocitas B, & acquiratur æqualis; <lb/>certè in fine BC e&longs;t velocitas CH dupla illius quæ commen&longs;uratur BI. <lb/>&longs;uniliter in parte CD remanebit vtraque, & accedet altera; 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>igitur cum &longs;patium acqui&longs;itum re&longs;­<lb/>pondeat exercitio huius velocitatis; &longs;itque in&longs;tanti B vt BI, & in&longs;tanti <lb/>C vt CH; certè tempore AB e&longs;t vt triangulum AIB; nam &longs;patium AIB <lb/>e&longs;t collectio omnium linearum, quæ duci po&longs;&longs;unt parallelæ in tempote <lb/>AB; idem dico de trapezo CBIH, qui e&longs;t triplus trianguli IBA; & de <lb/>trapezo GDCH, qui e&longs;t quintuplus; igitur triangulum HCA e&longs;t qua­<lb/>druplum IBA; quia hæc triangula &longs;unt vt quadrata laterum; 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; igitur vt quadratum <lb/>AB ad quadratum AC; 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 type="main"> <s>Hæc ratio ad &longs;peciem videtur e&longs;&longs;e demon&longs;tratiua, deficit tamen à ve­<lb/>ra demon&longs;tratione; primo, quia &longs;upponit in&longs;tantia infinita, quæ multi <lb/>pa&longs;&longs;im negabunt in tempore; immò aliquis vltrò demon&longs;trare tentaret <lb/>non e&longs;&longs;e infinita; 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; igitur <lb/>in&longs;tanti AC re&longs;pondeat velocitas, cuius men&longs;ura &longs;it ABCG; haud du­<lb/>biè in&longs;tanti CD re&longs;pondebit velocitas CH, &longs;cilicet dupla AB; nam re­<lb/>manet primus velocitatis gradus acqui&longs;itus primo in&longs;tanti: &longs;ed alter æ­<lb/>qualis acquiritur; igitur e&longs;t duplus prioris; igitur re&longs;pondet lineæ DK. <lb/>quæ tripla e&longs;t AB, & quarto lineæ FN, quæ e&longs;t quadrupla AB; igitur <lb/>cre&longs;cit &longs;patium, vt rectangula CB, DH, EK, FM; &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; quod alibi ex profe&longs;&longs;o tractamus: quippe <lb/>illa quæ&longs;tio pertinet ad Metaphy&longs;icam, non verò ad phy&longs;icun; nam vel <lb/>&longs;ingula aliquid addunt, vel nihil: aliquid addunt haud dubiè; igitur con­<lb/>&longs;iderantur tantùm 4. in&longs;tantia prima AC, CD, DE, EF, in &longs;ua &longs;crie; 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><lb/>Hinc quò rectangula CB, DH, &c. </s> <s>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"/>4.progre&longs;&longs;io longiùs di&longs;cedet à vera; vt &longs;uprà iam totius repetitum fuit: <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; nam AB, & GH &longs;unt æquales; 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>; vnde <lb/>tota velocitas in hypothe&longs;i Galilci acqui&longs;ita in 4.partibus temporis a&longs;­<lb/>&longs;umptis e&longs;t, vt triangulum AFN; 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 type="main"> <s>Si tamen diuidantur i&longs;tæ partes temporis in minores v. </s> <s>g. </s> <s>in 8. tunc <lb/>&longs;umma rectangulorum erit tantùm maior 1/8; &longs;i in 16. (1/16) &longs;i in 32. (1/32); &longs;i in <lb/>64.(11/64), cuius &longs;eliema 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. </s> <s>g. </s> <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>denique acqui&longs;itum in temporibus <lb/>inæqualibus, vt quadrata temporum v. </s> <s>g. </s> <s>acqui&longs;itum in prima parte ad <lb/>acqui&longs;itum in duabus, vt triangulum ACG ad triangulum ADI; id e&longs;t <lb/>quadratum CA ad quadratum DA; in no&longs;tra verò hypothe&longs;i, &longs;i velocitas <lb/>in tota prima parte AC ponatur vt CG æquabiliter; 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>tantùm 8. </s></p><p type="main"> <s>Iam verò diuidantur 4. partes temporis AF, in 8. æquales; 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; 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 type="main"> <s>Præterea diuidatur vlteriùs tempus AF in 16. æquales partes; 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; 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; &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. </s> <s>g. </s> <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"/><p type="main"> <s>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 <gap/>. </s> <s>&longs;i verò diuidantur in 8. <lb/>temporis partes, triangulum AFN continebit 64. triangula æqualia <lb/>AXQ: 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 type="main"> <s>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; &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; at &longs;i diuidatur in 16. partes, tempus AF, &longs;umma rectan­<lb/>gulorum e&longs;t minor præcedente 8. quadratis æqualibus Q 7., 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; &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; &longs;i diuidantur in 16. detrahitur 1/4 eiu&longs;dem diffe­<lb/>rentiæ; &longs;i diuidantur in 32.detrahitur 1/<gap/>, &longs;i in 64. (1/16); 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>in infinitum non fa­<lb/>ciunt vnum integrum; &longs;ed hæc &longs;unt facilia. </s></p><p type="main"> <s>Quarta ratio, quam afferunt aliqui, e&longs;t; quia &longs;i cum eadem velocita­<lb/>te acqui&longs;ita in fine temporis dati &longs;ine augmento nouo moueatur mobi­<lb/>le; haud dubiè acquiret duplum &longs;patium tempore æquali tempori dato; <lb/>v. </s> <s>g. </s> <s>&longs;it triangulum AFE; &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: certè &longs;i du­<lb/>catur velocitas EF in tempus AE, vel EL æquale; habebitur rectan­<lb/>gulum EK duplum trianguli AFE: &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 quabilis, quod erat demon&longs;trandum. </s> <s>Præterea &longs;i diai­<lb/>datur velocitas EF, & eius &longs;ubdupla ducatur in tempus AE; habebitur <lb/>rectangulum æquale triangulo AFE, vt con&longs;tat. </s> <s>Re&longs;pondeo facilè ex di­<lb/>ctis, hoc ip&longs;um etiam ex no&longs;tra hypothe&longs;i proxime &longs;equi; &longs;int enim duo <lb/>in&longs;tantia; haud dubie &longs;i non cre&longs;cit velocitas, &longs;ecundo in&longs;tanti æquale <lb/>&longs;patium percurretur; &longs;i vero &longs;ecundo in&longs;tanti cre&longs;cat, percurrentur illo <lb/>motu 3.&longs;patia; & 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. </s> <s>&longs;ecundo crunt 2. gradus; igi­<lb/>tur moueatur per duo in&longs;tantia motu æquabili veloci vt 2. percurrentur <lb/>4. &longs;patia; 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"/>tu naturaliter accelerato vt 4. ad 3. igitur continet illud 1. (11/3); &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 crunt in&longs;tantia <lb/>accedet propiùs ad rationem duplam, nunquam tamen ad illam perue­<lb/>niet. </s> <s>Ex dictis multa tumultuatim Corollaria congeri po&longs;&longs;unt; </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s>Etiam&longs;i non &longs;int partes infinitæ temporis; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s>Etiam &longs;i non &longs;int infiniti tarditatis gradus, vt con&longs;tat ex dictis, &longs;ed fi­<lb/>niti; in ordine tamen ad praxim eodem modo &longs;e habent, ac &longs;i e&longs;&longs;ent in­<lb/>&longs;initi; quia non pote&longs;t di&longs;tingui primus, & minimus ab omnibus <lb/>al is. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s>Licèt hypothe&longs;is Galilei &longs;it fal&longs;a in hypothe&longs;i in&longs;tantium finitorum; <lb/>nam &longs;ingulis in&longs;tantibus noua fit velocitatis acce&longs;&longs;io; phy&longs;icè tamen lo­<lb/>quendo eodem modo &longs;e habet, ac &longs;i e&longs;&longs;et vera; quia cum non po&longs;&longs;it pro­<lb/>bari, ni&longs;i in partibus temporis &longs;en&longs;ibilibus; certà, cùm quælibet pars <lb/>&longs;en&longs;ibilis innumera ferè in&longs;tantia contineat, in quibus fit progre&longs;&longs;io; <lb/>differentia vtriu&longs;que &longs;en&longs;ibilis e&longs;&longs;e non pote&longs;t; 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; &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><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; igitur hæc e&longs;to clauis huius difficul­<lb/>tatis; progre&longs;&longs;io &longs;implex principium phy&longs;icum habet, non experimen­<lb/>tum; progre&longs;&longs;io numerorum imparium experimentum non principium; <lb/>vtramque cum principio & experimento componimus; prima enim &longs;i. </s> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s>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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s>Similiter cognito &longs;patio quod percurrit 4. &longs;ecundis minutis, cogno­<lb/>&longs;ues &longs;patium, quod percurret 2. vel 1. v.g. </s> <s>percurrit 4. &longs;ecundis 192. pe-<pb pagenum="109"/>des; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s>Similiter cognito tempore cogno&longs;ci pote&longs;t &longs;patium decur&longs;um; quia <lb/>&longs;patia &longs;unt vt quadrata temporum; 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 type="main"> <s>Sed iam re&longs;tat, vt &longs;oluamus objectiones aliquas, quæ contra motus ac­<lb/>celerationem pugnare videntur. </s></p><p type="main"> <s>Prima objectio e&longs;t; &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: <lb/> &longs;it &longs;patium AB quod percurrit corpus graue primo in&longs;tanti; haud du­<lb/>biè AB, e&longs;t eius locus adæquatus; &longs;ecundo in&longs;tanti percurrit BC duplum <lb/>AB; 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 type="main"> <s>Hæc objectio impugnat omnem velocitatem; hoc e&longs;t, non modò eam, <lb/>quæ motui naturaliter accelerato competit; verùm etiam illam, quæ <lb/>ine&longs;t motui violento; igitur vt re&longs;pondeam faciliùs; &longs;uppono punctum <lb/>phy&longs;icum, mobile &longs;cilicet A; aut &longs;i mauis Angelum coëxten&longs;um quadra­<lb/>to A; qui &longs;cilicet moueatur motu accelerato, & primo in&longs;tanti acquirat <lb/>locum immediatum æqualem priori, &longs;cilicet AB; licèt enim po&longs;&longs;et ac­<lb/>quirere vibrationem participantem de priori; 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; hoc po&longs;ito, primo in&longs;tanti pun­<lb/>ctum A acquirit AB tanquam locum adæquatum, vt certum e&longs;t: certum <lb/>e&longs;t etiam loca BC, CD, e&longs;&longs;e adæquata: igitur &longs;imul', id e&longs;t eodem in­<lb/>&longs;tanti in vtroque e&longs;&longs;e non pote&longs;t; nam in&longs;tans &longs;imul totum e&longs;t; <gap/>ur <lb/>&longs;ecundo in&longs;tanti non percurrit BC, &longs;ed &longs;ecundo tempore æquali primo; <lb/>hoc enim &longs;ecundum tempus con&longs;tat duobus in&longs;tantibus, quod &longs;unul <lb/>vtrumque re&longs;pondet primo: quippe dari po&longs;&longs;unt in&longs;tantia phy&longs;ica; 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. </s></p><p type="main"> <s>Secunda objectio; 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>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; <lb/>tà tamen lege, vt vtrumque &longs;imul &longs;umptum &longs;it onminò equale in&longs;tanti, <pb pagenum="110"/>quo percurritur primum &longs;patíum AB; &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; &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 type="main"> <s>Ob&longs;eruabis po&longs;&longs;e velocitatem motus explicari duobus modis. </s> <s>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>Secundò &longs;i accipiantur &longs;patia æqualia & <lb/>tempora inæqualia, quod duobus modis fieri tantùm pote&longs;t. </s> <s>Primò &longs;i ac­<lb/>cipiantur &longs;patia æqualia primo &longs;patio, quod percurritur primo in&longs;tanti. </s> <s><lb/>Secundò &longs;i accipiantur &longs;patia æqualia alteri &longs;patio, quod in parte tempo­<lb/>ris &longs;en&longs;ibili percurritur; in qua verò proportione tempora fiant &longs;emper <lb/>minora, 'dicemus infrà; nec dicas durum e&longs;&longs;e dicere in&longs;tans e&longs;&longs;e po&longs;&longs;e <lb/>minus in&longs;tanti; nam equidem fateor in&longs;tanti mathematico nihil e&longs;&longs;e <lb/>po&longs;&longs;e minus; &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 type="main"> <s>Tertia objectio. </s> <s>Sed inquies, igitur &longs;ecundo tempore æquali primo <lb/>acquiruntur 2.gradus velocitatis, vel impetus; igitur tria &longs;patia &longs;ecun­<lb/>do tempore percurruntur, quod e&longs;t contra hypothe&longs;im; quippc duo gra­<lb/>dus impetus accedunt primo, &longs;imiliter tertio tempore producentur tres <lb/>gradus impetus; 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 type="main"> <s>Re&longs;pondço, equidem co in&longs;tanti, quo percurritur &longs;patium BD, pro­<lb/>duci aliquid impetus, & aliquid eo in&longs;tanti, quo percurritur &longs;patium <lb/>DC; 