s, repl. by , --> ]> Fabri, Honoré Tractatus physicus de motu locali 1646 Lyons la page 0000000026

TRACTATVS PHYSICVS DE MOTV LOCALI, IN QVO

EFFECTVS OMNES, QVI AD IMPETVM, Motum naturalem, violentum, & mixtum pertinent, explicantur, & ex principiis Phy&longs;icis demon&longs;trantur.

Auctore PETRO MOVSNERIO Doctore Medico:CVNCTA EXCERPTA

Ex prælectionibus R. P. HONORATI FABR, Societatis IESV.

LVGDVNI,Apud IOANNEM CHAMPION, in foro Cambij.

M. D C. XLVI.Cum Priuilegio Regis, & Approbatione Doctorum.

AMPLISSIMO, NOBILISSIMOQVE DOMINO,

D. PETRO DE SEVE, DOMINO DE FLECHERES, SANCTIORIS CONSILII REGIS Con&longs;iliario, in Lugdunen&longs;i Curia Prætori prima­rio, & &longs;ecundùm Mercatorum Præpo&longs;ito, &c.

PETRVS MOVSNERIVS,

TIBI alterum no&longs;træ Philo&longs;o­phiæ fœtum in&longs;cribo, cui iam primum in&longs;crip&longs;i (PRÆTOR AMPLISSIME) nempe idem e&longs;&longs;e debeo, quia tu &longs;emper idem es: non muta&longs;ti merita, non mu­tabo officia: multos non expo&longs;cam Patronos, qui iam omnium optimum, & meriti&longs;simum habeo; neo enim &longs;acra Philo&longs;ophiæ anathemata rudi, & ru­&longs;tico muro appendam, quæ ex &longs;acro tholo templi Themidos amœniter pendent: Nec leuem toti rei li­terariæ iniuriam inferrem, &longs;i alium illi, quàm li-teratum Mecænatem accer&longs;erem: & verò Tracta­tum hunc de Motu Locali, alteri quàm tibi in&longs;cri­bere non debui, cuius imperia Ludgunen&longs;is orbis, po­tiùs quàm vrbis, componunt: Tu prudens Intelli­gentia, huic orbi &longs;emper a&longs;si&longs;tis; ita motibus in­uigilas, vt quieti publicæ con&longs;ulas, remque ita pu­blicam admini&longs;tras, vt &longs;ingulis commoda procures: Cæterùm dubitare non po&longs;&longs;um, quin hunc meu&mtail; quantulumcumque conatum, fidemque meam ia&mtail; tibi &longs;emel oppigneratam, & nunc altero voto peni­tus ob&longs;trictam, æqui bonique &longs;is con&longs;ulturus, Val&etail;.

PRÆFATIO.

NIHIL habeo præfari (Beneuole Lector) in gratiam huius tractatus de Motu Locali, cuius amœnitatem & vtilitatem, rerum co­piam & &longs;yluam, tuo gu&longs;tui & iudicio re­linquo: Multi &longs;anè hactenus in hac mate­ria feliciter de&longs;udarunt; & quidem præ cæteris magnus ille Galileus, qui mirificâ, & ferè diuinâ ingenijacie, motum localem eò perduxit, quò mortalium nemo per­duxerat; quia tamen multa omi&longs;it, quæ ad motum &longs;pe­ctant, vt nemo ne&longs;cit; nec ex principijs Phy&longs;icis mira­biles illos effectus demon&longs;trauit, &longs;ed tantùm certis qui­bu&longs;dam proportionibus ex geometricis addixit; vt Phy­&longs;icæ con&longs;ulamus, aliam inimus viam: Geometriam qui­dem adhibemus, ad explicandas, exponenda&longs;que præ­dictas illas proportiones, quæ motibus in&longs;unt; &longs;ed effe­ctus illos prædictis proportionibus affixos ad principia Phy&longs;ica reducimus; id e&longs;t, cùm &longs;upponamus quòd &longs;int, propter quid &longs;int demon&longs;tramus: in votis erat motus omnes vno volumine complecti; id e&longs;t effectus omnes cuiu&longs;uis potentiæ motricis; tres enim agno&longs;cimus hu­iu&longs;modi potentias: primam naturalem voco, quæ e&longs;t grauium: alteram animalem, quæ e&longs;t animantium: ter­tiam mediam, quæ ten&longs;orum e&longs;t vel compre&longs;&longs;orum: In hoc tractatu tùm à motu progre&longs;&longs;iuo animantium, tùm ab alijs motibus, qui in animato corpore, neruorum & mu&longs;culorum opera fiunt, penitus ab&longs;tinemus; cùm &longs;ci­licèt eas notiones &longs;upponant, quæ huius loci e&longs;&longs;e non po&longs;&longs;unt, ab&longs;tinemus etiam à mirifica illa ten&longs;orum & compre&longs;&longs;orum vi, quæ mediæ illius virtutis e&longs;t; neque adhuc eò rem Phy&longs;icam adduximus; Sed hîc tantùm na­turam impetus con&longs;ideramus, motus naturalis affectio­nes, violenti, mixti ex rectis, reflexi, circularis, mixti ex circularibus, illius qui fit in planis inclinatis &longs;ur&longs;um & deor&longs;um, vibrationum funependuli, diuer&longs;arum im­pre&longs;&longs;ionum, centri percu&longs;&longs;ionis, &c. Fortè aliquis poten­tias machanicas de&longs;ideraret, lineas, motus, & cæle&longs;tes &longs;piras; &longs;ed hæ quidquid phy&longs;icum habent, &longs;ingulari tra­ctatui de corpore cæle&longs;ti, reliqua verò A&longs;tronomiæ con­cedunt: potentiæ mechanicæ ad Staticam pertinent, qua­re illarum tantùm phy&longs;icum principium in hoc tractatu explicamus, lineæ motus nihil phy&longs;icum habent. Quare ad vitandam confu&longs;ionem ad Mathe&longs;im illas remittimus, cuius non modicam facient acce&longs;&longs;ionem; igitur &longs;ecun­dum Tomum de motu locali non expectabis, qui ne cuncta quidem, quæ ad motum &longs;pectant comprehende­ret, &longs;ed huic &longs;tatim Metaphy&longs;icam demon&longs;tratiuam &longs;ub­necto. Cæterùm de &longs;ubtili&longs;&longs;imo i&longs;torum omnium inuen­torum auctore nihil dicam, qui cum ægrè tulerit paucula illa quæ in prima tractatu præfatus &longs;um, os mihi peni­tus ob&longs;truxit: omitto etiam quæ in me quidam iniquè certè rerum æ&longs;timatores iactarunt: reponere po&longs;&longs;em cum fænore; &longs;ed nos talem con&longs;uetudinem non habemus; de­dici hactenus pati iniurias, non inferre; quod non modò moralis Philo&longs;ophia, &longs;ed præ&longs;ertim Chri&longs;tiana Religio me docet.

Vnum e&longs;t, de quo te monitum velim (Amice Lector) opu&longs;culum i&longs;tud non &longs;ine aliquot erratis edi potui&longs;&longs;e, præ&longs;ertim cùm in a&longs;&longs;ignandis cuilibet figuræ &longs;uis chara­cteribus &longs;æpiùs peccatum &longs;it; operas excu&longs;abis in rebus Geometricis minimè ver&longs;atos: auctor tibi &longs;um, vt errata, quæ fideliter adnotaui ca&longs;tiges, vt deinde cum maiore gu&longs;tu Librum hunc perlegere po&longs;&longs;is.

SYNOPSIS LIBRORVM

huius tractatus.

TABELLE WAR HIER

SYNOPSIS AMPLIOR.

BREVISSIMAM huius operis Epitomem hîc habes (Amice Lector) quam ex The&longs;ibus no&longs;tri Philo&longs;ophi huc traduxi, quæ tibi ampli&longs;&longs;imi indicis loco erit.

De Impetu.

1. IMPETVS e&longs;t qualitas exigens motum &longs;ui &longs;ubiecti: datur impetus; quia non pote&longs;t e&longs;&longs;e alia cau&longs;a exi­gitiua motus: adde quòd, potentia motrix e&longs;t acti­ua; igitur aliquid producit, &longs;ed non aliud quàm impetum, vt con&longs;tat ex dictis de motu: e&longs;t aliquid di&longs;tinctum à &longs;ub&longs;tantia mobilis, quæ pote&longs;t e&longs;&longs;e &longs;ine impetu: non e&longs;t modus, quia di&longs;tinguitur ab effectu &longs;uo formali &longs;ecundario: impetus non producitur in eo mobili, quod moueri non pote&longs;t à potentia mo­trice applicata: & produci tantùm pote&longs;t, vel in omni parte, vel in nulla; alioquin e&longs;&longs;et fru&longs;trà; & gratis ponitur ne&longs;cio quis impe­tus inefficax.

2. Primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, ne &longs;imulim­petus &longs;it in duobus locis. Impetus productus ad extra non produci­tur à quantitate, nec virtute re&longs;i&longs;titiua, nec ab alio, quàm ab impe­tu, qui maximè e&longs;t cau&longs;a connaturalis alterius impetus: agit tan­tùm ad extra, vt tollat impedimentum: hinc, cùm pro diuer&longs;a applicatione &longs;it diuer&longs;um impedimentum, modò plùs, modò minùs agit; maximè verò, cum maximum e&longs;t impedimentum: hinc ictus per lineam perpendicularem forti&longs;&longs;imus e&longs;t: portò omnes partes impetus agunt ad extra actione communi.

3. Impetus inten&longs;us producere pote&longs;t remi&longs;&longs;um, minotis mobi­lis in maiore; & remi&longs;&longs;us inten&longs;um, maioris mobilis in minore, vt patet; æqualis æqualem, æqualis mobilis in æquali, modò &longs;it debi-ta applicatio, cum maximo impedimento, quod reuerâ tunc e&longs;t, cùm linea directionis connectit centra grauitatis vtriu&longs;que. Datur impetus alio impetu perfectior, & imperfectior, &longs;ine quo non po­te&longs;t explicari natura vectis: itaque dato quocunque dari pote&longs;t per­fectior, & imperfectior: quia dato quocunque motu pote&longs;t dari ve­locior, & tardior.

4. Propagatur impetus vniformiter tantùm, cùm omnes partes corporis mouentur moctu recto æquali: ibi enim e&longs;t æqualis cau&longs;a, vbi e&longs;t æqualis effectus: in motu circulari applicata potentia cen­tro vectis, producitur æqualis perfectionis versùs circunferentiam, & inæqualis numerus; applicata verò potentia circunferentiæ, pro­ducitur æqualis numerus, &longs;ed inæqualis perfectionis versùs cen­trum; quia potentia non pote&longs;t producere immediatè perfectiorem, & imperfectiorem in infinitum: eadem potentia nece&longs;&longs;aria æquali­bus temporibus, & ii&longs;dem cireun&longs;tantiis, producit æqualem impe­tum, & inæqualibus inæqualem: e&longs;t enim hæc ratio cau&longs;æ nece&longs;­&longs;ariæ.

5. Impetus innatus e&longs;t tantùm determinatus ad lineam perpen­dicularem deor&longs;um; alioquin &longs;i ad aliam determinari po&longs;&longs;et, primo e&longs;&longs;et æqualis motus per inclinatam, & perpendicularem; corpus graue mi&longs;&longs;um per lineam inclinatam ab eo non declinaret; imò im­petus &longs;emel productus (&longs;i liberum e&longs;&longs;et medium) non de&longs;trueretur: quæ omnia phy&longs;icis hypothe&longs;ibus repugnant: omnis alius impetus, etiam acqui&longs;itus motu naturali deor&longs;um, e&longs;t indifferens ad omnem lineam, ad vitanda infinita ferè naturæ incommoda.

6. Impetus indifferens determinatur ad lineam multis modis: primò, à potentia motrice: &longs;ecundò, ab impetu: tertiò, ab alio impe­tu concurrente; quartò, ab obice occurrente: quintò, ab ip&longs;o appli­cationis diuer&longs;o modo: quæ omnia clara &longs;unt: hinc duo impetus ad motum mixtum &longs;æpè concurrunt, quod &longs;emper fit, ni&longs;i determina­tiones &longs;int oppo&longs;itæ ex diametro. Impetus e&longs;t capax inten&longs;ionis; quia aliquando de&longs;truitur ex parte: eius exten&longs;io commen&longs;uratur exten&longs;ioni mobilis; quod etiam cæteris qualitatibus commune e&longs;t: impetus productus non con&longs;eruatur à cau&longs;a primò productiua, à qua etiam &longs;eparatus exi&longs;tit.

7. Impetus non e&longs;t contrarius alteri ratione entitatis; quia qui­libet cum quolibet in eodem &longs;ubiecto coëxi&longs;tere pote&longs;t: pugnat tamen vnus cum alio ratione determinationis: hinc vnus impetus pugnat cum alio ratione lineæ motus: hinc vnus videtur de&longs;trui ab alio; quanquam impetus tantùm de&longs;truitur, cùm e&longs;t fru&longs;trà: hinc, &longs;i e&longs;&longs;et tantùm vnicus in eodem mobili, & liberum e&longs;&longs;et medium, nunquam de&longs;trueretur nec vnquam dici po&longs;&longs;et functus &longs;uo mune­re; quod omninò gratis dicitur.

8. Hinc, &longs;i &longs;int tantùm duo impetus in eodem mobili æquales verbi gratia, vel ad eandem lineam determinantur, vel ad diver&longs;as; &longs;i ad eandem, nihil impetus de&longs;truitur, &longs;ed e&longs;t duplò velocior mo­tus; &longs;i ad diuer&longs;as, vel &longs;unt oppo&longs;itæ ex diametro, vel concurrentes faciunt angulum; &longs;i primum, vterque de&longs;truitur impetus; &longs;i &longs;e­cundum, de&longs;truitur aliquid illius, quod determinabimus in­frà. Impetus innatus nunquam de&longs;truitur: dici po&longs;&longs;et grauitas ab­&longs;oluta; &longs;altem nihil e&longs;t, quod di&longs;tingui ab illa probare po&longs;&longs;it. Porrò nunquam de&longs;truitur; quia nunquam e&longs;t fru&longs;trà; quippe eius finis, vel v&longs;us, non e&longs;t tantùm motus deor&longs;um, &longs;ed grauitatio, &longs;eu ni&longs;us quidam deor&longs;um. Sed de grauitate aliàs.

De motu naturali deor&longs;um.

1. DAtur motus naturalis grauium deor&longs;um ab intrin&longs;eco, quippe non pote&longs;t e&longs;&longs;e, vel à vi tractrice terræ vel fila­mentis quibu&longs;dam, vel materia quadam tenui expultrice. Eius finis e&longs;t globi terre&longs;tris compactio, &c. E&longs;t autem motus naturalis ab impetu: primò, quia eius acceleratio &longs;ine impetu explicari non po­te&longs;t: &longs;ecundò, quia, cùm graue deor&longs;um cadens imprimat impetum in corpore occcurrente, certè debet habere impetum: nec alio ar­gumento mihi probabis, Solem e&longs;&longs;e lucidum, ignem calidum.

2. Motus hic e&longs;t naturaliter acceleratus, &longs;cilicet, ab intrin&longs;eco; patet experientiâ. Ratio e&longs;t: quia, cùm in libero medio non impe­diatur motus, & impetus productus primo in&longs;tanti non con&longs;erue­tur &longs;ecundo à cau&longs;a primò productiua, &longs;ed ab alia, &longs;itque ip&longs;a mo­bilis &longs;ub&longs;tantia cau&longs;a nece&longs;&longs;atia; certè &longs;ecundo in&longs;tanti producit nouum impetum: idem dica de tertio, quarto, &c. igitur cre&longs;cit cau&longs;a motus; igitur & motus: quæ ratio clari&longs;&longs;ima e&longs;t: hinc æquali­bus remporibns æqualia acquiruntur velocitatis momenta; quia cau&longs;a nece&longs;&longs;aria æqualibus temporibus, æqualem effectum produ­cit: quid clarius?

3. Hinc non pote&longs;t cre&longs;cere hic impetus &longs;ecundùm porportio-nem duplicatam temporum, cùm cre&longs;cat &longs;ecundùm proportionem temporum, etïam ex mente Galilei: cre&longs;eit autem velocitas, vt im­petus; effectus, &longs;cilicet, vt cau&longs;a: idem dico de motu, ratione velo­citatis; quippe motus ip&longs;e e&longs;t &longs;ua velocitas: at verò ip&longs;a &longs;patia, quæ decurruntur illo motu, &longs;i cou&longs;ideretur crementum in in&longs;tan­&longs;tantibus, cre&longs;cunt iuxta progre&longs;&longs;ionem arithmeticam &longs;implicem, id e&longs;t, &longs;i primo in&longs;tanti, acquiritur vnum &longs;patium, &longs;ecundo acquiri­tur vnum &longs;patium, &longs;ecundo acquiruntur duo, tertio 3. quarto 4. at­que ita deinceps.

4. Hoc autem facilè pote&longs;t demon&longs;trari: quia, cùm velocitas cre&longs;­cat iuxta proportionem temporum, &longs;i primo in&longs;tanti &longs;it vnus gradus velocitatis, &longs;ecundo erunt duo, tertio tres, at que ita deinceps: igitur, &longs;i mobile cum vno gradu velocitatis acquirit vnum &longs;patium, certè cum duobus acquiret duo &longs;patia, cum tribus tria, atque ita dein­ceps: debet autem vera progre&longs;&longs;io crementorum a&longs;&longs;umi in &longs;ingulis in&longs;tantibus, quia reuerà &longs;ingulis in&longs;tantibus phy&longs;icis (nam de iis loquor) noua fit huius crementi acce&longs;&longs;io.

5. Quia tamen in&longs;tantia non &longs;unt &longs;en&longs;ibilia, vt Phy&longs;icæ con&longs;u­latur, quæ res &longs;en&longs;ibiles con&longs;iderat, a&longs;&longs;umi debent partes tempotis &longs;en&longs;ibiles, in quibus reuerâ progre&longs;&longs;io &longs;patiorum non e&longs;t arithmeti­ca &longs;implex; &longs;ed tam propè accedit ad hanc numerorum imparium, 1. 3. 5. 7. &c. quam Galileus excogitauit, vt &longs;ine &longs;crupulo hæc a&longs;­&longs;umi po&longs;&longs;it: hinc &longs;patia &longs;unt ferè vt temporum quadrata: dixi, ferè: nam e&longs;t paulò minor proportio, cùm tantùm finita &longs;int in&longs;tantia phy&longs;ica, quæ reuerà &longs;i infinita e&longs;&longs;ent in qualibet temporis &longs;en&longs;ibilis parte, haud dubiè &longs;patia e&longs;&longs;ent omninò in ratione duplicata tem­porum: &longs;ed, quia parum pro nihilo computatur, hanc progre&longs;&longs;io­nem Galilei deinceps v&longs;urpabimus in Phy&longs;ica.

6. Hinc ratio euidens maioris ictus inflicti à corpore graui, cùm ex maiori altitudine cadit. Sunt autem ictus, vt impetus; impetus, vt tempora; hæc demum, vt radices &longs;patiorum &longs;en&longs;ibi­liter quæ omnia con&longs;tant ex dictis. Impetus acqui&longs;itus in de&longs;cen&longs;u e&longs;t &longs;emper imperfectior, &longs;i a&longs;&longs;umantur &longs;ingula in&longs;tantia, quæ reuerâ &longs;unt &longs;emper minora; quia motus fit &longs;emper velocior: cùm graue de&longs;cendit in medio, quod re&longs;i&longs;tit, minùs aecuratè &longs;eruantur prædi­ctæ proportiones, quæ in vacuo modico accurati&longs;&longs;imè &longs;eruaren­tur.

7. Re&longs;i&longs;tentia medij non e&longs;t propter vllam formam improportio­nat am, qua&longs;i verò impetus &longs;it forma improportionata aëri: &longs;ed in duobus præ&longs;ertim con&longs;i&longs;tit; primò, eò quòd medium detrahat ali­quid grauitationis corporis grauis; &longs;ecundò, eò quòd partes medij aliquam implicationem habeant, quæ &longs;olui non pote&longs;t &longs;ine aliqua compre&longs;&longs;ione, vel ten&longs;ione; vtraque autem re&longs;i&longs;tit impetui: quod &longs;pectat ad primum, &longs;i medium &longs;it æqualis grauitatis cum ip&longs;o cor­pore, detrahitur tota grauitatio, &longs;i &longs;ubduplæ &longs;ubduplum, &c. de quo aliàs.

8. Hinc corpus graue per medium rarius, cæteris paribus, fa­cilè de&longs;cendit; non tamen ex re&longs;i&longs;tentia medij cognita, pote&longs;t co­gno&longs;ci proportio grauitatis vtriu&longs;que, propter &longs;ecundum caput, ex quo etiam petitur re&longs;i&longs;tentia. Idem corpus cum eodem medio comparatum, habet tres coniugationes: nam, vel e&longs;t grauius, vel­e&longs;t grauius, vel æquè graue, vel minùs. Sunt etiam tres aliæ con­iugationes, &longs;cilicet, eiu&longs;dem mobilis cum diuer&longs;is mediis, duorum mobilium cum eodem medio, duorum mobilium cum duobus mediis.

9. Figura corporis grauis deor&longs;um cadentis motum vel retardat vel accelerat; retardat quidem, &longs;i plures partes medij amouendæ &longs;unt vel pauciores velociori motu; accelerat è contratio: hinc idem corpus parallelipedum iuxta tres diuer&longs;os &longs;itus, triplici motu diuer­&longs;o de&longs;cendere pote&longs;t: hinc ratio, cur acuminata tam facilè de&longs;cen­dant. Cubus, qui de&longs;cendit, imprimit aëri velociorem motum, quàm ip&longs;e habeat; & quò maior e&longs;t eius &longs;uperficies, eò velociorem.

10. Duo globi, vel cubi eiu&longs;dem materiæ æquè velociter de&longs;­cendunt: ratio e&longs;t, quia, licèt maioris vires habeant maiorem pro­portionem ad molem aëris re&longs;i&longs;tentis, quàm vires minoris ad alte­ram aëris molem, quæ proprium illius motum retardat, cùm tamen aër, qui re&longs;i&longs;tit maiori cubo, debeat amoueri velociori motu, quàm aër, qui re&longs;i&longs;tit minori, &longs;itque eadem proportio re&longs;i&longs;tentiæ ratione motus, minoris ad maiorem, quæ e&longs;t ratione molis, maioris ad mi­norem; certè ratio compo&longs;ita vtriu&longs;que erit eadem in vtroque cu­bo: igitur æqualiter de&longs;cendet vterque.

11. Si tamen &longs;int diuer&longs;æ materiæ, hand dubiè, qui con&longs;tat leuio­ri materia, tardiùs de&longs;cendet; quia eius vires habent minorem proportionem ad re&longs;i&longs;tentiam. Corpu&longs;cula etiam ex graui&longs;&longs;ima ma­teria tardi&longs;&longs;imè de&longs;cendunt: tum, quia à filamentis illis, quibus par­tes aëris implicantur, facilè detinentur; analogiam habes in lapil­lo, qui ab araneæ tela intercipitur: tum, quia, cùm lati&longs;&longs;imam ali­quando habeant &longs;uperficiem pro modica mole, minimam habent proportionem virium ad re&longs;i&longs;tentiam: tùm denique, quia, cùm modico impetu agitari po&longs;&longs;int ab aëre mobili, vnus motus alium impedit.

12. Singulis in&longs;tantibus motus naturaliter accelerati cre&longs;cit re&longs;i&longs;tentia; quia, cùm motus cre&longs;cat, æqualibus temporibus, plures partes medij occurrunt; cre&longs;cunt tamen vires in eadem proportio­ne, &longs;cilicet, impetus: igitur non mutatur progre&longs;&longs;io motus. Hinc colligo, contra Galilæum, motum rectum ex naturaliter accelerato nunquam fieri æquabilem: dixi motum rectum; quia motus corpo­rum cœle&longs;tium ex accelerato factus e&longs;t æqualis.

De motu violento &longs;ur&longs;um.

1. MOtus violentus &longs;ur&longs;um vulgò dicitur e&longs;&longs;e à principio ex­trin&longs;eco. Triplici modo accidere pote&longs;t: primò, &longs;i reuerà imprimatur impetus ab extrin&longs;eco, vt, cùm mitto lapidem &longs;ur&longs;um: &longs;ecundò, &longs;i corpus deor&longs;um cadens deinde reflectatur &longs;ur&longs;um; tunc autem nihil e&longs;t ab extrin&longs;eco, ni&longs;i determinatio noua, quæ e&longs;t à cor­pore reflectente: tertiò, &longs;i terra vtrinque e&longs;&longs;et peruia; nam lapis haud dubiè non &longs;i&longs;teret in centro, &longs;altem po&longs;t primum de&longs;cen&longs;um; igitur a&longs;cenderet per eandem lineam; nullum tamen e&longs;t principium ex­trin&longs;ecum; igitur motus violentus dicit tantùm motum &longs;ur&longs;um corporis grauis.

2. Dari autem motum violentum, dubium e&longs;&longs;e non pote&longs;t, qui &longs;upponit impetum, vel impre&longs;&longs;um ab extrin&longs;eco, vel in de&longs;centu acqui&longs;itum, qui reuerâ ine&longs;t ip&longs;i mobili, cùm ip&longs;um medium hunc motum potiùs impediat, quàm iuuet: hinc, &longs;i nullus e&longs;&longs;et impetus extrin&longs;ecus, vel acqui&longs;itus, nullus e&longs;&longs;et motus violentus; quia im­petus innatus illius cau&longs;a e&longs;&longs;e non pote&longs;t. Portò hic motus non e&longs;t acceleratus, nec æqualis, alioquin nunquam rediret deor&longs;um mobile.

3. Hinc nece&longs;&longs;ariò e&longs;t retardatus: igitur de&longs;truitur impetus, non quidem ab ip&longs;a medij re&longs;i&longs;tentia; quippe idem medium non magis re&longs;i&longs;tit motui &longs;ur&longs;um, quàm motui deor&longs;um, vt patet: igitur de&longs;trui­tur ille impetus motus violenti ab impetu innato aliquo modo; non quidem vt à contrario ratione entitatis, &longs;ed ratione determinatio­nis: cùm enim impetus innatus exigat motum deor&longs;um, & alius &longs;ur­&longs;um: hic quidem præualet, attamen fru&longs;trà e&longs;t, ratione gradus æqualis impetui innato: igitur de&longs;truitur ille gradus illo in&longs;tanti.

4. Hinc &longs;ingulis temporibus æqualibus de&longs;truitur gradus impe­tui innato; e&longs;t enim eadem ratio pro omnibus: igitur temporibus æqualibus de&longs;truitur æqualis impetus: igitur amittit ille motus æqualia velocitatis momenta: igitur e&longs;t naturaliter retardatus: igi­tur iuxta eam proportionem decre&longs;cit motus violentus, iuxtaquam cre&longs;cit naturalis: igitur dici debent de hac progre&longs;&longs;ione retardatio­nis, quæ dicta &longs;unt de illa progre&longs;&longs;ione accelerationis.

5. Hinc impetus imperfectior initio de&longs;truitur: quia, cùm motus ille &longs;it velocior initio, in&longs;tantia &longs;unt minora: atqui minori tempore minùs retardatur: igitur inperfectior impetus de&longs;truitur; cùm è contrario in motu acceleratio initio acquiratur imperfectior, quia in&longs;tantia &longs;unt maiora: vnde vides, gradus impetus e&longs;&longs;e heteroge­neos, & principium illud etiam in impetu valere, &longs;cilicet, &longs;ubiectum ita compleri ab vna forma, vt alterius homogeneæ non &longs;it ampliùs capax, &longs;altem naturaliter.

6. Hinc vltimus gradus impetus violentie&longs;t omnium perfecti&longs;­&longs;imus, vt con&longs;tat. Quie&longs;ceret vno in&longs;tanti mobile iactum &longs;ur&longs;um, &longs;i gradus vltimus violenti e&longs;&longs;et æqualis perfectionis, cum impetu in­nato: vbi enim ventum e&longs;&longs;et ad in&longs;tans æqualitatis, neutrum præ­ualere po&longs;&longs;et: igitur in&longs;tanti &longs;equenti e&longs;&longs;et quies: cùm tamen &longs;int diuer&longs;æ perfectionis, perfectior præualet: vter autem &longs;it perfectior, dicemus infrà.

7. Cum mobile &longs;ur&longs;um reflectitur, vel terra perforata &longs;uam lineam motus &longs;ur&longs;um versus oppo&longs;itam cœli plagam promouet, vel aliud æqualis ponderis, vel maioris, &longs;ur&longs;um mouet, tunc certum e&longs;t, inna­tum e&longs;&longs;e perfectiorem: &longs;i verò imprimitur ab alia potentia motrice, tunc etiam imperfectior e&longs;t impetu innato; nam inæqualis e&longs;t; alio­quin, &longs;i e&longs;&longs;et æqualis, &longs;imul e&longs;&longs;ent in eodem &longs;ubiecto duo gradus homogenei: præ&longs;tat autem e&longs;&longs;e imperfectiorem, quàm perfectio­rem, vt plura impetus puncta à potentia imprimantur; quòd mul­tum facit ad mouenda maiora pondera: hinc nullo in&longs;tanti quie&longs;­cunt proiecta &longs;ur&longs;um.

8. Tandiu durat &longs;en&longs;ibiliter de&longs;cen&longs;us globi proiecti &longs;ur&longs;um, quandiu durauit a&longs;cen&longs;us; e&longs;t enim eadem ratio: &longs;agittæ verò mi­nùs durat a&longs;cen&longs;us, quàm de&longs;cen&longs;us propter mixtionem materiæ. Si motus violentus e&longs;&longs;et æquabilis, percurreret proiectum &longs;patium ferè duplum eo tempore, quo retardato percurrit &longs;ubduplum: hinc &longs;onus tam citò auditur; quia propagatur cum particulis aëris æqua­bili ferè motu: e&longs;&longs;e autem &longs;patium ferè duplum, probatur ex eo, quòd &longs;patium motu æquabili decur&longs;um re&longs;pondet rectangulo; de­cur&longs;um verò motu retardato, re&longs;pondet triangulo, &longs;ubduplo rectan­guli: a&longs;&longs;umpto &longs;cilicet, æquali tempore.

9. Vites potentiæ proiicientis toto ni&longs;u re&longs;pondent velocitati acqui&longs;itæ in toto de&longs;cen&longs;u corporis proiecti; tantumdem enim impetus in de&longs;cen&longs;u acquiritur, quantùm in a&longs;cen&longs;u deperditur. Impetus primo in&longs;tanti, quo e&longs;t, agit, &longs;i e&longs;t aliquod impedimen­tum; e&longs;t enim cau&longs;a nece&longs;&longs;aria: primo in&longs;tanti motus aliquid im­petus de&longs;truitur: &longs;iue præce&longs;&longs;erit motus violentus, &longs;iue non præce&longs;­&longs;erit, corpus graue æquali motu deor&longs;um cadit: re&longs;i&longs;tentia aëris e&longs;t quidem maior initio; &longs;ed etiam &longs;unt maiores vires.

De motu in planis inclinatis.

1. PLanum inclinatum e&longs;t &longs;ur&longs;um, vel deor&longs;um: in hoc de&longs;cen­dit corpus graue, ni&longs;i fortè retineatur ab a&longs;peritate, vel pro­pria, vel ip&longs;ius plani: impeditur autem motus naturalis in plano prædicto, quia impeditur eius linea: ideò e&longs;t tardior hic motus in plano inclinato, quàm in perpendiculari: in ea porrò porportione e&longs;t tardior, in qua perpendiculum e&longs;t minus linea inclinata, eiu&longs;dem &longs;cilicet, altitudinis; quippe eò tardior e&longs;t, quò magis impeditur, & magis impeditur, quò maius &longs;patium decurrendum e&longs;t, ad acqui­rendam eandem altitudinem: igitur eadem e&longs;t proportio impe­dimenti, quæ &longs;patij, &c.

2. Hinc motus &longs;unt vt lineæ permutando: hinc mobile de&longs;cendit per &longs;e in prædicto plano: licet enim motus impediatur, non tamen tous, impetus, qui acquiritur in eodem plano e&longs;t imperfectior ac­qui&longs;ito in perpendiculari in eadem proportione; nam impetus &longs;unt vt motus: hinc pote&longs;t perfectio impetus imminui in infinitum, cùm po&longs;&longs;it e&longs;&longs;e in infinitum linea magis, ac magis inclinata: igitur mo­tum imminui po&longs;&longs;e in infinitum, non tantùm ex vecte, &longs;ed etiam ex planis inclinatis haberi pote&longs;t.

3. Hinc producit impetum imperfectiorem impetus acqui&longs;itus in hoc eodem plano, quàm acqui&longs;itus in perpendiculari, æqualibus &longs;cilicet temporibus, quia cau&longs;a imperfectior imperfectiorem pro­ducit effectum: motus in plano inclinato deor&longs;um e&longs;t acceleratus iuxta eandem proportionem, iuxta quam acceleratur in perpendi-culo: tempora, quibus pereurruntur perpendiculum, & linea plani inclinati, &longs;unt vt lineæ; &longs;patia autem, quæ in prædictis lineis acqui­runtur æqualibus temporibus, &longs;unt vt motus, id e&longs;t, vt lineæ per­mutando, vt patet ex dictis.

4. Ex his concludo, nece&longs;&longs;ariò per plana omnia eiu&longs;dem altitu­dinis acquiri eandem velocitatem, quantumuis a&longs;&longs;umantur longi&longs;­&longs;ima, modò &longs;cilicet perpendicula &longs;int &longs;emper parallela. Hinc habes apud Galileum, per omnes chordas circuli erecti de&longs;cen&longs;um fieri æqualibus temporibus. Vires, quæ &longs;u&longs;tinent pondus in plano in­clinato per lineam plano parallelam, &longs;unt ad eas, quæ &longs;u&longs;tinent in per­pendiculo, vt lineæ permutando; quia debent adæquare impetum, qui producitur, tùm in plano inclinato, tùm in perpendiculo.

5. Porrò minùs grauitat in ip&longs;um planum inclinatum corpus gra­ue, quàm in planum horizontale: e&longs;t autem grauitatio in horizonta­li, &longs;eu Tangente, ad grauitationem in inclinata, &longs;eu &longs;ecante, vt ip &longs;æ lineæ permutando: quod facilè demon&longs;tramus. Proiicitur mobile faciliùs per inclinatum planum &longs;ur&longs;um, quàm per ip&longs;am perpendi­cularem: patet experientia: euius ratio e&longs;t, quia minùs re&longs;i&longs;tit im­petus innatus, cuius minor e&longs;t ni&longs;us per inclinatam, vt con&longs;tat ex dictis.

6. Illæ vires, quæ &longs;ufficiunt ad eum motum &longs;ur&longs;um in perpendi­culo, &longs;ufficiunt ad motum &longs;ur&longs;um in plano inclinato eiu&longs;dem alti­tudinis: quia illæ vires &longs;ufficiunt ad a&longs;cen&longs;um, quæ acquiruntur in toto de&longs;cen&longs;u: &longs;ed in de&longs;cen&longs;u inclinatæ, & perpendiculi acquirun­tur vires æquales, id e&longs;t, velocitas æqualis, vt dictum e&longs;t &longs;uprà. Om­nia puncta plani inclinati rectilinei, imò & horizontalis, &longs;unt di­uer&longs;æ inclinationis: in iis tamen planis inclinatis quæ vulgò a&longs;&longs;u­muntur, non mutatur &longs;en&longs;ibiliter inclinatio.

7. Hinc minùs de&longs;truitur impetus in plano inclinato &longs;ur&longs;um, quàm in perpendiculo; quia diutiùs durat: cùm enim minùs ac­quiratur in de&longs;cen&longs;u, vt dictum e&longs;t, minùs etiam de&longs;truitur in a&longs;­cen&longs;u: hinc accedit propriùs hic motus ad æquabilem: in eodem plano rectilineo pote&longs;t e&longs;&longs;e a&longs;cen&longs;us, & de&longs;cen&longs;us, versùs eandem partem: tale e&longs;&longs;et planum horizontale, in cuius vnico tantùm pun­cto nulla e&longs;t inclinatio: in quolibet puncto huius plani e&longs;t &longs;ingu­laris inclinatio, vt patet, quæ e&longs;t ad perpendiculum, vt Tangens ad &longs;ecantemé&longs;tque eadem proportio motuum.

8. Corpus graue in &longs;uperficie quadrantis caua, deor&longs;um cadit motu naturaliter accelerato; quia &longs;ingulis in&longs;tantibus accedit nouus impetus; non tamen æqualibus temperibus, acquiruntur æqualia velocitatis momenta; quia in &longs;ingulis punctis quadrantis, e&longs;t diuer­&longs;a tangens; igitur mutatur progre&longs;&longs;io accelerationis, quæ certè ma­jor e&longs;t initio, & &longs;ub finem minor; quia initio tangentes acce­dunt propriùs ad perpendiculum, & &longs;ub finem ad horizonta lem.

9. De&longs;cendie etiam in &longs;uper&longs;icie conuexa globi erecti motu ac­celerato; initio quidem, in minore proportione; &longs;ub finem, in maio­re; vnde e&longs;t inuer&longs;a prioris: pote&longs;t etiam de&longs;cendere corpus graue v&longs;que ad centrum terræ motu accelerato, in &longs;uperficie conuexa &longs;e­micirculi: &longs;i &longs;uperficies terræ e&longs;&longs;et læuigati&longs;&longs;ima, corpus proje­ctum moueretur in ea motu æquabili, nec de&longs;trueretut impetus im­pre&longs;&longs;us, vt con&longs;tat; pote&longs;t quoque de&longs;cendere per &longs;piralem: &longs;unt in­finita plana curua, in quibus faciliùs moueri pote&longs;t, quam in ho­rizontali recta.

De motu mixto ex rectis.

1. DAri motum mixtum ille non dubitat, qui di&longs;cum proiicit. Mixtus ex duobus rectis æquabilibus e&longs;t rectus, e&longs;t que diagonalis vtriu&longs;que: hinc de&longs;truitur aliquid impetus, iuxta pro­portionem differentiæ diagonalis, & vtriu&longs;que lateris &longs;imul &longs;ump­pti; quia, &longs;eilicet, e&longs;t fru&longs;trà: quò maior e&longs;t angulus, quem faciunt li­neæ determinationum, minor e&longs;t diagonalis; igitur plùs impetus de&longs;truitur, donectandem concurrant in oppo&longs;itas lineas, tunc enim totius impetus de&longs;truitur.

2. Qoum minor e&longs;t, vel acutior prædictus angulus, minùs impetus de&longs;truitur; quia diagonalis maior e&longs;t; donec tandem conueniant in eandem lineam, tunc enim nihil de&longs;truitur: datur de facto hic mo­tus in rerum natura; talis e&longs;t motus nauis à duobus ventis im pre&longs;­&longs;us; vel eiu&longs;dem partis aëris; imò & ip&longs;ius venti: motus mixtus ex duobus retardatis iuxta eandem progre&longs;&longs;ionem e&longs;t rectus; quia fit per hypothenu&longs;im triangulorum proporionalium: idem dico de duobus acceleratis.

3. Si mixtus &longs;it ex æquali, & accelerato, velex duobus accelera­tis in diuer&longs;a progre&longs;&longs;ione, vel ex duobus retardatis &longs;imiliter, fit per lineam curuam, vt patet: dum proiicitur corpus graue per horizon-talem in medio libero e&longs;t motus mixtus ex accelerato naturali, & retardato violento: e&longs;t enim acceleratus naturalis, cùm deor&longs;um deor&longs;um tendat qua&longs;i per gradus, &longs;eu diuer&longs;a plana inclinata.

4. Non tamen impetus acqui&longs;itus in eo motu e&longs;t eiu&longs;dem perfe­ctionis cum illo, qui acquiteretur in perpendiculari eiu&longs;dem longi­tudinis; &longs;ed tantùm eiu&longs;dem altitudinis: nam perinde cre&longs;cit ille impetus, atque cre&longs;ceret in diuer&longs;is planis inclinaris: impetus verò violentus in hoc motu retardatur; tùm, quia, &longs;i maneret idem, maior e&longs;&longs;er ictus &longs;ub finem iactus, quod e&longs;t ridiculum; nec e&longs;t, quòd aliqui dicant, ab aëre de&longs;trui, qui non minùs re&longs;i&longs;tit naturali, quàm vio­lento.

5. Adde, quòd e&longs;t duplex determinatio: igitur aliquid de&longs;trui de­bet, non acqui&longs;iti; igitur impre&longs;&longs;i: de&longs;trui autem non dicitur acqui­fitus, quòd, fcilicet, plùs de nouo accedat, quàm percat; e&longs;t enim ac­celeratus: adde, quòd non infligitur tantus ictus &longs;ub finem; igitur de&longs;truitur aliquid impetus, non acqui&longs;iti, eo modo, quo diximus; igitur impre&longs;&longs;i: ita tamen &longs;en&longs;im de&longs;truitur, vt pro æquabili per ali­quod &longs;patium qua&longs;i haberi po&longs;&longs;it.

6. Hinc mobile proiectum per horizontalem, ne primo quidem in&longs;tanti per horizontalem mouetur, alioqui non e&longs;&longs;et motus mix­tus: tardiùs cadit mobile ita proiectum in planùm horizontale &longs;ub­iectum, quàm cum &longs;ua &longs;ponte, ex eadem altitudine de&longs;cendit: cuius rei clari&longs;&longs;ima e&longs;t experientia: ratio e&longs;t; quia impetus acqui&longs;itus in hoc iactu non e&longs;t eiu&longs;dem perfectionis, cùm acqui&longs;ito in perpendi­cnlo: cùm proiicitur mobile per inclinatam &longs;ur&longs;um, mouetur motu mixto ex naturali æquabili, & violento retardato: patet prima pars; quia acceleratur tantùm naturalis deor&longs;um, &longs;altem in inclinata: &longs;e­cunda pars etiam patet; quia &longs;ub finem minor e&longs;t ictus.

7. Hinc linea motus e&longs;t curua: iuxta diuer&longs;am progre&longs;&longs;ionem de­&longs;truit ur hic impetus impre&longs;&longs;us: tùm pro diuer&longs;a inclinatione plani, cuius etiam hîc habetur ratio; nam &longs;ingulis in&longs;tantibus mutatur: tùm, quia modò plùs impetus e&longs;t fru&longs;trà, modò minùs; plùs certè, cùm linea determinationis impetus impre&longs;&longs;i facit obtu­&longs;iorem: atqui initio e&longs;t obtu&longs;ior; &longs;ub finem verò a&longs;cen&longs;us acu­tior.

8. A&longs;cen&longs;us proiecti per inclinatam diutiùs durat, quàm de&longs;­cen&longs;us, ratione ciu&longs;dem plani horizontalis; quia, &longs;cilicer, a&longs;­cen&longs;us longior e&longs;t, quàm de&longs;cen&longs;us: e&longs;t autem longior; quia, vt e&longs;&longs;et æqualis, nihil impetus impre&longs;&longs;i deberet de&longs;trui in a&longs;cen&longs;u porrò in de&longs;cen&longs;u e&longs;t motus mixtus ex accelerato naturali, & retardato violento, vt con&longs;tat ex dictis: iactus per incli­natam ad angulum 45. e&longs;t omnium maximus, ratione eiu&longs;dem plani horizontalis: clara e&longs;t experientia. Ratio e&longs;t: quia per verti­calem &longs;ur&longs;um, nihil acquiritur in plano horizontali, ex quo fit ia­ctus; nihil etiam per ip&longs;am horizontalem; igitur plùs acquiritur per illam, quæ maximè ab vtraque &longs;imul recedit.

9. Hæc ratio e&longs;t verè phy&longs;ica, geometrica nulla e&longs;t: hinc illi iactus æquale &longs;patium acquirunt in prædicto plano horizontali, qui fiunt per inclinatas æqualiter à prædicta inclinata ad ang. 45. di&longs;tantes. Cùm emitcitur mobile per inclinatm deor&longs;um, in libero medio, mouetur motu mixto ex naturali accelerato, & impre&longs;­&longs;o retardato, vt con&longs;tat ex dictis; ille autem primus accelera­tur per acce&longs;&longs;ionem impetus perfectionis quàm in iactu per ho­rizontalem; &longs;ed imperfectionis, quàm in perpendiculo: retarda­tur verò impetus minùs, quàm in iactu per horizontalem; plùs ve­rò, quàm in iactu per ip&longs;um perpendiculum, in quo nihil impetus de&longs;truitur.

10. Cùm è naui mobili &longs;ur&longs;um mittitur corpus graue, e&longs;t motus mixtus ex tribus, in a&longs;cen&longs;u, &longs;cilicet, ex naturali æquabili, ex verti­cali retardato, & horizontali æquabili: mouetur &longs;ur&longs;um per our­uam, &longs;empérque capiti iaculatoris imminet; quippe tantùm acqui­rit in horizontali, quantùm nauis: in de&longs;cen&longs;u verò e&longs;t motus mix­ex horizontali retardato, & naturali accelerato: quia tamen bre­ui&longs;&longs;imo illo tempore, retardatio illa horizontalis non e&longs;t &longs;en&longs;ibilis, ferè in ip&longs;ius iaculatoris caput de&longs;cendit; quod certè phænomenon ex no&longs;tris principiis euincitur.

11. Parum cautè Vfanus vniuer&longs;im a&longs;&longs;erit, iaculationem pilæ ex­tormento, maiorem e&longs;&longs;e ex naui in continentem, & minorem vi­ci&longs;&longs;im, cùm vtriu&longs;que differentia peti po&longs;&longs;it, vel à puluere tormen­tario, vel ab eius compre&longs;&longs;ione, vel humiditate, vel tormenti fabri­ca, vel ip&longs;ius demum nauigij motu, qui pilæ motum, vel accelerat, &longs;i versùs eandem partem e&longs;t, vel retardat è contrario: in plano ho­rizontali duro pote&longs;t e&longs;&longs;e motus mixtus ex duobus, tribus, qua­tuor, & pluribus aliis.

12. Cùm è naui mobili emittitur &longs;agitta per horizontalem, quæ fa­cit anglum rectum cum linea directionis nauis, fertur qua&longs;i per dia­gonalem vtriu&longs;que, &longs;altem per aliquod &longs;patium: cùm verò emitti-tur per horizontalem, quæ conueniat cum eadem linea directionis, &longs;actus e&longs;t longior toto illo &longs;patio, quod nauis decurrit, dum iactus durat; breuior tamen, &longs;i in partem oppo&longs;itam fiat iactus in hoc ca­&longs;u, &longs;i nauis æqualem impetum imprimeret, deor&longs;um rectà ferretur mobile motu naturali; imò &longs;agitta po&longs;&longs;et retorqueri in iaculatorem: &longs;i terra e&longs;&longs;et vtrimque peruia, lapis demi&longs;&longs;us per multa annorum millia libraretur; non tamen e&longs;&longs;et motuus perpetuus.

De motu reflexo.

1. MOtus reflexi vera cau&longs;a e&longs;t impetus prior, ad nouam li­neam determinatus ab occurrente obice; planum refle­ctens e&longs;t cau&longs;a nouæ determinationis &longs;uo modo; cau&longs;am enim di­co eam, ex qua aliquid &longs;equitur: ex gemina determinatione, noua, &longs;cilicet, per ip&longs;am perpendicularem erectam in puncto contactus, & priore per lineam incidentiæ, ab eodem puncto contactus pro­pagatam, fit determinatio mixta per lineam reflexionis; quæ omnia patent ex terminis: hinc nullus impetus producitur à plano refle­ctente; quippe prior pote&longs;t determinariad nouam lineam: adde, quòd planum, quod caret impetu, impetum producere non pote&longs;t.

2. Imò nihil impetus de&longs;truirur in reflexione pura per &longs;e; quia ni­hil impetus e&longs;t fru&longs;trà per &longs;e in pura reflexione; multus tamen im­petus de&longs;truitur per accidens, tùm ab ip&longs;o attritu tùm mollitie & ce&longs;&longs;ione, tùm pre&longs;&longs;ione: hinc &longs;uppo&longs;ito eodem iactu, perpendi­cularis reflexa e&longs;t omnium reflexarum minima; quia per eam li­neam maximus ictus infligitur; igitur maxima e&longs;t partium colli&longs;io, & pre&longs;&longs;io: hinc etiam corpora duriora longiùs reflectuntur, per ip&longs;am quoque perpendicularem, dum planum reflectens &longs;it æquè durum.

3. Determinatio noua dupla e&longs;t prioris, po&longs;ita linea incidentiæ perpendiculari, & po&longs;ito etiam plano reflectente immobili; quia alioquin anguli reflexionis non e&longs;&longs;ent æquales angulis incidentiæ: &longs;i globus reflectens &longs;it æqualisimpacto, æqualis e&longs;t ce&longs;&longs;io re&longs;i&longs;tenciæ cùm &longs;it æquale agens re&longs;i&longs;tenti, perid enim reflectens re&longs;i&longs;tit, per quod e&longs;t: igitur, &longs;i æqualis re&longs;i&longs;tit, & cedit, certè æqualiter ce­dit, & re&longs;i&longs;tit: hinc noua determinatio æqualis e&longs;t priori: hinc glo­bus impact is &longs;i&longs;tit immobilis; quia ex duabus determinationibus oppo&longs;itis neutra præualet.

4. Tantum e&longs;t ab æqualitate prædicta ce&longs;&longs;ionis, & re&longs;i&longs;tentiæ, ad nullam ce&longs;&longs;ionem, & notam re&longs;i&longs;tentiam, quantum e&longs;t ad nullam re&longs;i&longs;tentiam, & totam ce&longs;&longs;ionem: hinc, cùm à tota ce&longs;&longs;ione ad æqua­litatem prædictam acquiratur tantùm noua determinato æqualis priori; igitur ab eadem æqualitate ad nullam ce&longs;&longs;ionem tantun­dem acquiritur; igitur dupla prioris, vt iam &longs;uprà dictum e&longs;t; nulla e&longs;&longs;et re&longs;i&longs;tentia in vacuo; nulla e&longs;t ce&longs;&longs;io, cùm ip&longs;um corpus refle­ctens nullo modo mouetur ab ictu.

5. Determinatio noua per lineam obliquam, e&longs;t ad nouam per lineam perpendicularem, vt &longs;inus rectus anguli incidentiæ, ad &longs;i­num totum, in qualibet hypothe&longs;i; quia &longs;unt hæ, vt ictus, per vtran­que lineam; ictus verò vt grauitationes in horizontale planum, & in planum inclinatum, &longs;ub angulo complementi anguli incidentiæ: hinc noua determinatio per lineam obliquam, e&longs;t vt dupla &longs;inus re­cti anguli incidentiæ, ad &longs;inum totum: hinc &longs;upra angulum inci­dentiæ 30, noua e&longs;t maior priore, infrà minor; in ip&longs;o angulo 30. æqualis, &longs;uppo&longs;ita hypothe&longs;i plani reflectentis immobilis.

6. Ex hoc po&longs;itiuo principio demon&longs;tratur accurati&longs;&longs;imè æqua­litas anguli reflexionis, & incidentiæ, quod certè demon&longs;tratum non fuit ab Ari&longs;t. in problematis, &longs;ect. 17. problem. 4. & 13. quibus in locis fusè &longs;atis explicatur hoc Theorema, ducta comparatione, tùm à grauibus, quæ cadunt, tùm ab orbibus, quæ rotantur, rùm à &longs;peculis&longs;ed minimè demon&longs;tratur ex certis principiis &longs;ine petitio­ne principij. In puncto reflexionis, po&longs;ita hypothe&longs;i plani immo­bilis reflectentis, nulla datur quies; quia vnum tantùm e&longs;t conta­ctus in&longs;tans; &longs;ed eo in&longs;tanti e&longs;t motus, quo primo acquiritur locus.

7. Omnes lineæ reflexæ per &longs;e &longs;unt æqualis longitudinis, & ab eodem puncto contactus, ad communem peripheriam terminan­tur: &longs;i globus impactus &longs;it æqualis reflectenti, &longs;itque linea inciden­tiæ obliqua quælibet terminata ad idem punctum contactus, re­flectitur prædictus globus per lineam tangentem globum refle­ctentem in eodem puncto; quia hæc tangens e&longs;t diagonalis com­munis, & determinatio mixta communis omnibus lineis inciden­tiæ: e&longs;t tamen modò longior, modò breuior linea reflexa, é&longs;tque vvt &longs;inus complementi anguli incidentiæ, ad &longs;inum totum, qui &longs;it determinatio prior, vt facilè demon&longs;tramus.

8. Si globus impactus &longs;it minor corpore reflectente, reflectitur eriam per ip&longs;am perpendicularem, & determinatio noua e&longs;t dupla­prioris, minùs ratione globorum v. g. &longs;i globus impactus &longs;it &longs;ubdu-plus, determinatio noua e&longs;t dupla prioris, minùs vna quarta, &c. ratio e&longs;t, quia in ea proportione globus reflectens cedit, in qua mouetur, igitur tantùm detrahitur determinationis impacto globo, quantùm additur motus reflectenti: at verò noua determina­tio per lineam incidentiæ obliquam, e&longs;t ad nouam per ip&longs;am per­pendicularem, vt &longs;inus rectus anguli incidentiæ ad &longs;inum totum.

9. In hac hypothe&longs;i lineæ reflexæ omnes &longs;unt &longs;upra prædictam tangentem, &longs;eu &longs;ectionem plani, maiores, vel minores, pro diuer&longs;a men&longs;ura diagonalis: in &longs;uperiori verò hypothe&longs;i æqualium globo­rum, &longs;unt omnes in ip&longs;a &longs;ectione plani: &longs;i denique globus impactus &longs;it maior alio, omnes &longs;unt infra prædictam &longs;ectionem. Porrò in hac hypothe&longs;i vltima, determinatio noua per ip&longs;am perpendicularem e&longs;t minor priore: hinc non modò nulla fit reflexio in perpendicula­ri, &longs;ed linea directa vlteriùs propagatur; quia prior determinatio præualet.

10. Detrahitur priori portio æqualis rationi globorum; v. g. glo­bus reflectens e&longs;t &longs;ubduplus impacto de trahitur priori determina­tioni vna &longs;ecunda; e&longs;t &longs;ubquadruplus, vna quarta; atque ita dein­ceps: ratio patet ex dictis: in linea verò incidentiæ obliqua, deter­minatio e&longs;t ad determinationem in perpendiculari, vt &longs;inus rectus anguli incidentiæ ad &longs;inum totum: linea demum reflexa e&longs;t modò maior, modò minor pro diuer&longs;a diagonali.

11. Si duo globi æquales in &longs;e inuicem impingantur æquali mo­tu, per lineam connectentem centra, vterque æquali motu priori re­troagitur; quia æqualis in æqualis æqualem impetum imprimit: non e&longs;t tamen motus reflexus; quia totus prior impetus de&longs;truitur, vt patet ex dictis: &longs;i autem inæquali motu concurrant, retroaguntur ii&longs;dem motibus, permutando; quod etiam clarum e&longs;t: hinc egre­gium paradoxum, &longs;i quod aliud con&longs;equitur, &longs;cilicet, globum A, v. g. æqualem motum imprimere globo B, &longs;iue hic moueatur, &longs;iue quie&longs;cat.

12. Si verò linea incidentiæ &longs;it obliqua, vterque globus reflecte­tur pror&longs;us vt à plano immobili: hinc reflexio &longs;it ad angulos æqua­les, & lineæ omnes reflexionis &longs;unt æquales: ratio e&longs;t; quia, quantùm detrahit globus reflectens re&longs;i&longs;tendo, tantùm addit in partem op­po&longs;itam repellendo, po&longs;itiuo ni&longs;u, vel impetu: quòd &longs;i alter globus maiore, vel minore motu moueatur, vel &longs;i globi &longs;int inæquales, cum æquali motu, vel inæquali, resetiam determinari pote&longs;t ex præmi&longs;&longs;is.

13. Cum duo globi in &longs;e&longs;e inuicem impinguntur æquali motu, minor retroagitur velociore motu, quàm ante moueretur, vt clarum e&longs;t: maior verò, &longs;i duplus e&longs;t alterius, &longs;i&longs;tit immobilis in puncto contactus; &longs;i maior duplo &longs;uum iter pro&longs;equitur, &longs;ed tardiore mo­tu; &longs;i minor duplo, retroagitur: quæ omnia facilè ex dictis demon­&longs;trantur. Pote&longs;t impetus e&longs;&longs;e æqualis alteri, & præualere; pote&longs;t æqualem impetnm producere hoc in&longs;tanti, & &longs;tatim in&longs;tanti, quod &longs;equitur, totus de&longs;trui.

14. Pote&longs;t globus retroagi in plano horizontali, licèt in aliud cor­pus non incidat, ita vt initio tendat in ortum, verbi gratia: tùm deinde, licèt nihil pror&longs;us addatur, versùs occa&longs;um; quod accidit, cum globus vtroque motu, centri, &longs;cilicet, & orbis, mouetur, &longs;ed contrario; primùm enim motus centri plæualet, &longs;ed facilè cedit propter attritum maiorem partium. Nullus datur propriè motus refractus: licèt enim incuruetur linea motus, dum per aquam &longs;u­bit mobile; hæc tamen e&longs;t reflexionis &longs;pecies.

15. Globus reflectens, qui ab ictu alterius mouetur, non mouetur in&longs;tanti contactus; quia impetus primo in&longs;tanti, quo e&longs;t, non mo­uetur; producitur enim impetus primo in&longs;tanti contactus: &longs;i impe­tus e&longs;&longs;et tantùm determinatus ad vnam lineam, nulla fieri po&longs;&longs;et reflexio, &longs;ed tantùm repercu&longs;&longs;io; quia veri&longs;&longs;ima cau&longs;a reflexionis con&longs;i&longs;tit in noua determinatione: per reflexionem po&longs;&longs;unt colligi plures partes aëris &longs;onori ad Echometriam: &longs;agitta emi&longs;&longs;a per ho­rizontalem &longs;ursùm, tantillùm a&longs;cendit per arcum; quia tantillùm reflectitur ab aëre.

De motu circulari.

1. DAri motum circularem, probatur infinitis ferè experimen­tis: cuius ratio à priori e&longs;t, quòd po&longs;&longs;int extremitates eiu&longs;­dem cylindri in partes oppo&longs;itas pelli; vnde &longs;equitur nece&longs;&longs;ariò motus circularis; quem ij negare coguntur, qui ex punctis mathe­maticis quantitatem componunt. Motus circularis in &longs;ublunaribus oritur ex recto impedito; quia, &longs;cilicet, determinatur tantùm im­petus ad lineam rectam: hinc quidam motus circularis e&longs;t merè per accidens, vt cùm retinetur extremitas funependuli, &longs;eu fundæ, quæ &longs;i demittatur, &longs;equitur motus rectus: quidam tamen non e&longs;t merè peraccidens, vt cùm pellitur extremitas cylindri in plano horizontali; e&longs;t enim, iuxta in&longs;titutionem naturæ, ad facili­tatem motus.

2. Quippe tale e&longs;t naturæ in&longs;titutum, vt eo motu corpora mo­ueantur, quo faciliùs moueri po&longs;&longs;unt: atqui cùm pellitur altera cy­lindri extremitas, in plano horizontali putà innatantis, faciliùs mouetur, quàm recto, & qua&longs;i minore &longs;umptu, cùm minùs &longs;patij acquirat: æqualitempore: pote&longs;t dari motus circularis mixtus ex duobus rectis, quorum vnus &longs;it, vt &longs;inus recti, alius vt ver&longs;i; vix tamen hoc accidit vnquàm, &longs;ed tantùm oritur hic motus ex determinatione per tangentem impedita, ratione alicuius puncti immobilis.

3. Hinc, &longs;i tollatur impedimentum, &longs;tatim per tangentem or­bis fit motus, vt patet in funda: inæqualiter partes radij prædicti orbis mouentur, iuxta proportionem di&longs;tantiæ maioris, & minoris à centro: hinc propagatio impetus inæqualis, de qua iam &longs;uprà, &longs;ingulis in&longs;tantibus & punctis e&longs;t noua determinatio; quia, &longs;cilicet, &longs;ingulis punctis &longs;ua tangens re&longs;pondet: hinc, &longs;i imponatur rotæ aliud corpus, &longs;tatim abigitur, &longs;ine &longs;it in &longs;itu verticali, &longs;iue in &longs;itu ho­rizontali; hinc dum turbo rotatur, &longs;i vel aquæ guttula eius &longs;uper­ficies a&longs;pergitur, & &longs;tatim di&longs;pergitur.

4 Dari impetum in motu circulari certi&longs;&longs;imum e&longs;t: punctum phy­&longs;icum e&longs;t capax huius motus; cuius finis multiplex e&longs;t; corpus mo­uetur motu circulari circa centrum immobile cum motus centri impeditur non tamen motus orbis, ad quem impetus facilè deter­minatur, cùm &longs;it ad omnes lineas indifferens: adde v&longs;um vectis, trochleæ, aliorúmque organorum, qui &longs;ine motu circulari e&longs;&longs;e non pote&longs;t: omitto motum progre&longs;&longs;iuum, ipsúmque brachiorum, & ti­biarum v&longs;um, qui moru circulari carere non pote&longs;t.

5. Motus circularis rotæ in plano verticali e&longs;t æquabilis per &longs;e; quia nihil e&longs;t, quod impetum &longs;emel impre&longs;&longs;um de&longs;truat: licèt enim &longs;ingulis in&longs;tantibus &longs;it noua determinatio, nullus tamen impetus e&longs;t fru&longs;trà; quippe illud &longs;patium acquiritur in linea curua, quod in recta, &longs;i nullum e&longs;&longs;et impedimentum, percurreret: quemadmodum enim in reflexione, quæ fit à plano immobili, nullus de&longs;truitur im­petus; ita nullus hîc de&longs;truitur; tam enim centrum illud immobile ad &longs;e qua&longs;i trahit mobile, quàm planum immobile à &longs;e repellit; in quo e&longs;t perfectè analogia.

6. Hinc per &longs;e motus circularis integri orbis e&longs;t perpetuus; de­&longs;truitur tamen per accidens, &longs;cilicet, propter attritum axis: hinc tam diu durat hic motus: clari&longs;&longs;imum experimentum habes in tur­bine, cuius cu&longs;pis læuigati&longs;&longs;ima in plano læuigati&longs;&longs;imo rotatur; nec vnquam ce&longs;&longs;aret hic motus &longs;ine prædicto attritu, & partium a&longs;peri­tate: nec quidquam ob&longs;tat, quòd aliquæ partes rotæ, quæ in circu­lo verticali voluitur, a&longs;cendant; quia etiam aliquæ de&longs;cendunt: qua­re &longs;emper remanet perfectum æquilibrium, & harum de&longs;cen&longs;us, il­larum a&longs;cen&longs;um compen&longs;at. Quò diutiùs potentia motrix manet applicata manubrio axis rotæ, ita vt nouum &longs;emper producat im­petum, rotæ motus velocior e&longs;t, atque diutiùs durat: idem pror&longs;us dico de rota circulo horizontali parallela.

7. Cùm mouetur æquali ni&longs;u acus circa immobile centrum, tùm in plano horizorrali, tùm in verticali, &longs;iue &longs;it longior vna, &longs;iue breuior alia, per &longs;e plures gyros non de&longs;cribit vna, quàm alia; quia per &longs;e mouetur motu æquabili: per accidens tamen &longs;ecus accidit; quippe maior e&longs;t maioris attritus: dixi, cùm mouetur æquali ni&longs;u; nam &longs;æpè contingit, maiore ni&longs;u potentiam motricem agere circa maiorem; æquali tamen tempore numerus circuitionum minoris, e&longs;t ad nu­merum circuitionum maioris per &longs;e vt acuum quadrata permu­tando; &longs;unt enim motus vt &longs;patia, &longs;pacia vt quadrata.

8. Verbi gratia, &longs;it acus maior 2. minor 1. certè cùm tota area or­bis maioris &longs;it quadrupla minoris, &longs;itque area maioris, &longs;patium ma­ioris, & area minoris &longs;patium minoris, haud dubiè de&longs;cribet minor quatuor circuitiones, co tempore, quo maior decurret vnicam: li­cèt enim extremitas minoris, quæ impellitur, habeat tantùm du­plum impetum extremitatis maioris, &longs;itque impetus inten&longs;io in minore, dupla inten&longs;ionis impetus in maiore; e&longs;t tamen quadrupla illius, quæ e&longs;t in &longs;egmento maioris versùs centrum æquali minori acui: porrò motus circulares æquabiles in vtraque cum eodem impetu, &longs;unt vt motus recti.

9. Rota in plano verticali faciliùs mouetur, quàm in horizonta­li; quia in illo mouetur per minimam impetus, vel potentiæ acce&longs;­&longs;ionem; &longs;ecùs in i&longs;to; quippe per minimam acce&longs;&longs;ionem tollitur æquilibrium; imò moueri pote&longs;t in plano verticali, licèt nullus im­primatur impetus rotæ, v. g. per additionem minimi ponderis, vel momenti, vt patet; cùm tamen in plano horizontali moueri non po&longs;&longs;it, ni&longs;i impetus imprimatur.

10. Si cylindrus in plano horizontali læuigato in alteaa extremi­tate per tangentem impellatur, mouebitur motu circulati, &longs;cilicet, faciliori, cirra centrum, quod di&longs;tet ab altera extremitate vna quarta totius cylindri: ratio e&longs;t: quia faciliùs mouetur circa illud centrum, quàm circa alia puncta, quòd, &longs;cilicet, minùs &longs;patij decur­ratur, po&longs;ito eodem &longs;emper motu alterius extremitatis, cui appli­catur immediatè potentia motrix.

11. Cùm rota mouetur in verticali, atque præponderat alter &longs;emi­circulus, haud dubiè hic præponderans producit impetum in alio &longs;emicirculo: hinc fortè e&longs;t, quòd mirere, impetus determinatus deor&longs;um producit alium &longs;ur&longs;um: hinc impetus vnius partis mobi­lis pote&longs;t producere&longs;imilem in alia parte continua; quod tantùm in hoc ca&longs;u locum habet: quando corpus incumbit plano, quod mo­uetur motu recto æquabili, ab eo non &longs;eparatur; &longs;ecùs verò, &longs;i in­cumbat plano, quod mouetur motu circulari.

De motu funependuli.

1. FVnependulum de&longs;cendit per arcum motu naturaliter acce­lerato: experientia clari&longs;&longs;ima e&longs;t: cùm enim ex maiori &longs;ubli­mitate de&longs;cendit, maiorem ictum infligit. Ratio à priori e&longs;t quia priori impetui acqui&longs;ito nouus accedit: non acceleratur in eadem proportione, in qua &longs;uprà dictum e&longs;t accelerari in linea recta; quia in hac acceleratur vniformiter, id e&longs;t, æqualibus temporibus, æqualia acquiruntur velocitatis momenta; quia vel e&longs;t &longs;emper ea­dem inclinatio plani, vel idem perpendiculum: at verò in fune­pendulo in &longs;ingulis punctis e&longs;t noua tangens; igitur noua inclina­tio plani; igitur noua ratio motus.

2. Initio acceleratur motus per maiora crementa, &longs;ub finem per mi­nora; v.g. &longs;i dato tempore acqui&longs;iuit vnum gradum impetus initio, æquali deinde tempore acquiret minùs: ratio clara e&longs;t: quia, vt ac­quireret æqualem, deberet e&longs;&longs;e eadem plni inclinatio; &longs;ed &longs;empes cre&longs;cit Inclinatio; igitur &longs;emper imminuitur impetus æquali tempore acqui&longs;itus: acquiritur tamen æqualis velocitas in arcu, & in chor­da, &longs;eu plano inclinato, eiu&longs;dem altitudinis; igitur &longs;emper cre&longs;cit motus funependuli in de&longs;cen&longs;u, &longs;ed minoribus incrementis.

3. Hinc breuiore tempore de&longs;cendit per radium perpendicula­rem, quàm per quadrantis arcum eiu&longs;dem radij; tùm quia breuior e&longs;t linea; tùm, quia in perpendiculari acceleratur motus per maiora crementa. Vibrario maior eiu&longs;dem funependuli æquali ferè tem-pore cum minore perficitur: ratio quia, cùm ferè decurrantur arcus iuxta &longs;ubten&longs;arum proportionem, certè cùm &longs;ubten&longs;æ om­nes æquali tempore decurrantur, idem ferè fit in ip&longs;is arcubus: dixi ferè: nam reuerà minor vibratio citiùs, maior tardiùs perficitur, vt con&longs;tat experientia: neque dee&longs;t ratio, quam in analyticcam remittimus.

4. Non a&longs;cendit funependulum ad eam altitudinem, ex qua priùs de&longs;cenderat: clara e&longs;t experientia: neque ratio tantùm petitur ab 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 eo, quòd &longs;ingulis in&longs;tantibus &longs;it quædam pugna, inter impetum in­natum, & alium determinatum ad arcum &longs;ur&longs;um: quippe impetus innatus ad totum de&longs;cen&longs;um, &longs;ed nullo modo ad a&longs;cen&longs;um con­currit: hinc in maiori vibratione imminuitur motus, & &longs;patium in maiori proportione, quàm in minori; quia in hac lineæ &longs;ingulæ a&longs;­cen&longs;us qua&longs;i totidem inclinatæ &longs;unt inclinatiores; in illa verò minùs.

5. Hinc diu vibratur funependulum per minores arcus, quippe facilis e&longs;t a&longs;cen&longs;us per planum proximè ad horizontale accedens: hinc etiam in funependulo maiori dintiùs durant huiu&longs;modi vi­brationes, idque in arcubus paulò maioribus; quia &longs;ubten&longs;æ his arcubus &longs;unt inclinatiores: hinc refutabis eos, qui dicunt, vibra­tiones funependuli in vacuo fore perpetuas: arcus vibratio­nis a&longs;cen&longs;us fit motu naturaliter retardato, &longs;ed per imminu­tiones inæquales; quia pro diuer&longs;a inclinatione plani diuer&longs;imodè retardatur.

6. Vltimum punctum impetus acqui&longs;itus acqui&longs;itum in de&longs;cen&longs;u, nullo modo ad de&longs;cen&longs;um concurrit, &longs;ed ad a&longs;cen&longs;um, vnico tan­tùm in&longs;tanti; quippe e&longs;t omnium imperfecti&longs;&longs;imum; quod reuerà &longs;i e&longs;&longs;et eiu&longs;dem perfectionis cum innato, a&longs;cen&longs;us æqualis e&longs;t de&longs;cen­&longs;ui: &longs;i &longs;int funependula inæqualia, vibrationes non &longs;unt æquè diu­turnæ: ratio e&longs;t: quia, &longs;i a&longs;&longs;umamtur, v.g.duo quadrantes inæquales, &longs;unt eju&longs;dem inclinationis; igitur minor citiùs percurritur.

7. Porrò tempora vibrationum &longs;unt in ratione &longs;ubduplicata ar­cuum &longs;imilium, vel chordarum &longs;imilium, vel radiorum; id e&longs;t, vt radices &longs;patiorum &longs;imilium: verbi gratia, &longs;it quadruplus alterius, tempus vibrationis maioris e&longs;t duplum temporis vibrationis mino­ris; quod ita intelligendum e&longs;t, vt hæc proportio con&longs;ideretur in partibus temporis &longs;en&longs;ibilibus, vt iam dictum e&longs;t de motu natura­liter accelerato deor&longs;um in perpendiculo, & in planis inclinatis; nam progre&longs;&longs;io arithmetica; a&longs;&longs;umpta in &longs;ingulis in&longs;tantibus, tran­&longs;it in hanc, &longs;i a&longs;&longs;umantur partes temporis &longs;en&longs;ibiles, quarum &longs;ingu­læ infinitis ferè con&longs;tent in&longs;tantibus.

8. In maiori quadrante, circa &longs;upremam extremitatem, e&longs;t minor inclinatio, quàm in minore; hic enim &longs;tatim detorquetur à perpen­diculo, cum quo facit angulum maiorem: at verò circa infirmam extremitatem, e&longs;t maior inclinario in maiore, quàm in minore: hinc, &longs;i comparetur vibratio maioris, cum vibratione minoris in modico arcu, tempus illius e&longs;t paulò maius duplo, temporis huius; in maxi­mo arcu paulò minùs duplo, dum, &longs;cilicet, longitudinum ratio &longs;it quadrupla.

9. In de&longs;cen&longs;u funependuli velocitas acqui&longs;ita e&longs;t eadem cum ea, quæ in &longs;ubten&longs;a eiu&longs;dem arcus acquiritur: hinc &longs;unt ijdem ictus: numerus, vibrationum non e&longs;t infinitus, licèt in vacuo vibraretur funependulum; quia, cùm &longs;ingulæ imminuantur, & infinitis pun­ctis non con&longs;tent; tandem ad vltimam peruenitur: illa autem e&longs;t vl­tima, in cuius de&longs;cen&longs;u acquiritur tantùm vnum punctum impetus &longs;upra innatum; in ea tamen &longs;ententia, quæ vel infinitas partes actu, vel infinita puncta cogno&longs;cit, certè nunquam quie&longs;ceret funepen­dulum in vacuo vibratum.

10. Funependulum in fine a&longs;cen&longs;us non quie&longs;cit vno in&longs;tanti; quia impetui innato nunquam redditur æqualis acqui&longs;itus; po&longs;ita ta­men illa æqualirate, in&longs;tanti &longs;equenti e&longs;&longs;et quies: funependulum grauius citiùs de&longs;cendit; e&longs;t enim eadem ratio, quæ fuit pro mo­tu naturali; corpus oblongum &longs;olidum circa punctum immobile in circulo verticali rotatum vibratur adin&longs;tat funependuli; de&longs;­cendit tamen citiùs, quàm funependulum eiu&longs;dem longitudinis.

11. Ratio facilis e&longs;t; quia partes &longs;olidæ, quæ accedunt propiùs ad extremitatem immobilem, accelerant motum aliarum, quæ ad mobilem extremitatem accedunt; faciunt enim arcum mino­rem: hinc a&longs;cen&longs;us non peruenit ad tantam &longs;ublimitatem; quia, vt prædictæ partes accelerant motum aliarum in de&longs;cen&longs;u, ita retar­dant in de&longs;cen&longs;u: hinc citiùs quie&longs;cit hoc penduli genus, quàm aliud: ex hoc colligo paradoxon, &longs;cilicet, corpus moueri po&longs;&longs;e &longs;ua &longs;ponte velociùs in arcu deor&longs;um, quàm in perpendiculo; v.g. &longs;i iuxta extremitatem immobilem &longs;it nodus plumbeus, cuius vi, altera ex­tremitas longiùs di&longs;tans deor&longs;um rapiatur.

De motu mixto ex circulari.

1. ROta, quæ mouetur in &longs;uperficie plana, mouetur motu mixto ex recto centri, & circulari orbis: axis tantùm rotæ mouetur motu recto: punctum contactus rotæ mouetur motu tardi&longs;&longs;imo, quando motus centri, & &longs;uprema rotæ pars in eandem partem &longs;e­runtur; punctum verò oppo&longs;itum veloci&longs;&longs;imo, quia in motu huius rotus motus orbis additur motui centri; in motu verò illius, to­tus motus orbis, motui centri detrahitur: quod autem detrahit mo­tus orbis, nunquam æquale e&longs;t toti motui centri.

2. Hinc omnia puncta eiu&longs;dem circuli rotæ mobilis in plano hoc motu mixto mouentur in æquali motu: hoc etiam motu mo­uetur globus de&longs;cendens in plano inclinato, in quo reuerâ motu hæc habes: primò, non modò accelerari motum centri, verùm etiam motum orbis; &longs;ecundò, ita impetum propagari ab intrin&longs;eco, vt &longs;ingu­lis partibus eiu&longs;dem circuli, & plani in æqualiter di&longs;tribuatur, tertiò hoc motu motum rectum non impediri à circulari, & &longs;ed iuuari.

3. Cùm rota voluitur in &longs;uperficie connexa, mouetur motu mix­to ex duobus circularibus: &longs;imilis e&longs;t hic motus motui epicycli. Ca­lamus volatilis, cuius mi&longs;&longs;io frequens, & repercu&longs;&longs;io, ludi non in­grati copiam facit: mouetur motu mixto ex recto, & circulari: in hoc porrò motu præit calami caput, & &longs;equuntur pennæ; quia aër fortiùs re&longs;i&longs;tit pennis, quàm thecæ: hinc pennarum motum theca grauior accelerat, cuius motum pennæ retardant.

4. Hinc, &longs;i quando accidat, penas educi ex theca in libero medio; &longs;tatim theca velociori motu mouetur, cùm tamen pennæ ip&longs;æ &longs;i­&longs;tant: ex hac inæqualitate, ne impetus &longs;it fru&longs;trà, propter detortas in alteram partem pennas ab aëre re&longs;i&longs;tente totum iaculum defle­ctitur, agitúr que in orbem; hinc motus orbis traducitur ex theca in pennas, non contrà, vt aliquis fortè exi&longs;timaret, licèt pennarum tar­ditas, & obliqua deflexio, ratione cuius ab aëre re&longs;tente, in alteram partem qua&longs;i reflectentur, &longs;int nece&longs;&longs;aria conditio huius traductio­nis.

5. Hinc motu recto prædictum iaculum in vacuo tantùm mo­ueretur, vt patet: hinc: cùm pennæ &longs;unt explicatiores, tardiùs; cùm verò contractiores, velociùs mouetur, etiam motu orbis; cui non minùs aër re&longs;i&longs;tit, in pennis, &longs;cilicet, quàm motui axis: hinc, &longs;i theca &longs;it grauior, velociùs; &longs;i leuior, tardiùs iaculum fertur; etiam tenera plumarum lanugo tarditatem conciliat: porrò, &longs;i axis mouetur mo­tu recto, quod reuerà fit, cùm iaculum deor&longs;um demittitur in per­pendiculo, hic motus e&longs;t &longs;piralis cylindricus: ex his infinita ferè phænomena explicari po&longs;&longs;unt.