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; quia duo illa in&longs;tantia &longs;unul &longs;umpta faciunt tempus <lb/>æquale primo in&longs;tanti; 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 type="main"> <s>Quarta objectio; &longs;i impetus &longs;ingulis in&longs;tuitibus cre&longs;ceret, vel intende­<lb/>retur, augeretur grauitatio: quippe &longs;i grauitas primo in&longs;tanti producat <lb/>vnum gradum impetus; &longs;ecundo æqualem producet, & rertio, 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 type="main"> <s>Re&longs;pondeo nunquam impetum intendi, ni&longs;i &longs;it motus, qui e&longs;t illius fi­<lb/>nis; alioquin fru&longs;tra e&longs;&longs;et per plura in&longs;tantia; igitur de&longs;trui deberet; nec <lb/>dicas impetum naturalem etiam fru&longs;trà e&longs;&longs;e &longs;ine motu; quia cum mo­<lb/>tus non &longs;it eius finis adæquatus; non mirum e&longs;t &longs;i po&longs;&longs;it e&longs;&longs;e &longs;ine motu; <lb/>atqui iam diximus &longs;uprà habere duos fines, quorum alterum &longs;emper ha-<pb pagenum="111"/>bet; primus e&longs;t grauitatio, &longs;eu ni&longs;us ver&longs;us centrum; &longs;ecundus motus <lb/>deor&longs;um; cùm tamen impetus addititius motum tantùm pro fine habeat; <lb/>igitur &longs;i impeditur totus motus, non producitur hic impetus. </s></p><p type="main"> <s>Quinta objectio; &longs;i impetum &longs;uum intendit corpus graue; &longs;imiliter <lb/>Ignis diceretur intendere calorem; Sol lucem, &c. </s> <s>Re&longs;pondeo primò de <lb/>luce &longs;ingularem e&longs;&longs;e rationem; quia &longs;cilicet con&longs;eruatur à cau&longs;a &longs;ua pri­<lb/>mo productiua; quidquid &longs;it; &longs;i viderem effectum caloris, vel frigoris <lb/>perpetuò cre&longs;cere; haud dubiè dicerem etiam cau&longs;as ip&longs;as intendi; atqui <lb/>hoc ip&longs;um video in motu naturali, qui effectus impetus e&longs;t; adde quod <lb/>argumentum à pari debile e&longs;t; cum enim &longs;int diuer&longs;i naturæ fines, diuer­<lb/>&longs;æ &longs;unt viæ quibus &longs;uos fines con&longs;equítur; denique contrarietas caloris, <lb/>& frigoris impedit fortè, ne vlterius vtraque qualitas intendatur, de qua <lb/>fusè &longs;uo loco; 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 type="main"> <s>Sexta objectio; igitur &longs;i ex infinita di&longs;tantia lapis de&longs;cenderet, inten­<lb/>deret etiam &longs;uum motum. </s> <s>Re&longs;pondeo primò, non po&longs;&longs;e dari infinitam il­<lb/>lam di&longs;tantiam. </s> <s>Secundò etiani&longs;i daretur lapis, ex ea non caderet; fru&longs;trà <lb/>enim e&longs;&longs;et ille motus: Tertiò, &longs;i daretur motus infinitus, haud dubiè e&longs;&longs;et <lb/>æquabilis; qualis e&longs;t motus circularis corporum cœle&longs;tium; 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; vel &longs;u&longs;tineri vix po&longs;&longs;ent, &longs;i impetum innatum intontiorem habe­<lb/>rent; vtrum verò &longs;emper intendatur, & ex quacumque altitudine cadat <lb/>corpus graue, videbimus infrà. </s></p><p type="main"> <s>Ex dictis hactenus facilè refelluntur aliæ &longs;ententiæ de proportione <lb/>motus naturaliter accelerati. </s></p><p type="main"> <s>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; 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; interim quia faciliùs explicatur in motu recto per <lb/>rationem quadratorum quàm &longs;inuum, illam retinebimus; præ&longs;ertim cùm <lb/>vtraque ad no&longs;tram reducatur; modò progre&longs;&longs;io fiat in in&longs;tantibus. </s> <s><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: 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°. </s> <s>octuplam; 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; 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>&c. </s> <s>&longs;ed profectò vix vlla apparet rationis &longs;pecies, cùm nulla &longs;it cau&longs;a, <pb pagenum="112"/>quæ 3°in&longs;tanti, & 4°plùs agat <expan abbr="quã">quam</expan> primo, & &longs;ecundo; igitur e&longs;t peculiaris <lb/>cau&longs;a huius inæqualitatis rationum; quòd &longs;cilicet æqualibus temporibus <lb/>æqualia acquirantur velocitatis momenta; vt &longs;uprà demon&longs;trauimus; <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 type="main"> <s>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 type="main"> <s>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: 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/>interrruptam Theorematum &longs;eriem tandem repetimus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; probatur, quia &longs;e­<lb/>cundum &longs;patium æquale primo percurritur motu velociore, quàm pri­<lb/>mo, & tertium quam &longs;ecundo: ergo minori tempore per Def.2.l.1. &longs;ed <lb/>primum &longs;patium conficitur vno in&longs;tanti; igitur &longs;ecundum vno in&longs;tanti, <lb/>&longs;ed minore; idem dico de tertio. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s></p><p type="main"> <s><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>faciant &longs;emper tempora æqualia, quia temporibus æqualibus æ­<lb/>qualia acquiruntur velocitatis momenta? </s> <s>igitur &longs;i primo in&longs;tanti per­<lb/>curritur vnum &longs;patium; &longs;ecundo tempore æquali percurruntur duo &longs;pa­<lb/>tia æqualia primo, & tertio, tria; atque deinceps; &longs;ed vt &longs;uprà dictum e&longs;t <lb/>in re&longs;pon&longs;.ad obiect primam, vno, & <expan abbr="cod&etilde;">codem</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; igitur &longs;ingula <lb/>&longs;patia re&longs;pondent &longs;ingulis in&longs;tantibus licèt minoribus; &longs;ed &longs;ecundo tem­<lb/>pore æquali primo in&longs;tanti percurruntur duo &longs;patia æqualia primo &longs;pa­<lb/>tio; igitur &longs;ecundum, & tertium in&longs;tans debent &longs;imul &longs;umpta adæquare <pb pagenum="113"/>primum, &longs;ed non &longs;unt æqualia, vt con&longs;tat; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Non decre&longs;cunt illa in&longs;tantia &longs;ecundum lineam &longs;extam in extremam & <lb/>mediam rationrm propagatam; 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"/>; <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: Equidem <lb/>&longs;ecundum, & tertium adæquant primum; at verò quartum, quintum, <lb/>&longs;extum nullo modo adæquant; immò ne quidem eius &longs;ubduplum, & <lb/>multò minus 3. alij addito primo: 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;tar; igitur reiici debet hæc propo&longs;itio. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>In&longs;tans primum non e&longs;t ad &longs;ecundum vt numerus ad numerum; 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; &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. Equidem &longs;ecundum, & tertium adç­<lb/>quant primum; adde quod non pote&longs;t amplius &longs;eries propagari per nu­<lb/>meros rationales; &longs;it autem &longs;ecundum (6/11) 3. (5/11) cum tribus aliis 4/9 1/9 7/9; <lb/>cquidem &longs;i reducantur hæ 5. minutiæ, re&longs;pondebunt his (54/99) (45/99) (<gap/>4/99) (12/99) (26/99): <lb/>igitur &longs;ecunda crit maior quarta; 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>quia &longs;ecunda. </s> <s>& tertia non adæquant <lb/>primam idem dicendum e&longs;t potiori iure de tribus aliis; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s></p><p type="main"> <s><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>vt &longs;ecundus, & tertius adæquent primum, item <lb/>quartus, quintus, &longs;extus. </s> <s>item 4. alij, qui &longs;equuntur, item 5. item 6. &c. </s> <s><lb/>v. </s> <s>g. </s> <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; addi pote&longs;t quartum &longs;egmentum æquale AB; quod <lb/>&longs;ubdiuidetur in 4. partes, quarum prima &longs;it maior &longs;ecunda, & hçc tertia <lb/>& hæc quarta, & omnes minores FG; ita autem &longs;uperant primæ &longs;equen­<lb/>tes, vt differentia primæ, & &longs;ecundæ &longs;it maior differentia &longs;ecundæ, & <pb pagenum="114"/>tertiæ, & hæc maior differentia tertiæ, & quartæ; atque ita deinceps, nec <lb/>aliter res e&longs;&longs;e pote&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc partes, quo fiunt minores, accedunt propiùs ad æqualitatem,<emph.end type="italics"/> v.g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc qua preportione decre&longs;cunt instantia, decre&longs;cit etiam per&longs;ectio <lb/>impetus<emph.end type="italics"/>; quia temporibus æqualibus cadem cau&longs;a nece&longs;&longs;aria æqualem ef­<lb/>fectum producit per Ax. tertium igitur inæqualem inæqualibus, per Ax. </s> <s><lb/>13. num.4. igitur minorem minore tempore; igitur minorem in eadem <lb/>proportione, in qua tempus e&longs;t; igitur qua proportione, &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc vides quâm &longs;it nece&longs;&longs;aria illa diuer &longs;a perfestio 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; hinc e&longs;t eadem differentia impetuum, quæ <expan abbr="in&longs;tãtium">in&longs;tantium</expan>; hinc &longs;in­<lb/>gulis &longs;patiis æqualibus primo &longs;patio, quod percurritur primo in&longs;tanti; <lb/>re&longs;pondent &longs;ingula in&longs;tantia, & &longs;ingulis in&longs;tantibus &longs;inguli, & &longs;ingulares <lb/>impetus; hinc non e&longs;t quod primo in&longs;tanti dicantur produci plura pun­<lb/>cta impetus in eodem puncto corporis grauis; &longs;ed vnicum tantùm pun­<lb/>ctum talis perfectionis &longs;cilicet phy&longs;icum; cur enim potius duo puncta, <lb/>quam tria? </s> <s>&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 perfectie­<lb/>nis quàm alterius? </s> <s>quippe perfectio impetus &longs;equitur perfectionem in­<lb/>&longs;tantis quo producitur; hinc dicendum videtur omnia puncta impetus <lb/>e&longs;&longs;e diuer&longs;æ perfectionis, vel heterogenea; vt vulgò aiunt Philo&longs;ophi; <lb/>cuius rationem demon&longs;tratiuam afferemus lib. &longs;equenti cum de motu <lb/>violento; hinc vides duplicem progre&longs;&longs;ionem; primam &longs;cilicet, qua ex <lb/>&longs;uppo&longs;itis temporibus æqualibus acquiruntur &longs;patia inæqualia, de qua <lb/>fusè &longs;uprà; in hac enim velocitas eadem proportione cum impetu cre&longs;­<lb/>cit, & cum ip&longs;o tempore; hoc e&longs;t tempore triplo e&longs;t tripla, quadruplo <lb/>quadrupla; item impetus in duplo tempore e&longs;t duplus, in triplo triplus; <lb/>modò progre&longs;&longs;io fiat in temporibus primo in&longs;tanti æqualibus; &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; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; crun enim &longs;patia &longs;int vt quadrata <expan abbr="t&etilde;porum">temporum</expan> &longs;en&longs;ibiliter; <lb/>certè tempora &longs;unt, vt radices i&longs;torum quadtatorum, &longs;cilicet &longs;patiorum; <lb/>&longs;int enim quæcunque &longs;patia æqualia in linea AF; &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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quia &longs;cilicet nullum e&longs;&longs;e impedimentum; at verò &longs;i aliquod <lb/>intercedit impedimentum; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus naturalis addititius de&longs;truitur<emph.end type="italics"/>; patet experientiâ; 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>; &longs;ed de his fusè in primo libro <lb/>à Th.148. ad finem v&longs;que libri: 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; atqui &longs;i duplex e&longs;&longs;et oppo­<lb/>&longs;itus, pugnarent pro rata; igitur tàm pugnant duæ determinationes op­<lb/>po&longs;itæ in codem impetu, quàm duo impetus ad oppo&longs;itas lincas deter­<lb/>minati; igitur impetus naturalis aduentitius de&longs;truitur, &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus naturalis innatus nunquam de&longs;truitur<emph.end type="italics"/>; Probatur, quia nihil e&longs;te <lb/>quod exigat eius de&longs;tructionem, quia &longs;cilicet nunquam e&longs;t fru&longs;trà; 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 type="main"> <s>Dices lignum vi extrin&longs;eca in aqua immer&longs;um &longs;ua &longs;ponte a&longs;cendit; <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>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"/>libus, in quo o&longs;tendemus ideo lignum &longs;ur&longs;um emergere, quia ab aqua <lb/>extenditur, & idco corpora &longs;ur&longs;um ire, quia alia dcor&longs;um eunt. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quando lapis de&longs;cendit per medium aëra, impeditur aliquantulum eius <lb/>motus<emph.end type="italics"/>: Probatur primò experientiâ, quæ certa e&longs;t; 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; quomodo ve­<lb/>rò impediat, dicemus aliàs; &longs;ecundò corpus immobile, in quod mobile <lb/>impingitur, motum illius impedit; &longs;ed in diuer&longs;as partes aëris corpus <lb/>graue impingitur in de&longs;cen&longs;u; igitur aliquantulum impeditur eius <lb/>inotus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc etiam impetus producitur imperfectior<emph.end type="italics"/>; quia ex imperfectione ef­<lb/>fectus requiritur imperfectic cau&longs;æ per Ax. 13.l. </s> <s>1. & quâ proportione <lb/>e&longs;t tardior motus eâdem impetus e&longs;t imperfectior, per Ax. </s> <s>5. excipe ta­<lb/>men impetum innatum, qui &longs;emper habet cundem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quo medium den&longs;ius e&longs;t plus impedit motum deor&longs;um<emph.end type="italics"/>; Probatur, quia &longs;i <lb/>motum impedit; certè non totum; quis enim hoc dicat; &longs;ed eæ dumta­<lb/>xat partes, quibus incubat corpus graue; igitur quò &longs;unt plures huiu&longs;­<lb/>modi partes, maius e&longs;t impedimentum; &longs;ed in medio den&longs;iori plures &longs;unt <lb/>cum minore exten&longs;ione; hoc enim e&longs;t, quod voco den&longs;ius; igitur me­<lb/>dium den&longs;ius plùs impedit. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;erua e&longs;&longs;e aliqua corpora minus den&longs;a, quæ motum omninò im­<lb/>pediunt; quippe certum e&longs;t aquam e&longs;&longs;e den&longs;iorem ligno; atqui li­<lb/>gnum de&longs;cen&longs;um lapidis impedit, non verò aqua; quia &longs;cilicet lignum <lb/>non e&longs;t medium, vt aqua; 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>ratio <lb/>e&longs;t, quia partes ligni, vel alterius corpotis 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; igitur vt corpus ali­<lb/>quod vice medij defungatur, debet in co &longs;tatu e&longs;&longs;e, in quo eius partes <pb pagenum="117"/>modico ferè ni&longs;u &longs;eiungantur, & loco cedant; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc producitur impetus imperfectior in medio den&longs;iore:<emph.end type="italics"/> quia in co tar­<lb/>dior e&longs;t motus, ex cuius tarditate arguitur imperfectio impetus per Ax. </s> <s><lb/>13.num.4. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;erua den&longs;itatem medij cogno&longs;ci ex cius grauitate; illud enim <lb/>den&longs;ius e&longs;t, quod e&longs;t grauius & vici&longs;&longs;im; quod fusè explicabimus &longs;uo lo­<lb/>co; 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; <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 type="main"> <s>Ob&longs;eruabis ctiam hîc à me non di&longs;cuti, in quo con&longs;i&longs;tat den&longs;itas, vel <lb/>raritas, grauitas, vel leuitas; &longs;uppono tantùm graue illud e&longs;&longs;e, quod ten­<lb/>dit deor&longs;um; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Sub medium leuius corpus graue de&longs;cendit<emph.end type="italics"/>; certa e&longs;t hypothe&longs;is, ni&longs;i for­<lb/>tè aliquando per accidens &longs;ecus accidat; ratio porrò petitur ex ip&longs;a <lb/>grauitatis natura, quâ corpus graue tendit deor&longs;um; nihil enim aliud <lb/>grauitas e&longs;t, quidquid tandem illa &longs;it; quippe corpus graue de&longs;cendit, <lb/>quando medium liberum habet, idemque leuius, per quod de&longs;cendat; <lb/>quod certè &longs;i grauius e&longs;&longs;et, haud dubiè non de&longs;cenderet; &longs;ic ferrum, & <lb/>&longs;axum plumbo liquato innatant; cum tamen per mediam aquam de­<lb/>fcendant; fic lignum aquæ &longs;upernatat, quod per liberum aëra de&longs;cendit; <lb/>ratio e&longs;t, quia grauius de&longs;cendit &longs;ub medium leuius; cur autem id fiat <lb/>fusè alibi explicabo; id tantùm obiter indico. </s> <s>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>cum à natura acceperit vim illam intrin&longs;ecam, quâ in eum lo­<lb/>cum &longs;e&longs;e recipere pote&longs;t; 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; 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; cum enim corpus, quod e&longs;t &longs;uprà, &longs;u&longs;tineatur ab eo quod e&longs;t <lb/>infrà; illud certè infra e&longs;&longs;e connaturalius e&longs;t, quod aptius e&longs;t ad &longs;u&longs;tinen-<pb pagenum="118"/>dum; 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; v.g. </s> <s>certum <lb/>e&longs;t camdem aëris partem pluribus aquæ partibus re&longs;pondere; &longs;ed de hoc <lb/>alias fusè; hæc interim &longs;ufficiat indica&longs;&longs;e, vt vel aliqua ratio affulgeat; <lb/>cur &longs;cilicet corpus graue &longs;ub medium leuius &longs;ua &longs;ponte de&longs;cendat; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Sub medium grauius corpus leuius minimè de&longs;cendit, &longs;ed huic inna­<lb/>tat<emph.end type="italics"/>; v.g. </s> <s>lignum aquæ, ferrum plumbo liquato; certa e&longs;t hypothe&longs;is: 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; 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>v.g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Sub medium æquè graue corpus non de&longs;cendit, nec etiam &longs;upra a&longs;cendit<emph.end type="italics"/>; 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; igitur &longs;i nec &longs;it grauius nec <lb/>leuius, non e&longs;t quod a&longs;cendat vel de&longs;cendat; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Aër &longs;uam grauitatem habet<emph.end type="italics"/>; quod iam à nullo in dubium reuocari po­<lb/>te&longs;t; nam &longs;i comprimatur intra vas æneum v.g. </s> <s>etiam minimæ cra&longs;&longs;itu­<lb/>dinis; &longs;i deinde ponderetur, maius e&longs;t haud dubiè pondus, quo maior <lb/>e&longs;t aëris copia intru&longs;a; 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; verùm etiam vigecuplum; immò centuplum, & millecuplum <lb/>adhibita cochleâ, vel alio mechanico organo, & aucta va&longs;is cra&longs;&longs;itudine, <lb/>de quo aliàs: 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; hâc vero minorem <lb/>e&longs;&longs;e ob&longs;eruauimus, de quo aliàs; nec enim e&longs;t præ&longs;entis in&longs;tituti, pro <lb/>quo &longs;ufficiat modò, aëri aliquam ine&longs;&longs;e grauitatem; nec dicas aëra le­<lb/>uem e&longs;&longs;e; nam reuerâ leuis e&longs;t, &longs;i comparetur cum aqua; grauis autem &longs;i <lb/>comparetur cum a&longs;cendente halitu, vel fortè cum vacuo; 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; 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; nam de alio aëre non affirmo <pb pagenum="119"/>grauem e&longs;&longs;e, ni&longs;i tantùm de illo, quem &longs;piramus, in quo ambulamus, qui <lb/>nos ambit: 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Medium eiu&longs;dem grauitatis cum dato corpore graui detrahit totam eius <lb/>grauitationem &longs;ingularem; hoc e&longs;t corpus graue in medium æquè graue non <lb/>grauitat<emph.end type="italics"/>; quia &longs;i grauitaret de&longs;cenderet; &longs;ic pars aquæ in aliam partem <lb/>aquæ non grauitat, & &longs;i aqua ponderetur in aqua, nullius ponderis e&longs;t; <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Medium graue detrahit aliquid de &longs;ingulari grauit atione corporis grauio­<lb/>ris<emph.end type="italics"/>; certa e&longs;t hypothe&longs;is; nec enim plumbum e&longs;t eius ponderis &longs;ingula­<lb/>ris in aqua, cuius e&longs;t in aëre; dixi &longs;ingularis; nam &longs;i plumbum & ip&longs;a <lb/>aqua &longs;imul appendantur, haud dubiè totum habebis pondus plumbi, & <lb/>totum pondus aquæ; ratio verò huius effectus non e&longs;t huius loci; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Medium graue detrahit eam partem grauit ationis corporis grauioris, quæ <lb/>e&longs;t æqualis &longs;uæ grauitationi.<emph.end type="italics"/> v. </s> <s>g. </s> <s>&longs;i medij grauitas e&longs;t &longs;ubdupla, detrahit <lb/>&longs;ubduplum grauitationis; &longs;i &longs;ubdecupla, &longs;ubdecuplum, atque ita dein­<lb/>ceps; hoc iam olim &longs;uppo&longs;uit magnus Archim. </s> <s>&longs;upponunt etiam reliqui <lb/>omnes, præ&longs;ertim recentior Galileus; &longs;i enim æqualis &longs;uperat æqualem, <lb/>ergo inæqualis pro rata; &longs;cilicet &longs;ubdupla &longs;ubduplum &longs;ubtripla, &c. </s> <s>Præ­<lb/>terca, 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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; hinc etiam maioris &longs;unt ponderis in aëre quam in <lb/>aqua; 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); id e&longs;t, &longs;i penderet 100. <lb/>libras in vacuo, in aëre penderet 99.& eo tempore quo in vacuo decur­<lb/>teret 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; &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/<gap/>), & <lb/>co tempore quo decurreret in vacuo 10. palmos &longs;patij, in aqua decurre <pb pagenum="120"/>ret tantùm 9. po&longs;ito quod non &longs;it aliud quod re&longs;i&longs;tat; quanta verò &longs;it <lb/>grauitas omnium corporum tùm duriorum, qualia &longs;unt metall<gap/>, tùm li­<lb/>quidorum, tùm &longs;pirabilium, dicemus &longs;uo loco; illorum tabulas habes <lb/>apud Gethaldum, & Mer&longs;ennum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc, &longs;i nihil aliud de&longs;cen&longs;um corporum grauium impediret, cognito pen­<lb/>dere vtriu&longs;que, medij & corperis grauis, &longs;patio, quod in vno illorum conficit, <lb/>cogno&longs;cipo&longs;&longs;et &longs;patium, quod in alio conficeret æquali tempore<emph.end type="italics"/>, v. </s> <s>g. </s> <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; haud dubiè eo tempore, quo <lb/>conficit in aëre 799. &longs;patia, in aqua con&longs;iciet tantùm 400. quia in vacuo <lb/>conficeret 800. aër autem detrahit (<gap/>/800), & aqua <gap/>/2, vt con&longs;tat ex dictis; &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 hærco. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis dictum e&longs;&longs;e hactenus; &longs;i nihil aliud de&longs;cen&longs;um corporis <lb/>grauis impedit; 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; 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 codem tem­<lb/>pore; certè in vacuo ip&longs;o moueretur tardiùs quàm in aëre; 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; atqui in aëre codem tempore <lb/>conficit 48. pedes; igitur velociùs moueretur in aëre quàm in vacuo; <lb/>igitur e&longs;t aliquid aliud quod impedit motum; 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; in aqua <lb/>verò 4400. quod e&longs;t contra experientiam; nam &longs;patium, quod decurrit <lb/>in aëre e&longs;t maius &longs;patio, quod decurrit in aqua 3/4; quippe conficit 12. <lb/>pedes in aqua eodem tempore, quo in aëre conficir 48; igitur in aqua <lb/>amittit 3/4 &longs;uæ grauitationis, & &longs;ui motus; igitur 3600. partes; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc potiori iure reiicies illerum &longs;ententiam, qui volunt impediri motum <lb/>corporis de&longs;cendentis per diuer&longs;a media pro diuer&longs;aratione grauitatum vtriu&longs;­<lb/>que medy<emph.end type="italics"/>; quod certè fal&longs;um e&longs;t; 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"/>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 90.<emph.end type="center"/></s></p><p type="main"> <s><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 <gap/>iu&longs;dem medij,<emph.end type="italics"/> patet; quia vnum cor­<lb/>pus non pote&longs;t penetrari cum alio. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 91.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Totum medium loco non cedit in de&longs;cen&longs;u grauium<emph.end type="italics"/>; patet etiam, tùm <lb/>quia ad mouendum totum medium exigua vis corporis grauis non &longs;uffi­<lb/>cit; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s></p><p type="main"> <s><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.igitut <lb/>&longs;ecundum, in his certè non e&longs;t vlla difficultas. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 93.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Non po&longs;&longs;unt illæ partes loco cedere &longs;ine motu; 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 94.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 95.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc corpus grauius &longs;ustinetur à leuiore.<emph.end type="italics"/> v.g. </s> <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; 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; hinc corpora dura non po&longs;&longs;unt <lb/>e&longs;&longs;e medium; immò neque mollia, vt cera, argilla; &longs;ed vel liquida, vel <lb/>&longs;pirabilia. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 97.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; quia non modo partes <pb pagenum="122"/>medij amouendæ &longs;unt è &longs;uo loco; verùm etiam nexus ille partium &longs;ol­<lb/>uendus; igitur ex vtroque capite impeditur motus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 98.