6. Sunt infiniti propemodum motus mixti; v. g. cylindri ab alte­ra extremitate rotata emi&longs;&longs;i; longioris ha&longs;tæ, quæ &longs;ur&longs;um facta cir­cuitione emittitur; brachij, gladij, &c. &longs;ed poti&longs;&longs;imùm turbinis, qui vel &longs;eutica, vel funiculo in torto circumagitur, in quo clari&longs;&longs;i­mè apparet motus centri, & orbis: ratio motus orbis e&longs;t impe­tus impre&longs;&longs;us vtrique extremitati diametri va&longs;is in partes contra­rias; ratio verò motus centri e&longs;t, quia adducitur funiculo vel ex­ploditur, &longs;eu expellitur &longs;cutica: huius motus phænomena &longs;unt ferè infinita: &longs;ingula ex no&longs;tris principiis facilè explicantur.

De diuer&longs;is impre&longs;&longs;ionibus motus.

1. CVm &longs;u&longs;tinetur manus, &longs;eu brachium, in &longs;itu horizontali im­mobile, producitur nece&longs;&longs;aliò impetus æqualis impetui gra­uitationis; alioquin, &longs;i maior e&longs;&longs;et, &longs;ur&longs;um ferretur brachium; &longs;i verò minor, deor&longs;um: quia præualeret grauitatio, porrò hic impetus pro­ducitur tantùm à potentia motrice animantis, in &longs;ingulari organo; non verò in aliis partibus, etiam animatis, ni&longs;i quando mouentur; nec in ip&longs;o pondere, &longs;i aliquod &longs;u&longs;tinetur: &longs;ic men&longs;a in pondere &longs;u­per po&longs;ito impetum nullum producit. Si anima immediatè in toto corpore po&longs;&longs;et producere impetum, homo facilè volare po&longs;&longs;et.

2. Cùm &longs;u&longs;tinetur funependulum, nullus impetus producitur à &longs;u&longs;tinente in ip&longs;o globo, ne &longs;cilicet, &longs;it fru&longs;trà; &longs;ecùs verò, &longs;i attolla­tur: &longs;ic per quamlibet lineam corpus retineri pote&longs;t &longs;ine impetu in eo corpore producto per &longs;e: hinc, cùm duo &longs;e&longs;e inuicem trahunt ad­uer&longs;o ni&longs;u, neuter in altero producit impetum per &longs;e; &longs;ed per acci­dens, propter mollitiem, & ten&longs;ionem partium: cùm verò defertur aliquid coniunctum, producitur haud dubiè æqualis impetus; hinc &longs;eparari non pote&longs;t; quia æqualis e&longs;t motus latoris, & delati: exem­plum habes in naui.

3. Si verò nauis illicò &longs;i&longs;tat, vel tardiùs moueri pergat, tunc fit &longs;e­paratio: hinc liquida effunduntur, &longs;i dum feruntur, breuior quietis in va&longs;e intercedat morula. Vt feratur cylindrus humeris commodiùs debet &longs;u&longs;tineri in centro grauitatis, ad eleuationem anguli 49. quia tunc manui, & humero æqualiter pondus di&longs;tribuitur: ideò in circulo voluitur &longs;cyphus aqua plenus &longs;ine effu&longs;ione; quia impetus determi­natus per tangentem circuli aquam ip&longs;am à centro circuli remouet.

4. Cùm trahitur aliquod corpus impetus impre&longs;&longs;us in vna parte non producit impetum in alia, alioquim daretur proce&longs;&longs;us in in&longs;i­nitum; &longs;i chorda vtrinque trahatur, rumpetur in medio: &longs;i affixa extremitati immobili, trahatur à potentia applicata alteri extremi-tati, rumpetur iuxta primam illam extremitatem: &longs;i denique pon­ticulo &longs;uppo&longs;ito tendatur, vel pondere deprimente, in eo puncto rumpetur. Ratio communis i&longs;torum omnium e&longs;t: quia inter illas duas partes fieri debet diui&longs;io per &longs;e, quarum vna mouetur, &longs;ecùs alia; vel quarum vtraque in partes oppo&longs;itas mouetur.

5. Vt quodlibet pondus faciliùs trahatur, &longs;inguli equi trahere debent fune communi, potiùs quàm bigati; quia tunc nihil ferè pe­rit impetus: cùm plures idem pondus trahunt, agunt actione com­muni, alioqui &longs;inguli in toto pondere &longs;uum impetum producerent; igitur &longs;inguli &longs;eor&longs;im trahere? o&longs;&longs;ent, quod fal&longs;um e&longs;t: ideò currus paulò po&longs;t initium motus faciliùs mouetur; quia aliquid impetus priùs producti remanet: hinc etiam rupto fune, quo trahitur currus, currus ip&longs;e modicum tempus adhuc mouetur.

6. Si, dum quis trahit toto ni&longs;u magnum aliquod pondus, funis rumpatur, pronùs corruit: quia maiorem impetum in &longs;e producit, totum, &longs;cilicet, illum, quem in toto pondere produxi&longs;&longs;et eo in&longs;tan­ti, quo rumpitur finis, qui reuerà maior e&longs;t, propter impedimen­tum, ex præmi&longs;&longs;is principiis, maiorique applicatione potentiæ, ner­norum ten&longs;ione, &c. dum trahitur vnco annullus immobilis ver­sùs nauim, nauis fertur versùs littus; dum pellitur aduersùm littus, recedit à littore, quia pede, vel genu, imprimitur naui impetus in contrariam pattem.

7. Cùm trahitur cylindrus vtrinque æqualiter, qui neque flecti, neque tendi pote&longs;t, nullum impetum accipit; imò in tractione nul­lus impetus e&longs;t inutilis: brachium infligit maiorem ictum, cùm ma­iorem arcum de&longs;cribit &longs;uo motu; quia, &longs;cilicet, mouetur motu natu­raliter accelerato: hinc auer&longs;a manu validior impingitur colaphus, quàm aduer&longs;a; quia illa maiorem arcum de&longs;cribit: hinc longius bra­chium cæteris paribus grauiùs ferit: hinc diu qua&longs;i rotatur bra­chium, vt longiùs mittatur lapis.

8. Maiore fu&longs;te maior ictus infligitur; quia potentia toto ni&longs;u agens, diutiùs manet applicata maiori, quàm minori; &longs;untque ictus in ratione &longs;ubduplicata vtriu&longs;que fu&longs;tis; v. g. fu&longs;tis pendens vnam libram per maximum arcum impactus, infligit &longs;ubduplum ictum alterius, quem infligit fu&longs;tis quatuor pendens libras per eundem arcum impactus: idem dicatur de mi&longs;&longs;o lapide: principium huius veritatis pender iis, quæ diximus lib. 2. de motu naturali­ter accelerate, luxta progre&longs;&longs;ionem numerorum imparium, 1. 3. 5. &c.

9. Fu&longs;tis circa centrum immobile vibratus, maximum ictum in-fligit, non quidem in centro grauitatis, id e&longs;t, in medio, &longs;i &longs;it cy­lindrus, vel parallelipedum; nec in extremitate mobili; &longs;ed in eo puncto, in quo e&longs;t centrum impetus impre&longs;&longs;i, id e&longs;t, quod æqualem vtrinque dirimit impetum: ratio e&longs;t; quia tunc totus impetus agit, quantùm pote&longs;t; illud autem punctum Geometria demon&longs;trat e&longs;&longs;e terminum mediæ proportionalis, inter totum cylindrum, & &longs;ub­duplum; modò nulla ratio vectis habeatur alioquin centrum pro­cu&longs;&longs;ionis di&longs;tat 2/3 ab extremitate immobili.

10. Cùm fu&longs;tis inflectitur, reditque ad pri&longs;tinum &longs;tatum, vt videre e&longs;t in tudicula maiore, maior ictus imprimitur: quia non tantùm agit impetus extrin&longs;ecùs adueniens; verùm etiam potentia quædam media, quæ corpora compre&longs;&longs;a, vel ten&longs;a, ad pri&longs;tinum &longs;tatum reducit: hinc maximus e&longs;t ictus tudiculæ, cùm eo in&longs;tanti, quo reductum e&longs;t omninò manubrium priori rectitudini, infligitur ictus, quia tunc vis potentiæ mediæ e&longs;t maxima.

11. Rotato flagello ideò maxima vis ine&longs;t, quia diutiùs potentia manet applicata: hinc vides hoc principium e&longs;&longs;e vniuer&longs;ali&longs;&longs;imum, quod iactis, pul&longs;is, & impactis competit; de malleorum ictu idem pror&longs;us dicendum e&longs;t, quod de fu&longs;te; &longs;i autem mallei cadant ex eadem altitudine, motu naturali accelerato, ictus &longs;unt vt mallei, quia duplus malleus, v. g. duplum impetum acquirit: nam &longs;ingulæ partes &longs;eor&longs;im æqualem impetum acquirunt.

12. Si verò ex diuer&longs;a altitudine cadant, vel &longs;unt æquales, vel inæquales: &longs;i primum, ictus &longs;unt vt tempora, quibus cadunt: &longs;i &longs;ecundum, ictus &longs;unt in ratione compo&longs;ita temporum, & mal­leorum: &longs;i &longs;unt infinitæ, partes actu, nulla e&longs;t proportio percu&longs;&longs;ionis granuli cadentis, & rupis ingentis grauitantis; &longs;ed hoc vltimum fal­&longs;um e&longs;&longs;e con&longs;tat; non pote&longs;t tamen determinari proportio vitium grauitationis, & percu&longs;&longs;ionis, ni&longs;i numerus in&longs;tantium: quibus durat motus deor&longs;um cogno&longs;catur.

13. Leui&longs;&longs;imi lapides vix emittuntur ad modicam di&longs;tantiam; quia &longs;tatim &longs;eparantur à potentia: parallelipedum cadens de or­&longs;um in &longs;itu horizontali maximum ictum infligit in centro grauita­tis, id e&longs;t, in medio; quia tunc totus impetus agit, totus enim impe­ditur: in aliis punctis minor e&longs;t ictus, iuxta proportionem maioris di&longs;tantiæ à prædicto centro: &longs;i verò percutiatur cylindrus innatans, maxima erit vis, vel effectus ictus in centro grauitatis propter ean­dem rationem.

LIBER PRIMVS,

DE IMPETV.

TRACTATVM hunc de motu locali ab ip&longs;o impetu au&longs;picamur, ex cuius profectò cognitione tota res i&longs;ta de­pendet; cum enim impetus &longs;it cau&longs;a immediata motus, vt fusè demon&longs;tra­bimus infrà; & cum propter quid &longs;it res cogno&longs;ci non po&longs;&longs;it, ni&longs;i eius cau&longs;a cogno&longs;catur; dubium e&longs;&longs;e non pote&longs;t, quin præmittenda &longs;it tractatio illa, quæ e&longs;t de impetu, vt deinde affectiones ip&longs;ius motus per cau&longs;am eiu&longs;dem demon&longs;trentur; immò au&longs;im dicere ex vnius impetus cognitione, non modò mo­tum ip&longs;um, verùm etiam totam rem Phy&longs;icam pen­dere.

DEFINITIO I.

MOTVS localis e&longs;t tran&longs;itus mobilis è loco in locum continuo fluxu.Huius definitionis explicationem habebis in Metaphy&longs;icâ, quæ &longs;anè explicatio ad rem præ&longs;entem non facit.

Definitio II.

Motus velox e&longs;t quo percurritur maius &longs;patium æquali tempore, vet æquale &longs;patium minori tempore; contrà verò motus tardus.

Definitio III.

Impetus e&longs;t qualitas exigens motum, &longs;eu fluxum localem &longs;ui &longs;ubiecti, vel qua est cau&longs;a proxima motus illius mobilis, cui ine&longs;t, eo &longs;cilicet modo, quo pote&longs;t e&longs;&longs;e cau&longs;a motus.

Dico e&longs;&longs;e qualitatem &longs;iue di&longs;tincta &longs;it, &longs;iue non di&longs;tincta; quod hîc certè non di&longs;cutio; nec enim affirmo in hac definitione dari impetum; &longs;ed definio tantùm quid &longs;it impetus; qui reuera aliud non e&longs;t, &longs;i e&longs;t: quippe id tantùm concipio, cum impetum appello; &longs;iue &longs;it, &longs;iue non &longs;it, ne quis fortè initio &longs;tatim mihi litem intendat; quemadmodum definit circulum Geometra; licèt non a&longs;&longs;erat dari perfectum circulum; ita Phy­&longs;icus definit impetum, quamuis non affirmet dari impetum; quod tamen in &longs;exto Theoremate demon&longs;trabimus; itaque &longs;i e&longs;t impetus, haud dubiè nihil omninò præ&longs;tat in &longs;uo &longs;ubiecto ni&longs;i motum; quod quomodò fiat, explicabimus in&longs;rà in Theorematis.

Hypothe&longs;is I.

Datur motus localis; quis enim non videt volantem auem, natantem pi&longs;cem; currentem equum, rotatum globum; denique vnum corpus mi­grans è loco in locam? &longs;ed hoc e&longs;t moueri per Def. 1. igitur infinitis fe­rè experimentis nititur hæc hypothe&longs;is, quam veram e&longs;&longs;e nece&longs;&longs;e e&longs;t, &longs;i illa vera &longs;unt; &longs;ed illa certa &longs;unt phy&longs;icè, neque citra miraculum fallere po&longs;&longs;unt.

Diceret fortè aliquis etiam motum &longs;ube&longs;&longs;e oculorum fallaciæ; cùm è naui mobili littus ip&longs;um moueri, ip&longs;umque nauigium non moueri iudi­cemus. Quis enim oculos in Solem intendens, primo intuitu Solem &longs;ta­re non iudicet? cum tamen deinde pernici&longs;&longs;imo cur&longs;u rotari demon&longs;tre­mus; adde alias oculorum fallacias circa motum; &longs;ic rotata &longs;cintilla, vel carbo accen&longs;us immotum orbem de&longs;cribere videtur; &longs;ic nota inu&longs;ta trocho, dum celerrimè rotatur, orbem etiam immobilem de&longs;cribere iu­dicatur; &longs;ic &longs;tella cadens, vel exhalatio continenti &longs;ucce&longs;&longs;ione accen&longs;a moueri videtur; licet minimè moucatur; idem dicendum de puluere tormentario, vel alia qualibet materia; quæ continuata con&longs;ecutione accenditur; immò trochus ip&longs;e in orbem celerrimè agitatus, quie&longs;cere videtur; &longs;ic qui vertigine laborant, ea moueri exi&longs;timant, quæ quie&longs;cunt; idem exemplum habemus in ebrio&longs;is, iracundis, in iis qui ex graui febris ardore delirant, & in pueris qui diu in gyros eunt, vbi verti de&longs;ierint; &longs;ic corum quæ motu æquali feruntur, remotiora tardiùs moueri viden­tur; immò &longs;i per eandem lincam oculus, & mobile pari velocitate ince­dant, ip&longs;um mobile quie&longs;cere videtur, plura leges apud Opticos, de quibus agemus &longs;uo loco: Igitur ex his omnibus con&longs;tat minimè con&longs;ta­re dari motum, ex co quòd oculis aliquid moucri videatur.

Re&longs;pondeo equidem &longs;ateri me, vi&longs;um ip&longs;um plurimis &longs;ube&longs;&longs;e fraudi­bus; attamen &longs;i rectè oculus admoueatur, iu&longs;ta di&longs;tantià, nec vllum &longs;it impedimentum exterius nec interius; fieri non pote&longs;t, quin oculus mo­tum ob&longs;eruet; an fortè currentis calami motus oculum meum fallere po-te&longs;t? quidquid &longs;it, fateor vltrò hanc hypothe&longs;im in eo tantùm certitudi­nis gradu e&longs;&longs;e reponendam, in quo reponitur hæc cognitio, quâ modo cogno&longs;co me &longs;cribere, manu&longs;que, & calami motum ob&longs;eruo; &longs;iue id tan­tùm oculis fiat, &longs;iue intellectu ex oculis; quod aliàs di&longs;cutiemus; &longs;i quis fortè in Phy&longs;ica maiorem certitudinem po&longs;tularet, cum eo certè conuc­nire non po&longs;&longs;um.

Porrò quod &longs;pectat ad fallacias illas quæ &longs;upra adductæ &longs;unt; certum e&longs;t vel obiectum e&longs;&longs;e remotius, quam par &longs;it; vel moueri celeriùs, vel e&longs;&longs;e aliquod impedimentum interius; præ&longs;ertim in iis, qui &longs;eu vertigine, vel alio capitis morbo laborant; &longs;ed ne hîc opticum agere videar, harum fallaciarum certi&longs;&longs;imas cau&longs;as in &longs;uum locum remittimus.

Cæterùm licèt ad &longs;tatuendam, firmandamque hanc hypote&longs;im, Phy­&longs;ica experimenta rectè applicato &longs;en&longs;u comprobata &longs;ufficere po&longs;&longs;int; non de&longs;unt tamen rationes multæ à priori, vt vulgò aiunt, quibus euin­citur, non modò quid &longs;it motus, verùm etiam propter quid &longs;it.

Prima duci pote&longs;t à fine motus; cum enim res creatæ vbique &longs;imul e&longs;&longs;e non po&longs;&longs;int, certè, vt illo bono gaudeant, quo fortè carent, & vt coniungantur &longs;uo fini, motu locali opus e&longs;t; &longs;itit equus, abe&longs;t aqua, certè, ni&longs;i vel hæc propinetur, vel ille accedat, &longs;itim leuare non pote­rit; at neutrum &longs;ine motu haberi pote&longs;t: Lapis remouetur à &longs;uo centro, 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. Itaque ad cum finem res omnes creatæ in&longs;titutæ &longs;unt, quem &longs;ine motu a&longs;&longs;equi non po&longs;&longs;unt; igitur dari motum nece&longs;&longs;e e&longs;t, vt res creatæ cum lo­cum acquirant, in quo &longs;uo bono, &longs;uo fini, &longs;uæ perfectioni coniungan­tur; vel &longs;altem id muneris obeant, cui ab ipsâ naturâ de&longs;tinantur.

Secunda ratio ducitur à cau&longs;a efficiente; ni&longs;i enim daretur motus, fru&longs;trà daretur potentia motrix, tùm in animantibus, tùm in grauibus, de quâ aliàs.

Tertia petitur à cau&longs;a formali; cum enim detur impetus, vt demon­&longs;trabimus infrà, nece&longs;&longs;e e&longs;t dari motum.

Quarta petitur à termino motus; cum enim globus proiectus &longs;it in nouo loco in quo ante non erat; certè nouus locus qui &longs;uccedit alteri relicto, e&longs;t terminus motus citra miraculum; igitur &longs;i e&longs;t nouus locus, e&longs;t quoque motus.

Quinta ab v&longs;u; nec enim &longs;ine motu flueret aqua, caderet lapis, gyros agerent a&longs;tra, flaret ventus, volarent nubes, &c.

Sexta ab ip&longs;a Mechanica, quæ organa motui mini&longs;trat: quis enim ne­garet maius momentum e&longs;&longs;e cum maiori di&longs;tantiâ coniunctum; &longs;i verò maius momentum e&longs;t, nunquid præualebit; igitur deor&longs;um cadet, immò &longs;euerior Geometria, vt omittam A&longs;tronomiam, motum &longs;upponit, cum ex fluxu &longs;eu motu puncti infinitas fere lineas de&longs;cribat. Igitur certum e&longs;t dari motum localem.

Hypothe&longs;is II.

Datur quies, id e&longs;t priuatio motus. Hæc hypothe&longs;is etiam ceita e&longs;t, Quis enim neget &longs;edentem humi, vel decumbentem in lecto quie&longs;ceret con&longs;ule &longs;en&longs;us rectè applicatos; tam enim certus &longs;um me iam in cathe­dra quie&longs;cere, quam &longs;um certus Solem lucere; igitur ex certis experi­mentis certa hypothe&longs;is con&longs;equitur. Non de&longs;unt rationes à priori; nam primò res aliqua &longs;uo bono, &longs;eu fini coniuncta ab eo &longs;eparari non po&longs;tu­lat, igitur nec moueri. Secundò maximum incommodum e&longs;&longs;et, &longs;i res &longs;e­mel mota perpetuò moueretur. Tertiò, finis, &longs;eu terminus motus recti, e&longs;t quies; nam ideo lapis deor&longs;um cadit, vt in &longs;uo centro &longs;eu globo quie&longs;cat, id e&longs;t vt cum aliis partibus totum illud, &longs;eu globum componat, vt dicemus aliàs.

Diceret fortè aliquis &longs;ententias prædictas non valere in &longs;ententiâ Copernici, quæ terræ motum ad&longs;truit; præterea non modò falli &longs;en&longs;us circa motum, verùm etiam circa quietem.

Re&longs;pondeo primò illam Copernici &longs;ententiam e&longs;le fal&longs;i&longs;&longs;imam, vt &longs;uo loco o&longs;tendemus: &longs;ecundò, licèt terra moueretur &longs;ecundum Coperni­cum, Sol, & &longs;tellæ quie&longs;cerent.

Dices iuxta hypothe&longs;im Heraclidis Pontici, terra ip&longs;a, Sol etiam, & &longs;tellæ mouentur. Re&longs;pondeo primò hypothe&longs;un illam e&longs;&longs;e fal&longs;am, vt &longs;uo loco videbimus; &longs;ecundò etiam data illa hypothe&longs;i po&longs;&longs;et dari quies; &longs;i enim globus eodem ver&longs;us occa&longs;um impetu proiiceretur, quò ver&longs;us or­tum à terra ip&longs;a rapitur, haùd dubiè quie&longs;ceret: præterea iuxta hanc hy­pothe&longs;un, quietem appellarem vnius partis cum alia connexionem in ip­&longs;o toto &longs;eu globo, & quie&longs;cere dicerem lapidem, qui tantùm totius glo­bi motu mouetur, ex quo profectò tota &longs;oluitur difficultas.

Quod verò &longs;pectat ad fallacias oculi circa quietem; codem pror&longs;us modo &longs;oluendæ &longs;unt, quo iam &longs;upra &longs;olutæ &longs;unt aliæ circa motum: vtrùm verò motus, & quies dicant aliquid di&longs;tinctum à mobili, dice­mus infrà.

Hypothe&longs;is III.

Aliquid mouetur quod incœpit moueri. Video lapidem quie&longs;centem, qui deinde proiectus mouetur; igitur ante non mouebatur, igitur cum deinde mouetur, cœpit moueri; mille aliis experimentis hæc hypothe­&longs;is confirmari pote&longs;t.

Hypothe&longs;is IV.

Aliquid mouetur quod tandem de&longs;init moueri, vel incipit quie&longs;cere. Vi­deo rotatam pilam, quæ tandem quie&longs;cit, cadentem lapidem, qui tan­dem &longs;i&longs;tit, &c. igitur certa e&longs;t hæc hypothe&longs;is.

Hypothe&longs;is V.

Idem mouetur modò tardiùs, modò velociùs. Video rotatum globum, qui &longs;en&longs;im quie&longs;cit: &longs;entio ab codem globo modò maiorem, modò mi­norem ictum infligi, &c. igitur e&longs;t certa hypothe&longs;is.

Hypothe&longs;is VI.

Corpus proiectum etiam à potentiâ motrice &longs;eiunctum adhuc moue tur.Oculos omnium te&longs;tes appello.

Hypothe&longs;is VII.

Corpus proiectum, & in aliud impactum illud ip&longs;um impellit, & mouet.

Hypothe&longs;is VIII.

Ignis applicatus &longs;ubiectum aptum, cui rectè applicatur nece&longs;&longs;ariò calefa­cit, nix frigefacit, Sol illuminat, corpus in aliud impactum illud ip&longs;um im­pellit. Prædictæ omnes Hypothe&longs;es certi&longs;&longs;imis nixæ experimentis certi­tudinem phy&longs;icam habent, & citra miraculum fallere non po&longs;&longs;unt.

Axioma I.

Contradictoria &longs;imul e&longs;&longs;e non po&longs;&longs;unt, vel non e&longs;&longs;e. Hoc ip&longs;um iam ptæ­mi&longs;unus Logicæ no&longs;træ demon&longs;tratiuæ, complectiturque prima illa principia Metaphy&longs;icæ.

1. Impo&longs;&longs;ibile est idem &longs;imul e&longs;&longs;e, & non e&longs;&longs;e.

2. Quodlibet e&longs;t, vel non est.

3. De eodem alterum contradictoriorum verè affirmatur, & alterum verè negatur, non &longs;imul vtrumque.

Axioma II.

Maximum &longs;ignum di&longs;tinctionis realis in phy&longs;icis est &longs;eparabilitas, vel op­po&longs;itio. Nihil enim a &longs;e ip&longs;o &longs;eparari po&longs;t; quippe, vbi e&longs;t &longs;eparatio, &longs;eu diui&longs;io, e&longs;t pluralitas; cur enim nummus A & nummus B eiu&longs;dem ma­teriæ, formæ, ponderis, realiter di&longs;tinguuntur? quia &longs;cilicet vnus non e&longs;t alius inquies; & quare vnus non e&longs;t alius? quia vnus e&longs;t hic & alius non e&longs;t hic, vnum tango, & alium non tango, vnus e&longs;t meus, & alius non e&longs;t meus, &c. vides prædicata contradictoria, quæ cum eidem &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 e&longs;t.

Diceret fortè aliquis hominem reproductum in duobus locis e&longs;&longs;e po&longs;­&longs;e, & dum Romæ e&longs;t à &longs;e ip&longs;o Lugduni exi&longs;tente &longs;eiunctum e&longs;&longs;e; hoc ip&longs;um aliàs examinabimus, dum con&longs;tet modò id totum, &longs;i fiat, mira­culo tribuendum e&longs;&longs;e, cum tamen res phy&longs;icas citra miraculum con&longs;ide­temus.

Axioma III.

Vt dicatur aliquid exi&longs;tere, vel debet &longs;en&longs;u percipi, vel aliqua ratione probari. Qui enim a&longs;&longs;erit rem aliquam po&longs;itiuam exi&longs;tere, certè po&longs;i­tiuo argumento demon&longs;trare debet quod &longs;it; illud porrò argumentum duci pote&longs;t vel ab experimento certo; &longs;ic probo exi&longs;tere rem aliquam, quam video; vel ab aliqua ratione; &longs;ic ex eo quòd cau&longs;a &longs;it nece&longs;&longs;aria applicata &longs;ubiecto apto, probo effectum ip&longs;um produci; vel co quòd &longs;it effectus probo cau&longs;am e&longs;&longs;e vel ex nece&longs;&longs;itate, quâ aliquid e&longs;t nece&longs;&longs;a­ium ad aliquem finem à natura in&longs;titutum, quo natura ip&longs;a &longs;ine ab&longs;ur-do, vel graui&longs;&longs;imo incommodo carere non pote&longs;t, probo illud ip&longs;um e&longs;&longs;e; vel demùm ex aliqua reuelatione certa in rebus fidei; igitur hoc Axioma certum e&longs;t phy&longs;icè; quod ni&longs;i recipiatur à Philo&longs;ophis; cuique licebit impunè mentiri; &longs;i enim dicam extra mundi huius fines e&longs;&longs;e alios orbes, intra tuum mu&longs;æum, in quo &longs;olus fortè degis, e&longs;&longs;e quin­quaginta homines, e&longs;&longs;e mille Soles, & totidem Lunas in cœlo, &c. numquid &longs;tatim oppones Axioma i&longs;tud, qua ratio, qua experientia, qua nece&longs;&longs;itas, qua reuelatio? Quæ&longs;tio facti e&longs;t, producendi &longs;unt te&longs;tes: huc reuoea principium illud commune.

1. Non &longs;unt multiplicanda entia &longs;ine nece&longs;&longs;itate, quod certè non valet ni&longs;i addas, vel &longs;ine ratione, vel &longs;ine experientia.

2. Qui a&longs;&longs;erit aliquid po&longs;itiuè, debet argumento po&longs;itiuo probare.

Axioma IV.

Quidquid exi&longs;tit phy&longs;icè extra &longs;uas cau&longs;as ab omni alio &longs;eparatum, de­terminatum e&longs;t.

Hoc Axioma explicatione modicâ indiget: Determinatum illud apello, quod illud ip&longs;um e&longs;t, quod e&longs;t, & nihil aliud; quod e&longs;t hoc, id e&longs;t ab omni alio di&longs;tinctum; atqui quidquid productum e&longs;t, &longs;ingulare e&longs;t, id e&longs;t, e&longs;t hoc; &longs;i enim producitur, alicubi producitur, & ali­quando, ergo dici pote&longs;t, e&longs;t hîc, e&longs;t nune; igitur determinatum e&longs;t. Aliquis fortè &longs;tatim opponet mihi partes indeterminatas quantitatis: &longs;ed pro&longs;ectò nulla pars actu e&longs;t quæ non &longs;it hæc, & non alia; igitur quæ non &longs;it determinata, de quo aliàs; quidquid &longs;it, &longs;altem partes illæ fa­ciunt aliquod totum quod e&longs;t determinatum, quod mihi &longs;atis e&longs;t modò ad veritatem huius Axiomatis. Dices aliquid po&longs;&longs;e e&longs;&longs;e nullibi; has nugas refutabimus in Metaphy&longs;ica, quæ in mentem &longs;apientis viri ca­dere non po&longs;&longs;unt; nunc &longs;altem con&longs;tat id naturali modo fieri non po&longs;&longs;e.

Axioma V.

Quod vnum e&longs;t, determinatum e&longs;t. Quia quod vnum e&longs;t, e&longs;t hoc, & nihil aliud; nihil enim aliud e&longs;t vnum, ni&longs;i indiui &longs;um in &longs;e, & diui­&longs;um à quolibet alio: quipnè indifferentia, vel indeterminatio ibi tan­tum e&longs;t, vbi &longs;unt plura; &longs;i enim tantum vnum e&longs;t, certè non datur op­tio, &longs;i aliqua cau&longs;a e&longs;t indifferens ad effectum A & B, id e&longs;t &longs;i non e&longs;t, cur vnum potius quàm alium producat? plures e&longs;&longs;e nece&longs;&longs;e e&longs;t; &longs;i enim tantùm vnus e&longs;t, certè indifferens non e&longs;t.

Axioma VI.

Quidquid e&longs;t, fru&longs;trà non e&longs;t. Quidquid e&longs;t, id e&longs;t exi&longs;tit naturaliter &longs;cilicet, & citra miraculum, fru&longs;trà non e&longs;t, id e&longs;t propter aliquem fi­nem e&longs;t ab ip&longs;a natura in&longs;titutum; finem autem rei ex ip&longs;o v&longs;u cogno­&longs;cimus; v&longs;um verò ip&longs;o ferè &longs;en&longs;u: quod vt breui inductione confirme­mus, quidquid exi&longs;tit vel e&longs;t &longs;ub&longs;tantia, vel accidens; &longs;i &longs;ub&longs;tantia, vel incorporea, vel corporea; &longs;i incorporea, vel e&longs;t Deus, vel Angelus, vel Animarationalis; atqui nihil horum fru&longs;trà e&longs;t, vt con&longs;tat; &longs;i corporea, vel e&longs;t corpus, vel forma; &longs;i corpus, vel elementum, vel mixtum; vtrumque &longs;uum finem habet, & con&longs;tantem v&longs;um; &longs;i forma quamdiu e&longs;t principium actionum compo&longs;iti fru&longs;trà non e&longs;t; quippe ad cum finem e&longs;t in&longs;tituta; hinc optima ratio ducitur, cur forma materialis &longs;eparata exi&longs;tere non po&longs;&longs;it citra miraculum, quia &longs;cilicet fru&longs;trà e&longs;&longs;et; cum enim 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 anima rationalis, quæ aliquas actiones inorganicas habet, fru&longs;trà non e&longs;t etiam &longs;eparata, igitur immortalis e&longs;t: vtramque rationem &longs;uo loco fu­sè demon&longs;trabimus; &longs;i verò accidens e&longs;t, haud dubiè alteri ine&longs;&longs;e debet propter &longs;uum finem intrin&longs;ecum, quem alibi effectum formalem &longs;ecun­darium appellamus; quem &longs;cilicet præ&longs;tat in &longs;uo &longs;ubiecto, cui certè &longs;i ni­hil præ&longs;taret, in co fru&longs;trà e&longs;&longs;et; &longs;ic caloris effectus &longs;ecundarius e&longs;t rare­factio, vel re&longs;olutio partium &longs;ui &longs;ubiecti, vel aliquid aliud; impetus, motus &c. Igitur tunc effet fru&longs;trà accidens, cum &longs;uo illo effectu careret; hinc rationem contrarietatis aliquando petemus, certi&longs;&longs;imam quidem, licet nouam, & inde clari&longs;&longs;imè con&longs;tabit, cur, & quomodo vnum contra­rium ab alio de&longs;trui dicatur; &longs;ed non e&longs;t huius loci: cùm verò audis fi­nem: ne quæ&longs;o cogites aliquid morale, nec enim illum finem intelligo, ad quem ab agente rationabili de&longs;tinatur: &longs;ed eum dumtaxat, ad quem na­tura ip&longs;a, vel e&longs;&longs;entia rei &longs;pectat, &longs;ed de his &longs;atis.

Huc reuoca Principium illud, Deus & Natura nihil faciunt fru&longs;trà,id e&longs;t quod &longs;uo fine careat intrin&longs;eco.

Dices fortè, multa videri e&longs;&longs;e fru&longs;trà, quæ tamen exi&longs;tunt; ad quid enim vel tanta aquarum copia, vel tantus &longs;tellarum numerus, vel tot are­næ puncta? tot fluitantes atomi? tot in&longs;ecta? & vermiculi: Re&longs;pondeo quamlibet &longs;tellam, quodlibet in&longs;ectum, &longs;eu vermiculum &longs;uis pollere pro­prietatibus; igitur fru&longs;trà non e&longs; & quodlibet punctum, quamlibet ato­mum, & quamlibet guttulam aquæ e&longs;&longs;e partem huius vniuer&longs;itatis: quod enim dices de vna, dicam de omnibus; equidem pauciores e&longs;&longs;e po&longs;&longs;ent; attamen nulla e&longs;t fru&longs;trà, cum quælibet &longs;imul cum aliis totum hoc com­ponat.

Axioma VII.

Tunc ponenda e&longs;t forma distincta &longs;ub&longs;t antialis vel accidentalis, dum e&longs;t ali­qua proprietas &longs;en&longs;ibilis, quæ non pote&longs;t tribui ip&longs;i materiæ, hîc res tantùm naturales con&longs;idero, nec &longs;uper naturales attingo, quæ &longs;uas regulas diui­næ fidei debent, non &longs;en&longs;ibus.

Hoc Axioma omninò certum e&longs;t, & per Ax. 3. confirmatur, vt enim dicas aliquid di&longs;tinctum ab omni alio exi&longs;tere, vel debet id &longs;en&longs;u percipi, vel aliqua ratione probari quod &longs;it; atqui formam &longs;ub&longs;tantialem &longs;en&longs;u non percipis immediatè; igitur aliquem eius effectum &longs;en&longs;ibilem vel me­diatè, vel immediatè; qui certè &longs;i tribui po&longs;&longs;it materiæ, haud dubiè per il­lum &longs;ormam non probabis, ni&longs;i formæ ip&longs;ius e&longs;&longs;e antè demon&longs;tres; &longs;i ve­to e&longs;t forma accidentalis, quam &longs;en&longs;u percipis; certè id tantùm accidit ex aliqua affectione, quâ&longs;en&longs;um ip&longs;um afficit hæc forma, igitur ex effectu il­lo illam percipis, quod clarum e&longs;t.

Huc reuoca vulgare illud principium, Frustrà fit per plura, quod po­test fieri per pauciora, quod ad Tertium etiam reuocatur; quod ita in­telligi non debet, vt &longs;ine gutta aquæ Oceanus, &longs;ine &longs;tella cœlum, &longs;ine gra­nulo arenæ terra, &longs;ine altero oculorum homo &longs;tare non po&longs;&longs;int; quæ 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 &longs;it &longs;ub&longs;tantia, &longs;iue accidens, quod vel experientia certa euincit, vel nece&longs;­&longs;itas, vel ratio, vel diuina fides (immò & humana in rebus humanis, non tamen in &longs;cientiis.)

Igitur nunquam claudicat hic equus Omi, vt vulgò dicitur, &longs;i hoc fræno regatur, & præ&longs;cripto ambulet pa&longs;&longs;u.

Scholium.

Ob&longs;eruabis &longs;eptem præmi&longs;&longs;a Axiomata, licet metaphy&longs;ica &longs;altem ali­qua ex parte e&longs;&longs;e videantur, ita pertinere ad Phy&longs;icam, vt plurimæ phy­&longs;icæ affectiones &longs;ine illis explicari, & demon&longs;trari non po&longs;&longs;int.

Primum certum e&longs;t etiam certitudine metaphy&longs;ica, &longs;eu geometrica. Secundum, Quartum, & Quintum per Primum demon&longs;trari po&longs;&longs;unt. Tertium e&longs;t veluti communis po&longs;itio, &longs;eu commune po&longs;tulatum, in quo docti omnes conucniunt; quippe nihil &longs;ine ratione dici debet à philo&longs;o­pho; Sextum & Septimum probari po&longs;&longs;unt per Tertium; &longs;ed iam ad alia, quæ propiùs ad phy&longs;icam accedunt, veniamus.

Axioma VIII.

Quidquid primò e&longs;t, & antè non erat, habet cau&longs;am di&longs;tinctam. Id e&longs;t quid­quid 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 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, quod verò nece&longs;&longs;ariò e&longs;t, non e&longs;&longs;e non pote&longs;t, alioquin priùs e&longs;&longs;et, & po&longs;terius, priùs vt cau&longs;a, po&longs;teriùs vt effectus: præterea quidquid produci­tur aliquando producitur, & alicubi, vt certi&longs;&longs;imum e&longs;t; &longs;ed quia hoc ali­qui negant, contendo tantùm in hoc rerum ordine, & naturaliter lo­quendo, quidquid producitur alicubi produci, & aliquando, quod nemo negabit; Igitur &longs;i aliquid &longs;e producit; cur hîc potiùs quam illîc? cur nunc potius quam antè? cum enim antè nullibi e&longs;&longs;et, cur de&longs;init non e&longs;&longs;e hîc & non illîc, nunc & non antè? hinc quod à &longs;e e&longs;t, vbique, & &longs;emper e&longs;t, &longs;ed ne quis mihi litem intendat, licet hoc Axioma certitudi­nem geometricam habeat; &longs;ufficit modò habere phy&longs;icam, quod ex om­nibus hypothe&longs;ibus demon&longs;tratur; &longs;i enim aliquid de nouo produci­tur, quod certum e&longs;t, abalio produci video: calor ab igne mediatè vel immediatè, impetus à potentia motrice, vel ab alio impetu: cuncta hæc &longs;i reueraproducuntur de quo alibi, ab alio produci con&longs;tat; in Me­taphy&longs;ica hoc ip&longs;um geometricè demon&longs;trabimus; cum enim agere &longs;up­ponat e&longs;&longs;e; quippe omnis actio alicuius agentis e&longs;t; & cum agere termi­netur ad effectum, nam fieri e&longs;t alicuius fieri; certè agens, & terminus, cau&longs;a, & effectus di&longs;tinguuntur, igitur. Quidquid primo e&longs;t, &c.

Axioma IX.

Cau&longs;a debet exi&longs;tere vt immediatè agat. Hoc certum e&longs;t; quia agere &longs;upponit e&longs;&longs;e; quippe agere e&longs;t perfectio realis actu exi&longs;tens; igitur ali­cuius actu exi&longs;tentis; igitur certum e&longs;t etiam Geometricè, de quo in Metaph. Iam vero &longs;ufficiat certum e&longs;&longs;e phi&longs;icè, vt con&longs;tat ex omnibus hypoth. phy&longs;icis; nihil enim videmus agere, ni&longs;i quod e&longs;t; &longs;i enim age­ret quod non e&longs;t; cur potius hîc, & nunc quam alibi, & aliàs? cur in hoc &longs;ubiecto potius quàm in alio?

Dices, finis qui non e&longs;t influit; igitur agit; Re&longs;pondeo finem non agere, nec influere ni&longs;i obiectiuè; atqui quod non exi&longs;tit actu, id e&longs;t in &longs;tatu entatiuo, & reali, pote&longs;t e&longs;&longs;e in &longs;tatu obiectiuo; id e&longs;t quod non habet actum rei, pote&longs;t habere actum obiecti, id e&longs;t e&longs;&longs;e cognitum, & volitum, de quo aliàs; porrò hîc tantùm intelligimus cau&longs;am efficien­tem, &c.

Dices, cau&longs;a principalis pulli exclu&longs;i pote&longs;t non e&longs;&longs;e; hæc omnia di­&longs;cutiemus &longs;uo loco cum de generatione animalium; &longs;ufficiat dixi&longs;&longs;e non e&longs;&longs;e cau&longs;am immediatam, de qua hîc tantum loquimur; idem re&longs;pon&longs;um e&longs;to de rana vaga.

Axioma X.

Cau&longs;a debet e&longs;&longs;e applicata vt immediatè agat. Cur enim potiùs hîc quam illîc; in hoc &longs;ubiecto potiùs, quam in alio, in hac di&longs;tantia potiùs, quam in alia? quidquid &longs;it, certum e&longs;t phy&longs;icè; nec enim ignis, qui e&longs;t Romæ, calefacit Lugduni.

Dices dari fortè actionem in di&longs;tans; Re&longs;pondeo negando, quod de­mon&longs;trabimus in Metaph. præterea, licet daretur in productione quali­tatum occultarum, & &longs;impathicorum quorundam effectuum, quos exa­minabimus &longs;uo loco; nemo tamen dubitat quin productio caloris, lu­minis, impetus; de quibus hic tantùm agimus, debeat e&longs;&longs;e ab applicata cau&longs;a.

Dices impetum produci in extremitate perticæ, quæ non e&longs;t applica­ta, vel in globo tudiculario etiam non applicato; calorem & lucem produci à Sole in terra non applicata. Re&longs;pondeo, e&longs;&longs;e applicationem mediatam; nam &longs;i reuera hæ qualitates producuntur continuata propa­gatione, diffunduntur per medium, in quo non e&longs;t difficultas.

Dices etiam partes interiores cau&longs;æ v. g. Solis agunt, &longs;ed non agunt per totum medium; alioquin agerent in alias partes Solis, à quibus obteguntur. Re&longs;pondeo, diffu&longs;ionem vel propagationem actioms in­choari tantum ab ipsâ &longs;uperficie Solis; quippe omnes partes agunt actione communi, de quo infrà; atqui actio communis à communi m­dio incipit.

Dices ignem produci in parte medij remota interrupta propagatio­ne, vt con&longs;tat, &longs;i vitro per refractionem, vel &longs;peculo per reflectionem radios Solares colligas.

Re&longs;pondeo, ignem quidem accendi in data di&longs;tantia; at non &longs;ine aliqua applicatione, &longs;altem virtutis, in quo non e&longs;t difficultas; quomo­do vero ignis accendatur, & quid &longs;it ignem accendi, explicabimus &longs;uo loco; quidquid &longs;it, certum e&longs;t ad productionem impetus requiri ali­quam applicationem, vt patet etiam in magnte. e

Axioma XI.

Si cau&longs;a vniuoca applicata, & non impedita est &longs;ufficiens ad productionem effectus, non e&longs;t ponenda alia &longs;cilicet æquiuoca. Non dico omnem cau&longs;am e&longs;&longs;e vniuocam, &longs;ed tantùm vniuocam &longs;ufficientem, & applicatam e&longs;&longs;e cau&longs;am, v. g. calor e&longs;t cau&longs;a &longs;ufficiens caloris, vt con&longs;tat in aqua calida; igitur &longs;i calor e&longs;t applicatus &longs;ubiecto, in quo producitur calor non &longs;upe­rans vires caloris applicati; dicendum e&longs;t calorem illum ab hoc produ­ci; cum calor &longs;it cau&longs;a nece&longs;&longs;aria; igitur &longs;i &longs;it applicatus &longs;ubjecto apto, nece&longs;&longs;ariò agit; igitur quantum pote&longs;t; igitur effectus non e&longs;t tribuen­dus alteri cau&longs;æ, quam &longs;ufficientem e&longs;&longs;e ignoramus.

Ad hoc Axioma aliud reuoca. Si ex applicatione alicuius &longs;equitur &longs;em­per effectus aliquis, illud ip&longs;um cau&longs;a dici debet huius effectus; licet aliud &longs;it coniunctum, ex quo &longs;eor&longs;im &longs;umpto applicato non &longs;equitur effectus; v. g. ex applicatione aquæ calidæ &longs;equitur productio caloris; ex applicatione &longs;o­lius aquæ non &longs;equitur; igitur dicendum e&longs;t calorem hunc produci ab ip&longs;o calore, qui aquæ ine&longs;t, non verò ab ip&longs;a aquæ &longs;ub&longs;tantia; idem dico de ferro frigido, &c.

Dices non e&longs;&longs;e certum calorem produci; Re&longs;pondeo, negando; &longs;ed, quidquid &longs;it, loquor tantùm hypotheticè; dixi enim &longs;i producatur, à calore aquæ inhærente producitur.

Dices produci po&longs;&longs;e ab aliqua cau&longs;a ignota po&longs;ita dumtaxat tali, vel tali conditione. Re&longs;pondeo, hoc reuera geometricè non probari, &longs;ed tantùm phy&longs;icè; quidquid &longs;it, voco cau&longs;am id, ex cuius applicatione &longs;equitur &longs;emper effectus, & nunquam aliàs; nam phy&longs;icè loquendo, &longs;iue &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 certum e&longs;t phy&longs;icè ignem calefacere, Solem illuminare, quod &longs;atis e&longs;t.

Axioma XII.

Ca nece&longs;&longs;aria &longs;ubiecto apto applicata, & non impedita &longs;emper agit, & quantum pote&longs;t. Hoc Axioma duas partes habet; prima certa e&longs;t per hy­poth. 8. & per definitionem cau&longs;æ nece&longs;&longs;ariæ, quæ in hoc differt à libe­tâ: Secunda pars probatur; quia &longs;i partem effectus omitteret, quam ta­men ponere po&longs;&longs;et; haud dubiè non e&longs;&longs;et cau&longs;a nece&longs;&longs;aria contra hypoth. nam &longs;i vnam partem effectus omittat; cur vnam potiùs quam aliam? cur non duas? cur non omnes? denique video cau&longs;am eandem eidem &longs;ubiecto codem modo applicatam, eundem &longs;emper effectum producere per Hyp. 8.

Axioma XIII

Exten&longs;io cau&longs;a non intendit effectum ad intra. Quælibet pars maioris ignis non habet calorem inten&longs;iorem, quàm quælibet pars minoris; idem dico de grauitate plumbi, &c. nec enim libra plumbi coniuncta cum alia habet diuer&longs;am grauitatem ab eâ, quam habet &longs;eparata.

Dixi ad intra; quia ad extra multum iuuat exten&longs;io; &longs;ic maior ignis longiùs diffundit &longs;uum calorem; corpus grauiùs cadens majorem ictum infligit; Ad hoc Axioma reuocatur i&longs;tud.

1. Omnes partes eiu&longs;dem cau&longs;æ agunt ad extra actions communi, iuxta eum modum quo illam explicabimus in Metaph. nec punctum Solis &longs;e­paratum ad eandem di&longs;tantiam &longs;uam lucem, caloremque &longs;uum diffunde­ret; ad quam diffundit coniunctum cum aliis; idem dico de igne maiori, & minori; de quibus omnibus &longs;uo loco. Huc etiam reuoca dicta illa communia.

2. Plures partes cau&longs;a plures partes effectus producunt, & vici&longs;&longs;im.

3. Maior, & perfectior cau&longs;a maiorem effectum producit, & perfectiorem, & vici&longs;&longs;im.

4. Perfectior effectus, vel imperfectior arguit cau&longs;am perfectiorem, vel im­perfectiorem, &longs;uppo&longs;itâ eâdem applicatione; &longs;i enim maior e&longs;t applicatio &longs;ine ratione loci, &longs;iue ratione temporis; haud dubiè maior erit effectus, vt con&longs;tat.

Axioma XIV.

Quidquid de&longs;truitur non e&longs;t à &longs;e. Hoc Axioma geometricum e&longs;t; Quod enim e&longs;t à &longs;e, nece&longs;&longs;ariò e&longs;t; cùm à libertate &longs;eu voluntate alterius non pendeat; cum enim primo in&longs;tanti quo res e&longs;t, non &longs;it à &longs;e per Axiom. 8. de &longs;ecundo idem dici debet, quod de primo, vt patet: quippe id eo 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 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 nece&longs;&longs;ariò, igitur pendet ab alio, quod pote&longs;t facere vt non &longs;it.

Dices po&longs;&longs;e de&longs;trui &longs;ecundo in&longs;tanti ab aliquo contrario, à quo tamen non pendet per po&longs;itiuum influxum. Re&longs;pondeo, non videri quomo­do de&longs;trui po&longs;&longs;it, quod influxu po&longs;itiuo non indiget, vt &longs;it; quid enim faceret contrarium, quod tantùm exigere pote&longs;t contrarij de&longs;tructio­nem, quid e&longs;t porro de&longs;trui, ni&longs;i de&longs;inere con&longs;eruari? quæ omnia fusè in Metaphy&longs;ica demon&longs;trabimus; quidquid enim e&longs;t aliquo in&longs;tanti vel 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: quidquid &longs;it, hoc Axioma certum e&longs;t phy&longs;icè.

Huc reuoca Axiomata &longs;equentia, quæ ex hoc vno deducuntur.

1. Quidquid e&longs;t, & non e&longs;t à &longs;e, e&longs;t, &longs;eu pendet, &longs;eu con&longs;eruatur ab alio.Hæc enim &longs;unt idem, vt con&longs;tat.

2. Quidquid destruitur, ad exigentiam alicuius de&longs;truitur, &longs;altem totius natura, ne aliquid &longs;it fru&longs;trà. Hoc etiam ex hypothe&longs;ibus &longs;equitur; cum enim de&longs;trui &longs;it idem ac de&longs;inere con&longs;eruari; certè qui de&longs;init con&longs;er­uare in&longs;tanti A potiùs quam in&longs;tanti B, hoc facere non pote&longs;t ni&longs;i ali­quid hoc exigat; &longs;cilicet iuxta leges naturæ.

3. Tandiu aliquid con&longs;eruatur, quandiu nihil exigit eius de&longs;tructionem.Hoc &longs;equitur ex priori, id e&longs;t quandiu e&longs;t eadem ratio, cur &longs;it, & con­&longs;eruetur, quæ erat antè.

Axioma XV.

Contraria pugnant pro rata. Nec enim alia regula e&longs;&longs;e pote&longs;t; &longs;ic minot calor minùs de&longs;truit frigoris; minor impetus minùs de&longs;truit impetus contrarij (&longs;i contrarium habet) quæ omnia con&longs;tant ex hypothe&longs;ibus. Ratio e&longs;t, quia plùs vel minùs contrarij de&longs;truere, multam habet ex­ten&longs;ionem. v.g. &longs;int duo contraria A & B, &longs;it A vt 20. &longs;it B vt 5. certè &longs;i B de&longs;truat A &longs;upra ratam, vel &longs;upra id, quod &longs;ibi ex æquo re&longs;pondet, id e&longs;t &longs;upra 5. cur potius 6. quam 7. 8. &c. Si infra, cur potius 4. quam 3. 2. &c. Igitur cum plures &longs;int termini tùm infra, tùm &longs;upra 5. cur potius vnus quàm alius? atqui vnus tantùm ex æquo re&longs;pondet, &longs;cilicet 5. &longs;ed quod vnum e&longs;t determinatum e&longs;t, per Axioma 5. igitur pugnant pro rata. Nec dicas A totum de&longs;trui à B, quòd e&longs;t contra hypothe&longs;im, nam modicum caloris non de&longs;truit totum frigus: in impetu res e&longs;t clari&longs;&longs;ima; adde quod minor cau&longs;a minùs agit per Ax. 13. num. 3. igitur minùs exi­git; porrò cum dico vnum ab alio de&longs;trui, intelligo tantùm ex applica­tione vnius &longs;equi de&longs;tructionem alterius &longs;alrem ex parte.

Ob&longs;eruabis hæc Axiomata &longs;altem maiori ex parte e&longs;&longs;e metaph. quæ nos fusè in Theorematis metaph. explicabimus, & demon&longs;trabimus; &longs;ed nobis hoc loco &longs;atis e&longs;t, &longs;i parem cum phy&longs;icis &longs;upponas habere cer­titudinem, quod nemo negabit; con&longs;tátque ex hypothe&longs;ibus, licèt ma­iorem etiam habeant, de qua &longs;uo loco.

Ob&longs;eruabis prætereà nos diutiùs hæ&longs;i&longs;&longs;e in præmittendis huic libro Axiomatis, quod tamen in aliis libris non faciemus.

Postulatum,

Liceat datum corpus impellere, proiicere, deor&longs;um cadens excipere, motus durationem &longs;en&longs;ibilem, &longs;patiumque &longs;en&longs;ibile, metiri, comparare, &c.

Theorema 1.

Motus e&longs;t aliquid realiter di&longs;tinctum à mobili. Demon&longs;tratur; Motus e&longs;t in mobili, in quo antè non erat per hypoth. 3. & de&longs;init e&longs;&longs;e in mobili, in quo antè erat per hypoth.4. igitur mobile e&longs;t, & non e&longs;t motus; igi­tur à motu &longs;eparatum; igitur realiter di&longs;tinctum per Ax. 2. præterea moueri, & non moueri &longs;unt prædicata contradictoria, vt con&longs;tat; igi­tur cidem &longs;imul ine&longs;&longs;e non po&longs;&longs;unt per Ax. 1. igitur cum co non &longs;unt idem; alioquin &longs;imul e&longs;&longs;ent; igitur alterum illorum e&longs;t di&longs;tinctum à mobili; non quies, vt con&longs;tat, quæ e&longs;t tantùm negatio motus, &longs;eu per­&longs;euerantia in codem loco; igitur nullam dicit mutationem; at verò motus mutationem dicit, per Def. 1. hoc Theorema fusè demon&longs;trabo in Metaph.

Theorema 2.

Motus non pote&longs;t dici propriè preductus immediatè, vel effectus immedia­tus cau&longs;æ efficientis. Demon&longs;t. Motus e&longs;t mutatio, &longs;eu tian&longs;itus ex loco in locum per Def. 1. &longs;ed mutatio propriè non producitur; quppè pro­ductio tantùm terminatur ad ens; nihil enim ni&longs;i ens produci pote&longs;t; atqui nulla mutatio dicit tantùm ens; præ&longs;ertim hæc, quæ tantùm dicit terminum à quo, ide&longs;t locum relictum; & terminum ad quem, id e&longs;t lo­cum immediatum acqui&longs;itum; nam &longs;eparato quocunque alio ab ip&longs;o mobili; modo &longs;imul, id e&longs;t eodem in&longs;tanti relinquat primum locum, & nouum acquirat, omninò mouetur, &longs;ed concretum illud ex loco relicto, & acqui&longs;ito produci non pote&longs;t; illud autem e&longs;t motus, qui certè non dicit tantùm locum relictum &longs;ine acqui&longs;ito; alioqui &longs;i mobile de&longs;true­retur, diceretur moueri; nec etiam locum acqui&longs;itum &longs;ine priori relicto: alioqui &longs;i mobile primò produceretur, diceretur moueri localiter; igitur motus neutrum dicit &longs;eor&longs;im; &longs;i primum, diceretur de&longs;tructus; &longs;i &longs;ecun­dum, diceretur aliquo modo productus, vel potiùs acqui&longs;itus; at vtrum­que coniunctim, &longs;imulque e&longs;&longs;entialiter dicit motus; nec enim conci­pio aliud, dum concipio motum: porrò vtrumque &longs;imul &longs;umptum indi­ui&longs;ibiliter non pote&longs;t dici, vel de&longs;tructum propriè, vel productum; Dixi propriè; nam impropriè dici pote&longs;t motus productus.

Dices Motus e&longs;t ens, non à &longs;e; igitur ab alio; igitur motus e&longs;t pro­ductus. Re&longs;pondeo Motum non e&longs;&longs;e ens ab&longs;olutum, &longs;ed e&longs;&longs;e mutatio­nem entis, quæ mutatio e&longs;t concretum quoddam ex ente & non ente; quòd certè non pote&longs;t dici propriè productum, &longs;ed re&longs;ultans, vt relatio; nam producatur, &longs;i fieri pote&longs;t; certè e&longs;t aliquid, quod tam facilè de­&longs;trui pote&longs;t, quam produci; igitur de&longs;truatur, & remaneat tantùm en­titas mobilis, quæ, quo in&longs;tanti priorem locum relinquit, nouum acqui­rat; certè dicitur adhuc moueri, & tamen non erit motus ex &longs;uppo&longs;itio­ne, quod ab&longs;urdum e&longs;t.

Dices potentia motrix e&longs;t actiua; igitur agit; igitur producit, &longs;ed ni­hil ni&longs;i motum. Re&longs;p. potentiam motricem e&longs;&longs;e actiuam vt dicemus, & ab eâ produci impetum, qui deinde exigit motum, vt dicemus infrà.

Nec e&longs;t quod aliqui ita mirentur hæc à me dici; cum certum &longs;it effe­ctus &longs;ormales &longs;ecundarios principum ferè qualitatum tales e&longs;&longs;e, vt mini­mè producantur; &longs;ed qua&longs;i re&longs;ultent ab exigentia; v. g. effectus calo­ris in &longs;uo &longs;ubiecto e&longs;t eiu&longs;dem &longs;ubiecti rarefactio, quæ reuerâ non producitur, vt con&longs;tat.

Theorema 3.

Motus e&longs;t ab alio di&longs;tincto in aliquo genere cau&longs;æ. Demon&longs;tratur, quia motus, qui non erat, incipit e&longs;&longs;e per hypothe&longs;im tertiam; &longs;ed quod huiu&longs;modi e&longs;t, habet cau&longs;am di&longs;tinctam per Ax.8.

Scholium.

Ob&longs;eruabis motum localem e&longs;&longs;e duplicis generis; primum genus mo­tus e&longs;t actio potentiæ motricis, quæ reuerà mouet, & cuius exercitium dicitur motus, &longs;eu latio, &longs;eu motio, &longs;eu actio, qua reuerâ agit, produ­citque impetum, non motum; cum etiam &longs;ine motu defatigetur, vt cum quisalium pellit, à quo pellitur æquali ni&longs;u; patet etiam in manu &longs;u­ine nte aliquod pondus, quæ non mouetur; licet reuerâ etiam &longs;ummo conatu agat: immò &longs;i potentia motrix produceret motum primum, non impetum in corpore proiecto; nulla deinde e&longs;&longs;et cau&longs;a applicata ad pro­ducendum impetum: Itaque hic motus primi generis, &longs;i comparetur cum potentia motrice, e&longs;t verè in&longs;luxus, vel actio; &longs;i cum termino, e&longs;t eius fieri, &longs;eu dependentia; &longs;i cum &longs;ubiecto, &longs;eu mobili e&longs;t pa&longs;&longs;io; nec propriè dicitur produci, ni&longs;i vt quo (vt vulgò loquuntur) nec enim actio e&longs;t terminus, vel effectus, in quo &longs;i&longs;tat cau&longs;a; &longs;ed e&longs;t via, qua ten­dit ad terminum. Motus &longs;ecundi generis e&longs;t mutatio, &longs;eu tran&longs;itus ex vno loco in alium; hoc e&longs;t finis, vel effectus formalis &longs;ecundarius, quem exigit impetus; & fru&longs;trà ponitur alia entitas, quæ tantùm e&longs;&longs;et in&longs;tituta ad exigendam i&longs;tam loci mutationem; Igitur &longs;i &longs;ufficienter exigatur ab ip&longs;o impetu, de quo infrà, certè fru&longs;tra ponitur quodcun­que aliud per Ax.3. & 7.

Theorema 4.

Cau&longs;ailla immediata motus, quæ non est efficiens, potest tantùm e&longs;&longs;e exi­gens, quæ reducitur ad formalem, quæ &longs;uum effectum formalem &longs;ecundarium, id est &longs;uum finem intrin&longs;ecum exigit. Sic calor exigit rarefactionem, vel re&longs;olutiouem, impetus motum; cum enim non &longs;it cau&longs;a efficiens per Th. 2. &longs;it tamen cau&longs;a per Th.3. nec &longs;it materialis, nec finalis, vt con&longs;tat, de­bet e&longs;&longs;e formalis, vel exigens, &longs;eu exigitiua; vt patet ex ip&longs;a cau&longs;arum enumeratione; non e&longs;t materialis, quia non recipit motum, ni&longs;i ab alio; nec finalis, quæ &longs;upponit alias; cum ip&longs;a non &longs;it dum ponitur effectus.

Theorema 5.

Entitas &longs;eu &longs;ubstantia mobilis non e&longs;t cau&longs;a immediata motus, Sit enim lapis proiectus per Po&longs;tul. haud dubiè &longs;ub&longs;tantia lapidis non e&longs;t cau&longs;a huius motus; quia lapis tandem &longs;i&longs;tit per hypoth.4. igitur non e&longs;t cau&longs;a motus, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria; igitur &longs;emper cau&longs;aret per Ax.12. præ­terea potentia motrix proiicientis verè agit, cum etiam defatigetur; igi­turaliquid producit, non motum immediatè, qui produci non pote&longs;t pro prièper Th. 2. Adde quod motus &longs;ecundi generis habet tantùm caulam immediatam exigentem, &longs;ed potentia motrix non exigit; quia primò non defatigaretur exigendo; &longs;ecundò quia lapis &longs;eparatus à manu etiam mouetur, &longs;ed non ad exigentiam potentiæ motricis, vt patet; quia &longs;tatim po&longs;t &longs;eparationem pote&longs;t illa potentia de&longs;trui, licèt lapis longo pò&longs;t temporemoueatur; &longs;ed quod non e&longs;t, nihil exigit.

Aliquis fortè diceret potentiam motriœm exigere primam partem motus, quæ deinde &longs;ecundam exigit, & &longs;ecunda tertiam, tertia quar­tam, &c. Sed contra; quæro quid &longs;it prima illa pars motus; nec enim aliud agno&longs;co ni&longs;i primam mutationem loci, quæ mutatio non pote&longs;t exigere ni&longs;i quando e&longs;t; atqui quando e&longs;t, nihil reale e&longs;t actu ni&longs;i mo­bile, & nouus locus acqui&longs;itus, mobile ip&longs;um non exigit, vt demon&longs;tra­tum e&longs;t, & conce&longs;&longs;um, nec etiam locus de nouo acqui&longs;itus, in quo &longs;cilicet mobile &longs;i&longs;tere pote&longs;t: quidquid pones aliud, impetum appellabo.

Dices cum graue aliquod mouetur deor&longs;um, vel leue &longs;ur&longs;um, vel corpus animatum &longs;e ip&longs;um mouet, dici pote&longs;t &longs;ub&longs;tantia corporis cau&longs;a immediata motus. Re&longs;p. negando, tùm quia omnis potentia motrix agit; igitur producit aliquid aliud, quod e&longs;t cau&longs;a motus: præterea po­tentia motrix corporis animati, agit v&longs;que ad defatigationem, &longs;udorem, licèt non &longs;it motus, igitur aliud producit, de corpore graui probabi­mus infrà.

Theorema 6.

Datur impetus. Demon&longs;tro, Sub&longs;tantia mobilis non e&longs;t cau&longs;a imme­diata motus, per Th.5. ergo aliquid aliud; igitur impetus, nam quod di­&longs;tinctum e&longs;t à &longs;ub&longs;tantia mobilis, & exigit motum, e&longs;t impetus per Def.3. &longs;ed quia hoc Theorema e&longs;t veluti princeps huius tractatus cardo, in eo paulò diutius hærendum e&longs;t, igitur.

Demon&longs;tro primò dari impetum: Quidquid e&longs;t, & antè non erat, non 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 tertiam; igitur habet cau&longs;am, &longs;ed non aliam, quam impetum, quod pro­bo: Lapis cadens, vel impactus in alium lapidem mouet illum per hy­poth.7. &longs;ed &longs;ub&longs;tantia lapidis in alium impacti non e&longs;t cau&longs;a huius mo­tus, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt patet; igitur applicata eundem effe­ctum produceret per Ax.12. &longs;ed etiam applicata immediata non agit, vt con&longs;tat experientia; igitur per idem Axioma non e&longs;t cau&longs;a.

Scio e&longs;&longs;e aliquas re&longs;pon&longs;iones, quas infrà refellemus; nunc &longs;ufficiat dixi&longs;&longs;e lapidem impactum non producere motum, qui propriè non pro­ducitur per Th.2. nec exigere, vt con&longs;tat ex &longs;ecunda probatione Th. 5. igitur &longs;i aliquid exigit, vel producit, voco impetum.

Secundò probatur; potentia motrix e&longs;t actiua, quia defatigatur, quis hoc neget? igitur aliquid producit; non motum, qui propriè non pro­ducitur per Th.2. igitur aliquid aliud; voco impetum; adde quod etiam &longs;ine motu agit, & defatigatur vt iam dictum e&longs;t; igitur habet alium effe­ctum immediatum; denique mouere, pellere, trahere, proiicere, percu­tere, nihil ni&longs;i actionem &longs;onant.

Tertiò probatur; pila di&longs;iuncta à manu proiicientis diu adhuc mo­uetur per hypoth.6. igitur hic motus habet cau&longs;am per Ax. 8. quælibet enim pars motus de nouo e&longs;t, neque duæ illius partes &longs;imul e&longs;&longs;e po&longs;&longs;unt. atqui potentia motrix non e&longs;t can&longs;a per Ax.10. immò pote&longs;t e&longs;&longs;e de&longs;tru­cta; igitur non e&longs;t cau&longs;a per Ax. 9. Non e&longs;t etiam cau&longs;a &longs;ub&longs;tantia pilæ mobilis per Th.5.5. nec priores pattes motus per re&longs;p. ad primam in­&longs;tantiam Th 5. igitur aliquid aliud; voco impetum.

Quartò probatur; pila proiecta &longs;en&longs;im &longs;ine &longs;en&longs;u tardiore motu mouetur; donec tandem moueri omnino de&longs;inat per hypoth. 5. igitur non e&longs;t &longs;emper æqualis, & eadem cau&longs;a huius motus per Ax. 12. & 13. num.3. igitur cau&longs;a huius motus eodem modo debilitatur, &longs;eu remitti­tur, quo ip&longs;e motus; &longs;ed decre&longs;cit &longs;ub&longs;tantia mobilis, nec potentia mo-trix, vel corpus prius impactum; ergo e&longs;t alia cau&longs;a præ&longs;ens, quæ mi­nuitur; voco impetum.

Quintò corpus graue deor&longs;um cadens accelerat &longs;uum motum, vt patet experientia; quæ maximè clara e&longs;t in funependulis, de qua in &longs;equen­tibus libris; igitur debet e&longs;&longs;e cau&longs;a huius motus velocioris; non e&longs;t au­tem &longs;ub&longs;tantia lapidis, nec grauitas per Ax. 12. nec aliud quidpiam ex­trin&longs;ecum, vt videbimus &longs;uo loco; igitur aliquid aliquid inttin&longs;ecum, voco impetum. Igitur certum e&longs;t dari impetum; qui certè tribui non pote&longs;t, vel vlli connotationi, vel alteri exigentiæ, vt con&longs;tat ex dictis.

Diceret fortè alius hæc omnia e&longs;&longs;e dubia; nam fieri pote&longs;t vt Deus tantùm moueat; quod &longs;ine impetu fieri po&longs;&longs;e certum e&longs;t; Re&longs;p. equi­dem per miraculum hoc fieri po&longs;&longs;e; &longs;ed quemadmadum certum e&longs;t phy­ficè ignem applicatum calefacere, niuem frigefacere, & modò calamum à me hæc &longs;cribente moueri, ita certum o&longs;t phy&longs;icè &longs;agittam à &longs;agittario emitti, & pilam à proiiciente, &c. adde quod Deus, vt auctor naturæ e&longs;t, agit tantùm; vel de&longs;init agere iuxta exigentiam cau&longs;arum &longs;ecunda­rum; denique cau&longs;am phy&longs;icè appello, ex cuius applicatione nunquam non &longs;equitur effectus per Ax.11. num.1.

Dicerent alij hoc totum prouenire à corpu&longs;culis; vel atomis, vel fila­mentis &longs;ine vlla actione; equidem non reiicio corpu&longs;cula, & perennia corporum effluuia: Dico tamen primò globum quie&longs;centem humi ha­bere &longs;altem aliquas partes quie&longs;centes, vel immobiles; quis hoc neget? immò maximam &longs;uarum partium partem; igitur cum deinde proiicitur idem globus, illæ partes mouentur; dari igitur debet cau&longs;a huius motus per Ax.8, igitur impetus: nec dicas moueri illas partes à corpu&longs;culis; quia antè erant eadem, immò plura corpu&longs;cula; & tamen non mouebant: igi­tur non &longs;unt cau&longs;a huius motus per Ax.12. Dices excitari; &longs;ed quid hoc e&longs;t excitari? vel enim mutantur, vel non mutantur; &longs;ecundum dici non pote&longs;t; quia vt excitentur, ex non excitatis mutari debent; igitur per aliquid: deinde quid e&longs;t illa excitatio, ni&longs;i impul&longs;io; igitur &longs;i mouen­tur illa corpu&longs;cula, & excitantur à potentia motrice, etiam partes prius immobiles moucbuntur, & excitabuntur per Ax.12. quia &longs;unt applicatæ cau&longs;æ nece&longs;&longs;ariæ.

Dico &longs;ecundò minimum ex his corpu&longs;culis non &longs;emper moueri; po­te&longs;t enim &longs;i&longs;tere; quis hoc neget? igitur &longs;i modò mouetur, modò quie&longs;­cit, motus ab eo di&longs;tinguitur per Th.1. igitur mouetur per impetum, de quo infrà.

Igitur datur nece&longs;&longs;ariò impetus, &longs;ine quo non po&longs;&longs;unt explicari prædi­ctæ omnes hypothe&longs;es, contra quem &longs;unt quidem graui&longs;&longs;imæ difficultates, quas &longs;en&longs;im in &longs;equentibus Theorematis, in quibus explicantur pro­prietates huius impetus, di&longs;cutiemus.

Diceret aliquis lapidem impul&longs;um ab aëre deinde propelli; &longs;ed aër po­tius re&longs;i&longs;tit motui; vt con&longs;tat experientiâ; &longs;ed hoc &longs;oluemus infrà.

Theorema 7.

Impetus est aliquid distinctum à &longs;ubstantiâ mobilis. Demon&longs;tratur. Quia &longs;ub&longs;tantia mobilis non e&longs;t cau&longs;a exigens motum per Th. 5. Impe­tus e&longs;t cau&longs;a exigens per Def. 3. & Th. 6. de codem contradictoria dici non po&longs;&longs;unt per Ax. 1. n. 3. Igitur impetus non e&longs;t idem cum &longs;ub&longs;tantià mobilis; igitur di&longs;tinctus; dcinde &longs;eparari pote&longs;t à &longs;ub&longs;tantia mobilis per Hypoth. 4. igitur e&longs;t di&longs;tinctus per Ax. 2.

Theorema 8.

Impetus est accidens; Quippe non e&longs;t corpus, nec forma &longs;ub&longs;tantia­lis; quia omne corpus, & omnis forma &longs;ub&longs;tantialis moueri pote&longs;t, & non moucri, vt con&longs;tat ex po&longs;t. & ex Hypoth. 3. & 4. igitur di&longs;tingui­tur à motu; igitur & ab impetu per Ax. 2. igitur impetus non e&longs;t &longs;ub­&longs;tantia; igitur accidens.

Theorema 9.

Impetus non e&longs;t modus. Modus duplicis generis e&longs;&longs;e pote&longs;t: Modus primi generis e&longs;t entitas quædam diminuta, vt vulgò loquuntur, di&longs;tin­cta quidem modaliter, vt aiunt, à re, cui adhæret; ac proinde ab ca &longs;e­parari pote&longs;t, non tamen exi&longs;tere &longs;eparata. Modus &longs;ecundi generis non e&longs;t entitas quidem di&longs;tincta; e&longs;t tamen &longs;tatus quidam corporis; &longs;ic &longs;e&longs;&longs;ie&longs;t modus, conden&longs;atio, compre&longs;&longs;io, &c. His po&longs;itis Impetus non e&longs;t mo­dus primi generis; nihil enim probat impetum e&longs;&longs;e modum, quod etiam non probet calorem, & luccm e&longs;&longs;e modos; dicere autem omnia acci­dentia e&longs;&longs;e modos non debemus, de quo &longs;uo loco; modus enim ita à na­turâ comparatus e&longs;t, vt &longs;ine &longs;ubiecto actuali &longs;eu fulcro non exi&longs;tere mo­dò, &longs;ed ne concipi quidem po&longs;&longs;it; v. g. actio non pote&longs;t concipi ni&longs;i &longs;it alicuius actio; nec fieri &longs;ine facto; nec via &longs;ine termino; nec dependen­tia &longs;ine dependente; at verò po&longs;&longs;um concipere calorem, & impetum &longs;ine alio, quod &longs;it actu; licèt enim calor exigat re&longs;olutionem partium &longs;ui &longs;ubiecti, &longs;eu rarefactionem, & impetus motum; nihil tamen impe­dit, quin per miraculum calor, & impetus con&longs;eruari po&longs;&longs;int &longs;ine eo. quod exigunt, hoc e&longs;t &longs;ine &longs;uo &longs;ine; igitur &longs;ine &longs;ubiecto; non e&longs;t etiam modus &longs;ecundi generis vt patet, &longs;ed de modis in Metaphy&longs;ica; vix enim hoc Theorema ad rem Phy&longs;icam quicquam facit.

Theorema 10.

Impetus e&longs;t qualitas Phy&longs;ica. Sequitur ex dictis; cum nec &longs;it motus. nec &longs;ub&longs;tantia, nec modus, nec quidquam negatiuum, alioquin exige­ret; igitur e&longs;t aliud accidens; vocetur qualitas.

Theorema 11.

Impetus est qualitas Phy&longs;ica. Quia impetus e&longs;t di&longs;tinctus realiter à &longs;ue &longs;ubiecto per Th. 7. E&longs;t enim &longs;eparabilis per Hypoth. 3. & 4. igitur di­&longs;tinctus per Ax. 2. &longs;ed qualitatem realiter di&longs;tinctam apello Phy&longs;icam; præ&longs;ertim cum nec moralis &longs;it, nec Logica, &c.

Theorema 12.

Impetus est qualitas permanens. Quia lapis proiectus etiam &longs;eparatus mouetur aliquandiu per Hyp. 6. igitur durat eius cau&longs;a, &longs;cilicet impe­tus; igitur e&longs;t qualitas permanens.

Diceret fortè aliquis lapidem proiectum pelli ab aëre à tergo in&longs;tan­te, vt voluit Ari&longs;toteles pluribus in locis; &longs;ed præ&longs;ertim 8. Ph.c.vlt.& 7. cap.2. 3.de Cœlo, cap. 3. Re&longs;pondeo hoc dici non po&longs;&longs;e; Primò quia non modò non iuuat aër; &longs;ed ctiam impedit motum proiecti, quod de omni medio nece&longs;&longs;ariò dicendum e&longs;t, vt patet experientiâ; vnde quo cra&longs;&longs;ius, &longs;eu den&longs;ius e&longs;t medium, motum potentiùs ratardat, vt videmus in proiectis per aquam; rationem à priori afferemus infrà, cum de re&longs;i&longs;tentia medij: Secundò, quis dicat pilam rotatam in &longs;olo moueri aëris appul&longs;u? cum alia corpora, quæ pila rotata præterlambendo qua&longs;i allambit, nullo mo­do moueantur; præ&longs;ertim granula pulueris. Tertiò, an fortè aër id præ­&longs;tare pote&longs;t &longs;ine vi impre&longs;&longs;a; igitur non minus ip&longs;i pilæ proiectæ, quam aëri ambienti imprimi poterit: Quartò, nullus aër à tergo pellitur; &longs;ed potius ip&longs;a pila aducr&longs;us aëra pellit, dum emittitur manu; igitur &longs;i aër &longs;uccedit à tergo, id totum accidit, vel metu vacui, vel ne aër compri­matur, vt videbimus infrà. Quintò denique, cum diu moueatur cadem pars aëris, haud dubiè in ca manet vis impre&longs;&longs;a; igitur impetus erit ad­huc qualitas permanens.

Ad id quod obiicitur ex Ari&longs;totele; aliqui putant inclina&longs;&longs;e in cam &longs;en­tentiam; cùm tam en no&longs;tram tencant illu&longs;tres Peripatetici, quorum no­minibus parco, ne tot citationes paginas impleant; vide apud Conim­bric. l. 7. Phy&longs;. cap. 2. Aliqui excu&longs;ant ip&longs;um Ari&longs;torelem, putantque non e&longs;&longs;e locutum ex propriâ &longs;ententiâ: Alij dicunt Ari&longs;totelem quidem tribui&longs;&longs;e aliquam vim extrin&longs;ecam aëri; non tamen nega&longs;&longs;e intrin&longs;ecam impetus; quidquid &longs;it, ip&longs;a verba Ari&longs;totclis demon&longs;trant ip&longs;um agno­ui&longs;&longs;e vim motricem impre&longs;&longs;am aëri, hoc e&longs;t impetum (potentia enim (in­quit) &longs;cilicet motrix, quâ pollet proijciens qua&longs;i vim impre&longs;&longs;am tradit vtrique) id e&longs;t aëri &longs;ur&longs;um, &deor&longs;um; quid porrò e&longs;t illa vis motrix, ni&longs;i impetus.

Theorema 13.

Impetus non producit motum. Probatur, quia motus non dicitur pro­ductus per Th. 2. Adde &longs;i vis rationem metaphy&longs;icam; quia nihil cogit dicere accidens aliquod, ex iis &longs;cilicet, quæ &longs;en&longs;u percipimus, agere ad intra; quod videtur e&longs;&longs;e proprium &longs;ub&longs;tantiæ, &longs;altem naturaliter; vt demon&longs;trabimus in Metaph.

Theorema 14.

Impetus exigit motum, id est fluxum mobilis in loco; quia cau&longs;a imme­diata motus e&longs;t tantum exigens, per Th. 4. &longs;ed impetus e&longs;t cau&longs;a motus immediata per Th. 5. & 6. igitur e&longs;t cau&longs;a exigens, adde quod id tantùm accidens &longs;en&longs;ibile præ&longs;tare pote&longs;t in &longs;uo &longs;ubiecto, vt aliquam illius mu­tationem præ&longs;tet, vel exigat; quæ vel e&longs;t localis, hoc e&longs;t fluxus quidam: per &longs;patium loci; vel alteratiua, vt vulgò vocatur; quà &longs;cilicet vel re­&longs;oluuntur partes, vel rare&longs;iunt, vel lique&longs;cunt, vel concre&longs;cunt &c. vel demùm mutant &longs;en&longs;ibilem &longs;tatum; vel e&longs;t perfectiua aliquo modo, qua­tenus &longs;ubiectum nouam aliquam habitudinem acquirit ad &longs;en&longs;us; &longs;ic lumen illuminando obiectum reddit illud vi&longs;ibile. &c. de quibus aliàs.

Theorema 15.

Motus e&longs;t effectus formalis &longs;ecundarius impetus. Cum enim &longs;it cau&longs;a exigens per Th. 121. Voco effectum formalem &longs;ecundarium, quem in mobili exigit impetus; quippe, vt iam dictum e&longs;t, cau&longs;a exigens redu­citur ad formalem; nec enim cau&longs;at aliquid producendo, quod &longs;pectat ad efficientem; nec mouendo, quod &longs;pectat ad finalem; nec determinando, quod &longs;pectat ad obiectiuam; nec recipiendo, quod &longs;pectat ad materia­lem; nec dirigendo, quod &longs;pectat ad idæalem, vel exemplarem; &longs;ed exigendo; quatenus &longs;cilicet ad id à natura e&longs;t in&longs;tituta, vt ex eius in &longs;ubiecto præ&longs;entia talis affectio, vel mutatio con&longs;equatur; vocatur au­tem effectus formalis &longs;ecundarius; non verò primarius, qui e&longs;t tantùm concretum ex ip&longs;a formâ, & &longs;ubiecto.

Theorema 16.

Motus e&longs;t finis intrin&longs;ecus impetus. Dum finem audis intrin&longs;ecum, cogita quæ&longs;o aliquid phy&longs;icum; e&longs;t enim id, propter quod talis, vel ta­lis forma in&longs;tituta e&longs;t: quid enim aliud e&longs;&longs;e pote&longs;t; finem enim rerum naturalium ex ip&longs;o v&longs;u cogno&longs;cimus; immò idem e&longs;t finis cum ip&longs;o v&longs;u; cum igitur impetus illum tantùm v&longs;um habeat, quem in ip&longs;o mobili præ&longs;tare cernimus, &longs;cilicet motum; dicendum e&longs;t motum e&longs;&longs;e finem in­trin&longs;ecum impetus; adde quod cum fru&longs;trà &longs;it impetus ille, qui non præ­&longs;tat motum mediatè &longs;altem in &longs;uo &longs;ubiecto; quid enim aliud in &longs;uo &longs;ub­iecto præ&longs;taret, quem effectum, quam mutationem? certè &longs;i fru&longs;trà e&longs;t, non 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 maximum eius bonum e&longs;t, igitur finis, quem natiuâ vel innatâ veiut appetentiâ concupi&longs;cit, vel exigit. Dixi mediatè, vel immediatè; num reuera datur fortè aliquis impetus, vt dicemus infrà; &longs;cilicet primus na­turalis, qui &longs;cilicet duos fines habet di&longs;iunctiuè; quorum alter e&longs;t gra­uitatio, alter motus deor&longs;um.

Theorema 17.

Ni&longs;i e&longs;&longs;et motus non e&longs;&longs;et impetus. Probatur quia motus e&longs;t finis intrin­&longs;ecus impetus per Th. 16. igitur &longs;i nullus motus e&longs;&longs;e po&longs;&longs;et, &longs;uo finc ca­reret impetus; igitur non e&longs;&longs;et, vt patet, igitur non e&longs;&longs;et; quia quod 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 petu naturali primo vel innato (&longs;ic enim deinceps appellabimus vtrecti di&longs;tinguamus ab acqui&longs;ito quem vocabimus impetum accelerationis) qui &longs;ine motu con&longs;eruatur in corpore grauitante; quia ni&longs;i po&longs;&longs;ibilis e&longs;­&longs;et motus deor&longs;um nulla e&longs;&longs;et grauitatio; quippe grauitare e&longs;t dcor­&longs;um inclinari, motumque inclinationis impediri; hinc dicemus in &longs;ecundo libro impetum innatum &longs;æpiùs e&longs;&longs;e &longs;ine motu; cum &longs;cilicet impeditur à corpore &longs;u&longs;tinente? immò dicemus infrà primo in&longs;tanti, quo e&longs;t impetus, nondum e&longs;&longs;e motum.

Ob&longs;eruabis autem certi&longs;&longs;imam regulam; &longs;cilicet ex impo&longs;&longs;ibilitate effectus formalis, &longs;equi impo&longs;&longs;ibilitatem cau&longs;æ formalis, huiu&longs;que po&longs;&longs;i­bilitatem ex illius po&longs;&longs;ibilitate.

Theorema 18.

Ni&longs;i e&longs;&longs;et impetus, non e&longs;&longs;et naturaliter motus. Quia ni&longs;i e&longs;&longs;etcau&longs;a, non e&longs;&longs;et naturaliter effectus per Ax. 8. Impetus enim e&longs;t cau&longs;a motus per Th.15. Deinde omnis motus e&longs;t ab aliqua potentia motrice, vt patet ex omni hypothe&longs;i; &longs;iue &longs;it naturalis in grauibus, & leuibus, &longs;iue &longs;evitalis in viuentibus; &longs;iue &longs;it media in compre&longs;&longs;is, & dilatatis; &longs;iue alia quæli­bet: &longs;ed omnis potentia motrix e&longs;t actiua, quia mouet; ergo agit, &longs;ed motum non producit per Th. 2. Igitur impetum, qui deinde exigit mo­tum per Th. 14. Dixi naturaliter; quia non e&longs;t dubium, quin Deus &longs;ine impetu aliquo modo mouere po&longs;&longs;it; ide&longs;t, facere &longs;ine impetu, vt corpus mutet locum: nec dicas Deum non po&longs;&longs;e &longs;upplere vices cau&longs;æ formalis; nam concedo id quidem pro effectu formali primario; nec enim Deus pote&longs;t facere, vt aliquid &longs;it calidum &longs;ine calore; cum e&longs;&longs;e calidum &longs;it idem, ac e&longs;&longs;e habens calorem; id tamen nego pro effectu &longs;ecundario, quem &longs;cilicet cau&longs;a formalis exigit: Etenim &longs;icut pote&longs;t &longs;ummo iure non &longs;atisfacere exigentiæ; ita pote&longs;t id conferre &longs;ine exigentiâ, quòd cum exi­gentia conferre pote&longs;t; &longs;ic pote&longs;t corpus re&longs;oluere &longs;ine calore, mouere fine impetu &c. quanquam vt verum fatear non e&longs;&longs;et propriè motus, &longs;ed qua&longs;i continuæ reproductionis modus; nam motus dicit aliquam pa&longs;­&longs;ionem; &longs;cilicet actum entis in potentiâ, vt aiunt.

Theorema 19.

Si e&longs;&longs;et motus naturaliter &longs;ine impetu, corpus per &longs;e ip&longs;um moueretur, id e&longs;t, exigeret motum per &longs;uam entitatem; quia nullus impetus exigeret; ergo aliquid aliud, nihil di&longs;tinctum, alioquin e&longs;&longs;et impetus; ergo ip&longs;a corpo­ris entitas; quanquam non e&longs;&longs;etmotus, vt iam dictum e&longs;t, quia non e&longs;­&longs;et pa&longs;&longs;io.

Theorema 20.

Corpus illud æquali &longs;emper motu ferretur per &longs;e; Quia e&longs;&longs;et &longs;emper ca­dem cau&longs;a nece&longs;&longs;aria motus, id e&longs;t, ip&longs;a entitas corporis; igitur idem effectus per Axioma 12. igitur idem, vel æqualis motus: dixi per &longs;e pro­pter diuer&longs;um medium.

Theorema 21.

Si e&longs;&longs;et aliquod corpus e&longs;&longs;entialiter mobile, impetu non indigeret. Probatur; quia in tantum indiget mobile impetu vt impetus exigat motum; &longs;ed corpus illud per &longs;uam e&longs;&longs;entiam exigeret motum; igitur non indigeret impetu; po&longs;&longs;et tamen impediri cius motus, vt patet; immò e&longs;&longs;et capax recipiendi impetus., &longs;iue quem in ip&longs;o produceret, &longs;iue quem ab a lia cau&longs;a extrin&longs;eca acciperet.

Theorema 22.

Si e&longs;&longs;et aliquod corpus e&longs;&longs;entialiter immobile, e&longs;&longs;et incapax impetus. Pro­batur; quia, ni&longs;i e&longs;&longs;et motus, non e&longs;&longs;et impetus per Th. 17. igitur &longs;ubic­ctum incapax motus e&longs;t incapax impetus.

Theorema 23.

Si e&longs;&longs;et aliquod &longs;ubiectum incapax impetus, e&longs;&longs;et incapax motus. Quia vbi non pote&longs;t e&longs;&longs;e cau&longs;a formalis, ibi non pote&longs;t e&longs;&longs;e effectus forma­lis, quod certum e&longs;t.

Theorema 24.

Omne corpus, quod e&longs;t capax motus, e&longs;t capax impetus, & vici&longs;&longs;im.Probatur 1. pars; quia impetus in eo non e&longs;&longs;et fru&longs;trà; haberet enim fuum effectum formalem, & finem intrin&longs;ecum. Probatur 2.pars; quia in eo impetus non e&longs;&longs;et fru&longs;trà per Ax. 6. igitur haberet &longs;uum effectum; igitur motum.

Theorema 25.

Omne corpus finitum e&longs;t capax motus, & impetus. Probatur 1. pars; quia non e&longs;t vbique, igitur pote&longs;t transferri è loco in locum; cur enim non po&longs;&longs;et? Dices fortè quia affixum e&longs;&longs;et e&longs;&longs;entialiter tali, vel tali lo­co, &longs;ed contra; quia de&longs;truantur omnia, præter ip&longs;um corpus; certè nulli affixum manet. Dices &longs;patio imaginario; apage i&longs;tas nugas: de i&longs;to &longs;patio plura demon&longs;trabimus in Metaphy. Probatur 2. pars; quia &longs;i e&longs;t capax motus, e&longs;t capax impetus per Th. 24. Quod dixi de corpo­re; dicendum e&longs;t de omni re creata finita permanente.

Theorema 26.

Quod dur at tantùmvno in&longs;tanti, e&longs;t incapax motus, & impetus. Pro­batur, quia non e&longs;t moueri, ni&longs;i relinquat locum, & acquirat alium; &longs;ed 1. acquirere locum, e&longs;t 1. e&longs;&longs;e in illo loco; & relinquere locum e&longs;t, 1. non e&longs;&longs;e in co loco; nec &longs;imul e&longs;t in vtroque, quia in duobus locis idem &longs;imul e&longs;&longs;e non pote&longs;t; vt demon&longs;tramus in Metaphy&longs;ica; & phy­ficè certum e&longs;t ex omni hypothe&longs;i; igitur moueri nunc, id e&longs;t, hoc in­&longs;tanti, id e&longs;t, 1. acquirere nouum locum, & 1. relinquere priorem, &longs;upponit nece&longs;&longs;ariò antè fui&longs;&longs;e in loco nunc relicto; &longs;ed quod durat tantùm in in&longs;tanti, non habet antè, neque po&longs;t; igitur quod durat tan­tùm vno in&longs;tanti, moucri non pote&longs;t; igitur e&longs;t incapax motus; igitur & impetus.

Theorema 27.

Deus e&longs;t incapax motus, & impetus: Tum quia vbique, e&longs;t igitur nouum locum acquirere non pote&longs;t; igitur nec moueri per Deffinitio­nem 1. tùm quia æternitas Dei tota &longs;imul e&longs;t; igitur nec fuit antè, ne­que po&longs;t in ca; igitur non pote&longs;t dici antè habui&longs;&longs;e locum, quo nunc caret: & nunc non habere illum quo caret; tùm quia immutabilitas Dei hoc prohibet; nam moucri, e&longs;t affici intrin&longs;ecè; quia etiam de­&longs;tructis omnibus extrin&longs;ecis creatis moueri po&longs;&longs;em, & fru&longs;trà recurres ad partes virtuales immen&longs;itatis Dei, quas ferè animus abhorret; apa­ge partes in Deo: quis hoc ferre po&longs;&longs;it? præterea &longs;i &longs;unt, &longs;unt e&longs;&longs;entia­liter immobiles; igitur valet &longs;emper ratio allata; igitur Deus e&longs;t inca­pax motus; igitur & impetus.

Diceret aliquis Deum quantumuis Immen&longs;um in orbem conuolui po&longs;&longs;e; igitur 1. ratio non probat de omni motu. Re&longs;pondeo adhuc va­lere, quia etiam in orbem conuolui non pote&longs;t, ni&longs;i mutetur intrin&longs;e­cè; atqui &longs;i e&longs;t immen&longs;us, non pote&longs;t mutari intrin&longs;ecè per motum; quia nullum locum de nouo acquireret; &longs;ed de hoc motu aliàs, cum de infinito; vel de puncto phy&longs;ico mobili; quidquid &longs;it. valet &longs;altem 1. ratio pro motu recto, & aliæ duæ pro omni motu.

Theorema 28.

Motus ip&longs;e mouerinon pote&longs;t. Quia cum tantùm dicat mutationem loci; certè mutatio non e&longs;t in loco; dicit enim tantùm locum relictum eo in&longs;tanti, quo nouus acquiritur. Præterea quod e&longs;t in loco dicit tan­rùm ens phy&longs;icum; &longs;ed mutatio dicit etiam non ens; Hinc egregium pa­radoxum; illud non mouetur per quod cuncta mouentur, quæ mouentur.

Theorema 29.

Duratio moueri non pote&longs;t. Cum enim &longs;it &longs;ucce&longs;&longs;iua, fluit per partes, igitur quælibet illius pars, &longs;eu quod durat vna in&longs;tanti tantùm e&longs;t inca­pax motus, per Th. 26.

Theorema 30.

Hinc actio moueri non pote&longs;t; cum enim actio per quam res con&longs;erua­tur, &longs;it eius duratio; vt con&longs;tabit ex iis, quæ demon&longs;trabimus in Me­taphy&longs;ica, & cum duratio moucri non po&longs;&longs;it, per Th. 29. certè neque actio moueri pote&longs;t.

Corollarium 1.

Hinc in tanta rerum creatarum multitudine &longs;unt tantùm duæ, quæ &longs;unt e&longs;&longs;entialiter immobiles; &longs;cilicet motus, & actio; quorum ille cum &longs;it mutatio non e&longs;t adæquatè aliquid po&longs;itiuum; &longs;ecus actio.

Corollarium 2.

Hinc &longs;unt tantùm duo adæquatè po&longs;itiua, quæ moucri non po&longs;&longs;uat; &longs;cilicet Deus, & actio; Deus, qui &longs;emper e&longs;t; actio, quæ tantùm vno in&longs;tanti e&longs;t; Deus vbique e&longs;&longs;entialiter; actio hic tantum e&longs;&longs;entialiter; Deus primum ens; actio infinitum ens; e&longs;t enim modus; Deus primum mouens; actio ip&longs;e motus; &longs;cilicet primi generis, de quo in &longs;ect. Th.3.

Corollarium 3.

Hinc &longs;i res aliqua creata per actionem tantæ perfectionis, quæ mille annis e&longs;&longs;entialiter re&longs;ponderet, con&longs;eruaretur; certè per totum illud tempus moucri non po&longs;&longs;et; e&longs;&longs;et enim vnicum in&longs;tans, hoc e&longs;t duratio tota &longs;imul; &longs;ed codem in&longs;tanti in pluribus locis e&longs;&longs;e non pote&longs;t; igitur nec moueri; adde quod per cam actionem &longs;um in loco, per quam &longs;um in tempore; igitur &longs;i hæc e&longs;t &longs;emper eadem, illam eandem e&longs;&longs;e nece&longs;&longs;e e&longs;t; &longs;ed hæc &longs;unt metaphy&longs;ica, quæ obiter tantùm attingo, aliàs fusè de mon&longs;trabo.

Scolium.

Ob&longs;eruabis primò ex dictis præclarum naturæ in&longs;titutum; cum enim corpus moueri &longs;emper non debeat, (quippe hoc e&longs;&longs;et maximè incom­modum) certè per &longs;uam entitatem moucri non exigit; alioquin &longs;emper moueretur; igitur per aliud ab entitate di&longs;tinctum, id e&longs;t per impetum; itaque licet per &longs;uam entitatem exigat fluxum in tempore, id e&longs;t con&longs;er­nari, & durare; id e&longs;t nouam &longs;emper actionem con&longs;eruatiuam; quia maximum eius bonum e&longs;t durare vel exi&longs;tere; Igitur per &longs;e ip&longs;um illud exigit; quia &longs;emper exigit, non tamen per &longs;e ip&longs;um exigit fluxum in loco, id e&longs;t motum; quia moueri non &longs;emper e&longs;t bonum.

Ob&longs;eruabis &longs;ecundò, cum idem corpus aliquando velociùs, tardiùs aliquando moueri exigat; &longs;i per &longs;uam entitatem moueri exigeret, eo­dem &longs;emper ferretur motu; quia cadem &longs;emper e&longs;&longs;et exigentia; igitur debet e&longs;&longs;e aliquid aliud; illud autem e&longs;t impetus, qui aliquando maior &longs;eu perfectior, aliquando verò minor e&longs;t; igitur maiorem &longs;eu velcio­rem motum aliquando exigit, aliquando minorem, &longs;eu tardiorem; cum enim motus &longs;it eius finis intrin&longs;ecus, vt re&longs;olutio e&longs;t finis caloris vel rarefactio; quemadmodum maior calor maiorem exigit, &longs;eu præ­&longs;tat re&longs;olutionem; ita & maior, &longs;eu perfectior impetus maiorem, &longs;eu velociorem motum exigit.

Ob&longs;eruabis tertiò aliud naturæ in&longs;titutum, quo &longs;cilicet in eo tan­tùm &longs;ubiecto recipi pote&longs;t cau&longs;a formalis, in quo recipi pote&longs;t cius effe­ctus formalis &longs;ecundarius: nec alia regula, præter eam excogitari pote&longs;t; cum enim aliqua forma ad talem, vel talem finem à natura in&longs;tituta e&longs;t; certè propter illum finem e&longs;t, igitur in co non e&longs;t, in quo &longs;uum finem 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, in quo fru&longs;trà non e&longs;t; cum &longs;cilicet in eo &longs;uum finem con&longs;equatur; ad­de quod finis ille intrin&longs;ecus phy&longs;icus &longs;cilicet, non moralis, aliquis no­uus effectus e&longs;t; atqui nouus effectus &longs;ine &longs;ua cau&longs;a e&longs;&longs;e non pote&longs;t, neque cau&longs;a nece&longs;&longs;aria &longs;ine e&longs;&longs;ectu; igitur ibi, &longs;cilicet in hoc &longs;ubiecto, in quo 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, e&longs;t inquam citra miraculum.

Ob&longs;eruabis quartò egregiam rationem; propter quam res eadem in pluribus locis naturaliter e&longs;&longs;e non pote&longs;t; quippe cum res fuerit primo producta in aliquo loco, illa certè nouum locum acquirere non pote&longs;t naturaliter; ni&longs;i per motum, atqui motus dicit nece&longs;&longs;ario priorem lo­tum relictum, & nouum acqui&longs;itum; igitur cum tot acquirantur loca per motum, quot relinquuntur; &longs;i ante motum vnus tantùm erat eiu&longs;­dem rci locus, po&longs;t motum etiam vnus e&longs;t: quod autem producatur tan-tùm res in vno loco patet; vel enim à cau&longs;a prima vel ab aliqua 2. pro­ducitur; &longs;i à 2. ergo ab aliqua aplicata; igitur ex &longs;uppo&longs;itione quòd il­la cau&longs;a 2. in vno tantùm loco producta &longs;it, vni tantum applicari po­te&longs;t; quod autem cau&longs;a 1. in pluribus locis naturaliter eundem effectum non producat, certum e&longs;t, & demon&longs;trabimus in Metaphy&longs;. quia &longs;in­gulis effectibus &longs;ingulæ &longs;ufficiunt actiones; &longs;ingulis terminis &longs;ingulæ viæ; immò hoc requiri videtur, &longs;eu &longs;pectare ad huius vniuer&longs;itatis or­dinem; quippe &longs;i res eadem in pluribus locis e&longs;&longs;et; cur potius in duo­bus quam in tribus? deinde multiplex iure po&longs;&longs;et exi&longs;timari; denique quod vnum e&longs;t in entitate creata, &longs;eu dependente ab cadem cau&longs;a, vnum e&longs;t etiam in dependentia; quæ e&longs;t actio, per quam dependet; &longs;ed de his aliàs.

Theorema 31.

Impetus non producitur in eo mobili, quod moueri non pote&longs;t à potentia motrice applicata, licèt à fortiori moueri po&longs;&longs;it. Probatur, quia impetus e&longs;t tantùm propter motum, qui eius effectus e&longs;t, & finis, per Th. 15. & 16. Igitur vbi non e&longs;t motus, fru&longs;trà e&longs;t impetus; &longs;ed quod fru&longs;trà 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. Exci­pio tamen impetum naturalem innatum, qui nunquam e&longs;t fru&longs;trà, vt dictum e&longs;t &longs;uprà in Theorem. 17. adde quod non pote&longs;t cogno&longs;oi impetus, ni&longs;i vel ex motu, vel ex ictu, vel ex contrario ni&longs;u, vel impul&longs;u; &longs;ed nihil horum cernitur in rupe quam ferio; Igitur non e&longs;t dicendum in ea produci impetum, cuius rationem afferemus infrà; nunc&longs;atis e&longs;t Ax. 3. id manife&longs;tè probari; nam qui diceret in rupe im­mobili impetum imprimi; certè po&longs;itiuo argumento probare tenere­tur, quod tantùm duci pote&longs;t, vel ab experimento; atqui hîc nullum e&longs;t; vel à nece&longs;&longs;itate, quæ nulla e&longs;t; vel ex alio quocumque capite, quod nullum excogitari pote&longs;t; &longs;ed maiorem lucem huic Th. 3. ex proximè &longs;equentibus accer&longs;emus; nec e&longs;t quòd aliqui dicant produci impetum inefficacem; qui cum fru&longs;trà &longs;it, &longs;i e&longs;t, ex nullo capite probari pote&longs;t: ad­de quòd de&longs;truitur impetus, ne &longs;it fru&longs;trà; Igitur non producitur, ne &longs;it fru&longs;trà; nam con&longs;eruatio e&longs;t vera actio, vt dicemus &longs;uo loco; Igitur &longs;i hæc non ponitur, ne aliquid &longs;it fru&longs;trà; etiam 1. productio poni non debet; vnde commentum illud impetus inefficacis pror&longs;us inefficax e&longs;t.

Theorema 32.

Ideo potentia motrix non producit impetum in pradicta rupe. v.g. quia de­bilior e&longs;t. Probatur, & explicatur; quippedebilior potentia minorem ef­fectum producit per. Ax. 13. num. 2. igitur pauciores partes impetus æquales vni certæper idem num. 1. igitur &longs;i &longs;int plures partes &longs;ubiecti mobilis, &longs;eu rupis, quàm impetus; cum vna pars impetus duobus parti­bus &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; non e&longs;t mirum &longs;i nullus impetus producatur; cum non po&longs;&longs;int tot partes illius produci, quot e&longs;&longs;ent nece&longs;&longs;ariæ; vt &longs;altem &longs;ingulæ &longs;ingulis &longs;ubie­cti, &longs;eu rupis partibus di&longs;tribuerentur.

Ob&longs;eruabis autem nouum quoddam genús re&longs;i&longs;tentiæ; nam &longs;ingulæ partes rupis ab applicata potentiâ aptæ &longs;unt loco moueri per impre&longs;­&longs;um impetum, & maior potentia &longs;imul omnes loco moueret; at verò omnes &longs;imul, & coniunctim con&longs;ideratæ; quatenus &longs;cilicet vna pars non pote&longs;t moueri &longs;ine alia, & comparatæ cum illa potentia debili di­cuntur habere prædictam re&longs;i&longs;tentiam, quæ &longs;uperat potentiæ vires; quòd &longs;cilicet à maiori moueri tantùm po&longs;&longs;int; quia plures partes im­petus po&longs;tulantur, quam &longs;int eæ, quæ à prædictâ potentiâ po&longs;&longs;unt pro­duci.

Theorema 33.

Vel producitur impetus in omnibus &longs;ubiecti partibus vnitis, vel in nulla; modò nulla fiat &longs;eparatio, neque compre&longs;&longs;io: Certum e&longs;t enim in ijs omni­bus partibus, quæ auolant ab ictu, produci impetum. Probatur igitur 1. quia &longs;i non producatur in omnibus partibus, & nulla &longs;eparetur ab alijs; certè nulla mouetur, vt certum e&longs;t; igitur nulla habet impetum; quia ibi non e&longs;t cau&longs;a formalis, vbi non e&longs;t effectus formalis; alicquin e&longs;&longs;et fru&longs;trà, contra Ax. 6.2. Tu dicis produci impetum in aliquot parti­hus; hoc dicis, hoc proba? an potes digno&longs;cere impetum ni&longs;i ex motu? vel con&longs;eruaretur hîc impetus &longs;equentibus in&longs;tantibus, vel &longs;tatim &longs;ecun­do in&longs;tanti de&longs;trucretur. Primum dicere ab&longs;urdume&longs;t; quia &longs;i hoc e&longs;&longs;et multisictibus repetitis tandem moueretur totum mobile; &longs;i verò de­ftrui dicatur. Secundo in&longs;tanti; eadem ratio probat non produci. Pri­mo in&longs;tanti, quæ probt de&longs;trui. Secundo nam ideo de&longs;truitur. Secun­do quia e&longs;t fru&longs;trà, &longs;ed non minus e&longs;t fru&longs;trà. Primo igitur non produ­citur. Primo 4. probatur; quia cum non &longs;ufficiant partes impetus, quas dixi produci, vt omnibus partibus &longs;ubiecti di&longs;tribuantur; certè non e&longs;t vlla ratio, cur potiùs his quàm illis di&longs;tribui dicantur; cum vna &longs;it tan­tùm immediatè applicata. Igitur certum e&longs;tvel produci in omnibus, vel in nulla, ni&longs;i forte aliquæ auolent, &longs;ed tunc &longs;eparantur.

Obiiciet aliquis 1. e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam applicatam &longs;ubiecto ap­to: igitur agit per Ax. 12. Re&longs;pondeo e&longs;&longs;e impeditam; nam re&longs;i&longs;tentia &longs;ubiecti &longs;uperat vires potentiæ vt dictum e&longs;t; immò in ip&longs;o motu re­torqueo argumentum; licèt enim &longs;it applicata cau&longs;a nec&longs;&longs;aria mouens, non tamen mouet.

Obiiciet 2. Ignis applicatus agit in nonnullas partes fubiecti, licèt non agat in omnes; igitur & potentia motrix. Re&longs;pondeo non e&longs;&longs;e pa­ritatem; quia vna pars pote&longs;t calefieri, & re&longs;olui &longs;ine alia, vt con&longs;tat non tamen vna moueri &longs;ine alia, cui coniuncta e&longs;t, ni&longs;i &longs;eparetur; igi­tur nec recipere impetum &longs;ine alia.

Obiiciet. 3. &longs;int duo trahentes idem mobile; ita vt &longs;eor&longs;im neuter trahere po&longs;&longs;it, coniunctim verò vterque po&longs;&longs;it; certè &longs;i alter non pro­ducit impetum &longs;cor&longs;un, nec etiam coniunctim producet; nec enim au­gentur eius vires ab altero: Re&longs;pondeo vtrunque agere actione com­muni; igitur non e&longs;t mirum &longs;i effectus maior e&longs;t, quem tamen neuter &longs;eor&longs;im producere pote&longs;t.

Dices &longs;i vterque coniunctim producit effectum: &longs;int v. g. 100. par­tes impetus; Igitur &longs;inguli producunt tantùm 50. Igitur cur potiùs in in his partibus &longs;ubiecti, quàm in alijs, cum vtriu&longs;que potentia eidem &longs;ubiecti parti po&longs;&longs;et e&longs;&longs;e applicata? Re&longs;pondeo &longs;ingulos producere 100. actione &longs;cilicet communi indiui&longs;ibiliter; &longs;int enim duo trahentes A. & B. A. producit 100. &longs;ed non &longs;olus; B. producit ea&longs;dem 100. &longs;ed non &longs;o­lus; &longs;ed explicabimus hunc modum actionis communis in Metaphys. quod autem agant actione communi patet per Ax. 13.

Obiicies 4. producitur &longs;onus &longs;i ferias rupem; igitur & impetus; Re&longs;­pondeo ad &longs;onum &longs;olam aëris colli&longs;ionem &longs;ufficere, quam fieri certum e&longs;t à prædicto ictu; deinde mallej motus impacti in rupem facit &longs;onum; quidquid tandem &longs;it &longs;onus, de quo hîc non di&longs;puto: adde qnod in ru­pe &longs;unt &longs;emper aliquæ partes trem ulæ, quæ modico tantùm, coque flexi­bili nexu cum alijs partibus copu lantur; adde aliquam compre&longs;&longs;ionem, ex qua modicæ vibrationes &longs;equuntur.

Obiicies 5. Quando aliquæ partes auolant ab ictu, haud dubiè auo­lant propter impetum impre&longs;&longs;um: Igitur prius e&longs;t imprimi impetum, quàm auolare; igitur productus e&longs;t impetus in nonnullis partibus, & non in aliis, cum quibus illæ &longs;unt coniunctæ. Re&longs;pondeo equidem im­petum produci in illis partibus antequam auolent; &longs;ed ideo produci vt deinde auolent nam tota ratio cur non producatur, e&longs;t ne &longs;it fru&longs;trà; &longs;ed &longs;i auolent aliquæ partes: certè in ijs non e&longs;t fru&longs;trà, in quibus habet &longs;uum effectum, id e&longs;t, motum.

Dices; igitur primo in&longs;tanti impetus ille e&longs;t fru&longs;trà; in quo non habet &longs;uum effectum; Re&longs;pondeo nunquam primo in&longs;tanti e&longs;&longs;e fru&longs;trà, modò &longs;it motus &longs;ecundo cum etiam primo in&longs;tanti, quo e&longs;t impetus, non po&longs;&longs;it e&longs;&longs;e motus, vt demon&longs;trabo infrà; immò ideo ponitur im­petus primo vt &longs;it motus &longs;ecundo exigendo pro i n&longs;tant i &longs;equenti, de cum impetus ponat tantùm motum quo aliàs.

Dices; &longs;ed potentia motrix ne&longs;cit an po&longs;&longs;it pars aliqua mobilis &longs;epa­rari; igitur non e&longs;t quòd aliquando producat impetum, aliquando non producat. Re&longs;pondeo non &longs;tare per cau&longs;am nece&longs;&longs;ariam, quin &longs;em­per agat; &longs;ed per &longs;ubiectum, quod &longs;i aptum e&longs;t, & capax e&longs;&longs;ectus; haud dubiè co ip&longs;o cau&longs;a nece&longs;&longs;aria applicata in ip&longs;um aget; &longs;i verò ineptum. haud dubiè non aget; nam ad hoc vt producatur effectus in &longs;ubiecto; non &longs;atis e&longs;t cau&longs;am po&longs;&longs;e producere, ni&longs;i etiam &longs;ubiectum po&longs;&longs;it recipe­re; igitur cum &longs;it talis ordo à natura in&longs;titutus, ne aliquid &longs;it fru&longs;trà; certè &longs;i impetus producibilis &longs;it futurus fru&longs;trà, hauddubiè non produ­cetur; &longs;ecus verò &longs;i fru&longs;trà non &longs;it futurus, in quo non e&longs;t difficultas.

Scholium.

Ob&longs;eruabis 1. vix fieri po&longs;&longs;e quin &longs;emper aliquæ partes &longs;eparentur, comprimantur, vel dilatentur, vt patet experientiâ.

Ob&longs;eruabis 2. etiam maximam corporis molem à debili potentia mi-nimo etiam ictu moucri; quod etiam ob&longs;eruauit Galileus in &longs;uis dialo­gis de motu; quem certè motum ob&longs;eruabis etiam in&longs;en&longs;ibilem, tùm operâ radij luminis repercu&longs;&longs;i, & ad aliquod interuallum proiecti; tùm operâ &longs;eu pi&longs;orum in tympani membranâ tremulo qua&longs;i motu &longs;ub&longs;ul­tantium; quâ etiam arte deprehenditur in arce ob&longs;e&longs;&longs;a, &longs;ub quam muri partem cuniculi agantur.

Corollarium 1.

Hinc egregia ratio erui pote&longs;t, cur ingens corporis moles à debili po­tenria loco moueri non po&longs;&longs;it; cum enim tot &longs;altem requirantur partes impetus, quot &longs;unt partes &longs;ubiecti: quia vel in omnibus, vel in nulla producitur; certè cum &longs;int plures partes &longs;ubiecti, quàm vt in &longs;ingulis ab ea dumtaxat potentiâ impetus produci po&longs;&longs;it; quid mirum e&longs;t, &longs;i mo­ueri non po&longs;&longs;it.

Collorarium 2.

Hinc certa ratio alterius vulgaris effectus potentiæ motricis, quæ lapi­dem 40. librarum tardo tantùm motu impellit, etiam cum &longs;ummo ni&longs;u, cum tamen &longs;axo vnius libræ velocioremmotum imprimat; quia &longs;cilicet partes impetus producti di&longs;tribuuntur pluribus partibus &longs;ubiecti in ma­iori lapide, & paucioribus in minori; igitur &longs;ingulæ partes minoris habent plures partes impetus, vt manife&longs;tè coa&longs;tat; ergo ille impetus inten&longs;ior e&longs;t; igitur maiorem exigit &longs;eu perfectiorem motum per Ax. 13. num.2.

Collorarium 3.

Hinc &longs;ublata ratione diuer&longs;æ re&longs;i&longs;tentiæ medij, dato pondere mobilis vtriu&longs;que, datoque ni&longs;u communi potentiæ, pote&longs;t de­terminari certus velocitatis gradus vtriu&longs;que; nam ratio velocitatum e&longs;t inur&longs;a ponderum v. g. &longs;it pondùs 4. librarum; fit etiam 2. librarum &longs;it impetus impre&longs;&longs;us vtrique &longs;uppo&longs;ito communi, & æquali ni&longs;u potentiæ, & æquali tempore; haud dubiè velocitas mobilis 2. libra­rum erit dupla velocitatis mobilis 4. librarum; quia cum &longs;int duplo plures partes &longs;ubiecti in hoc mobili quàm in illo (accipio enim vtrum­que eiu&longs;dem marcriæ, vt omnes lites fugiam) igitur in minori e&longs;t duplo inten&longs;ior impetus: Igitur duplo velocior motus; dixi, &longs;i fiat æquali ni&longs;u, & æquali tempore; quia reuerâ non fit in tempore æquali, &longs;ed inæquali, &longs;i &longs;upponatur idem arcus brachij v. g. iacientis; nam tempo­ra &longs;unt in ratione &longs;ubduplicata ponderum; vt demon&longs;trabimus lib. 10. & velocitates &longs;unt vt tempora permutando.

Collorarium 4.

Hinc facilè determinari pote&longs;t proportio impetus impre&longs;&longs;i cognitâ grauitate mobilium; v. g. &longs;it mobile grauc vt4. & aliud graue vt 2. haud dubiè vt moucatur æquali gradu velocitatis, debet produci duplo maior impetus in maiori mobili, hoc e&longs;t, iuxta rationem maioris ad mi­nus, quod clari&longs;&longs;imè &longs;cquitur ex dictis; vt enim tot&longs;int gradus impetus in qualibet parte minoris, quot &longs;unt in qualibet parte minoris; haud dubiè impetus maioris habet eandem rationem ad impetum minoris; quam habet maius ad minus.

Collorarium 5.

Hinc quoque ducitur manife&longs;ta ratio &longs;eu re&longs;pon&longs;io ad illud præcla­rum certè quorundam philo&longs;ophorum commentum, qui volunt ex mini­ma ponderis acce&longs;&longs;ione totam terræ molem inclinari, vt in nouo æqui­librio &longs;tatuatur; quod omninò fal&longs;um e&longs;t; nam ex &longs;uppotione quòd terra non grauitet (vt vulgò dicitur, & aliàs à nobis demon&longs;trabitur) illa certè moueri non pote&longs;t ni&longs;i producantur tot partes impetus quot &longs;unt partes &longs;ubiecti in tota terra; quæ certè maximas potentiæ vires po&longs;tulant.

Theorema 34.

Primo in&longs;tanti, quo est impetus, non est ille motus, cuius hic impetus e&longs;t cau&longs;a. Probatur; quia non pote&longs;t e&longs;&longs;e motus, ni&longs;i &longs;it locus prior reli­ctus, & nouus acqui&longs;itus, igitur &longs;i eodem in&longs;tanti, quo e&longs;t impetus, haberet motum, codem in&longs;tanti e&longs;&longs;et in duobus locis, quod dici non pote&longs;t; & iam diximus in Th. 26. igitur impetus primo in&longs;tanti quo e&longs;t non habet &longs;uum motum.

Theorema 35.

Immò nihil e&longs;t, quod primo in&longs;tanti, quo e&longs;t, moueri po&longs;&longs;it. Quia non pote&longs;t moueri, ni&longs;i acquirat nouum locum, & priorem relinquat; igitur, vel &longs;i­mul in vtroque e&longs;t, quod dici non pote&longs;t; vel in relicto antè fuit; igitur non e&longs;t primum in&longs;tans, contra &longs;uppo&longs;itionem.

Theorema 36.

Potest impetus aliquo in&longs;tanti non moueri quo mouetur ip&longs;um mobile, in quo est. Nam moueatur mobile quodlibet; & dum mouetur, impella­tur, factâ &longs;cilicet acce&longs;&longs;ione noui impetus; haud dubiè hoc primo in­&longs;tanti, quo producitur impetus in dato mobili non mouetur per Th. 35. quo tamen in&longs;tanti mouetur prædictum mobilc.

Collorarium 1.

Hinc egregium paradoxon; Pote&longs;t alique in&longs;tanti moueri &longs;ubiectum, licèt non moueantur illa omnia, que eidem &longs;ubiecto reuerâ in&longs;unt.

Corollarium 2.

Hinc etiam aliud paradoxon; Impetus primo in&longs;tanti, quo e&longs;t, non habet &longs;uum finem, nec habere pote&longs;t; patet, quia primo in&longs;tanti non habet motum.

Corollarium 3.

Hinc pote&longs;t aliquid dato in&longs;tanti carere &longs;uo fine; licèt non &longs;it fru&longs;trà; fru&longs;trâ enim tantùm dicitur ille impetus, qui pro in&longs;tanti &longs;equenti non pote&longs;t haberemotum.

Theorema 37.

Impetus pars recepta in parte &longs;ubiecti non exigit motum aliarum partiumeiu&longs;dem &longs;ubiecti, licèt coniunctarum. Probatur 1. quia alioquin vna pars impetus &longs;ufficeret ad mouendam ingentem rupem; quod ab&longs;urdum e&longs;t. 2. &longs;icut vna pars caloris non re&longs;oluit alias partes &longs;ubiecti; ita nec im­petus. 3. Ratio à priori e&longs;t; quia impetus non e&longs;t cau&longs;a efficiens motus per Th. 13. &longs;ed tantùm cau&longs;a formalis per Th. 15. Igitur præ&longs;tat tantùm &longs;uum effectum formalem in eo &longs;ubiecto, in quo e&longs;t.

Corollarium 1.

Hinc partes impetus non cau&longs;ant motum in &longs;uo &longs;ubiecto actione, vel exigentia communi; quia quælibet pars impetus exigit tantùm motum &longs;ui &longs;ubiecti; id e&longs;t illius partis, quàm afficit; quod etiam probatur per Ax. 13.

Corollarium 2.

Hinc corpus grauius per&longs;e, &longs;altem eiu&longs;dem materiæ, non cadit velo­ciùs, quàm leuius, vti globus plumbeus 100. librarum, quàm globus vnius libræ plumbeus; quia &longs;cilicet impetus vnius partis non iuuat mo­tum alterius: præterca tam facilè 2, partes impetus in 2. partibus &longs;ubie­cti receptæ ca&longs;dem mouent, quàm 100. alias 100. dixi per &longs;e; nam di­uer&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.

Theorema 38.

Impetus recipitur tantùm in ip&longs;a &longs;ub&longs;tantia &longs;ubiecti naturaliter. v. g. &longs;i mobile &longs;it ferrum calidum, recipitur in ip&longs;a &longs;ub&longs;tantia ferri; non verò in ip&longs;o calore (ex &longs;uppo&longs;itione quod calor &longs;it accidens, vt aliàs demon­&longs;trabimus; nec in alijs accidentibus, &longs;i quæ &longs;unt, in eodem &longs;ubiecto; pro­batur 1. quia &longs;i produceretur etiam impetus in accidentibus, quo plu­ra e&longs;&longs;ent accidentia in aliquo &longs;ubiecto; plures quoque partes impetus producendæ e&longs;&longs;ent; igitur maiori potentiâ opus e&longs;&longs;et per Ax. 13. n. 4. Igitur difficiliùs mouerentur, quod e&longs;t ab&longs;urdum. Diceret fortè ali­quis eundem impetum recipi &longs;imul in &longs;ub&longs;tantia & in ip&longs;is accidenti­bus; &longs;ed contra, nam reuera, &longs;i hoc e&longs;&longs;et, dum proijcitur ferrum cali­dum, & &longs;tatim frigefit, de&longs;trueretur totus impetus, de&longs;tructo &longs;cilicet eius &longs;ubiecto: 2. qui hoc diceret, deberet probare; nam codem modo mouetur corpus &longs;iue afficiatur pluribus accidentibus, &longs;iue paucioribus; igitur non euincit experientia recipiin illis impetum, nec etiam ratio, vt dicam paulò po&longs;t. Ratio à priori e&longs;&longs;e pote&longs;t; quia accidens cum &longs;uo &longs;ubiecto coniunctum exigit &longs;emper e&longs;&longs;e præ&longs;ens &longs;ubiecto, cum natura­liter extra &longs;ubiectum exi&longs;tere non po&longs;&longs;it; igitur cum exigat con&longs;erua­ri, & exi&longs;tere; co tantùm modo, quo pote&longs;t naturaliter con&longs;eruari & exi&longs;tere; certè exigit con&longs;eruari, & ine&longs;&longs;e &longs;ubiecto; igitur exi&longs;tere in eo loco, in quo exi&longs;tit &longs;ubiectum, vt patet; igitur, &longs;i &longs;ubiectum mutet locum etiam accidens cnm co coniu nctum mutare debet.

Dices, igitur &longs;imiliter dici pote&longs;t non recipi impetum in omni­bus partibus &longs;ubiecti mobilis, &longs;ed in vnâ dumtaxat; cui cum aliæ &longs;int vnitæ, exigunt moueri &longs;ine impetu ad illius motum? cum hoc ip&longs;um ad omnem vnionem &longs;pectare videatur; Re&longs;pondeo vnam partem plumbi ita coniungi cum alia, vt etiam &longs;eparata naturaliter exi&longs;tere po&longs;&longs;it; igitur non e&longs;t par ratio; præterea vna pars plumbi non e&longs;t in loco alterius; nec enim inuicem penetrantur cum &longs;it compene­tratio accidentium cum &longs;ubiecto; deinde, quò plures &longs;unt partes vnitæ, maior e&longs;t re&longs;i&longs;tentia, quæ ip&longs;o etiam &longs;en&longs;u percipitur; denique non vide­rur cur potius produceretur in vna parte, quam in alia; quæ omnia iam &longs;uprà Th. 33. demon&longs;trauimus.

Adde quod &longs;i impetus produceretur in ip&longs;is accidentibus, etiam in ip&longs;o impetu prius producto alius impetus produceretur; cum &longs;cilicet noua fit impetus acce&longs;&longs;io; quod &longs;atis ridiculum e&longs;t; qua&longs;i verò impetus indigeat impetu &c. hîc loquor tantùm de accidentibus in &longs;ubiecto; non verò de Euchari&longs;ticis, quæ à &longs;ubiecto per miraculum &longs;eparata etiam moueri po&longs;&longs;unt per impre&longs;&longs;um impetum.

Corollarium 1.

Hinc manife&longs;tè patet, quid dicendum &longs;it de anima bruti, quæ mouc­tur etiam &longs;ine impetu; quia exigit &longs;emper e&longs;&longs;e coniuncta corpori, à quo di&longs;iuncta naturaliter exi&longs;tere non pote&longs;t, vt &longs;uo loco dicemus; igi­tur ad motum corporis, &longs;eu &longs;ubiecti moueri deber.

Corollarium 2.

Idem quoque de Anima rationali dicendum e&longs;&longs;e videtur; licèt enim à corpore &longs;eparata naturaliter exi&longs;tere po&longs;&longs;it; tandiù tamen cum corpore manet coniuncta, quandiu agere pote&longs;t in organis corporcis; ac proinde exigit con&longs;eruari in corpore ip&longs;o, quandiu &longs;uas operatio­nes organicas in co exercere pote&longs;t.

Corollarium 3.

Hinc patet ratio manife&longs;ta ad quæ&longs;itum illud; quomodo &longs;cilicet po­tentia motrix materialis v.g. Taurus &longs;uo cornu hominem ventilare po&longs;­&longs;it; nec vlla &longs;upere&longs;t difficultas, dum dicas impetum non produci in anima.

Scolium.

Ob&longs;eruabis primò In hoc Theoremate dictum e&longs;&longs;e naturaliter; quia per miraculum accidens &longs;eparatum ab omni &longs;ub&longs;tantia, dum &longs;it impe­netrabile, per impetum &longs;ibi impre&longs;&longs;um moueri pote&longs;t.

Ob&longs;eruabis &longs;ecundò de anima bruti per miraculum &longs;eparatâ, idem pror&longs;us dicendum e&longs;&longs;e.

Ob&longs;eruabis tertiò etiam Animam rationalem &longs;eparatam, modò &longs;it cum impenetrabilitate coniuncta, capacem e&longs;&longs;e impetus; quem etiam à potentia motrice corporea recipere pote&longs;t; idem dictum e&longs;to de An­gelo; &longs;ed de vtroque aliàs.

Theorema 39.

Quando corpus pellitur ab alio corpore per impetum impre&longs;&longs;um; band da­biè impetus ille impre&longs;&longs;us ab aliqua cau&longs;a efficiente producijur; patet per Ax. 8.

Theorema 40.

Ille impetus non producitur à &longs;ub&longs;tantia corporis in aliud impacti. Proba­tur; quia &longs;i produceretur, e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt clarum e&longs;t; igitur appli­cata, & non impedita ageret per Ax. 32. quod e&longs;t contra experientiam. Dicunt aliqui requiri motum præuium, vt agat; &longs;ed contra; nam motus præuius non requiritur vt cau&longs;a, vt patet; quia cau&longs;a vt agat debet exi­&longs;tere per Ax. 9. Igitur requiritur, vt conditio, quod dici non pote&longs;t; quia primo etiam conditio debet e&longs;&longs;e præ&longs;ens; &longs;ed motus præuius de nihil pre&longs;enti e&longs;t &longs;ecundo quia non pote&longs;t excogitari aliud munus con­ditionis; ni&longs;i vel vt tollat impedimentum, vel vt applicet cau&longs;am &longs;ubie­cto apto; præterea motus præuius non e&longs;t; igitur codem modo &longs;e habet, ac &longs;i nunquam extiti&longs;&longs;et; & &longs;i co in&longs;tanti quo corpus impa­ctum primo tangit, amitteret totum impetum, ita vt expræterito motu nihil reliquum e&longs;&longs;et, haud dubiè corpus aliud non pelleret.

Diceret alius impetum e&longs;&longs;e tantùm conditionem, quæ &longs;emper e&longs;t de præ&longs;enti: ad hanc in&longs;tantiam non valet &longs;uperior re&longs;pon&longs;io; & certè &longs;i eo ip&longs;o in&longs;tanti contactus noua fieret impetus acce&longs;&longs;io; haud dubiè maior e&longs;&longs;et ictus; licèt cum codem motu præuio, & tamen idem e&longs;&longs;et corpus impactum, Igitnr ad hanc in&longs;tantiam alio modo re&longs;pondeo, ex appli­catione impetus &longs;emper &longs;equitur productio alterius impetus; dum &longs;cili­cet &longs;ubiectum, cui applicatur &longs;it capax motus; ex applicatione corporis &longs;eu &longs;ubiecti ip&longs;ius non &longs;emper &longs;equitur; igitur dicendum e&longs;t impetum ip&longs;um e&longs;&longs;e cau&longs;am alterius per Ax. 11. n. 1. voco enim illud cau&longs;am, ex cuius applicatione &longs;emper &longs;equitur &longs;imilis effectus; alioquin &longs;i hoc neges; proba mihi aliter ignem accendi ab alio igne; dicam enim tibi ignem applicatum e&longs;&longs;e tantùm conditionem, & produci à cœlo; proba mihi aliter calorem produci à calore? quo enim medio, vel argu­mento id euinces? quo etiam non euincam impetum produci ab im­petu: Deinde affer rationem à priori, propter quam &longs;ub&longs;tantia corporis producat impetum &longs;ur&longs;um? v. g. cum non exigat à &longs;e ip&longs;a mo­tum &longs;ursùm, qui violentus e&longs;t corpori graui; numquid certum e&longs;t, vt dicemus infrà impetum produci ad extra, vt tollatur impedimentum motus? igitur illius e&longs;t tollere impedimentnm, cuius e&longs;t exigere motum, corpus ip&longs;um graue non exigit motum &longs;ur&longs;um, &longs;ed impetus; igitur im­petus e&longs;t tollerc impedimentum &longs;ui effectus; igitur producere impetum, quo vno tolli tantùm pote&longs;t: En tibi rationem à priori, cutum nullam habeas: Præterea, cur negas impetum e&longs;&longs;e cau&longs;am &longs;ufficientem alterius impetus, cum ex eius applicatione ip&longs;o &longs;en&longs;u percipiamus produci alium impetum? quæ ratio? quid inde ab&longs;urdi, quid incommodi: Igitur tàm certum e&longs;t, immò certius impetum produci ab alio impetu, quàm calo­rem à calore. Dices impetum iam habere alium effectum &longs;cilicet mo­tum; bella profecto ratio! &longs;ed numquid motus e&longs;t effectus formalis im­petus? prætereà e&longs;t-ne effectus ad extra? deinde idem dico de calore; qui reuera habet effectum formalem &longs;ecundarium ad intra, &longs;cilicet rare­factionem, quæ e&longs;t mutatio exten&longs;ionis; quemadmodum motus e&longs;t mu­tatio loci, vel vbicationis; igitur cum hoc |non ob&longs;tante, calor pro­ducat calorem ad extra; cur impetus non producit impetum? cuius pro­ductionem concedis virtuti corporum re&longs;i&longs;titiuæ, id e&longs;t vnioni, impe­netrabilitati, & cæteris huiu&longs;modi modorum &longs;uperfluorum qui&longs;quiliis; de quibus plurimi tecum contendunt.

Scholium.

Ob&longs;eruabis nonnullas e&longs;&longs;e difficultates, quæ communes &longs;unt etiam illi &longs;ententiæ, quam &longs;equuntur ij, qui exi&longs;timant impetum ad extra produci à corpore impacto; quas tamen facilè &longs;oluemus infrà in conti­nuata no&longs;trorum Theorematum &longs;erie.

Theorema 41.

Aliquis impetus non producitur ab alio impetu. Probatur, quia aliquis impetus producitur ad intra à potentia motrice, vt patet. 2. cum non detur progre&longs;&longs;us in infinitum, nec impetus idem producatur à &longs;e ip&longs;o, ad aliquem tandem vltimum &longs;eu primum deueniendum e&longs;t, qui ab alio im­petu non producatur.

Theorema 42.

Impetus producitur &longs;emper ad extra ab alio impetu. Quia cum &longs;emper ad illius productionem requiratur applicatio alterius impetus; certè non e&longs;t ponenda alia cau&longs;a per Ax. 11.

Theorema 43.

Hinc impetus habet duplex munus cau&longs;æ; &longs;cilicet cau&longs;æ exigentis ad intra & efficientis ad extra; vtrumque patet ex dictis.

Theorema 44.

Impetus agit tantùm ad extra, vt tollat impedimentum motus; cum enim motus &longs;it finis intrin&longs;ecus impetus; certè &longs;i nihil impediret motum, haud dubiè gauderet impetus &longs;uo fine; igitur fru&longs;trà quidquam aliud de&longs;ideraret; præterea licèt applicetur à tergo aliud mobile; non tamen propterea in eo producit, vt con&longs;tat experientiâ; denique cum tan­tùm impetum cogno&longs;camus per motum; cum nequidem e&longs;&longs;et impetus, &longs;i non e&longs;&longs;et motus, per Th. 17.ce rtè totus e&longs;t impetus propter motum qui e&longs;t eius finis; igitur non agit ni&longs;i propter motum: &longs;ed non pote&longs;t excogitari, quid faciat propter motum, dum agit, ni&longs;i dicamus ideo tantùm agere, vt tollatur impedimentum; cum certum &longs;it corpus im­mobile, in quod impingitur aliud mobile, impedire eius motum.

Theorema 45.

Hinc non &longs;imul agit impetus in orbem &longs;ed tantùm per lineam &longs;ui motus; cui &longs;i nullum corpus occurrit reuerà non agit, Ratio e&longs;t; quia li­cèt aliud corpus mobili admoueatur in alia linea; cum non impediat eius motum, vt &longs;uppono; cum agat tantùm impetus ad extra, vt tollat, impedimentum motu &longs;ui &longs;ubiecti, in eo non agit, quod non impedit; & cum impediatur tantùm in vna linea, in ca tantùm agit; igitur non agit in orbem.

Scholium.

Ob&longs;eruabis primò, hanc primam e&longs;&longs;e difficultatem; cum in hoc im­petus maximè differat ab alijs qualitalibus &longs;i quæ &longs;unt, quæ agunt in or­bem, vt dicemus &longs;uo loco.

Ob&longs;eruabis &longs;ecundò, hanc etiam e&longs;&longs;e communem illorum &longs;ententiam, qui dicunt imptum ad extrà produci ab ip&longs;o mobili, &longs;ed ita vt ab illis vix &longs;olui po&longs;&longs;it; cum tamen à nobis facilè &longs;oluatur.

Ob&longs;eruabis tertiò, impetum in vtroque muncre cau&longs;æ &longs;ube&longs;&longs;e tantùm vni lineæ; &longs;cilicet exigit motum per vnam lineam; cum per plures &longs;i­mul motus e&longs;&longs;e non po&longs;&longs;it; ne idem mobile &longs;imul e&longs;&longs;et in pluribus lo­cis; & producit impetum per vnam lincam; cum producat tantùm pro­pter motum.

Ob&longs;eruabis quartò, alias qualitates, &longs;i quæ &longs;unt, non agere ad extra, vt tollant impedimentum &longs;ui effectus ad intra; qui &longs;cilicet ab impedi­mento extrin&longs;eco impediri non pote&longs;t; vt accidit in ip&longs;o impetu; etenim corpus non pote&longs;t moucri ni&longs;i nouum locum acquirat: neque nouum locum acquirere ab alio corpore occupatum, ni&longs;i corpus hoc loco ce­dat, neque hoc loco cedere pote&longs;t &longs;ine motu, vel moueri &longs;ine impetu, igitur cum impediat motum amoucri debet, accepto dumtaxat impetu ab alio mobili.

Ob&longs;eruabis quintò nonnullos e&longs;&longs;e, qui volunt motum vnius corporis transferri in aliud corpus; &longs;ed mera e&longs;t metaphora; nihil cnim pror&longs;us e&longs;t quod ab vno in aliud tran&longs;eat, &longs;eu transferatur; nec aliud dici po­te&longs;t, ni&longs;i quod dictum e&longs;t, impetum &longs;cicilet nouum produci.

Hinc etiam reiicies commentum illorum, qui dicunt ideo vnum corpus ab alio moueri, quia ab vno in aliud deriuantur corpu&longs;cula illa, quæ faciunt lumen, & calorem; quia lumen, & calor &longs;unt veræ qualita­tates, non corpu&longs;cula, vt demon&longs;trabimus in 5. tractatu: Adde quod li­cet ferrum candens aliud frigidum impellat, etiam veloci&longs;&longs;imè; hoc ip­&longs;um æquè frigidum manet; denique in cra&longs;&longs;is tenebris nix &longs;eu glacies frigidi&longs;&longs;ima pernici&longs;&longs;imè moueri pote&longs;t: &longs;ed apage i&longs;ta commenta.

Theorema 46.

Omnes partes impetus mobilis agunt ad extra actione communi. Probatur per Ax. 13. n. 1. ni&longs;i enim agerent actione communi &longs;ed quælibet &longs;uam produceret; cur potius in hac parte &longs;ubiecti, quam in alia, demde ap­pìicatur tantùm vna immediatè; Igitur agunt omnes actione commu­ni; omnes inquam illæ, quæ impediuntur; cum enim impetus agat tantùm ad extrà vt tollat impedimentum &longs;ui motus; ille pro&longs;ectò age­re non debet, cuius motus vel effectus non impeditur.

Theorema 47.

Hinc maiora corpora putà onerariæ naues, licèt tardi&longs;&longs;imo motu ferantur, cum in aliud corpus impinguntur maxima vi illud impellunt. Ratio e&longs;t; quia cum &longs;int plures partes impetus in pluribus partibus &longs;ubiecti, & omnes agant actione communi, non mirum e&longs;t &longs;i maiorem effectum producant, per Ax. 13. n. 2.

Scholium.

Vides primò in hoc ca&longs;u compen&longs;ari inten&longs;ionem ab exten&longs;ione; quippe quod præ&longs;tarent plures partes impetus in minore corporis mole inten&longs;æ; hoc idem præ&longs;tare po&longs;&longs;unt exten&longs;æ in maiore mole.

Secundò &longs;icut maior moles aptior e&longs;t ad motum imprimendum, & mi­nùs apta ad recipiendum ita minor contrà aptior e&longs;t ad recipiendum, & minùs apta ad imprimendum.

Tertiò, Hinc corpora illa, quorum partes vel nullo vel modico nexu copulantur, minimo ferè impul&longs;u commouentur; &longs;ic aër & aqua mini­mo flante vento agitantur, nubes pelluntur; hinc tot procellæ tempe­&longs;tate&longs;que cientur; nec vlla e&longs;t alia ratio, cur minima ferè venti vis, cui modicum &longs;axum re&longs;i&longs;tit, tantam aquæ, vel aëris molem commoueat, ni­&longs;i quia cum partes illorum corporum nullo ferè nexu coniunctæ &longs;int vna &longs;ine alia moucri pote&longs;t, quod in aqua gelu concreta minimè accidit.

Quartò, Hinc &longs;i maxima rupes ita comminueretur vt tota in pulue­rem &longs;eu &longs;abulum abiret, minima vis impre&longs;&longs;a particulas illas moueret.

Quintò, Hinc diuino penè con&longs;ilio factum e&longs;t, vt partes terre&longs;tris globi arctiore fibula copulentur; ne, &longs;i di&longs;iunctæ e&longs;&longs;ent, minimo flatu di&longs;pergerentur: vt videre e&longs;t in puluere etiam graui&longs;&longs;imo, qui ab aura flant e di&longs;pergitur.

Theorema 48.

Impetus, cuius motus non impeditur, non agit ad extrà. Probatur per Th. 44. hinc &longs;i aliud corpus affigas mobili à tergo, nullum impetum in eo producet, cuius effectus, qui certè impetui &longs;ingularis e&longs;t, alia ratio e&longs;&longs;e non pote&longs;t; tam enim corpus e&longs;t applicatum à tergo, quam in ip&longs;a fronte; & nihil e&longs;t in vno, quod non &longs;it in alio, ni&longs;i quod in fronte impedit motum, à tergo verò non impedit.

Corollarium 1.

Hinc egregium paradoxon erui pote&longs;t; quod &longs;cilicet cau&longs;a nece&longs;&longs;aria etiam immediatè applicata, & non impedita in &longs;ubiecto apto non agit; quod videtur e&longs;&longs;e contra Ax. 12. vnde vt agat cau&longs;a nece&longs;&longs;aria, debet applicari debito modo; &longs;i agat in orbem, omnis applicatio &longs;ufficiens e&longs;t: &longs;i verò agat tantùm per vnam lineam; certè applicari debet in ca linea; alioquin non aget defectu debitæ applicationis.

Corollarium 2.

Hinc etiam aliud paradoxon non minus iucundum; cau&longs;a nece&longs;&longs;aria appllcata, & non impedita non agit; at verò agit impedita; &longs;cilicet impetus qui tantùm agit, vt tollat impedimentum; igitur, &longs;i non impediatur non agit.

Theorema 49.

Quo minùs impeditur impetus, minùs agit ad extra, & contrà; quo plùs impeditur, plùs agit. Cum enim ideò agat ad extra, vt tollat impedi­mentum; certè &longs;i nullum e&longs;t, nihil agit, &longs;i minùs, minùs agit; igitur agit pro rata, id e&longs;t, pro diucr&longs;a impedimenti ratione.

Theorema 50.

Si linea motus, quam directionis appellant, ducatur per centrum vriu&longs;que corporis, maximum est impedimentum, vt patet. &longs;int enim duo globi, A mobilis, & B. occurrens ip&longs;i A, &longs;itque linea directionis DE ducta per centrum vtriu&longs;que AB, & punctum contactus &longs;it C; certè glo­bus B maximum ponit impedimentum, quod ab co poni po&longs;&longs;it; Igitur impetus globi A agit quantùm pote&longs;t in globum B; vt &longs;cilicet maxi­mum impedimentum remoucat.

Theorema 51.

Si linea motus vel ip&longs;ius parallela cadat perpendiculariter in extremam diametrum globi immobilis: haud dubiè nihil impedit; &longs;it enim globus mobilis A, Immobilis B, linea directionis &longs;it GA, ip&longs;i parallela FC; certè globus B. non impedit motum globi A. cum nihil loci globi B occupari debeat à globo A; Igitur impetus A non agit in globum B per Th. 48.

Theorema 52.

Si linea motus &longs;it inter vtramque; est minus impedimentum. &longs;it globus immobilis BA; &longs;it linea motus GC cum impedimento, de qua in Th. 50. &longs;it alia KB cum nullo impedimento, de qua in Th. 51. &longs;int aliæ HD, IE; certè minus e&longs;t impedimentum in contactu D, quàm in C; quia ca­dit obliquè in D, perinde atque &longs;i caderet in tangentem NO; Igitur minus impeditur; in qua vero proportione, dicemus aliàs, cum de re­flexione, & de motu mixto.

Theorema 53.

Hinc producitur in contactu C, totus impetus; in contactu D, minùs; contactu E adhuc minùs; in B nihil; quia in ca proportione producitui plùs vel minùs impetus, quo plùs e&longs;t, vel minùs impedimenti per Th. 49. &longs;ed minùs e&longs;t impedimentum in E, quàm in C; & in E, quàm in D, per Th. 52; Igitur in D producitur minùs impetus, quàm in C, & minùs in E, quàm in D.

Theorema 54.

Hinc eadem cau&longs;a nece&longs;&longs;aria etiam immedia&longs;e applicata diuer&longs;um impetum producit; vt patet in impetu, non tamen est eodem modo applicata, id e&longs;t in eadem linea.

Theorema 55.

Hinc ratio multorum effectuum phy&longs;icorum e. ui potest; cur &longs;cilicet cor­pus incidens in aliud perpendiculariter maximum ictum infligat; quia &longs;cilicet maximum impetum producit, qui po&longs;&longs;it ab eo produci; cur idem corpus obliquè incidens in aliud minorem ictum infligat; cuius rei alia ratio e&longs;&longs;e non pote&longs;t. Huc etiam reuoca tormenta bellica, quæ vel directo, vel obliquo ictu muros verberant; hinc perpendicularis forti&longs;&longs;ima e&longs;t; licèt cadem ratio pro motu corporum non valeat, quæ valet pro diffu&longs;ione, &longs;eu propagatione qualitatum.

Theorema 56.

Hinc pote&longs;t determinari quota pars impetus producatur, & quantus &longs;it ictus; cognito &longs;cilicet & &longs;uppo&longs;ito co impetus gradu, qui producitur, cum totus producitur, vt fit in perpendiculari; quippe tota men&longs;ura impetus continetur in arcu CB; quam proportionem nos infrà demon­&longs;trabimus.

Theorema 57.

Si linea directionis ducatur per centrum vtriu&longs;que globi, mobilis &longs;cilicet & immobilis, impetus producit totum impetum quem pote&longs;t producere &longs;iue in maiori globo, &longs;iue in minori, &longs;iue in æquali; patet experientia; cuius ratio e&longs;t; quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria; Igitur idem impetus codem mo­do applicatus æquali tempore, æqualem &longs;emper effectum producit, per Ax. 12. igitur cum impetus agat tantùm, vt tollat impedimentum per Th. 44. & cum in prædicta linea agat quantum pote&longs;t per Th. 50. cer­tè æqualem effectum producat nece&longs;&longs;e e&longs;t; &longs;iue in maiori &longs;iue in mino­ri, &longs;iue in æquali globo immobili.

Theorema 58.

Hinc impetus remi&longs;&longs;us potest producere inten&longs;um; & hæc e&longs;t altera diffcul­tas; cum &longs;cilicet maior globus in minorem impingitur; cum enim omnes partes impetus maioris globi agant actione communi per Th. 46. & cum agant quantùm maximè po&longs;&longs;unt; in minore globo, tot partes pro­ducunt impetus, quot in maiore, vt patet; igitur in minore globo pau­cioribus partibus &longs;ubiecti di&longs;tribuuntur plures partes impetus; crgo in qualibet parte &longs;ubiecti &longs;unt plures; &longs;ed hoc e&longs;t e&longs;&longs;e inten&longs;um, vt con&longs;tat, igitur impetus remi&longs;&longs;us producit inten&longs;um; quod e&longs;t paradoxon egre­gium.

Theorema 59.

Hinc etiam impetus inten&longs;us producit remi&longs;&longs;um, cum &longs;cilicet minor globus in maiorem incidit; quia &longs;cilicet pauciores partes impetus di&longs;tribuun­tur pluribus partibus &longs;ubiecti; igitur quælibet &longs;ubiecti pauciores impe­tus habet; quæ omnia con&longs;tant ex dictis.

Scholium.

Ob&longs;eruabis primò, &longs;ingularem impetus proprictatem, quæ alijs qua­litatibus minimè competit; nam aliæ qualitates v. g. calor; lumen in cadem di&longs;tantia effectum &longs;emper æquè inten&longs;um producunt; &longs;ecus verò impetus, qui pro maiori vel minori obice maiorem, vel minorem, hoc e&longs;t inten&longs;iorem, vel remi&longs;&longs;iorem impetum in eadem di&longs;tantia producit; cuius ratio ex eo capite petitur; quòd impetus agat tantùm ad extra propter &longs;uum effectum ad intra, vt &longs;cilicet tollat impedimentum; igi­tur in totum, quod impedit, agit; igitur non habet certam, & deter­minatam &longs;phæram; cum tantùm agat in obicem, &longs;iue &longs;it maior, &longs;iue minor: Quia verò e&longs;t cau&longs;a nece&longs;&longs;aria, æqualem effectum producit, id e&longs;t tot partes impetus in maiore, quot in minore, ergo, cum in mino­re &longs;int pauciores partes &longs;ubiecti, & plures in maiore; haud dubiè quæli­bet pars minoris habebit plures partes effectus, & quælibet pars maio­ris pauciores; igitur effectus erit inten&longs;ior in minore, & remi&longs;&longs;ior in maiorc.

Prætereà, cum dixi omnes partcs mobilis actione communi agere ad extra; ita primò intelligi debet, vt omnes illæ partes moueantur: &longs;ecun­dò, vt linea motus, &longs;eu directionis per centra grauitatis vtriu&longs;que glo­bi v, g. ducatur; alioquin, vel omnes actione communi non agunt, vel minus agunt, de quo infrà; &longs;ufficit verò iuxta præ&longs;ens in&longs;titutum, vt globus ita impellat alium vel æqualem, vel inæqualem, vt linea dire­ctionis ducatur per centrum grauitatis alterius; vide figuram. in qua linea directionis e&longs;t DE.

Theorema 60.

Impetus globi impacti in alium globum eo modo, quo diximus, id est, linea­directionis ducta per centra grauitatis vtriu&longs;que producit in eo æqualem; Pro­batur, quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria, quæ tunc agit quantum pote&longs;t per Th. 57. &longs;ed æqualis pote&longs;t producere æqualemi. Probatur primò, exemplo aliarum qualitatum; &longs;ecundò, quia ideo agit vt tollat impedi­mentum, hoc e&longs;t vt corpus illud amoueat loco; igitur æquali motu per &longs;e; alioquin ni&longs;i æquali motu amoueret, non tolleret impedimentum, vt pater; tertiò &longs;int 30. partes impetus, certèvel producent plures vel pauciores, vel totidem, non plures; cur enim potius 31. quam 32. nec etiam pauciores; cur enim potius 20. quam 18, &c. Igitur totidem; quia cum &longs;int plures numeri plurium partium &longs;upra 30. & pauciorum infra vt patet; &longs;itque tantùm vnicus numerus æqualium; certè quod vnum e&longs;t, determinatum e&longs;t, per Ax. 5. hæc ratio licèt videatur negati­ua e&longs;t tamen potenri&longs;&longs;ima: quartò, quia actus &longs;ecundus, re&longs;pondet actui primo, id e&longs;t, effectus productus virtuti cau&longs;æ producentis; itaque cum virtus agendi impetus &longs;it eius entitas, vt patet, certè impetus productus e&longs;t per &longs;e æqualis impetui producenti per &longs;e; id e&longs;t remoto omni impedimento, & facto co contactu iuxta modum prædictum, ca quo-que lege, vt impetu agat quantum pote&longs;t, & omnes partes mobilis moueantur æquali motu.

Corollarium 1.

Hinc reijcics illos, qui volunt à globo æquali produci in æquali &longs;ub­duplum impetum; in &longs;ubduplo &longs;ubtriplum; in &longs;ubquadruplo &longs;ubquin­tuplum; ratio illorum e&longs;t; quia duo globi æquales in&longs;tanti contactus perinde &longs;e habent, atque &longs;i conflarent vnum corpus; &longs;ed &longs;i conflarent vnum corpus quilibet &longs;ubduplum impetum haberet; &longs;i verò globus cum alio &longs;ubduplo faceret vnum mobilc; haud dubiè minor, id e&longs;t, &longs;ubduplus haberet tantùm &longs;ubtriplum impetum; atque ita deinceps; hoc totum fal&longs;i&longs;&longs;imum e&longs;t; nam primò &longs;i globus æqualis acciperet tantùm &longs;ubdu­plum impetum ab alio, &longs;ubduplo tantùm motu ferretur; igitur &longs;ubdu­plum &longs;patium decurreret, quod e&longs;t contra experientiam, & Th. 47. Se­cundò, ratio propo&longs;ita nulla e&longs;t; quia quando globus impactus impellit alium, e&longs;t veluti potentiâ, quæ cum tota&longs;ua vi, & cum impetu agit, cuius nulla pars transfertur in alium globum; nec enim migrat de de &longs;ubiecto in &longs;ubiectum, &longs;ed producit &longs;ibi æqualem: equidem &longs;i duo globi æquales e&longs;&longs;ent vel coniuncti, vel contigui in linea directionis, quilibet pro rata acciperet impetus producti partem à potentia applica­ta; &longs;i e&longs;&longs;ent æquales, qui&longs;que &longs;ubduplum: &longs;i alter &longs;ubduplus &longs;ubtri­plum, &c. &longs;ed hæc &longs;unt &longs;atis facilia.

Obijci fortè po&longs;&longs;et ab aliquo primò experientia; videmus enim &longs;æpè globum impul&longs;um in ludo Tudiculario moueri tardiùs globo impellen­te; re&longs;pondeo id &longs;æpè accidere; tùm quia linea directionis non connec­tit centra vtriu&longs;que globi; igitur minor e&longs;t ictus per Th 52. tùm quia globus impellens, vel impul&longs;us deficiunt à perfecta &longs;phæra; tùm quia non e&longs;t perfecta æqualitas globorum; adde quod quò accuratiùs prædi­ctæ leges ob&longs;eruantur, ip&longs;i motus ad æqualitatem propiùs accedunt, vt con&longs;tat experientia.

Obiici po&longs;&longs;et &longs;ecundò de&longs;trui aliquid impetus globi impellentis ab ip&longs;o ictu, vt con&longs;tat experientia; igitur illa pars impetus, quæ de&longs;truitur, non producit nouum impetum in globo impul&longs;o; Re&longs;pondco de&longs;truiquidem aliquid impetus in globo impacto, vt videbimus infrà; cum tamen de­&longs;truatur tantùm &longs;equenti po&longs;t ictum in&longs;tanti; certè cum exi&longs;tat adhuc ip&longs;o in&longs;tanti contactus, nece&longs;&longs;ariò agit, quippe aliquid vltimo in&longs;tanti pote&longs;t agere; adde quod illud ip&longs;um repugnat manife&longs;tæ experientiæ; licèt enim aliquando de&longs;truatur totus impetus in globo impacto, quod &longs;æpè accidit in ludo Tudiculario, nam illicò &longs;i&longs;tit pila eburnea; alius tamen globus velociter mouetur, cuius effectus rationem infrà addu­cemus.

Obijci po&longs;&longs;et tertiò inde &longs;equi progre&longs;&longs;um in infinitum, nam globus A impactus in globum B impellet cum æquali motu, & B in C etiam æquali, C in D, atque ita deinceps; modò illi globi ita &longs;tatuantur, vt linea directionis per omnium centra rectà ducatur; Re&longs;pondco, vel il-los omnes globos ita e&longs;&longs;e contiguos, vt mutuo contactu &longs;e inuicem tan­gant; vel aliquod &longs;patium inter &longs;ingulos intercipi; &longs;i primum, produci­tur impetus à potentia motrice in omnibus, &longs;i &longs;ufficiens e&longs;t; non verò vnus globus in alio, vt con&longs;tat; &longs;icut duo pondera &longs;imul attollo, quorum vnum alteri incumbit: &longs;i verò non &longs;e tangant, dico antequam A im­pingatur in B, dum &longs;patium illud interiectum percurrit, amittere aliquid impetus: idem dico de B, & C, vnde &longs;i nihil impetus in co primo motu periret & linea directionis omnium centra perfectè connecteret; ita vt omnium ictus illi omnino &longs;ine vlla deflexione re&longs;ponderent; haud du­biè non po&longs;&longs;ent e&longs;&longs;e tot globi, quin po&longs;&longs;et alius addi, qui ab vltimo pelleretur; &longs;ed vix illa omnia de quibus &longs;uprà po&longs;&longs;unt ob&longs;eruari; Hinc tamen facilè vna pars aëris aliam pellit, quod di&longs;tinctè videmus in aqua; &longs;ed de his aliàs, &longs;ufficiat modò propo&longs;itam obiectionem inde manere &longs;olutam.

Theorema 61.

Globus maior impactus in minorem imprimit illi inten&longs;iorem impetum, & velociorem motum per Th. 48. & 47. Nec e&longs;t quod aliqui opponant Prin­cipium illud mechanicum; id e&longs;t, nullum corpus po&longs;le maiorem veloci­tatis gradum alteri corpori imprimere; co &longs;cilicet gradu, quem ip&longs;um habet; nec enim inuenio Principium illud apud eos Mechanicos, qui mechanica momenta &longs;uarum demon&longs;trationum momentis confirmant; quî porro fieri pote&longs;t, vt principium illud admittatur, quod manife&longs;tæ experientiæ repugnat? Quis enim non vidit vel maius &longs;axum in aliud etiam tardo motu impactum maiorem motum, & impetum imprimere? quis non vidit maiores illas onerarias naues etiam pigro, & tardo motu labentes maximum impetum minori occurrenti cymbæ ctiam impri­mere? Rationem habes in Th. 47. &longs;ed dices; igitur aliquis velocitatis gradus nullam habet cau&longs;am; igitur e&longs;t à nihilo, quod dici non pote&longs;t. Re&longs;pondeo, plures partes impetus non produci in minore globo, quàm &longs;int in maiore; igitur nulla pars e&longs;t impetus minoris globi, quæ &longs;ui cau&longs;am &longs;ufficientem non habeat; &longs;ed cum partes impetus maioris globi di&longs;tribuantur pluribus partibus &longs;ubiecti, faciunt remi&longs;&longs;um impetum, igi­tur & tardum; cum &longs;cilicct impetus vnius partis non iuuet motum alte­rius per Th. 37. at verò cum partes impetus producti in minore globo di&longs;tribuantur paucioribus partibus &longs;ubiecti, faciunt inten&longs;iorem im­petum; igitur velociorem motum, quippe omnes producuntur ab omnibus illis actione communi per Ax. 17. num. 1. quid clarius.