<emph.end type="center"/></s></p><p type="main"> <s><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; 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; &longs;ic videmus multùm aquæ hauriri, dum <lb/>arctioribus retibus attollitur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ob&longs;eruabis vmonem 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>aliquando in vacui metu; &longs;ic aqua, vt &longs;uo va&longs;i adhæreat, <lb/>a&longs;cendit, vel &longs;ur&longs;um attollitur, ne detur vacuum; aliquando in coitione <lb/>quadam magnetica; 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: 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/>aurri po&longs;&longs;e, &longs;eu perrumpi. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 99.<emph.end type="center"/></s></p><p type="main"> <s><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 græ­<lb/>uitas &longs;it dedecupla grauitatis aquæ, verùm etiam propter re&longs;istentiam petitam <lb/>ex alio capite aliquid adhuc detrabere pote&longs;t<emph.end type="italics"/>; &longs;cilicet quia partes aquæ non <lb/>po&longs;&longs;unt amoueri, ni&longs;i ab aliis &longs;eparentur; atqui maiore vi opus e&longs;t ad­<lb/>&longs;oiuendum &longs;trictiorem nexum; immò licèt partes aquæ nullo penitus <lb/>nexu vniantur, &longs;ed tantùm vel vacui metu, vel alio modo, quod alibi ex­<lb/>plicabimus; omninò detraherent adhuc plumbo (1/12) motus; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 100.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; hinc <lb/>tenui nexu partes aëris copulantur; partes verò aquæ firmiori; hinc aër <lb/>minùs re&longs;i&longs;tit etiam motibus violentis; hinc vix pote&longs;t qui&longs;piam in aqua <lb/>currere propter maiorem aquæ re&longs;i&longs;tentiam; hinc pote&longs;t dici quota parte <lb/>firmior &longs;it nexus vnius corporis quàm alterius; hinc non tantùm copu­<lb/>lantur partes metu vacui; alioquin æquè re&longs;i&longs;terent partes aëris, ac par­<lb/>tes aquæ ratione nexus; hinc videntur guttulæ illæ &longs;phericæ inuolui te­<lb/>nui qua&longs;i membranula, &longs;eu &longs;uperficie, cuius analogiam videmus in aqua<gap/><pb pagenum="123"/>feruente; in bullis, quæ ex guttis pluuiæ re&longs;ilientibus na&longs;ci videntur; in <lb/>bullis etiam illis &longs;aponariis, quas leui calamo pueri inter ludendum in­<lb/>flant; 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; &longs;ic bullæ il­<lb/>læ ad minimum etiam contactum di&longs;&longs;ipantur; hinc ip&longs;a &longs;uperficies <lb/>aquæ plus videtur re&longs;i&longs;tere quod multis experimentis comprobatur; &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; immò cum ab ip&longs;a naui qua&longs;i &longs;ulcatur, <lb/>idem &longs;tridor auditur, maximè in iis tractibus; in quibus nullis fluctibus <lb/>agitata læuigati&longs;&longs;imam faciem præfert; habes analogiam in illa cru&longs;ta, <lb/>quæ concre&longs;cit in &longs;uperficie liquorum, &longs;ed præ&longs;ertim offarum:adde quod <lb/>aër paulò compre&longs;&longs;ior vndique guttulam premens æquali ni&longs;u eam miri­<lb/>ficè tornat: 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 101.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; cum enim tantus relinquatur <lb/>locus retrò, quantus acquiritur antè, nulla opus e&longs;t compre&longs;&longs;ione; &longs;ed <lb/>partes à fronte pul&longs;æ factâ circuitione retror&longs;um eunt, non certè tramite <lb/>recto; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 102.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; igitur maiore vi opus e&longs;t, igitur maiore grauitate; &longs;ed in medio <lb/>den&longs;iore ab codem mobili plures &longs;eparantur quàm in rariore; 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; igitur in medio den&longs;iore idem mobile ma­<lb/>jorem re&longs;i&longs;tentiam inuenit, quàm in rariore; licèt vtriu&longs;que partes <lb/>æquali nexu &longs;eu fibula copulentur; quia &longs;cilicet plures &longs;unt diuidendæ <lb/>in den&longs;iore; quia plures &longs;cilicet in æquali &longs;patio occurrunt, quàm in ra­<lb/><gap/>iore; igitur maiore vi grauitatis opus e&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 103.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; Primum e&longs;t in <lb/>grauitate, vel den&longs;itate, nam reuerâ ex maiori den&longs;itate maiorem gra­<lb/>uitatem reducimus; Secundum e&longs;t in maiori, vel minori partium nexu, <lb/>ex quo 4. &longs;equuntur combinationes 2.mediorum; nam vel &longs;unt eiu&longs;dem <lb/>grauitatis, & mollitiei; vel eiu&longs;dem grauitatis & diuer&longs;æ mollitiei; vel <lb/>ciu&longs;dem mollitici, & diuer&longs;æ grauitatis; vel diuer&longs;æ grauitatis, & ciu&longs;­<lb/>dem mollitici; mollius autem illud appello, cuius partes laxiori nexu <lb/>copulantur; porrò 4.i &longs;tæ combinationes &longs;upponunt <expan abbr="id&etilde;">idem</expan> mobile <expan abbr="invtroq;">invtroque</expan> <lb/>medio; &longs;i &longs;it prima combinatio, motus e&longs;t æqualis in vtroque; &longs;i &longs;ecunda <pb pagenum="124"/>maior e&longs;t in molliori; &longs;i tertia maior in grauiori; &longs;i verò quarta &longs;ubdi­<lb/>uidi pote&longs;t in duas; nam vel grauius e&longs;t conjunctum cum maiori molli­<lb/>tie, vel leuius; &longs;i leuius, haud dubiè maior e&longs;t motus in leuiore; &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; ita &longs;e habet mollities illius ad mollitiem <lb/>huius, æqualis e&longs;t in vtroque; &longs;i &longs;ecus, pro rata; 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; maior quidem, &longs;i maior &longs;it ratio mollitici gra­<lb/>uioris ad mollitiem leuioris, quàm grauitatis ad grauitatem; minor ve­<lb/>rò, &longs;i maior &longs;itratio grauitatis ad grauitatem, quàm mollitici ad molli­<lb/>tiem; æqualis denique &longs;i æqualis ratio; & his regulis cuncta facilè ex­<lb/>plicari po&longs;&longs;unt; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 104.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Sunt tres combinationes mobilis cum medio<emph.end type="italics"/>; prima, &longs;i &longs;it idem mobile <lb/>cum diuer&longs;is mediis; &longs;ecunda, &longs;i idem medium cum diuer&longs;is mobilibus; <lb/>tertia &longs;i diuer&longs;a mobïlia cum diuer&longs;is mediis; de primâ actum e&longs;t iam <lb/>fuprà; &longs;ecunda &longs;ube&longs;t 4. combinationibus. </s> <s>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>Quarta diuer&longs;æ materiæ & figuræ; &longs;i prima & &longs;ecunda, vel &longs;unt <lb/>figuræ æquales, vel inæquales; &longs;i primum &longs;unt eiu&longs;dem grauitatis; &longs;i &longs;e­<lb/>cundum diuer&longs;æ; quippe figuræ &longs;imiles po&longs;&longs;unt e&longs;&longs;e æquales, vel inæ­<lb/>quales; & figuræ æquales po&longs;&longs;unt e&longs;&longs;e &longs;imiles, vel di&longs;&longs;imiles; &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; &longs;i figuræ &longs;unt æquales, &longs;emper e&longs;t diuer&longs;a grauitas; &longs;i <lb/>inæquales pote&longs;t e&longs;&longs;e vel eadem, vel tertia; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 105.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si mobilia duo eiu&longs;dem materiæ, figuræ, & grauitatis in eodem medio de­<lb/>&longs;<gap/>endant, æquali motu feruntur<emph.end type="italics"/> dem. </s> <s>vbi e&longs;t eadem proportio cau&longs;æ & re&longs;i­<lb/>&longs;bentiæ ibi e&longs;t idem effectus, per Ax. 5. &longs;ed in hoc ca&longs;u eadem e&longs;t illa pro­<lb/>portio; nam e&longs;t æqualis cau&longs;a, &longs;cilicet grauitas; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 106.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Maior e&longs;t re&longs;istentia <gap/>iu&longs;dem medij ratione &longs;cilicet partium, cum plupes <lb/>cius 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 107.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Plures partes re&longs;istunt, quando plures pelluntur à mobili deor&longs;um<emph.end type="italics"/>; quip­<lb/>pe in tantum re&longs;i&longs;tunt, in quantum ab aliis &longs;eparantur; atqui in tantum <lb/>&longs;eparantur, in quantum amouentur è &longs;uo loco; &longs;ed ideo amouentur è <lb/>&longs;uo loco, in quantum pelluntur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 108.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <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 i<gap/><lb/>fronte. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 109.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><arrow.to.target n="note2"></arrow.to.target><lb/>GK, nam in gratiam corum 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. </s> <s>g. </s> <s>facies AI in quad. </s> <s>AE. EC. EG. EI. idem fiat in aliis <lb/>5. faciebus, quarum duæ hîc tantum apparent; &longs;cilicet AK. KL; &longs;ed <lb/>tribus aliis parallelis; his tribus cædem diui&longs;iones re&longs;pondent; 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>1.2.4.8. </s></p><p type="margin"> <s><margin.target id="note2"></margin.target>a <emph type="italics"/>Fig.<emph.end type="italics"/>26 <lb/><emph type="italics"/>Tab.<emph.end type="italics"/>1.</s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 110.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc plùs minuitur &longs;olidum in diuer&longs;ione cubi quam facies, & plùs facies <lb/>quàm latus<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 111.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc plùs minuitur grauitas, quàm re&longs;i&longs;tentia minoris cubi<emph.end type="italics"/>; quia grauitas <lb/>re&longs;pondet &longs;olido, & re&longs;i&longs;tentia primç facici; re&longs;i&longs;tentia <expan abbr="inquā">inquam</expan> ratione par­<lb/>tium medij; &longs;ed &longs;olidum plus miuuitur quàm facies, vt dictum e&longs;t; igitur <lb/>plus minuitur grauitas, quæ e&longs;t cau&longs;a virium quàm hæc re&longs;i&longs;tentia; crgo <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 type="main"> <s>Hinc concludit Galileus duos cubos eiu&longs;dem materiæ, &longs;ed inæquales <lb/>de&longs;cendere inæquali motu; maiorem &longs;cilicet velociùs minori; demon­<lb/>ftrare 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 type="main"> <s>Scio non dee&longs;&longs;e multos viros doctos qui acriter in hanc &longs;ententiam <pb pagenum="126"/>in&longs;urgant: Obiicient fortè primò, experientiam e&longs;&longs;e contrariam; &longs;i enim <lb/>accipiantur duo cubi maior, & minor eiu&longs;dem materiæ, & dimittantur <lb/>ex eadem altitudine codem pror&longs;us momento terram ferient; 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>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 ctiam &longs;en&longs;ibilem motuum differentiam videbîs, atqui <lb/>e&longs;t cadem r<gap/>tio de omni minore. </s></p><p type="main"> <s>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>Re&longs;p. videri po­<lb/>te&longs;t equidem æquali motu de&longs;cendere quia &longs;unt veluti partes ciu&longs;dem <lb/>corporis, & grauitant grauitatione communi, neque minor habet &longs;ingu­<lb/>larem re&longs;i&longs;tentiam &longs;uperandam; immò &longs;i &longs;uperponatur minor maiori, <lb/>vel maior minori, motus e&longs;t velocior quàm e&longs;&longs;et &longs;olius maioris; 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 type="main"> <s>Tertiò obiicient; e&longs;t eadem &longs;pecie grauitas; igitur eadem grauitatio, <lb/>idemque motus deor&longs;um; 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 type="main"> <s>Quartò obij:igitur motus po&longs;&longs;et e&longs;&longs;e velocior, & velocior in infini­<lb/>tum; &longs;i enim maior cubus de&longs;cenderet velociùs; igitur &longs;i detur maior ad­<lb/>huc velociùs, atque ita deinceps: Re&longs;p. inanem pror&longs;us e&longs;&longs;e difficulta­<lb/>tem; 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>v. </s> <s>g. </s> <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/<gap/> (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>igitur obiectiones illæ non cuertunt Gali­<lb/>lei &longs;ententiam. </s></p><p type="main"> <s>Inde idem Galilcus 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 type="main"> <s>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; certè ex­<lb/>tabit maior toto rectangulo CA æquali CF, & minor toto rectangulo <lb/>KH æquali KM; 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 type="main"> <s>Re&longs;ponderi pote&longs;t e&longs;&longs;e maximam di&longs;paritatem inter grauitationem, & <pb pagenum="127"/>motum; &longs;it