Theorema 62.

Globus minor imprimit maiori remi&longs;&longs;iorem impetum & tardiorem motum & æqualis, aqitali æqualem; hæc omnia probantur per Th. 60. & præ-, cedentia.

Scholium.

Ob&longs;eruabis primò, vtrumque globum e&longs;&longs;e eiu&longs;dem materiæ; &longs;i enim &longs;int diuer&longs;æ materiæ, &longs;ecùs accidit, quàm diximus; &longs;i v. g. æncus mi­nor pellatur ab cburneo maiore, maiorem motum hic illi non impri­met; licèt enim &longs;it maior exten&longs;io eburnci; e&longs;t tamen minus pondus; igitur pauciores partes.

Secundò, cos globos accipiendos e&longs;&longs;e, quorum partes, vel non auo­lent ab ictu, vel non comprimantur; comprimuntur in plumbeis, æncis, & auolant in vitreis; cum enim &longs;it compre&longs;&longs;io, vel partium di­ui&longs;io, de&longs;truitur multùm impetus.

Tcrtiò reiice commentum illorum, qui dicunt corpus illud e&longs;&longs;e ma­joris vclocitatis capax, quod plures habet partes materiæ &longs;ub eadem quantitate; nam &longs;uppo&longs;ita cadem re&longs;i&longs;tentiæ ratione, omne corpus e&longs;t capax illius vclocitatis, cuius alind e&longs;t capax; cum nullus &longs;it motus, quo non po&longs;&longs;it dari velocior, & tardior, vt dicemus infrà; immò &longs;it glo­bus plumbeus 12. librarum, &longs;it eburncus eiu&longs;dem diametri 2. librarum, v. g. haud dubiè cadem potentia producet inten&longs;iorem impetum in cburnco, vt patet experientia, & ratio con&longs;tat ex dictis; qua&longs;i verò &longs;it aliqua materiæ inertia, quæ motum re&longs;puat; licèt fortè maior &longs;it pro­portio re&longs;i&longs;tentiæ medij comparatz cum globo cburneo, quàm compa­ratæ cum plumbeo; &longs;ed de re&longs;i&longs;tentia de percu&longs;&longs;ionc, & de &longs;patio age­mus infra.

Theorema 63.

Omnis globus, qui in alium, qui mouetur impingitur, dum hie mouetur, vt­lociùs mouetur eo &c. in quem impingi patet; alioquin numquam a&longs;&longs;cqui po&longs;&longs;et, quod ex ip&longs;is terminis con&longs;tat.

Theorema 64.

Ex hac hype&longs;i globus impactus producit in alie mouas partes impetus; quia impeditur cius motus, igitur vt tollat impedimentum, agit ad cxtra per Th. 44.

Theorema 65.

Hie impetus neuus preductus miner e&longs;t eo qus preduceretur in codem glob­immobili: ratio c&longs;t; quia &longs;i &longs;i&longs;tcret, maius e&longs;&longs;et impedimentum, quia totum motum impediret, cuius tantùm partem impedit, dum mouctur, licèt paulò tardius; igitur minus agit ad cxtra per Th. 49.

Theorema 66.

Mobile adbarens alseri mobili à terge; dum vtrque æqu velociter feratur nullum preducis in ce ipesum. Probatur, quia mobile quod præit, non impedit motum &longs;ub&longs;equentis; igitur nullum impetum ab co acci pit per Th. 48.

Theorema 67.

Hinc paradoxon egregium &longs;i quod aliud; globus percu&longs;&longs;us ab alio eadem &longs;emper velocitate mouetur, &longs;iue moueretur in&longs;tanti percu&longs;&longs;ionis, &longs;iue &longs;i­&longs;teret. v. g. &longs;it globus A quie&longs;cens, cui unprimantur ab alio B 40. gra­dus velocitatis: id e&longs;t æqualis impetus impetui percutientis, iam verò moueatur A, cum 20. grad. vclocitatis, & B, qui mouetur cum 40. impingatur, certè cum impediatur tantùm &longs;ubduplum motus, produce­tur tantùm &longs;ubduplum impetus, id c&longs;t 20. qui &longs;i addantur 20. grad. erunt 40. quæ omnia con&longs;tant per Th.49.48.&c.

Corollarium 1.

Hinc æquale &longs;emper &longs;patium percu&longs;&longs;us globus conficit, &longs;iue ante per­cu&longs;&longs;ionem moueretur, &longs;iue quie&longs;ceret.

Corollarium 2.

Hinc &longs;i &longs;ecundò percutiatur idem globus, &longs;patium totum, quod per­currit tùm à primò, tùm à &longs;ecundo ictu e&longs;t maius co, quod à primo ictu confeci&longs;&longs;et, &longs;i non fui&longs;&longs;et &longs;ecundò percu&longs;&longs;us; maius inquam &longs;egmento &longs;pa­tij interiecto inter primum & &longs;ecundum ictum.

Corollarium 3.

Hinc reiicies aliquos, quorum &longs;ententiam habes apud Doctum Mer­&longs;emium, in prop. 20. phæn. mech. quorum &longs;unt hæc verba; &longs;i malleus pcurrentem eodem, ac anteà modo percutiat, nonam &longs;ui motus partem; &longs;i verò currentem tertia vice percutiat, vnam vige&longs;imam &longs;eptimam &longs;ui motus par­tem ei tribuet, atque ita deinceps. Supponit primò hæc &longs;ententia mal­leum e&longs;&longs;e duplum pilæ percu&longs;&longs;æ. Secundò, mallcum imprimere pilæ &longs;ub­duplæ &longs;ubtriplum motum; quod fal&longs;um e&longs;t, vt con&longs;tat ex Th 6. & Co­roll. 1. Præterea, licètin primà percu&longs;&longs;ione imprimeret tantùm prædi­ctæ pilæ &longs;ubtriplum impetum, in &longs;ecunda percu&longs;&longs;ione maiorem impri­meret po&longs;t longiorem motum, vbi iam ad quictem propiùs accedit; mi­norem verò paulò po&longs;t initium motus, vt con&longs;tat ex dictis, & exip&longs;a ex­perientia; potc&longs;t quidem in aliquo puncto &longs;ui motus &longs;ecunda vice per­cuti, in quo &longs;ubtriplum tantùm motum imprimet; hoc e&longs;t eo in&longs;tanti­quo tantùm ami&longs;it tertiam fui impetus partem; tum deindc in tertia percu&longs;&longs;ione pote&longs;t tantùm (1/27) motus partem illi tribuere; co &longs;cilicet in­&longs;tanti, quo tantùm ami&longs;it (1/27) &longs;ui impetus partem; &longs;ed in alijs cemporis punctis longè alia crit impetus producti ratio; Igitur tota hæc progre&longs;­&longs;io gratis omninò fuit excogitata.

Corollarium 4.

Hinc ctiam po&longs;t &longs;ecundam percu&longs;&longs;ioncm æquale &longs;patium conficier al­teri, quod iam confecit po&longs;t primam æqualibus temporibus; igitur æqua­lis e&longs;t velocitas vtriu&longs;que motus; quia &longs;cilicet, &longs;i e&longs;t æqualis impetus, e&longs;t qualis motus: Ex his maximam carum dubitationum partem &longs;oluere po­teris quæ in eadem Mer&longs;enni propo&longs;itione courinentur reliquas vero ex dicendis infrà.

Corollarium 5.

Ex dictis ctiam colliges diuer&longs;as percu&longs;&longs;ionum rationes &longs;uppo&longs;ita di­uer&longs;a ratione ponderum globi percutientis, & percu&longs;&longs;i; cum enim impe­tus productus &longs;it æqualis per &longs;e impetui producenti, per Th.60. modò debita fiat applicatio, de qua in Th.50. &longs;i percutiens &longs;it duplus percu&longs;&longs;i, &longs;uppo&longs;ita eadem materia, motus percu&longs;&longs;i erit duplò velocior; quia im­petus erit duplò inten&longs;ior, vt con&longs;tat ex Th. 61. &longs;i verò &longs;it quadruplus, quadruplo, &c. Igitur velocitates motuum &longs;unt in ratiòne ponderum permutando.

Theorema 68.

Si corpus percu&longs;&longs;um &longs;it oblongum, & percu&longs;&longs;io fiat in centro grauitatis eiu&longs;­dem corpois; producitur impetus in percu&longs;&longs;io æqualis impetui percutientis; &longs;ed opus e&longs;t aliqua figura: Sit corpus AD, parallelipedum; diuidatur æqua­liter in E ita vt E &longs;it centrum grauitatis; &longs;i percu&longs;&longs;io &longs;iat in E per lineam perpendicularem HE, producetur impetus in corpore AD æqualis im­petui corporis percutientis; quia &longs;cilicet à corpore AD non pote&longs;t maius e&longs;&longs;e impedimentum; igitur agit quantùm pote&longs;t impetus corporis per­cutientis per Th.50. igitur producit æqualem per Th.69.

Theorema 69.

Si percu&longs;&longs;io fiat in F per lineam perpendicularem IF, minus erit impedi­mentum, quàm per HE, Quia &longs;i per HE, moueri tantùm pote&longs;t motu recto, &longs;i per IF, etiam motu circulari circa aliquod centrum; &longs;ed hic motus e&longs;t facilior quam ille; igitur minus e&longs;t impedimentum; (&longs;uppono autem cylindrum BC vtroque modo moueri po&longs;&longs;e ab applicata potentia) igitur minùs impetus producitur, &longs;i percu&longs;&longs;io fiat per IF, quàm &longs;i fiat per LK: In qua verò proportione &longs;it minus impedimentum, & minori opus impetu, po&longs;ito eodem potentiæ ni&longs;u, determinabimus facilè aliàs; vt etiam demon&longs;trabimus circa quod centrum hic circularis motus ficri debeat.

Scholium.

Ex duobus capitibus minus e&longs;&longs;e pote&longs;t impedimentum; primum e&longs;t, quod petitur à puncto contactus, &longs;ecundum à linea incidentiæ; v. g. &longs;i accipiatur punctum E, in quo e&longs;t centrum grauitatis corporis AD, & in eo &longs;iat percu&longs;&longs;io; maximum e&longs;t impedimentum ratione puncti conta­ctus, in quo fit percu&longs;&longs;io; &longs;i verò percu&longs;&longs;io fiat per lincam perpendicu­larem HE, maximum e&longs;t impedimentum, ratione lineæ; &longs;i autem ex vtroque capite &longs;imul accidat impedimentum, maximum e&longs;t omnium; iam verò &longs;i accipiatur punctum E, & linea percu&longs;sionis ME; minor e&longs;t percu&longs;sio ratione lineæ non puncti; accipiatur punctum N, & linea percu&longs;sionis MN, minor e&longs;t percu&longs;sio ratione puncti non lineæ, acci­piatur punctum N, & linea IN, minor e&longs;t percu&longs;sio ratione vtriu&longs;que; &longs;i demum accipiatur punctum E, & linea ME, minor e&longs;t percu&longs;sio ra-tione lineæ non puncti; accipiatur punctum N linea percu&longs;&longs;ionis MN, minor e&longs;t percu&longs;&longs;io ratione puncti non lineæ; &longs;i accipiatur punctum N, & linea IN, minor e&longs;t percu&longs;&longs;io ratione vtriu&longs;que: &longs;i demum accipia­tur punctum E & linea HE, maior e&longs;t percu&longs;&longs;io ratione vtriu&longs;que; igi­tur &longs;unt quatuor coniugationes; &longs;eu quatuor cla&longs;&longs;es diucr&longs;arum percu&longs;­&longs;ionum.

Hinc compen&longs;ari pote&longs;t ratione vnius quod dee&longs;t ratione alterius, v. g. &longs;i fiat percu&longs;&longs;io in puncto E per lineam ME, pote&longs;t &longs;ciri punctum inter ED, in quo percu&longs;&longs;io per lineam perpendicularem &longs;it æqualis percu&longs;&longs;ioni per lineam ME; &longs;ed de his infrà in lib. 10. cum de percu&longs;­&longs;ione, determinabimus enim vnde proportiones i&longs;tæ petendæ &longs;, & demon&longs;trabimus totam i&longs;tam rem, quæ multùm curio&longs;itatis habet, & vtilitatis.

Determinabimus etiam dato puncto percu&longs;&longs;ionis F v.g. cum &longs;equatur motus vectis, quodnam &longs;it centrum vectis &longs;eu huius motus.

Hinc demum &longs;equitur, ne hoc omittam, data minimâ percu&longs;&longs;ione per lineam MN dari po&longs;&longs;e adhuc minorem per lineam IN, & alias incli­natas; & data percu&longs;&longs;ione per lineam quantumuis inclinatam, po&longs;&longs;e da­ri æqualem per lineam perpendicularem; & data per lineam perpendi­cularem extra centrum grauitatis E, po&longs;&longs;e dari æqualem; & in qualibet data ratione per aliquam inclinatam, quæ cadat in E, &longs;ed de his fusè &longs;uo loco.

Theorema 70.

Corpus oblongum parallelipedum percutiens aliud corpus, putà globu&mtail;, motu recto per lineam directionis, quæ producta à puncto contactus ducitur per centrum globi, dum fiat contactus in centro grauitatis parallelipedi, maximum ictum infligit, &longs;eu agit quæntùm pote&longs;t. v. g. &longs;it parallelipedum EB; quod moueatur motu recto parallclo, lineis CD, HG, &c. &longs;itque globus in D; haud dubiè agit quantùm pote&longs;t, quia &longs;cilicet e&longs;t maximum impedi­mentum per Th.68. Tam enim globus in D impedit motum parallcli­pedi, quàm parallclip edum motum globi impacti per lineam ID; impedit inquam ratione oppo&longs;itionis; quia centra grauitatis vtriu&longs;que con­currunt in cadem linea; igitur &longs;i maximum e&longs;t impedimentum, agit quantùm pote&longs;t Th. 50. hinc producitur impetus æqualis per Th.60.

Theorema 71.

Si percu&longs;&longs;io fiat in G, id e&longs;t &longs;i globus e&longs;&longs;et in G, producetur minor impetus, & in M adhuc minor; vt con&longs;tat ex dictis in &longs;uperioribus Theorematis; in qua vero proportionc determinabimus aliàs.

Theorema 72.

Si corpus percutiens non &longs;it par allelipedum, &longs;ed alterius &longs;iguræ v.g. trigo­non, ADE, &longs;itque maioris facilitatis gratia Orthonium; eiu&longs;que motus &longs;it parallelus lineis ED, BC: &longs;it autem DA dupla DE; &longs;itque diui&longs;a to­ta DA æqualiter in C, in C non erit maximus ictus; quia in C non e&longs;t centrum grauitatis, vt patet; vt autem habeatur centrum impre&longs;&longs;io­nis; a&longs;&longs;umatur AN media proportionalis inter totam AD, & &longs;ubdu­plum AC; certè cum triangulum ANO &longs;it &longs;ubduplum totius ADE, vt con&longs;tat ex Geometria, & æquale trapezo ND EO; erit impetus in vtroque æqualis; igitur in N erit centrum impre&longs;&longs;ionis, vel impetus; vt autem habeatur centrum percu&longs;&longs;ionis; in quo &longs;cilicet maximus ictus in­fligitur, inueniatur centrum grauitatis H, ducaturque KHI parallela DE, centrum percu&longs;&longs;ionis erit in I; quippe in I totus impeditur impetus grauitatis vtrimque, cum &longs;it in æquilibrio; quomodo verò inueniatur punctum H facilè habetur ex Archimede, ductis &longs;cilicet AF, DB, quæ diuidant bifariam æqualiter DE, EA; vel a&longs;&longs;umpta AI dupla ID, quod demon&longs;trabimus in Mechan.

Theorema 37.

Si circa centrum immobile rotetur corpus parallelipedum CA, diuer&longs;a e&longs;t ratio percu&longs;&longs;ionum ab ea, quàm &longs;uprà propo&longs;uimus; moucatur enim circa centrum C, fitque CA diui&longs;a bifariam in B, haud dubiè punctum A faciet arcum AE eo tempore, quò punctum B faciet BD &longs;ubduplum AE; igitur punctum A duplò velociùs mouetur quàm B, vt con&longs;tat; igi­tur habet duplò maiorem impetum; cum effectum habeat duplò maio­rem per Ax. 13. n. 4. igitur cum totus motus &longs;egmenti AB &longs;it ad to­tum motum &longs;egmenti BC, vt &longs;patia acqui&longs;ita; certè &longs;patia acqui&longs;ita &longs;unt vt arcus; igitur & trapezus BAED, continet 3/4 totius CAE, vt con&longs;tat; &longs;unt enim &longs;ectores &longs;imilis in ratione duplicata radiorum; igi­tur totus motus &longs;egmenti BC &longs;ubquadruplus motus totius CA;, gitur & impetus; vt autem habeatur centrum impre&longs;&longs;ionis, vel impetus; &longs;it &longs;e­ctor CHI, &longs;ubduplus totius CAE quod quomodo fiat, patet ex Geo­metria; accipiatur tantùm &longs;ubdupla diagonalis quadrati lateris CA, igi­tur in puncto H e&longs;t centrum impre&longs;&longs;ionis, &longs;eu media proportionalis in­ter totam CA, & &longs;ubduplam CB: vt autem habeatur percu&longs;&longs;ionis, a&longs;­&longs;umatur CY dupla YA; Dico punctum Y e&longs;&longs;c centrum percu&longs;&longs;ionis; quia perinde &longs;e habet, atque &longs;i e&longs;&longs;et trianguli cadentis ictus, vt demon­&longs;trabimus aliàs nunc tantùm indica&longs;&longs;e &longs;ufficiat.

Corollarium 1.

Hinc etiam&longs;oluctur, quod proponunt aliqui; &longs;eu potiùs quærunt; in quà &longs;cilicet parte maiorem ictum infligat en&longs;is; &longs;i enim &longs;it ciu&longs;dem cra&longs;&longs;itici in omnibus &longs;uis partibus, idem dicendum e&longs;t quod de cylin­dro CA; &longs;i verò in mucronem de&longs;inat, inueniemus etiam centrum percu&longs;&longs;ionis.

Corollarium 2.

Huc etiam reuoca clauarum ictus, vel aliorum corporum, quæ ad in &longs;tar &longs;eu conorum, &longs;eu pyidum ver&longs;us mucronem maiora &longs;unt, vel den&longs;iora; quippe ex iacto &longs;uprà principio i&longs;torum omnium effectuum rationes demon&longs;trabimus.

Corollarium 1.

Colligemus etiam quid dicendum &longs;it de malleorum ictu; &longs;it enim malleus F æqualis malleo G (in his vna fere manubrij longitudinis ha­betur ratio) ducatur arcus NM, itemque OG; ictus mallei G e&longs;t ferè &longs;ubduplus alterius, dum vterque malleus &longs;it æqualis; dixi ferè, quia motus totius mallei G non e&longs;t omninò &longs;ubduplus motus mallci F, quia &longs;cilicet trapezus OD e&longs;t minor &longs;ubduplo alterius NE; quotâ vero parte &longs;it minor facilè pote&longs;t &longs;ciri opera Geometriæ: &longs;ed hæc omnia determi­nabimus.

Theorema 74.

Si daretur potentia motrix, quæ &longs;emper agere po&longs;&longs;et, impetus po&longs;&longs;et intendi in infinitum; pater, quia quocumque dato motu pote&longs;t dari velocior in infinitum; igitur pote&longs;t dari impetus inten&longs;ior, & inten&longs;ior in infinitum.

Scholium.

Hîc ob&longs;erua nouum di&longs;crimen, quod intercedit inter impetum, & alias qualitates; quæ fortè non po&longs;&longs;unt intendi in infinitum, ratio di&longs;­criminis e&longs;t, quia totus calor exten&longs;us in maiore &longs;ubiecto non pote&longs;t produci in minore, in quo eadem cau&longs;a cumdem &longs;emper effectum pro­ducit; quia &longs;cilicet agit vniformiter difformiter; at verò impetus exten­&longs;us in magno den&longs;oque malleo pote&longs;t producere æqualem in maximâ ferè pilâ.

Theorema 75.

Impetus &longs;imilis, id e&longs;t, ad eamdem lineam determinatus, & æqualis in in­ten&longs;ione, non pote&longs;t intendere alium &longs;imilem; Probatur, quia agit tantùm ad extra, vt tollat impedimentum per Th. 44. &longs;ed eorum mobilium, quæ ver&longs;us eamdem partem pari velocitate mouentur, neutrum impedit al­terius motum, vt con&longs;tat; igitur impetus &longs;imilis, &c.

Scholium.

Ob&longs;erua de impetu &longs;imili id tantùm dici; &longs;imili inquam id e&longs;t non modò eiu&longs;dem inten&longs;ionis; &longs;ed etiam eiu&longs;dem lineæ: &longs;i enim alterum de&longs;it, haud dubiè &longs;imilis impetus non e&longs;t; &longs;ic impetus quatuor grad. in­tendere pote&longs;t impetum duorum graduum; licèt vterque ad eamdem li­neam &longs;it determinatus; &longs;i verò ad diuer&longs;as lineas determinentur; etiam impetus vt duo pote&longs;t intendere impetum vt quatuor.

Ob&longs;eruabis præterea hoc Theorema ita e&longs;&longs;e intelligendum, vt impe­tus mobilis præeuntis nullo modo impediatur; alioquin mobile &longs;ucce­dens omninò aliud vrgeret, vt con&longs;tat.

Corollarium.

Hinc &longs;imile pote&longs;t in aliquo ca&longs;u agere in &longs;imile; vnde rectè colligo id tantùm dictum e&longs;&longs;e ab Ari&longs;totele de qualitatibus alteratiuis; quid verò accidat, cum mobile grauc mobili alteri &longs;uperponitur; dicemus infrà

Theorema 76.

Exten&longs;io impetus respondet extentioni &longs;ui &longs;nbiecti, &longs;cilicet mobilis; cum enim extra &longs;ubjectum e&longs;&longs;e non po&longs;&longs;it, cum &longs;it qualitas; certè ibi e&longs;t, vbi &longs;ubjectum e&longs;t; nam penetratur accidens cum ip&longs;o &longs;ujecto.

Scolium.

Ob&longs;eruabis qualitatem omnem ita &longs;uo &longs;ubjecto coëxtendi, vt æqua­lem omnino quodlibet eius punctum, &longs;eu pars extentionem habeat ex­tentioni puncti, &longs;eu partis &longs;ui &longs;ubjecti; nec enim alliud e&longs;t, vnde po&longs;&longs;it determinari extentio qualitatum, præter ip&longs;am exten&longs;ionem &longs;ubjecti; quod maximè in impetu videre e&longs;t, cuius partes in mobili den&longs;o minori extentioni &longs;ubjacent, quàm in mobili raro; cum ex maiore ictu &longs;eu per­cu&longs;&longs;ione in mobili den&longs;o plures impetus agentis partes e&longs;&longs;e con&longs;tet; quia &longs;cilicet &longs;unt plures partes &longs;ubiecti.

Theorema 77.

Datur impetus altero impetu perfectior &longs;ecundum entitatem; dixi &longs;ecun­dum entitatem; quia iam dictum e&longs;t &longs;uprà dari perfectiorem &longs;ecundum inten&longs;ionem; huius Theorematis veritas mihi maximè demon&longs;tranda e&longs;t, ex quo tàm multa infrà deducemus; &longs;ic autem probamus; Quotie&longs;­cunque mouetur corpus, producuntur &longs;altem tot partes impetus quot &longs;unt partes mobilis per Th. 33. Quotie&longs;cunque producuntur in mobili tot partes impetus quot &longs;unt in mobili partes &longs;ubjecti, mouetur mobile, modó non impediatur; quia po&longs;ita cau&longs;a nece&longs;&longs;aria, & non impedita per Ax. 11. ponitur effectus, quod de omni cau&longs;a, &longs;ed de fotmali poti&longs;&longs;imum dicidebet; præterea-datur aliquod pondus, quod data potentia &longs;ine me­chanico organo moucre non pote&longs;t, licèt cum organo facilè moueat; hæc hypothe&longs;is certa e&longs;t; igitur cum mouet, producit tot partes impetus quot &longs;unt nece&longs;&longs;ariæ, vt omnibus partibus mobilis di&longs;tribuantur per idem Th. 33. cum verò non mouet, non producit tot partes impetus vt con&longs;tat ex dictis; igitur producit plures cnm organo in mobili, quàm &longs;ine organo; igitur imperfectiores, quod demon&longs;tro: &longs;it enim vectis BF, cuius cen­trum &longs;eu fulcrum &longs;it in A, potentia in B, pondus G, quod attollitur in F; plures partes impetus produci po&longs;&longs;unt in F, vel in E, quàm in B, &longs;cilicet in ip&longs;o pondere; quia pondus quod non pote&longs;t attolli in B, attollitur in E, vel in F, vt patet ex dictis; præterea punctum F mouetur tardius, quàm B; quia motus &longs;unt vt arcus, atcus vt &longs;emidiametri, hæ demum vt AF, ad AB; igitur motus puncti F, e&longs;t tardior, vel imperfectior; igitur im­petus puncti F, e&longs;t imperfectior impetu puncti B, per Ax. 13 num.4. atqui non e&longs;t imperfectior ratione numeri partium, igitur ratione entiratis, quæ imperfectior e&longs;t; igitur datur impetus altero impetu imperfectior.

Scholium.

Ob&longs;eruabis primò multa hîc &longs;upponi &longs;eu de&longs;iderari, quæ pertinent ad propagationem impetus, de quibus infrà; Secundò hoc Theorema per Axioma illud Metaph. probari, Data quacumque creatura dari potest perfectior, vel imperfectior.

Tertiò, &longs;i dato quocunque motu pote&longs;t dari tardior: igitur dato quo­cunque impetu pote&longs;t dari imperfectior.

Quartò, &longs;i daretur punctum impetus in inten&longs;ione: non po&longs;&longs;et dari motus tatdior in infinitum &longs;ine diuer&longs;is gradibus perfectionis.

Quintò, &longs;ine hac diuer&longs;a impetus perfectione nonp o&longs;&longs;et explicari productio continua impetus, quæ &longs;it temporibus inæqualibus, neque de­&longs;tructio ciu&longs;dem impetus; nec motus in diuer&longs;is planis inclinatis, vel in­diuer&longs;is lineis citra perpendicularem, &longs;ed de his omnibus &longs;uo loco.

Sextò, Denique ratio propo&longs;ita rem i&longs;tam euincit; cum enim in motu vectis plures partes producantur ver&longs;us centrum, &longs;cilicet, in maiori pon­dere, quod attollitur; & cum hæ habeant motum tardiorem, &longs;equitur ne­ce&longs;&longs;ariò e&longs;&longs;e imperfectiores.

Theorema 78.

Dato quocumque impetu dari pote&longs;t imperfectior, & imperfectior, quia da­to quocumque motu dari pote&longs;t tardior, ergo dato quocumque impetu imperfectior.

Theorema 79.

Non pote&longs;t explicari tarditas motus &longs;ine diuer&longs;a perfectione impetus, per pauciores &longs;cilicet eiu&longs;dem impetus partes. Primò, quia cum retardari po&longs;&longs;it hic motus, & de&longs;trui &longs;ucce&longs;&longs;inè hic impetus; cumque in&longs;tantia motus velocioris &longs;int breuiora; certè initio motus, breuiori &longs;cilicet tempore imperfectior impetus de&longs;trui tantùm pote&longs;t; cum enim æqualis æquali­bus temporibus; certè inæqualis inæqualibus. Secundò quia vix explica­ri pore&longs;t quomodo duæ formæ homogeneæ in eodem &longs;ubiecti puncto exi&longs;tere po&longs;&longs;int, quod ctiam in commune e&longs;t calori, lumini, &c.

Theorema 80.

Cum applicatur potentia centro vectis, non producitur æqualis impetus ver­&longs;us circumferentiam in omnibus partibus, &longs;ed maior ver&longs;us eandem circumfe­rentiam, quia e&longs;t maior motus.

Corollarium 1.

Hinc difficiliùs attollitur pertica CA ex puncto C motu circulari, quàm ex puncto B motu recto; quia &longs;cilicet, cum motu recto ex puncto B attollitur, omnes partes mouentur motu æquali; igitur impetus æqualiter omnibus di&longs;tribuitur; igitur modò producantur tot partes impetus, quot &longs;unt partes in mobili; haud dubiè attolletur: at verò, cum motu circulari ex puncto C attollitur, omnes partcs inæquali motu attolluntur; igitur plures &longs;unt nece&longs;&longs;ariæ, vt attollatur motu circulari; igitur difficiliùs iuxta experimentum; adde quod cum applicatur potentia in C, punctum A, maius momentum habet, de quo aùàs.

Corollarium 2.

Hinc ratio cuidens illius experimenti, quo manife&longs;tè con&longs;tat perti-cam CA, ex A, facilius attolli motu recto, quàm circulari; cum &longs;ci­licct cuiu&longs;dam qua&longs;i reflexionis opera eodem tempore vtraque extremi­tas æquali motu attollitur.

Theorema 81.

Fig.7. Tab.1.

Si verò applicetur potentia extra centrum vectis v. g. in F, po&longs;ito centro inA, producitur impetus minor ab F, ver&longs;us A; ab verò ver&longs;us E, producitur eiu&longs;dem perfectionis proportionaliter, cuius e&longs;t ab F, ver&longs;us A; denique ab E, ver&longs;us B, producitur quidem vnum punctum, vel vnus gradus impetus eiu&longs;dem perfectionis cum eo, qui productus e&longs;t in F, & in E (&longs;upponi­turenim ex. gr. vnus tantùm gradus in F, & in E, productus) at verò producuntur alij imperfectiones. v.g. in D, præter æquè perfectum pro­ducuntur 3. alij adæquantes perfectionem prioris; in C verò, præter 4. &longs;imiles ijs, qui &longs;unt in D, producuntur 5. alij adæquantes prioris perfe­ctionem in B7; atque ita deinceps per numeros impares, & quadrata, nullus tamen producitur perfectioris entitatis.

Theorema 82.

Determinatur hæc diuer&longs;a perfectio impetus à diuer&longs;a perfectione motus, quatenus fit tali modo; quæ non pote&longs;t explicari per impetum remi&longs;&longs;io­rem, vel inten&longs;iorem; nam cum &longs;it tantùm impetus in&longs;titutus propter motum; certè ille tantùm impetus produci pote&longs;t, ex quo pote&longs;t &longs;equi motus; igitur &longs;i tali tantùm motu data pars mobilis moueri pote&longs;t; haud dubiètalis tantùm impetus, ex quo &longs;equitur talis motus, in ea produ­cetur, & tali modo.

Theorema 83.

Perfectio impetus non petitur tantùm à perfectione motus &longs;i con&longs;ideretur &longs;eor&longs;im entitas eiu&longs;dem impetus; &longs;ed debet comparari tota collectio omniu&mtail; partium impetus, quæ in&longs;unt datæ parti &longs;ubiecti, cum tota collectione partium quæ alteri porti mobilis in&longs;unt; quippe plures partes impetus po&longs;&longs;unt ha­bere eum motum, vel potius eam motus perfectionem, quam pauciores haberent; igitur perfectio illarum e&longs;t ab ip&longs;o motu, quatenus cum ip&longs;o partium numero comparatur.

Theorema 84.

Impetus perfectus producere pote&longs;t imperfectum; patet in vecte; nam po­tentia, &longs;en pondus extremitati appen&longs;um producit in &longs;e impetum, à quo deinde impetus in toto vecte producitur per Th.42. &longs;ed impetus pon­deris appen&longs;i e&longs;t ciu&longs;dem perfectionis cum impetu producto in ip&longs;a ve­ctis extremitate, ex qua pendet; cum &longs;it vrriu&longs;que æqualis motus; &longs;ed ver&longs;us centrum ciu&longs;dem vectis producitur impetus imperfectior per Th.82. igitur imperfectus à perfecto producitur.

Theorema 85.

Impetus perfectus nunquam producitur ab imperfecto, per Ax. 3. num. 2. adde quod nunquam effectus perfectio &longs;uperat perfectionem cau&longs;æ; dixi perfectum ab imperfecto; &longs;cilicet &longs;i con&longs;ideretur perfectio ratione en-titatis; cum reuerâ, vt dictum e&longs;t &longs;uprà, remi&longs;&longs;us producat inten&longs;um, quod in vecte clari&longs;&longs;unum e&longs;t; quippe momentum applicatum in F, quod tardiùs mouetur deor&longs;um, quàm B, &longs;ur&longs;um, vt patet, habet impetum re­mi&longs;&longs;iorem, qui tamen producit in B, inten&longs;iorem: Pro quo, ob&longs;eruabis impetum imperfectum cum alio perfecto actione communi agentem po&longs;&longs;e concurrere ad producendum perfectum, vt patet; non tamen in ratione cau&longs;æ totalis: &longs;imiliter plures imperfecti &longs;imul concurrentes po&longs;&longs;unt producere perfectum; quia plures imperfecti conjunctim adæ­quant perfectionem alterius perfectioris &longs;inguli &longs;eor&longs;im.

Ob&longs;eruabis &longs;ecundò præclarum naturæ in&longs;titutum, quo factum e&longs;t; vt cum vires hominum maiora pondera leuare non po&longs;&longs;int, &longs;i &longs;eor&longs;un con&longs;iderentur; cum organis tamen mechanicis conjunctæ nullum pon­dus quantumuis immane leuare non po&longs;&longs;int; quod certè nullo modo ac­cideret, ni&longs;i plures partes impetus producerent neque plures producere po&longs;&longs;ent, ni&longs;i minoris perfectionis e&longs;&longs;ent; quia faciliùs producitur effe­ctus imperfectus, quam perfectus per Ax. 13.num.4.

Tertiò hinc optimè à natura proui&longs;um e&longs;t, vt motus tardior in infi­nitum 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 alio imperfectior.

Quartò, hinc quoque benè explicatur diuer&longs;itas impetus, quæ oritur tum à diuer&longs;o medio, tùm à plano inclinato, tùm ab aliis impedimentis, tùm à diuer&longs;o ni&longs;u eiu&longs;dem potentiæ, tùm maximè à diuer&longs;o applicatio­nis modo; de quibus aliàs.

Quintò, &longs;i potentia applicata mobili immediatè illud moueat motu recto, vel in &longs;ingulis punctis mobilis producitur vnum punctum impe­tus, vel plura; &longs;i primum, erit primus tantùm gradus maximæ perfectio­nis; ita vt perfectiorem producere non po&longs;&longs;it, ad quem e&longs;t determinata potentia; imperfectiorem tamen impetu innato, de quo infrà; &longs;i verò &longs;ecundum, producet in &longs;ingulis paitibus eumdem gradum perfecti&longs;&longs;i­mum cum aliis pluribus, vel paucioribus heterogeneis, & imperfectio­ribus.

Theorema 86.

Potentia naturalis grauium producit tantùm vno in&longs;tanti ad intra vnicum punctum impetus in quolibet puncto &longs;ubiecti; &longs;i tamen impetum producit, quod definiam lib. 20. & &longs;i dentur puncta &longs;ubiecti, quod ad præ&longs;ens in&longs;titutum non pertinet; Probatur, quia fru&longs;trà e&longs;&longs;ent plura puncta impetus; nec enim &longs;unt multiplicandæ formæ &longs;ine nece&longs;&longs;itate, ratione &c. per Ax. 7. & 3. n. 1. Præterea non e&longs;t, cur potius produceret 2. quàm 3. 4. &c. atqui quod vnum e&longs;t, determinatum e&longs;t per Ax. 5.

Theorema 87.

Potentia motrix animantium etiam vno in&longs;tanti plura puncta, &longs;en partes impetus in eadem parte &longs;ubiecti producere potest; Probatur in proiectis, quorum impetus aliquando plùs, aliquando minùs durat licèt &longs;en&longs;im &longs;ingulis in&longs;tantibus aliquid illius de&longs;truatur; determinatur autem numerus punctorum, &longs;eu partium ab ea potentia, cui &longs;ube&longs;t potentia motrix; quia modò maior e&longs;t ni&longs;us, modò minor.

Theorema 88.

Eadem potentia inæqualibus temporibus impetum inæqualem in perfectio­ne producit; accipiatur enim totum illud tempus, quo vnicum tantùm punctum impetus producit (vocetur in&longs;tans) de quo in Th. 86; certè &longs;i in minori tempore agat, minùs aget, per Ax. 13. num. 4. &longs;ed non pote&longs;t minùs agere ratione numeri, vt patet; igitur ratione perfectio­nis.

Scholium.

Ob&longs;eruabis &longs;ine hoc Theoremate explicari non po&longs;&longs;e accelerationem motus naturalis, vel augmentum impetus, vt videbimus.

Theorema 89.

Impetus violenti, qui &longs;en&longs;im de&longs;truitur in proiectis, po&longs;itis ij&longs;dem circum­&longs;tantiis medij, & re&longs;i&longs;tentiæ, minori tempore minùs de&longs;truitur; plus verò ma­jori: Quia hæc de&longs;tructio habet cau&longs;am; nam quidquid de&longs;truitur, ad exigentiam alicuius de&longs;truitur, per Ax. 14. num. 2. igitur minori tempore minùs de&longs;truitur per Ax. 13. 4. alioquin totus &longs;imul debe­ret de&longs;trui.

Scholium.

Ob&longs;eruabis etiam &longs;ine hoc Theoremate non po&longs;&longs;e explicari de&longs;tru­ctionem impetus violenti, vt videbimus infrà.

Corollarium 1.

Hinc, quò potentia diutiùs manet applicata (putà malleo) percu&longs;&longs;io ma­ior e&longs;t.

Corollarium 2.

Hinc, quò impedimentum diutiùs manet applicatum, illa de&longs;tructio e&longs;t maior.

Corollarium 3.

Hinc præclara eruitur ratio, cur maior lapis, quàm minor impactus maiorem ictum infligat; licèt tot partes impetus eodem in&longs;tanti produ­cantur in vno, quot in alio: quia &longs;cilicet diutiùs manet applicatus po­tentiæ; &longs;ed hanc rationem explicabimus fusè lib. 10. cum de percu&longs;­&longs;ione.

Theorema 90.

Impetus propagatur nece&longs;&longs;ariò per totum corpus impul&longs;um, &longs;eu proiectum.

Probatur; quia cum omnes eius partes moueantur, nec vlla &longs;ine im­petu moueri po&longs;&longs;it per Th. 18. & 33. cum etiam potentia motrix non &longs;it omnibus immediatè applicata, vt con&longs;tat; certè &longs;ine propagatione, vel diffu&longs;ione non pote&longs;t explicari productio huius motus.

Scholium.

Ob&longs;eruabis propagationem impetus, vel alterius qualitatis e&longs;&longs;e tan­tùm continuatam eiu&longs;dem productionem, quæ incipit ab ea parte, cui potentia e&longs;t immediatè applicata, & propagatur, &longs;eu diffunditur per omnes alias donec ad vltimam perueniat eo modo, quo iam definio.

Theorema 92.

Illa progatio non fit per motum localem, ita vt pars impetus producta in prima parte &longs;ubiecti tran&longs;eat ad &longs;ecundam, patet; quia cum impetus &longs;it ac­cidens per Th. 8. de &longs;ubiecto in &longs;ubiectum tran&longs;ire non pote&longs;t per deff. accidentis; de qua in Metaphy&longs;icâ; nec e&longs;t quod aliqui dicant &longs;e non po&longs;&longs;e concipere, quomodo id fiat &longs;ine motu locali; cum ip&longs;is etiam oculis qua&longs;i ocrnatur; cum enim percutis corpus oblongum AE, & cadit ictus in extremitatem A, corpus ip&longs;um totum &longs;unul moues; igitur pars impe­tus, quæ recipitur in A, non migrat in E, &longs;ed hæc producitur in A, & alia in B, alia in C, atque ita deinceps.

Scholium.

Ob&longs;eruabis ex hac propagatione impetus per analogiam rectè om­ninò explicari propagationem luminis, & aliarum qualitatum, de qui­bus &longs;uo loco.

Theorema 92.

In propagatione impetus prima pars A v. g. non producit partem B, & hæc C; hæc verò D, atque ita deinceps; Probatur. Primò, quia &longs;i hoc e&longs;&longs;et, omne corpus po&longs;&longs;et moueri à qualibet potentia; nam modò po&longs;&longs;et pro­duci vnum punctum impetus, hoc etiam aliud produceret, & hoc aliud, atque ita deinceps. Secundò, Minimum granum &longs;uperpo&longs;itum rupi, to­tam ip&longs;am rupem mouere po&longs;&longs;et. Tertio, Quia vel in omnibus, vel in nulla parte impetus producitur per Th.33. Quartò, quia impetus mobi­lis projecti intenderetur; nam impetus vnius partis impetum alterius intenderet. Quintò, quia impetus partis B, tàm ageret in A, trahendo, quàm in C pellendo; cum impetus vtroque modo propagetur. Sextò, &longs;i applicaretur potentia in C, non video, cur impetus partis C, ageret po­tius versùs E, quàm versùs A? alioquin cadem pars impetus plures pro­ducere po&longs;&longs;et; igitur impetus potentiæ motricis &longs;ufficiens erit cau&longs;a ad producendum totum alium. Septimò, tractionis impetus explicari non pote&longs;t, &longs;i impetus vnius partis producat in alia impetum; alioquin dare­tur mutua actio infinities repetita, vt con&longs;ideranti patebit. Octauò, &longs;i impetus vnius partis producit in alia; &longs;int duo globi contigui; igitur il­le, qui impellit alium, reflecti po&longs;&longs;et, quod nunquam accidit quando &longs;unt contigui.

Ob&longs;eruabis illud quidem verum e&longs;&longs;e in motu recto, &longs;ecus in circulari; nam cum cylindrus circa alteram extremitatem vibratus deor&longs;um cadit; partes, quæ propiùs ad extremitatem immobilem accedunt iuuant mo­tum aliarum, quæ longiùs ab eadem recedunt.

Theorema 93.

Impetus propagatur eodem in&longs;tanti, id e&longs;t, &longs;ine temporis &longs;ucce&longs;&longs;ione. Proba­tur; &longs;it enim applicata potentia in A, dico &longs;imul produci impetum in BCDE; quia &longs;i primo in&longs;tanti produceretur in A, & &longs;ecundo in B, vel A moueretur ante B, vel impetus in A e&longs;&longs;et fru&longs;trà; vtrumque e&longs;t ab&longs;ur­dum; nam totum AE, &longs;imul mouetur.

Theorema 94.

Tribus tantùm modis propagari pote&longs;t impetus ratione inten&longs;ionis. Primò &longs;i æqualiter omnibus partibus &longs;ubjecti di&longs;tribuatur; id e&longs;t vniformiter. Secundò, &longs;i plùs partibus propioribus, & minùs remotioribus. Tertiò, è contra, &longs;i plùs remotioribus, & minùs propioribus; tribus etiam ratione perfectionis eo modo, quo diximus de inten&longs;ione; at verò nouem mo­dis propagari pote&longs;t ratione vtriu&longs;que; patet ex regula combinationum; &longs;i enim 3. ducantur in 3. habebis 9. Iam &longs;upere&longs;t, vt videamus, an reue­rà omnibus i&longs;tis modis impetus re ip&longs;a propagetur; quod licèt difficile &longs;it, & vix hactenus explicatum: Audeo tamen polliceri meum &longs;uper hac re conatum non pror&longs;us inutilem fore.

Theorema 95.

Impetus propagatur vniformiter in mobili, cuius omnes partes mouentur æquali motu; probatur, quia impetus non cogno&longs;citur ni&longs;i per motum; igitur vbi e&longs;t æqualis motus, debet e&longs;&longs;e æqualis impetus in omnibus par­tibus, id e&longs;t æqualis graduum heterogeneorum collectio, in quo non e&longs;t difficultas.

Scholium.

Ob&longs;eruabis illud mobile moueri motu æquali &longs;ecundum omnes &longs;ui partes, quod mouetur motu recto; quippe fieri non pote&longs;t, quin omnes partes, quæ mouentur motu recto &longs;implici, motu etiam æquali mouean­tur.

Theorema 96.

Cum duo corpora &longs;e&longs;e mutuò tangunt, impetus in vtroque propagatur &longs;int v. g. globi A & B, æquales &longs;ibi inuicem contigui in C, &longs;it applicata po­tentia in D, non modò producet impetum in globo A, &longs;ed etiam in B: probatur primò, quia &longs;e habent per modum vnius, vt patet ex re&longs;i&longs;ten­&longs;tia, nec enim A moueri pote&longs;t &longs;ine B per lineam DE, quod certè cla­ri&longs;&longs;imum e&longs;t; probatur &longs;ecundò quia &longs;i A produceret impetum in B, duo globi, vel 3. vel 5. vel infiniti tantùm re&longs;i&longs;terent, quantùm vnicus glo­bus, quod fal&longs;um & ab&longs;urdum e&longs;t. Tertiò, Ratio à priori e&longs;t; quia idco producitur, & propagatur impetus in toto A; quia vna pars non pote&longs;t moueri &longs;ine alia per Th. 33. &longs;ed non pote&longs;t A moueri ni&longs;i moucatur B; igitur in vtroque &longs;imul, & æqualiter propagatur impetus.

Corollarium 1.

Hinc ratio manife&longs;ta cur maior &longs;it re&longs;i&longs;tentia duorum quàm vnius.

Corollarium. 2.

Hinc eadem vis requiritur ad &longs;u&longs;tinenda duo pondera; &longs;iue vtrum­que &longs;eor&longs;im humeris incubet, &longs;iue alterum alteri &longs;uperponatur.

Corollarium 3.

Hinc percu&longs;&longs;io vel ictus globi B, cui alter A à tergo immediatè in­&longs;t&longs;tit maior e&longs;t.

Corollarium 4.

Hinc pondus alteri &longs;uperpo&longs;itum actione communi cum alio graui­tat in &longs;uppo&longs;itam manum. v. g.

Corollarium 5.

Hinc potentia applicata in D, minùs impetus &longs;ingulis imprimit.

Corollarium 6.

Hinc demum licèt impetus ratione inten&longs;ionis &longs;it æqualis in vtroque globo; attamen, &longs;i accipiatur numerus partium vtriu&longs;que impetus, im­petus &longs;unt vt globi v. g. &longs;i B e&longs;t æqualis A impetus productus in B e&longs;t æqualis producto in A, &longs;i B &longs;it &longs;ubduplus, vel &longs;ubtriplus, impetus e&longs;t &longs;ubtriplus, vel &longs;ubduplus; quorum omnium rationes patent ex Th.96.

Corollarium 7.

Hinc etiam colligi pote&longs;t manife&longs;tum di&longs;crimen, quod intercedit inter propagationem impetus, & aliarum qualitatum, quæ (vt vulgò dicitur) vniformiter difformiter propagantur, id e&longs;t, æqualiter in æquali di&longs;tantia, & inæqualiter inæquali.

Corollarium 8.

Hinc demum colligi pote&longs;t non modò impetum produci in globo B v. g. verùm etiam in aëre ambiente, cui &longs;cilicet globus contiguus e&longs;t; qui reuera aër facilè amouetur; tùm quia propter raritatem pauci&longs;&longs;imæ partes mouendæ &longs;unt; tùm quia facilè diuiduntur, de quibus alias; tùm quia, ne detur vaçuum, &longs;patium à tergo relictum occupare debet, quod reuerà præ&longs;tat breui peracto circuitu, vt videre e&longs;t in aqua; nec enim totus aër agitari debet; quis enim id con&longs;equi po&longs;&longs;et; tum denique, quia aër non grauitat in aëre, igitur cum non re&longs;i&longs;tat vlla grauitatio, facilè moueri pote&longs;t.

Theorema 97.

Cum applicatur potentia centro motus circularis, ita propagatur impetus, vt plures partes impetus continuò producantur ver&longs;us circumferentiam; &longs;it enim cylindrus CA, fig. Th. 73. &longs;it centrum motus C; haud dubiè plures partes impetus producuntur in B, quàm in C, & plures in A, quam in B; quia, cum pars B moueatur velociùs, quàm C, & A quàm B; certè, vbi e&longs;t maior motus, vel effectus, ibi debet e&longs;&longs;e maior impetus, vel cau&longs;a per Ax. 13. n. 4. quod autem &longs;it maior motus, con&longs;tat ex maioribus &longs;patiis, vel arcubus æquali tempore confectis; quod verò &longs;it impetus inten&longs;ior versùs circumferentiam, non perfectior, patet per Th. 8.

Theorema 98.

Inten&longs;io impetus propagati iuxta hunc modum &longs;e habet, vt distantia à cen­tro motus; &longs;int enim punctum B, & pnnctum A: ita &longs;e habet inten&longs;io impetus puncti A ad inten&longs;ionem impetus puncti B, vt di&longs;tantia AC ad BC. Probatur, quia cum impetus &longs;int vt motus, motus vt &longs;patia, &longs;patia verò &longs;int arcus AE. BD; arcus &longs;unt, vt &longs;emidiametri AC, BC; igitur vt di&longs;tantiæ quòd erat demon&longs;trandum.

Corollarium 1.

Hinc &longs;i di&longs;tantia CA e&longs;t dupla di&longs;tantiæ CB, impetus in A e&longs;t du­plus impetus in B: at verò impetus &longs;egmenti e&longs;t ad impetum alterius, vt diximus in Th. 73.

Corollarium 2.

Hinc hæc propagatio fit iuxta progre&longs;&longs;ionem arithmeticam id e&longs;t, &longs;i in primâ parte ver&longs;us centrum producitur impetus vt 1. in &longs;ecunda pro­ducitur vt duo, in tertiâ vt tria, atque ita deinceps; quia proportio arithmetica e&longs;t laterum, &longs;eu linearum.

Corollarium 3.

Hinc hæc propagatio e&longs;t omninò inuer&longs;a illius, quæ aliis qualitatibus competit, vt patet.

Corollarium 4.

Hinc etiam manife&longs;ta ratio &longs;equitur illius experimenti, quod propo­&longs;uimus corol. 2. Th. 80.

Corollarium 5.

Hinc &longs;i tantùm habeatur ratio impetus, facilè pote&longs;t determinari in qua proportione cylindrus faciliùs moueatur motu recto, quàm motu circulari; po&longs;ito &longs;cilicet centro motus in altera extremitate, cui applica­tur potentia; quippe impetus propagatus in motu circulari e&longs;t &longs;umma terminorum; propagatus verò in motu recto e&longs;t vltimus terminorum, v.g. &longs;int &longs;ex puncta &longs;ubiecti; in quolibet producatur impetus vt vnum; haud dubiè erit motus rectus; vt verò &longs;it motus circularis in primo puncto; producatur vt 1. in &longs;ecundo vt 2. in tertio, vt 3. atque ita dein­ceps; &longs;umma erit 21. cum tamen in motu recto e&longs;&longs;ent tantùm 6. igitur vt &longs;e habent 21. ad 6. ita &longs;e habet facilitas motus recti ad facilitatem motus circularis.

Dixi, &longs;i tantùm habeatur ratio impetus; quia &longs;i addatur ratio graui­tationis, &longs;eu momenti; haud dubiè maior erit adhuc difficultas, de quo infrà in Schol.

Corollarium 6.

Hinc quò longior e&longs;t cylindrus, v. g. cre&longs;cit proportio maioris illius facilitatis, vt patet inductione; nam &longs;i &longs;int tantùm 2. puncta, proportio 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. &longs;i 7. 28. ad 7; &longs;i 8. 36. ad 8; &longs;i 9. 45. ad 9; atque ita deinceps; ex quibus primò vides cre&longs;cere &longs;emper proportionem. Secundò inter duplam, & triplam rationem, &longs;cilicet 6. ad 3. & 15. ad 5. intercedere 2 1/2; inter triplam & quadruplam intercedere 3. 1/2; inter quadruplam & quintuplam inter­cedere 4 1/2; atque ita deinceps.

Corollarium 7.

Colligo denique po&longs;&longs;e in motu recto cum maiore ni&longs;u produci inten­&longs;iorem impetum in data ratione; &longs;it enim cylindrus AB, qui moueatur circa centrum A, percurrátque B, arcum BD; qui accipiatur vt recta, quæ à minimis arcubus &longs;en&longs;u di&longs;tingui non pote&longs;t; haud dubiè &longs;i eo tempore, vel æquali, quo AB tran&longs;it in AD; eadem AB, vel æqualis motu recto tran&longs;eat in FD, Dico impetum huius motus e&longs;&longs;e duplò in­ten&longs;iorem impetu illius; quia impetus &longs;unt vt motus; motus verò vt &longs;patia, quæ percurruntur æqualibus temporibus; &longs;ed &longs;patium rectanguli AD, e&longs;t duplum trianguli ADB; igitur & motus; igitur & impetus; &longs;i verò AB tran&longs;eat in EL, ita vt AF, &longs;it dupla AE; impetus erunt æquales; quia rectangulum AC, e&longs;t æquale triangulo ABD.

Dixi arcum BD, accipi vt lineam rectam; Si enim accipiatur vt ar­cus; haud dubiè motus cylindri AB, dum transfertur in FD, e&longs;t ad mo­tum eiu&longs;dem AB, dum transfertur in AD, vt rectangulum AD, ad &longs;e­ctorem, cuius arcus &longs;it æqualis rectæ BD, & radius ip&longs;i AB.

Scholium.

Ob&longs;eruabis primò, id quod &longs;uprà dictum e&longs;t ita e&longs;&longs;e intelligendum, vt momentum grauitationis nullo modo con&longs;ideretur, & prædictus cylindrus cen&longs;eatur potiùs moueri in plano horizontali, à quo &longs;u&longs;tinea­tur, quàm in circulo verticali, in quo libera &longs;it eius libratio, &longs;eu gra­uitatio.

Secundò, non po&longs;&longs;e &longs;u&longs;tineri cylindrum horizonti parallelum, ni&longs;i aliqua eius portio &longs;eu manu, &longs;eu forcipe, vel alio quouis modo accipia­tur, v.g. &longs;it cylindrus AG horizonti parallelus; vt in hoc &longs;itu reti­neatur, debet aliqua eius portio putà AB, manu teneri, alioqui ne à po­tentiâ quidem infinita &longs;u&longs;tineri po&longs;&longs;et.

Tertiò, &longs;i &longs;upponatur fulcitus in B; vt retineatur in æquilibrio, debet addi momentum in A; &longs;eu debet retineri ab ip&longs;a potentiâ applicata in A.

Quartò, pondus in G &longs;e habet ad idem pondus in A, &longs;tatuto centro in B, vt &longs;egmentum GB, ad BA, id e&longs;t, vt 5. ad 1.

Quintò, &longs;i proprio pondere frangeretur BG, haud dubiè in B frange­retur; e&longs;t autem momentum ponderis BG, vt &longs;ubduplum eiu&longs;dem BG po&longs;itum in G, vt demon&longs;trat Galileus prop.1.de re&longs;i&longs;tentia corp.&longs;it enim BG, duarum librarum, &longs;itque BG, diui&longs;a bifariam in H; haud dubiè pondus in H, facit momentum &longs;ubduplum eiu&longs;dem in G, vt patet; &longs;unt enim vt di&longs;tantiæ; igitur cum &longs;egmentum HG tantùm addat momenti &longs;upra H, quantùm detrahit HB; certè momentum totius ponderis BG, e&longs;t tantùm &longs;ubduplum eiu&longs;dem po&longs;iti in G; itaque &longs;it BG, 10. librarum, æquiualet 5. libris &longs;tatutis in G, & AB, vni libræ po&longs;itæ in A; &longs;ed hæc libra in A, habet tantùm &longs;ubquintuplum momentum eiu&longs;dem in G, igi­tur 5. libræ in A, æquiualent vni in G; igitur vt &longs;tatuatur æquilibrium, debent e&longs;&longs;e 24. libræ in A, &longs;eu vires æquiualentes; quibus adde pondus ab&longs;olutum 12. librarum; erunt 36. igitur re&longs;i&longs;tentia ad motum circula­rem verticalem ex triplici capite oritur. Primò ex ip&longs;o pondere ab&longs;olutè &longs;umpto, quæ communis e&longs;t motui propagationis. Secundò, ex momento eiu&longs;dem ponderis; Tertiò, ex tali genere propagationis, de quo &longs;uprà; quæ omnia &longs;unt apprimè tenenda, ne quis error &longs;ubrepat.

Theorema 99.

Cum applicatur potentia circumferentiæ motus circularis; ita propagatur impetus, vt plures partes ver&longs;us centrum motus producantur in pondere, quod attollitur; &longs;it enim idem cylindrus CA; &longs;itque applicata potentia in A, dico ver&longs;us C, plures partes produci in pondere, Probatur, quia attol­litur pondus in C, quod moueri non pote&longs;tin A, operâ vectis AC, vt con­&longs;tat ex certa hypothe&longs;i; igitur plures partes impetus producuntur per rationem 6. & 7. Th.77,

Scholium.

Scio quidem hoc ip&longs;um à nemine hactenus, quod &longs;ciam, explicatum e&longs;&longs;e; atque fore vt à multis tanquam nouum, & in&longs;olens minùs fortè probetur: quamquam illa hypothe&longs;is hoc ip&longs;um euincit, vulgaris certè, & nemini qua&longs;i non nota; qua nempè dicimus in omnibus partibus mo­bilis, quod actu mouetur, impetum produci; & &longs;i quando accidat corpo­ris ingentem molem ab applicata potentia non po&longs;&longs;e moueri, illud e&longs;&longs;e tantùm, quòd non po&longs;&longs;int produci tot partes impetus, quot &longs;unt nece&longs;&longs;a­riæ, vt omnibus partibus &longs;ubjecti di&longs;tribuantur; igitur ex hac hypothe­&longs;i, quæ ex manife&longs;tis ducitur experimentis, nece&longs;&longs;ariò dicendum e&longs;t plu­res partes impetus versùs centrum vectis produci in pondere, quod at­tollitur, cuius propagationis proportionem infrà demon&longs;trabimus.

Theorema 100.

Impetus, qui producitur ver&longs;us centrum vectis in pondere, licèt cre&longs;cat nu­mero, decre&longs;cit tamen in perfectione. Probatur per Th.81. ex motu imper­ctiore, cui re&longs;pondet impetus imperfectior per Ax. 17.num.4. non ratio­ne numeri, qui maior e&longs;t per Th.99. igitur ratione entitatis, &longs;eu perfe­ctionis entitatiuæ.

Theorema 101.

Tota collectio impetus, quæ in pondere ex dato puncto vectis producitur, e&longs;t ad aliam collectionem alterius puncti in perfectione, vt distantia illius puncti à centro, ad di&longs;tantiam huius: probatur, quia perfectio vnius collectionis e&longs;t ad perfectionem alterius, vt motus ad motum; motus verò &longs;unt vt &longs;patia, &longs;patia vt arcus, arcus vt &longs;emediametri, hæ demum, vt di&longs;tantiæ.

Theorema 102.

Impetus in ip&longs;o vecte &longs;ine pondere addito ita propagatur, vt &longs;it imperfectior ver&longs;us centrum vectis; probatur, quia pondus ver&longs;us centrum mouetur minore motu, vt con&longs;tat; igitur ab imperfectiore impetu; &longs;ed non e&longs;t imperfectior tantùm ratione numeri, id e&longs;t, pauciorum partium impe­tus; quia &longs;i hoc e&longs;&longs;et, &longs;it vectis AC, motus B, e&longs;t &longs;ubduplus motus A; igitur &longs;i e&longs;t impetus eiu&longs;dem perfectionis entitatiuæ, vt &longs;ic loquar; ita &longs;e habet numerus partium impetus in B, ad numerum partium in A, vt motus B, ad motum A; & hic vt arcus BD, ad arcum AE; & hic vt BC, ad AC; igitur e&longs;t &longs;ubduplus; igitur æqualis omninò producitur impetus ab eadem potentia in vecte AC, &longs;iue applicetur centro C, &longs;iue circumferentiæ A; igitur æquè facilè; quod e&longs;t contra experientiam; probatur &longs;ecundò, quia &longs;i hoc e&longs;&longs;et, pondus idem tàm facilè attolleretur in A, quàm in B; quia idem impetus produceretur, quod e&longs;t contra ex­perientiam.

Theorema 103.

Ex hoc facilè intelligitur, cur impetus propagetur faciliùs à circumferen­tia ad centrum, quàm à centro ad circumferentiam, & cur longior vectis ab eadem potentia moueri po&longs;&longs;it primo modo, non &longs;ecundo, quod clarum est.

Theorema 104.

Decre&longs;cit impetus ver&longs;us centrum iuxta rationem distantiarum; probatur quia decre&longs;cit iuxta rationem motuum; & hæc iuxta rationem di&longs;tan­tiarum.

Theorema 105.

Non decre&longs;cit numerus partium impetus à circumferentia ad centrum; probatur, quia cum à circumferentia ad centrum ita propagetur impe­tus, vt vnicum tantùm punctum producatur in ip&longs;a extremitate mobilis; certè non pote&longs;t minùs impetus produci ver&longs;us centrum ratione nume­ri; igitur non decre&longs;cit numerus; hinc producitur nece&longs;&longs;ariò imperfe­ctior ver&longs;us centrum.

Theorema 106.

Non producuntur plures partes impetus in vecte ver&longs;us centrum, id est, non &longs;unt plures in punclo vectis propiùs ad centrum accedente, quàm in co; quod longiùs distat: Probatur primò, quia fru&longs;trà e&longs;&longs;ent plures. Secundò, cur potiùs in vna proportione, quàm in alia?

Theorema 107.

Ex his constat produci impetum æqualem numero in omnibus punctis vectis a circumferentia ad centrum, cum &longs;cilicet applicatur potentia circumferentiæ; probatur, quia non producitur numerus minor per Th.105. neque maior per Th. 106. igitur æqualis; adde quod res explicari non pote&longs;t per ma­iorem, neque per minorem; ita vt &longs;cilicet pondera, quæ à data potentia leuantur, &longs;int vt di&longs;tantiæ, de quo &longs;uprà.

Scholium.

Ob&longs;eruabis, quod aliquando in mentem venerat; &longs;cilicet, ver&longs;us cen­trum produci maiorem numerum in ratione di&longs;tantiarum permutando; & imperfectiorem in ratione duplicata earumdem di&longs;tantiarum, etiam permutando, v. g. &longs;it idem vectis AC &longs;ectus bifariam in B; in puncto B producitur numerus duplus producti in A; at verò perfectio impetus in B e&longs;t ad perfectionem impetus in A, vt quadratum BC ad quadra­tum AC; vel in ratione &longs;ubquadrupla, licèt tota collectio impetus B &longs;it tantùm &longs;ubdupla perfectione collectionis impetus A; &longs;ed hoc profe­ctò dici non pote&longs;t; nam &longs;int in A 4. partes impetus; igitur in B erunt 8. applicetur autem pondus in B. Primò producentur in co partes 8. impetus perfectionis &longs;ubquadruplæ; &longs;i comparentur cum partibus A, tum producentur 16. quæ æquiualent 4 A; igitur 24. at verò in A pro­ducentur primò 4. tum deinde 2. quæ æquiualent 8. productis in B; igitur 6. igitur pondus, quod leuari pote&longs;t in B, e&longs;t ad pondus, quod leuari pote&longs;t in A, vt 24. ad 6.id e&longs;t, in ratione quadrupla quod omninò fal&longs;um e&longs;t.

Theorema 108.

Iam facilè explicatur ex dictis, quomodo, & cuius rationis pondera attol­lantur ex diuer&longs;is punctis vectis; &longs;it enim idem vectis AC, & producan­tur.v.g. in &longs;ingulis punctis vectis &longs;ingula puncta impetus, &longs;ed diuer&longs;æ perfectionis; haud dubiè plures partes impetus imperfecti po&longs;&longs;unt face­re impetum æqualem in perfectione alteri, qui con&longs;tat paucioribus, &longs;ed perfectioribus; igitur cum impetus B &longs;it imperfectior duplò quàm im­petus in A, duplò plures partes impetus producentur in B, quàm in A, er­go duplò maius pondus moucbitur; atque ita deinceps; eum enim ap­ponitur pondus in B, producuntur in eo partes impetus omnes eiu&longs;dem perfectionis; quæ &longs;cilicet re&longs;pondet B, id e&longs;t, quæ e&longs;t &longs;ubdupla perfectio­nis impetus A; igitur plures partes producuntur, quàm &longs;i e&longs;&longs;ent perfe­ctionis A; &longs;ed pauciores quàm &longs;i e&longs;&longs;ent perfectionis O, quæ minor e&longs;t; quippe eadem potentia, &longs;eu cau&longs;a, quæ agit quantum pote&longs;t (quod &longs;up­pono modò) producit æqualem effectum in perfectione, per Ax. 13. n. 4. &longs;ed æqualis perfectio pote&longs;t con&longs;tare pluribus, vel paucioribus parti­bus perfectionis, nam 4. pattes perfectionis vt 4. faciunt æqualem effe­ctum alteri qui con&longs;tat 8. partibus perfectionis vt 2. quod certum e&longs;t; &longs;ed de his plura aliàs.

Theorema 109.

Perfectio decre&longs;cit ver&longs;us centrum iuxta diuer&longs;am rationem longitudinum vectis, &longs;eu distantiarum. v.g.&longs;it idem vectis AC, ita decre&longs;cit ab A ver&longs;us centrum C; vt impetus puncti B &longs;it &longs;ubduplus in perfectione, puncti R fubtriplus: iam verò &longs;it vectis &longs;ubduplus prioris BC, &longs;ectus bifariam in Z; &longs;i impetus productus in B, qu&ecedil; e&longs;t extremitas minoris vectis B &longs;it æqua­lis perfectionis cum impetu producto in A (& reuera &longs;unt æquales) &longs;i æquali æmpore percurrant arcus æquales, &longs;cilicet AV, & BD) certè im-pertus productus in Z e&longs;t æqualis producto in B, cum B pertinet ad ma­iorem vectem; quia vt AC totus maior vectis e&longs;t ad BC ita BC ad ZC: igitur decre&longs;cit perfectio versùs centrum iuxta rationem longi­tudinum.

Theorema 110.

Minima potentia est illa, quæ in extremitate vectis, quæ procul recedit à centro, vnam tantùm partem, vel vnum punctum impetus producit; nihil enim minùs produci pote&longs;t, po&longs;ito quod potentia applicata ad talem gra­dum perfectionis &longs;it determinata, id e&longs;t ad producendum impetum talis perfectionis in ea parte &longs;ubjecti, cui applicatur immediatè, vt &longs;uprà di­ctum e&longs;t.

Theorema 111.

Si &longs;int tantum duo puncta vel duæ partes vectis, illa potentia ad illum mo­uendum &longs;ufficiens motu circulari est ad aliam &longs;ufficientem ad illum mouen­dum motu recto, vt 1/2 ad 2. &longs;i &longs;int tria puncta vt 2. ad 3. &longs;i 4. vt 2. 1/2 ad 4. &longs;i 5. vt 3. ad 5. &longs;i 6. vt 3. 1/2 ad 6. atque ita deinceps iuxta hanc propor­tionem in quo non e&longs;t difficultas, cum hoc totum &longs;equatur ex Th. 109.

Scholium.

Ob&longs;erua tamen quacumque data potentia po&longs;&longs;e dari minorem; quia quocumque dato motu, etiam recto, pote&longs;t dari tardior; igitur quocum­que impetu imperfectior; igitur quando appellaui potentiam minimam; intellige illam quæ comparatur cum vnico puncto impetus talis perfe­ctionis; hæc enim reuera minima e&longs;t illarum omnium, quæ po&longs;&longs;unt pro­ducere impetum talis perfectionis, &longs;i verò comparetur cum impetu im­perfectiore, haud dubiè minima non e&longs;t.

Ob&longs;erua præterea &longs;uppo&longs;itum e&longs;&longs;e hactenus in extremitate vectis &longs;iue maioris, &longs;iue minoris, produci impetum ciu&longs;dem perfectionis, ciu&longs;que vnicum punctum, &longs;eu partem, vnde potentia quæ applicatur maiori vecti conuenit quidem cum ea, quæ applicatur minori in eo, quòd vtraque in extremitate &longs;ui vectis producat vnum punctum impetus eiu&longs;dem perfe­ctionis; differt tamen in eo, quòd illa, quæ applicatur maiori vecti, &longs;it maior iuxta rationes prædictas in Theoremate. v. g. illa, quæ applicatur vecti. 2. punctorum e&longs;t ad eam, quæ applicatur vecti trium punctorum, &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. &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 enim vectis 2. punctorum AB, in puncto A, quod e&longs;t extremitas, produ­catur punctum impetus datæ perfectionis, in B producetur aliud, cuius perfectio e&longs;t &longs;ubdupla prioris per Th. 109. igitur caracter, &longs;eu momen­tum totius impetus e&longs;t 1. 1/2. &longs;it porrò vectis 4. punctorum CDEF, in C, quod e&longs;t extremitas; producatur vnum punctum impetus ciu&longs;dem perfectionis cum eo, quod productum e&longs;t in A; certè in D producetur aliud cuius perfectio erit ad priorem vt 3.ad 4. per idem Th. &longs;ic autem notetur 1/4, in E 2/4, in F 3/4, in C vero 4/4; perfectiones enim &longs;unt vt lon-gitudines; quæ &longs;i colligantur, habebis characterem totius impetus, 2 1/2: igitur totus impetus productus in minore vecte, qui con&longs;tat 2. punctis, e&longs;t ad impetum, qui producitur in maiore con&longs;tante 4.punctis, vt 1. 1/2 ad 2. 1/2; igitur vectis maior maiorem potentiam ad mouendum ip&longs;um ve­ctem requirit; non certè in de&longs;cen&longs;u; quippe &longs;uo pondere de&longs;cendit, &longs;ed in plano horizontali; ni&longs;i enim potentia po&longs;&longs;it mouere vectem; haud dubiè nullum pondus vecte mouebit.

At verò &longs;i potentia &longs;it tantùm dupla minimæ, quæ datum vectem mo­uere po&longs;&longs;it; haud dubiè dato illo vecte datum ferè quodcumque pondus mouere poterit; cum ip&longs;e vectis con&longs;tet ferè infinitis punctis in longi­tudine, vt patet ex dictis, & con&longs;ideranti patebit.

Ob&longs;eruabis demum in mechanicis nullam ferè haberi rationem pon­deris ip&longs;ius vectis; parum enim pro nihilo computatur: Ex his tamen erui po&longs;&longs;unt veri&longs;&longs;imæ rationes Phy&longs;icæ proportionum vectis AH; &longs;ia­que A extremitas, H centrum; &longs;itque BH 1/2. CH 1/4, DH 1/2, EH (1/16), FH (1/32), GH (1/64) pondus I applicetur in A, & moueatur; certè in B moue­bitur pondus K duplum I; quia, cum impetus productus in B, &longs;it &longs;ubdu­plus in perfectione illius, qui producitur in A; vt æqualis producatur in B, & in A, debent produci in B duplò plures partes impetus; igitur du­plò maius pondus mouebit; at verò in C mouebitur pondus L quadru­plum I, in D octuplum, atque ita deinceps; donec tandem in G mot ea­tur pondus, quod &longs;it ad I vt 64. ad 1. & cum adhuc po&longs;&longs;int accipi inter GH, partes aliquotæ minores, & minores ferè in infinitum, non mirum e&longs;t &longs;i pondus maius po&longs;&longs;it adhuc moueri.

Ob&longs;eruabis etiam in omni vecte ab&longs;trahendo ab eius pondere, & ap­plicata cadem potentia, hoc e&longs;&longs;e commune; vt po&longs;&longs;it quodcumque pon­dus attolli, licèt difficiliùs in minore; quia hic non pote&longs;t in tam mul­tas partes aliquotas &longs;en&longs;ibiliter diuidi, in medio tamen vecte duplum femper pondus mouetur; &longs;iue ip&longs;e vectis &longs;it maior, &longs;iue minor.

Ob&longs;eruabis deinde, &longs;i centrum vectis non &longs;it in altera extremitate, &longs;ed. v.g. in C; haud dubiè producitur in H, & in B impetus æqualis; quia æqualiter di&longs;tat vtrumque punctum à centro C; igitur æquale pondus mouebitur in B, & in H; propagatur tamen nouo modo à C ver&longs;us H, de quo iam &longs;uprà dictum e&longs;t.

Ob&longs;eruabis denique triplicem propagationem impetus e&longs;&longs;e legiti­mam. Prima e&longs;t in motu recto, cum propagatur per partes æquales, tùm in perfectione, tùm in numero in &longs;ingulis partibus &longs;ubjecti per gradus, &longs;cilicet heterogencos. Secunda e&longs;t in motu circulari, applicata &longs;cilicet potentia centro; cum propagatur per partes æquales in perfectione, & inæquales in numero. Tertia e&longs;t in vecte, cum propagatur per partes æquales in numero, & inæquales in perfectione.

Theorema 112.

Impetus debet determinari ad aliquam lineam motus; probatur, quia non pote&longs;t e&longs;&longs;e impetus, ni&longs;i exigat motum per Th.14. nec exigere mo-tum, ni&longs;i per aliquam lineam, vt patet; &longs;ed hoc e&longs;t impetum e&longs;&longs;e de­terminatum ad aliquam lineam motus; præterea &longs;i non e&longs;t determina­tus ad aliquam lineam; igitur indeterminatus, & indifferens per Ax.1. &longs;ed indifferens manere non pote&longs;t; cur enim potius haberet motum per vnam lineam, quàm per aliam? igitur debet determinari.

Theorema 113.

Impetus ad plures lineas &longs;eor&longs;im indifferens e&longs;t: Probatur, quia idem im­petus pilæ in aliam impactæ producit in ea impetum, qui pro diuer&longs;o contactu ad diuer&longs;am lineam determinari pote&longs;t; præterea corpus graue in diuer&longs;is planis inclinatis de&longs;cendit; igitur per diuer&longs;as lineas; deinde pila reflectitur propter impetum priorem, qui tantùm mutat lineam, vt dicemus infrà; adde quod funependuli vibrati impetus &longs;ine reflexione mutat lineam motus; igitur idem impetus ad plures lineas &longs;eor&longs;im e&longs;t indifferens.

Theorema 114.

Hinc idem impetus ad plures lineas potest determinari &longs;eor&longs;im; quia ad eas pote&longs;t determinari, ad quas e&longs;t indifferens, vt patet; &longs;ed ad multas e&longs;t indifferens per Theorema 113. igitur ad multas pote&longs;t determi­nari.

Scholium.

Ob&longs;eruabis primò determinationem hanc nihil e&longs;&longs;e aliud, ni&longs;i ip&longs;um impetum cum tali linea comparatum, &longs;eu coniunctum; vnam verò li­neam differre ab alia ratione terminorum v. g. illa quæ tendit ver&longs;us ortum differt ab ea, quæ tendit ver&longs;us au&longs;trum, vel occa&longs;um, &longs;cilicet ratione terminorum, &longs;unt enim duo termini, nempè à quo, & ad quem; 4. autem modis differunt termini lineæ, vel enim neuter communis e&longs;t vt AB. DC, vel terminus à quo vtrique lineæ communis e&longs;t, vt BA. BE, vel terminus ad quem vt AB, EB; vel denique vici&longs;&longs;im commu­tantur termini, vt BE, EB, & hæc terminorum coniugatio facit oppo­&longs;itionem maximam, id e&longs;t diametralem.

Secundò ob&longs;eruabis aliquando videri e&longs;&longs;e vtrumque terminum com­munem licèt differant lineæ; &longs;it linea recta BE, habet communes ter­minos cum curua BFE, licèt omninò differat ab illa; at profectò licèt BE videatur e&longs;&longs;e vnica &longs;implex linea duobus terminis clau&longs;a; con&longs;tat ramen ex pluribus aliis continuata, rectáque &longs;erie iunctis; vnde, vt linea dicatur eadem e&longs;&longs;e cum alia, debet vna cum aliâ conuenire; ita vt alteri &longs;uperpo&longs;ita nec excedat, nec deficiat.

Tertiò linea motus non differt ab ip&longs;o motu continuo tractu, &longs;eu fluxu qua&longs;i labenti: Porrò vnus motus differt ab alio, vel ratione velo­citatis, vel ratione terminorum; &longs;ed hæc parum difficultatis habent.

Theorema 115.

Impetus aliquis ad vnam tantùm lineam pote&longs;t e&longs;&longs;e determinatus; v. g. impetus naturalis innatus, de quo in Th. 17. nam de acqui&longs;ito certum e&longs;t adplures determinari po&longs;&longs;e, vt videbimus cum de motureflexo; probatur quia motus deor&longs;um e&longs;t finis huius impetus; quia ideo corpus graue produ­cit in &longs;e impetum (&longs;i tamen producit) vt tendat deor&longs;um, vt certum e&longs;t; tàm enim omne graue non impeditum tendit deor&longs;um, quàm omnis ignis e&longs;t calidus; igitur &longs;i e&longs;t proprietas omnis ignis e&longs;&longs;e calidum, quia omni competit; ita omni graui competit tendere infrà leuius, modò non impediatur; igitur e&longs;t eius proprietas; igitur ille impetus e&longs;t de­terminatus ad lineam quæ tendit deor&longs;um; &longs;ed de hoc impetu naturali innato fusè agemus infrà in &longs;ecundò libro; nunc &longs;ufficiat dixi&longs;&longs;e po&longs;&longs;e dari aliquem impetum ita determinatum ad certam lineam, vt ad aliam determinari non po&longs;&longs;it naturaliter, nulla e&longs;t enim repugnantia.

Theorema 116.

Impetus determinatur aliquando ad lineam motus à potentia motrice; pro­batur, quia primus impetus ab ip&longs;a potentia productus &longs;ine impedimen­to ab alio determinari non pote&longs;t; potentia porrò motrix vel e&longs;t gra­uium, vel leuium, vel animantium, vel proiectorum, vel compre&longs;&longs;o­rum, &c.

Theorema 117.

Potentia verò motrix determinatur vel à &longs;uo fine intrin&longs;eco, vel potius ab ip&longs;a &longs;ua natura; &longs;ic grauitas &longs;eu potentia motrix grauium determinata e&longs;t ad motum deor&longs;um perpendicularem, dum in medio libero corpus graue mouetur; vel à plano inclinato; pro cuius diuer&longs;a inclinatione diuer&longs;a e&longs;t linea motus deor&longs;um; vel ab ip&longs;a via, &longs;eu exitu patefacto; &longs;ic potentia motrix compre&longs;&longs;orum &longs;uas vires exerit, & mobile ip&longs;um agit, quâ patet viâ, &longs;ur&longs;um, deor&longs;um &c. vel ab appetitu &longs;eu libero, &longs;eu &longs;en&longs;itiuo; &longs;ic potentia progre&longs;&longs;iua animantium cò corpus agit, quò iu­bet appetitus, vel ab aliqua affectione intrin&longs;eca intrin&longs;ecùs vel extrin­&longs;ecùs adueniente; &longs;ic dilatatur pupilla, vel contrahitur pro diuer&longs;a lu­minis appul&longs;i vi, vel obiecti di&longs;tantia: Huc reuoca motus illos natura­les, qui animalibus competunt v. g. tu&longs;&longs;is, &longs;ingultus, &longs;ternuationis, &c. de quibus fusè &longs;uo loco.

Theorema 118.

Impetus determinatur aliquando ad lineam ab alio impetu producente; &longs;ic impetus corporis proiecti determinatur ab impetu vel organi vel manus proiicientis; quia nihil e&longs;t aliud à quo determinari po&longs;&longs;it, vt patet; adde figuram organi, di&longs;po&longs;itionem &longs;eu &longs;itum mobilis, quod ma­nu tenetur; impedimenti etiam habetur ratio v. g. corpus oblongum proiici pote&longs;t, vel motu recto ad in&longs;tar teli, vel motu mixto ex recto & circulari; cum &longs;cilicet diuer&longs;imodè vibratur: &longs;i enim altera extremi­tas adhuc hæreat in manu, dum altera mouetur, vt cum quis baculo ferit; tunc certè e&longs;t aliquòd impedimenti genus, ex quo oritur talis li­nea motus; illud autem impedimentum emergit ex diuer&longs;a applicatione diuer&longs;aque brachij vibratione, quæ omnia &longs;unt &longs;atis clara.

Theorema 119.

Impetus determinatus ad vnam lineam pote&longs;t ad aliam in &longs;uo fluxu deter­minatu; vt patet in corpore reflexo; nec enim dici pote&longs;t totum prio­rem impetum in ip&longs;o reflexionis puncto de&longs;trui, vt demon&longs;trabimus aliàs. Probatur etiam ex impetu proiectorum, quæ mutant lineam mo­tus manente adhuc priore impetu &longs;altem ex parte.

Theorema 120.

Corpus proiectum in aliud ita illud impellit, vt determinet lineam motus ratione puncti contactus; Sit enim, ne mnltiplicemus figuras, globus, cuius linea directionis &longs;it DC, punctum contactus C, ita globus A im­pellet globum B, vt linea motus, ad quam determinatur, &longs;it CB, id e&longs;t ducta à puncto contactus ad centrum globi impul&longs;i; &longs;it etiam globus P impactus in globum A punctum contactus &longs;it D, linea motus, ad quam determinatur, e&longs;t DA, quæ &longs;cilicet à puncto contactus ducitur per centrum grauitatis corporis impul&longs;i: experientia huius rei certa e&longs;t, nec ignorant qui in ludo minoris tudiculæ ver&longs;ati &longs;unt; ratio au­tem inde tantùm duci pote&longs;t, quod &longs;cilicet ab ip&longs;o puncto contactus ita diffunditur impetus, vt hinc inde æqualiter in vtroque hemi&longs;phærio diffundatur; coniungitur autem vtrumque hemi&longs;phærium circulo A, vel B, in priore figura, e&longs;tque vtriu&longs;que communis &longs;ectio; cum autem vtrimque &longs;it æqualis impetus, nulla e&longs;t ratio, cur linea directionis in­clinet potiùs in vnum hemi&longs;phærium, quàm in aliud: præterea cum motus orbis globi determinetur à motu centri; cum &longs;cilicet globus in globum impingitur; haud dubiè non pote&longs;t e&longs;&longs;e alius motus centri, ni&longs;i qui determinatur à puncto contactus, à quo vnica tantùm linea ad cen­trum duci pote&longs;t, vt con&longs;tat; & hæc ratio veri&longs;&longs;ima e&longs;t, & totam rem ip&longs;am cuincit.

Theorema 121.

Hinc licèt diuer&longs;æ &longs;int linea motus globi impellentis, &longs;i tamen &longs;it idem pun­ctum contactus ad eamdem lineam globus impul&longs;us determinabitur, v. g. li­cet globus P. ciu&longs;dem figuræ tangat globum A in D per lineam PD &longs;iue per lineam HD &longs;iue per quamlibet aliam, globus A mouebitur &longs;emper per lineam directionis DA propter rationem propo&longs;itam, quod etiam mille experimentis conuincitur.

Theorema 122.

Determinatur impetus corporis proiecti impacti in corpus reflectens ad no­uam lineam; patet experientiâ in pilâ reflexâ; reflexionis autem ratio­nem afferemus in lib. de motu reflexo.

Theorema 123.

Non determinatur tantùm ratione puncti contactus. Probatur, quia cum eodem puncto contactus pote&longs;t e&longs;&longs;e determinatio ad diuer&longs;am lineam, vt manife&longs;tum e&longs;t; &longs;it enim reflexio per angulum æqualem incidentiæ, &longs;ed diuer&longs;i anguli po&longs;&longs;unt in idem punctum coire, vt patet.

Theorema 124.

Non determinatur noua linea in motu reflexo â priore tantùm linea incidentiæ; probatur, quia pote&longs;t e&longs;&longs;e eadem linea incidentiæ cum di­uer&longs;is lineis motus reflexi, vt patet.

Theorema 125.

Non determinatur noua linea motus reflexi ratione tantùm plani reflecten­tis: Probatur, quia cum eodem plano reflectente diuer&longs;æ lineæ motus reflexi e&longs;&longs;e po&longs;&longs;unt, vt con&longs;tat.

Theorema 126.

Determinatur noua linea motus reflexi ratione lineæ prioris incidentiæ com­paratæ cum plano reflectente, e&longs;t enim angulus reflexionis æqualis angu­lo incidentiæ, cuius effectus rationem aliàs afferemus, cum de motu reflexo; & verò multa hîc cur&longs;im tantùm per&longs;tringimus, quæ in libro de motu reflexo accurati&longs;&longs;imè demon&longs;trabimus; Hìc tantùm dixi&longs;&longs;e &longs;uf­ficiat determinari mobile in reflexionis puncto ad nouam lineam motus, quod nemo in dubium reuocare pote&longs;t, & propter quid fiat loco citato demon&longs;trabimus.

Theorema 127.

Quando globus in globum æqualem ita impingitur, vt linea directionis per centra vtriu&longs;que ducatur, determinatio noua e&longs;t æqualis priori; Patet ex­perientia in pilis illis eburneis, quas de&longs;iderat ludus minoris tudiculæ; nec e&longs;t vlla ratio, cur determinatio &longs;it maior potiùs, quàm minor, cum vtraque pila &longs;it æqualis; &longs;i enim maior e&longs;&longs;et, vel minor; cur potiùs vno gradu, quàm duobus? quàm tribus? Præterca, cum re&longs;i&longs;tens, vel im­pediens e&longs;t æquale agenti; cere &longs;icut agens refundit in pa&longs;&longs;um totum id, quod habet, id e&longs;t æqualem impetum in inten&longs;ione, & æquè velo­cem motum per Th. 60. Ita re&longs;i&longs;tens, vel impediens refundit æquale impedimentum, quod tantùm &longs;umi pote&longs;t ex æqualitate mobilium; &longs;ed ex æquali impedimento duci tantùm pote&longs;t æqualis determinatio priori; denique pote&longs;t dari determinatio noua æqualis priori, vt con&longs;tat, &longs;ed aliunde duci non pote&longs;t quàm ex ip&longs;a mobilium æqualitate, modò fiat contactus per lineam connectentem centra.

Theorema 128.

Hinc ratio manife&longs;ta illius mirifici effectus, &longs;cilicet quietis pilæ impactæ; quippe hæc quie&longs;cet illicò ab ictu; quia &longs;cilicet, cum noua determina­tio &longs;it æqualis priori, non e&longs;t vlla ratio, cur alterutra præualeat; nec etiam pote&longs;t e&longs;&longs;e determinatio communis, &longs;eu mixta; cur enim potius dextror&longs;um quam &longs;ini&longs;tror&longs;um? de quo infrà.

Theorema 129.

Quæ linea directionis globi impacti non connectit centra vtriu&longs;qu&etail; globi, determinatur noua linea motus tùm à priore linea incidentiæ, tùm à connectante centra, quæ &longs;cilicet per punctum contactus à centro impacli globiad centrum alterius ducitur; quippe nihil e&longs;t aliud à quo determinari. po&longs;&longs;it, vt patet; non determinatur etiam ab alterutra &longs;eor&longs;im, vt con­&longs;tat, igitur ab vtraque conjunctim; in qua verò proportione dicemus, & demon&longs;trabimus in libro de motu reflexo; &longs;unt enim mirificæ quæ­dam reflexionum proportiones, quas ibidem explicabimus.

Theorema 130.

Hinc globus &longs;ic impactus nunquam quie&longs;cit; ratio e&longs;t, quia vtraque linea determinationis cum angulum faciat, in communem lineam abit; nam ex duabus lineis motus minimè oppo&longs;itis ex diametro, fit alia tertia me­dia pro rata; hîc etiam latent my&longs;teria, de quibus loco citato.

Theorema 131.

Si globus minor in maiorem impingatur per quamcumque lineam directio­nis, determinatur ad nouam lineam motus reflexi; experientia clara e&longs;t; ra­tio e&longs;t, quia maior globus maius e&longs;t impedimentum, hinc nunquam quie&longs;cit minor globus impactus.

Theorema 132.

Si globus major in minorem impingatur per lineam directionis, quæ conne­ctat centra, &longs;eruat eamdem lineam; patet etiam experientiâ, cuius ratio e&longs;t minor re&longs;i&longs;tentia minoris globi; &longs;i verò &longs;it alia linea directionis, omni­nò reflectitur &longs;uo modo; id e&longs;t mutat lineam; &longs;ed de his omnibus fusè aliàs; hîc tantùm &longs;ufficiat indica&longs;&longs;e; (&longs;uppo&longs;ita linea directionis cen­trali &longs;eu connectente centra, &longs;ic enim deinceps eam appellabimus, in quo ca&longs;u duplex determinatio tertiam mediam conflare non pote&longs;t) in­dica&longs;&longs;e inquam &longs;ufficiat nouam determinationem, vel e&longs;&longs;e æqualem prio­ri, vel maiorem, vel minorem; &longs;i æqualis e&longs;t, globus impactus &longs;i&longs;tit; &longs;i maior, reflectitur; &longs;i minor, eamdem lineam, &longs;ed lentiùs pro rata pro­&longs;equitur.

Theorema 133.

Si &longs;it duplex impetus æqualis ad diuer&longs;as lineas determinatus in eodem mo­bili, &longs;ique illæ &longs;int ex diametro oppo&longs;itæ &longs;i&longs;tere debet mobile; patet; &longs;it enim globus vtrimque gemino malleo percu&longs;&longs;us æquali ictu; haud dubiè &longs;i&longs;tit; cur enim potiùs in vnam partem quam in aliam? cum &longs;imul in vtramque moueri non po&longs;&longs;it.

Theorema 134.

Si verò alter impetus &longs;it inten&longs;ior, po&longs;ito eodem ca&longs;u, haud dubiè eius de­terminatio præualebit pro rata; patet etiam experientià; ratio e&longs;t, quia im­petus fortior debiliorem vincit; pugnant enim pro rata per Ax. 15. hinc &longs;i &longs;it duplò inten&longs;ior, &longs;ubduplum &longs;uæ velocitatis amittet, &longs;i triplè &longs;ubtriplum, &c. de quo aliàs.

Theorema 135.

Siduo globi projecti &longs;ibi inuicem occurrant in lineæ directionis connectente centra, reflectitur vterque æquali motu, quo antè. Probatur; &longs;unt enim globi A & B, & A feratur per lineam DE, & B per lineam ED, punctum con­tactus &longs;it C, haud dubiè globus A impactus in B amittit totum &longs;uum im­petum per Th.127. & 128. B, item impactus in A amittit totum &longs;uum per camdem rationem; globus A producit impetum in B æqualem &longs;uo per Th.60. item B producit in A æqualem per idem Th. igitur tantùm perit impetus quantùm accedit; igitur in vtroque globo remanet æqualis im­petus priori; igitur æquali motu vterque mouetur, quod erat dem. & hæc e&longs;t ratio veri&longs;&longs;ima toties probatæ experientiæ.

Theorema 136.

Hinc æquale &longs;patium conficiet regrediendo po&longs;t reflexionem, quem confeci&longs;­&longs;et motu directo, &longs;i propagatus fui&longs;&longs;et &longs;ine obice; nam æquali motu æquali tempore in eodem plano &longs;eu medio idem &longs;patium decurritur; quid verò accidat in aliis punctis contactus dicemus infrà, cum de reflexione.

Theorema 137.

Si in eodem mobili duplex impetus producatur, quorum vterque &longs;eor&longs;im ad duas lineas &longs;it determinatus quæ conjunctæ faciant angulum, determinatur vterque ad tertiam lineam mediam; &longs;it enim mobile in A. v. g. globus, cui &longs;imul imprimatur impetus determinatus ad lineam AD, in plano horizontali AF; &longs;i vterque &longs;it æqualis, ad nouam lineam determinabi­tur AE; quippe tantùm debet acquirere in horizontali AB, vel in eius parallela DE, quantum acquirit in alia horizontali AD, vel in eius pa­rallela BE; igitur debet ferri in E; igitur per diagonalem AE; clara e&longs;t omninò experientia; cuius ratio à priori hæc e&longs;t, quòd &longs;cilicet impetus po&longs;&longs;it determinari ad quamlibet lineam ab alio impetu per Th.118.119. igitur in eodem mobili pro rata quilibet alium determinat; igitur &longs;i vterque æqualis e&longs;t, vterque æqualiter; igitur debet tantum &longs;patij acqui­ri in linea vnius, quantum in linea alterius.

Si verò impetus per AC &longs;it duplus impetus per AD; accipiatur AC dupla AD, ducatur DF æqualis & parallela AC; linea motus noua erit diagonalis AF, quia vtraque determinatio concurrit ad nouam pro rata; igitur debet &longs;patium acqui&longs;itum in AC e&longs;&longs;e duplum acqui&longs;iti in AD.

Theorema 138.

Si &longs;it duplex impetus in eodem mobili ad eamdem lineam determinatus, non mutabitur linea; &longs;ed cre&longs;cet motus & &longs;patium Imprimatur impetus in A, per AB, quo dato tempore percurratur &longs;patium AB; deinde produca­tur &longs;imul alius impetus æqualis priori in eodem mobili per lineam AB; Dico quod eodem tempore percurretur tota AE, dupla &longs;cilicet AB; quia &longs;cilicet dupla cau&longs;a non impedita duplum effectum habet per Ax. 13. num.1. duplus impetus duplum motum; igitur duplum &longs;patium; &longs;i verò &longs;it triplus impetus, triplum erit &longs;patium, &c.

Theorema 139.

Si lineæ duplicis impetus, faciunt angulum acutiorem, longius erit &longs;patiumacqui&longs;itum: &longs;int duæ lineæ IK IL, mobili &longs;eilicet &longs;tatuto in I; haud dubiè noua linea erit IM; & quo angulus KIL, erit acutior (&longs;up­po&longs;itis æqualibus &longs;emper lateribus IK IL) Diagonalis IM, erit ma­ior; donec tandem IL & IK coeant in eandem lineam; tunc enim li­nea erit dupla IK per Th. &longs;uperius: quandiu verò e&longs;t aliquis angulus in I quantumuis acutus, linea motus erit minor dupla IK, ad quam tamen propiùs &longs;emper accedit; quæ omnia con&longs;tant ex elementis.

Theorema 140.

Si lineæ duplicis impetus faciunt angulum obtu&longs;um, &longs;patium acqui&longs;itum erit breuius, & eò breuius quò angulus e&longs;t obtu&longs;ior; &longs;int enim c duæ lineæ AD AB mobili &longs;tatuto in A, noua linea erit AC per Th. 137. & &longs;i accipia­tur angulus obtu&longs;ior HEF; noua linea erit EG, eo rectè breuior, quò angulus e&longs;t obtu&longs;ior, non tamen iuxta rationem angulorum; donec tandem de&longs;inat angulus, & ED EF coëant in vnam lineam; tunc enim nullum erit &longs;patium, quia &longs;i&longs;ter omninò mobile per Th.133.quæ omnia ip&longs;a luce clariora e&longs;&longs;e con&longs;tat; quippe quæ cum certis experimentis, & clari&longs;&longs;imis principiis con&longs;entiant; &longs;ed de his plura infrà.

Theorema 141.

Ex his nece&longs;&longs;aria ducitur ratio, cur impetus duplus ad diuer&longs;as lineas de­terminatus non habeat motum duplum, & con&longs;equenter &longs;patium duplum; nec enim AE e&longs;t dupla AB, vt con&longs;tat; nam &longs;i lineæ &longs;int oppo&longs;itæ ex diametro vt BA BE totus de&longs;truitur impetus, per Th.133. &longs;i verò vna in eamdem lineam coëat cum aliâ, nihil impetus de&longs;truitur, nec impedi­tur per Th.138. igitur quà proportione propiùs accedet ad oppo&longs;itas; plùs de&longs;truetur, & minus erit &longs;patium; & quâ proportione accedent propiùs ad coëuntes, minùs de&longs;truetur, & maius erit &longs;patium, vt con&longs;tat ex dictis.

Theorema 142.

Hinc impetus ad diuer&longs;as lineas determinati it a pugnant pro rata, vt mi­nùs pugnent, quorum lineæ propiùs accedunt ad coëuntes; plùs verò, quorum lineæ propiùs accedunt ad oppo&longs;itas, idque iuxta proportiones Diagonalium,quod totum &longs;equitur ex dictis.

Scholium.

Ob&longs;eruabis vt faciliùs concipias duos impetus ad duas lineas deter­minatos; finge tibi nauim à diuer&longs;is ventis impul&longs;am, &longs;eu lapidempro­jectum è naui mobili; &longs;ed de his plura in lib.4. cum de motu mixto.

Theorema 143.

Impetus &longs;emel productus, quamdiu durat motus, con&longs;eruatur. Probatur, 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 mo­tus, e&longs;t impetus.

Theorema 144.

Impetus non con&longs;eruatur à cau&longs;a primò productiua. Probatur; quia proii-ciatur mobile per Po&longs;tulatum, etiam mouetur &longs;eparatum à potentia mo­trice per hypoth. 6. igitur non con&longs;eruatur à potentia motrice per Ax. 10. igitur nec à causâ primò productiua.

Theorema 145.

Hinc ab alia causâ con&longs;eruari nece&longs;&longs;e e&longs;t impetum: Probatur, quia impe­tus non e&longs;t à &longs;e, quia de&longs;truitur aliquando per Ax. 14. igitur con&longs;eruatur ab alio per Ax.14. num. 1. non à cau&longs;a primò productiua per Th.144.igi­tur ab alia, eaque applicata per Ax. 10. quæcumque tandem illa &longs;it, ali­quando cau&longs;am primam e&longs;&longs;e demon&longs;trabimus; nunc verò &longs;ufficiat dixi&longs;­&longs;e dari aliquam cau&longs;am reuerâ applicatam, quæ ip&longs;um con&longs;eruat impe­tum; immò ex hac ip&longs;a rerum con&longs;eruatione argumentum aliquando ducemus, quo Deum ip&longs;um exi&longs;tere demon&longs;trabimus.

Theorema 146.

Si impetus con&longs;eruaretur à cau&longs;a primò productiua, nunquam de&longs;truere­tur, quamdiu e&longs;&longs;et applicata. Demon&longs;tratur, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria (nam de hac ip&longs;a loquor) igitur &longs;emper ageret, igitur &longs;emper con­&longs;eruaret, quod e&longs;t contra experientiam; nam reuerâ impetus pro­ductus deor&longs;um à corpore graui motu naturaliter accelerato de&longs;truitur, vt patet; præterea &longs;i corpus graue con&longs;eruaret impetum primò produ­ctum, non produceret nouum contra experientiam; quippe cau&longs;a ne­ce&longs;&longs;aria non plùs agit vno in&longs;tanti quàm alio, per Ax.12. adde quod im­petus de&longs;truitur ad exigentiam alterius, quidquid tandem illud &longs;it per Ax.14. num.2. & 3. &longs;ed cau&longs;a primò productiua impetus non nouit rerum exigentiam; igitur illi facere &longs;atis non pote&longs;t; ex hoc etiam capite cau­&longs;æ primæ exi&longs;tentiam&longs;uo loco demon&longs;trabimus.

Scholium.

Ob&longs;eruabis primò rem quamlibet ideo de&longs;trui, quia ce&longs;&longs;at cau&longs;a con­&longs;eruans illam con&longs;eruare; quippe quod de&longs;truitur eo in&longs;tanti dicitur de­&longs;trui, quo primò non e&longs;t, &longs;eu quo incipit primò non e&longs;&longs;e; atqui incipit primò non e&longs;&longs;e &longs;eu de&longs;init e&longs;&longs;e, cum de&longs;init con&longs;eruari.

Secundò ob&longs;eruabis præclarum naturæ in&longs;titutum, quod etiam ex ip&longs;is hypothe&longs;ibus con&longs;tat, quo fit vt qualitates quæ carent contrario à cau&longs;a primò productiua con&longs;eruentur, vt lumen; ne &longs;i ab alia con&longs;eruarentur, de&longs;truerentur vmquam; cum earum de&longs;tructionem nihil exigeret per Ax.14.n.2. & 3. at verò qualitates, quæ contrarias habent: &longs;i quæ &longs;unt, à cau&longs;a primò productiua minimè con&longs;eruantur; cum enim ideo con­trarium dicatur de&longs;truere contrarium, quia exigit eius de&longs;tructionem, id e&longs;t, ne con&longs;eruetur amplius; certè vt cau&longs;a con&longs;eruans ce&longs;&longs;et con&longs;eruare, debet no&longs;&longs;e illam exigentiam; atqui nulla cognitione pollent cau&longs;æ illæ motrices naturales, de quibus e&longs;t quæ&longs;tio.

Theorema 147.

Tamdiu con&longs;eruatur impetus, quamdiu nihil exigit eius destructionem; quia de&longs;truitur tantùm ad exigentiam alicuius, quidquid tandem illud &longs;it, de quo infrà, per Ax.14.num.2. certè tamdiu non de&longs;truitur, quamdiu nihil e&longs;t, quod exigat eius de&longs;tructionem; igitur tamdiu con&longs;eruatur per Ax. 14.num.3.

Corollarium 1.

Inde certa ducitur ratio, cur mobile etiam &longs;eparatum à manu mouea­tur; quia &longs;cilicet ip&longs;i adhuc ine&longs;t impetus, qui e&longs;t cau&longs;a motus; quippe &longs;uppo&longs;ui iam antè de hac hypothe&longs;i quod &longs;it, non tamen propter quid &longs;it; igitur hæc e&longs;t germana illius ratio & cau&longs;a.

Corollarium. 2.

Hinc etiam rationem ducemus æquè præclaram in lib.2. motus natu­raliter accelerati.

Theorema 148.

Impetus productus aliquando de&longs;truitur; Probatur, quia mobile, quod antè mouebatur, de&longs;init tandem moueri per hyp. 4. igitur de&longs;truitur impetus; alioqui &longs;i remaneret, e&longs;&longs;et cau&longs;a nece&longs;&longs;aria &longs;ine effectu contra Ax.12. ideo porrò de&longs;truitur, quia aliquid exigit eius de&longs;tructionem, quippe hæc e&longs;t vnica de&longs;tructionis ratio per Ax.14. num.2.

Theorema 149.

In lineis oppo&longs;itis impetus de&longs;truitur ab impetu &longs;uo modo; &longs;it enim globus proiectus ver&longs;us au&longs;trum; cui deinde imprimatur nouus impetus ver­&longs;us Boream; de&longs;truitur prior vt con&longs;tat, igitur ad exigentiam alicuius, &longs;ed nihil e&longs;t quod po&longs;&longs;it exigere, ni&longs;i nouus impetus, &longs;cilicet mediatè; nihil enim aliud e&longs;t applicatum, igitar nihil aliud exigit per Ax. 10. hæc porrò exigentia non e&longs;t immediata, &longs;ed mediata, vt dixi.

Theorema 150.

Impetus naturalis innatus exigit de&longs;tructionem alterius, qui ab extrin&longs;eco ad diuer&longs;am lineam corpori graui impre&longs;&longs;us e&longs;t &longs;cilicet mediatè, experientia certa e&longs;t in proiectis, quæ tandem quie&longs;cunt; igitur ad exigentiam ali­cuius, &longs;ed illud tantùm e&longs;t impetus innatus; nec enim e&longs;t &longs;ub&longs;tantia corporis; tùm quia qualitas &longs;ub&longs;tantiæ non opponitur; tùm quia nulla e&longs;&longs;et ratio, cur &longs;ub&longs;tantia de&longs;trueret potiùs vno in&longs;tanti vnum gradum, quàm duos, quàm tres; adde quod ex duobus violentis oppo&longs;itis alte­rum de&longs;truit; igitur impetus e&longs;t cau&longs;a &longs;ufficiens de&longs;tructiua impetus, igitur non e&longs;t ponenda alia, eo &longs;cilicet modo, quo diximus.

Theorema 151.

In reflexione de&longs;truitur aliquid impotus &longs;altem per accidens; patet expe­rientia, &longs;iue propter nouam determinationem, &longs;iue propter attritum, vel pre&longs;&longs;ionem partium, de quo infrà.

Theorema 152.

Hinc &longs;i excipias tantùm impetum natur alem innatum, qui per &longs;uam de­terminationem nece&longs;&longs;ariam, & quam nunquam mutat, pugnat cum omniextrin&longs;eco ad aliam lineam determinato, & cum ip&longs;o acqui&longs;ito, quando mu­tat lineam perpendicularem deor&longs;um, de quo infrà; &longs;i hunc igitur excipias, omnes aly pugnant tantùm ratione diuer&longs;æ lineæ, &longs;eu determinationis, in eodem mobili: Vnde ille idem, qui modo pugnat probè conueniet, &longs;i ad ean­dem lineam determinetur.

Scholium.

Ob&longs;eruabis primò, præclarum naturæ in&longs;titutum, quo fit, vt impe­tus perennis non &longs;it; vnde certè infinita propemodum emergerent ab­&longs;urda, & incommoda.

Secundò, faciliorem modum de&longs;tructionis impetus in&longs;titui non po­tui&longs;&longs;e, immò nec excogitari po&longs;&longs;e; quàm enim facilè, vel impetus op­po&longs;itus in mobili producitur, vel corpus durum opponitur &c.

Tertiò, præcipuam rationem huius de&longs;tructionis ducendam e&longs;&longs;e ex Ax.6. in quo dicimus nihil e&longs;&longs;e fru&longs;trà, cumque ordinem à natura e&longs;&longs;e in&longs;titutum, vt potiùs aliquid de&longs;truatur, & de&longs;inat e&longs;&longs;e, quàm fru&longs;trà &longs;it, & dicimus de&longs;trui ad exigentiam totius naturæ.

Quartò, cum impetus &longs;uo fine caret, fru&longs;trà e&longs;t; finis impetus e&longs;t mo­tus, vt &longs;æpè diximus, &longs;ic cum globus impactus in alium æqualem &longs;tatim ab ictu &longs;i&longs;tit immobilis; certe ne fru&longs;trà &longs;it impetus, de&longs;truitur per Ax.6. & per Ax. 14. num.2. cum verò determinatio altera maior e&longs;t, certè præ­ualet tantùm pro rata; igitur minor e&longs;t motus; igitur, ne aliqui gradus impetus &longs;int fru&longs;trà, de&longs;truuntur, cum verò &longs;unt duo impetus in eodem mobili, vt in naui mobili ad lineas oppo&longs;itas determinati; haud dubiè maior impetus præualet pro rata per Ax. 15. Igitur non modò totus impetus minor perit, ne &longs;it fru&longs;trà; &longs;ed etiam aliquot gradus maioris, ne &longs;int etiam fru&longs;trà; nec enim in communem lineam coïre po&longs;&longs;unt.

Denique quando &longs;unt duo impetus ad lineas diuer&longs;as determinati, &longs;ed non oppo&longs;itas ex diametro, pugnant pro diuer&longs;o oppo&longs;itionis gradu, vt &longs;uprà fusè dictum e&longs;t. Igitur cum totus impetus non habeat totum motum, quod duplex illa determinatio impedit, ne aliqui gradus &longs;int fru&longs;trà, de&longs;truuntur; igitur vides impetum impre&longs;&longs;um ab ex­trin&longs;eco de&longs;trui tantùm ne &longs;it fru&longs;trà; faceret enim vt e&longs;&longs;et fru&longs;trà vel nouus impetus, vel determinato noua, & in hoc &longs;en&longs;u dicitur impetus de&longs;trui ab impetu.

Quintò, &longs;i de&longs;trueretur mobile, etiam de&longs;trueretur impetus per idem Ax. 6. quia e&longs;&longs;et fru&longs;trà &longs;eparatum; immò ex hoc vno principio demon­&longs;tramus accidentia & formas &longs;ub&longs;tantiales materiales non po&longs;&longs;e natura­liter con&longs;eruari extra &longs;uum &longs;ubiectum, quia &longs;cilicet e&longs;&longs;ent fru&longs;trà; quip­pe finem &longs;uum habent in &longs;ubiecto.

Sextò, Impetus naturalis innatus nunquam de&longs;truitur; quia nunquam e&longs;t fru&longs;trà; quippe &longs;emper habet alterum &longs;uorum effectuum formalium, id e&longs;t vel motum deor&longs;um, vel grauitationem, adde quod fru&longs;trà de­&longs;trueretur, cum &longs;it &longs;emper applicata potentia, id e&longs;t ip&longs;a grauitas, &longs;ed de his infrâ fusè.

Septimò, Impetus &longs;ur&longs;um de&longs;truitur etiam, quia e&longs;t fru&longs;trà; quippe naturalis detrahit aliquid &longs;patij pro rata; igitur ne aliquid impetus &longs;it fru&longs;trà, de&longs;truitur; idem dico de impetu per inclinatam &longs;ur&longs;um, licèt minùs de&longs;truatur quàm in perpendiculari &longs;ur&longs;um; idem de impetu per inclinatam deor&longs;um, &longs;ed minùs adhuc, &longs;ed hæc acuratiori medita ioni &longs;unt relinquenda; quod reuerâ præ&longs;tabimus in lib.4. de motu mixto; quidquid &longs;it, con&longs;tat ex dictis per idem Principium probari po&longs;&longs;e de­&longs;tructionem impetus, &longs;cilicet ne &longs;it fru&longs;trà; &longs;ed de his aliàs fusè.

Theorema 153.

Impetus productus ab extrin&longs;eco e&longs;t tantùm contrarius ratione diuer&longs;æ de­terminationis, &longs;eu diuer&longs;æ lineæ; Probatur primò, quia vterque ad omnem lineam e&longs;t indifferens per Th.113. igitur vnus non c&longs;t alteri contrarius ratione entitatis; cùm vterque &longs;imilem motum, immò eumdem habere po&longs;&longs;it, vt patet ex dictis: Igitur ratione tanuùn lineæ vnus alteri e&longs;t contrarius; hinc minùs e&longs;t contrarietatis, quo minùs e&longs;t oppo&longs;itionis inter lineas & contrà.

Theorema 154.

Impetus naturalis acqui&longs;itus e&longs;t tantùm contrarius alteri extrin&longs;eco ratio­ne lineæ. Probatur eodem modo; quia determinari pote&longs;t ad omnem li­neam, vt patet ex reflexione grauis cadentis.

Theorema 155.

Impetus naturalis innatus non e&longs;t tantùm contrarius ratione lineæ; quia &longs;cilicet non pote&longs;t determinari ad omnem lineam, patet, alioquin cor­pus graue, quod &longs;ur&longs;um po&longs;t ca&longs;um reflectitur non de&longs;cenderet amplius, de quo aliàs, hæc enim cur&longs;im tantùm per&longs;tringo, ne quid aliis libris detrahatur.

Theorema 156.

Impetus ex naturali acqui&longs;ito pote&longs;t fieri violentus; vt patet in motu re­flexo grauium; ratio e&longs;t. quia mutatur linea.

Theorema 157.

Impetus ex non contrario eidem fit contrarius; vt patet in eodem ca&longs;u; nam impetus naturalis innatus, qui in de&longs;cen&longs;u non erat contrarius ac qui&longs;ito, in inotu &longs;ur&longs;um reflexo fit contrarius.

Theorema 158.

Impetus deor&longs;um ab extrin&longs;eco non e&longs;t contrarius naturali innato ratione lineæ, quia &longs;cilicet e&longs;t determinatus ad eandem lineam, &longs;i tamen e&longs;t con­trarius, id tantùm e&longs;t ratione propagationis impetus acqui&longs;iti, vel ac celerationis motus; quod reuerà multa, & benè longâ explicatione indi­get, quam con&longs;ule in lib.4.

Scholium.

Ob&longs;eruabis cogno&longs;ci tantùm contrarietatem qualitatum ex mutua de­&longs;t ructione; cur verò vna qualitas dicatur de&longs;truere aliam, & cur illam de&longs;tructionem exigat; maximum my&longs;terium e&longs;t, quod alibi enucleabi­mus; quàm multa enim &longs;uper hac re tacuere Philo&longs;ophi!

Theorema 159.

Impetus &longs;ibi ip&longs;i pote&longs;t reddi contrarius, vt reuerâ accidit in reflexione, in qua de&longs;truitur impetus ex parte propter diuer&longs;as determinationes; cum &longs;cilicet corpus reflectens mouetur; igitur impetus prout determina­tus ad lineam incidentiæ e&longs;t aliquo modo &longs;ibi ip&longs;i contrarius, prout e&longs;t determinatus ad lineam reflexionis.

Iam ferè tumultuatim, &longs;i quæ &longs;unt reliqua, Theoremata congeremus.

Theorema 160.

Impetus violentus intendi pote&longs;t à naturali, & vici&longs;&longs;im; patet in projectis deor&longs;um.

Theorema 161.

Idem impetus pote&longs;t eumdem alium aliquando plùs, aliquando minùs intendere. v. g. 4. gradus impetus additi aliis 4. per eamdem lineam iidem ei&longs;dem, minùs intendunt, vt iam &longs;uprà &longs;atis fusè dictum e&longs;t.

Theorema 162.

Impetus dici pote&longs;t propriè de&longs;trui ad exigentiam totius naturæ per Ax.14. num.2. vt con&longs;tat ex multis Theorematis &longs;uperioribus.

Theorema 163.

Omnis dici debet incipere, & de&longs;inere intrin&longs;ecè, & extrin&longs;ecè; quod enim hoc in&longs;tanti primo e&longs;t, immediatè antecedenti vltimo non fuit, & quod primo non e&longs;t hoc in&longs;tanti, immediatè antè vltimo fuit, nec pote&longs;t e&longs;&longs;e immediatè pò&longs;t, ni&longs;i &longs;it immediatè antè, & vici&longs;&longs;im.

Theorema 164.

Ideo producitur hic impetus numero potiùs, quàm alius omninò &longs;imilis; quia potentia motrix e&longs;t determinata ad tale indiuiduum &longs;iue à &longs;e, &longs;iue ab alio; idem enim de illa dicendum e&longs;t, quod de aliis cau&longs;is naturalibus; porrò idem dici debet de de&longs;tructione, quod de productione.

Scholium.

Ob&longs;eruabis breuiter aliqua, quæ fortè in no&longs;tris Theorematis fuere omi&longs;&longs;a.

Primò qualitates, quæ à cau&longs;a primò productiua con&longs;eruantur, ab ea intendi non po&longs;&longs;e; quia &longs;ingulis in&longs;tantibus nouum effectum non pro­ducit; exemplum habes in luce; &longs;ecus vero de iis dicendum e&longs;t, quæ à cau&longs;a primò productiua non con&longs;eruantur.

Secundò qualitates, quæ contrarias habent, etiam de&longs;trui po&longs;&longs;e ab alio, quam ab iis, &longs;cilicet ad exigentiam totius naturæ; ne &longs;cilicet &longs;int fru&longs;trà.

Tertiò aliqua carere contrario, non tamen con&longs;eruari à cau&longs;a primò productiua. v.g. anima bruti, quæ de&longs;truitur ad exigentiam totius natu­ræ, nç &longs;it fru&longs;trà.

Quartò, impetum inten&longs;iorem in projectis diutiùs durare; quia cum &longs;en&longs;im de&longs;truatur; certè plures partes maiori tempore de&longs;truuntur, quàm pauciores.

Quintò, &longs;i totus impetus de&longs;trueretur vno in&longs;tanti, minima re&longs;i&longs;tentia &longs;ufficeret ad motum impediendum: adde quod contraria pugnant pro rata per Ax.15.

Sextò, ob&longs;eruabis plurima in hoc libro qua&longs;i obiter e&longs;&longs;e indicata, quæ in aliis fusè explicata maiorem lucem accipient.

Septimò, denique totam rem i&longs;tam, quæ pertinet ad impetum paulò fu&longs;ius pertractatam in hoc primo libro; quòd &longs;cilicet ab ea reliqua ferè omnia pendeant, quæ in hoc tractatu habentur; &longs;ed de his &longs;atis.

LIBER SECVNDVS, DE MOTV NATVRALI.

MOtus localis naturalis latè &longs;umptus e&longs;t, qui ab aliqua causâ naturali ponitur; &longs;trictè verò &longs;umitur pro motu grauium deor&longs;um, à principio intrin&longs;eco &longs;altem &longs;en&longs;ibiliter; In hoc vltimo &longs;en&longs;u mo­tum naturalem v&longs;urpabo; &longs;it ergo.

DEFINITIO 1.

MOtus localis naturalis e&longs;t, qui e&longs;t à grauitate deor&longs;um. hæc defini­tio vix aliqua explicatione indiget; dicitur e&longs;&longs;e à grauitate, quidquid &longs;it grauitas, &longs;iue qualitas di&longs;tincta, &longs;iue non.

Definitio 2.

Motus æquabilis e&longs;t, quo æqualibus quibu&longs;cumque temporibus æqualia per­curruntur &longs;patia ab eodem mobili.

Definitio 3.

Motus naturaliter acceleratus e&longs;t, quo &longs;ecundo tempore æquali primo ma­ius &longs;patium acquiritur, & tertio, quàm &longs;ecundo, & quarto quàm tertio, atque ita deinceps; nulla &longs;cilicet addita vi ab extrin&longs;eco &longs;altem &longs;en&longs;ibiliter.

Definit aliter hunc motum Galileus; dicit enim eum e&longs;&longs;e, qui æquali­bus temporibus æqualia acquirit velocitatis momenta; &longs;ed profectò non conuenit hæc definitio omni motui naturaliter accelerato, v. g. motui de&longs;cen&longs;us funependuli, vel in orbe cauo, vel etiam in plano decliui ma­ximæ longitudinis; definitio no&longs;tra clarior e&longs;t.

Hypothe&longs;is 1.

Corpus graue cadit deor&longs;um, & cadens ex maiori altitudine maioremictum infligit quam &longs;i caderet ex minore; &longs;i quis hoc neget hoc probet, patet ma­nife&longs;ta experientia.

Hypothe&longs;is 2.

Arcus maior & minor eiu&longs;dem funependuli æqualibus ferè temporibus, percurruntur; hæc etiam &longs;æpiùs probata e&longs;t, & &longs;i quis fidem detrectat, probare conetur.

Hypothe&longs;is 3.

Globus per planum inclinatum læuigatum de&longs;cendens &longs;ecundum &longs;pa­tium citiùs percurrit, quàm primum; quod etiam &longs;en&longs;u percipi pote&longs;t, & tam &longs;æpè probatum e&longs;t, vt nemo iam negare audeat motus naturalis accelerationem.

Hypothe&longs;is 4.

Omne tempus &longs;en&longs;ibile non e&longs;t; idem dico de &longs;patio, quod nemo etiam negare au&longs;it; alioquin &longs;i quis negaret, dicat mihi quæ&longs;o quot &longs;int in mi­nuto horæ in&longs;tantia? quot in apice acus puncta?

Axioma 1.

Impetus additus alteri, & determinatus ad eamdem lineam, facit maiorem & inten&longs;iorom impetum; patet, & vici&longs;&longs;im, & detractus alteri minorem facit, & vici&longs;&longs;im.

Axioma 2.

Quâ proportione cre&longs;cit cau&longs;a, eâdem cre&longs;cit effectus, & vici&longs;&longs;im, &longs;i eodem modo eidemque &longs;ubjecto &longs;it applicata, probatur per Ax.12. l. 1. & quâ pro­portione illa decre&longs;cit, hic decre&longs;cit, & vici&longs;&longs;im.

Axioma 3.

Eadem cau&longs;a nece&longs;&longs;aria non impedita &longs;ubjecto ap applicata æqualibus temporibus æquælem effectum producit, & contrà. Probatur per Ax.12.l. 1. & vici&longs;&longs;im æqualis effectus &longs;upponit æqualem cau&longs;am.

Axioma 4.

Ille effectus, qui non producitur à causâ primâ, & ad cuius productionem nulla cau&longs;a extrin&longs;eca e&longs;t applicata, producitur ab intrin&longs;eco; probatur, quia habere debet aliquam cau&longs;am per Ax.8.

Axioma 5.

Illa cau&longs;a plus agit proportionaliter quæ habet minorem re&longs;istentiam; minùs verò, quæ maiorem, quæ demum æqualem, æquali proportione agit. v.g. cau&longs;a, cuius virtus, vel actiuitas e&longs;t vt 20. & re&longs;i&longs;tentia vt 10. agit in maiori proportione, quàm illa cuius actiuitas e&longs;t 30. & re&longs;i&longs;tentia 20. in minori verò quàm ea, cuius actiuitas e&longs;t vt 3. & re&longs;i&longs;tentia vt 1. in æquali de­nique cum illa, cuius actiuitas e&longs;t vt 4. & re&longs;i&longs;tentia vt 2.

Hoc Axioma certi&longs;&longs;imum e&longs;t; quippe 20. faciliùs &longs;uperabunt 10. quàm 30. 20. & difficiliùs quam 3. 1. & æquè facilè, ac 4. 2. In motu locali res e&longs;t clari&longs;&longs;ima; quippe vires vt 12. tam facilè mouebunt 12. libras, quàm vires vt 4. 4.libras; &longs;ed faciliùs, quàm vires vt 20. 30.libras, & dif­ficiliùs quàm vires vt 4. 3. libras; quid clarius? Igitur illa cau&longs;a faciliùs &longs;uperat re&longs;i&longs;tentiam impedimenti, quæ habet maiorem proportionem virium cum re&longs;i&longs;tentia, quàm quæ minorem.

Scholium.

Si quando appellandum erit aliquod Axioma vel Theorema lib. 1.ci­tabitur Liber.

Theorema 1.

Datur motus localis naturalis, i&longs;que ab intrin&longs;eco. Probatur; corpus gra­ue mouetur localiter deor&longs;um per hypoth. hic motus e&longs;t ab intrin&longs;eco, quod probatur; non e&longs;t ab vllâ causâ extrin&longs;ecâ; igitur e&longs;t ab intrin&longs;eca per Ax.4. antecedens probatur inductione factâ omnium extrin&longs;ecorum. Primò non e&longs;t à cau&longs;a prima, vt aliquis fortè minùs prudenter, & magis piè, quàm par &longs;it, diceret; quia ille effectus tribui tantùm debet cau&longs;æ primæ, qui nullam habere pote&longs;t cau&longs;am &longs;ecundam applicatam, vt patet; &longs;ed hic effectus pote&longs;t habere cau&longs;am &longs;ecundam applicatam, quam a&longs;&longs;i­gnabimus infrà; deinde cau&longs;a prima agit tantùm naturaliter iuxta exi­gentiam cau&longs;arum &longs;ecundarum; igitur ideo moueret corpus graue deor­&longs;um; quia tunc motum corpus graue exigeret; &longs;ed lioc milri &longs;u&longs;ficit, vt dicatur hic motus e&longs;&longs;e ab intrin&longs;eco; præterea, &longs;i dicatur Deus mouere corpus graue deor&longs;um iuxta illius exigentiam, dicetur etiam tùm cale­facere, tùm illuminare, ad exigentiam ignis; quippe tàm mihi &longs;en&longs;ibile e&longs;t corpus graue de&longs;cendere &longs;ine vi impre&longs;&longs;a ab extrin&longs;eco, quàm ignem calefacere, & &longs;olem lucere &longs;ine vi extrin&longs;eca; adde quod illud &longs;olenne e&longs;t naturæ in&longs;titutum, vt id, quod exigit res aliqua ad finem &longs;uum con&longs;e­quendum, per virtutem intrin&longs;ecam po&longs;&longs;it ponere, &longs;i dumtaxat excipias concur&longs;um diuinum, & ip&longs;am con&longs;oruationem; &longs;ic animal exigit vide­re, audire, &longs;entire, moueri; igitur habet virtutem intrin&longs;ecam, per quam videat, audiat, & moueatur; &longs;ic ignis exigit calefacere, lucere; aër, vel aqua frigefacere, quidquid tandem &longs;int i&longs;tæ qualitates, de quibus alibi; &longs;ic demum corpus graue exigit moueri deor&longs;um; quis enim neget corpori graui tàm natiuum e&longs;&longs;e tendere deor&longs;um, cum &longs;cilicer corpus leuius &longs;ub­e&longs;t, quàm &longs;it animali progredi, vrere igni, lucere, &c.

Denique &longs;atis e&longs;t mihi, vt dicatur aliquid cau&longs;a, Phy&longs;icè loquendo, &longs;i ex illius applicatione &longs;emper &longs;equatur effectus; nam non nego po&longs;&longs;e fie­ri effectus omnes, qui no&longs;tris &longs;en&longs;ibus &longs;ubiiciuntur, &longs;altem extrin&longs;ecos, e&longs;&longs;e à cau&longs;a prima, quippe &longs;i &longs;emper ex ignis applicatione Deus diffun­deret lucem, & calorem, quem &longs;olus ip&longs;e produceret, igne ip&longs;o inerte re­licto, nullam pror&longs;us mutationem perciperemus; & nemo e&longs;&longs;et, qui non exi&longs;timaret lucom hanc & calorem ltunc e&longs;&longs;e ab igne; igitur Phy&longs;icè lo­quendo cau&longs;am appellamus id, ex cuius applicatione &longs;emper &longs;equitur effectus, vt iam diximus in Ax. 11.l.1. n.1. Igitur cum ex corpore graui po&longs;ito in aëre libero &longs;equatur motus deor&longs;um; dicendum e&longs;t, Phy&longs;icè lo­quendò, e&longs;&longs;e huius motus cau&longs;am, id e&longs;t in ordine ad Phy&longs;icam, perinde omninò &longs;e habere, atque &longs;i e&longs;&longs;et cau&longs;a, licèt cau&longs;a non e&longs;&longs;et.

Secundò hic motus non e&longs;t ab aëre ambiente; probatur, ruderet aër deor&longs;um corpus graue, quia leuior e&longs;t, id e&longs;t ne &longs;uprà &longs;e corpus grauins haberet; &longs;ed eâdem ratione corpus graue debet remouere &longs;ur&longs;um aëra, id e&longs;t corpus leue, ne infrà &longs;e habeat corpus leuius; e&longs;t enim par omni­nò ratio: Præterea &longs;i aër trudit deor&longs;inn corpus graue, quia ip&longs;i loco cedit; certè ip&longs;e aër mouetur, igitur ab intrin&longs;eco; &longs;i enim vna pars aë­ris pellit aliam, & hæc aliam, tandem ad aliquam peruenitur, quæ &longs;e ip­&longs;am mouet; igitur motus illius e&longs;t ab intrin&longs;eco; igitur motus natura­lis; deinde non modò lapis de&longs;cendit per aëra, &longs;ed per mediam aquam; igitur &longs;i ab aëre truditur deor&longs;um, idem dicendum e&longs;t de aquâ, a qui haud dubiè maiore vi truderetur; nam corpus den&longs;um maiore vi pellit, quàm rarum, vt con&longs;tat exprientiâ; cum tamen corpus graue per me­dium den&longs;ius difficiliùs de cendat; igitur medium ip&longs;um re&longs;i&longs;tit motui, quis hoc neget? igitur non e&longs;t cau&longs;a motus, quem impedit.

Denique &longs;i corpus graue non tendit, fertur que deor&longs;um &longs;uá &longs;ponte, &longs;ed ab aëre extru&longs;um; igitur dum vix &longs;u&longs;tineo manu; o. libras ferri, &longs;eu plumbi; hæc vis illata manui, quam probè &longs;entio, e&longs;t ab aëre impel­lente plumbum, quod e&longs;t ridiculum, cnm eadem quantitas aëris incu­ber, & &longs;ub&longs;it manui, &longs;iue &longs;u&longs;tineat plumbum, &longs;iue &longs;it vacua; ex hoc, ni fallor, euincitur pondus ip&longs;um &longs;ui &longs;ponte deor&longs;unr tendere.

Tertiò non de&longs;unt, qui dicant corpus graue trahi ab ip&longs;a vi quadam magneticâ, quod triplici modo fieri pote&longs;t; Primò per qualitatem quamdam diffu&longs;am, quod dici non pote&longs;t; quia capillus traheretur faci­liùs, quàm ingens &longs;axum, quàm ma&longs;&longs;a, &longs;eu lamina; & faciliùs eadem po­tentia motrix minus pondus moueret quàm maius, cæteris paribus; præ­terea manum meam æqualiter traheret, &longs;iue &longs;it cum aliquo pondere con­iuncta, &longs;iuc &longs;it nuda &longs;ine pondere; deinde illa virtus tractrix ita diffun­ditur, vt in maiori di&longs;tantia &longs;it infirmior, fortior in minori; alioqui diffunderetur in infinitum, quod dici non pote&longs;t; igitur &longs;i idem lapis demittatur ex maiore altitudine, tum ex minore; haud dubiè morus ille primus initio e&longs;&longs;et tardior i&longs;to contra experientiam; deinde in &longs;pecu al­ti&longs;&longs;ima &longs;ubterranea trahi po&longs;&longs;et corpus virdequaque, &longs;icut in magnete; quæ omnia intelligi non po&longs;&longs;unt; denique virtutes illas &longs;eu qualitates tractrices refellemus &longs;uo loco.

Secundò, aliqui dicunt hoc totum fieri per vim quamdam &longs;ympathi­cam, quod etiam fal&longs;i&longs;&longs;imum e&longs;t; tùm quia hæc &longs;ympathia explicari non pote&longs;t; tùm quia vel terip&longs;a producit aliquid in corpore graui, quod in aëre libratur; vel corpus in &longs;e ip&longs;o; &longs;i primum; refellitur ii&longs;­dem omninò rationibus, quibus ip&longs;am vim terræ tractricem &longs;uptà expu­gnauimus; &longs;i verò &longs;ccundum, hoc ip&longs;um e&longs;t, quod &longs;uprà diximus.

Tertiò, Dixere aliqui &longs;ubtiliùs profectò quàm veriùs, corpus graue trahi deor&longs;um, non vi quadam occultâ, vt &longs;uprà dictum e&longs;t; &longs;ed filamen­tis quibu&longs;dam, &longs;eu ductili terræ profluuio, quod illius capillitium vo­cant; idque tantùm fieri probant ducta ab electro analogiâ, quod pa­am & minutiora corpu&longs;cula hac câdem arte trahit; &longs;ed profectò gra-uiores &longs;unt di&longs;&longs;icultates, quam vt illis fieri &longs;atis queat; nam primò cor­pus leuius ab his filamentis abripi faciliùs po&longs;&longs;et, vt con&longs;tat in electro; igitur citiùs de&longs;cenderet.

Secundò, corpus vicinius etiam faciliùs abriperetur.

Tertiò, numquid flante vento, vel imbre cadente di&longs;&longs;ipantur hæc &longs;i­lamenta? quod etiam videmus in electro.

Quartò, manum meam æquè facilè traheret terra his funiculis &longs;eu pondere grauatam, &longs;eu vacuam.

Quintò, quemadmodum electrum ex omni parte trahit, ita terra ip&longs;a per omnem lineam traheret; immò etiam &longs;ur&longs;um in &longs;ubterranea &longs;pecu, quod e&longs;t ab&longs;urdum.

Sextò, hæc filamenta, quæ deinde reducuntur, debent habere cau­&longs;am huius reductionis non extrin&longs;ecam; igitur intrin&longs;ecam; igitur datur motus naturalis.

Soptimò, hæc filamenta per mediam flammam non traherent, quod etiam fieri videmus in electro.

Quartò, motus naturalis non e&longs;t à virtute quadam pellente, quam cælo quidam affingunt; nam vel ab omni parte cæli deor&longs;um trudere­tur, vel ab vnâ; &longs;i ab vna; igitur in omni cæli plaga corpus non fertur deor&longs;um; &longs;i ab omni, ergo cum pellatur corpus per plures lineas etiam oppo&longs;itas moueri non pote&longs;t: Præterea debilior e&longs;&longs;et hæc vis in maiori di&longs;tantiâ; denique vapores, & alia minutiora corpu&longs;cula in aëre fluitan­tia faciliùs deor&longs;um truderentur, contra experientiam.

Sed non e&longs;t omittendum, quod aliqui putant ex illis filamentis con­texi po&longs;&longs;e legitimam rationem, cur atomi etiam plumbeæ materiæ non ita facilè de&longs;cendant; quòd &longs;cilicet propter &longs;uam tenuitatem ab illis fi­lamentis non ita intercipi vel implicari po&longs;&longs;int; &longs;ed qua&longs;i pi&longs;ces per fo­ramina retium euadant; &longs;ed profectò longè alia ratio e&longs;t, qaàm &longs;uo loco afferemus, nam etiam plumæ, fe&longs;tucæ, paleæ, & alia corpu&longs;cula longio­ra, &longs;ed leui&longs;&longs;ima iis filamentis implicarentur, vt videre e&longs;t in electro.

Quintò, aliqui recentiores exi&longs;timant corpora deor&longs;um trudi ab ip&longs;a luce, quæ nihil e&longs;t aliud, quam motio æthereæ cuiu&longs;dam &longs;ub&longs;tan­tiæ per poros aëris traductæ, vt ip&longs;i volunt; &longs;ed neque hoc probari po­te&longs;t. Primò quia de nocte corpora æquali motu deor&longs;um feruntur; pe­rinde atque de die, nec minùs in ob&longs;curi&longs;&longs;imo cclaui, quàm &longs;ub dio, vel aperto cælo. Secundò, in &longs;ubterraneis locis etiam grauia æquè veloci­ter de&longs;cendunt; licèr eò lumen non penetret; quod &longs;i aliquis ob&longs;tinatè, id a&longs;&longs;ereret; haud dubiè per medium aëra maior huius materiæ copia diffunditur, quàm per medias rupes, quis hoc neget; igitur pauci&longs;&longs;imi radij v&longs;que ad interius & inferius antrum perueniunt. Tertiò, manum meam &longs;iue ponderi coniunctam &longs;iue ab eo &longs;eparatam æqualis portio illius materiæ deor&longs;um pelleret, vt patet; igitur æquali motus vi. Quartò, cor­pus diaphanum, per cuius poros facilè traiicitur hæc materia, e&longs;&longs;et leuius alio quod tamen fal&longs;um e&longs;t, vt videre e&longs;t in vitro, cry&longs;tallo, adamante, glacie. Quintò maxima huius materiæ copia collecta &longs;eu &longs;peculi opera &longs;eu vitri, maiore vi corpora deor&longs;um truderet; quia maior cau&longs;a maio­rem effectum producit per Ax.2. Sextò po&longs;t refractionem lineam mutat radius luminis; igitur deor&longs;um rectà non pelleret. Septimò radij traie­cti per vitrum maiore vi deor&longs;um pellerent quàm per lignum, vel &longs;pon­giam; quippè per hæc corpora traiecti &longs;ecundum authores huius &longs;enten­tiæ di&longs;trahuntur propter obliquitatem pororum. Octauò denique radij profecti à Sole iuxta ortum, vel occa&longs;um &longs;unt valdè obliqui; igitur non truderent deor&longs;um rectà.

Nec e&longs;t quod prædicti àuthores confugiant ad experientiam, qua &longs;cilicet videmus tripudiantes atomos in radio &longs;olari immer&longs;as; igitur agitantur ab ip&longs;o radio, quod maximè accidit in linea v&longs;toria, cuius effectus veri&longs;&longs;imam rationem &longs;uo loco afferemus, cum de lumine.

Sextò, &longs;unt denique multi, iique ex &longs;euerioribus Peripateticis, qui exi&longs;timant grauia moueri deor&longs;um à generante, quod expre&longs;&longs;rs verbis traditum e&longs;t ab Ari&longs;totele l.8. phy&longs;. cap.4. iuxta principium illud vniuer­fali&longs;&longs;imum; Quidquid mouetur; ab alio mouetur; &longs;ed profectò ij ip&longs;i, qui motum grauium generanti tribuunt, tanquam principi cau&longs;æ, non ne­gant ine&longs;&longs;e grauibus grauitatem, quæ &longs;it principium actiuum minus principale motus; ad quem etiam, vt ip&longs;i exi&longs;timant, forma &longs;ub&longs;tantialis concurrit; In hoc quippe conueniunt omnes tùm &longs;ectarum Principes, tùm recentiores: quidquid &longs;it etiam ex iis ip&longs;is datur motus naturalis, qui e&longs;t à virtute proxima intrin&longs;eca; hoc ip&longs;um etiam &longs;en&longs;it Ari&longs;toteles lib.4. de cælo cap. 3. t. 25. vbi ait grauibus & leuibus ine&longs;&longs;e principium actiuum &longs;uorum motuum; immò &longs;i totum cap.4. l.8. phy&longs;. attentè lega­tur, vbi dicit moueri à generante, haud dubiè intelligetur nihil aliud in­tendi&longs;&longs;e Ari&longs;totelem quàm grauia à generante, in&longs;tanti, quo generan­tur, accipere actum primum huius motus; id e&longs;t virtutem, à qua po&longs;­&longs;int reduci ad actum &longs;ecundum, id e&longs;t ad ip&longs;um motum, de cuius rei ve­ritate iam mihi non e&longs;t laborandum.

Igitur non mouetur corpus graue à cau&longs;a primâ, licèt hæc concurrat cum aliâ ad eius motum, nec ab aëre, nec à virtute magnetica, quæ in­&longs;it terræ, nec adductis, reducti&longs;que filamentis, nec à cælo pellente, nec à vi &longs;ympathicâ, nec à generante proximè & immediatè; quia fortè iam interiit, nec ab vllo alio extrin&longs;eco, vt con&longs;tat inductione; igitur ab ali­quâ vi intrin&longs;ecâ, quidquid &longs;it, de qua alibi: hæc omnia paulò fu&longs;iùs tractauimus, quia in hoc vno Theoremate totam motus naturalis rem verti iudicamus.

Theorema 2.

Motus naturalis est aliquid distinctum realiter à mobili: Probatur; mobile ip&longs;um aliquando quie&longs;cit per hypoth.4.lib.1. igitur e&longs;t &longs;ine mo­tu; igitur &longs;eparatum à motu; igitur realiter di&longs;tinctum per Ax.2. lib.1. hoc etiam probatus per Th. 1.lib. 1. Et certè mirari &longs;atis non po&longs;&longs;um aliquos recentiores non po&longs;&longs;e concipere, vt ip&longs;i aiunt, motum e&longs;&longs;e ali­quid ab ip&longs;o mobili di&longs;tinctum; nam quotie&longs;cunque duo prædicata, vel attributa contradictoria, quorum &longs;cilicet vnum negat aliud, eidem &longs;ub­jecto diuer&longs;is temporibus ine&longs;&longs;e dicuntur, haud dubiè alterum &longs;altem ab eo di&longs;tingui realiter nece&longs;&longs;e e&longs;t; alioqui &longs;i vtrumque idem e&longs;&longs;e cum vno tertio vere dicitur; mouetur, non monetur, quæ &longs;unt prædicata contradi­ctoria; igitur vel moueri, vel non moueri dicit di&longs;tinctum realiter à mo­bili; Secundum e&longs;t mera negatio; nam eo ip&longs;o, quod mobile e&longs;t &longs;ine vllo addito, non mouetur; igitur &longs;uprà ip&longs;um mobile dicit puram putam ne­gationem motus; igitur moueri, dicit aliquid di&longs;tinctum.

Præterea quotie&longs;cunque prædicatum aliquod tribuitur in propo&longs;i­tione affirmatiua falsâ; certè prædicatum illud non ine&longs;t &longs;ubiecto; alio­quin e&longs;&longs;et vera, vt patet; igitur di&longs;tinguitur à &longs;ubiecto realiter; &longs;ed hæc propo&longs;itio, lapis mouetur, dum ip&longs;e quie&longs;cit, e&longs;t fal&longs;a; igitur motus non ine&longs;t mobili, igitur ab eo di&longs;tinguitur realiter, &longs;eu modaliter, quæ e&longs;t di&longs;tinctio realis minor.

Theorema 3.

Metus naturalis non e&longs;t immediatè ab entitate mobilis, ita vt nihil &longs;it aliud vnde &longs;it hic motus: Probatur; lapis cadens ex maiore altitudine maiorem ictum infligit perhypoth. 1. maior e&longs;t effectus, igitur maior cau&longs;a, id e&longs;t motus; igitur cau&longs;a motus per Ax.2. &longs;ed e&longs;t eadem entitas mobilis, vt patet; igitur non e&longs;t cau&longs;a immediata motus; Præterea globus per pla­num inclinatum deuolutus &longs;uum motum accelerat per hypotl. 3. & fune­pendulum &longs;uam vibrationem per hypoth. 2. igitur debet e&longs;&longs;e cau&longs;a huius maioris, &longs;eu velocioris motus per Ax.8. lib. 1. hæc porrò non e&longs;t &longs;ub­&longs;tantia ip&longs;ius corporis, quæ &longs;emper eadem e&longs;t, tùm initio, tùm in fine motus per Ax.2.

Theorema 4.

Motus naturalis non e&longs;t immediatè ab ip&longs;a grauitate. Probatur, &longs;int enim eædem hypoth.1.2.3. igitur maior ictus in fine motus, & velocior motus debent habere cau&longs;am; &longs;ed hæc grauitas non e&longs;t, quæ &longs;emper ea­dem e&longs;t, vt patet, vtrum verò di&longs;tinguatur grauitas ab ip&longs;a corporis &longs;ub&longs;tantia di&longs;cutiemus in tractatu &longs;equenti. Fuit aliquis non infunæ no­tæ Philo&longs;ophus, qui diceret maiorem illum ictum e&longs;&longs;e ab ipsâ corporis &longs;ub&longs;tantiâ; &longs;ed hoc iam refellimus Theoremate 4. lib.1. Adde quod im­petu, ad extra producitur ab alio impetu per Th.42.lib.1. Dicebat etiam velociorem motum e&longs;&longs;e ab ipsâ grauitate connotante præuium motum, quod etiam refellemus infrà.

Theorema 5.

Hinc motus naturalis e&longs;t ab impetu. Probatur; e&longs;t ab aliqua cau&longs;a per Ax.8. lib.1. ab aliqua intrin&longs;eca per Th. 1. non à &longs;ub&longs;tantia corporis grauis per Th. 3. non à grauitace per Th. 4. igitur ab impetu, quia nihil aliud e&longs;&longs;e pote&longs;t intrin&longs;ecum, à quo &longs;it motus pet definitionem 3. lib. 1.

Theorema 6.

Ille impetus ab aliqua cau&longs;a producitur. Probatur, quia quidquid de no­uo e&longs;t, habet cau&longs;am per Ax.8. lib. 1.

Theorema 7.

Producitur ab aliqua cau&longs;a intrin&longs;eca, quia non producitur ab aliqua extrin&longs;eca; alioquin motus naturalis e&longs;&longs;et ab extrin&longs;eco contra definitio­nem primam, & Th.1.

Theorema 8.

Hinc produci tantùm pote&longs;t ab ip&longs;a &longs;ubstantia corporis grauis; nam graui­tas e&longs;t ip&longs;e impetus innatus, de qua infrà: probatur; quia nihil e&longs;t aliud in­trin&longs;ecum, à quo produci po&longs;&longs;it; quòd autem non produc tur ab alio im­petu ad intra, patet per Th.41. lib. 1.

Theorema 9.

Impetus productus primo instanti durat proximè &longs;equenti. Probatur pri­mò; quia &longs;emper habet &longs;uum effectum formalem; vel grauitationis, &longs;i impeditur; vel motus in medio libero; igitur non e&longs;t fru&longs;trà; igitur non de&longs;truitur per Th.162.lib.1. nihil enim exigit de&longs;tructionem; non tota natura, quia non e&longs;t fru&longs;trà per Ax. 6. non à contrario impetu, qui &longs;æpè abe&longs;t, vt cum liberè mouetur corpus graue in aëre, vel &longs;u&longs;tinetur, v.g. glans plumbea ab ingenti rupe: adde quod, licèt producatur in cor­pore graui impetus violentus &longs;ur&longs;um, non de&longs;truitur, tamen innatus; alio­quin nihil e&longs;&longs;et, quod de&longs;trueret violentum per Th.150. & Schol. Th. 152.num.6.lib.1.

Theorema 10.

Impetus ille innatus, qui durat &longs;ecundo instanti, con&longs;eruatur ab aiiqua cau­&longs;a; e&longs;t certum per Ax. 14.lib.1.num.1.

Theorema 11.

Non con&longs;eruatur à cau&longs;a primò productiua. Probatur per Th.144. lib.1. alioquin non po&longs;&longs;et intendi ab eadem cau&longs;a per Th. 146. lib 1. quippè con&longs;eruatio nihil e&longs;t aliud, quàm repetita productio, vt con&longs;tat; nam cau&longs;a con&longs;eruans verè influit; igitur &longs;i e&longs;t can&longs;a nece&longs;&longs;aria primo, & &longs;e­cundo in&longs;tanti æquali ni&longs;u influit; influit enim quantum pote&longs;t per Ax. 12.lib.1.quòd autem impetus intendatur, demon&longs;trabimus infrà; con&longs;ule Schol.Th.146.lib.1.in quo habes rationem præclari natura in&longs;tituti; quo &longs;cilicet factum e&longs;t, vt qualitates, quæ contrario carent à causâ primò pro­ductiua; aliæ verò, quæ contrarium habent, ab alia causà con&longs;er­uentur.

Corollarium.

Hinc ab aliâ causâ con&longs;eruari nece&longs;&longs;e e&longs;t, vt patet, eáque aplicatâ per Ax.10.lib.1. quæcumque tandem illa &longs;it; nos aliquando cau&longs;am primam e&longs;&longs;e dicemus.

Theorema 12.

Quando graue e&longs;t in medio libero, per quod &longs;cilicet de&longs;cendere pote&longs;t, &longs;ecun­do instanti producitur nouus impetus, itemque tertio, quarto, quinto. &c. Pro­batur primò; quia &longs;ecundo in&longs;tanti e&longs;t eadem cau&longs;a quæ primo non ma­gis impedita, eáque nece&longs;&longs;aria; igitur nece&longs;&longs;ariò agit per Ax. 12. lib.1. igitur aliquem effectum producit; &longs;ed hic effectus non e&longs;t impetus pro­ductus primo in&longs;tanti, quia non con&longs;eruatur à cau&longs;a primò productiua per Th.11. igitur e&longs;t nouus. Probatur &longs;ecundò; cre&longs;cit motus grauium in libero medio per hypoth. 1.2.3. igitur cre&longs;cit impetus; quia cum motus naturalis &longs;it ab impetu per Th.5. quâ proportionc cre&longs;cit effectus, &longs;cilicet &longs;ormalis, & exigentiæ; &longs;ic enim motus e&longs;t effectus impetus per Th. 15. lib.1.eàdem cre&longs;cit cau&longs;a per Ax.2. Probatur tertiò, quia corpus graue ex maiore altitudine cadens maiorem quoque ictum infligit per hypoth.1. igitur maior impetus imprimitur in corpore percu&longs;&longs;o; &longs;ed impetus ad ex­tra producitur ab alio impetu per Th.42.lib.1. igitur &longs;i cre&longs;cit productus inpetus, cre&longs;cit impetus producens.

Corollarium.

Hinc reiicies quorumdam placitum, qui volunt cau&longs;am velociotis motus e&longs;&longs;e grauitatem ip&longs;am, quatenus connotat motum præuium; quia &longs;cilicet grauitas non producit illum maiorem impetum ad extra, vt con­&longs;tat; nec &longs;ub&longs;tantia ip&longs;ius corporis grauis per Th.40.lib.1.igitur ip&longs;e im­petus: præterea &longs;i hoc e&longs;&longs;et, fru&longs;trà requireretur impetus contra Th. 5. Denique motus præuius nihil e&longs;t amplius, cum alius &longs;uccedit: Vide Th. 40.lib.1. vbi hæc fusè di&longs;cu&longs;&longs;imus.

Theorema 13.

Impetus productus &longs;ecundo instanti in medio libero con&longs;eruatur tertio, & productus tertio con&longs;eruatur, quarto, atque ita deinceps; quia &longs;cilicet nec con­antur à cau&longs;a primo productiua per Th.144.libri: nec aliquid exigit de&longs;tructionem; non contrarius impetus, quia nullus e&longs;t applicatus, vt con&longs;tat; non re&longs;i&longs;tentia medij, quæ quidem alicuius momenti e&longs;t; &longs;ed non tanti, vt impedire po&longs;&longs;it motum omninò, vt con&longs;tat; nam &longs;uppono liberum medium, igitur nec de&longs;truere impetum; cum tamdiu duret cau­&longs;a quamdiu durat effectus, vt patet; igitur nihil e&longs;t quod exigat impe­ms huius de&longs;tructionem; igitur non de&longs;truitur per Ax. 14. lib.1. qunanta verò &longs;it, & quid &longs;it cuiu&longs;libet medij re&longs;i&longs;tentia, dicemus infrà.

Theorema 14.

Si impetus innatus impeditur, ita vt moueri non po&longs;&longs;it corpus graue, &longs;e­cundo instanti non producitur nouus impetus. Probatur primò, non cre&longs;cit corporis grauis &longs;eu grauitas, &longs;cu grauitatio, vt con&longs;tat experientiâ; igitur non cre&longs;cit impectus; alioquin &longs;i cre&longs;ceret cau&longs;a, cre&longs;ceret effectus per Ax.2. igitur de re, quòd &longs;it, certum e&longs;t, atque cuidens; iam demon&longs;tratur propter quid &longs;it; impetus &longs;ecundo in&longs;tanti productus e&longs;&longs;et fru&longs;trà; careret enim &longs;uo fine, vel effectu formali, id e&longs;t motu; igitur e&longs;&longs;et fru&longs;trà, &longs;ed quod fru&longs;trà e&longs;t, non e&longs;t per Ax.6.l.1.

Scholium.

Ob&longs;erua quæ&longs;o, quod iam &longs;uprà indicattun e&longs;t, e&longs;&longs;e tres veluti &longs;pecies impetus. Prima e&longs;t impetus naturalis innati. Secunda naturalis acqui&longs;iti. Tertia violenti; innatus e&longs;t qui vel à generante &longs;imul cum corpore graui productus e&longs;t; qui&longs;quis tandem &longs;it generans, de quo aliàs; vel ab ip&longs;o graui qua&longs;i profunditur, id e&longs;t, producitur in &longs;e ip&longs;o &longs;tatim initio, quo e&longs;t; porrò cum in corpore graui duplex qua&longs;i proprietas &longs;en&longs;ibilis e&longs;&longs;e videatur, &longs;cilicet grauitas, &longs;eu pondus & motus deor&longs;um; certè de­bet e&longs;&longs;e in co aliquid per quod tùm cogno&longs;ci po&longs;&longs;it eins pondus, tùm in­cipiat moueri deor&longs;um; quippe maximè corpora ex pondere cogno&longs;ci­mus, vnumque ab alio di&longs;tinguimus; igitur debet e&longs;&longs;e aliquid, quod &longs;en­&longs;um afficiat, vt cogno&longs;ci po&longs;&longs;it; atqui illud ip&longs;um non e&longs;t &longs;ub&longs;tantia cor­poris; nam corpus graue meæ manui &longs;u&longs;tinenti impetum imptimit; immò vim alterius impetus infringit; igitur operâ alterius per Th. 40. & 42.lib.1. Præterea illud ip&longs;um, quod agit, &longs;eu deor&longs;um pellit &longs;u&longs;tinen­tem manum, e&longs;t illud ip&longs;um quod inclinat corpus graue deor&longs;um imme­diatè, &longs;eu quod exigit motum naturalem deor&longs;um; illud autem quod immediatè præ&longs;tat hunc motum, nec e&longs;t &longs;ub&longs;tantia corporis grauis per Th.3. igitur ip&longs;e impetus per Th.5. adde quod primo in&longs;tanti, quo e&longs;t im­petus, non e&longs;t motus ille, quem exigit per Th.34. lib.1. igitur præexi&longs;tit &longs;emper impetus, qui ne &longs;it fru&longs;trà, habet primum effectum &longs;uum forma­lem, id e&longs;t grauitationem: Ex his dicendum e&longs;t hunc impetum natiuum nunquam de&longs;trui, quia nunquam e&longs;t fru&longs;trà, habet enim &longs;emper aliquem effectum, primum quidem &longs;i caret &longs;ecundo; &longs;ecundum verò &longs;i caret pri­mo; quippe vtrumque &longs;imul habere non pote&longs;t; nam corporis pondus cogno&longs;ci non pote&longs;t, dum fertur deor&longs;um accelerato motu, quot verò, & quanta commoda ex cognitione ponderis cuiu&longs;libet materiæ proce­dant, vix explicari pote&longs;t.

Ex his verò concludendum &longs;upere&longs;t impetum innatum e&longs;&longs;e proprie­tatem quarto modo, vt vulgò aiunt, corporis grauis; ac proinde ab illo in&longs;eparabilem; quid verò fiat de illo, cum corpus graue fit leuc; &longs;i tamen hoc aliquando accidit, dicemus cum de grauitate, & grauitatione, iam verò &longs;atis e&longs;t ad præ&longs;ens in&longs;titutum impetum innatum ab ip&longs;o corport graui nunquam &longs;eparari, quandiu remanet graue.

Impetus naturalis acqui&longs;itus producitur ab codem principio intrin­&longs;eco; hinc dicitur naturalis: dicitur verò acqui&longs;itus, quia non e&longs;t inna­tus; &longs;ed &longs;eparatur à corpore graui; quod &longs;emper co caret, quandiu quie&longs;cit: &longs;ed innato tantùm accedit ad motus accelerationem, & ad alia quamplurima, quæ ex ea &longs;equuntur; putà maiorem percu&longs;&longs;ionem, re&longs;i­&longs;tentiam, vim, & ad tollendum totius naturæ languiorem; quo certè af­ficeretur, &longs;i corpus grauc tardi&longs;&longs;imo motu deor&longs;um ferretur, de quo in­frà; Porrò impetus acqui&longs;itus in multis differt ab innato; primò quia de&longs;truitur à corpore re&longs;i&longs;tente co modo, quo diximus, & dicemus infrà. Secundò, quia determinari pote&longs;t ad omnem lincam.

Impetus violentus e&longs;t, qui e&longs;t ab extrin&longs;eco, de quo agemus infrà, & iam &longs;uprà in lib.1. multa &longs;unt de eo demon&longs;trata.

Theorema 15.

Impetus naturalis corporis grauis intenditur dum hoc ip&longs;um de&longs;cendit in medio libero; demon&longs;tratur, Impetus nouus producitur in &longs;ecundo, ter­tio, quarto, &c. in&longs;tantibus per Th.12. &longs;ed productus in primo con&longs;er­uatur &longs;ecundo, per Th.9. productus &longs;ecundo con&longs;eruatur tertio, produ­ctus tertio con&longs;eruatur quarto per Th.13. igitur &longs;ecundus additur tertio, tertius primo, &longs;ecundo, quartus primo, &longs;ecundo, & tertio, &c.&longs;ed impetus additus alteri facit inten&longs;iorem impetum per Ax.1. igitur impetus natu­ralis intenditur, quod crat demon&longs;trandum.

Theorema 16.