enim cubus BD qui de&longs;cendat per totam AH; 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; &longs;it autem cubus BG; 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 type="main"> <s>Dices ergo &longs;patia &longs;unt vt latera: Re&longs;ponderi pote&longs;t hoc reuerâ per &longs;e <lb/>e&longs;&longs;e debere; &longs;ed quia cubus DM vt extrudatur, maiorem debet facere cir­<lb/>cuitionem, vt à fronte retrò eat, velociori motu extrudi debet; igitur vi­<lb/>res &longs;uas in co con&longs;umit maiori ex parte cubus AE; hinc compen&longs;atio e&longs;&longs;e <lb/>videtur. </s></p><p type="main"> <s>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>Primò itaque certum e&longs;t <lb/>partes medij, quæ prius in fronte erant, retroire; 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; &longs;ic enim &longs;æpè a&longs;&longs;urgunt, & intume&longs;cunt &longs;upra labra va&longs;is; cur verò <lb/>continui penè circulares limbi dilatentur: Re&longs;p. nullo flante vento <lb/>vix aliquem circulum huiu&longs;inodi 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 type="main"> <s>Secundò certum e&longs;t numerum partium, quas impellit maior cubus A <lb/>E; e&longs;&longs;e quadruplum numeri partium, quas impellit cubus BG: &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 type="main"> <s>Sit cubus maior AF octuplus cubi GI, vt iam dictum e&longs;t; 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; &longs;ed &longs;unt octuplæ; igitur faci­<lb/>liùs vincetur re&longs;i&longs;tentia; igitur amouebitur aër faciliùs; &longs;it autem aër <lb/>expre&longs;&longs;us in globulis EFB, &c. </s> <s>cuius &longs;uperficies cum relinquatur retrò <lb/>ver&longs;us AB, & occupetur illa quæ e&longs;t in fronte EF; haud dubiè partes <lb/>hinc inde diuiduntur in D, & fegmentum NB tran&longs;it in locum relicti <lb/>loci BC, FN tran&longs;it in NB, & DF, in FN; idem dico de &longs;egmentis oppo­<lb/>fitis; idem pror&longs;us dico de minori globo; nam MH tran&longs;it in HQ, & H <lb/>Q in QG, & QG in GL, idem dico de &longs;egmentis oppo&longs;itis; 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 grauc, quod de&longs;cendit. </s></p><p type="main"> <s>In hoc porrò ob&longs;erua &longs;egmentum MH moueri tardiùs quàm DF; quia <lb/>conficit &longs;ubduplum &longs;patium, co tempore, quo DF conficit duplum; <pb pagenum="128"/>nam DF & FN &longs;unt duplæ MH & & <expan abbr="Hq;">Hque</expan> igitur dupla vi motrice opus <lb/>e&longs;t; &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; certè maior moueretur motu velociori; vt autem moueantur par­<lb/>tes DF duplò velociore motu, quàm partes MH; debent vires, quæ mo­<lb/>nent DF, e&longs;&longs;e in ratione dupla ad illas, quæ mouent MH, id e&longs;teo 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 reperlitur, vt &longs;ur&longs;um ver&longs;us <lb/>FN repellatur. </s></p><p type="main"> <s>Equidem tota &longs;uperficies aëris DF, cum pluribus partibus con&longs;tet, <lb/>non pote&longs;t &longs;imul tran&longs;ire in FN; quia pars D antequam perueniat ad F <lb/>tran&longs;it per medium DF; igitur &longs;ucce&longs;&longs;iuè permea ad illud &longs;patium DF, <lb/>quo tempore quie&longs;ceret globus AF, quod ridiculum e&longs;t. </s></p><p type="main"> <s>Quare fit nece&longs;&longs;ariò aliqua circuitio, & partium aëris commixtio, <lb/>&longs;eu conflictus; ita vt retroeant pul&longs;æ prius & repercu&longs;&longs;æ; non quidem <lb/>tramite recto, &longs;ed cum aliqua circuitione; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 112.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; probatur, quia licèt &longs;it maior proportio actiuitatis minus ad <lb/>&longs;uam re&longs;i&longs;tentiam, quàm alterius; illud tamen compen&longs;atur; eóque par­<lb/>tes aëris velociùs moueri debeant inxta rationem laterum, vt patet ex <lb/>dictis; vnde nece&longs;&longs;ariò &longs;equitur motus æqualis in vtroque cubo; igitur <lb/>licèt maioris cubi vires habeant maiorem proportionem ad molem, <lb/>quæ præcipuum illius motus retardat; 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; certè ratio <lb/>compo&longs;ita vtriu&longs;què erit cadem in vtroque cubo; igitur æquè velociter <lb/>vterque de&longs;cendet: hinc &longs;atís facilè &longs;oluitur ratio Galilci, 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; <lb/>ideo corpu&longs;cula illa diu fluitare in aëre, tùm quòd minimo ferè tenuis <lb/>auræ flatu agitentur; &longs;ic pulueris nubes medius ventus agit; quis enim <lb/>ne&longs;cit aëris partes agitari perpetuò; immò & aquæ inter &longs;e mi&longs;ceri; igi­<lb/>tur ab agitationis veluti impre&longs;&longs;ione fluitant illa corpu&longs;cula, cum mini­<lb/>musferèimpetus extrin&longs;ecus illa commouere po&longs;&longs;it; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 113.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Duo globi eiu&longs;dem materiæ, & diuer&longs;æ diametri de&longs;cendunt etiam æquali <lb/>motu propter eamdem rationem<emph.end type="italics"/>; immò e&longs;t perfectior æqualitas in globis, <lb/>quàm in cubis; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 114.<emph.end type="center"/></s></p><p type="main"> <s><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; quia breuiore circuitu partes re­<lb/>trocunt; 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; atqui hoc ip­<lb/>&longs;um e&longs;t quod accidit mobili vtrimque aucto; nam linea curua DBA, <lb/>quam percurrit de&longs;criptum mebile, non e&longs;t multò longior; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 115.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Idem corpus diuer&longs;o motu de&longs;cendere pote&longs;t,<emph.end type="italics"/> v. </s> <s>g. </s> <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; 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; hinc cylindrus eiu&longs;dem grauitatis de&longs;cendet <lb/>velociùs quàm parallelipedum, vt patet ex dictis; 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>Porrò ob&longs;cruabis omne <lb/>corpus difficiliùs pelli per lineam perpendicularem quàm per obliquam; <lb/>hinc globus pellit tantùm vnicum punctum perpendiculariter; idem di­<lb/>co de cono; cylindrus verò vnam lineam, cubus integrum planum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 116.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 117.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 118.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Globi æquales diuer&longs;æ materiæ inæqualiter de&longs;cendunt<emph.end type="italics"/>; quia &longs;cilicet alte­<lb/>rum e&longs;t grauius, quod &longs;uppono; igitur æqualis e&longs;t re&longs;i&longs;tentia, & vires <lb/>inæquales; 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 119.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Globi otiam inæquales diuer&longs;æ materiæ inæqualiter de&longs;cendunt<emph.end type="italics"/>; quod de­<lb/>mon&longs;tro; quia globi ciu&longs;dem materiæ inæqualiter de&longs;cendunt per Th. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 120.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Globus materiæ leuioris pote&longs;t de&longs;cendere velociori motu quam parallelipe­<lb/>dum grauioris<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 121.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 122.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Globi æquales diuer&longs;æ materiæ,<emph.end type="italics"/> v. </s> <s>g. </s> <s>ligneus, & plumbeus de&longs;cendunt <lb/>mæqualiter iuxta proportionem grauitatis, & re&longs;i&longs;tentiæ medij compa­<lb/>ratæ cum vtroque, v.g. </s> <s>plumbo detrahitur (1/4800); ligno verò (8/300) v. </s> <s>g. </s> <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>reducantur in digitos erunt 400. <lb/>in lineas 4800. igitur detrahetur vna linea &longs;patij plumbeo globo; ligneo­<lb/>verò vnus digitus cum 4. lineis; &longs;ed quis hoc ob&longs;eruet? </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 123.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Corpus graue &longs;pongio&longs;um longè tardiùs de&longs;cendit<emph.end type="italics"/>; 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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>Ex his con&longs;tat quid dicendum &longs;it de motu corporum grauium in <lb/>medio, &longs;iue &longs;int eiu&longs;dem maieriæ, & &longs;imilis figuræ, maioris vel minoris, <lb/>vel æqualis; 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"/>tunc enim de&longs;cendunt inæqualiter, &longs;iue diuer&longs;æ materiæ & diuer&longs;æ fi­<lb/>guræ; tunc enim de&longs;cendunt modò æqualiter, modò inæqualiter; æquali­<lb/>ter certè, cum figura compen&longs;at materiam; cum verò non compen&longs;at, <lb/>inæqualiter pro rata; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 124.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>In modico vacuo omnia æquè velociter de&longs;cenderent<emph.end type="italics"/>: 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; igitur nulla proportio; igitur nulla ratio motus inæqualis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 125.<emph.end type="center"/></s></p><p type="main"> <s><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"/>: probatur, quia &longs;ingulis in&longs;tantibus plures partes medij &longs;unt <lb/>&longs;uperandæ; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 126.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cre&longs;cit re&longs;istentia iuxta rationem &longs;patiorum,<emph.end type="italics"/> probatur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 127.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc cre&longs;cit re&longs;i&longs;tentia iuxta rationem velocitatum &longs;ingulis instantibus<emph.end type="italics"/>; <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; <lb/>nam codem modo cre&longs;cit velocitas, quo cre&longs;cunt numeri prædicti; &longs;ed <lb/>eodem modo cre&longs;cunt &longs;patia, &longs;i dumtaxat accipiantur in &longs;ingulis in&longs;tan­<lb/>tibus; re&longs;i&longs;tentia cre&longs;cit iuxta rationem &longs;patiorum; igitur iuxta ratio­<lb/>nem velocitatum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; 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; <emph type="italics"/>æqualibus temporibus æqualia acquiruntur velocita­<lb/>tis momenta<emph.end type="italics"/>; &longs;i verò tempus con&longs;tat ex finitis in&longs;tantibus æqualibus, nul­<lb/>la datur progre&longs;&longs;io motus naturaliter accelerati; quia motus accelerari <lb/>non pote&longs;t; 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"/>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 128.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>127. cre&longs;cit etiam in ratione im­<lb/>petuum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 129.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc cre&longs;cit re&longs;istentia medij in eademratione, in qua cre&longs;cunt vires mobi­<lb/>lis<emph.end type="italics"/>; demon&longs;tr. </s> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 130.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <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; certè 2. gradus impetus vno <lb/>in&longs;tanti &longs;uperabunt re&longs;i&longs;tentiam vt 2. dum con&longs;icit mobile 2. &longs;patia; at­<lb/>que ita deinceps. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 132.<emph.end type="center"/></s></p><p type="main"> <s><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; &longs;ed hæc ratio nulla e&longs;t; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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><pb pagenum="133"/><figure></figure><p type="main"> <s><emph type="center"/>LIBER TERTIVS,<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>DE MOTV VIOLENTO <lb/>&longs;ur&longs;um Perpendiculariter.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s>Dixi à principio extrin&longs;eco, &longs;iue cunjuncto, 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; &longs;iue non &longs;it <lb/>principium effectiuum, &longs;ed tantùm determinans, vt cum mobilc quod <lb/>cadit deor&longs;um, &longs;ur&longs;um deinde repercutitur; nec enim corpus repercu­<lb/>tiens producit impetum nouum, vt dicemus cum de motu re&longs;lexo; 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 type="main"> <s>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"/>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 type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quidquid destruitur, ad exigentiam alicuius destruitur, &longs;altem totius na­<lb/>turæ.