Hinc motus naturalis deor&longs;um acceleratur; hoc ip&longs;um &longs;uppo&longs;ui &longs;uprà Quod e&longs;&longs;et in hyp.1.2.3. iam verò demon&longs;tro propter quid e&longs;t; &longs;ie cnim hypothe&longs;is in Theorema conuerti pote&longs;t, vt fæpè monuimus in metho­do; igitur probatur hoc Theorema facilè; cre&longs;cit impetus in corpore gra­ui, quod tendit deor&longs;um in libero medio per T. 15. igitar cre&longs;cit cau&longs;a motus; nam impetus e&longs;t can&longs;a immediata motus naturalis per Th. 51. &longs;ed quâ proportione cre&longs;cit cau&longs;a, debet cre&longs;cere effectus per Ax.2. igi­tur motus naturalis deor&longs;um cre&longs;cit, id e&longs;t acceleratur, id e&longs;t fit velo­cior, quod erat dem: nec e&longs;t quod aliquis exi&longs;timet hic à me committi vitio&longs;um argumentationis circulum; quippe probaui &longs;uprà cre&longs;cere im­petum, quia cre&longs;cit motus; iam verò probo cre&longs;cere motum, quia cre&longs;­cit impetus; nam primò probaui produci nouum impetum in Th.12. co quod &longs;ecundo in&longs;tanti. v.g. &longs;it cadem cau&longs;a nece&longs;&longs;aria applicata non im­pedita, igitur tàm debet agere &longs;ecundo quàm primo in&longs;tanti, hæc fuit mea probatio à priori; &longs;ecundò verò probaui ex hypothe&longs;i certa; quia &longs;cilicet cre&longs;cit motus, cuius veritatem cogno&longs;co &longs;en&longs;ibiliter in &longs;e, vnde &longs;uppono tantùm de illa quod &longs;it; igitur nullus committitur circulus; nam diuer&longs;a e&longs;t omninò cognitio. Prima &longs;cilicet qua cogno&longs;co de motu na­turaliter accelerato quod &longs;it, quæ mihi, & ru&longs;tico communis e&longs;t. Secun­da verò qua non modò cogno&longs;co de motu illo quod &longs;it acceleratus, ve­rùm propter quid &longs;it acceleratus, id e&longs;t cau&longs;am huius accelerationis, id e&longs;t propter quam attributum hoc ine&longs;t &longs;ubiecto, & hæc e&longs;t vera demon­&longs;tratio à priori; porrò in Phy&longs;ica de effectu &longs;en&longs;ibili &longs;upponi debet quod &longs;it, hoc enim percipitur &longs;en&longs;u. v. g. &longs;upponam in Phy&longs;ica quod &longs;it motus acceleratus, quod ignis &longs;it calidus, Sol lucidus, nix candida, vinum ru­brum, &c. at verò demon&longs;trabo propter quid hæc &longs;int, &longs;ed de his &longs;atis.

Scholium.

Ob&longs;eruabis etiam aliud naturæ in&longs;titutum, quo &longs;cilicet factum e&longs;t, vt corpora grauia motu naturali accelerato deor&longs;um ferantur; &longs;i enim motu ferrentur æquabili, vel e&longs;&longs;et æqualis illi quem initio &longs;ui de&longs;cen&longs;us ha­bent, qui e&longs;t tardi&longs;&longs;imus, vt con&longs;tat ex ip&longs;a ictuum differentia; atque ita infinitum ferè tempus ponerent grauia in minimo etiam de&longs;cen&longs;u, quod e&longs;&longs;et maximè incommodum; &longs;i verò motus ille e&longs;&longs;et æqualis mo­tui v.g. quem acqui&longs;iuit in fpatio 3. vel 4. perticarum, pondera corpo­rum cre&longs;cerent in immen&longs;um, ide&longs;t in ea proportione, qua ictus, qui in­fligitur à corpore graui confecto 4. perticarum &longs;patio maior e&longs;t ictu, qui infligitur po&longs;t decur&longs;um minimum omnium &longs;patiorum, quod valdè in­commodum e&longs;&longs;et.

Theorema 17.

Æqualibus temporibus æqualis impetus producitur, &longs;i &longs;it eadem applica­tio, idemque impedimentum; probatur, quia cau&longs;a huius impetus e&longs;t ne­ce&longs;&longs;aria; &longs;ed cadem cau&longs;a nece&longs;&longs;aria æqualibus temporibus æqualem impetum producit per Ax.3.

Theorema 18.

Qua proportione cre&longs;cit impetus acceleratur motus; quia quæ proportio­ne cre&longs;cit cau&longs;a, ctiam cre&longs;cit effectus per Ax.2.

Theorema 19.

Hinc æqualibus temporibus in de&longs;cen&longs;u corpus graue acquirit aqualia ve­locitatis, vel acce&longs;erationis momenta; hoc ip&longs;um e&longs;t quod definitionis lo­co Galileus in dialogo tertio de motu naturali a&longs;&longs;umit; quod tamen meo iudicio fuit antè demon&longs;trandum quàm &longs;upponendum; quare &longs;ic demon&longs;tramus, quâ proportione cre&longs;cit impetus, cre&longs;cit motus per Th. 18. &longs;ed temporibus æqualibus acquiruntur æquales impetus gradus per Th.17. igitur æqualia velocitatis momenta, vel incrementa.

Theorema 20.

Spatia que per curruntur motu aquabili æqualibus temporibus &longs;unt æqualia; Probatur per Def.2.

Theorema 21.

Duo motus æquabiles, qui durant æqualibus temporibus, &longs;unt vt &longs;patia; patet; cùm enim impetus &longs;int vt motus per Ax. 2. motus &longs;unt vt &longs;patia; quippe vt ex impetu &longs;equitur motus, ita ex motu confectum &longs;pa­tium.

Theorema 22.

Duo motus æquabiles, quibus percurruntur &longs;patia æqualia &longs;unt vt tempora permutande;, patet, quia velocior e&longs;t, quò percurritur &longs;patium æquale minori tempore per Def.2. l. 1. Igitur eò veiocior, quò minori tem­pore.

Theorema 23.

Spatium, quod percurritur maiori tempore motu æquabili, est maius eo, quod percurritur minori æquè veloci motu in ea ratione, qua vnum tempusest maius alio; patet, quia æqualia &longs;unt æqualibus temporibus per Th. 20. igitur inæqualibus inæqualia iuxta rationem temporum; item &longs;pa­tium, quod idem percurritur minori tempore minus e&longs;t.

Theorema 24.

Tempus quo maius &longs;patium percurritur eodem motu æquabili, e&longs;t maius eò quò minus conficitur iuxta rationem &longs;patiorum: Si enim &longs;patia &longs;unt vt tem­pora, igitur tempora &longs;unt vt &longs;patia; item tempus, quo minus &longs;patium percurritur e&longs;t minus co, quo maius.

Theorema 25.

Spatium, quod conficitur motu velociore, e&longs;t maius eo, quod percur­ritur æquali certè tempore, &longs;ed tardiore motu, vt con&longs;tat per def. 2. l. 1. imò e&longs;t maius iuxta rationem velocitatis maioris, item e&longs;t minus iuxta rationem tarditatis maioris.

Theorema 26.

Tempus, quo conficitur &longs;patium æquale &longs;ed uelociore motu, est minus eo quo conficitur tardiore; Probatur per def.2. & per Th.22. idque in ratio­ne velocitatum permutando; item tempus quo conficitur &longs;patium æqua­le tardiore motu e&longs;t maius eo, quo conficitur velociore, patet.

Theorema 27.

Si datum mobile eodem motu æquabili duo percurrat &longs;patia, tempora mo­tuum erunt vt &longs;patia, & vici&longs;&longs;im &longs;patia vt tempora. Probatur per Th. 24. & 23.

Theorema 28.

Si idem mobile temporibus æqualibus pereurrat duo &longs;patia motu æquabili, &longs;ed inæquali velocitate; &longs;patia erunt vt velocitates, & hæ vt illa; imò &longs;i &longs;patia &longs;unt vt velocitates, tempora erunt æqualia; pater etiam per Th.25.

Theorema 29.

Si percurrantnr à mobili æqualia &longs;patia, &longs;ed inæquali velocitate, ip&longs;æ ve­locitates erunt in ratione permutata temporum, ide&longs;t maior velocitas re&longs;pon­debit minori tempori, & minor maiori; Probatur per Th.23.

Theorema 30.

Si duo mobilia mouentur motu æquabili, &longs;ed inæquali velocitate, & libus temporibus, &longs;patia &longs;unt in ratione compo&longs;ita ex ratione temporum, & ex ratione velocitatum, &longs;i enim æqualia &longs;int tempora, &longs;patia erunt vt velo­citates per Th.25. &longs;i æquales &longs;int velocitates, &longs;patia crunt vt tempora, per Th.29. igitur &longs;i nec æquales velocitates, nec æqualia tempora, erit ratio &longs;patiorum compo&longs;ita ex ratione temporum, & ex ratione velocitatum; &longs;it ratio temporum 3/2 ratio velocitatum 2/3 compo&longs;ita ex vtraque crit 6/2 &longs;eu 3. vt con&longs;tat ex ip&longs;is elementis.

Theorema 31.

Si duo mobilia ferantur motu æquabili per diuer&longs;a &longs;patia, & diuer&longs;a velo­eitate, tempora erunt in ratione compo&longs;ita ex ratione &longs;paliorum & ratioue velocitatum permutata; probatur eodem modo quo &longs;uperius Th. 30. &longs;it ratio &longs;patiorum 4/1, velocitatum 4/2; permutetur hæc 1/4; componetur ex vtraque 4/1, ide&longs;t 1/2, quæ e&longs;t ratio temporum.

Theorema 32.

Si duo mobilia æquabili motu ferantur per diuer&longs;a &longs;patia, & inæqualibus temporibus; ratio velocitatum erit compo&longs;ita ex ratione &longs;patiorum, & ex ra­tione temporum permutata; Probatur eodem modo; &longs;it ratio &longs;patiorum 4/2 temporum 1/2, permutetur 2/1, compo&longs;ita ex vtraque crit 2/2, ide&longs;t 4. quæ e&longs;t ratio velocitatum.

Scholium.

Ob&longs;eruabis hæc omnia à vige&longs;imo Theoremate maiori ex parte tradi à Galileo &longs;uo modo, optimo quidem, &longs;ed fortè longiore quàm par &longs;it, nulla habita ratione cau&longs;arum phy&longs;icarum.

Theorema 33.

In motu naturaliter accelerato impetus nouus acquiritur &longs;ingulis in&longs;tanti­bus; Probatur quia &longs;ingulis in&longs;tantibus e&longs;t eadem cau&longs;a nece&longs;&longs;aria, igi­tur &longs;ingulis in&longs;tantibus aliquem effectum producit, per Ax. 12. l.1. &longs;ed priorem non con&longs;eruat, vt dictum e&longs;t &longs;uprà, igitur nouum producit.

Theorema 34.

Hinc &longs;ingulis in&longs;tantibus æqualibus nouus impetus æqualis acquiritur, quip­pe e&longs;t æqualis, imò cadem cau&longs;a, igitur æqualem effectum producit per Ax.12. l.1.

Theorema 35.

Hinc &longs;ingulis in&longs;tantibus intenditur impetus in hoc motu; cum &longs;ingulis in&longs;tantibus producatur nouus, & prior con&longs;eruetur, cui cum addatur, intenditur per Ax. 1.

Theorema 36.

Hinc &longs;ingalis in&longs;tantibus æqualiter cre&longs;cit & intenditur impetus per Th. 34. igitur æqualiter etiam &longs;ingulis in&longs;tantibus cre&longs;cit velocitas motus per Ax.2.

Scholium

Ob&longs;eruabis dictum e&longs;&longs;e &longs;uprà instantibus æqualibus, quia temporis natura aliter explicari non pote&longs;t, quàm per in&longs;tantia finita, vt demon&longs;trabimus in Metaphy&longs;ica; quid quid &longs;it, voco in&longs;tans totum illud tempus, quo res aliqua &longs;imul producitur, &longs;iue &longs;it maius, &longs;iue minus, &longs;iue &longs;it pars maior, vel minor, quod ad rem no&longs;tram nihil facit penitus; nam dato quocun­que tempore finito pote&longs;t dari maius & minus, quod certum e&longs;t; igitur totum illud tempus, quo producitur primus impetus acqui&longs;itus, vo-co in&longs;tans primum motus; cui æqualia deinde &longs;uccedunt tem­pora.

Theorema 37.

Hinc cre&longs;cit impetus iuxta progre&longs;&longs;ionem arithmeticam; cum &longs;ingula in­&longs;tantia æqualem impetum addant; &longs;i primo in&longs;tanti &longs;it vnus gradus, crunt duo; productus &longs;cilicet alteri additus qui con&longs;eruatur, tertio erunt;. quarto 4. quinto 5. &c. igitur cre&longs;cit &longs;ecundum progre&longs;&longs;ionem arith­meticam.

Theorema 38.

Eodem modo cre&longs;cit velocitas, quia &longs;ingulis in&longs;tantibus æqualia acquirun­tur velocitatis momenta per Ax.2. & per Th.36.

Theorema 39.

Maius &longs;patium acquiritur &longs;ecundo in&longs;tanti, quàm primo, quia &longs;ecundoin&longs;tanti motus e&longs;t velocior per Th.36. igitur maius conficitur &longs;patium, tempore &longs;cilicet æquali per Def. 2. l. 1. idem dico de tertio, quar­to, &c.

Theorema 40.

Spatium quod acquiritur &longs;ecundò instanti e&longs;t ad &longs;patium quod acquiritur primo vt velocitas, quæ e&longs;t &longs;ecundo ad velocitatem, quæ e&longs;t primo. Patet per Th.28. quia cum tempora illa &longs;int æqualia, &longs;patia &longs;unt nece&longs;&longs;ariò vt ve­locitates; quippe æquali velocitati æquale &longs;patium re&longs;pondet tempore æquali, igitur inæquale inæquali, igitur maius maiori, idem dico de aliis in&longs;tantibus.

Theorema 41.

Hinc &longs;patium qucd acquiritur &longs;ecundo in&longs;tanti e&longs;t duplum illius, qaod ac­quiritur primo. Probatur, quia velocitas e&longs;t dupla per Th 38. igitur &longs;pa­tium duplum, & triplum tertio, quadruplum quarto, &c.

Theorema 42.

Hinc quodlibet &longs;patium cre&longs;cit æqualiter &longs;ingulis in&longs;tantibus æqualibus; quia &longs;patia cre&longs;cunt vt motus, &longs;eu vt velocitates; hæ cre&longs;cunt æqualiter &longs;ingulis in&longs;tantibus æqualibus per Th.36. igitur æqualiter cre&longs;cunt &longs;in­gula &longs;patia per Th.40.

Theorema 43.

Hinc &longs;patia cre&longs;cunt &longs;ingulis in&longs;tantibus æqualibus &longs;ecundùm pregre&longs;&longs;io­nem arithmeticam; quia cre&longs;cit vt velocitas per Th.40. hæc vt impetus per Th.38. hic demum iuxta progre&longs;&longs;ionem arithmeticam per Th. 37. igitur &longs;i &longs;patium acqui&longs;itum primo in&longs;tanti &longs;it 1. acqui&longs;itum &longs;ecundo crit 2. tertio 3. quarto 4. &c. hinc &longs;patia acqui&longs;ita &longs;ingulis in&longs;tantibus &longs;unt vt &longs;eries numerorm, qui componunt progre&longs;&longs;ionem &longs;implicem, &longs;cilicet 1.2.3.4.5.6. &c. dixi &longs;ingulis in&longs;tantibus æqualibus, quod e&longs;t apprimè tenendum; &longs;i enim a&longs;&longs;umantur partes temporis maiores, perturbatur hæc progre&longs;&longs;io, de quo infrà.

Theorema 44.

Hinc pete&longs;t dici cre&longs;cere velocitatem quolibet in&longs;tanti iuxta rationem &longs;patij quod illo in&longs;tanti decurritur; quod certè verum e&longs;t, dum intelligatur legi­timus horum verborum &longs;en&longs;us; quidquid reclamet Saluiatus apud Galil. dialogo 3. modò a&longs;&longs;umatur progre&longs;&longs;io incrementi in &longs;ingulis in­&longs;tantibus, in quibus reuerà fit; cur enim potiùs in vno quàm in alio? quippe &longs;i comparetur velocitas vnius in&longs;tantis cum velocitate alterius; haud dubiè erit eadem vtriu&longs;que ratio, quæ &longs;patiorum; &longs;i enim vno in­&longs;tanti percurritur vnum &longs;patium cum vno velocitatis gradu; certè in­&longs;tanti æquali acquiritur duplum &longs;patium cum duobus velocitatis gradi­bus, nec obe&longs;t, quod obiicit Galileus tunc motus e&longs;&longs;e æquabiles; quia motus qui fit in in&longs;tanti debet con&longs;iderari vt æquabilis; appello enim in&longs;tans totum illud tempus, quo &longs;imul acquiritur aliquid impetus, ali­quid enim &longs;imul acquiri nece&longs;&longs;e e&longs;t; nec demum ob&longs;tat quod dicit, dari non po&longs;&longs;e motum in&longs;tantaneum, quod multi haud dubiè negabunt; ego in Metaphy&longs;ica explicabo quonam pacto dari po&longs;&longs;it motus in&longs;tanta­neus, qui reuerà datur actu, non potentiâ; quia quacunque duratione data pote&longs;t dari minor; igitur quocunque dato motu pote&longs;t dari minor.

Scholium.

Ob&longs;eruabis primò hanc &longs;patiorum rationem, quæ e&longs;t eadem cum ra­tione velocitatum a&longs;&longs;umendam tantùm e&longs;&longs;e in iis &longs;patiis, quæ acquirun­tur &longs;ingulis in&longs;tantibus; &longs;i enim accipiantur partes temporis maiores, quæ conflentur ex multis in&longs;tantibus; haud dubiè maior erit ratio &longs;patio­rum, quàm velocitatum.v.g.&longs;i primo in&longs;tanti acquiratur primo &longs;patium, &longs;ecundo, 2.tertio, 3.quarto 4.igitur &longs;i comparetur velocitas primi in&longs;tantis cum velocitate quarti æqualis erit, vt ratio &longs;patiorum, id e&longs;t, vt 1. ad 4. At verò &longs;i accipiatur pars temporis con&longs;tans duobus in&longs;tantibus, hæc 4. in&longs;tantia conflabunt tantùm 2. partes temporis æquales; in prima ac­quirentur 3.&longs;patia, in &longs;ecunda 7.vt patet: &longs;ed quia velocitas primæ par­tis temporis non e&longs;t æquabilis, nec etiam velocitas &longs;ecundæ; addantur velocitates primi & &longs;ecundi in&longs;tantis, itemque &longs;eor&longs;im velocitates tertij, & quarti; certè ratio collectorum crit vt ratio &longs;patiorum; &longs;i enim velo­citas &longs;ecundi in&longs;tantis comparetur cum velocitate quarti e&longs;t tantùm 1/2 cum tamen primum &longs;patium &longs;it ad &longs;ecundum in ratione 3/7.

Secundò, &longs;i comparentur &longs;patia cum temporibus e&longs;t alia ratio v.g.&longs;pa­tium acqui&longs;itum vno in&longs;tanti &longs;e habet ad &longs;patium acqui&longs;itum in duobus in&longs;tantibus, vt 1, ad 3.in tribus vt 1.ad 6.in 4. vt 1. ad 10.

Tertiò ob&longs;eruabis, non po&longs;&longs;e &longs;en&longs;u percipi in&longs;tans, imò neque tempo­ris partem ex mille in&longs;tantibus conflatam; nec etiam &longs;patium quod ac­quiritur primo in&longs;tanti; adhibenda &longs;unt tamen in&longs;tantia nece&longs;&longs;ariò ad explicandam proportionem huius accelerationis, quæ fit in &longs;ingulis in­&longs;tantibus; vt verò rem i&longs;tam reuocemus ad &longs;en&longs;ibilem praxim, a&longs;&longs;ume­mus proportionem aliam &longs;en&longs;ibilem, quæ proximè ad veram accedit, nec ferè &longs;en&longs;ibiliter fallere pote&longs;t, de qua infrà.

Theorema 40.

Collectio &longs;patiorum e&longs;t &longs;umma terminorum huius progre&longs;&longs;ionis arithmeticæ; Gùm enim ratio &longs;patiorum &longs;it vt ratio velocitatum; dum &longs;cilicet hæc progre&longs;&longs;io accipitur in in&longs;tantibus, & ratio velocitatum vt ratio incre­menti impetuum; vt con&longs;tat ex dictis, & hæc &longs;equatur &longs;implicem progre&longs;&longs;ionem 1. 2. 3. 4. &c. certè collectio &longs;patiorum e&longs;t &longs;umma ter­minorum.

Theorema 41.

Hinc cognito primo termino, & vltimo, id e&longs;t &longs;patio quod per curritur primo in&longs;tanti & &longs;patio quod percurritur vltimo instanti, cogno&longs;citur &longs;umma, id e&longs;t collectio &longs;patiorum, id e&longs;t, totum &longs;patium confectum. v.g.&longs;i primus terminus, &longs;ecundus S.igitur &longs;umma e&longs;t 36. quippe vltimus terminus indicat nume­rum terminorum, quia primus e&longs;t &longs;emper vnitas, & progre&longs;&longs;iuus etiam vnitas.

Theorema 42.

Hinc cognita &longs;umma & vltimo termino cogno&longs;citur etiam numerus in&longs;tan­tium æqualium, qui &longs;emper est idem cum numero terminorum, cogno&longs;citur etiam primus terminus, id e&longs;t &longs;patium quod primo instanti percurritur, cogno­&longs;cuntur etiam gradus velocitatis; quippe hæc omnia &longs;unt in eadem ratio­ne; quæ omnia con&longs;tant ex regulis arithmeticis præter alia multa data, quæ lubens omitto; tùm quia Phy&longs;icam non &longs;apiunt, tùm quia hypothe­&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&longs;tinguere vnum temporis in&longs;tans, vel &longs;patij punctum? licèt recen&longs;enda fuerit hæc accelerati motus proportio in in&longs;tantibus, vt ad &longs;ua phy&longs;ica principia reduceretur.

Theorema 43.

Data &longs;umma progre&longs;&longs;ionis huius &longs;implicis, inuenietur numerus terminorum, &longs;i inueniatur numerus, per quem diuidatur, qui &longs;uperet tantùm vnitate du­plum quotientis; quippe habebis in duplo quotientis numerum termino­rum v.g. &longs;it &longs;umma 10. diui&longs;or &longs;it 5. quotiens 2. duplus 4. hic e&longs;t nume­rus terminorum datæ &longs;ummæ; &longs;it alia &longs;umma 21. diui&longs;or &longs;it 7.quotiens 3. numerus terminorum 6. &longs;it alia &longs;umma 36. dini&longs;or &longs;it 9. quotiens 4. nu­merus terminorum 8. &longs;it alia &longs;umma 45. partitor &longs;it 10. quotiens 4 1/2, numerus terminorum 9. quomodo verò hic partitor inueniri po&longs;&longs;it, vi­derint Arithinetici; nec enim e&longs;t huius loci, quamquam datâ &longs;ummâ huius progre&longs;&longs;ionis &longs;implicis facilè cogno&longs;ci pote&longs;t numerus termino­rum; duplicetur enim, & radix 9. neglecto re&longs;iduo dabit numerum ter­minorum v.g. &longs;it &longs;umma 21. duplicetur, erit 42. rad. 9. 6. dat numerum terminorum; &longs;it &longs;umma 36. duplicetur, erit 72.rad.9.8. dabit numerum terminorum.

Theorema 44.

Semper decre&longs;cit proportio incrementi velocitatis, id est maior est proportio velocitatis &longs;ecundi in&longs;tantis ad primum quàm tertij ad &longs;ecundum, & maiortertij ad &longs;ecundum quàm quarti ad tertium, atque ita deinceps; &longs;it enim primo in&longs;tanti velocitas vt 1.&longs;ecundo erit, vt 2.tertio, vt 3.quarto, vt 4. &longs;ed maior e&longs;t proportio 2.ad 1.quàm 3.ad 2. & hæc maior quàm 4. ad 3. atque ita deinceps; &longs;imiliter maior e&longs;t proportio &longs;patij quod percurritur &longs;ecundo in&longs;tanti ad &longs;patium, quod percurritur primo, quàm &longs;patij, quod percurritur &longs;ecundo in&longs;tanti ad &longs;patium, quod percurtitur primo quàm &longs;patij quod percurritur tertio ad &longs;patium, quod percurritur &longs;ecundo, at­que ita deinceps; e&longs;t enim eadem ratio &longs;patiorum quæ &longs;ingulis in&longs;tanti­bus re&longs;pondent, quæ velocitatum, vt demon&longs;tratum e&longs;t &longs;uprà.

Theorema 45.

Minor e&longs;t proportio totius &longs;patij, quod acquiritur duobus instantibus ad to, tum &longs;patium, quod acquiritur vno, quàm &longs;it illius, quod acquiritur quatuor in­&longs;tantibus ad aliud, quod acquiritur duobus; patet-ex dictis; &longs;i enim primo in&longs;tanti acquiritur vnum &longs;patium, &longs;ecundo acquiruntur 2.igitur duobus &longs;imul acquirantur 3. igitur proportio e&longs;t vt 3.ad 1.Sed &longs;i duobus acqui­runtur 3. &longs;patia; certè 4.in&longs;tantibus acquiruntur 10. igitur proportio e&longs;t vt 10.ad 3. &longs;ed proportio 10/3 e&longs;t maior 3/1, erit adhuc maior proportio &longs;pa­tij quod acquiretur 6. in&longs;tantibus ad illud quod acquiritur tribus; e&longs;t enim (21/6) vt patet.

Theorema 46.

Si componatur æquabilis motus ex &longs;ubdupla velocitate maxima, & mini­ma, æquali tempore, idem &longs;patium percurretur hoc motu naturaliter aceclera­to; &longs;it enim maxima velocitas vt 6. minima vt 1. motu naturaliter acce­lerato percurrentur &longs;patia 21. cuius &longs;ummæ termini &longs;unt 6.igitur 6. in­&longs;tantibus con&longs;tat hic motus; accipiatur &longs;ubduplum maximæ, & minimæ velocitatis, &longs;cilicet 3 1/2. sítque velocitas motus æquabilis in&longs;tantium 6. haud dubiè &longs;i ducantur 3 1/2 in 6 erunt 21.ratio ex eo petitur quod &longs;cili­cet, vt habeatur &longs;umma progre&longs;&longs;ionis arithmeticæ, debet addi primus terminus maximo, & a&longs;&longs;umi &longs;ubduplum totius; illudque ducere in nu­merum terminorum per regulam arithmeticam; atqui eadom e&longs;t ratio velocitatum, quæ &longs;patiorum; vt dictum e&longs;t &longs;uprà; &longs;cilice, in &longs;ingulis in&longs;tantibus.

Theorema 47.

Si a&longs;&longs;umantur partes temporis majores; quæ &longs;cilicet pluribus in&longs;tantibus constent, &longs;erueturque eadem accelerationis progre&longs;&longs;io arithinetica, &longs;patium quod ex &longs;umma huius progre&longs;&longs;ionis re&longs;ultabit, erit minus vero, &longs;int enim 6.in­&longs;tantia, & cuilibet iuxta progre&longs;&longs;ionem prædictam &longs;uum &longs;patium re&longs;pon­deat, haud dubiè &longs;patium &longs;ecundi erit duplum &longs;patij primi, & tertium triplum, &c. vt con&longs;tat ex dictis; igitur erunt &longs;patia 21. iam verò a&longs;&longs;u­mantur 3. partes temporis, quarum quælibet ex 2. con&longs;tet in&longs;tantibus; primæ parti tria ex prædictis &longs;patiis re&longs;pondeant; certè &longs;i &longs;eruetur pro­gre&longs;&longs;io arithmetica, &longs;ecundæ re&longs;pondebunt 6. & tertiæ 9. igitur totum &longs;patium erit 18. minus vero quod erat 21. &longs;i verò a&longs;&longs;umantur tantùm 2. partes, quarum quælibet tribus in&longs;tantibus con&longs;tet; primæ parti re&longs;pon-debunt 6. &longs;ecundæ 12. igitur &longs;umma erit 18. minor vero &longs;patio &longs;cilicet 21.hinc vides &longs;uppo&longs;ito eodem in&longs;tantium numero &longs;patium e&longs;&longs;e &longs;empet æquale, &longs;iue a&longs;&longs;umantur partes maiores temporis, &longs;iuc minores, v. g. &longs;up­po&longs;itis 6.in&longs;tantibus, ex quibus totum &longs;patium 21.con&longs;equitur, &longs;iue a&longs;&longs;u­mantur tres partes, quarum quælibet con&longs;tet 2. in&longs;tantibus, &longs;iue duæ, quarum quælibet con&longs;tet tribus, &longs;patium quod ex illis re&longs;ultat, e&longs;t &longs;em­per idem &longs;cilicet 18. a&longs;&longs;umptis verò 8. in&longs;tantibus, & totali &longs;patio, quod illis re&longs;pondet 36. &longs;patium quod ex partibus re&longs;ultabit erit 30. &longs;iue &longs;int duæ partes, quarum quælibet con&longs;tet 4. in&longs;tantibus, &longs;iue &longs;int 4. quarum quælibet con&longs;tet duobus: hinc rur&longs;us vides a&longs;&longs;umpto maiori in&longs;tantium numero &longs;patium verum habere maiorem rationem ad non verum, quàm a&longs;&longs;umpto minori in&longs;tantium numero, v.g.a&longs;&longs;umantur 4.in&longs;tantia, &longs;umma &longs;patiorum erit 10. &longs;i verò a&longs;&longs;umantur 2.partes temporis, quarum quæli­bet duobus in&longs;tantibus re&longs;pondeat; &longs;umma &longs;patij erit 9.igitur ratio ve­ri &longs;patij ad non verum e&longs;t (10/9). a&longs;&longs;umantur 6. in&longs;tantia &longs;patij veri, &longs;umma erit 21.non veri 18. igitur ratio (21/18) &longs;eu 7/6 quæ maior e&longs;t priori: denique a&longs;&longs;umantur 8. in&longs;tantia &longs;patij veri, &longs;umma erit 36. non veri 30 igitur ra­tio (36/30) &longs;eu 6/3 quæ maior e&longs;t prioribus, atque ita deinceps.

Theorema 48.

Datis duabus partibus temporis, & cognito &longs;patio quod percurritur in prima, matius &longs;patium re&longs;pondebit &longs;ecundæ quo vtraque in plures partes minores diui­detur, &longs;uppofita &longs;emper eadem proare&longs;&longs;ione arithmetica in ip&longs;o incremento; &longs;int enim duæ partes temporis &longs;en&longs;ibiles æquales AG. GH. & &longs;pa­tium quod percurritur prima parte temporis AG &longs;it HI; in &longs;ecunda percurretur IO, id e&longs;t, duplum HI; at verò diuidatur pars temporis AG in duas æquales AF, FG, & con&longs;equenter totum tempus AH in 4. æquales; haud dubiè in prima AF percurretur NP &longs;ubtripla HI, & in &longs;ecunda FG percurretur PK dupla NP; igitur in 4. partibus temporis AH percurretur &longs;patium decuplum PN, &longs;ed HO e&longs;t tantùm nouecupla NP; igitur re&longs;ultabit maius &longs;patium in 4.partibus temporis, quam in dua­bus; licèt duæ æquiualeant 4. iuxta progre&longs;&longs;ionem arithmeticam.

Similiter AF diuidatur bifariam in E. & tota AH in 8. æquales AE; certè primis 4.percurretur idem &longs;patium ML æquale NK & HI; igitur in prima AE percurretur MR. cuius ML &longs;it decupla; nam 4. terminis re&longs;pondet &longs;umma 10. &longs;ed 8. terminis id e&longs;t 8.partibus temporis re&longs;pon­det &longs;umma; 6. æqualium RM; &longs;ed HO tripla ML e&longs;t tantum 30. æqualium MR; igitur in 8.partibus re&longs;ultabit maius &longs;patium, quàm in 4.quæ æquiualent 8.

Ex quibus etiam con&longs;tat quo plures accipientur partes temporis ma­ius &longs;patium re&longs;ultare, donec tandem perueniatur ad vltima in&longs;tantia, ex quibus re&longs;ultat maximum; & &longs;i accipias AG partes temporis AG. GH. habebitur HO; &longs;i verò 4.æquales AF, cre&longs;cet &longs;patium &longs;eu &longs;umma 1/9 HO; &longs;i autem 8. æquales AE cre&longs;cet 1/5 HO; &longs;i porrò 16. æquales AD cre&longs;­cet (22/108) &longs;i 32. æquales AC cre&longs;cet (120/408); &longs;i 64. æquales AB cre&longs;cet (496/1584).

Theorema 49.

In progre&longs;&longs;ione arithmetica &longs;i diuidatur numerus terminorum bifariam æ­qualiter nunquam &longs;umma po&longs;terioris &longs;egmenti e&longs;t tripla prioris; &longs;ed &longs;i acci­piantur 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(0/25), &longs;i 12. (57/21), &longs;i 14. (77/28), atque ita deinceps.

Ex quo ob&longs;erua mirabilem con&longs;equutionem; quippe &longs;i a&longs;&longs;umantur tantùm duo termini, & diuidantur bifariam, &longs;umma po&longs;terioris medie­tatis e&longs;t tripla primæ minùs vnitate; &longs;i accipiantur 4. e&longs;t tripla minùs 2. &longs;i 6. minùs 3. &longs;i 8. minùs 4. &longs;i 10. minùs 5. &longs;i 12. minùs 6. &longs;i 14. mi­nùs 7. atque ita deinceps; vnde &longs;umma po&longs;terioris medietatis e&longs;t &longs;emper tripla minùs numero &longs;uorum terminorum, vel quod clarum e&longs;t minùs &longs;ubduplo vltimi, &longs;eu maximi termini, vel numeri terminorum totius progre&longs;&longs;ionis, quod probè omninò tenendum e&longs;t, vt omnes experientiæ explica ri po&longs;&longs;int, quod infrà faciemus.

Theorema 50.

Ex dictis hactenus facilè redditur ratio maioris ictus eiu&longs;dem corporis im­pacti quod cadit ex maiori altitudine; fuit hyp. 1. &longs;ed ideò e&longs;t maior ictus, quia maior imprimitur impetus, vt patet, at ideò maior impetus impri­mitur, quia maior e&longs;t imprimens per Ax. 2. cre&longs;cit enim impetus, vt con&longs;tat ex dictis.

Theorema 51.

Hinc quoque ratio maximæ percu&longs;&longs;ionis ex &longs;olo pondere cadentis illius arie­tis inflictæ; quâ &longs;cilicet altè infiguntur lignei pali, quibus in mediis aquis tanquam iacto &longs;undamini &longs;uperædificatur ingens &longs;æpè ædificij moles.

Theorema 52.

Hinc ex minima altitudine cadens corpus graue minimum ferè ictum in­fligit; quia primus impetus valdè debilis e&longs;t, qui tamen deinde &longs;acta acce&longs;&longs;ione maximus ferè euadit.

Theorema 53.

Hinc ratio, cur tanta &longs;it differentia impetus grauit ationis, & percu&longs;&longs;ionis ab eodem mobili; quia &longs;cilicet quantumuis tempore breui&longs;&longs;imo mouea­tur, plurimis tamen cius motus durat in&longs;tantibus; atqui quolibet in&longs;tan­ti motus acquiritur impetus æqualis primo impetui grauitationis, vt con&longs;tat ex dictis. v. g. &longs;it mobile quod moueatur per mille in&longs;tantia (modicum certè tempus & minimè &longs;en&longs;ibile) po&longs;t hunc motum impetus erit millecuplus; igitur effectus etiam millecuplus; quæ omnia con&longs;tant ex dictis.

Theorema 54.

Hinc percu&longs;&longs;io quæ fit in primo in&longs;tanti contactus cre&longs;cit vt tempus; quia cùm &longs;ingulis in&longs;tantibus cre&longs;cat impetus per partes æquales, & cùm per­cu&longs;&longs;io &longs;it vt impetus; etiam erit vt tempus; igitur percu&longs;&longs;io, quæ fit po&longs;t duo in&longs;tantia motus eiu&longs;dem corporis grauis deor&longs;um cadentis e&longs;t du-plaillius, quæ &longs;it po&longs;t vnum in&longs;tans motus, & quæ fit po&longs;t tria tripla, po&longs;t 4. quadrupla, atque ita deinceps; cùm enim æqualibus temporibus æqua­lia acquirantur velocitatis momenta, id e&longs;t æquales impetus, impetus erunt vt tempora, percu&longs;&longs;iones vt impetus, igitur percu&longs;&longs;iones vt tem­pora.

Dixi in primo in&longs;tanti contactus; nam reuerâ &longs;ecundò in&longs;tanti con­tactus, ni&longs;i fiat reflexio, augetur vis ictus, quia cau&longs;a nece&longs;&longs;aria e&longs;t ap­plicata.

Theorema 55.

Hinc po&longs;&longs;unt comparari duæ percu&longs;&longs;iones duorum grauium inæqualium dum cadunt deor&longs;um; &longs;i enim cadunt æqualibus temporibus, percu&longs;&longs;io­nes erunt vt corpora &longs;eu grauitates, vt patet v.g. corpus 2. librarum po&longs;t 2. in&longs;tantia motus infligit duplam percu&longs;&longs;ionem illius, quam infligit cor­pus vnius libræ po&longs;t 2. in&longs;tantia motus; &longs;i verò tempora motus &longs;unt inæ­qualia, & grauitates æquales, percu&longs;&longs;iones erunt vt tempora; &longs;i demum grauitates inæquales, & tempora motus inæqualia, percu&longs;&longs;iones erunt in ratione compo&longs;ita ex ratione grauitatum & temporum, quæ omnia patent ex dictis in Th. &longs;uperioribus, v. g. &longs;it corpus duarum librarum, & alterum trium librarum; primum moueatur per 5. in&longs;tantia, & &longs;ecun­dum 2.per 5. ratio grauitatum e&longs;t 3/2; ratio temporum e&longs;t 7/5; compo&longs;ita ex vtraque erit (21/10); & hæc e&longs;t ratio percu&longs;&longs;ionum.

Theorema 56.

Hinc pote&longs;t &longs;ciri ratio percu&longs;&longs;ionis. & grauitationis eiu&longs;dem mobilis in pri­mo in&longs;tanti vtriu&longs;que, &longs;i cogno&longs;catur numerus in&longs;tantium motus; cum enim &longs;ingulis in&longs;tantibus æqualis impetus accedat, vt &longs;æpè dictum e&longs;t; certè erit percu&longs;&longs;io ad grauitationem, vt numerus in&longs;tantium motus ad vnita­tem, v.g. grauitatio &longs;it vt 4.&longs;it&qacute;ue motus eiu&longs;dem corporis per 8. in&longs;tan­tia; percu&longs;&longs;io erit ad grauitationem, vt 32. ad 4.vel vt 8.ad 1.quæ om­nia con&longs;tant ex dictis.

Theorema 57.

Hinc data percu&longs;&longs;ione, &longs;i cogno&longs;ceretur probè numerus in&longs;tantium motus, dari po&longs;&longs;et grauitatio ip&longs;i æqualis; v.g. &longs;it percu&longs;&longs;io dati corporis cadentis per 8.in&longs;tantia, eius percu&longs;&longs;io e&longs;t octupla grauitationis eiu&longs;dem per Th. 56. igitur &longs;i detur grauitatio octupla huius, erit æqualis datæ percu&longs;­&longs;ioni; dabitur autem grauitatio octupla, &longs;i detur corpus eiu&longs;dem mate­riæ octuplò grauius, vt con&longs;tat.

Theorema 38.

Hinc primo in&longs;tanti grauit ationis nullum ferè &longs;entitur pondus, quia mini­ma vis e&longs;t, quæ con&longs;equentibus in&longs;tantibus augetur, hinc licèt corpus breui tempore quis &longs;u&longs;tineat, paulò po&longs;t tamen ponderi cedit, ratio e&longs;t elara ex dictis.

Scholium.

Ob&longs;eruabis primò numerum in&longs;tantium non po&longs;&longs;e à quoquam &longs;en&longs;u percipi, nec in calculos vocari, vt patet; vnde Theoremata non po&longs;&longs;unt ad praxim reduci defectu huius cognitionis; quam &longs;upra adhibui hypo­the&longs;eos loco.

Secundò non pote&longs;t ad amu&longs;&longs;im tempus cum tempore componi ad æqualitatem, vel aliam datam rationem; licèt enim vnum tempus &longs;en&longs;i­bile haberet mille in&longs;tantia &longs;upra aliud; illa tamen inæqualitas &longs;en&longs;u minimè perciperetur; idem dico de aliis rationibus, in quo, ni fallor, maximè peccant, qui temporum æqualitatem perfectam ob&longs;eruari po&longs;&longs;e contendunt.

Tertiò, idem dico de percu&longs;&longs;ionum ratione; quippe non pote&longs;t &longs;en&longs;u percipi inæqualitas duarum percu&longs;&longs;ionum, licèt vires vnius præualeant mille punctis &longs;eu gradibus in&longs;en&longs;ibilibus; quippe non pote&longs;t di&longs;tingui ab alia ni&longs;i vel ex &longs;patio; atqui di&longs;cerni non pote&longs;t, an vnum &longs;patium &longs;uperet aliud mille punctis; vel ex &longs;ono; atqui &longs;onus pote&longs;t diuidi in in­finitos ferè gradus &longs;en&longs;u minimè perceptibiles; igitur nulla hypothe&longs;is in his experimentis &longs;tatui pote&longs;t, quibus æqualitas vel temporum, vel &longs;patiorum cogno&longs;ci dicatur; nec dicas aliquot in&longs;tantia parùm di&longs;cti­minis importare, nam cùm &longs;ingulis in&longs;tantibus fiat æqualis impetus ac­ce&longs;&longs;io, mille in&longs;tantia reddunt percu&longs;&longs;ionem millecuplam grauitationis; hinc certum e&longs;t ex numero in&longs;tantium cognito cogno&longs;ci tantùm po&longs;&longs;e numerum punctorum, & vici&longs;&longs;im; at certè neuter &longs;en&longs;u percipi pote&longs;t; ne­que tanti e&longs;t hoc &longs;cire.

Theorema 59.

Hinc &longs;i corpus graue de&longs;cenderet motu æquabili eoque æquali motui primi m&longs;tantis; certè vix modicum &longs;patium post multos annos decurreret; &longs;uppo­namus enim quod plures habent, licèt accuratè experimento &longs;ubii­ci non po&longs;&longs;it, &longs;cilicet vno &longs;ecundo minuto temporis decurri à corpore graui deor&longs;um 12. pedes &longs;patij; in &longs;ecundo minuto &longs;upponamus e&longs;&longs;e mille in&longs;tantia, quamuis infinita penè contineat; &longs;itque in primo in­&longs;tanti motus vnus gradus impetus; &longs;ic enim vocetur illa pars impetus, que producitur primo in&longs;tanti; certè po&longs;t mille in&longs;tantia motus, erunt mille gradus impetus; iam vcrò &longs;i accipiatur &longs;ubduplum maximæ & minimæ velocitatis; id e&longs;t vnius gradus, & mille graduum, &longs;cilicet 500. 1/2 tri­buaturque motui æquabili; haud dubiè vno fecundo minuto percur­rentur 12. pedes &longs;patij per Th. 46. Igitur &longs;i cum velocitate vt 500, 1/2 percurrentur 12. pedes 1.minuto, cum velocitate vt 1. percurrentur 12. pedes 500.&longs;ecundis minutis, &; 30. tertiis; &longs;i verò accipiantur plura in&longs;tantia, v.g. 1000000.in&longs;tantia, percurrentur 12. pedes 500000. &longs;e­cundis minutis; &longs;i verò 1000000000000. percurremur 500000000000. &longs;ecundis, id e&longs;t 8333333333. minutis, id e&longs;t 138888888. horis id e&longs;t 5787037. diebus id e&longs;t 89031. annis, omitto minutias; atqui lon­gè adhuc plura in vno minuto continentur in&longs;tantia.

Theorema 60.

Si corpus graue de&longs;cenderet motu æquabili, eoque æquali motui vltimi in­stantis, duplum ferè &longs;patium æquali tempore conficeret illius quod conficit motu accelerato, duplum inquam ferè &longs;cilicet panlò minùs; quia conficit idem motu æquabili; cuius velocitas e&longs;t &longs;ubdupla maximæ & minimæ; &longs;ed minima velocitas primi in&longs;tantis pro nihilo reputatur; igitur acci­piatur tantùm &longs;ubduplum maximæ, igitur cum velocitate æquali maxi­mæ, eodem tempore duplum &longs;patium percurretur; igitur in vno minuto &longs;ecundo, v.g. 24. pedes; igitur in vno minuto primo codem motu æqua­bili 1440. pedes percurrentur; igitur in vna hora 86400. pedes; hinc non e&longs;t quod aliqui adeo mirentur, &longs;eu potiùs reiiciant hanc motus accelerationem quod ex ea tùm tardi&longs;&longs;imus motus, tùm veloci&longs;&longs;imus con&longs;equatur.

Theorema 61.

Motus naturaliter acceleratus non propagatur per omnes tarditatis gra­dus; quia tot &longs;unt huius propagationis gradus, quot &longs;unt in&longs;tantia, quibus durat hic motus, cum &longs;ingulis in&longs;tantibus noua fiat impetus ac­ce&longs;&longs;io, &longs;ed non &longs;unt infinita in&longs;tantia, vt demon&longs;trabimus in Metaphy­&longs;ica; prætereà licèt e&longs;&longs;ent infinita in&longs;tantia, non fieret adhuc per omnes tarditatis gradus hæc propagatio; quia daretur aliquis gradus tarditatis, quem non comprehenderet hæc graduum &longs;eries; nam incipit moucri tardiùs in plano inclinato quàm in libero medio rectà deor&longs;um, vt con­&longs;tat, & in medio den&longs;o quàm in raro v.g. in aqua quàm in aëre; igitur hic tarditatis gradus, quo incipit moueri in plano tantillùm inclinato, non continetur inter illos, quibus mouetur rectà deor&longs;um.

Hinc duplici nomine reiice Galilæum qui hoc a&longs;&longs;erit. Primò, quia fru&longs;trà ponit infinita in&longs;tantia &longs;ine nece&longs;&longs;itate; &longs;ecundò, quia ratio, quam habet, non conuincit; vocat enim quietem tarditatem infinitam; à qua dum recedit mobile, haud dubiè per omnes tarditatis gradus propagari pote&longs;t eius motus; &longs;ed contrà primò, nam reuerà quies non e&longs;t tarditas, quæ motui tantùm ine&longs;&longs;e pote&longs;t. Secundò, quia tàm ex quiete &longs;equi po­te&longs;t immediatè velox motus, quàm tardus, vt patet in proiectis. Tertiò, quia motus incipit; igitur per aliquid &longs;ui, igitur ille primus motus à quiete infinitè non di&longs;tat; denique rationes &longs;uprà propo&longs;itæ rem i&longs;tam euincunt.

Scholium.

Ob&longs;eruabis con&longs;ideratum e&longs;&longs;e hactenus hunc motum nulla habita ratione re&longs;i&longs;tentiæ medij, quæ haud dubiè hanc propo&longs;itionem motus accelerati tantillùm impedit, &longs;ed de re&longs;i&longs;tentià medij agemus infrà.

Corollarium 1.

Ex dictis facilè reiicies primò &longs;ententiam illorum, qui negant mo-m naturalem accelerari, quos non ratio modò euidenti&longs;&longs;ima, &longs;ed adeò &longs;en&longs;ibile experimentum omninò conuincere pote&longs;t.

Corollarium. 2.

Secundò reiicies illos, qui volunt accelerationem motus e&longs;&longs;e, vel à vi magnetica, quâ terra trahit ad &longs;e omnia grauia; vel ab alia vi occulta, quâ cœlum pellit deor&longs;um; vel à cœle&longs;ti illa, imò potiùs fabulosâ mate­riâ; vel demum ab ip&longs;a vi fympathicâ, quâ corpus &longs;uo centro propiùs factum totas &longs;uas vires exerit, vt ei &longs;e conjungat; quæ omnia gratis di­cuntur, & ex dictis plu&longs;quam efficaciter refelli po&longs;&longs;unt, nefru&longs;trà tempus in iis iterum refellendis teramus.

Corollarium 3.

Tertiò reiicies, qui volunt motum accelerari ex aëris à tergo impel­lentis appul&longs;u, quod ridiculum e&longs;t: licèt enim Ari&longs;toscles videatur illud &longs;en&longs;i&longs;&longs;e de projectis, quod examinabimus &longs;uo loco; nunquam tamen hoc dixit de motu naturali; quin potiùs antiquorum fuit omnium hic &longs;en­&longs;us, fieri acce&longs;&longs;ionem mobili alicuius, vnde reddatur motus velocior; hinc dictum illud vulgare, vire&longs;que acquirit eundo; nihil porrò intelligi pote&longs;t nomine virium, ni&longs;i id, ex quo maior ictus, &longs;eu percu&longs;&longs;io &longs;equitur illud autem e&longs;&longs;e impetum con&longs;tat.

Corollarium. 4.

Quartò ex his &longs;ententia Ari&longs;totelica de motu accelerato optunè vin­dicatur; quòd &longs;cilicet grauia &longs;ub finem &longs;ui motus velociùs &longs;erantur ver­sùs centrum; quod ex dictis, & &longs;implici&longs;&longs;imis, cerimi&longs;que principiis demon&longs;tratum fuit.

Corollarium. 5.

Quintò reiicies etiam illorum &longs;ententiam, qui hanc accelerationem tribuunt vel medio minùs re&longs;i&longs;tenti, vel grauitatis augmento, vel impe­tui violento priùs impre&longs;&longs;o dum corpus graue attollitur, quod meo iudi­cio ridiculum e&longs;t; qua&longs;i verò fru&longs;tum rupis deci&longs;um, deor&longs;umque ruens impetum violentum aliquando habuerit.

Corollarium 6.

Sextò reiicies illorum &longs;ententiam, qui volunt accelerationem motus naturalis ita fieri, vt &longs;patia temporibus æqualibus acqui&longs;ita &longs;equantur &longs;e­riem numerorum imparium 1.3.5.7.9.11.13. &c. & &longs;patia &longs;int vt quadrata temporum v. g. &longs;i primo in&longs;tanti acquiritur 1.&longs;patium: &longs;ecundo acquiruntur 3. tertio 5. quarto 7. &c. fique vno in&longs;tanti acquiritur 1. &longs;patium, duobus acquiruntur 4. tribus 9. quatuor 16. atque ita deinceps per quadrata, quæ omnia ex dictis falfa e&longs;&longs;e con&longs;tat; quippe &longs;i æqualibus temporibus acquiruntur æqualia velocitatis momenta; igitur &longs;i primo in&longs;tanti e&longs;t 1.gradus, &longs;ecundo erunt 2. igitur &longs;ecundo tempore cum duo­bus gradibus velocitatis vel impetus percurrentur duo tantùm &longs;patia, &longs;i primò in&longs;tanti æquali cum vno gradu percurritur vnus, &longs;ed de his fusè infrà.

Corollarium 7.

Septimò reiicies etiam aliquos recentiores, qui volunt fieri hanc pro­gre&longs;&longs;ionem &longs;patiorum æqualibus temporibus re&longs;pondentium &longs;ecundùm progre&longs;&longs;ionem Geometricam, duplam, &longs;cilicet iuxta hos numeros 1. 2. 4. 8. 16. 32. &c. quod etiam ex eadem ratione facilè confutatur: reiicies etiam alium recentiorem, qui vult hanc progre&longs;&longs;ionem &longs;umi ex linea proportionaliter &longs;ectâ, id e&longs;t in mediam & extremam rationem; &longs;ed de his omnibus in di&longs;&longs;ertatione &longs;equenti fusè di&longs;putamus; quippe rem hanc tanti e&longs;&longs;e putamus, vt nihil omittendum &longs;it, quod ad cius pleni&longs;&longs;imam confirmationem pertineat.

DISSERTATIO

De Motu naturaliter accelerato.

DVæ &longs;unt poti&longs;&longs;imùm in hac materia celebres &longs;ententiæ; Prima e&longs;t Galilei, & ferè omnium recentiorum, qui po&longs;t Galileum de motu &longs;crip&longs;erunt; inter quos, ne omittam Genuen&longs;em Patricium, BalianumDoctus Mer&longs;ennus, & eruditus Ga&longs;&longs;endus primum locum obtinent; quorum ille hanc &longs;ententiam multis in locis, &longs;cilicet in &longs;uis quæ&longs;tioni­bus Phy&longs;icis, in &longs;ua Galilei ver&longs;ione, in harmonia vniuer&longs;ali, & demum in &longs;ua Bali&longs;tica pa&longs;&longs;im, tùm fusè proponit, & explicat, tùm etiam &longs;uis ra­tionibus confirmat; Galileus verò illam habet tùm in gemino &longs;y&longs;tema­te, tùm in dialogo tertio de motu locali.

Secunda &longs;ententia no&longs;tra e&longs;t, de qua non &longs;emel di&longs;putandum fuit à Magi&longs;tro, tùm verbis tùm etiam litteris &longs;criptis; &ne quid fortè di&longs;&longs;imu­lem, illa e&longs;t &longs;ententia quam anonimo Philo&longs;ophe (quem non &longs;ine laudappellat idem Mer&longs;ennus) tribuit. prop.18.&longs;uæ Bali&longs;ticæ &longs;ub finem; illa e&longs;t inquam &longs;ententia, quam hactenus meo iudicio &longs;atis luculenter de­mon&longs;trauimus.

Sunt tres aliæ &longs;ententiæ, quæ ab eodem Mer&longs;enno referuntur; prima e&longs;t quæ progre&longs;&longs;ionem &longs;patiorum eamdem e&longs;&longs;e vult cum eâ, quæ e&longs;t &longs;i­nuum ver&longs;orum, centro quadrantis po&longs;ito in centro terræ, & altero ex­tremo &longs;inus totius in co punctò, in quo incipit motus. Secunda e&longs;t quo­rumdam, qui volunt progre&longs;&longs;ionem &longs;patiorum, quæ &longs;ingulis temporibus re&longs;pondent, e&longs;&longs;e in progre&longs;&longs;ione geometrica dupla iuxta hos numeros, 1.2.4.8.32. Tertia e&longs;t alicuius, qui voluit e&longs;&longs;e iuxta proportionem lineæ &longs;ectæ in mediam, & extremam rationem.

Tres vltimæ &longs;ententiæ nullo pror&longs;us nituntur fundamento; igitur vel inde maximè confutantur, quòd gratis &longs;ine vllo pror&longs;us vel rationis vel experimenti momento excogitatæ &longs;int. Igitur in hac di&longs;&longs;ertatione duæ tantùm primæ di&longs;cutiendæ &longs;unt Sententiæ Galilei &longs;chema hic habes in linea AF, in qua a&longs;&longs;umitur AB, &longs;patium &longs;cilicet, quod dato tempore corpus graue &longs;uo motu percurrit; & &longs;ecundo tempore æquali BC, quæ tripla e&longs;t AB, tertio CD quintupla quarto DE &longs;eptupla, quinto EF nouecupla; vides primò &longs;eriem numerorum imparium 1. 3. 5. 7. 9.atque ita deinceps. Secundò vides &longs;patia e&longs;&longs;e in ratione duplicata temporum, hoc e&longs;t vt temporum quadrata. v.g. &longs;i accipiatur &longs;patium AB primo tem­pore peractum, & &longs;patium AC duobus temporibus confectum: ratio hu­ius ad illud e&longs;t vt 4.ad 1.id e&longs;t vt quadratum 2.ad quadratum 1. &longs;imiliter, &longs;i accipiatur &longs;patium AD confectum tribus temporibus, erit 9.id e&longs;t qua­dratum 3, &longs;patium AE confectum 4.temporibus erit 16.id e&longs;t quadratum 4. & AF 25. quadratum 5.

Hæc &longs;ententia ingeniosè à Galileo excogitata ex duplici capite à &longs;uis auctoribus confirmatur; primò experientiâ, &longs;ecundò ratione. Experien­tia tribus poti&longs;&longs;imum experimentis fulcitur; primum e&longs;t in motu deor­&longs;um per lineam perpendicularem. v. g. in linea AF; nam reuerà multi &longs;unt, iique graui&longs;&longs;imi auctores in rebus tùm philo&longs;ophicis, tùm mathe­maticis ver&longs;ati&longs;&longs;imi, qui &longs;æpiùs &longs;en&longs;u ip&longs;o probarunt, repetitis v&longs;que ad nau&longs;eam experimentis, tempore vnius &longs;ecundi minuti corpus graue in libero aëre 12. pedes &longs;patij motu naturali deor&longs;um percurrere; in 2.ve­rò &longs;ecundis 48. in 3.&longs;ecundis 108.&longs;ed &longs;patia i&longs;ta &longs;unt vt temporum qua­drata, vt con&longs;tat.

Secundum experimentum e&longs;t in plano inclinato, in quo corpus graue de&longs;cendit iuxta prædictam progre&longs;&longs;ionem, quod expre&longs;&longs;is verbis te&longs;tatur Galileus à &longs;e fui&longs;&longs;e probatum &longs;æpiùs, nec vnquam à vero ne tantillùm quidem aberra&longs;&longs;e. &longs;ed in perpendiculari deor&longs;um eadem proportione cre&longs;cit motus, quâ in plano inclinato; licèt in plano inclinato tardior &longs;it motus, vt demon&longs;trabimus aliàs.

Tertium experimentum petitur ex funependulis; in quibus &longs;æpiùs ob&longs;eruatum e&longs;t longitudinem funis, & con&longs;equenter arcum quadrantis longioris funependuli e&longs;&longs;e ad longitudinem, &longs;eu quadrantem alterius breuioris, vt quadratum temporis, quo perficitur vibratio maioris ad quadratum temporis, quo perficitur vibratio minoris.v.g.&longs;it longitudo funependuli maioris, CG minoris verò &longs;ubquadrupla CF; eleuetur vter­que funis, cui pondus æquale &longs;it appen&longs;um v&longs;que ad horizontalem CDE & alterum ex D; alterum verò ex E demi&longs;&longs;um cadat deor&longs;um; haud dubiè funependulum CE duplum temporis collocabit in decurrendo quadrante EG, & funependulum ED &longs;ubduplum. v. g. &longs;i CD conficit &longs;uam vibrationem DF vno &longs;ecundo, EG conficiet &longs;uam EG duobus, vt centies ob&longs;eruatum e&longs;t; &longs;ed EG e&longs;t quadruplus DF, vt patet; igitur EG & DF &longs;unt vt quadrata temporum, quibus percurritur EG & DF &longs;ed vt de&longs;cendit graue per DF & EG, ita de&longs;cendit per CF & CG, quippe DF & EG habent rationem plani inclinati deor&longs;um.

Adde quod, vt &longs;e habet tempus, quo de&longs;cendit per totum quadrantem DF, ad tempus, quo de&longs;cendit per totum quadrantem EG. &longs;ic &longs;e habet tempus, quo de&longs;cendit per arcum DL &longs;ubduplum DF ad tempus, quo de&longs;cendit per arcum EI &longs;ubduplum EG; item tempus, quo de&longs;cendit per arcum DM &longs;ubquadruplum DF.ad tempus, quo de&longs;cendit per arcum EK &longs;ubquadruplum EG; denique vt tempus, quo per minimum ar­cum quadrantis DF, ad tempus, quo de&longs;cendit per alium proportiona­lem, &longs;cilicet quadruplum in quadrante EG; atqui tam parui arcus po&longs;­&longs;unt a&longs;&longs;umi, vt &longs;int ad in&longs;tar lineæ rectæ deor&longs;um tangentis &longs;cilicet in D & in E; igitur in his rectis de&longs;cendunt grauia iuxta progre&longs;&longs;ionem præ­dictam; id e&longs;t, cum arcus minimus a&longs;&longs;umptus ab E, qui æquiualet rectæ, &longs;it quadruplus arcus minimi a&longs;&longs;umpti à puncto D, tempus, quo percurri­tur ille primus, e&longs;t ad tempus, quo percurritur hic &longs;ubquadruplus, vt tem­pus, quo percurritur EG ad tempus, quo percurritur DF vt dictum e&longs;t; &longs;ed tempus, quo percurritur EG e&longs;t duplum illius, quo percurritur DF; igitur tempus, quo percurritur minimus arcus a&longs;&longs;umptus ab E, & qui e&longs;t ad in&longs;tar rectæ, e&longs;t duplum temporis quo percurritur minimus arcus a&longs;­&longs;umptus à puncto D &longs;ubquadruplus prioris, & qui e&longs;t etiam ad in&longs;tar re­ctæ; igitur &longs;patia &longs;unt vt temporum quadrata.

Quod autem tempus, quo percurritur EG &longs;it duplum illius, quo per­curritur DF, patet experientiâ; nam &longs;i numerentur ducentæ vibrationes funependuli CD; eodem tempore numerabuntur centum vibrationes maioris CE; igitur vibrationum minoris numerus e&longs;t duplus numeri vi­brationum maioris, dum &longs;imul vibrantur; igitur eo tempore, quo fiunt 100.maioris, fient 200. minoris; nam ommes vibrationes eiu&longs;dem fune­penduli &longs;unt æquò diuturnæ, licèt fiant per arcus inæquales ciu&longs;dem. quadrantis, vt &longs;æpè ob&longs;eruatum e&longs;t. In his tribus poti&longs;&longs;imum experimen­tis fundatur hæc hypothe&longs;is Galilei, quæ nec clariùs meo. iudicio, nec &longs;inceriùs exponi po&longs;&longs;unt.

Antequam rationes, quæ pro hac &longs;ententia facere videntur, propona­mus, refellamu&longs;que; oftendo primò quomodo cum his experimentis &longs;tare po&longs;&longs;it no&longs;tra hypothe&longs;is; igitur ex iis hypothe&longs;is Galilei rectè de­duci non pote&longs;t: quippe hæc e&longs;t certi&longs;&longs;ima regula, quam nemo Philo&longs;o­phus negare au&longs;it: Quotie&longs;cumque aliquod experimentum tale e&longs;t, vt cum co &longs;tare po&longs;&longs;int contrariæ hypothe&longs;es; ex eo certè neutra deduci po­te&longs;t; igitur ex propo&longs;itis experimentis &longs;uam hypothe&longs;im Galileus non legitimè deducit, quod vt clari&longs;&longs;imè o&longs;tendam.

Suppono, quando dicitur &longs;ecundum &longs;patium e&longs;&longs;e triplum primi &longs;up­po&longs;itis æqualibus temporibus, non ita Geometricè, certaque, & acuratâ a&longs;&longs;ertione hoc dici; quin vel aliqua puncta in &longs;patiis, vel in&longs;tantia in temporibus de&longs;int, vel &longs;uper&longs;int; &longs;i enim quis diceret &longs;patium e&longs;&longs;e tri­plum primi minus 100000. punctis, vel &longs;ecundum tempus e&longs;&longs;e maius primo 100000. in&longs;tantibus; quis hanc, vel &longs;patij, vel temporis differen­tiam &longs;en&longs;u percipiat? cum tamen experimentum omne phy&longs;icum &longs;en&longs;ui &longs;ube&longs;&longs;e po&longs;&longs;it; nec e&longs;t quod aliquis dicat hoc idem toties ob&longs;eruatum e&longs;&longs;e, tam multis locis temporibus, totque ac tantis etiam te&longs;tibus, vt mi­nimè fraus aliqua, vel error &longs;ubrepere potuerit; nam cum parua &longs;it, & in&longs;en&longs;ibilis tùm &longs;patiorum, tùm temporum differentia, maius vel minus æquali tempus, pro æquali, maius.vel minus triplò &longs;patium pro triplo facilè accipi pote&longs;t, cum nullum di&longs;erimen &longs;en&longs;ibile e&longs;t.

Adde quod non de&longs;unt viri graui&longs;&longs;uni qui dicant &longs;e vix ob&longs;eruare po­tui&longs;&longs;e hanc &longs;patiorum progre&longs;&longs;ionem; plures appellare po&longs;&longs;em; vnus Ga&longs;&longs;endus e&longs;t in&longs;tar omnium; qui &longs;anè in ob&longs;eruando fuit acurati&longs;&longs;imus, qui literis &longs;criptis, quas ego vidi, expre&longs;&longs;is verbis a&longs;&longs;erit progre&longs;&longs;ionem hanc non e&longs;&longs;e omninò iuxta hos numeros 1.3.5.7. &longs;ed &longs;ingulis addendas e&longs;&longs;e &longs;uas minutias, quas ip&longs;e habet; &longs;ed ego omitto, quia etiam &longs;ua incer­titudine laborant; igitur nullo experimento ad amu&longs;&longs;im concludes, vel æqualitatem vel aliam accuratam tùm temporum tùm &longs;patiorum pro­portionem: Equidem &longs;en&longs;u percipio practicam hanc e&longs;&longs;e maiorem pede; at tot lineis vel punctis &longs;uperare ne Argus quidem certò, ac di&longs;tinctè cer­neret: Sed efficaciter, meo iudicio, hanc Galilei hypothe&longs;im refello; &longs;int 2.partes temporis æquales AE, EF, eæque &longs;en&longs;ibiles; nec enim aliæ a&longs;­&longs;iuni po&longs;&longs;unt; &longs;intque minintæ omnium &longs;en&longs;ibilium; haud dubiè con&longs;tant &longs;ingulæ infinitis ferè aliis in&longs;en&longs;ibilibus, vt patet; igitur &longs;ic ratiocinatur Galileus; in prima parte temporis AE corpus graue percurrit &longs;patium GH, & in &longs;ecunda æquali EF percurrit &longs;patium HL triplum prioris; igitur &longs;patia &longs;unt vt quadrata temporum, rectè; &longs;ed antequam vlterius progrediar; Quæro vel à Galileo, vel à quolibet alro, vtrum &longs;patium HL &longs;it omnino triplum? & &longs;i aliquis contenderet dec&longs;&longs;e (1/1000000) GH vtrum experimento præ&longs;enti conuinci po&longs;&longs;it? nemo, vt puto, id a&longs;&longs;erere au&longs;it; hoc po&longs;ito, a&longs;&longs;umptaque progre&longs;&longs;ione arithmetica quam no&longs;tra &longs;en­tentia in &longs;patiis ad&longs;truit; &longs;i prima parte temporis AE percurratur &longs;pa­tium GH, &longs;ecunda EF. percurretur tantùm HK duplum GH; igitur minus e&longs;t hoc &longs;patium vero &longs;patio 1/4. &longs;cilicet tota KL; res pror&longs;us de­mon&longs;trata e&longs;&longs;et, &longs;i termini proportionis vnius e&longs;&longs;ent tantùm 2. id e&longs;t, &longs;i progre&longs;&longs;io fieret in partibus temporis &longs;en&longs;ibilibus; at po&longs;ito quod &longs;int plures termini, vt reuerâ &longs;unt; nam in totidem terminis fit progre&longs;&longs;io, in quibus fit augmentum impetus, vel accelorationis acce&longs;&longs;io; atqui hæc fit in &longs;ingulis in&longs;tantibus, licèt finitis, igitur & progre&longs;&longs;ro; Quare duæ partes temporis AE, EF diuidantur in 4. æquales AD; certè in duabus primis percurretur &longs;patium. VQ æquale GH; igitur duabus vltimis per­curretur QK, quæ &longs;it ad QV vt 7. ad 3. nam prima parte percurritur 1. &longs;patium. &longs;ecunda 2. igitur QV continet tria &longs;patia; tertia verò 3. quarta 4.ergo hæ duæ vltimæ 7. &longs;ed QM e&longs;t dupla QV; igitur continet 6. igi­tur MK e&longs;t 1/3 VQ, vel KL; igitur KM e&longs;t (1/12) GL; igitur 12. L (1/10), vel 1/6, igitur VK e&longs;t ad GL vt 10.ad 12. igitur totum &longs;patium VK e&longs;t mi­nus vero 1/6. Præterea 2. partes temporis AE EF diuidantur in 8. partes æquales AE; haud dubiè 4. primis percurretur &longs;patium XT æqualc GH, quod debet diuidi in 10. &longs;patia; nam 4. terminis, &longs;eu temporibus re&longs;pondent &longs;paria 10. quibus æqualia &longs;unt 40. in teta GL, cuius XT e&longs;t (1/14), &longs;ed &longs;i in 4.primis acquiruntur 10. 4. vltimis EF acquiruntur 26.&longs;cili­cet T 5; igitur tota X 5. e&longs;t 6. igitur e&longs;t ad GL vt 36. ad 40. &longs;eu 9. ad 10. igitur X 5. e&longs;t &longs;patium minus vero (1/10).

Prærerea diuidatur tempus AF in 16. partes æquales AB; haud dubiè 8 primis acquiritur &longs;patium YS æquale GH; quod debet diuidi in &longs;pa­tiola 36, quæ re&longs;poudent 8. temporibus, &longs;eu terminis huius progre&longs;&longs;io­nis, quibus æqualia &longs;unt 144. in GL, cuius YS e&longs;t 1/4, &longs;ed &longs;i in 8. primis acquiruntur 36. in 8. vltimis acquirentur 100. igitur S 6. e&longs;t 100. igitur 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 &longs;patium totale minus vero (1/18).

Deinde diuidatur adhuc tempus AF in partes 32. æquales, 16. pri­mis acquiritur ZR æquale GH, quod debet diuidi in &longs;patiola 136.quæ re&longs;pondent 16. temporibus quibus æqualia &longs;unt 544. in tota GL, cuius ZR e&longs;t 1/4 &longs;ed &longs;i in 16. primis temporibus acquiruntur 136. in vltimis 16. acquiruntur 392. igitur R 7. e&longs;t 392. & ZR 136. igitur Z 7.528. igitur Z 7. e&longs;t ad GL, vt 528. ad 544. &longs;eu vt 33. ad 34. igitur Z 7 e&longs;t &longs;patium minus verò (1/34)

Denique &longs;i diuidatur tempus AF in partes 64.&longs;patium acqui&longs;itum erit minus vero, a&longs;&longs;umpto &longs;cilicet tota HL (1/66), &longs;i diuidatur in 128. partes, erit minus (1/130) &longs;i diuidatur in 256. partes, erit minus (1/258) &longs;ed temporis par­tes 2.AE. EF minimè &longs;en&longs;ibilium diuidi po&longs;&longs;unt in infinita ferè in&longs;tan­tia; &longs;int tantùm ex.g. 1000000. igitur &longs;patium tunc acqui&longs;itum erit mi­nus &longs;uppo&longs;ito vero HL (1/1000002), quæ &longs;i de&longs;it tantùm &longs;patio KL vt &longs;it 1/4 totius GL, quis hoc di&longs;cernat? igitur etiam &longs;uppo&longs;ita progre&longs;&longs;ione arith­metica, quæ fiat in finitis in&longs;tantibus; &longs;i ob&longs;eruetur acurati&longs;&longs;imè &longs;patium, quod percurritur in vna parte temporis &longs;en&longs;ibili v. g. &longs;patium GH in parte temporis AE; &longs;patium, quod acquiretur in tempore &longs;ecundo æqua­li tàm propè accedet ad &longs;patium HL, id e&longs;t ad triplum prioris GH, vt nullus mortalium di&longs;cernere po&longs;&longs;it; igitur cum hoc experimento tàm pote&longs;t &longs;tare no&longs;tra hypothe&longs;is, quàm alia Galilei, igitur neutra ex co tan­tùm euinci pote&longs;t.

Hinc obiter ob&longs;erua progre&longs;&longs;ionem differentiarum; quippe &longs;i &longs;int 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. (1/34); &longs;i 64.(1/66) nam primò denominator fractionis &longs;uperat tantùm binario numerum partium temporis; &longs;ecundò differentiæ denominatorum &longs;unt in progre&longs;&longs;ione geometrica dupla numerorum 2. 4. 8. 16. 32. 64. 128. &c.

Eodem modo &longs;oluendum e&longs;t &longs;ecundum experimentum rotati globi in plano decliui; præ&longs;ertim cum globus ab incur&longs;u a&longs;periorum partium tùm globi, tùm plani &longs;altuatim de&longs;cendat; quod dubium e&longs;&longs;e non pote&longs;t, & quò decliuius erit, faciliùs re&longs;iliet a plano, vt patet; &longs;ed de motu in planis inclinatis fusè agemus infrà libro integro.

Quod &longs;pectat ad tertium experimentum; multa in eo &longs;upponuntur vel fal&longs;a, vel &longs;altem dubia: vel ea quæ cum no&longs;tra hypothe&longs;i optimè con­ueniant. Primum e&longs;t, quando dicuntur omnes vibrationes eiu&longs;dem fune­penduli, &longs;iue maiores, &longs;iue minores e&longs;&longs;e æquediuturnæ, quod manife&longs;tis experimentis repugnat; quippe vibratio maior plùs temporis; minor ve­rò minùs in &longs;uo de&longs;cen&longs;u ponit; dimittantur enim duo funependula æ­qualia; alterum quidem ex altitudine 90.graduum, alterum ex altitudine 10. vel 15.graduum; ita vt &longs;imul vibrationes &longs;uas incipiant; numerentur vibrationes vtriu&longs;que, vbi 100. è minoribus numeratç fuerint, numera­buntur circiter 97. è maioribus, quod &longs;æpiùs ob&longs;eruaui te&longs;tibus etiam adhibitis; hoc ip&longs;um etiam ob&longs;eruarunt alij; atque adeo ip&longs;e P.Mer&longs;en­nus, qui L. 2. &longs;uæ ver&longs;ionis, Ar.17. Galileum arguit parùm acurati &longs;tu­dij in his ob&longs;eruationibus adhibiti: rationem huius effectus in libro de funependulis explicabimus; immò &longs;i omnes vibratìones maiores primæ vibrationi 90. grad. e&longs;&longs;ent æquales, & aliæ minores alterius funependu­li &longs;en&longs;un, vt &longs;it, minuerentur; vix 90. maiores numerare po&longs;&longs;es, iam enu­meratis 100. ex minoribus; &longs;ed de his omnibus &longs;uo loco; in vna tamen vel altera vibratione vix aliquod di&longs;crimen ob&longs;eruatur; quod tamen ob­&longs;eruari facilè po&longs;&longs;et in maioribus funependulis.

Secundum, quod &longs;upponitur, e&longs;t quod longitudines funependulorum &longs;int pror&longs;us, vt quadrata temporum, quibus vibrationes &longs;ingulorum fiunt, v.g. funependulum longitudinis 4. pedum facere vnam vibratio­nem eo tempore, quo funependulum longitudinis vnius pedis facit duas; quod primò in multis vibrationibus non tàm accuratè ob&longs;eruatur; &longs;ecun­dò licèt ob&longs;eruaretur &longs;en&longs;ibiliter, id emre&longs;ponderi debet, quod &longs;uprà in &longs;ingulis vibrationibus e&longs;&longs;e tantùm di&longs;crimen; uod etiam in multis &longs;en&longs;i­bile non e&longs;t; &longs;i enim di&longs;crimen primarum vibrationem v.g.&longs;it (1/100000000) certè vltimarum adhuc in&longs;en&longs;ibile erit.

Tertium &longs;uppo&longs;itum fuit, minimum arcum minoris quadrantis a&longs;&longs;um­ptum, & alium minoris quadrantis e&longs;&longs;e ad in&longs;tar perpendicularium; cùm tamen diuer&longs;a &longs;it inclinatio minoris, & maioris quadrantis: quippe principium maioris accedit propiùs ad perpendicularem; facit enim angulum contingentiæ minorem; alia verò extremitas accedit propiùs ad horizontalem propter rationem prædictam; hinc illa extremitas ma­ioris, vnde e&longs;t initium motus, planum decliuius facit; altera verò minùs decliue; &longs;ed hæc fusè pro&longs;equar &longs;uo loco.

Quartum, quod &longs;upponitur e&longs;t, accelerationem motus fieri in qua­drante in ea ratione, in qua fit per plana chordarum inclinata, quod etiam fai&longs;um e&longs;t; quia in eodem plano inclinato &longs;upponitur eadem inclinatio; &longs;ecus in quadrante, cuius &longs;ingula puncta nouam faciunt in­clinationem: adde quod quarta pars quadrantis maioris EK non facit eandem inclinationem, quam totus quadrans minor DF ip&longs;i EK æqua­lis; quamquam hoc ip&longs;i vltrò concedent aduer&longs;arij.

Præterea, &longs;it ita vt &longs;upponitur; ita vt &longs;en&longs;ibiliter differentia huius progre&longs;&longs;ionis percipi non po&longs;&longs;it, &longs;intque numeratæ omnes vibrationes &longs;en&longs;ibiles dati funependuli ex altitudine 90, grad. demi&longs;&longs;i; quæ vix e&longs;&longs;e po&longs;&longs;unt 1800; &longs;int autem plures &longs;cilicet 2000. dicis confectas e&longs;&longs;e 2000 minoris funependuli eo tempore, quo 1000. tantùm in quadruplo fune­pendulo nnmerantur; annuo quidem, &longs;i res tantùm &longs;en&longs;ibiliter con&longs;ideretur; &longs;in verò &longs;ecùs, id pernego; &longs;ed dico dee&longs;&longs;e v. g. 1000000. puncta &longs;patij, quæ di&longs;cerni non po&longs;&longs;unt, ita vt primæ vibrationi 1000. puir­cta &longs;ecundæ, 2000. tertiæ 3000. &c. vltimæ verò, &longs;eu mille&longs;im1000000. quæ omnia &longs;unt in&longs;en&longs;ibilia, neque maiorem habent diffi­cultatem, quàm in motu perpendiculari, de quo &longs;uprà; etiam conce&longs;&longs;is vltrò omnibus experimétis propo&longs;itis. Igitur &longs;uppo&longs;itâ progre&longs;&longs;ione &longs;pa­tiorum arithinetica in in&longs;tantibus, tàm propè accedit ad aliam, quàm Galileus ponit, &longs;iue in perpendiculari deor&longs;um, &longs;iue in quadrante fune­penduli; a&longs;&longs;umptis &longs;cilicet partibus temporis &longs;en&longs;ibilibus, vt differentia di&longs;cernit non po&longs;&longs;it; immò nec duplum diffetentiæ, nec centuplum, nec millecuplum; &longs;ed de his &longs;atis quæ ex dictis &longs;uprà facilè intelligi po&longs;&longs;unt: quare veniemus iam ad rationes.