<emph.end type="italics"/></s><s> Hoc Axioma idem e&longs;t cum Axiom. </s> <s>14. l. </s> <s>1. n. </s> <s>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>l. </s> <s>1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Datur motus violentus<emph.end type="italics"/>; demon&longs;tro; corpus proiicitur per lineam ver­<lb/>ticalem per hyp. </s> <s>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. </s> <s>1.n. </s> <s>1. &longs;ed ex applicatione potentiæ extrin&longs;ecæ v. </s> <s>g. </s> <s>arcus, manus, &c. </s> <s><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><lb/>Secundò probatur, quia mobile projectum &longs;ursùm mouetur adhuc &longs;epa­<lb/>ratum à potentia motrice per hyp. </s> <s>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 type="main"> <s>Diceret fortè aliquis produci hunc motum ab ip&longs;o mobili; &longs;ed con­<lb/>trà; igitur &longs;emper produceret, quod ab&longs;urdum e&longs;t: dicet, ad hoc vt pro­<lb/>ducat determinari debere ab aliquo, &longs;ed contrà; 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>1. nam reuera non &longs;emper mo­<lb/>uetur. </s></p><p type="main"> <s>Diceret fortè alius excitari quædam corpu&longs;cula, à quibus mouetur <lb/>corpus graue &longs;ursùm; &longs;ed contrà; nam vel &longs;unt in ip&longs;o mobili illa cor­<lb/>pu&longs;cula, vel extra mobile; &longs;i primum; igitur hic motus &longs;emper crit ab <lb/>extrin&longs;eco mediatè, cum ab extrin&longs;eco excitentur; &longs;ed hoc &longs;ufficit ad <lb/>hoc; vt motus dicatur violentus per def. </s> <s>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 type="main"> <s>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; 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 cadem naui <pb pagenum="135"/>&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 type="main"> <s>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; &longs;cilicet centrum terræ, idem punctum &longs;uperfi­<lb/>ciei terræ, & ip&longs;am pilam; 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>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><lb/>quis credat produci frigus ab igne? </s> <s>calorem à niue? </s> <s>lucem à tenebris? </s> <s><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 type="main"> <s>Illa quoqne 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 type="main"> <s>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 type="main"> <s>Vnum fortè e&longs;t, quod difficilius obiici pote&longs;t; &longs;it enim linea vertica­<lb/>lis AC, &longs;itque globus in A æqualiter impul&longs;us per lineas AD & AB; <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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus violentus habet caufam<emph.end type="italics"/>; quia de nouo e&longs;t, & tandem de&longs;init per <lb/>hypoth. </s> <s>1. igitur habet cau&longs;am per Ax.8.l.1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>I&longs;te impetus debet e&longs;&longs;e in mobili projecto &longs;ur&longs;um<emph.end type="italics"/>; 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 co e&longs;t etiam <lb/>impetus: 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 type="main"> <s>Diceret aliquis ab aëre extrin&longs;ecùs ambiente mobile ip&longs;um propelli; <lb/>&longs;ed contra, nam aër, & omne aliud medium re&longs;i&longs;tit potiùs quàm iuuet, vt <lb/>demon&longs;trauimus l. </s> <s>&longs;ecundo Th. 1. Nec dicas fui&longs;&longs;e mentem Ari&longs;totelis, <lb/>cum nobiles Peripatetici contrâ &longs;entiant; Albertus Magnus, Toletus, <lb/>Scaliger, Suarius, & recentiores; neque hoc negauit vnquam Ari&longs;tote-<pb pagenum="136"/>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>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; 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 type="main"> <s>Diceret fortè alius primam partem motus produci à potentiâ motri­<lb/>ce, &longs;ecundam verò ab entitate ip&longs;ius corporis; &longs;ed contrà; igitur corpus <lb/>e&longs;&longs;et cau&longs;a nece&longs;&longs;aria; igitur &longs;emper produceret. </s> <s>Dices &longs;emper producere <lb/>&longs;i determinetur, &longs;ed contrà; à quo determinatur ad producendam &longs;ecun­<lb/>dam partem? </s> <s>nihil e&longs;t enim applicatum, à quo determinari po&longs;&longs;it; Dices <lb/>accepi&longs;&longs;e determinationem; &longs;ed contrà; quid e&longs;t illa determinatio? </s> <s><lb/>Dices e&longs;&longs;e modum; igitur permanentem; igitur e&longs;t cau&longs;a motus per Ax. </s> <s><lb/>1. l. </s> <s>1. n. </s> <s>1. igitur e&longs;t impetus per def. </s> <s>3. l. </s> <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>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. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ille impetus e&longs;t vera qualitas Phy&longs;ica ab&longs;oluta<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>I&longs;te impetus producitur ab aliqua cau&longs;a<emph.end type="italics"/>; Probatur, quia e&longs;t de nouo; igi­<lb/>tur non e&longs;t à &longs;e per Ax. 8. l. </s> <s>1. igitur e&longs;t ab alio; igitur ab aliqua <lb/>cau&longs;a. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Producitur ab aliqua cau&longs;a extrin&longs;eca<emph.end type="italics"/>; 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; igitur e&longs;t cau­<lb/>&longs;a, per Ax. 11. l. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Producitur ab alio impetu<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus impre&longs;&longs;us mobili &longs;ur&longs;um con&longs;eruatur per aliquod tempus<emph.end type="italics"/>; Probatur, <pb pagenum="137"/>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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Con&longs;eruatur ab aliqua cau&longs;a extrin&longs;eca applicata<emph.end type="italics"/>; vt patet ex dictis, non <lb/>ab aëre; igitur à nullo corpore; igitur ab alia causâ in&longs;en&longs;ibili; igitur <lb/>illam e&longs;&longs;e oportet, & no&longs;&longs;e rerum omnium exigentias, & po&longs;&longs;e cuncta <lb/>producere; quippe con&longs;eruatio e&longs;t repetita productio; immò con&longs;erua­<lb/>re per actionem, per quam &longs;it res in tali loco, & tali tempore; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc multa colligi po&longs;&longs;unt.<emph.end type="italics"/></s><s> 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>Secundò, &longs;i nullus e&longs;&longs;et Deus, nullus e&longs;&longs;et motus <lb/>violentus; immò nec vllus naturaliter acceleratus. </s> <s>Tertiò, &longs;i impetus e&longs;­<lb/>&longs;et fluens vt motus, nullus e&longs;&longs;et motus violentus. </s> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; Probatur, quia &longs;i e&longs;&longs;ent plures natura­<lb/>lis deorsùm, quàm &longs;int vielenti &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s></p><p type="main"> <s><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>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus violentus non e&longs;t æquabilis<emph.end type="italics"/>; 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>cur ab hoc puncto &longs;patij potiùs, quàm ab alio? </s></p><pb pagenum="138"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc non con&longs;eruatur intactus impetus<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc nece&longs;&longs;e e&longs;t aliquid impetus destrui<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s></p><p type="main"> <s><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>quippe illa ratio, quæ probat de vno probat <lb/>de &longs;ingulis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 19.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s><lb/>3. l. </s> <s>2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 20.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Illa cau&longs;a non e&longs;t tantùm aër ambiens vt velunt aliqui<emph.end type="italics"/>; 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;truera; 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; e&longs;&longs;et enim eadem cau&longs;a applicata; igitur idem <lb/>& æqualis effectus; probatur &longs;ec<gap/>dò, 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/><gap/>men e&longs;t imperfecti&longs;&longs;unus; igitur non e&longs;t &longs;ufficiens ad de&longs;truendum im­<lb/>petum violentum, ni&longs;i longo tempore. </s> <s>Tertiò, globus &longs;ursùm projectus <lb/>a&longs;cendit, & deinde de&longs;cendit æquali tempore; igitur &longs;altem &longs;ingulis in­<lb/>&longs;tantibus de&longs;truitur vnus gradus impetus violenti æqualis primo gradui <lb/>innato; 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>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. </s> <s>& multis aliis pa&longs;&longs;im; atqui con&longs;er­<lb/>natur &longs;emper impetus naturalis innatus per Sch. </s> <s>Th.152.n.6.l.1.per Th. </s> <s><lb/>9. & Schol.Th.14. & Th.73.l.2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Illa cau&longs;a non e&longs;t eutitas corporis mobilis, vel ip&longs;a grauitas, di&longs;tincta &longs;cili­<lb/>cet ab impetu muato&longs;i quæ e&longs;t de quæ alias,<emph.end type="italics"/> probatur, quia non e&longs;&longs;et potior <lb/>ratiocur 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"/>ade&longs;t contrarius impetus de&longs;tructiuus co modo, quo explicuimus l. </s> <s>1. non <lb/>e&longs;t ponenda alia cau&longs;a de&longs;tructiua. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc nece&longs;&longs;e e&longs;t impetum violentum de&longs;trui ab impetu naturali innato<emph.end type="italics"/>; 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; &longs;ed mobilis entitas non e&longs;t cau&longs;a <lb/>de&longs;tructiua; nec etiamip&longs;a grauitas per Th.21. igitur impetus naturalis <lb/>innatus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc etiam ratio cur &longs;ingulis instantibus, &longs;eu æqualibus temporibus æqua­<lb/>liter de&longs;truatur<emph.end type="italics"/>; 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 cius <expan abbr="de&longs;truction&etilde;">de&longs;tructionem</expan>, nam perinde e&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>In cadem proportione retardatur motus violentus, in qua naturaiis accele­<lb/>ratur<emph.end type="italics"/>: probatur quia &longs;ingulis in&longs;tantibus æqualibus acquiritur æqualis <lb/>gradus impetus, vt &longs;æpè dictum e&longs;t &longs;uprà; 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>certè cum impetus innatus pugnet cum vio­<lb/>lento pro rata; 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"/><p type="main"> <s>Hinc inuertenda e&longs;t progre&longs;&longs;ionis linca; 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; <lb/>in illa primo in&longs;tanti decurritur primum &longs;patium AB, &longs;ecundo tempore <lb/>æquali BC, tertio CD, quarto DE: in hac vero prima parte acquiritur <lb/>&longs;patium FG &longs;ecunda æquali primæ GH, tertia HI, quarta IK; 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 acceletati; <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; vt con­<lb/>&longs;tat ex præcedèatibus 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 temperis &longs;en&longs;ibilibus iuxta hos 7.5. 3. 1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 28.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>&longs;it vnus gradus im­<lb/>petus innati; 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 erunr 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Si accipiantur &longs;patia æqualiain hac progre&longs;&longs;ione retardationis, e&longs;t inuer&longs;a <lb/>illius, quàm tribuimus &longs;uprà acceberationi, a&longs;&longs;umptis &longs;cilicet &longs;patiis æqualibus; <lb/>tiem &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"/>; quippe quemadmodum in progre&longs;&longs;ione <lb/>accelerationis decre&longs;cunt tempora; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc instantia initio huius metus &longs;unt minora &longs;icut initio met menatur alis <lb/>&longs;unt maiora; & &longs;ub finem in motu violente &longs;unt maiora, in naturali &longs;unt mi­<lb/>nora<emph.end type="italics"/>; quia &longs;cilicet hic acceleratur, ille retardatur: igitur velo­<lb/>catas accelerati cre&longs;cit; igitur &longs;i accipiantur &longs;patia æqualia, decre&longs;cit tem­<lb/>pus; at verò velocitas retardati decre&longs;cit, igitur a&longs;&longs;umptis &longs;patiis æquali­<lb/>bus, cre&longs;cit tempus; igitur &longs;i accipiatur &longs;patium, quod percurritur primo <lb/>in&longs;tanti huius motus, & deinde alia huic æqualia; 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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s></p><p type="main"> <s><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; quia cadem 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 type="main"> <s>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>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. </s></p><p type="main"> <s>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 ci­<lb/>dem parti mobilis in&longs;unt, communi qua&longs;i actione, vel exigentia indi­<lb/>ui&longs;ibiliter exigunt motum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc gradus omnes producti in eadem parte &longs;ubiecti &longs;unt inæquales in­<lb/>perfectione<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; immò, ni&longs;i hoc e&longs;&longs;et, nulla e&longs;&longs;et huiu&longs;modi <lb/>naturalis retardatio; nam producantut, &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>cur autem potiùs vnum dices quàm aliud? </s> <s>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s></p><p type="main"> <s><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>g. </s> <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;truitut tertio, atque ita deinceps. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc perfecti&longs;&longs;imus omnium gruduum ille e&longs;t qui de&longs;truitur vltimo in&longs;tan­<lb/>ti, de quo infrá<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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/>taum, &longs;itque vnum qua&longs;i principium perfectionis totius impetus; tùm in <pb pagenum="142"/>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; <lb/>producitur imperfectior ab cadem cau&longs;a in minoribus temporibus, & <lb/>de&longs;truitur perfectior ab eadem cau&longs;a in maioribus temporibus; & 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; & cum <lb/>idem impetus innatus &longs;it qua&longs;i principium crementi, vel accelerationis, <lb/>&longs;icut e&longs;t principium retardationis; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc finis motus naturalis omninò conuenit cum principio motus violenti; <lb/>& finis huius cum principio illius<emph.end type="italics"/>; quæcumque tandem progre&longs;&longs;io accipia­<lb/>tur; &longs;iue temporum æqualium in &longs;patiis inæquaiibus; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s></p><p type="main"> <s><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: Primo <lb/>in&longs;tanti, quo percurtit &longs;patium AB; haud dubiè, quando perueuit ad pun­<lb/>ctum G, habet 7. gradus impetus æquales, quia ante motum AB habebat <lb/>innatum; &longs;ed in motu illo fluunt 6. tempora æqualia, vt dictum e&longs;t; igitur <lb/>6. acquirit gradus impetus, quorum quidem vltimò acqui&longs;itus nullum <lb/>adhuc habuit motum; &longs;ed haud dubiè haberet, &longs;i vlteriùs hic motus pro­<lb/>pagaretur: his po&longs;itis imprimantur mobili in O 7.gradus impetus æqua­<lb/>les prioribus &longs;ursùm motu violento, per lineam OH; certè primo in&longs;tan­<lb/>ti motus, &longs;eu tempore æquali prioribus percurret ON, id e&longs;t 6. &longs;patiola; <lb/>quia licèt &longs;int 7.gradus; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 38.<emph.end type="center"/></s></p><p type="main"> <s><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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc &longs;i motus violentus, & naturalis durent æqualibus temporibus, &longs;patia <lb/>vtriu&longs;que erunt æqualia<emph.end type="italics"/>; con&longs;tat etiam ex dictis v.g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc, vt &longs;patia vtroque motu diner&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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 41.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; e&longs;t enim æquale &longs;patium; igitur æquale <lb/>tempus; igitur æqualis vtrobique impetus. </s> <s>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; &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 type="main"> <s>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; 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; 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; 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: præterea <lb/>mucro ferreus, quo &longs;agitta armatur, &longs;emper præire debet, cuius rei ratio­<lb/>nem afferemus infrà; 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, alteta tamen extremitas, quæ mouetur mo­<lb/>motu contrario alterius motum retardat; ad &longs;ecundam obiectionem <lb/>re&longs;pondebo Th.44. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; <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; igitur pote&longs;t tantùm a&longs;&longs;u-<pb pagenum="144"/>mi &longs;ubduplum maximæ; igitur velocitas motus &longs;it æqualis ma<gap/>d <lb/>dubiè &longs;patium duplum percurretur. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc benè à naturâ in&longs;titutum fuit impetum naturalem innatum &longs;emper <lb/>con&longs;eruari<emph.end type="italics"/>; alioqui violentus e&longs;&longs;et æquabilis, igitur nunquam de&longs;ineret: <lb/>quantum ab&longs;urdum! quale incommodum &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <s>g. <lb/></s> <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 eamdem <lb/>altitudinem a&longs;cenderet; 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; 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: 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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; 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; vult <lb/>enim motum acceleratum tran&longs;ire in æquabilem, cuius contrarium de­<lb/>mon&longs;trauimus &longs;uprà Th. 131, l. </s> <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 type="main"> <s>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; igitur præualet tandem re&longs;i&longs;tentia aëris: &longs;ed re&longs;pondeo id <lb/>rantùm 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theoroma<emph.end type="italics"/> 46.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; v.g. </s> <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; 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"/>con&longs;tat e<gap/> dictis; igitur rationes ictuum non &longs;unt, vt rationes &longs;egmen­<lb/>torum integri &longs;patij. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Vt in praxi determinentur rationes ictuum<emph.end type="italics"/>; 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; &longs;ecunda percurratur ED. 7. partium, tertia <lb/>DC 5. quarta CB 3; quinta BA 1. hoc po&longs;ito facilè erit determinare <lb/>rationes ictuum; nam in de&longs;cen&longs;u ictus &longs;unt vt velocitates, & hæ vt tem­<lb/>pora; igitur &longs;i AB percurritur in dato tempore, & AC in duobus prio­<lb/>ri æqualibus; 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>Igitur in a&longs;cen&longs;u ictus in F erit quintuplus, <lb/>ictus in E quadruplus in D triplus, &c. </s> <s>igitur ictus &longs;unt in ratione dupli­<lb/>cata &longs;patiorum facto &longs;patij initio à &longs;ummo puncto A. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; v. </s> <s>g. <lb/></s> <s>proiiciatur corpus graue ad altitudinem 48. pedum; vires &longs;unt iis æqua­<lb/>les, quas acquirit in de&longs;cen&longs;u ciu&longs;dem altitudinis 48. pedum; &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. </s> <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. </s> <s>48. ad v.g. </s> <s>100. id e&longs;t ferè vt 7. <lb/>ad 10. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cognitis etiam &longs;patiis cogno&longs;cetur tempus<emph.end type="italics"/>; &longs;it enim decur&longs;um idem &longs;pa­<lb/>tium 48. pedum motu violento &longs;ur&longs;um; idque v. </s> <s>g. </s> <s>tempore 2. &longs;ecundo­<lb/>rum, quod ferè cum experientia con&longs;entit; &longs;it aliud &longs;patium 100. tempus <lb/>primi motus e&longs;t ad tempus &longs;ecundi vt v. </s> <s>g. </s> <s>48. ad v. </s> <s>g. </s> <s>100. quia &longs;patia <lb/>&longs;unt vt quadrata temporum; igitur tempora vt radices 4. hinc vires &longs;unt <lb/>in ratione temporum; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 150.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>In vltimo contactu motus violenti nullus e&longs;t ictus, v. </s> <s>g. </s> <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; probatur <lb/>quia non tendit vlteriùs; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ex his &longs;atis facilè comparari po&longs;&longs;unt rationes pereu&longs;&longs;ionis,<emph.end type="italics"/> quæ infliguntur <pb pagenum="146"/>tùm ex ca&longs;u corporis grauis cadentis, tùm <gap/>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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s></p><p type="main"> <s><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/>ctror&longs;um? </s> <s>igitur non concurrunt ad communem motum, ni&longs;i dicatur <lb/>impetus innatus veleo 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc ad motum violentum impetus ab exteriore potentia mobili impre&longs;&longs;us <lb/>tantùm concurrit<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s></p><p type="main"> <s><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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 55.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; probatur, quia &longs;ecundo in&longs;tanti exigit motum &longs;ui &longs;ub­<lb/>jecti; igitur tolli etiam impedimentum; igitur per motum medij; igitur <lb/>priori in&longs;tanti in eodem mobili debet e&longs;&longs;e impetus; igitur produci ab <lb/>impetu organi; igitur & in organo ab impetu manus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s></p><p type="main"> <s><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; e&longs;t enim par vtrobique ratio; præterea agit in ip&longs;am <lb/>manum. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Impetus violentus producitur minor, quàm produceretur vno dunt ax at gra­<lb/>du aquali ip&longs;i impetui innato<emph.end type="italics"/>; quippe &longs;icut de&longs;truit &longs;ingulis in&longs;tantibu: <lb/>æqualibus vnum gradum; quia pugnat pro rata; ita pror&longs;us impedit, ne <pb pagenum="147"/>producatur vnus gradus &longs;ibi æqualis primo in&longs;tanti; cur enim duo po­<lb/>tiùs, quàm tres? </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Secundo &longs;tatim in&longs;tanti de&longs;truit alterum gradum<emph.end type="italics"/>: quippe e&longs;t cau&longs;a ne­<lb/>ce&longs;&longs;aria; igitur &longs;tatim primo in&longs;tanti exigit de&longs;tructionem; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc optimaratio illius insti<gap/>ti naturæ, quo factum e&longs;t, vt impetus innatus <lb/>numquam destruatur<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc corpus quod non grauitat, facilè proijcitur, vel impellitur<emph.end type="italics"/>: &longs;ic na­<lb/>uis aquis innatans, nubes in aëre libratæ; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc etiam è contrario corpus grauius difficiliùs &longs;ur&longs;um proijcitur<emph.end type="italics"/>: 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Concurrunt omnes illæ, quæ in&longs;unt eidem parti &longs;eu puncto mobilis comm<gap/><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Non ponitur tamen totus motus &longs;ecundo instanti, quem c<gap/>gunt pr<gap/><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc ille gradus motus quinon ponitur &longs;ecundo instanti respondet gradus <lb/>im petus qui destruitur<emph.end type="italics"/>; cum vterque habeat eamdem men&longs;uram, &longs;cilicet <lb/>im petum innatum. </s></p><pb pagenum="148"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; v.g. </s> <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; vnde dixi cau&longs;am partialem, quia <lb/>etiam exigitur ab aliis gradibus impetus, qui non de&longs;truuntur exigentiâ <lb/>communi; quippe impetus non exigit ni&longs;i pro &longs;ecundo in&longs;tanti; 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; 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 type="main"> <s>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; 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 per&longs;ectioni <lb/>cau&longs;æ; immò numquam de&longs;trueretur totus motus violentus, vt con&longs;tat; <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 &longs;ru&longs;trà &longs;i non exigeret, & &longs;ecundo <lb/>in&longs;tanti aliquis gradus impetus de&longs;truitur, ne &longs;it fru&longs;trà codem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ideo de&longs;truitur potiùs vnus gradus impetus quàm alius &longs;ecundo in&longs;tanti, <lb/>tertioque, &c. </s> <s>quia talis e&longs;t per&longs;ectionis<emph.end type="italics"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ille gradus impetus qui de&longs;truitur &longs;ecundo in&longs;tanti non concurrit ad motuns <lb/>tertij in&longs;tantis<emph.end type="italics"/>; quia non pote&longs;t concurrere ad motum ni&longs;i exigendo; at­<lb/>qui exigere tantùm pote&longs;t, quando e&longs;t; quod enim non e&longs;t non exigit, <lb/>&longs;ed motus tertij in&longs;tantis exigitur &longs;ecundo; &longs;ic enim tota res motus pro­<lb/>cedit vt impetus primo in&longs;tanti exigat motum pro &longs;ecundo; & &longs;ecundo <lb/>pro tertio; & tertio pro quarto, atque ita deinceps; igitur impetus ille <lb/>qui de&longs;tru&longs;tur; &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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 70.