Prima ratio, quam affert Galileus e&longs;t; quia cum natura in &longs;uis opera­tionibus adhibeat &longs;implici&longs;&longs;ima media; & cum acceleratio motus natu­ralis non po&longs;&longs;it fieri iuxta faciliorem, vel &longs;impliciorem progre&longs;&longs;ionem, quàm &longs;it-ea quæ fit per quadrata; non e&longs;t dubium, quin iuxta illam pro­gre&longs;&longs;io motus naturaliter accelerati fieri debeat; præ&longs;ertim cùm omni­bus experimentis con&longs;entiat, & in ea omnia phænomena explicari po&longs;&longs;int.

Re&longs;p. Primò progre&longs;&longs;ionem arithmeticam &longs;implicem iuxta hos nu­meros 1.2.3.4. longè &longs;impliciorem e&longs;&longs;e alia quæ fit iuxta illos 1.3.5.7.vt nemo non iudicabit. Secundò cum accidit duas hypothe&longs;es conuenire cum omnibus experimentis &longs;eu phænomonis, debet e&longs;&longs;e aliqua ratio, cur ad­hibeatur vna potiùs quàm alia; &longs;ed nulla e&longs;t ratio, cur Galileus adhibeat &longs;uam, vti videbimus; nos verò ratione demon&longs;tratiuâ probamus no&longs;tram; igitur no&longs;tra e&longs;t præferenda pro theorica rei veritate; quia verò alia in temporibus &longs;en&longs;ibilibus proximè ad verum accedit eam adhibendam e&longs;&longs;e decernemus infrà ad praxim, & communem i&longs;torum motuum men­&longs;uram.

Secunda ratio e&longs;t; quia, &longs;i accipiatur &longs;ubduplum maximæ, & minimæ velocitatis; &longs;itque ex his qua&longs;i conflata velocitas motus æquabilis, hoc motu æquabili æquali tempore pèrcurretur &longs;patium idem, quod antè motu naturaliter accelerato v.g. &longs;int numeri datæ progre&longs;&longs;ionis 1.3.5.7. 9.11. certè &longs;umma terminorum &longs;eu totum &longs;patium erit 36. accipiatur &longs;ubduplum primi 1/2 & &longs;exti 5. 1/2 habebitur velocitas vt 6. igitur cum velocitate vt 6. æquali tempore percurretur &longs;patium 36. quod rectè de­mon&longs;trauit Galileus.

Re&longs;pondeo non minùs no&longs;tram hypothe&longs;im cum hoc ip&longs;o &longs;tare, quàm &longs;tet hypothe&longs;is Galilei: &longs;int enim 6. in&longs;tantia, & &longs;ingulis &longs;ua tribuantur &longs;patiola more dicto 1 2 3 4 5 6. &longs;umma &longs;patiom e&longs;t 21. a&longs;&longs;umatur &longs;ub­duplum velocitatis primi in&longs;tantis 1/2, & &longs;ubduplum &longs;exti in&longs;tantis, &longs;cili­cet 3. conflatum ex vtroque 3 1/3; ducatur in 6.id e&longs;t in numerum termi­norum, vel in&longs;tantium; &longs;umma erit 21. igitur quod tribuit Galilcus &longs;uæ progre&longs;&longs;ioni, etiam no&longs;træ competit.

Tertia ratio petitur ex mathe&longs;i &longs;it enim linea AE diui&longs;a in qautuor partes æquales, quæ nobis repre&longs;entent 4. partes temporis æquales; haud dubiè, cùm acquirantur temporibus æqualibus æqualia velocitatis momenta; haud dubiè, inquam, his 4. temporibus AB, BC, CD, DE, ac-quirentur æquales velocitatis gradus; &longs;it autem BI, men&longs;ura velocitatis, quam acquirit mobile cadens ex &longs;ua quiete in fine primæ partis tempo­ris AB; certè in fine &longs;ecundæ partis temporis BC acquiret velocitatem, quæ coniuncta cum priore BI faciet duplam CH, & in fine tertiæ par­tiæ CD triplam DG; denique in fine quartæ DE quadruplam EF; quip­pe cum in parte BC remaneat tota velocitas B, & acquiratur æqualis; certè in fine BC e&longs;t velocitas CH dupla illius quæ commen&longs;uratur BI. &longs;uniliter in parte CD remanebit vtraque, & accedet altera; igitur e&longs;t ve­locitas DG tripla BI, & EF e&longs;t quadrupla: Similiter ita &longs;e ratio habet cuiu&longs;libet alterius partis inter AB ad aliam alterius partis inter BC, vt lineæ ductæ parallelæ BICH, &c. igitur cum &longs;patium acqui&longs;itum re&longs;­pondeat exercitio huius velocitatis; &longs;itque in&longs;tanti B vt BI, & in&longs;tanti C vt CH; certè tempore AB e&longs;t vt triangulum AIB; nam &longs;patium AIB e&longs;t collectio omnium linearum, quæ duci po&longs;&longs;unt parallelæ in tempote AB; idem dico de trapezo CBIH, qui e&longs;t triplus trianguli IBA; & de trapezo GDCH, qui e&longs;t quintuplus; igitur triangulum HCA e&longs;t qua­druplum IBA; quia hæc triangula &longs;unt vt quadrata laterum; igitur &longs;pa­tium acqui&longs;itum temporibus AB, BC, e&longs;t ad &longs;patium acqui&longs;itum tempo­re AB, vt triangulum HCB ad triangulum IBA; igitur vt quadratum AB ad quadratum AC; igitur vt quadratum temporis AB ad quadra­tum temporis AC; igitur &longs;patia diuer&longs;is temporibus decur&longs;a &longs;unt vt qua­drata temporum, quibus &longs;ingula decurruntur.

Hæc ratio ad &longs;peciem videtur e&longs;&longs;e demon&longs;tratiua, deficit tamen à ve­ra demon&longs;tratione; primo, quia &longs;upponit in&longs;tantia infinita, quæ multi pa&longs;&longs;im negabunt in tempore; immò aliquis vltrò demon&longs;trare tentaret non e&longs;&longs;e infinita; itaque ex &longs;uppo&longs;itione quod &longs;int tantùm finita in&longs;tan­tia a&longs;&longs;umantur 4. æqualia AC, CD, DE, EF, certè cum in&longs;tans &longs;it to­rum &longs;imul, velocitatem habet æquabilem &longs;ibi toti re&longs;pondentem; igitur in&longs;tanti AC re&longs;pondeat velocitas, cuius men&longs;ura &longs;it ABCG; haud du­biè in&longs;tanti CD re&longs;pondebit velocitas CH, &longs;cilicet dupla AB; nam re­manet primus velocitatis gradus acqui&longs;itus primo in&longs;tanti: &longs;ed alter æ­qualis acquiritur; igitur e&longs;t duplus prioris; igitur re&longs;pondet lineæ DK. quæ tripla e&longs;t AB, & quarto lineæ FN, quæ e&longs;t quadrupla AB; igitur cre&longs;cit &longs;patium, vt rectangula CB, DH, EK, FM; &longs;ed hæc cre&longs;cunt iuxta progre&longs;&longs;ionem numerorum 1.2.3.4. nec aliter res e&longs;&longs;e pote&longs;t ex &longs;uppo&longs;i­tione quod &longs;int in&longs;tantia finita; quod alibi ex profe&longs;&longs;o tractamus: quippe illa quæ&longs;tio pertinet ad Metaphy&longs;icam, non verò ad phy&longs;icun; nam vel &longs;ingula aliquid addunt, vel nihil: aliquid addunt haud dubiè; igitur con­&longs;iderantur tantùm 4. in&longs;tantia prima AC, CD, DE, EF, in &longs;ua &longs;crie; certè non po&longs;&longs;unt aliam progre&longs;&longs;ionem facere quàm eam, quæ e&longs;t iuxta hos numeros 1.2.3.4.vnde non fit per triangula &longs;ed per rectangula minima; igitur linea AF præcedentis figuræ non e&longs;t recta, &longs;ed denticulata, qualis e&longs;&longs;et ABGHIKLMN, &longs;ed longè minoribus gradibus, &longs;eu denticulis. Hinc quò rectangula CB, DH, &c. fient maiora in partibus &longs;cilicet tem­poris &longs;en&longs;ibilibus, &longs;eruata &longs;cilicet in illis progre&longs;&longs;ione numerorum 1.2.3. 4.progre&longs;&longs;io longiùs di&longs;cedet à vera; vt &longs;uprà iam totius repetitum fuit: quippe hæc progre&longs;&longs;io in puris in&longs;tantibus fieri tantùm pote&longs;t, cum &longs;in­gulis in&longs;tantibus noua fiat acce&longs;&longs;io velocitatis, in hoc enim e&longs;t error, quòd in tota parte temporis AC ponatur æquabilis velocitas, eiu&longs;que principium A, &longs;it æquale fini C; nam AB, & GH &longs;unt æquales; cùm ta­men &longs;it minor velocitas in A, quàm in C, ni&longs;i AC &longs;it tantùm in&longs;tans; vnde tota velocitas in hypothe&longs;i Galilci acqui&longs;ita in 4.partibus temporis a&longs;­&longs;umptis e&longs;t, vt triangulum AFN; acqui&longs;ita verò in no&longs;tra hypothe&longs;i e&longs;t vt &longs;umma rectangulorum CB, CI, EK, EN, quæ &longs;umma e&longs;t ad triangulum AFN, vt 10, ad 8. vel vt 5.ad 4. igitur maior 1/4; nam prima pars tempo­ris addit triangulum ABG, &longs;ecunda GHI. &c.

Si tamen diuidantur i&longs;tæ partes temporis in minores v. g. in 8. tunc &longs;umma rectangulorum erit tantùm maior 1/8; &longs;i in 16. (1/16) &longs;i in 32. (1/32); &longs;i in 64.(11/64), cuius &longs;eliema hîc habes; &longs;int enim 3.partes temporis &longs;en&longs;ibiles A CDFE, & &longs;patium vt triangulum AFN, &longs;patia verò acqui&longs;ita in &longs;ingulis partibus, vt portiones trianguli prædicti, quæ ip&longs;is re&longs;pondent v. g. ac­qui&longs;itum in prima parte ad acqui&longs;itum in &longs;ecunda tantùm, vt triangu­lum ACG ad trapezum GCDI &c. denique acqui&longs;itum in temporibus inæqualibus, vt quadrata temporum v. g. acqui&longs;itum in prima parte ad acqui&longs;itum in duabus, vt triangulum ACG ad triangulum ADI; id e&longs;t quadratum CA ad quadratum DA; in no&longs;tra verò hypothe&longs;i, &longs;i velocitas in tota prima parte AC ponatur vt CG æquabiliter; haud dubiè &longs;patium acqui&longs;itum in prædictis 4. temporibus erit, vt &longs;umma rectangulorum C B, CI, EK, EN, quæ maior e&longs;t toto triangulo, AFN, 4. triangulis ABG, GHI, IKL, LMN, ie e&longs;t 1/4 totius trianguli AFN; atque ita &longs;umma re­ctangulorum continet 10. quadrata æqualia quadrato CB, & triangu­lum AFN, continet. tantùm 8.

Iam verò diuidantur 4. partes temporis AF, in 8. æquales; in &longs;enten­tia Galilei totum &longs;patium erit &longs;emper triangulum AFN, id e&longs;t vt &longs;ubdu­plum quadrati &longs;ub AF; quæ cùm &longs;it 8. quadratum erit 64.& &longs;ubduplum quadrati 32. at verò &longs;umma rectangulorum e&longs;t 36. id e&longs;t continet 36. quadrata æqualia quadrato XA; cùm tamen triangulum AFN, conti­neat tantùm 32. igitur &longs;umma prædicta e&longs;t ad triangulum AFN, vt 36. ad 32. id e&longs;t vt 9.ad 8. igitur &longs;umma e&longs;t maior triangulo 1/8, quæ omnia con&longs;tant.

Præterea diuidatur vlteriùs tempus AF in 16. æquales partes; qua­dratum 16. cum &longs;it 256. accipiatur &longs;ubduplum id e&longs;t 128. & erit trian­gulum AFN, cui &longs;emper re&longs;pondet totum &longs;patium acqui&longs;itum in &longs;enten­tia Galilei; at verò &longs;umma rectangulorum erit 136. igitur &longs;umma e&longs;t ad &longs;ummam vt 136.ad 128.id e&longs;t vt 17.ad 16. igitur e&longs;t maior &longs;umma trian­gulo (1/16) atque ita deinceps; &longs;i vlteriùs diuidas prædictum tempus in par­tes minores: quot porrò erunt, antequam fiat tota re&longs;olutio in in&longs;tan­tia, &longs;int enim v. g. in tempore AF in&longs;tantia 1000000. &longs;umma quæ re&longs;­pondet no&longs;træ progre&longs;&longs;ioni, erit maior altera, quæ re&longs;pondet progre&longs;&longs;io­ni Galilei (1/1000000)quis hoc percipiat?

Si verò in no&longs;tra hypothe&longs;i &longs;patium, quod re&longs;pondet primæ parti tem­poris AC &longs;it idem cum illo, quod re&longs;pondet eidem parti in &longs;ententia Galilei, id e&longs;t æquale triangulo CAG, &longs;umma &longs;patiorum erit minor in no&longs;tra hypothe&longs;i triangulo AFN &longs;ex triangulis æqualibus triangulo ACG; igitur erit vt 10.ad 16. igitur minor . &longs;i verò diuidantur in 8. temporis partes, triangulum AFN continebit 64. triangula æqualia AXQ: at verò &longs;umma quæ re&longs;pondet no&longs;træ hypothe&longs;i 36.igitur minor (7/16). denique &longs;i diuidantur in 16. partes, triangulum AFN continebit 256. triangula æqualia AYZ; at verò &longs;umma no&longs;tra 136. igitur minor (15/52) &longs;ed nunquam erit minor 1/2.

Ob&longs;eruabis obiter dictum e&longs;&longs;e &longs;uprà &longs;ummam rectangulorum CB CI EK EN e&longs;&longs;e maiorem triangulo AFN, 2.quadratis æqualibus CB; &longs;i verò diuidatur tempus in 8. partes, &longs;umma rectangulorum e&longs;t minor præ­cedenti &longs;ummâ, toto quadrato æquali CB, id e&longs;t 4.quadratis æqualibus XB, id e&longs;t 1/2 primæ differentiæ, quæ e&longs;t &longs;umma duorum quadratorum æqualium CB; at &longs;i diuidatur in 16. partes, tempus AF, &longs;umma rectan­gulorum e&longs;t minor præcedente 8. quadratis æqualibus Q 7., vel &longs;ubdu­plo quadrati CB, id e&longs;t 1/4 primæ differentiæ quæ e&longs;t &longs;umma duorum quadratorum æqualium CB; &longs;i 4. partes temporis diuidantur in 8. de­trahitur 1/2 differentiæ, quæ e&longs;t inter &longs;ummam primam rectangulorum, & triangulum AFN; &longs;i diuidantur in 16. detrahitur 1/4 eiu&longs;dem diffe­rentiæ; &longs;i diuidantur in 32.detrahitur 1/, &longs;i in 64. (1/16); atque ita deinceps, & nunquam hæ minutiæ &longs;ubtractæ in infinitum totam differentiam ex­haurient; hinc minutiæ i&longs;tæ 1/2 1/4 1/8 (1/16) (1/32) (1/64) &c. in infinitum non fa­ciunt vnum integrum; &longs;ed hæc &longs;unt facilia.

Quarta ratio, quam afferunt aliqui, e&longs;t; quia &longs;i cum eadem velocita­te acqui&longs;ita in fine temporis dati &longs;ine augmento nouo moueatur mobi­le; haud dubiè acquiret duplum &longs;patium tempore æquali tempori dato; v. g. &longs;it triangulum AFE; &longs;itque velocitas acqui&longs;ita EF in 4. parti­bus temporis AE, vt iam &longs;uprà dictum e&longs;t, ne cogar repetere: certè &longs;i du­catur velocitas EF in tempus AE, vel EL æquale; habebitur rectan­gulum EK duplum trianguli AFE: &longs;ed triangulum AFE e&longs;t &longs;umma &longs;patiorum motus accelerati tempore AE, & rectangulum EK e&longs;t &longs;um­ma &longs;patiorum motus æquabilis cum velocitate EF; igitur duplum e&longs;t &longs;patium motus quabilis, quod erat demon&longs;trandum. Præterea &longs;i diai­datur velocitas EF, & eius &longs;ubdupla ducatur in tempus AE; habebitur rectangulum æquale triangulo AFE, vt con&longs;tat. Re&longs;pondeo facilè ex di­ctis, hoc ip&longs;um etiam ex no&longs;tra hypothe&longs;i proxime &longs;equi; &longs;int enim duo in&longs;tantia; haud dubie &longs;i non cre&longs;cit velocitas, &longs;ecundo in&longs;tanti æquale &longs;patium percurretur; &longs;i vero &longs;ecundo in&longs;tanti cre&longs;cat, percurrentur illo motu 3.&longs;patia; & cùm velocitas &longs;ecundi in&longs;tantis &longs;it dupla velocitatis primi in&longs;tantis, primo in&longs;tanti &longs;it 1.gradus v.g. &longs;ecundo crunt 2. gradus; igi­tur moueatur per duo in&longs;tantia motu æquabili veloci vt 2. percurrentur 4. &longs;patia; igitur totum &longs;patium, quod percurritur motu veloci vt 2. per 2.in&longs;tantia e&longs;t ad totum &longs;patium, quod percurritur æquali tempore mo-tu naturaliter accelerato vt 4. ad 3. igitur continet illud 1. (11/3); &longs;i verò &longs;int 3. in&longs;tantis continet illud, 1/2; &longs;i 4. continet 1. 3/5, &longs;i 5. continet 1.2/3 &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 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 accedet propiùs ad rationem duplam, nunquam tamen ad illam perue­niet. Ex dictis multa tumultuatim Corollaria congeri po&longs;&longs;unt;

Corollarium 1.

Etiam&longs;i non &longs;int partes infinitæ temporis; in ordine tamen ad praxim eodem modo &longs;e habent, ac &longs;i e&longs;&longs;ent infinitæ; quia licèt finitæ &longs;int, nume­rari tamen non po&longs;&longs;unt.

Corollarium 2.

Etiam &longs;i non &longs;int infiniti tarditatis gradus, vt con&longs;tat ex dictis, &longs;ed fi­niti; in ordine tamen ad praxim eodem modo &longs;e habent, ac &longs;i e&longs;&longs;ent in­&longs;initi; quia non pote&longs;t di&longs;tingui primus, & minimus ab omnibus al is.

Corollarium 3.

Licèt hypothe&longs;is Galilei &longs;it fal&longs;a in hypothe&longs;i in&longs;tantium finitorum; nam &longs;ingulis in&longs;tantibus noua fit velocitatis acce&longs;&longs;io; phy&longs;icè tamen lo­quendo eodem modo &longs;e habet, ac &longs;i e&longs;&longs;et vera; quia cum non po&longs;&longs;it pro­bari, ni&longs;i in partibus temporis &longs;en&longs;ibilibus; certà, cùm quælibet pars &longs;en&longs;ibilis innumera ferè in&longs;tantia contineat, in quibus fit progre&longs;&longs;io; differentia vtriu&longs;que &longs;en&longs;ibilis e&longs;&longs;e non pote&longs;t; igitur linea denticulata 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;ic­que progre&longs;&longs;io arithmetica in multis terminis-reducitur &longs;en&longs;ibiliter ad Geometriam in paucioribus terminis; immò in communi illa &longs;ententia. in qua dicitur tempus con&longs;tare ex partibus actu infinitis, progre&longs;&longs;io Ga­lilei tantùm locum habere pete&longs;t; igitur hæc e&longs;to clauis huius difficul­tatis; progre&longs;&longs;io &longs;implex principium phy&longs;icum habet, non experimen­tum; progre&longs;&longs;io numerorum imparium experimentum non principium; vtramque cum principio & experimento componimus; prima enim &longs;i. a&longs;&longs;umantur partes temporis &longs;en&longs;ibiles tran&longs;it in &longs;ecundam, &longs;ecunda in primam, &longs;i vltima a&longs;&longs;umantur in&longs;tantia.

Corollarium 4.

Cognito &longs;patio quod percurritur in data parte temporis &longs;en&longs;ibili, co­gno&longs;ci pote&longs;t &longs;patium quod in duabus æqualibus vel 3.vel 4.&c.percurri pote&longs;t.v.g. multi probarunt &longs;æpiùs primo &longs;ecundo minuto corpus graue percurrere 12. pedes; igitur duobus percurreret 48. accipe enim 9. 2. id e&longs;t 4. & in 4. duces 12. vt habeas 48. 4. verò minutis percurret 192. nam accipe 9. 4. id e&longs;t 16. & in 16. duces 12.vt habeat 192. res omninò facilis.

Corollarium 5.

Similiter cognito &longs;patio quod percurrit 4. &longs;ecundis minutis, cogno­&longs;ues &longs;patium, quod percurret 2. vel 1. v.g. percurrit 4. &longs;ecundis 192. pe-des; accipe 9.4. id e&longs;t 16. & per 16. diuide 192. quotíens dabit 12. pro primo &longs;ecundo: accipe 9.2. id e&longs;t, 4. & diuide 192. per 4.quotiens dabit 48. pro duobus minutis, atque ita deinceps.

Corollarium 6.

Similiter cognito tempore cogno&longs;ci pote&longs;t &longs;patium decur&longs;um; quia &longs;patia &longs;unt vt quadrata temporum; vel cognito &longs;patio cogno&longs;ci pote&longs;t tempus; quia tempora &longs;unt, vt radices &longs;patiorum, hæc elementa &longs;altem Arithmetices de&longs;iderant.

Sed iam re&longs;tat, vt &longs;oluamus objectiones aliquas, quæ contra motus ac­celerationem pugnare videntur.

Prima objectio e&longs;t; &longs;i motus acceleratio fieret in in&longs;tantibus, &longs;ecundo in&longs;tanti idem corpus e&longs;&longs;et in duobus locis adæquatis quod &longs;ic o&longs;tendo: &longs;it &longs;patium AB quod percurrit corpus graue primo in&longs;tanti; haud du­biè AB, e&longs;t eius locus adæquatus; &longs;ecundo in&longs;tanti percurrit BC duplum AB; igitur eodem in&longs;tanti re&longs;pondet loco BD, & DC, quorum vterque e&longs;t æqualis AB; igitur &longs;ecundo in&longs;tanti e&longs;t in duobus locis, &longs;cilicet BD & DC, quod dici non pote&longs;t.

Hæc objectio impugnat omnem velocitatem; hoc e&longs;t, non modò eam, quæ motui naturaliter accelerato competit; verùm etiam illam, quæ ine&longs;t motui violento; igitur vt re&longs;pondeam faciliùs; &longs;uppono punctum phy&longs;icum, mobile &longs;cilicet A; aut &longs;i mauis Angelum coëxten&longs;um quadra­to A; qui &longs;cilicet moueatur motu accelerato, & primo in&longs;tanti acquirat locum immediatum æqualem priori, &longs;cilicet AB; licèt enim po&longs;&longs;et ac­quirere vibrationem participantem de priori; quia tamen acquireret tandem non participantem, id e&longs;t, quæ tota &longs;it extra illam, cui e&longs;t imme­diata, qualis e&longs;t AB. &longs;uppono hîc acquiri vibrationem non participan­tem de priori, id e&longs;t &longs;patium AB, æquale priori, in quo erat A, & pror­&longs;us extra illud po&longs;itum licèt immediatum; hoc po&longs;ito, primo in&longs;tanti pun­ctum A acquirit AB tanquam locum adæquatum, vt certum e&longs;t: certum e&longs;t etiam loca BC, CD, e&longs;&longs;e adæquata: igitur &longs;imul', id e&longs;t eodem in­&longs;tanti in vtroque e&longs;&longs;e non pote&longs;t; nam in&longs;tans &longs;imul totum e&longs;t; ur &longs;ecundo in&longs;tanti non percurrit BC, &longs;ed &longs;ecundo tempore æquali primo; hoc enim &longs;ecundum tempus con&longs;tat duobus in&longs;tantibus, quod &longs;unul vtrumque re&longs;pondet primo: quippe dari po&longs;&longs;unt in&longs;tantia phy&longs;ica; igitur primum in&longs;tans quo percurritur AB e&longs;t æquale duobus aliis, quibus percurruntur BD, & CD; vnde quando dixi primo in&longs;tanti acquiri &longs;pa­tium duplum primi, idem e&longs;t, ac &longs;i dixi&longs;&longs;em &longs;ecundo tempore æquali pri­mo, quod reuerà tempus con&longs;tat 2. in&longs;tantibus, quorum alterum re&longs;pon­det &longs;patio BC, & alterum &longs;patio DC.

Secunda objectio; Sed inquiet aliquis, igitur non e&longs;t continua acce­leratio motus; nam in&longs;tans quo percurritur &longs;ecundum &longs;patium BD, cùm &longs;it æquale in&longs;tanti quo percurritur tertium &longs;patium DC, in vtroque &longs;pa­tio e&longs;t æquabilis motus. Re&longs;pondeo in&longs;tans quo percurritur &longs;ecundum &longs;patium BD, e&longs;&longs;e maius in&longs;tanti, quo percurritur tertium &longs;patium DC; tà tamen lege, vt vtrumque &longs;imul &longs;umptum &longs;it onminò equale in&longs;tanti, quo percurritur primum &longs;patíum AB; &longs;imiliter totum &longs;patium CG ita percurritur tertio tempore, vt &longs;ingula &longs;patia CE. EI. FG. &longs;ingulis in­&longs;tantibus percurrantur; &longs;ed hæc tria in&longs;tantia &longs;imul &longs;umpta &longs;unt æqualia primo in&longs;tanti, quo percurritur &longs;patium; licèt primum quo percurritur CE &longs;it maius &longs;ecundo, quo percurritur EF, & hoc maius tertio, quo per­curritur FG, atque ita deinceps.

Ob&longs;eruabis po&longs;&longs;e velocitatem motus explicari duobus modis. Primò, &longs;i a&longs;&longs;umantur tempora æqualia, & &longs;patia inæqualia in ea progre&longs;&longs;ione, quam hactenus explicuimus. Secundò &longs;i accipiantur &longs;patia æqualia & tempora inæqualia, quod duobus modis fieri tantùm pote&longs;t. Primò &longs;i ac­cipiantur &longs;patia æqualia primo &longs;patio, quod percurritur primo in&longs;tanti. Secundò &longs;i accipiantur &longs;patia æqualia alteri &longs;patio, quod in parte tempo­ris &longs;en&longs;ibili percurritur; in qua verò proportione tempora fiant &longs;emper minora, 'dicemus infrà; nec dicas durum e&longs;&longs;e dicere in&longs;tans e&longs;&longs;e po&longs;&longs;e minus in&longs;tanti; nam equidem fateor in&longs;tanti mathematico nihil e&longs;&longs;e po&longs;&longs;e minus; &longs;ecus verò in&longs;tanti phy&longs;ico, quod e&longs;t diui&longs;ibile potentiâ, vt dicemus aliàs; nomine in&longs;tantis phy&longs;ici intelligo durationem indiui&longs;i­bilem, hoc e&longs;t, cuius entitas tota &longs;imul e&longs;t.

Tertia objectio. Sed inquies, igitur &longs;ecundo tempore æquali primo acquiruntur 2.gradus velocitatis, vel impetus; igitur tria &longs;patia &longs;ecun­do tempore percurruntur, quod e&longs;t contra hypothe&longs;im; quippc duo gra­dus impetus accedunt primo, &longs;imiliter tertio tempore producentur tres gradus impetus; qui &longs;i iungantur tribus præcedentibus, erunt 6. Igitur percurrentur tertio tempore 6. &longs;patia, & quarto 10.quinto 15. quia &longs;in­gulis in&longs;tantibus debet produci impetus; e&longs;t enim cau&longs;a nece&longs;&longs;aria ap­plicata.

Re&longs;pondço, equidem co in&longs;tanti, quo percurritur &longs;patium BD, pro­duci aliquid impetus, & aliquid eo in&longs;tanti, quo percurritur &longs;patium DC; ita vt tamen totus ille impetus, qui producitur his duobus in&longs;tan­tibus, &longs;it æqualis illi, qui producitur primo in&longs;tanti, quo &longs;cilicet percurri­tur &longs;patium AB; quia duo illa in&longs;tantia &longs;unul &longs;umpta faciunt tempus æquale primo in&longs;tanti; atqui temporibus æqualibus eadem cau&longs;a nece&longs;­&longs;aria non impedita æqualem effectum producit per Ax.3.hinc vides &longs;in­gulis in&longs;tantibus eadem proportione decre&longs;cere impetum in perfectio­ne, qua tempus e&longs;t breuius, &longs;eu velocior motus; &longs;ed de hoc infrà.

Quarta objectio; &longs;i impetus &longs;ingulis in&longs;tuitibus cre&longs;ceret, vel intende­retur, augeretur grauitatio: quippe &longs;i grauitas primo in&longs;tanti producat vnum gradum impetus; &longs;ecundo æqualem producet, & rertio, atque ita deinceps, quod e&longs;&longs;et ab&longs;urdum; alioqui minima atomus quodlibet cor­pus graue adæquaret, quod e&longs;t ab&longs;urdum.

Re&longs;pondeo nunquam impetum intendi, ni&longs;i &longs;it motus, qui e&longs;t illius fi­nis; alioquin fru&longs;tra e&longs;&longs;et per plura in&longs;tantia; igitur de&longs;trui deberet; nec dicas impetum naturalem etiam fru&longs;trà e&longs;&longs;e &longs;ine motu; quia cum mo­tus non &longs;it eius finis adæquatus; non mirum e&longs;t &longs;i po&longs;&longs;it e&longs;&longs;e &longs;ine motu; atqui iam diximus &longs;uprà habere duos fines, quorum alterum &longs;emper ha-bet; primus e&longs;t grauitatio, &longs;eu ni&longs;us ver&longs;us centrum; &longs;ecundus motus deor&longs;um; cùm tamen impetus addititius motum tantùm pro fine habeat; igitur &longs;i impeditur totus motus, non producitur hic impetus.

Quinta objectio; &longs;i impetum &longs;uum intendit corpus graue; &longs;imiliter Ignis diceretur intendere calorem; Sol lucem, &c. Re&longs;pondeo primò de luce &longs;ingularem e&longs;&longs;e rationem; quia &longs;cilicet con&longs;eruatur à cau&longs;a &longs;ua pri­mo productiua; quidquid &longs;it; &longs;i viderem effectum caloris, vel frigoris perpetuò cre&longs;cere; haud dubiè dicerem etiam cau&longs;as ip&longs;as intendi; atqui hoc ip&longs;um video in motu naturali, qui effectus impetus e&longs;t; adde quod argumentum à pari debile e&longs;t; cum enim &longs;int diuer&longs;i naturæ fines, diuer­&longs;æ &longs;unt viæ quibus &longs;uos fines con&longs;equítur; denique contrarietas caloris, & frigoris impedit fortè, ne vlterius vtraque qualitas intendatur, de qua fusè &longs;uo loco; porrò dicemus Tomo &longs;exto calorem con&longs;eruari à cau&longs;a &longs;ua primo productiua; quo po&longs;ito ce&longs;&longs;at difficultas; quod licèt alicui durum videri po&longs;&longs;it, demon&longs;trabo tamen.

Sexta objectio; igitur &longs;i ex infinita di&longs;tantia lapis de&longs;cenderet, inten­deret etiam &longs;uum motum. Re&longs;pondeo primò, non po&longs;&longs;e dari infinitam il­lam di&longs;tantiam. Secundò etiani&longs;i daretur lapis, ex ea non caderet; fru&longs;trà enim e&longs;&longs;et ille motus: Tertiò, &longs;i daretur motus infinitus, haud dubiè e&longs;&longs;et æquabilis; qualis e&longs;t motus circularis corporum cœle&longs;tium; at verò motus naturalis deor&longs;um corporum grauium debet e&longs;&longs;e acceleratus ne vel de&longs;cenderent tardiùs, &longs;i cum primo tantùm velocitatis gradu de&longs;cen­derent; vel &longs;u&longs;tineri vix po&longs;&longs;ent, &longs;i impetum innatum intontiorem habe­rent; vtrum verò &longs;emper intendatur, & ex quacumque altitudine cadat corpus graue, videbimus infrà.

Ex dictis hactenus facilè refelluntur aliæ &longs;ententiæ de proportione motus naturaliter accelerati.

Et primò quidem illa, quæ vult fieri &longs;ecundum rationem &longs;inuum ver&longs;orum, licèt initio tàm propè accedat ad proportionem Galilei, vt di&longs;cerni &longs;en&longs;ibiliter ab ea non po&longs;&longs;it; quare tutò &longs;atis a&longs;&longs;umi po­terit, &longs;i quando &longs;it opus illius loco, quod nos in explicandis motibus cœ­le&longs;tibus præ&longs;tabimus; interim quia faciliùs explicatur in motu recto per rationem quadratorum quàm &longs;inuum, illam retinebimus; præ&longs;ertim cùm vtraque ad no&longs;tram reducatur; modò progre&longs;&longs;io fiat in in&longs;tantibus. Secundò reiicitur &longs;ententia illorum qui volunt hanc progre&longs;&longs;ionem fie­ri iuxta proportionem geometricam, quam vides in his numeris 1.2.4.8. 16. quæ licèt initio minùs recedat à vera, in progre&longs;&longs;u tamen multùm aberrat, nec e&longs;t vlla ratio quæ pro illa faciat: Et verò nulla in mentem venire pote&longs;t; ni&longs;i fortè dicatur, cùm &longs;ecundo in&longs;tanti &longs;it dupla velocitas, tertio ponendam e&longs;&longs;e quadruplam, & 4°. octuplam; quia vt velocitas pri­mi in&longs;tantis e&longs;t ad velocitatem &longs;ecundi, ita velocitas huius ad velocita­tem tertij, & velocitas huius ad velocitatem quarti; igitur &longs;equitur pro­gre&longs;&longs;ionem rationis geometricæ duplæ; cur enim e&longs;&longs;et maior ratio pri­mi in&longs;tantis ad &longs;ecundum quàm &longs;ecundi ad tertium tertij ad quartum? &c. &longs;ed profectò vix vlla apparet rationis &longs;pecies, cùm nulla &longs;it cau&longs;a, quæ 3°in&longs;tanti, & 4°plùs agat quam primo, & &longs;ecundo; igitur e&longs;t peculiaris cau&longs;a huius inæqualitatis rationum; quòd &longs;cilicet æqualibus temporibus æqualia acquirantur velocitatis momenta; vt &longs;uprà demon&longs;trauimus; quippe id præ&longs;tari debet in explicandis inæqualitatibus motuum recto­rum naturalium, quod præ&longs;tant A&longs;tronomi in explicanda inæqualitate motuum cæle&longs;tium; qui &longs;emper æqualitatem aliquam &longs;upponunt, nec e&longs;t quòd hanc &longs;ententiam nonnullis experimentis ictuum qui&longs;quam con­firmet, in quibus multa fraus &longs;ube&longs;&longs;e pote&longs;t.

Tertiò reiicitur illa quoque &longs;ententia, quæ proportionem lineæ &longs;ectæ in mediam, & extremam rationem huic lineæ tribuit, quam ferè in his numeris vides 1.2.3.5.8, 13. 21. 34. 55. quæ &longs;ub finem etiam longi&longs;&longs;imè aberrat, vt videre e&longs;t, quare ii&longs;dem rationibus impugnatur, quibus iam aliam impugnauimus.

Scio e&longs;&longs;e alias multas rationes, quibus aliqui recentiores motus natu­ralis accelerationem explicare nituntur, &longs;ed iam &longs;uprà &longs;atis &longs;uperque re­iectæ fuerunt, vel profectò eæ &longs;unt, quæ ne quidem inter fabulo&longs;a poë­tarum commenta locum aliquem habere po&longs;&longs;int: Et verò ni&longs;i me ani­mus fallit in re clari&longs;&longs;ima, rationem huius effectus ex communibus principiis deductam cum ip&longs;is etiam experimentis con&longs;entire hactenus ita demon&longs;trauimus, vt iam vix vllus dubitationi locus relinquatur; &longs;ed interrruptam Theorematum &longs;eriem tandem repetimus.

Theorema 62.

Si accipiantur &longs;patia æqualia primo &longs;patio, quod vno in&longs;tanti percurritur, in&longs;tantia &longs;unt inæqualia in motu natur aliter accelerato; probatur, quia &longs;e­cundum &longs;patium æquale primo percurritur motu velociore, quàm pri­mo, & tertium quam &longs;ecundo: ergo minori tempore per Def.2.l.1. &longs;ed primum &longs;patium conficitur vno in&longs;tanti; igitur &longs;ecundum vno in&longs;tanti, &longs;ed minore; idem dico de tertio.

Theorema 63.

In ea proportione decre&longs;cunt hæc instantia, vt primum &longs;it maius &longs;ecundo, &longs;ecundum tertio, tertium quarto, quartum quinto, quintum &longs;exto, atque ita deinceps; ita vt &longs;ecundum & tertium &longs;imul &longs;umpta, item quar­tum, quintum, &longs;extum, &longs;eptimum, item octauum, nonum, decimum, &longs;imul &longs;umpta adæquent primum, hoc e&longs;t vt vnum, duo, tria, quatuor, quinque, &longs;ex, &c. faciant &longs;emper tempora æqualia, quia temporibus æqualibus æ­qualia acquiruntur velocitatis momenta? igitur &longs;i primo in&longs;tanti per­curritur vnum &longs;patium; &longs;ecundo tempore æquali percurruntur duo &longs;pa­tia æqualia primo, & tertio, tria; atque deinceps; &longs;ed vt &longs;uprà dictum e&longs;t in re&longs;pon&longs;.ad obiect primam, vno, & codem in&longs;tanti non pote&longs;t idem cor­pus percurrere duo &longs;patia, ne &longs;imul e&longs;&longs;et in duobus locis; igitur &longs;ingula &longs;patia re&longs;pondent &longs;ingulis in&longs;tantibus licèt minoribus; &longs;ed &longs;ecundo tem­pore æquali primo in&longs;tanti percurruntur duo &longs;patia æqualia primo &longs;pa­tio; igitur &longs;ecundum, & tertium in&longs;tans debent &longs;imul &longs;umpta adæquare primum, &longs;ed non &longs;unt æqualia, vt con&longs;tat; alioquin duobus illis in&longs;tanti bus motus e&longs;&longs;et æquabilis; igitur &longs;ecundum e&longs;t maius tertio, ita vt tamen ex vtroque tempus fiat æquale primo in&longs;tanti.

Theorema 64.

Non decre&longs;cunt illa in&longs;tantia &longs;ecundum lineam &longs;extam in extremam & mediam rationrm propagatam; ita vt primum &longs;it ad &longs;ecundum, vt &longs;ecundum ad tertium, tertium ad quartum, quartum ad quintum at que ita deinceps; &longs;it enim aliqua &longs;eries numerorum, qui aliquo modo accedant ad prædi­ctam proportionem 1.2.3.5.8.13.21.34.55. &longs;itque primum in&longs;tans vlti­mus numerus 55. &longs;ecundum 34.tertium 21. atque ita deinceps: Equidem &longs;ecundum, & tertium adæquant primum; at verò quartum, quintum, &longs;extum nullo modo adæquant; immò ne quidem eius &longs;ubduplum, & multò minus 3. alij addito primo: immò &longs;i linea data duodecies propor­tionaliter diuidatur, vltimum &longs;egmentum vix e&longs;&longs;et &longs;ubcentuplum primi, vt con&longs;tar; igitur reiici debet hæc propo&longs;itio.

Theorema 65.

In&longs;tans primum non e&longs;t ad &longs;ecundum vt numerus ad numerum; nec ad tertium, quartum, quintum, &longs;extum, &c. probatur, quia nullus numerus excogitari pote&longs;t quo de&longs;ignari po&longs;&longs;it quantitas, &longs;eu perfectio, &longs;eu va­lor i&longs;torum in&longs;tantium; &longs;it enim primum in&longs;tans &longs;ecundum &longs;it 3/5. tertium 2/5 quartum 4/9 quintum 2/9 &longs;extum 2/9. Equidem &longs;ecundum, & tertium adç­quant primum; adde quod non pote&longs;t amplius &longs;eries propagari per nu­meros rationales; &longs;it autem &longs;ecundum (6/11) 3. (5/11) cum tribus aliis 4/9 1/9 7/9; cquidem &longs;i reducantur hæ 5. minutiæ, re&longs;pondebunt his (54/99) (45/99) (4/99) (12/99) (26/99): igitur &longs;ecunda crit maior quarta; at prima &longs;uperat &longs;ecundam (9/999) &longs;ecunda tertiam (1/99) tertia quartam (11/99) quarta quintam (12/99). Cur porrò hæc inæqua­litas, igitur numeri po&longs;&longs;unt a&longs;&longs;ignari; non po&longs;&longs;unt etiam poni in &longs;erie geometrica &longs;ubdupla 1. 1/2 1/4 1/8 &c. quia &longs;ecunda. & tertia non adæquant primam idem dicendum e&longs;t potiori iure de tribus aliis; nec etiam in &longs;e­rie arithmetica &longs;implici 1. 1/2 1/3 1/4 2/5 1/6; quia &longs;ecunda, & tertia &longs;unt mi­nores prima 1/6, vt quarta, quinta, &longs;exta &longs;unt minores prima (26/74).

Theorema 66.

Datur aliquis &longs;eries numerorum irrationabilium, &longs;eu &longs;urdorum minorum, & minorum; quorum primus ita &longs;uperet &longs;ecundum, &longs;ecundus tertium, tertius quartum, &c. vt &longs;ecundus, & tertius adæquent primum, item quartus, quintus, &longs;extus. item 4. alij, qui &longs;equuntur, item 5. item 6. &c. v. g. pote&longs;t dari linea AG con&longs;tans tribus partibus æqualibus, &longs;cilicet AB, BC, CG, & &longs;ecunda BC duabus BD maiore, & DC minore, & ter­tia tribus prima CE minore ED, &longs;ed maiore EF, &longs;ecunda EF maiore F G, atque ita deinceps; addi pote&longs;t quartum &longs;egmentum æquale AB; quod &longs;ubdiuidetur in 4. partes, quarum prima &longs;it maior &longs;ecunda, & hçc tertia & hæc quarta, & omnes minores FG; ita autem &longs;uperant primæ &longs;equen­tes, vt differentia primæ, & &longs;ecundæ &longs;it maior differentia &longs;ecundæ, & tertiæ, & hæc maior differentia tertiæ, & quartæ; atque ita deinceps, nec aliter res e&longs;&longs;e pote&longs;t.

Theorema 67.

Hinc partes, quo fiunt minores, accedunt propiùs ad æqualitatem, v.g. BD, & DC accedunt propiùs ad æqualitatem quàm AB, BD, & DC, CE, pro­piùs quàm CD, DB, & CE, EF, quàm EC, CD, atque ita deinceps, vt patet; hinc po&longs;t aliquot in&longs;tantia motus, æqualia ferè redduntur in&longs;tantia, vt con&longs;tat.

Theorema 68.

Hinc qua preportione decre&longs;cunt instantia, decre&longs;cit etiam per&longs;ectio impetus; quia temporibus æqualibus cadem cau&longs;a nece&longs;&longs;aria æqualem ef­fectum producit per Ax. tertium igitur inæqualem inæqualibus, per Ax. 13. num.4. igitur minorem minore tempore; igitur minorem in eadem proportione, in qua tempus e&longs;t; igitur qua proportione, &c.

Theorema 69.

Hinc vides quâm &longs;it nece&longs;&longs;aria illa diuer &longs;a perfestio impetus, quam indi­cauimus lib.1. hinc impetus productus &longs;ecundo, & tertio in&longs;tanti adæ­quat impetum productum primo, quem etiam adæquat productus quar­to, quinto, &longs;exto, item productus &longs;eptimo, octauo, nono; decimo, atque ita deinceps; hinc e&longs;t eadem differentia impetuum, quæ in&longs;tantium; hinc &longs;in­gulis &longs;patiis æqualibus primo &longs;patio, quod percurritur primo in&longs;tanti; re&longs;pondent &longs;ingula in&longs;tantia, & &longs;ingulis in&longs;tantibus &longs;inguli, & &longs;ingulares impetus; hinc non e&longs;t quod primo in&longs;tanti dicantur produci plura pun­cta impetus in eodem puncto corporis grauis; &longs;ed vnicum tantùm pun­ctum talis perfectionis &longs;cilicet phy&longs;icum; cur enim potius duo puncta, quam tria? &longs;ed quod vnum e&longs;t determinatum e&longs;t per Ax. 5. lib. 1. hinc optima ratio cur potius tali in&longs;tanti producatur impetus talis perfectie­nis quàm alterius? quippe perfectio impetus &longs;equitur perfectionem in­&longs;tantis quo producitur; hinc dicendum videtur omnia puncta impetus e&longs;&longs;e diuer&longs;æ perfectionis, vel heterogenea; vt vulgò aiunt Philo&longs;ophi; cuius rationem demon&longs;tratiuam afferemus lib. &longs;equenti cum de motu violento; hinc vides duplicem progre&longs;&longs;ionem; primam &longs;cilicet, qua ex &longs;uppo&longs;itis temporibus æqualibus acquiruntur &longs;patia inæqualia, de qua fusè &longs;uprà; in hac enim velocitas eadem proportione cum impetu cre&longs;­cit, & cum ip&longs;o tempore; hoc e&longs;t tempore triplo e&longs;t tripla, quadruplo quadrupla; item impetus in duplo tempore e&longs;t duplus, in triplo triplus; modò progre&longs;&longs;io fiat in temporibus primo in&longs;tanti æqualibus; &longs;ecunda progre&longs;&longs;io e&longs;t qua ex &longs;uppo&longs;itis &longs;patiis æqualibus tempora fluunt inæ­qualia, hoc e&longs;t minora & minora; quibus etiam re&longs;pondet impetus im­perfectior in eadem proportione temporum; prima fit per differentias æquales, & proportiones inæquales, &longs;ecunda verò per differentias inæ­quales, & proportiones inæquales.

Theorema 70.

Si a&longs;&longs;umantur &longs;patia &longs;en&longs;ibilia æqualia, tempora &longs;unt ferè in ratione &longs;ubdu­plicata &longs;patiorum; crun enim &longs;patia &longs;int vt quadrata temporum &longs;en&longs;ibiliter; certè tempora &longs;unt, vt radices i&longs;torum quadtatorum, &longs;cilicet &longs;patiorum; &longs;int enim quæcunque &longs;patia æqualia in linea AF; &longs;intque &longs;patia AC 4. AE 16. radix quadr.4. e&longs;t 2.16. verò 4. igitur tempora &longs;unt vt 4.2.&longs;i ve­rò accipiatur primum &longs;patium, quod vno tempore percurritur; tempus quo percurruntur duo &longs;patia æqualia primum e&longs;t v.2.quo percurruntur tria v.3.quo percurruntur 4.&longs;patia, 2. atque ita deinceps; igitur in praxi quæ tantùm fit in &longs;patiis &longs;en&longs;ibilibus hæc progre&longs;&longs;io adhibenda e&longs;t, il­lamque deinceps, &longs;i quando opus e&longs;t, adhibebimus.

Theorema 71.

In vacuo &longs;i corpus graue de&longs;cenderet, prædictæ proportiones accurati&longs;&longs;imè &longs;eruarentur; quia &longs;cilicet nullum e&longs;&longs;e impedimentum; at verò &longs;i aliquod intercedit impedimentum; haud dubiè non &longs;eruantur accuratè; e&longs;t autem aliquod impedimentum in medio, quantumuis liberum e&longs;&longs;e videatur, quæ omnia con&longs;tant.

Theorema 72.

Impetus naturalis addititius de&longs;truitur; patet experientiâ; quippe pila deor&longs;um cadens tandem quie&longs;cit, licèt à terra reflectatur ratione impe­dimenti, ex quo re&longs;ultat duplex determinatio, ratione cuius idem im­petus &longs;ibi aliquo modo redditur contrarius; &longs;ed de his fusè in primo libro à Th.148. ad finem v&longs;que libri: nam reuerâ duæ determinationes op­po&longs;itæ pugnant pro rata per Ax. 15.l.1. & quotie&longs;cunque idem impetus e&longs;t ad lineas oppo&longs;itas determinatus eodem modo &longs;e habet, ac &longs;i duplex e&longs;&longs;et, & quilibet &longs;uæ &longs;ube&longs;&longs;et determinationi; atqui &longs;i duplex e&longs;&longs;et oppo­&longs;itus, pugnarent pro rata; igitur tàm pugnant duæ determinationes op­po&longs;itæ in codem impetu, quàm duo impetus ad oppo&longs;itas lincas deter­minati; igitur impetus naturalis aduentitius de&longs;truitur, &c.

Theorema 73.

Impetus naturalis innatus nunquam de&longs;truitur; Probatur, quia nihil e&longs;te quod exigat eius de&longs;tructionem, quia &longs;cilicet nunquam e&longs;t fru&longs;trà; nam vel habet motum deor&longs;um, vel grauitationis effectum, vel de&longs;truit impe­tum extrin&longs;ecum in motu violento; igitur nunquam e&longs;t fru&longs;trà, cum &longs;em­per habeat aliquem effectum.

Dices lignum vi extrin&longs;eca in aqua immer&longs;um &longs;ua &longs;ponte a&longs;cendit; igitur ille gradus impetus grauitationis de&longs;truitur, & alius producitur; hæc quæ&longs;tio ad præ&longs;ens in&longs;titutum non pertinet, &longs;ed ad librum de gra­uitate, & leuitate. Igitur breuiter re&longs;pondeo illum impetum nunquam de&longs;trui, quandiu mobile grauitat, vel grauitatione &longs;ingulari, (&longs;ic corpus grauitat in manum &longs;u&longs;tinentis,) vel grauitatione communi, (&longs;ic lignum humori innatans grauitat, non quidem in aquam, at &longs;imul cum aqua;) &longs;ed de grauitate, & grauitatione in Tomo de &longs;tatibus corporum &longs;en&longs;ibi-libus, in quo o&longs;tendemus ideo lignum &longs;ur&longs;um emergere, quia ab aqua extenditur, & idco corpora &longs;ur&longs;um ire, quia alia dcor&longs;um eunt.

Theorema 74.

Quando lapis de&longs;cendit per medium aëra, impeditur aliquantulum eius motus: Probatur primò experientiâ, quæ certa e&longs;t; tàm enim aër impe­dit motum deor&longs;um, quàm &longs;ur&longs;um, vt videre e&longs;t in mobili leuiore &longs;eu ra­riore, quod etiam flante vento ob&longs;eruare omnes po&longs;&longs;unt; quomodo ve­rò impediat, dicemus aliàs; &longs;ecundò corpus immobile, in quod mobile impingitur, motum illius impedit; &longs;ed in diuer&longs;as partes aëris corpus graue impingitur in de&longs;cen&longs;u; igitur aliquantulum impeditur eius inotus.

Theorema 75.

Hinc motus naturalis deor&longs;um aliquantulum retardatur, quia nihil aliud præ&longs;tare pote&longs;t huiu&longs;modi impedimentum, ni&longs;i aliquam retardationem; igitur motus inde redditur tardior.

Theorema 76.

Hinc etiam impetus producitur imperfectior; quia ex imperfectione ef­fectus requiritur imperfectic cau&longs;æ per Ax. 13.l. 1. & quâ proportione e&longs;t tardior motus eâdem impetus e&longs;t imperfectior, per Ax. 5. excipe ta­men impetum innatum, qui &longs;emper habet cundem effectum grauitatio­nis, vel &longs;ingularis, quâ grauitas cum ip&longs;o medio, &longs;i reuerâ medium gra­uitat, de quo aliàs.

Theorema 77.

Quo medium den&longs;ius e&longs;t plus impedit motum deor&longs;um; Probatur, quia &longs;i motum impedit; certè non totum; quis enim hoc dicat; &longs;ed eæ dumta­xat partes, quibus incubat corpus graue; igitur quò &longs;unt plures huiu&longs;­modi partes, maius e&longs;t impedimentum; &longs;ed in medio den&longs;iori plures &longs;unt cum minore exten&longs;ione; hoc enim e&longs;t, quod voco den&longs;ius; igitur me­dium den&longs;ius plùs impedit.

Theorema 78.

Hinc tardiùs de&longs;cendit mobile per mediam aquam, quàm per medium aëra, quia aqua e&longs;t den&longs;ior aëre.

Scholium.

Ob&longs;erua e&longs;&longs;e aliqua corpora minus den&longs;a, quæ motum omninò im­pediunt; quippe certum e&longs;t aquam e&longs;&longs;e den&longs;iorem ligno; atqui li­gnum de&longs;cen&longs;um lapidis impedit, non verò aqua; quia &longs;cilicet lignum non e&longs;t medium, vt aqua; vt enim aliquod corpus &longs;it medium, debet e&longs;&longs;e liquidum, vt, aqua & alij liquores; vel &longs;pirabile vt aër, vapor, &c. ratio e&longs;t, quia partes ligni, vel alterius corpotis durioris, ita &longs;unt inter &longs;e con­junctæ, vel implicatæ, vt omnem tran&longs;itum intercludant, ni&longs;i corpus ip­&longs;um graue valido ictu vel impetu &longs;ibi viam aperiat; igitur vt corpus ali­quod vice medij defungatur, debet in co &longs;tatu e&longs;&longs;e, in quo eius partes modico ferè ni&longs;u &longs;eiungantur, & loco cedant; &longs;ed de his &longs;tatibus cor­porum fusè agemus Tomo 5. adde quod ad medium &longs;ufficit vacuum &longs;i motus in vacuo e&longs;&longs;e pote&longs;t, de quo alibi; quod certè e&longs;t omnium me­diorum optimum, cum nullo modo re&longs;i&longs;tar mobili.

Theorema 79.

Hinc producitur impetus imperfectior in medio den&longs;iore: quia in co tar­dior e&longs;t motus, ex cuius tarditate arguitur imperfectio impetus per Ax. 13.num.4.

Scholium.

Ob&longs;erua den&longs;itatem medij cogno&longs;ci ex cius grauitate; illud enim den&longs;ius e&longs;t, quod e&longs;t grauius & vici&longs;&longs;im; quod fusè explicabimus &longs;uo lo­co; e&longs;t enim grauitas quædam den&longs;itas, vt ait Philo&longs;ophus tùm l.4.pb.c.9.t.85. & 86. den&longs;um & rarum, inquit, &longs;unt lationis efficientia, & paulò &longs;u­periùs; e&longs;t autem den&longs;um graue, rarum verò leue, & l.8.c.7.t.55. hæc habet, graue & leue; molle & durum den&longs;itates quædam e&longs;&longs;e, & raritates videntur,quæ adnotare volui, vt vel inde con&longs;tet doctrinam hanc cum Peripate­tica optimè con&longs;entire.

Ob&longs;eruabis ctiam hîc à me non di&longs;cuti, in quo con&longs;i&longs;tat den&longs;itas, vel raritas, grauitas, vel leuitas; &longs;uppono tantùm graue illud e&longs;&longs;e, quod ten­dit deor&longs;um; leue illud, quod tendit &longs;ur&longs;um &longs;iue pellatur à grauiori, &longs;iue non, den&longs;um verò e&longs;&longs;e id quod multùm materia habet &longs;ub parua exten­&longs;ione, rarum è contrario; quorum omnium cau&longs;as, & rationes &longs;uo loco explicabimus.

Theorema 80.

Sub medium leuius corpus graue de&longs;cendit; certa e&longs;t hypothe&longs;is, ni&longs;i for­tè aliquando per accidens &longs;ecus accidat; ratio porrò petitur ex ip&longs;a grauitatis natura, quâ corpus graue tendit deor&longs;um; nihil enim aliud grauitas e&longs;t, quidquid tandem illa &longs;it; quippe corpus graue de&longs;cendit, quando medium liberum habet, idemque leuius, per quod de&longs;cendat; quod certè &longs;i grauius e&longs;&longs;et, haud dubiè non de&longs;cenderet; &longs;ic ferrum, & &longs;axum plumbo liquato innatant; cum tamen per mediam aquam de­fcendant; fic lignum aquæ &longs;upernatat, quod per liberum aëra de&longs;cendit; ratio e&longs;t, quia grauius de&longs;cendit &longs;ub medium leuius; cur autem id fiat fusè alibi explicabo; id tantùm obiter indico. Omnis motus, qui fit à principio intrin&longs;eco per lineam rectam propter locum e&longs;t, vt patet; quis enim neget corpus graue ideo de&longs;cendere &longs;ub leuius, vt occupet aliquem locum quo prius carebat, qui tamen illi connaturalis e&longs;t in hoc rerum ordine? cum à natura acceperit vim illam intrin&longs;ecam, quâ in eum lo­cum &longs;e&longs;e recipere pote&longs;t; quam certè vim intrin&longs;ecam nunquam à na­tura rebus creatis in&longs;itam e&longs;&longs;e con&longs;tat, ni&longs;i ad eum finem con&longs;equendum, cui à natura de&longs;tinantur; cur verò locus connaturalis corporis grauio­ris &longs;it ille, in quo leuiori &longs;ube&longs;t, non diu hærebit animus, quin &longs;tatim ra­tio affulgeat; cum enim corpus, quod e&longs;t &longs;uprà, &longs;u&longs;tineatur ab eo quod e&longs;t infrà; illud certè infra e&longs;&longs;e connaturalius e&longs;t, quod aptius e&longs;t ad &longs;u&longs;tinen-dum; atqui den&longs;um aptius e&longs;t ad id munus, quia plures partes &longs;u&longs;tinentis pauciores &longs;u&longs;tinent alterius leuioris, &longs;eu rarioris, vt con&longs;tat; v.g. certum e&longs;t camdem aëris partem pluribus aquæ partibus re&longs;pondere; &longs;ed de hoc alias fusè; hæc interim &longs;ufficiat indica&longs;&longs;e, vt vel aliqua ratio affulgeat; cur &longs;cilicet corpus graue &longs;ub medium leuius &longs;ua &longs;ponte de&longs;cendat; adde quod cum omne corpus graue tendat deor&longs;um, tunc vnum infra aliud de­&longs;cendit, cum &longs;unt plures partes pellentis, quàm pul&longs;i; denique per va­cuum modicum &longs;ine vlla re&longs;i&longs;tentia de&longs;cenderet.

Theorema 81.

Sub medium grauius corpus leuius minimè de&longs;cendit, &longs;ed huic inna­tat; v.g. lignum aquæ, ferrum plumbo liquato; certa e&longs;t hypothe&longs;is: ratio e&longs;t, quia ideo de&longs;cendit graue &longs;ub medium, quia grauius &longs;eu den&longs;ius e&longs;t medio; igitur, &longs;i den&longs;ius e&longs;t ip&longs;um medium, non de&longs;cendet; clarum e&longs;t; cur verò a&longs;cendat &longs;upra medium. v.g. cur lignum aquæ immer&longs;um tan­dem emergat hîc non di&longs;cutio, &longs;ed tantùm indico ab ip&longs;a aqua &longs;ur&longs;um extendi; quanta verò parte lignum emergat, dicemus aliàs, cum de in­natantibus humido.

Theorema 82.

Sub medium æquè graue corpus non de&longs;cendit, nec etiam &longs;upra a&longs;cendit; ra­tio e&longs;t, quia ideo de&longs;cendit &longs;ub medium, quia medium leuius e&longs;t, ideo a&longs;cendit &longs;upra, quia medium grauius e&longs;t; igitur &longs;i nec &longs;it grauius nec leuius, non e&longs;t quod a&longs;cendat vel de&longs;cendat; nihil tamen illius &longs;upra primam medij &longs;uperficiem extare poterit; alioqui e&longs;&longs;et leuius medio, contra &longs;uppo&longs;itionem.

Theorema 83.

Aër &longs;uam grauitatem habet; quod iam à nullo in dubium reuocari po­te&longs;t; nam &longs;i comprimatur intra vas æneum v.g. etiam minimæ cra&longs;&longs;itu­dinis; &longs;i deinde ponderetur, maius e&longs;t haud dubiè pondus, quo maior e&longs;t aëris copia intru&longs;a; atqui non modo triplum totius aëris, qui ante compre&longs;&longs;ionem totam va&longs;is capacitatem occupabat intrudi pote&longs;t, vel decuplum; verùm etiam vigecuplum; immò centuplum, & millecuplum adhibita cochleâ, vel alio mechanico organo, & aucta va&longs;is cra&longs;&longs;itudine, de quo aliàs: quanta verò &longs;it grauitas aëris comparata cum grauitate aquæ, cen&longs;et Galileus e&longs;&longs;e ferè vt 1. ad 400. Mer&longs;ennus verò vt 1. ad 1356. vel &longs;altem vt 1.ad 1300. Nos maiorem illà; hâc vero minorem e&longs;&longs;e ob&longs;eruauimus, de quo aliàs; nec enim e&longs;t præ&longs;entis in&longs;tituti, pro quo &longs;ufficiat modò, aëri aliquam ine&longs;&longs;e grauitatem; nec dicas aëra le­uem e&longs;&longs;e; nam reuerâ leuis e&longs;t, &longs;i comparetur cum aqua; grauis autem &longs;i comparetur cum a&longs;cendente halitu, vel fortè cum vacuo; nec e&longs;t quod aliquis fortè metuat, ne &longs;i aër &longs;it grauis, ab eo tandem opprimatur, nam etiam&longs;i aqua &longs;it grauis non tamen opprimit vrinatores, cuius rei veri&longs;&longs;i­mam rationem &longs;uo loco afferemus; denique non e&longs;t quod aliqui &longs;atis incautè re&longs;pondeant, ip&longs;um aëra non e&longs;&longs;e grauem, &longs;ed tantùm &longs;entiri ali­quod pondus cra&longs;&longs;ioris vaporis immixti; nam de alio aëre non affirmo grauem e&longs;&longs;e, ni&longs;i tantùm de illo, quem &longs;piramus, in quo ambulamus, qui nos ambit: adde quod Ari&longs;toteles l.4. de Cœlo, c.5.t.36. tribuit aëri gra­uitatem his verbis; quapropter inquit, aër, & aqua habent & leuitatem, & grauitatem.

Theorema 84.

Medium eiu&longs;dem grauitatis cum dato corpore graui detrahit totam eius grauitationem &longs;ingularem; hoc e&longs;t corpus graue in medium æquè graue non grauitat; quia &longs;i grauitaret de&longs;cenderet; &longs;ic pars aquæ in aliam partem aquæ non grauitat, & &longs;i aqua ponderetur in aqua, nullius ponderis e&longs;t; cum enim nulla &longs;it ratio cur vna &longs;it infrà potiùs, quàm alia, vna certè al­terius locum non ambit; igitur caret grauitatione &longs;ingulari.

Theorema 85.

Medium graue detrahit aliquid de &longs;ingulari grauit atione corporis grauio­ris; certa e&longs;t hypothe&longs;is; nec enim plumbum e&longs;t eius ponderis &longs;ingula­ris in aqua, cuius e&longs;t in aëre; dixi &longs;ingularis; nam &longs;i plumbum & ip&longs;a aqua &longs;imul appendantur, haud dubiè totum habebis pondus plumbi, & totum pondus aquæ; ratio verò huius effectus non e&longs;t huius loci; quid­quid &longs;it, &longs;i æqualis grauitas medij tollit totam æqualem alterius corpo­ris; certè maiorem alterius corporis totam non tollit per Th. 80. &longs;ed tantùm aliquid illius, quod quomodo fiat, dicemus Tomo quinto cum de graui, & leui.

Theorema 86.

Medium graue detrahit eam partem grauit ationis corporis grauioris, quæ e&longs;t æqualis &longs;uæ grauitationi. v. g. &longs;i medij grauitas e&longs;t &longs;ubdupla, detrahit &longs;ubduplum grauitationis; &longs;i &longs;ubdecupla, &longs;ubdecuplum, atque ita dein­ceps; hoc iam olim &longs;uppo&longs;uit magnus Archim. &longs;upponunt etiam reliqui omnes, præ&longs;ertim recentior Galileus; &longs;i enim æqualis &longs;uperat æqualem, ergo inæqualis pro rata; &longs;cilicet &longs;ubdupla &longs;ubduplum &longs;ubtripla, &c. Præ­terca, cum detrahat aliquam partem grauitationis maioris per Th.85.nec detrahat inæqualem maiorem, per Th.80.nec inæqualem minorem; cur enim potius vnam minorem quam aliam? certè æqualem tantùm detrahere pote&longs;t, quod &longs;uo loco per Principium po&longs;itiuum demon&longs;tra­bimus.

Theorema 87.

Hinc ratio cur grauia de&longs;cendant tardius in aqua, quàm in aëre, & in aëre, quàm in vacuo; hinc etiam maioris &longs;unt ponderis in aëre quam in aqua; hinc &longs;i grauitas alicuius corporis &longs;it ad grauitatem aëris vt 100. ad 1. haud dubiè decre&longs;cet eius pondus in aëre (1/100); id e&longs;t, &longs;i penderet 100. libras in vacuo, in aëre penderet 99.& eo tempore quo in vacuo decur­teret 100. pa&longs;&longs;us, in aëre decurreret 99. &longs;i nulla &longs;it aliunde re&longs;i&longs;tentia, qualis reuerâ e&longs;t, vt dicam infrà; &longs;imiliter &longs;i grauitas alicuius corporis &longs;it ad grauitatem aquæ, vt 10.ad 1. decre&longs;cet eius pondus in aqua (1/), & co tempore quo decurreret in vacuo 10. palmos &longs;patij, in aqua decurre ret tantùm 9. po&longs;ito quod non &longs;it aliud quod re&longs;i&longs;tat; quanta verò &longs;it grauitas omnium corporum tùm duriorum, qualia &longs;unt metall, tùm li­quidorum, tùm &longs;pirabilium, dicemus &longs;uo loco; illorum tabulas habes apud Gethaldum, & Mer&longs;ennum.

Theorema 88.

Hinc, &longs;i nihil aliud de&longs;cen&longs;um corporum grauium impediret, cognito pen­dere vtriu&longs;que, medij & corperis grauis, &longs;patio, quod in vno illorum conficit, cogno&longs;cipo&longs;&longs;et &longs;patium, quod in alio conficeret æquali tempore, v. g. &longs;uppona­mus grauitatem aquæ e&longs;&longs;e ad grauitatem aëris vt 400.ad 1.&longs;itque corpus, cuius grauitas &longs;it dupla grauitatis aquæ; haud dubiè eo tempore, quo conficit in aëre 799. &longs;patia, in aqua con&longs;iciet tantùm 400. quia in vacuo conficeret 800. aër autem detrahit (/800), & aqua /2, vt con&longs;tat ex dictis; &longs;i­militer cognitis &longs;patiis in vtroque medio confectis, & grauitate vtriu&longs;que medij cogno&longs;ceretur grauitas corporis de&longs;cendentis; quia tamen e&longs;t alia re&longs;i&longs;tentiæ ratio, hîc non hærco.

Scholium.

Ob&longs;eruabis dictum e&longs;&longs;e hactenus; &longs;i nihil aliud de&longs;cen&longs;um corporis grauis impedit; nam certè aliud e&longs;t, de quo infrà, ex cuius ignoratione plures haud dubiè in inue&longs;tigandis grauitatum medij rationibus hallu­cinarentur; cum enim ob&longs;eruatum &longs;it globum plumbeum, cuius graui­tas e&longs;t ferè dodecupla grauitatis aquæ, conficere in libero aëre 48. pedes &longs;patij tempore duorum &longs;ecundorum, in aqua verò 12. pedes codem tem­pore; certè in vacuo ip&longs;o moueretur tardiùs quàm in aëre; quia eo tem­pore, quo conficit in aqua 12.pedes in vacuo conficeret (13 1/21), &longs;i tantùm detrahitur (1/12) grauitationis, & de&longs;cen&longs;us; atqui in aëre codem tempore conficit 48. pedes; igitur velociùs moueretur in aëre quàm in vacuo; igitur e&longs;t aliquid aliud quod impedit motum; vt enim optimè monet Mer&longs;ennus, &longs;i grauitas aquæ &longs;it ad grauitatem aëris vt 400 ad 1.& graui­tas plumbi ad grauitatem aquæ vt 12. ad 1.eadem grauitas plumbi e&longs;t ad grauitatem aëris vt 4800. igitur &longs;i &longs;patium, quod decurrit plumbum in vacuo diuidatur in 4800. partes, decurret in aëre 4799. partes; in aqua verò 4400. quod e&longs;t contra experientiam; nam &longs;patium, quod decurrit in aëre e&longs;t maius &longs;patio, quod decurrit in aqua 3/4; quippe conficit 12. pedes in aqua eodem tempore, quo in aëre conficir 48; igitur in aqua amittit 3/4 &longs;uæ grauitationis, & &longs;ui motus; igitur 3600. partes; igitur plumbi grauitas e&longs;&longs;et ad grauitatem aquæ vt 4.ad 3.& ad grauitatem aë­ris vt 3600. ad 1. atqui vtrumque fal&longs;um e&longs;&longs;e con&longs;tat; igitur e&longs;t aliquid aliud, quod etiam impedit motum; nec ex motu diuer&longs;o per diuer&longs;a me­dia cogno&longs;ci pote&longs;t eorum grauitas.

Theorema 89.

Hinc potiori iure reiicies illerum &longs;ententiam, qui volunt impediri motum corporis de&longs;cendentis per diuer&longs;a media pro diuer&longs;aratione grauitatum vtriu&longs;­que medy; quod certè fal&longs;um e&longs;t; nam aqua &longs;it ad grauitatem aëris vt 400. ad 1. deberet omne corpus de&longs;cendere velociùs in aëre quadrin-gente&longs;ies, quàm in aqua, quod fal&longs;um e&longs;t; cum aliquod corpus nullo mo­do de&longs;cendat in aqua, quod de&longs;cendit in aëre, vt lignum.

Theorema 90.

Non pote&longs;t corpus graue per medium corporeum de&longs;cendere, ni&longs;i vel totum medium loco cedat, vel aliquæ partes iu&longs;dem medij, patet; quia vnum cor­pus non pote&longs;t penetrari cum alio.

Theorema 91.

Totum medium loco non cedit in de&longs;cen&longs;u grauium; patet etiam, tùm quia ad mouendum totum medium exigua vis corporis grauis non &longs;uffi­cit; tùm quia tàm facilè per medium durum eiu&longs;dem grauitatis de&longs;cen­deret; denique patet manife&longs;tâ experientiâ.

Theorema 92.

Hinc aliqua tantùm partes medij loco cedunt; probatur, quia vel totum medium, vel aliquæ eius partes, per Th.90.non primum per Th.91.igitut &longs;ecundum, in his certè non e&longs;t vlla difficultas.

Theorema 93.

Non po&longs;&longs;unt illæ partes loco cedere &longs;ine motu; nec moueri &longs;ine impetu, nec habere impetum, ni&longs;i producatur in illis à cau&longs;a aliqua applicata; quæ certè alia none&longs;t quàm impetus corporis de&longs;cendentis, vt con&longs;tat ex iis, quæ dixi­mus primo lib.

Theorema 94.

Illæ partes, quæ loco cedunt de&longs;cendenti corpori graui, nece&longs;&longs;ariò ab aliis &longs;eparantur, & &longs;uo appul&longs;u, vel impul&longs;u alias multas impellunt, ac &longs;eparant,atqui &longs;eparari non po&longs;&longs;unt ab aliis, ni&longs;i &longs;oluatur vnio, &longs;eu nexus, quo cum aliis deuinciuntur; quidquid tandem &longs;it illa vnio, de qua aliàs.

Theorema 95.

Hinc quò arctior e&longs;t ille nexus, difficilius &longs;oluitur; igitur maiore vi, vel impetu opus e&longs;t, vt &longs;olui po&longs;&longs;it, vt con&longs;tat.

Theorema 96.

Hinc corpus grauius &longs;ustinetur à leuiore. v.g. plumbum à ligno propter arctiorem nexum partium ligni, qui ab impetu plumbi quantumuis gra­ui&longs;&longs;imi &longs;uperari non pote&longs;t; hinc corpus illud, medium tantùm appello in quo po&longs;&longs;int corpora moueri, cuius nexus &longs;uperari pote&longs;t à corpore grauiori in aliqua &longs;altem figura, vel &longs;itu; hinc corpora dura non po&longs;&longs;unt e&longs;&longs;e medium; immò neque mollia, vt cera, argilla; &longs;ed vel liquida, vel &longs;pirabilia.

Theorema 97.

Hinc ducitur euidens ratio, cur medium impediat motum &longs;i dumtaxat ha­beat arctiorum partium implicationem & nexum; quia non modo partes medij amouendæ &longs;unt è &longs;uo loco; verùm etiam nexus ille partium &longs;ol­uendus; igitur ex vtroque capite impeditur motus.

Theorema 98.

Quo &longs;ubtiliores &longs;unt partes difficilius inter &longs;e implicari po&longs;&longs;unt &longs;eu ligari quibu&longs;dam filamentis, con&longs;tat; igitur cum aëris partes &longs;int magis lubricæ, quàm partes aquæ, & faciliùs per obuia quæque foramina irrepere po&longs;­&longs;int, non po&longs;&longs;unt ita contineri; &longs;ic videmus multùm aquæ hauriri, dum arctioribus retibus attollitur; immò dum aquam manu &longs;tringimus, ali­quam re&longs;i&longs;tentiam &longs;en&longs;u percipimus; quæ certè nulla e&longs;t, dum aëra &longs;trin­gimus.

Scholium.

Ob&longs;eruabis vmonem continuatiuam corporum aliquando po&longs;itam e&longs;&longs;e in plexu, vel implicatione partium, vt videmus in fune, ligno, carne, o&longs;&longs;ibus, &c. aliquando in vacui metu; &longs;ic aqua, vt &longs;uo va&longs;i adhæreat, a&longs;cendit, vel &longs;ur&longs;um attollitur, ne detur vacuum; aliquando in coitione quadam magnetica; porrò hic plexus con&longs;tat ex infinitis ferè tenui&longs;&longs;i­morum filamentorum voluminibus, vel aduncis &longs;iue hamatis partibus, &longs;eu corpu&longs;culis: Vtrum verò præter hæc requiratur alius vnionis mo­dus, di&longs;cutiemus fusè Tomo 5. quidquid &longs;it; certum e&longs;t medium illud, cuius partes arctiori maiorique nexu copulantur, longè difficiliùs per­aurri po&longs;&longs;e, &longs;eu perrumpi.

Theorema 99.

Hinc non modò aqua detrahit plumbo (1/22) &longs;ui motus, quod &longs;cilicet plumbi græ­uitas &longs;it dedecupla grauitatis aquæ, verùm etiam propter re&longs;istentiam petitam ex alio capite aliquid adhuc detrabere pote&longs;t; &longs;cilicet quia partes aquæ non po&longs;&longs;unt amoueri, ni&longs;i ab aliis &longs;eparentur; atqui maiore vi opus e&longs;t ad­&longs;oiuendum &longs;trictiorem nexum; immò licèt partes aquæ nullo penitus nexu vniantur, &longs;ed tantùm vel vacui metu, vel alio modo, quod alibi ex­plicabimus; omninò detraherent adhuc plumbo (1/12) motus; igitur, &longs;i præter illud impedimentum, quod petitur à comparatione grauitatis corporis mobilis cum grauitate medij, addatur aliud longè robu&longs;tius; non mirum e&longs;t, &longs;i maior inde &longs;equatur effectus, id e&longs;t maior imminutio motus, qui qua&longs;i frangitur ab impedimento.

Theorema 100.

Hinc petitur ratio illius experimenti, &longs;i verum e&longs;t, duobus &longs;ecundis per­currere plumbeam pilam in aëre 48. &longs;patij pedes, in aqua verò 12. pedes; hinc tenui nexu partes aëris copulantur; partes verò aquæ firmiori; hinc aër minùs re&longs;i&longs;tit etiam motibus violentis; hinc vix pote&longs;t qui&longs;piam in aqua currere propter maiorem aquæ re&longs;i&longs;tentiam; hinc pote&longs;t dici quota parte firmior &longs;it nexus vnius corporis quàm alterius; hinc non tantùm copu­lantur partes metu vacui; alioquin æquè re&longs;i&longs;terent partes aëris, ac par­tes aquæ ratione nexus; hinc videntur guttulæ illæ &longs;phericæ inuolui te­nui qua&longs;i membranula, &longs;eu &longs;uperficie, cuius analogiam videmus in aquaferuente; in bullis, quæ ex guttis pluuiæ re&longs;ilientibus na&longs;ci videntur; in bullis etiam illis &longs;aponariis, quas leui calamo pueri inter ludendum in­flant; hinc ex minimo ferè contactu guttula &longs;pargitur, ni&longs;i fortè cum multo a&longs;per&longs;a puluere cru&longs;tam quamdam induit &longs;olidiorem; &longs;ic bullæ il­læ ad minimum etiam contactum di&longs;&longs;ipantur; hinc ip&longs;a &longs;uperficies aquæ plus videtur re&longs;i&longs;tere quod multis experimentis comprobatur; &longs;ed illo maximè, quo videmus findi à remo cum quodam qua&longs;i &longs;tridulo cre­pitu re&longs;i&longs;tentiæ maioris te&longs;te; immò cum ab ip&longs;a naui qua&longs;i &longs;ulcatur, idem &longs;tridor auditur, maximè in iis tractibus; in quibus nullis fluctibus agitata læuigati&longs;&longs;imam faciem præfert; habes analogiam in illa cru&longs;ta, quæ concre&longs;cit in &longs;uperficie liquorum, &longs;ed præ&longs;ertim offarum:adde quod aër paulò compre&longs;&longs;ior vndique guttulam premens æquali ni&longs;u eam miri­ficè tornat: hæc tantùm tumultuatim conge&longs;ta alibi fusè pertractabi­mus, & ex &longs;implici&longs;&longs;imis principiis demon&longs;trabimus; plura hîc de graui­tate crant dicenda, & de grauitatione, quæ tantùm indica&longs;&longs;e &longs;ufficiat, vt deinde Tomo quinto fusè explicentur.