<emph.end type="center"/></s></p><p type="main"> <s><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; 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; igitur aliquid cau&longs;æ fru&longs;trà e&longs;t; igitur de&longs;trui debet; hinc <pb pagenum="149"/>de&longs;truitur etiam hic impetus per principium commune, ne aliquid &longs;it <lb/>fru&longs;trà. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Linea motus &longs;ur&longs;um determinatur à potentia motrice<emph.end type="italics"/>; probatur, quia hæc <lb/>determinat impetum productum in manu vel in organo; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; cum <lb/>enim detrahatur &longs;emper pars aliquota alicuius totius, tandem perueni­<lb/>tur ad vltimam; igitur &longs;int 100. gradus impetus violenti, quorum quili­<lb/>bet &longs;it æqualis impetui innato; certè cum temporibus æqualibus æqua­<lb/>lis gradus impetus de&longs;truatur; 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; igitur &longs;upere&longs;t vnus; igitur <lb/>duo illi impetus perueniunt tandem ad æqualitatem. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Vbivterque perueni&longs;&longs;et ad æqualitatem, non e&longs;&longs;et potior ratiocur mobile mo­<lb/>ueretur &longs;ursùm quàm deor&longs;um in&longs;tanti &longs;equenti<emph.end type="italics"/>; 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>1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 74.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Igitur in&longs;tanti quietis nullus e&longs;&longs;et ampliùs impetus violantus<emph.end type="italics"/>; cum enim <lb/>&longs;ingulis in&longs;tantibus de&longs;truatur vnus gradus, v. </s> <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; nec <lb/>pote&longs;t vel plùs, vel minùs de&longs;trui; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quies illa duraret tantùm vno in&longs;t anti,<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; igitur pro <pb pagenum="150"/>in&longs;tanti &longs;equenti moueritur; &longs;ed pro alio antecedente mouebatur; igi­<lb/>tur quies illa durat tantùm vno in&longs;tanti. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Quies illa non fit propter æliquam reflexionem, vt aliqui dicunt<emph.end type="italics"/>; quia nul­<lb/>la pror&longs;us e&longs;t reflexio, vbi nullum e&longs;t reflectens; atqui nullum e&longs;t refle­<lb/>ctens, vt patet, quia nullum e&longs;t corpus impediens motus propagationem; <lb/>licèt enim medium impediat, non tamen per modum reflectentis pro­<lb/>priè; immo vt dicemus infrà in puncto reflexionis nulla datur quies; &longs;ed <lb/>motus reflexus &longs;ibi vendicat librum &longs;ingularem. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 80.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc reiicies aliquos æpud 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 type="main"> <s>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>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>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>Tertiò, quia violentus e&longs;t cuminnato 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Corpus quod non grauitat proiicitur &longs;ur&longs;um motu æquabili per &longs;e<emph.end type="italics"/>; patet, quia <lb/>nihil e&longs;t quod de&longs;truat ip&longs;um impetum; igitur &longs;emper moueretur, ni&longs;i <lb/>per accideens ab ip&longs;o medio eius motus retardaretur; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 82.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Noncre&longs;c it 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>ad quartam obiect. </s> <s>in di&longs;­<lb/>&longs;ert.l.2. adde quod tardiùs a&longs;cenderet, quàm de&longs;cenderet; 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"/><expan abbr="etiã">etiam</expan> vulgaribus repugnant; immò & cunctis ferè præmi&longs;&longs;is Theorematis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 83.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; Primò, quia non &longs;unt infinita in&longs;tantia, &longs;ed retarda­<lb/>tur tantùm &longs;ingulis in&longs;tantibus; Secundò in medio den&longs;iore minùs du­<lb/>rat; igitur non tran&longs;it per tot gradus tarditatis; 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 type="main"> <s>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; <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; nec e&longs;t quod ali­<lb/>qui confugiant ad di&longs;iunctiones; 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; 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;inodi <lb/>prædicata æqualitatis, vel inæqualitatis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 84.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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; nunquam certè <lb/>verum e&longs;&longs;et dicere fluxi&longs;&longs;e infinita; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s></p><p type="main"> <s><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. </s> <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"/><p type="main"> <s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>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; quod probant in pila <lb/>ex fi&longs;tula ænea &longs;ur&longs;um emi&longs;&longs;a, quæ <expan abbr="moior&etilde;">moiorem</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 type="main"> <s>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 type="main"> <s>Ob&longs;erua tertiò, non tantùm propter grauitationem con&longs;eruari impe­<lb/>rum 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 type="main"> <s>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 type="main"> <s>Ob&longs;erua quintò, finem motus violenti e&longs;&longs;e multiplicem, nullum ta­<lb/>men à natura in&longs;titutum; quippe potentia motrix, quæ agit ex appetitu <lb/>elicito, (vt vulgò aiunt,) &longs;eu cum cognitione, finem &longs;ibi proponit ad libi­<lb/>tùm; 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="fig13"></figure></s></p><pb pagenum="153"/><figure></figure><p type="main"> <s><emph type="center"/>LIBER QVARTVS,<emph.end type="center"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>DE MOTV MIXTO EX <lb/>duobus, vel pluribus rectis.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s>MOTVM mixtum eum e&longs;&longs;e non dico, qui <lb/>ex pluribus aliis motibus componatur; <lb/>&longs;eu mi&longs;ceatur; nec enim plures motus <lb/>&longs;imul e&longs;&longs;e po&longs;&longs;unt in eodem mobili; cùm <lb/>tantùm e&longs;&longs;e po&longs;&longs;it vno dumtaxat in&longs;tan­<lb/>ti vnica migratio ex loco in locum; nec plura loca <lb/>naturali virtute &longs;imul acquiri po&longs;&longs;unt; Igitur nec &longs;i­<lb/>mul e&longs;&longs;e duo motus; Itaque motus mixtus &longs;implex <lb/>e&longs;t, &longs;i con&longs;ideretur ratio, & linea motus; 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 eumdem 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 type="main"> <s><emph type="center"/><emph type="italics"/>DEFINITIO 1.<emph.end type="italics"/><emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>MOtus mixtus e&longs;t, qui &longs;equitur ex multiplici impetu ad eamdem, vel di­<lb/>uer&longs;as lineas determinato, vel eodem ad diuer&longs;as<emph.end type="italics"/>; hæc definitio cla­<lb/>ra e&longs;t; ob&longs;eruabis tantùm ad motum mixtum &longs;ufficere duplicem impe-<pb pagenum="154"/>tum ad eamdem lineam determinatam, deor&longs;um, v.g. </s> <s>in mobili proiecto; <lb/>nec enim e&longs;t motus purè naturalis, nec etiam violentus, vt con&longs;tat; igi­<lb/>tur mixtus. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Cum proiicitur corpus per lineam horizont alem, 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 type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><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"/>: <lb/>hæc hypothe&longs;is mille &longs;altem nititur experimentis; modò &longs;ufficiat quod <lb/>&longs;it; nam propter quid &longs;it, demon&longs;trabo. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Hypothe&longs;is<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; hæc hypothe&longs;is centies probata fuit; <lb/>nec in dubium reuocari pote&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><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"/>; 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>Ax.2. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Ille impetus qui cum alio ad eumdem 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 type="main"> <s><emph type="center"/><emph type="italics"/>Axioma<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Licèt &longs;int plures impetus in eodem mobili, non &longs;unt tamen plures &longs;imul li­<lb/>ueæ motus<emph.end type="italics"/>; ne mobile &longs;it &longs;imul in pluribus locis. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Po&longs;tulatum<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Licedt a&longs;&longs;ismere quamlibet coniugætionem motuum,<emph.end type="italics"/> v. </s> <s>g. </s> <s>vel duorum æ­<lb/>quabilium, vel alterius æquabilis, & alterius retardati, vel alterius æqua­<lb/>bilis, & alterius accelerati, vel alterius reterdati, & alterius accelera­<lb/>ti, &c. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Po&longs;tulatum<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Illa linea vocetur curua quæ con&longs;tat infinitis prope lateribus polygoni.<emph.end type="italics"/></s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus mixtus ex duobus æquabilibus æquælibus e&longs;t rectus<emph.end type="italics"/>; &longs;it enim mo-<pb pagenum="155"/>bile in A, &longs;itque impetus per AB, & alter æqualis per AD, motus mixtus <lb/>fiet per AE, a&longs;&longs;umpra fcilicet DE æquali, & parallela AB, quod probatur <lb/>per Th.137.l.1. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Linea AE e&longs;t diagonalis quadrati, quotie&longs;cumque vterque impetus e&longs;t æ­<lb/>qualis, & liueæ determinationum decu&longs;&longs;antur ad angulos rectos<emph.end type="italics"/>; probatur per <lb/>idem Th.137. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc de&longs;truitur aliquid impetus<emph.end type="italics"/>; 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; 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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Motus mi<gap/>s ex duobus æquabilibus inæqualibus est etiam rectus<emph.end type="italics"/>; &longs;it <lb/>enim mobile in A cadem 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. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Linea AF e&longs;t diagonalis rectanguli, quotie&longs;cunqne lineæ deterninationum <lb/>decu&longs;&longs;antur ad angulos rectos<emph.end type="italics"/>; probatur per idem Th.137. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s></p><p type="main"> <s><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 probatnm e&longs;t. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Hinc determinari pote&longs;t portio vtriu&longs;que impetus destructi,<emph.end type="italics"/> v.g. </s> <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. </s> <s><lb/>AF & compo&longs;itæ ex AC.AD. </s></p><p type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s></p><p type="main"> <s><emph type="italics"/>Aliquando impetus qui remanet in motu mixto est rationalis<emph.end type="italics"/>; 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; ad vtrumque v.g. </s> <s>&longs;i alter impetuum &longs;it 8.alter 6. <lb/>haud dubiè linea motus mixti erit 10. ad neuttum vt in diagonali qua­<lb/>drati, & in multis aliis; ad alterum denique v. </s> <s>g. </s> <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 type="main"> <s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s></p><p type="main"> <s><emph type="