Theorema 101.

Non re&longs;istit medium propter compre&longs;&longs;ionem partium inferiorum, quas nullo modo comprimi nece&longs;&longs;e e&longs;t, vel in&longs;en&longs;ibiliter; cum enim tantus relinquatur locus retrò, quantus acquiritur antè, nulla opus e&longs;t compre&longs;&longs;ione; &longs;ed partes à fronte pul&longs;æ factâ circuitione retror&longs;um eunt, non certè tramite recto; &longs;i enim frons ip&longs;ius lata &longs;it, haud dubiè partes pul&longs;æ alias pellunt, & hæ vici&longs;&longs;im alias longo circuitu, vt patet experientia; nulla tamen, vel modica fieri videtur compre&longs;&longs;io.

Theorema 102.

Hinc quo &longs;unt plures partes diuidendæ, quæ antè uniebantur, maior e&longs;t re&longs;i­&longs;tentia; igitur maiore vi opus e&longs;t, igitur maiore grauitate; &longs;ed in medio den&longs;iore ab codem mobili plures &longs;eparantur quàm in rariore; quia &longs;ci­licet corpus den&longs;um plures habet &longs;ub minori exten&longs;ione, & rarum è con­trario, vt videbimus &longs;uo loco; igitur in medio den&longs;iore idem mobile ma­jorem re&longs;i&longs;tentiam inuenit, quàm in rariore; licèt vtriu&longs;que partes æquali nexu &longs;eu fibula copulentur; quia &longs;cilicet plures &longs;unt diuidendæ in den&longs;iore; quia plures &longs;cilicet in æquali &longs;patio occurrunt, quàm in ra­iore; igitur maiore vi grauitatis opus e&longs;t.

Theorema 103.

Hinc medium pote&longs;t comparari cum alio in 2. capitibus; Primum e&longs;t in grauitate, vel den&longs;itate, nam reuerâ ex maiori den&longs;itate maiorem gra­uitatem reducimus; Secundum e&longs;t in maiori, vel minori partium nexu, ex quo 4. &longs;equuntur combinationes 2.mediorum; nam vel &longs;unt eiu&longs;dem grauitatis, & mollitiei; vel eiu&longs;dem grauitatis & diuer&longs;æ mollitiei; vel ciu&longs;dem mollitici, & diuer&longs;æ grauitatis; vel diuer&longs;æ grauitatis, & ciu&longs;­dem mollitici; mollius autem illud appello, cuius partes laxiori nexu copulantur; porrò 4.i &longs;tæ combinationes &longs;upponunt idem mobile invtroque medio; &longs;i &longs;it prima combinatio, motus e&longs;t æqualis in vtroque; &longs;i &longs;ecunda maior e&longs;t in molliori; &longs;i tertia maior in grauiori; &longs;i verò quarta &longs;ubdi­uidi pote&longs;t in duas; nam vel grauius e&longs;t conjunctum cum maiori molli­tie, vel leuius; &longs;i leuius, haud dubiè maior e&longs;t motus in leuiore; &longs;i gra­uius & mollities compen&longs;et grauitatem, id e&longs;t, &longs;i vt &longs;e habet grauitas gra­uioris ad leuitatem leuioris; ita &longs;e habet mollities illius ad mollitiem huius, æqualis e&longs;t in vtroque; &longs;i &longs;ecus, pro rata; hinc pote&longs;t e&longs;&longs;e æqualis motus in grauiore & leuiore medio, & in æquè graui pote&longs;t e&longs;&longs;e maior in grauiore; & minor; maior quidem, &longs;i maior &longs;it ratio mollitici gra­uioris ad mollitiem leuioris, quàm grauitatis ad grauitatem; minor ve­rò, &longs;i maior &longs;itratio grauitatis ad grauitatem, quàm mollitici ad molli­tiem; æqualis denique &longs;i æqualis ratio; & his regulis cuncta facilè ex­plicari po&longs;&longs;unt; hîc porrò &longs;uppono idem mobile, quod per vtrumque me­dium de&longs;cendere po&longs;&longs;it, id e&longs;t, quod &longs;it vtroque grauius, medium autem appello illud, per quod mobile grauius per &longs;e de&longs;cendit; dixi per &longs;e quia nonnunquam accidit, vt vel ratione figuræ, vel alterius impedimenti non de&longs;cendat.

Theorema 104.

Sunt tres combinationes mobilis cum medio; prima, &longs;i &longs;it idem mobile cum diuer&longs;is mediis; &longs;ecunda, &longs;i idem medium cum diuer&longs;is mobilibus; tertia &longs;i diuer&longs;a mobïlia cum diuer&longs;is mediis; de primâ actum e&longs;t iam fuprà; &longs;ecunda &longs;ube&longs;t 4. combinationibus. Prima &longs;i mobilia &longs;int eiu&longs;­dem materiæ, &longs;ed diuer&longs;æ figuræ; Secunda eiu&longs;dem figuræ & diuer&longs;æ materiæ. Quarta diuer&longs;æ materiæ & figuræ; &longs;i prima & &longs;ecunda, vel &longs;unt figuræ æquales, vel inæquales; &longs;i primum &longs;unt eiu&longs;dem grauitatis; &longs;i &longs;e­cundum diuer&longs;æ; quippe figuræ &longs;imiles po&longs;&longs;unt e&longs;&longs;e æquales, vel inæ­quales; & figuræ æquales po&longs;&longs;unt e&longs;&longs;e &longs;imiles, vel di&longs;&longs;imiles; &longs;i &longs;it tertia combinatio, in qua &longs;int eiu&longs;dem figuræ, & diuer&longs;æ materiæ, diuer&longs;æ in­quam in grauitate; &longs;i figuræ &longs;unt æquales, &longs;emper e&longs;t diuer&longs;a grauitas; &longs;i inæquales pote&longs;t e&longs;&longs;e vel eadem, vel tertia; in quarta combinatione di­uer&longs;a compen&longs;atio fieri pote&longs;t; idem dicendum e&longs;t de tertia combinatio­ne diuer&longs;orum mobilium, & mediorum, de quibus omnibus &longs;eor&longs;im iam dicemus.

Theorema 105.

Si mobilia duo eiu&longs;dem materiæ, figuræ, & grauitatis in eodem medio de­&longs;endant, æquali motu feruntur dem. vbi e&longs;t eadem proportio cau&longs;æ & re&longs;i­&longs;bentiæ ibi e&longs;t idem effectus, per Ax. 5. &longs;ed in hoc ca&longs;u eadem e&longs;t illa pro­portio; nam e&longs;t æqualis cau&longs;a, &longs;cilicet grauitas; idem medium æqualiter vtrique re&longs;i&longs;tens, cum non plures medij partes re&longs;i&longs;tant vni, quam alteri; igitur æqualis proportio.

Theorema 106.

Maior e&longs;t re&longs;istentia iu&longs;dem medij ratione &longs;cilicet partium, cum plupes cius partes re&longs;istunt quàm cum pauciores; patet, quia maior effectus re­&longs;pondet pluribus partibus cau&longs;æ per Ax.13.l.1. num.2.

Theorema 107.

Plures partes re&longs;istunt, quando plures pelluntur à mobili deor&longs;um; quip­pe in tantum re&longs;i&longs;tunt, in quantum ab aliis &longs;eparantur; atqui in tantum &longs;eparantur, in quantum amouentur è &longs;uo loco; &longs;ed ideo amouentur è &longs;uo loco, in quantum pelluntur; igitur cum plures pelluntur tunc plures re&longs;i&longs;tunt; igitur tunc maior e&longs;t re&longs;i&longs;tentia.

Theorema 108.

Plures pelluntur à maiori &longs;uperficie, quàm à minori, quæ tendit deor&longs;um parallela horizonti. v.g. à &longs;uperficie cubi maioris, quàm minoris; quippe tot pelluntur quot re&longs;pondent primæ faciei, &longs;eu primo plano, quod e&longs;t ifronte.

Theorema 109.

Si diuidatur cubus in cubos minores, ratio &longs;uperficierum erit duplicat a la­terum, & ratio &longs;olidorum triplicata, con&longs;tat ex Geometria, &longs;it enim cubus

GK, nam in gratiam corum qui Geometriam ignorant hoc ip&longs;um ocu­lis &longs;ubiiciendum e&longs;&longs;e videtur; diuidantur 6. eius facies in 4. quadrata æqualia v. g. facies AI in quad. AE. EC. EG. EI. idem fiat in aliis 5. faciebus, quarum duæ hîc tantum apparent; &longs;cilicet AK. KL; &longs;ed tribus aliis parallelis; his tribus cædem diui&longs;iones re&longs;pondent; haud dubiè erunt cubi minores, quorum latus &longs;it æquale AB, & quælibet fa­cies æqualis quadrato AE, &longs;ed facies maior AI, e&longs;t quadrupla minoris AE, ergo AI e&longs;t ad AE vt quadratum lateris AG ad quadratum lateris AD; &longs;ed hæc e&longs;t ratio duplicata laterum 1. 2. 4. &longs;imiliter cubus maior GK e&longs;t octuplum minoris DN, igitur vt cubus lateris AG ad cubum lateris AD. &longs;ed hæc e&longs;t ratio triplicata. 1.2.4.8.

a Fig.26 Tab.1.

Theorema 110.

Hinc plùs minuitur &longs;olidum in diuer&longs;ione cubi quam facies, & plùs facies quàm latus; patet ex dictis, nam latus minoris cubi e&longs;t tantùm &longs;ubdu­plum lateris maioris, & facies &longs;ubquadrupla; &longs;olidum verò &longs;ub­octuplum.

Theorema 111.

Hinc plùs minuitur grauitas, quàm re&longs;i&longs;tentia minoris cubi; quia grauitas re&longs;pondet &longs;olido, & re&longs;i&longs;tentia primç facici; re&longs;i&longs;tentia inquam ratione par­tium medij; &longs;ed &longs;olidum plus miuuitur quàm facies, vt dictum e&longs;t; igitur plus minuitur grauitas, quæ e&longs;t cau&longs;a virium quàm hæc re&longs;i&longs;tentia; crgo decre&longs;cunt vires in maiore proportione quàm hæc re&longs;i&longs;tentia, quod be­nè ob&longs;eruauit Galileus in dìalogis.

Hinc concludit Galileus duos cubos eiu&longs;dem materiæ, &longs;ed inæquales de&longs;cendere inæquali motu; maiorem &longs;cilicet velociùs minori; demon­ftrare videtur, quia maior habet maiorem proportionem virium ad re­&longs;i&longs;tentiam, quàm minor; igitur maiorem habet effectum per Ax. 5. igi­tur maiorem, & velociorem motum.

Scio non dee&longs;&longs;e multos viros doctos qui acriter in hanc &longs;ententiam in&longs;urgant: Obiicient fortè primò, experientiam e&longs;&longs;e contrariam; &longs;i enim accipiantur duo cubi maior, & minor eiu&longs;dem materiæ, & dimittantur ex eadem altitudine codem pror&longs;us momento terram ferient; Re&longs;ponde­ri pote&longs;t momentum illud &longs;en&longs;u percipi non po&longs;&longs;e; &longs;i enim dicam ma­iorem tangere terram 1000. in&longs;tantibus ante minorem, an fortè &longs;en&longs;u hoc percipies, vi&longs;u &longs;cilicet vel auditu? igitur in maxima altitudine hæc &longs;patiorum inæqualitas, & temporum &longs;en&longs;u percipi po&longs;&longs;et, quæ in minori &longs;ub &longs;en&longs;um non cadit: præterea accipe pulueris granulum eiu&longs;dem ma­teriæ, tuncque ctiam &longs;en&longs;ibilem motuum differentiam videbîs, atqui e&longs;t cadem rtio de omni minore.

Secundò obiicient, &longs;i &longs;uperponatur cubus minor maiori in &longs;uo motu nunquam &longs;eparantur; igitur æquali motu de&longs;cendunt. Re&longs;p. videri po­te&longs;t equidem æquali motu de&longs;cendere quia &longs;unt veluti partes ciu&longs;dem corporis, & grauitant grauitatione communi, neque minor habet &longs;ingu­larem re&longs;i&longs;tentiam &longs;uperandam; immò &longs;i &longs;uperponatur minor maiori, vel maior minori, motus e&longs;t velocior quàm e&longs;&longs;et &longs;olius maioris; quia cum non &longs;it maior re&longs;i&longs;tentia, maiores illi vires opponuntur; igitur fa­ciliùs &longs;uperatur.

Tertiò obiicient; e&longs;t eadem &longs;pecie grauitas; igitur eadem grauitatio, idemque motus deor&longs;um; Re&longs;ponderi po&longs;&longs;et concedendo antecedens, vnde in vacuo omnia grauia æquè velociter de&longs;cenderent, &longs;i in eo mo­tus e&longs;&longs;et; at verò altera duarum cau&longs;arum eiu&longs;dem &longs;peciei, quæ habet mi­norem proportionem actiuitatis ad re&longs;i&longs;tentiam, profectò minùs agit, quod certum e&longs;t.

Quartò obij:igitur motus po&longs;&longs;et e&longs;&longs;e velocior, & velocior in infini­tum; &longs;i enim maior cubus de&longs;cenderet velociùs; igitur &longs;i detur maior ad­huc velociùs, atque ita deinceps: Re&longs;p. inanem pror&longs;us e&longs;&longs;e difficulta­tem; quia cubus ille quantumuis maximus in vacuo de&longs;cendit velociùs quàm in aliquo medio v.g.in aëre, igitur nunquam augmentum veloci­tatis infinitum e&longs;t; quippe inter duos gradus velocitatis infiniti &longs;unt po&longs;&longs;ibiles. v. g. &longs;it velocitas, quam habet in vacuo vt 2. illa verò quàm habet in aëre vt 1. &longs;i cre&longs;cat velocitas iuxta has minutias &longs;ingulis in&longs;tan­tibus 1/2 1/4 1/ (1/16) (1/32), atque ita deinceps; quàm porrò multæ &longs;unt huiu&longs;modi progre&longs;&longs;iones 1/3 1/6 (1/12) (1/24) &c. igitur obiectiones illæ non cuertunt Gali­lei &longs;ententiam.

Inde idem Galilcus o&longs;tendere videtur cur atomi materiæ etiam gra­ui&longs;&longs;imæ, &longs;eu granula pulueris motu tardi&longs;&longs;imo de&longs;cendant in aëre vel in aqua; quia &longs;cilicet per illam diui&longs;ionem ita imminutæ &longs;unt vires graui­tatis, vt iam re&longs;i&longs;tentiam medij &longs;uperare non po&longs;&longs;int.

Sed videtur e&longs;&longs;e graui&longs;&longs;ima difficultas, &longs;int enim duo cubi, maior B F, minor GM, & vterque innatet medio liquido duplo grauiori; certè ex­tabit maior toto rectangulo CA æquali CF, & minor toto rectangulo KH æquali KM; igitur e&longs;t eadem proportio grauitatis maioris ad re&longs;i­&longs;tentiam medij in grauitatione, quæ e&longs;t minoris; igitur & in motu.

Re&longs;ponderi pote&longs;t e&longs;&longs;e maximam di&longs;paritatem inter grauitationem, & motum; &longs;it enim cubus BD qui de&longs;cendat per totam AH; haud dubiè cum &longs;patium DI, contineat 3. cubos medij æquales DB, eos debet remo­uere in &longs;uo de&longs;cen&longs;u; &longs;it autem cubus BG; haud dubiè, cum &longs;it eadem pro­portio cubi AE ad cubum medij DM, quæ e&longs;t cubi BG ad cubum me­dij FL, eodem tempore vterque cubum medij &longs;uppo&longs;iti è &longs;uo loco extru­det; igitur eo tempore, quo AE expellet 3. DI, FL extrudet 3. EO, ergo æquabili tempore inæquale &longs;patium percurrunt.

Dices ergo &longs;patia &longs;unt vt latera: Re&longs;ponderi pote&longs;t hoc reuerâ per &longs;e e&longs;&longs;e debere; &longs;ed quia cubus DM vt extrudatur, maiorem debet facere cir­cuitionem, vt à fronte retrò eat, velociori motu extrudi debet; igitur vi­res &longs;uas in co con&longs;umit maiori ex parte cubus AE; hinc compen&longs;atio e&longs;&longs;e videtur.

Vt &longs;olui po&longs;&longs;it præ&longs;ens difficultas, quæ cettè maxima e&longs;t, totam rem i&longs;tam paulò fu&longs;iùs e&longs;&longs;e explicandam iudico. Primò itaque certum e&longs;t partes medij, quæ prius in fronte erant, retroire; hoc ip&longs;um videmus in naui quæ &longs;ulcat aquas, hoc ip&longs;um accidit in omni corpore natante etiam immobili, quippe partes aquæ retinentur ab illa membranula, de qua &longs;u­prà; &longs;ic enim &longs;æpè a&longs;&longs;urgunt, & intume&longs;cunt &longs;upra labra va&longs;is; cur verò continui penè circulares limbi dilatentur: Re&longs;p. nullo flante vento vix aliquem circulum huiu&longs;inodi in &longs;uperficie aquæ apparere à fronte, &longs;ed tantùm à tergo, & lateribus, qua&longs;i ad in&longs;tar pyramidis; &longs;ed de his aliàs fusè.

Secundò certum e&longs;t numerum partium, quas impellit maior cubus A E; e&longs;&longs;e quadruplum numeri partium, quas impellit cubus BG: &longs;int autem v.g.8. partes re&longs;i&longs;tentes cubo maiori, &longs;unt duæ re&longs;i&longs;tentes cubo minoris; &longs;ed vires cubi maioris &longs;unt ad vires cubi minoris vt 8. ad 1. igitur vires vt 8. &longs;uperabunt faciliùs re&longs;i&longs;tentiam vt 8. quam vires vt 1. re&longs;i&longs;tentiam vt 2.vnde duplò velociùs moueretur, ni&longs;i aër duplò velociori motu amo­uendus e&longs;&longs;et, quod vt clarius explicetur;

Sit cubus maior AF octuplus cubi GI, vt iam dictum e&longs;t; haud dubiè aër qui &longs;ub&longs;tat cubo AF e&longs;t quadruplus aëris, qui &longs;ub&longs;tat cubo GI, vnde &longs;i vires cubi AF e&longs;&longs;ent quadruplæ virium cubi GI, e&longs;&longs;et æqualis proportio in vtroque virium, & re&longs;i&longs;tentiæ; &longs;ed &longs;unt octuplæ; igitur faci­liùs vincetur re&longs;i&longs;tentia; igitur amouebitur aër faciliùs; &longs;it autem aër expre&longs;&longs;us in globulis EFB, &c. cuius &longs;uperficies cum relinquatur retrò ver&longs;us AB, & occupetur illa quæ e&longs;t in fronte EF; haud dubiè partes hinc inde diuiduntur in D, & fegmentum NB tran&longs;it in locum relicti loci BC, FN tran&longs;it in NB, & DF, in FN; idem dico de &longs;egmentis oppo­fitis; idem pror&longs;us dico de minori globo; nam MH tran&longs;it in HQ, & H Q in QG, & QG in GL, idem dico de &longs;egmentis oppo&longs;itis; igitur hæc e&longs;t circuitio partium medij, quàm &longs;uprà indicauimus; hinc aër, qui amo­uetur à corpore graui de&longs;cendente moueri debet nece&longs;&longs;ariò velociùs quàm ip&longs;um corpus grauc, quod de&longs;cendit.

In hoc porrò ob&longs;erua &longs;egmentum MH moueri tardiùs quàm DF; quia conficit &longs;ubduplum &longs;patium, co tempore, quo DF conficit duplum; nam DF & FN &longs;unt duplæ MH & & Hque igitur dupla vi motrice opus e&longs;t; &longs;ed vires cubi AF &longs;unt ad vires cubi GI, vt 8. ad 1. partes verò aëris, quas impellit AF, &longs;unt ad partes aëris, quas impellit GI, vt 4.ad 1. igitur &longs;i partes aëris mouerentur æquali motu cum ip&longs;is cubis, à quibus mo­uentur; certè maior moueretur motu velociori; vt autem moueantur par­tes DF duplò velociore motu, quàm partes MH; debent vires, quæ mo­nent DF, e&longs;&longs;e in ratione dupla ad illas, quæ mouent MH, id e&longs;teo tem­pore, quo vires vt 8.mouebunt mobile vt 4. motu vt 2. vires vt 1.moue­bunt mobile vt 1. motu vt 1. licèt enim &longs;uperficies aëris EF moueatur deor&longs;um; attamen ab alio aëere inferiore ita reperlitur, vt &longs;ur&longs;um ver&longs;us FN repellatur.

Equidem tota &longs;uperficies aëris DF, cum pluribus partibus con&longs;tet, non pote&longs;t &longs;imul tran&longs;ire in FN; quia pars D antequam perueniat ad F tran&longs;it per medium DF; igitur &longs;ucce&longs;&longs;iuè permea ad illud &longs;patium DF, quo tempore quie&longs;ceret globus AF, quod ridiculum e&longs;t.

Quare fit nece&longs;&longs;ariò aliqua circuitio, & partium aëris commixtio, &longs;eu conflictus; ita vt retroeant pul&longs;æ prius & repercu&longs;&longs;æ; non quidem tramite recto, &longs;ed cum aliqua circuitione; quod certè facilè concipi po­te&longs;t, quæ circuitio eò maior e&longs;t, quo latera cuborum &longs;unt maiora; ita­que cum hæc &longs;atis fusè videantur e&longs;&longs;e explicata, &longs;it.

Theorema 112.

Duo cubi eiu&longs;dem materiæ, & diuer&longs;æ grauitatis æquali motu per &longs;e de&longs;­cendunt; probatur, quia licèt &longs;it maior proportio actiuitatis minus ad &longs;uam re&longs;i&longs;tentiam, quàm alterius; illud tamen compen&longs;atur; eóque par­tes aëris velociùs moueri debeant inxta rationem laterum, vt patet ex dictis; vnde nece&longs;&longs;ariò &longs;equitur motus æqualis in vtroque cubo; igitur licèt maioris cubi vires habeant maiorem proportionem ad molem, quæ præcipuum illius motus retardat; tum tamen aër, qui re&longs;i&longs;tit maiori cubo debeat amoueri, vt dictum e&longs;t velociore motu quam aër, qui re&longs;i­&longs;tit minori, &longs;itque eadem proportio re&longs;i&longs;tentiæ ratione motus minoris ad maiorem, quæ e&longs;t ratione molis maioris ad minorem; certè ratio compo&longs;ita vtriu&longs;què erit cadem in vtroque cubo; igitur æquè velociter vterque de&longs;cendet: hinc &longs;atís facilè &longs;oluitur ratio Galilci, quam multi parum cauti pro demon&longs;tratione venditarunt, ad aliam verò rationem, quam ex minuto puluere ducere videtur, etiam facilè re&longs;ponderi pote&longs;t; ideo corpu&longs;cula illa diu fluitare in aëre, tùm quòd minimo ferè tenuis auræ flatu agitentur; &longs;ic pulueris nubes medius ventus agit; quis enim ne&longs;cit aëris partes agitari perpetuò; immò & aquæ inter &longs;e mi&longs;ceri; igi­tur ab agitationis veluti impre&longs;&longs;ione fluitant illa corpu&longs;cula, cum mini­musferèimpetus extrin&longs;ecus illa commouere po&longs;&longs;it; tùm etiam quòd à filamentis illis, quibus partes aëris implicantur facilè detineantur; ana­logiam habes in lapillo, qui ab araneæ tela intercipitur.

Theorema 113.

Duo globi eiu&longs;dem materiæ, & diuer&longs;æ diametri de&longs;cendunt etiam æquali motu propter eamdem rationem; immò e&longs;t perfectior æqualitas in globis, quàm in cubis; quia perfectior fit circuitio, vt con&longs;ideranti patebit; hinc globus eiu&longs;dem materiæ, & grauitatis cum cubo de&longs;cendit velociùs quia &longs;cilicet aër in de&longs;cen&longs;u globi faciliùs agitur retrò, vt con&longs;tat.

Theorema 114.

Corpus vtrimque in mucronem de&longs;inens faciliùs adhuc de&longs;cendit, quâm globus eiu&longs;dem materiæ; ratio e&longs;t; quia breuiore circuitu partes re­trocunt; quippe tunc maxima e&longs;t facilitas in pellendo aëre, qui e&longs;t à fron­te mobilis, cum velociùs moueri non debet ip&longs;o mobili; atqui hoc ip­&longs;um e&longs;t quod accidit mobili vtrimque aucto; nam linea curua DBA, quam percurrit de&longs;criptum mebile, non e&longs;t multò longior; at verò in quadrato &longs;uperiori AF maiori e&longs;t duplò; in circulo quidem minor dia­meter &longs;emiperipheriæ, &longs;ed non duplò.

Theorema 115.

Idem corpus diuer&longs;o motu de&longs;cendere pote&longs;t, v. g. parallipedum A, &longs;i re­ctangulum BF &longs;it in fronte tardiùs de&longs;cendet, quàm &longs;i in fronte &longs;it re­ctangulum CE, vel rectangulum FH; hinc tribus motibus diuer&longs;is de&longs;­cendere pote&longs;t idem parallipedum, modò habeat &longs;emper alteram facie­rum horizonti parallelam; hinc cylindrus eiu&longs;dem grauitatis de&longs;cendet velociùs quàm parallelipedum, vt patet ex dictis; ex quibus facilè intel­ligi pote&longs;t, quænam corpora faciliùs quàm alia de&longs;cendant; quippe illa regula e&longs;t certi&longs;&longs;ima quàm &longs;uprà attulimus. Porrò ob&longs;cruabis omne corpus difficiliùs pelli per lineam perpendicularem quàm per obliquam; hinc globus pellit tantùm vnicum punctum perpendiculariter; idem di­co de cono; cylindrus verò vnam lineam, cubus integrum planum.

Theorema 116.

Hinc duo corpora eiu&longs;dem grauitatis, &longs;ed quorum alterum faciem, quæ e&longs;t in fronte, habet maiorem, inæquali motu de&longs;cendunt; patet ex dictis; quia in vtroque &longs;unt æquales vires, &longs;ed diuer&longs;a re&longs;i&longs;tentia.

Theorema 117.

Hinc tenues illæ &longs;uperficies corporum etiam materiæ graui&longs;&longs;imæ, vel in aëre fluitant, vel aquis innatant; ratio e&longs;t, quia re&longs;i&longs;tentia &longs;uperat vires.

Scholium.

Ob&longs;eruabis primam &longs;uperficiem aquæ habere maiorem quamdam re­&longs;i&longs;tentiam propter illam, qua&longs;i membranulam, de qua &longs;uprà; vnde a&longs;&longs;ur­git quiddam lymbus in margine bracteæ ferri, vel auri innatantis; vel etiam globuli paulò grauioris aquâ, igitur vt immergatur corpus debet e&longs;&longs;e grauius totâ illâ aquâ, quæ capacitatem illam non cauam occu­paret.

Theorema 118.

Globi æquales diuer&longs;æ materiæ inæqualiter de&longs;cendunt; quia &longs;cilicet alte­rum e&longs;t grauius, quod &longs;uppono; igitur æqualis e&longs;t re&longs;i&longs;tentia, & vires inæquales; igitur non e&longs;t eadem proportio actiuitatis: & re&longs;i&longs;tentiæ; igi­tur non e&longs;t æqualis motus per Ax.5.

Theorema 119.

Globi otiam inæquales diuer&longs;æ materiæ inæqualiter de&longs;cendunt; quod de­mon&longs;tro; quia globi ciu&longs;dem materiæ inæqualiter de&longs;cendunt per Th. 113. &longs;ed duo globi æquales diuer&longs;æ materiæ de&longs;cendunt inæqualiter per Th.118. igitur, & inæquales; quod dico de globis', dicatur de cubis, & aliis figuris &longs;imilibus.

Theorema 120.

Globus materiæ leuioris pote&longs;t de&longs;cendere velociori motu quam parallelipe­dum grauioris; con&longs;tat experientia; ratio e&longs;t, quia cum globus ferreus de&longs;­cendat velociùs, quàm ligneus per Th. 118. in data ratione, putà (1/100) haud dubiè bractea ferri non modo (1/100) tardiùs de&longs;cendet, verùm etiam (20/100) in quo non e&longs;t difficultas.

Theorema 121.

Hinc &longs;i mutetur figura po&longs;&longs;unt grauia diuer&longs;æ materiæ ita de&longs;cendere, vn vel grauius, vel leuius, vel grauioris materiæ, vel leuioris velociùs de&longs;cendat; vt con&longs;tat ex regulis præ&longs;criptis.

Theorema 122.

Globi æquales diuer&longs;æ materiæ, v. g. ligneus, & plumbeus de&longs;cendunt mæqualiter iuxta proportionem grauitatis, & re&longs;i&longs;tentiæ medij compa­ratæ cum vtroque, v.g. plumbo detrahitur (1/4800); ligno verò (8/300) v. g. &longs;i grauitas ligni &longs;it ad grauitatem aëris vt 300.ad 1. & plumbi vt 4800. ad 1. &longs;it enim altitudo 33. pedum 4. digit. reducantur in digitos erunt 400. in lineas 4800. igitur detrahetur vna linea &longs;patij plumbeo globo; ligneo­verò vnus digitus cum 4. lineis; &longs;ed quis hoc ob&longs;eruet?

Theorema 123.

Corpus graue &longs;pongio&longs;um longè tardiùs de&longs;cendit; quia aër in perexigua illa foramina inten&longs;us frangitur, re&longs;ilit, ac proinde motum impedit; talis e&longs;t medulla &longs;ambuci, &longs;pongia, &longs;tupa, &c. immò a&longs;perum corpus tardiùs de&longs;cendit, quòd &longs;cilicet aër ab a&longs;perioribus illis &longs;alebris re&longs;iliens mo­tum retardet, hinc &longs;ibilus ille auditur &c.

Scholium.

Ex his con&longs;tat quid dicendum &longs;it de motu corporum grauium in medio, &longs;iue &longs;int eiu&longs;dem maieriæ, & &longs;imilis figuræ, maioris vel minoris, vel æqualis; tunc enim de&longs;cendunt æqualiter contra Galileum, &longs;iue &longs;int diuer&longs;æ materiæ, & &longs;imilis figuræ, æqualis, vel inæqualis, tunc enim de&longs;cendunt inæqualiter, &longs;iue diuer&longs;æ materiæ & diuer&longs;æ fi­guræ; tunc enim de&longs;cendunt modò æqualiter, modò inæqualiter; æquali­ter certè, cum figura compen&longs;at materiam; cum verò non compen&longs;at, inæqualiter pro rata; denique &longs;i comparentur duo corpora cum diuer&longs;is mediis; primo inuenienda e&longs;t proportio motuum vtriu&longs;que in eodem tùm &longs;ingulorum in diuer&longs;is mediis, vt &longs;uprà dictum e&longs;t.

Theorema 124.

In modico vacuo omnia æquè velociter de&longs;cenderent: Probatur, quia tota diuer&longs;itas vel inæqualitas mediorum petitur à diuer&longs;a proportione acti­uitatis cum re&longs;i&longs;tentia medij per Ax. 5. &longs;ed in vacuo nulla e&longs;t re&longs;i&longs;ten­tia; igitur nulla proportio; igitur nulla ratio motus inæqualis.

Theorema 125.

In motu natur aliter accelerato deor&longs;um cre&longs;cit re&longs;istentia medij &longs;ingulis in­&longs;tantibus: probatur, quia &longs;ingulis in&longs;tantibus plures partes medij &longs;unt &longs;uperandæ; cre&longs;cunt enim &longs;patia, vt con&longs;tat ex dictis; igitur cre&longs;cit re&longs;i­&longs;tentia &longs;ingulis in&longs;tantibus.

Theorema 126.

Cre&longs;cit re&longs;istentia iuxta rationem &longs;patiorum, probatur; quia cre&longs;cit iux­ta rationem plurium partium medij, quæ temporibus æqualibus percur­runtur; &longs;ed eæ cre&longs;cunt iuxta rationem &longs;patiorum, vt con&longs;tat.

Theorema 127.

Hinc cre&longs;cit re&longs;i&longs;tentia iuxta rationem velocitatum &longs;ingulis instantibus; quæ ratio &longs;equitur progre&longs;&longs;ionem arithmeticam &longs;implicem numerorum 1.2.3.4.5.6. ex &longs;uppo&longs;itione quòd tempus con&longs;tet ex partibus finitis actu; nam codem modo cre&longs;cit velocitas, quo cre&longs;cunt numeri prædicti; &longs;ed eodem modo cre&longs;cunt &longs;patia, &longs;i dumtaxat accipiantur in &longs;ingulis in&longs;tan­tibus; re&longs;i&longs;tentia cre&longs;cit iuxta rationem &longs;patiorum; igitur iuxta ratio­nem velocitatum.

Scholium.

Ob&longs;eruabis, &longs;i tempus con&longs;tet ex infinitis actu partibus, ita vt &longs;ingu­læ partes motus &longs;ingulis partibus temporis & infinitæ infinitis re&longs;pon­deant; non pote&longs;t e&longs;&longs;e alia progre&longs;&longs;io, in qua fiat acceleratio motus na­turalis, quàm illa Galilei iuxta hos numeros 1. 3. 5. 7. vt con&longs;tat ex dictis per illud Principium; æqualibus temporibus æqualia acquiruntur velocita­tis momenta; &longs;i verò tempus con&longs;tat ex finitis in&longs;tantibus æqualibus, nul­la datur progre&longs;&longs;io motus naturaliter accelerati; quia motus accelerari non pote&longs;t; ne &longs;cilicet eodem in&longs;tanti mobile &longs;it in pluribus locis adæ­quatis; denique &longs;i tempus con&longs;tat ex finitis in&longs;tantibus actu, & infinitis potentiâ, non pote&longs;t e&longs;&longs;e alia progre&longs;&longs;io huius accelerationis, quam hæc no&longs;tra iuxta numeros toties repetitos 1.2.3.4.5. attamen quia illa finita in&longs;tantia &longs;unt ferè innumera in qualibet parte &longs;en&longs;ibili temporis, in praxi &longs;ine &longs;en&longs;ibili errore in partibus temporis &longs;en&longs;ibilibus po&longs;&longs;umus adhibere priorem progre&longs;&longs;ionem Galilei, & in hoc cardine tota verri­tur, meo iudicio, propo&longs;itæ quæ&longs;tionis difficultas.

Theorema 128.

Hinc cre&longs;cit re&longs;istentia iuxta rationem crementi impetus; cum enim cre­&longs;cant impetus in ratione velocitatum, vt con&longs;tat, & cre&longs;cat re&longs;i&longs;tentia medij in eadem ratione per Theor. 127. cre&longs;cit etiam in ratione im­petuum.

Theorema 129.

Hinc cre&longs;cit re&longs;istentia medij in eademratione, in qua cre&longs;cunt vires mobi­lis; demon&longs;tr. quia cre&longs;cunt vires, vt cre&longs;cit impetus; nam impetus e&longs;t vis illa, quâ mobile &longs;uperat re&longs;i&longs;tentiam medij vt con&longs;tat, &longs;ed re&longs;i&longs;ten­tia cre&longs;cit vt impetus per Th. 128. igitur cre&longs;cit in ratione virium.

Theorema 130.

Si cre&longs;cit re&longs;i&longs;tentia in eadem ratione in qua cre&longs;cunt vires, non mutatur progre&longs;&longs;io effectuum. v.g. primo in&longs;tanti impetus &longs;it vt 1.&longs;itque 1.&longs;patium, 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 2. &longs;patiis vt 2. haud dubiè &longs;i vno in&longs;tanti vnus gradus impetus &longs;uperat re&longs;i&longs;tentiam vt 1. dum percurrit 1.&longs;patium; certè 2. gradus impetus vno in&longs;tanti &longs;uperabunt re&longs;i&longs;tentiam vt 2. dum con&longs;icit mobile 2. &longs;patia; at­que ita deinceps.

Theorema 132.

Hinc certè concludo contra Galileum, & alios quo&longs;dam motum grauium po&longs;t aliquod &longs;patium decur&longs;um ex naturaliter accelerato non fieri æquabilem,quia in tantum fieret æquabilis in quantum tanta e&longs;&longs;et re&longs;i&longs;tentia, vt no­uam accelerationem impediret; &longs;ed hæc ratio nulla e&longs;t; quia in eadem ratione cre&longs;cit re&longs;i&longs;tentia, in qua cre&longs;cunt vires per Th. 129. igitur non mutatur progre&longs;&longs;io motuum per Th. 130. igitur nec acceleratio; igitur motus naturalis ex accelerato non fit æquabilis: Equidem, vt iam &longs;uprà dictum e&longs;t, in minori &longs;emper ratione cre&longs;cit velocitas, itémque ip&longs;a re&longs;i­&longs;tentia quod in omni progre&longs;&longs;ione arithmetica iuxta numeros 1.2.3.4.5.

Scholium.

Ob&longs;eruabis remitti à nobis motum leuium &longs;ur&longs;um in 5. Tomum, in cu­ius tertio libro agemus de graui, & leui; quia ideo corpus a&longs;cendit, quia ab alio de&longs;cendente truditur &longs;ur&longs;um.

LIBER TERTIVS,

DE MOTV VIOLENTO &longs;ur&longs;um Perpendiculariter.

OMnis certè motus, qui e&longs;t à principio ex­trin&longs;eco, violentus appellari pote&longs;t, attamen hîc non ago de omni violento, &longs;ed dumta­xat de illo, qui fit &longs;ursùm per lineam verticalem; quia &longs;cilicet ex diametro opponitur motui naturali, qui fit deorsùm perpendiculariter; igitur cum de hoc ip&longs;o in &longs;ecundo Libro egerimus, de illo in hoc non agemus.

DEFINITIO 1.

MOtus violentus e&longs;t, quo corpus graue mouetur &longs;ursùm per li­neam verticalem à principio extrin&longs;eco mediatè, vel immediatè vt plurimùm.

Dixi à principio extrin&longs;eco, &longs;iue cunjuncto, vt cum manu attollo &longs;ur­&longs;um corpus graue, &longs;iue non conjuncto, vt cum quis proiicit lapidem &longs;ur­sùm, &longs;iue &longs;it verum principium effectiuum, vt cum impetus, quem poten­tia motrix producit in manu, producit alium in mobili; &longs;iue non &longs;it principium effectiuum, &longs;ed tantùm determinans, vt cum mobilc quod cadit deor&longs;um, &longs;ur&longs;um deinde repercutitur; nec enim corpus repercu­tiens producit impetum nouum, vt dicemus cum de motu re&longs;lexo; quin potiùs producti partem de&longs;truit per accidens, & quidquid illius &longs;upere&longs;t, ad nouam lineam determinat; quod quomodo fiat fusè &longs;uo loco expli­cabimus, igitur licèt corpus reflectens &longs;it tantùm principium nouæ de­terminationis, non verò alicuius impetus producti, dici pote&longs;t princi­pium huius motus violenti.

Dixi vt plurimùm, nam &longs;i terra ducto per centrum foramine e&longs;&longs;et peruia, haud dubiè lapis demi&longs;&longs;us versùs centrum iret motu naturaliter accelerato, tùm deinde propter impetus acqui&longs;iti vim, à centro versùs oppo&longs;itum circumferentiæ punctum iret, motu certè violento, qui ta­men ab extrin&longs;eco non e&longs;&longs;et.

Hypothe&longs;is 1.

Corpus graue projectum &longs;ur&longs;um tandem redit; Hæc hypothe&longs;is certa e&longs;t, & nemo e&longs;t qui eam in dubium vocet.

Axioma 1.

Quidquid erat, & de&longs;init e&longs;&longs;e de&longs;truitur; Hoc Axioma certum e&longs;t, quip­pe de&longs;trui hoc tantùm dicitur, quod de&longs;init e&longs;&longs;e.

Axioma 2.

Quidquid destruitur, ad exigentiam alicuius destruitur, &longs;altem totius na­turæ. Hoc Axioma idem e&longs;t cum Axiom. 14. l. 1. n. 2. vnde alia expli­catione minimè indiget; hoc ip&longs;um etiam demon&longs;traui in Th.147.149. 150,&c. l. 1.

Theorema 1.

Datur motus violentus; demon&longs;tro; corpus proiicitur per lineam ver­ticalem per hyp. 1. &longs;ed hic motus e&longs;t à principio extrin&longs;eco, igitur e&longs;t violentus per def.1. probatur minor; Primò, quia illud e&longs;t principium, &longs;eu cau&longs;a motus, ex cuius applicatione &longs;emper &longs;equitur motus per Ax.11. l. 1.n. 1. &longs;ed ex applicatione potentiæ extrin&longs;ecæ v. g. arcus, manus, &c. ad lineam &longs;ur&longs;um &longs;emper &longs;equitur motus &longs;ur&longs;um; igitur e&longs;t illius cau&longs;a. Secundò probatur, quia mobile projectum &longs;ursùm mouetur adhuc &longs;epa­ratum à potentia motrice per hyp. 6.l.1. igitur potentia motrix impre&longs;­&longs;it aliquid mobili, vi cuius deinde mouetur, igitur hic motus e&longs;t à prin­cipio extrin&longs;eco.

Diceret fortè aliquis produci hunc motum ab ip&longs;o mobili; &longs;ed con­trà; igitur &longs;emper produceret, quod ab&longs;urdum e&longs;t: dicet, ad hoc vt pro­ducat determinari debere ab aliquo, &longs;ed contrà; illud à quo determina­tur vel e&longs;t extrin&longs;ecum, vel intrin&longs;ecum, &longs;i primum, ergo hic motus e&longs;t &longs;emper à principio extrin&longs;eco, quod &longs;atis e&longs;t e&longs;&longs;e determinans per def.1. &longs;i verò e&longs;t intrin&longs;ecum; igitur &longs;emper e&longs;&longs;et hic motus, quamdiu e&longs;&longs;et ip&longs;um mobile, quod e&longs;t contra hyp. 1. nam reuera non &longs;emper mo­uetur.

Diceret fortè alius excitari quædam corpu&longs;cula, à quibus mouetur corpus graue &longs;ursùm; &longs;ed contrà; nam vel &longs;unt in ip&longs;o mobili illa cor­pu&longs;cula, vel extra mobile; &longs;i primum; igitur hic motus &longs;emper crit ab extrin&longs;eco mediatè, cum ab extrin&longs;eco excitentur; &longs;ed hoc &longs;ufficit ad hoc; vt motus dicatur violentus per def. 1. &longs;i verò &longs;unt extra mobile; igitur motus ille e&longs;t &longs;emper ab extrin&longs;eco, idque duplici nomine.

Denique diceret alius ex &longs;uppo&longs;itione, quod terra moueatur non po&longs;­&longs;e corpus graue proiici &longs;ursùm per lineam verticalem, ni&longs;i tantùm ad &longs;peciem; vt &longs;i quis è naui mobili &longs;ur&longs;um proiiceret pilam rectà omni­nò, quoad eius fieri po&longs;&longs;it; videbitur enim iis, qui vehuntur cadem naui &longs;ur&longs;um ferri per lineam verticalem, aliis verò in&longs;tantibus videbitur cla­ri&longs;&longs;imè ferri per lineam nouam inclinatam.

Re&longs;pondeo etiam admi&longs;&longs;a &longs;uppo&longs;itione dici à me motum illum &longs;ur­&longs;um e&longs;&longs;e per lineam verticalem, quando eadem linea recta connectit &longs;emper hæc tria puncta; &longs;cilicet centrum terræ, idem punctum &longs;uperfi­ciei terræ, & ip&longs;am pilam; ad illud verò quod dicitur de naui, non diffi­teor verum e&longs;&longs;e; &longs;ed dico non e&longs;&longs;e propriè motum violentum, de quo hîc tantùm e&longs;t quæ&longs;tio, &longs;ed e&longs;&longs;e motum mixtum, de quo fusè &longs;uo loco. Ob&longs;er­uabis autem hîc me ab&longs;tinere à refellendis ab&longs;urdis illis &longs;uppo&longs;itioni­bus, quibus præmi&longs;&longs;æ objectiones innituntur; nam, cui quæ&longs;o in men­tem venire pote&longs;t ab ip&longs;a entitate corporis grauis produci motum in &longs;e? quis credat produci frigus ab igne? calorem à niue? lucem à tenebris? quæ porrò fabulæ, quæ commenta, quæ &longs;omnia excogitari po&longs;&longs;unt, quæ non vile&longs;cant &longs;i cum his comparentur.

Illa quoqne corpu&longs;cula excitata leuiora &longs;unt, quàm vt aliquod præfe­rant rationis momentum; cum mera &longs;int philo&longs;ophiæ ludibria.

Denique illa hypothe&longs;is de terræ motu nullis demon&longs;trationibus fir­mata e&longs;t, vt videbimus &longs;uo loco.

Vnum fortè e&longs;t, quod difficilius obiici pote&longs;t; &longs;it enim linea vertica­lis AC, &longs;itque globus in A æqualiter impul&longs;us per lineas AD & AB; haud dubiè &longs;i anguli DAC, BAC &longs;int æquales: certè mobile feretur per lineam verticalem AC, vt con&longs;tat ex dictis. Re&longs;pondeo motum illum e&longs;&longs;e violentum; e&longs;t enim à principio extrin&longs;eco, coque gemino, &longs;eu mix­to, in quo non e&longs;t difficultas.

Theorema 2.

Motus violentus habet caufam; quia de nouo e&longs;t, & tandem de&longs;init per hypoth. 1. igitur habet cau&longs;am per Ax.8.l.1.

Theorema 3.

I&longs;te motus &longs;upponit impetum; quia ni&longs;i e&longs;&longs;et impetus non e&longs;&longs;et natura­liter motus per Th.18.l.1.

Theorema 4.

I&longs;te impetus debet e&longs;&longs;e in mobili projecto &longs;ur&longs;um; quia ibi e&longs;t cau&longs;a, vbi e&longs;t effectus formalis, &longs;ed motus e&longs;t effectus formalis &longs;ecundarius impe­tus per Th.15.l.1. igitur cum motus &longs;it in projecto &longs;ur&longs;um, in co e&longs;t etiam impetus: præterea &longs;ecunda pars motus non ponitur à potentia motrice; quia illa non e&longs;t applicata mobili cum ponitur noua pars motus, igitur ab alia cau&longs;a applicata, &longs;ed nulla e&longs;t extrin&longs;eca, vt patet, nulla intrin&longs;eca præter impetum.

Diceret aliquis ab aëre extrin&longs;ecùs ambiente mobile ip&longs;um propelli; &longs;ed contra, nam aër, & omne aliud medium re&longs;i&longs;tit potiùs quàm iuuet, vt demon&longs;trauimus l. &longs;ecundo Th. 1. Nec dicas fui&longs;&longs;e mentem Ari&longs;totelis, cum nobiles Peripatetici contrâ &longs;entiant; Albertus Magnus, Toletus, Scaliger, Suarius, & recentiores; neque hoc negauit vnquam Ari&longs;tote-les, &longs;ed in hoc non multùm laboramus; nec dicas hinc &longs;equi motum violentum e&longs;&longs;e à principio intrin&longs;eco contra def. 1. nam e&longs;t quidem à principio intrin&longs;eco formali, non tamen à principio intrin&longs;eco mouen­te vel agente; nec enim impetus e&longs;t cau&longs;a efficiens motus &longs;ui &longs;ubjecti; &longs;ed cau&longs;a formalis vt &longs;æpè explicuimus.

Diceret fortè alius primam partem motus produci à potentiâ motri­ce, &longs;ecundam verò ab entitate ip&longs;ius corporis; &longs;ed contrà; igitur corpus e&longs;&longs;et cau&longs;a nece&longs;&longs;aria; igitur &longs;emper produceret. Dices &longs;emper producere &longs;i determinetur, &longs;ed contrà; à quo determinatur ad producendam &longs;ecun­dam partem? nihil e&longs;t enim applicatum, à quo determinari po&longs;&longs;it; Dices accepi&longs;&longs;e determinationem; &longs;ed contrà; quid e&longs;t illa determinatio? Dices e&longs;&longs;e modum; igitur permanentem; igitur e&longs;t cau&longs;a motus per Ax. 1. l. 1. n. 1. igitur e&longs;t impetus per def. 3. l. 1. Dices determinari à priori parte motus; &longs;ed contrà primò, nam reuerâ non e&longs;t illa pars cum deter­minatur corpus. Secundò, quid e&longs;t illa prima pars motus, ni&longs;i migratio è loco in locum, quæ reuerâ à potentia motrice produci propriè non po­te&longs;t per Th.2. l. 1. &longs;ed de his iam fusè actum e&longs;t in toto ferè libro primo, &longs;ed præ&longs;ertim in Th.6.

Theorema 5.

Ille impetus e&longs;t vera qualitas Phy&longs;ica ab&longs;oluta; hoc iam &longs;uprà demon­&longs;tratum e&longs;t, &longs;cilicet phy&longs;icè; immò ex motu violento maximè probatur dari impetum, & vix quidquam e&longs;t in rerum naturâ, quod clariùs euin­cat aliquid de nouo produci.

Theorema 6.

I&longs;te impetus producitur ab aliqua cau&longs;a; Probatur, quia e&longs;t de nouo; igi­tur non e&longs;t à &longs;e per Ax. 8. l. 1. igitur e&longs;t ab alio; igitur ab aliqua cau&longs;a.

Theorema 7.

Producitur ab aliqua cau&longs;a extrin&longs;eca; Probatur primò, quia aliquis motus violentus e&longs;t à cau&longs;a extrin&longs;eca per def.1. Secundò, e&longs;t ab aliqua cau&longs;a applicata, &longs;ed e&longs;t tantùm applicata potentia motrix; igitur e&longs;t cau­&longs;a, per Ax. 11. l. 1. nec enim producitur hic impetus ab entitate corpo­ris projecti, quod plu&longs;quàm certum e&longs;t ex dictis; hîc enim tantùm e&longs;t quæ&longs;tio de illo motu, qui extrin&longs;ecùs aduenit, non vero de reflexo &longs;ursùm, &c.

Theorema 8.

Producitur ab alio impetu; quia potentia motrix non agit ad extra ni&longs;i per impetum productum in organo, vt patet; præterea &longs;i e&longs;t cau&longs;a vni­uoca &longs;ufficiens applicata, non e&longs;t ponenda æquiuoca per Ax.11.l.1. adde quod impetus producitur &longs;emper ad extra ab alio impetu per Th. 42. l.1.nec in his hactenus propo&longs;itis vlla e&longs;t difficultas.

Theorema 9.

Impetus impre&longs;&longs;us mobili &longs;ur&longs;um con&longs;eruatur per aliquod tempus; Probatur, quia mobile &longs;eparatum à potentia motrice adhuc mouetur per hyp.6.l.1, igitur ille motus habet cau&longs;am, vt &longs;æpè dictum e&longs;t; non aliam, quàm im­petum per Th.4. non productum de nouo, quippe nulla e&longs;t cau&longs;a mobili applicata per Th. 7. & 8. igitur iam antè productam; igitur con&longs;er­uatur.

Theorema 10.

Con&longs;eruatur ab aliqua cau&longs;a extrin&longs;eca applicata; vt patet ex dictis, non ab aëre; igitur à nullo corpore; igitur ab alia causâ in&longs;en&longs;ibili; igitur illam e&longs;&longs;e oportet, & no&longs;&longs;e rerum omnium exigentias, & po&longs;&longs;e cuncta producere; quippe con&longs;eruatio e&longs;t repetita productio; immò con&longs;erua­re per actionem, per quam &longs;it res in tali loco, & tali tempore; illa porrò cau&longs;a in&longs;en&longs;ibilis incorporea, quæ vbique e&longs;t, & &longs;emper, Deus e&longs;t: Nec puta po&longs;&longs;e exi&longs;tentiam cau&longs;æ primæ probari &longs;en&longs;ibiliori, vt &longs;ic loquar, argumento, quàm eo, quod petitur ex motu projectorum, quorum motus durat etiam&longs;i à potentia motrice mobile ip&longs;um &longs;it &longs;eparatum.

Theorema 11.

Hinc multa colligi po&longs;&longs;unt. Primò, &longs;i nullus e&longs;&longs;et impetus extrin&longs;ecus, vel acqui&longs;itus, nullus e&longs;&longs;et motus violentus, ni&longs;i tantùm motus reflexus cadentium deorsùm. Secundò, &longs;i nullus e&longs;&longs;et Deus, nullus e&longs;&longs;et motus violentus; immò nec vllus naturaliter acceleratus. Tertiò, &longs;i impetus e&longs;­&longs;et fluens vt motus, nullus e&longs;&longs;et motus violentus. Quartò, &longs;i &longs;ingulæ par­tes motus produci debent ab aliquâ causâ efficiente, nullus etiam e&longs;&longs;et motus violentus.

Theorema 12.

Vt &longs;it motus violentus debent produci plures partes impetus violenti quàm &longs;int partes impetus naturalis; Probatur, quia &longs;i e&longs;&longs;ent plures natura­lis deorsùm, quàm &longs;int vielenti &longs;ur&longs;um, corpus tenderet deor&longs;um; &longs;ed tardiùs per Th.134.l.1. & &longs;i tot e&longs;&longs;ent vnius, quot alterius, mobile ip&longs;um non moueretur per Th.133.l.1.

Theorema 13.

Motus violentus non e&longs;t acceleratus; probatur primò experientiâ, quæ certa e&longs;t. Secundò, quia &longs;i &longs;emper cre&longs;ceret, numquam rediret mobile contra hyp.1. nec enim ab vllo reflectitur; &longs;i enim reflecteretur ab aëre inten&longs;us, multò magis remi&longs;&longs;us.

Theorema 14.

Hinc impetus in mobili &longs;ur&longs;um projecto non intenditur, quia non inten­ditur effectus per Th.13. igitur nec cau&longs;a per Ax.2.l.2.

Theorema 15.

Motus violentus non e&longs;t æquabilis; quia mobile tandem redit per hyp.1. &longs;ed numquam rediret, &longs;i e&longs;&longs;et æquabilis; cur enim potiùs hoc in&longs;tanti quàm alio? cur ab hoc puncto &longs;patij potiùs, quàm ab alio?

Theorema 16.

Hinc non con&longs;eruatur intactus impetus; quia &longs;i e&longs;&longs;et intactus, e&longs;&longs;et &longs;em­per æqualis; igitur haberet &longs;emper æqualem motum per Ax.3.l.2. igitur motus e&longs;&longs;et æquabilis, contra Th.15.

Theorema 17.

Hinc nece&longs;&longs;e e&longs;t aliquid impetus destrui; cum enim non remaneat inta­ctus, & æqualis; nec fiat maior per Th.14. certè fit minor, igitur detra­ctione aliqua per Ax.1.l.2.

Theorema 18.

Singulis in&longs;tantibus aliquid de&longs;truitur impetus impre&longs;&longs;i; probatur quia cur potiùs vno quam alio? quippe illa ratio, quæ probat de vno probat de &longs;ingulis.

Theorema 19.

Hinc nece&longs;&longs;ariè eadem vel aqualis cau&longs;a de&longs;tructionis debet e&longs;&longs;e applicata; probatur, quia æqualis effectus æqualem cau&longs;am &longs;upponit, per Ax. 3. l. 2.

Theorema 20.

Illa cau&longs;a non e&longs;t tantùm aër ambiens vt velunt aliqui; quia licèt re&longs;i­&longs;tat motui, &longs;eu potius mobili, non tamen e&longs;t ea re&longs;i&longs;tentia, quæ po&longs;&longs;it impetum tam citò de&longs;truera; probatur primò, quia &longs;i hoc e&longs;&longs;et, de&longs;true­retur æquali tempore per omnem lineam &longs;ur&longs;um, quod e&longs;t contra expe­rientiam, vt dicemus infrà; e&longs;&longs;et enim eadem cau&longs;a applicata; igitur idem & æqualis effectus; probatur &longs;ecdò, quia non de&longs;truit aër primum il­lum gradum impetus naturalis acqui&longs;iti, vt con&longs;tat in motu deor&longs;um, qui men e&longs;t imperfecti&longs;&longs;unus; igitur non e&longs;t &longs;ufficiens ad de&longs;truendum im­petum violentum, ni&longs;i longo tempore. Tertiò, globus &longs;ursùm projectus a&longs;cendit, & deinde de&longs;cendit æquali tempore; igitur &longs;altem &longs;ingulis in­&longs;tantibus de&longs;truitur vnus gradus impetus violenti æqualis primo gradui innato; atqui aër non pote&longs;t vno in&longs;tanti de&longs;truere impetum æqualem primo innato; alioqui non intenderetur motus naturalis. Quartò, & hæc e&longs;t ratio à priori, quotie&longs;cumque &longs;unt in eodem mobili duo impetus ad oppo&longs;itas lineas determinati, pugnant pro rata, vt demon&longs;trauimus l.1. Th. 149. 150. 152. & in toto Schol. & multis aliis pa&longs;&longs;im; atqui con&longs;er­natur &longs;emper impetus naturalis innatus per Sch. Th.152.n.6.l.1.per Th. 9. & Schol.Th.14. & Th.73.l.2.

Theorema 21.

Illa cau&longs;a non e&longs;t eutitas corporis mobilis, vel ip&longs;a grauitas, di&longs;tincta &longs;cili­cet ab impetu muato&longs;i quæ e&longs;t de quæ alias, probatur, quia non e&longs;&longs;et potior ratiocur vno in&longs;tanti de&longs;truerentur duo gradus impetus, quàm 3. 4. 5. quippe grauitas exigeret de&longs;tructionem omnium: præterea omnis impe­tus de&longs;truitur ne &longs;it fru&longs;trà per Schol, Th.152. & Th.162.l.1. denique &longs;i ade&longs;t contrarius impetus de&longs;tructiuus co modo, quo explicuimus l. 1. non e&longs;t ponenda alia cau&longs;a de&longs;tructiua.

Theorema 22.

Hinc nece&longs;&longs;e e&longs;t impetum violentum de&longs;trui ab impetu naturali innato; pro­batur, quia nulla e&longs;t cau&longs;a extrin&longs;eca de&longs;tructiua &longs;altem adæquatè per hT. 20.igitur e&longs;t intrin&longs;eca per Ax.4. l.2. &longs;ed intrin&longs;eca vel e&longs;t mobilis enti­tas, vel grauitas, vel impetus innatus; &longs;ed mobilis entitas non e&longs;t cau&longs;a de&longs;tructiua; nec etiamip&longs;a grauitas per Th.21. igitur impetus naturalis innatus.

Theorema 23.

Hinc vera ratio cur &longs;ingulis in&longs;tantibus aliquid de&longs;truatur, quia &longs;ingulis in&longs;tantibus e&longs;t cau&longs;a de&longs;tructiua applicata, igitur &longs;ingulis in&longs;tantibus de­&longs;truit per Ax. 12. l. 1.

Theorema 24.

Hinc etiam ratio cur &longs;ingulis instantibus, &longs;eu æqualibus temporibus æqua­liter de&longs;truatur; quia &longs;ingulis in&longs;tantibus e&longs;t eadem cau&longs;a de&longs;tructiua ap­plicata; igitur &longs;ingulis in&longs;tantibus æqualiter de&longs;truit per Ax.3.l.2.porrò in tantum de&longs;truit in quantum efficit, vt aliquid &longs;it fru&longs;trà, vt fusè di­ctum e&longs;t lib.1.vel in quantum exigit cius de&longs;tructionem, nam perinde e&longs;t.

Theorema 25.

Hinc etiam petitur ratio, propter quam talis portio impetus violenti de­&longs;truatur vne in&longs;tanti; quia &longs;cilicet contraria pugnant prorata per Ax.15. & per Th.134.l.1.

Theorema 26.

Hinc illa inuer&longs;a communis dicti, æqualibus temporibus æqualia de&longs;truun­tur velocitatis momenta in motu violento; quippe eadem cau&longs;a eidem &longs;ub­jecto applicata æqualibus temporibus æqualem effectum producit per Ax.3.l.2. &longs;ed impetus innatus e&longs;t cau&longs;a de&longs;tructiua impetus violenti per Th. 22. igitur æqualibus temporibus, &c.

Theorema 27.

In cadem proportione retardatur motus violentus, in qua naturaiis accele­ratur: probatur quia &longs;ingulis in&longs;tantibus æqualibus acquiritur æqualis gradus impetus, vt &longs;æpè dictum e&longs;t &longs;uprà; atqui &longs;ingulis in&longs;tantibus de­&longs;truitur vnus gradus impetus violenti per Th.24. &longs;ed ille gradus re&longs;pon­det impetui innato per Th. 25. igitur æqualibus temporibus tantùm de­&longs;truitur violenti, quantùm acquiritur naturalis; cum enim primo in­&longs;tanti &longs;it impetus naturalis, & &longs;ecundo tempore æquali acquiratur æqua­lis, item tertio, quarto, &c. certè cum impetus innatus pugnet cum vio­lento pro rata; nec &longs;it potior ratio cur maiorem portionem quàm mino­rem de&longs;truat, æqualem certè de&longs;truit, itemque &longs;ecundo in&longs;tanti æqua­lem, item tertio, quarto; igitur in eadem proportione decre&longs;cit violentus, &longs;eu retardatur, in qua naturalis acceleratur.

Hinc inuertenda e&longs;t progre&longs;&longs;ionis linca; quippe linea AE repræ&longs;en­tat nobis progre&longs;&longs;ionem motus accelerati, quæ fit in in&longs;tantibus, & li­nea FK progre&longs;&longs;ionem motus, quæ fit in partibus temporis &longs;en&longs;ibilibus; in illa primo in&longs;tanti decurritur primum &longs;patium AB, &longs;ecundo tempore æquali BC, tertio CD, quarto DE: in hac vero prima parte acquiritur &longs;patium FG &longs;ecunda æquali primæ GH, tertia HI, quarta IK; igitur &longs;i ac­cipiatur linea AE, progrediendo ab A ver&longs;us E, vel linea FK progre­diendo ab F ver&longs;us K habebitur progre&longs;&longs;io motus naturaliter acceletati; &longs;i verò accipiatur EA, vel KF, progrediendo &longs;cilicet ab E ver&longs;us A, vel à K ver&longs;us F, erit progre&longs;&longs;io motus violenti naturaliter retardati; vt con­&longs;tat ex præcedèatibus Theorematis; & quemadmodum progre&longs;&longs;io acce­lerationis in in&longs;tantibus finitis fit iuxta &longs;eriem i&longs;torum numerorum 1.2. 3.4. in partibus verò temporis &longs;en&longs;ibilibus iuxta &longs;eriem i&longs;torum 1.3.5.7. ita fit omninò progre&longs;&longs;io retardationis in in&longs;tantibus iuxta hos nume­ros 4.3.2.1. in partibus temperis &longs;en&longs;ibilibus iuxta hos 7.5. 3. 1.

Theorema 28.

Motus violentus durat tot in&longs;tantibus &longs;cilicet æquiualentibus quot &longs;unt ij gradus impetus quibus violentus &longs;uperat innatum, v.g. &longs;it vnus gradus im­petus innati; producantur 5. gradus violenti, quorum &longs;inguli &longs;int æqua­les innato etiam æquiualenter, motus durabit 4. in&longs;tantibus etiam æqui­ualenter id e&longs;t 4. temporibus, quorum &longs;ingula erunr æqualia primo in­&longs;tanti motus naturalis, probatur, cum &longs;ingulis in&longs;tantibus æqualibus de­&longs;truatur vnus gradus; certè 4. in&longs;tantibus durat motus.

Theorema 29.

Si accipiantur &longs;patia æqualiain hac progre&longs;&longs;ione retardationis, e&longs;t inuer&longs;a illius, quàm tribuimus &longs;uprà acceberationi, a&longs;&longs;umptis &longs;cilicet &longs;patiis æqualibus; tiem &longs;i accipiantur &longs;patia æqualia prime &longs;patie quod decurritur prime in&longs;tan­ti metus naturalis, tum &longs;i accipiantur &longs;patia æqualia date &longs;patie quod in par­te temporis &longs;en&longs;ibili percurritur; quippe quemadmodum in progre&longs;&longs;ione accelerationis decre&longs;cunt tempora; &longs;ic in progre&longs;&longs;ione retardationis cre&longs;cunt, a&longs;&longs;umptis &longs;cilicet &longs;patiis æqualibus; quare ne iam dicta hic re­petam, con&longs;ule quæ diximus lib.2. de hac progre&longs;&longs;ione.

Theorema 30.

Hinc instantia initio huius metus &longs;unt minora &longs;icut initio met menatur alis &longs;unt maiora; & &longs;ub finem in motu violente &longs;unt maiora, in naturali &longs;unt mi­nora; quia &longs;cilicet hic acceleratur, ille retardatur: igitur velo­catas accelerati cre&longs;cit; igitur &longs;i accipiantur &longs;patia æqualia, decre&longs;cit tem­pus; at verò velocitas retardati decre&longs;cit, igitur a&longs;&longs;umptis &longs;patiis æquali­bus, cre&longs;cit tempus; igitur &longs;i accipiatur &longs;patium, quod percurritur primo in&longs;tanti huius motus, & deinde alia huic æqualia; haud dubiè, cum &longs;e­cundo in&longs;tanti motus &longs;it tardior, &longs;itque a&longs;&longs;umptum æquale &longs;patium; haud dubiè inquam in&longs;tans &longs;ecundum erit maius primo, & tertium &longs;ecundo, atque ita deinceps.

Theorema 31.

Hinc primo in&longs;tanti motus violenti de&longs;truitur minor gradus impetus quàm &longs;ecundo, quod demon&longs;tro; quia cadem cau&longs;a breuiore tempore minùs agit per Ax.3.l.2. & Ax. 13.l.1. num.4. igitur minùs impetus de&longs;truitur pri­mo, quàm &longs;ecundo, & minùs &longs;ecundo quàm tertio, atque ita deinceps; idem enim dici debet de cau&longs;a de&longs;tructiua, quod de productiua.

Dices, igitur idem impetus de&longs;truitur primo in&longs;tanti, quo e&longs;t, &longs;i de&longs;trui­tur primo in&longs;tanti motus. Re&longs;pondeo negando; quia primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus per Th.34.l.1.

Dices, igitur impetus ille e&longs;t fru&longs;trà, quia nullus effectus, &longs;eu motus ex eo &longs;equitur; Re&longs;pondeo negando; nam omnes gradus impetus qui ci­dem parti mobilis in&longs;unt, communi qua&longs;i actione, vel exigentia indi­ui&longs;ibiliter exigunt motum.

Theorema 32.

Hinc gradus omnes producti in eadem parte &longs;ubiecti &longs;unt inæquales in­perfectione; cum enim &longs;inguli &longs;ingulis in&longs;tantibus de&longs;truantur, vt dictum e&longs;t; quippe e&longs;t tantùm vnus gradus impetus innati, & cum &longs;ingula in­&longs;tantia &longs;int inæqualia, etiam &longs;inguli gradus illius impetus &longs;unt inæquales in perfectione.

Theorema 33.

Hinc redditur optima ratio, cur tot producantur potiùs quàm plures, quæ alioquin minimè afferri pote&longs;t; immò, ni&longs;i hoc e&longs;&longs;et, nulla e&longs;&longs;et huiu&longs;modi naturalis retardatio; nam producantut, &longs;i fieri pote&longs;t, omnes æquales, &longs;int­que v.g.20. nunquid po&longs;&longs;unt e&longs;&longs;e 40. perfectionis &longs;ubduplæ, vel 10. du­plæ, vel 5. quadruplæ &c. cur autem potiùs vnum dices quàm aliud? at verò optimam inde reddo rationem quòd cum &longs;int omnes inæquales, cò plures &longs;unt, quò maior e&longs;t ni&longs;us; pauciores verò, quò minor.

Theorema 34.

Hinc &longs;unt inæquales in eâdem proportione, in quæ in&longs;tantia &longs;unt inæqualiav. g. quà proportione primum in&longs;tans e&longs;t minus &longs;ecundo, & &longs;ecundum tertio, ita ille gradus impetus, qui de&longs;truitur primo in&longs;tanti, e&longs;t minor vel imperfectior co, qui de&longs;truitur &longs;ecundo, & qui de&longs;truitur &longs;ecundo imperfectior co, qui de&longs;truitut tertio, atque ita deinceps.

Theorema 35.

Hinc perfecti&longs;&longs;imus omnium gruduum ille e&longs;t qui de&longs;truitur vltimo in&longs;tan­ti, de quo infrá; quod &longs;equitur ex dictis nece&longs;&longs;ariò: vtrùm verò ille &longs;it æ­qualis omninò in perfectione impetui naturali innato, dicemus infrà.

Scholium.

Hic ob&longs;eruabis mirabilem &longs;anæ naturæ prouidentiam, quæ motus omnes cum ip&longs;o naturali ita compo&longs;uit, vt &longs;it veluti regula omnium mo­taum, &longs;itque vnum qua&longs;i principium perfectionis totius impetus; tùm in motu naturali, in cuius progre&longs;&longs;ione producitur &longs;emper imperfectior, tùm in violento, in cuius progre&longs;&longs;ione de&longs;truitur &longs;emper perfectior; producitur imperfectior ab cadem cau&longs;a in minoribus temporibus, & de&longs;truitur perfectior ab eadem cau&longs;a in maioribus temporibus; & cum impetus innatus &longs;it cau&longs;a de&longs;tructiua impetus violenti, habet inæqualem proportionem cum &longs;uo effectu pro temporibus inæqualibus; & cum idem impetus innatus &longs;it qua&longs;i principium crementi, vel accelerationis, &longs;icut e&longs;t principium retardationis; certè pro inæqualitate temporum e&longs;t diuer&longs;a proportio crementorum; quo nihil clarius in hac materia meo iudicio dici pote&longs;t.

Theorema 36.

Hinc finis motus naturalis omninò conuenit cum principio motus violenti; & finis huius cum principio illius; quæcumque tandem progre&longs;&longs;io accipia­tur; &longs;iue temporum æqualium in &longs;patiis inæquaiibus; &longs;iue &longs;patio­rum æqualium in temporibus inæqualibus, &longs;iue a&longs;&longs;umantur in&longs;tan­tia in progre&longs;&longs;ione arithmetica &longs;implici iuxta hos numeros 1.2.3.4. &longs;iue a&longs;&longs;umantur temporis partes &longs;en&longs;ibiles in progre&longs;&longs;ione Galilei iuxta hos numeros 1.3.5.7. quæ omnia ex dictis nece&longs;&longs;ariò con&longs;equuntur.

Theorema 37.

Nec modò conuenit principium vnius cum alterius fine, & vici&longs;&longs;im, &longs;ed etiam aliæ partes motus in di&longs;tantiis æqualibus &longs;it enim linea AG, quam percurrit mobile demi&longs;&longs;um ex puncto A deor&longs;um motu naturaliter ac­celerato, & moueatur per 6. in&longs;tantia, &longs;eu 6. tempora æqualia: Primo in&longs;tanti, quo percurtit &longs;patium AB; haud dubiè, quando perueuit ad pun­ctum G, habet 7. gradus impetus æquales, quia ante motum AB habebat innatum; &longs;ed in motu illo fluunt 6. tempora æqualia, vt dictum e&longs;t; igitur 6. acquirit gradus impetus, quorum quidem vltimò acqui&longs;itus nullum adhuc habuit motum; &longs;ed haud dubiè haberet, &longs;i vlteriùs hic motus pro­pagaretur: his po&longs;itis imprimantur mobili in O 7.gradus impetus æqua­les prioribus &longs;ursùm motu violento, per lineam OH; certè primo in&longs;tan­ti motus, &longs;eu tempore æquali prioribus percurret ON, id e&longs;t 6. &longs;patiola; quia licèt &longs;int 7.gradus; attamen impetus innatus corporis grauis detra­hit vnum &longs;patium, &longs;imulque de&longs;truit vnum gradum, &longs;ecundo tempore percurret NM 5. tertio ML 4. quarto LK 3. quinto KI 2. &longs;exto IH 1. igitur primum violenti ON re&longs;pondet vltimo naturali FG &longs;eu &longs;ecun­dum illius quinto huius, tertium illius quarto huius, quartum tertio, quintum &longs;ecundo &longs;extum primo, & vici&longs;&longs;im; idem pror&longs;us in progre&longs;&longs;ione Galilei accidit, a&longs;&longs;umptis &longs;cilicet partibus temporis &longs;en&longs;ibilibus.

Theorema 38.

Hinc ad eam altitudinem a&longs;cendit motu violento cum iis gradibus impe­tus, quos habuit ab eadem altitudine decidens motu naturali; con&longs;tat ex dictis.

Theorema 39.

Hinc &longs;i motus violentus, & naturalis durent æqualibus temporibus, &longs;patia vtriu&longs;que erunt æqualia; con&longs;tat etiam ex dictis v.g. corpus graue, motu naturali in libero aëre tempore duorum &longs;ecundorum percurrit 48. pe­des, igitur &longs;i moueatur &longs;ur&longs;um æquali tempore percurret 48. pedes per &longs;e, dico per &longs;e; quippe ratione figuræ corporis &longs;ecus accidere pote&longs;t, vt plurimùm etiam accedit ratione motus mixti ex motu centri recto, & motu orbis circulari, de quo infrà.

Theorema 40.

Hinc, vt &longs;patia vtroque motu diner&longs;a &longs;unt æqualia, ita tempora quibus de­curruntur &longs;unt æqualia, & impetus acqui&longs;itus in fine naturalis cum in­nato e&longs;t æqualis impetui producta in principio violenti.

Theorema 41.

Hinc tandiu durat de&longs;cen&longs;us mobilis proiecti &longs;ursùm motu violento, quan­diu durat eiu&longs;dem a&longs;cen&longs;us, & tot habet gradus impetus in fine de&longs;cen&longs;us, quot habet in principio a&longs;cen&longs;us; e&longs;t enim æquale &longs;patium; igitur æquale tempus; igitur æqualis vtrobique impetus. Sed hîc duo obiici po&longs;&longs;unt, primò &longs;agittam per lineam verticalem vibratam po&longs;ui&longs;&longs;e tantùm in a&longs;­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&longs;enno; &longs;ecundò, &longs;i eodem tempore corpus graue &longs;ursùm proiectum motu violento a&longs;cenderet, quo deinde de&longs;cendit, in fine de&longs;cen&longs;us æqualis e&longs;&longs;et ictus, &longs;eu percu&longs;&longs;io vtriu&longs;que; cum tamen illa &longs;it maior, quæ infli­gitur motu violento, vt con&longs;tat multis experimentis.

Re&longs;pondeo ad primum etiam te&longs;te Mer&longs;enno globum ferreum trium aut 4. librarum &longs;ur&longs;um explo&longs;um è breuiore tormento &longs;ed latiore, æqua­le 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 acci­dit &longs;agittæ, cuius differentia a&longs;cen&longs;us, & de&longs;cen&longs;us &longs;en&longs;u etiam percipi pote&longs;t; tùm quia lignea materia multò leuior e&longs;t ferro, tùm quia leui&longs;&longs;i­mæ illæ pennæ, quibus in&longs;truitur, motum retardant in de&longs;cen&longs;u; quod maximè confirmatur ex eo quod pluma facilè anhelitu &longs;ur&longs;um pellatur &longs;atis veloci motu, quæ deinde tardi&longs;&longs;imo &longs;ua &longs;ponte de&longs;cendit: præterea mucro ferreus, quo &longs;agitta armatur, &longs;emper præire debet, cuius rei ratio­nem afferemus infrà; igitur cum in a&longs;cen&longs;u præeat, vt præeat in de&longs;cen­&longs;u, altera extremitas &longs;emicirculum &longs;uo motu facere debet, qui certè ad naturalem motum pertinet, alteta tamen extremitas, quæ mouetur mo­motu contrario alterius motum retardat; ad &longs;ecundam obiectionem re&longs;pondebo Th.44.

Theorema 42.

Si motus violentus e&longs;&longs;et æquabilis, &longs;patium e&longs;&longs;et ferè duplum illius, quod percurritur motu naturaliter retardato, a&longs;&longs;umptis &longs;cilicet temporibus æqualibus; cum enim motu æquabili compo&longs;ito ex &longs;ubdupla velocitate maximæ, & minimæ motus accelerati æquali tempore percurratur æquale &longs;patium, &longs;ubduplum minimæ pro nihilo ferè habetur; igitur pote&longs;t tantùm a&longs;&longs;u-mi &longs;ubduplum maximæ; igitur velocitas motus &longs;it æqualis mad dubiè &longs;patium duplum percurretur.

Theorema 43.

Hinc benè à naturâ in&longs;titutum fuit impetum naturalem innatum &longs;emper con&longs;eruari; alioqui violentus e&longs;&longs;et æquabilis, igitur nunquam de&longs;ineret: quantum ab&longs;urdum! quale incommodum &c.

Theorema 44.

Eadem e&longs;t ratio &longs;eu proportio ictuum, & percu&longs;&longs;ionum, quæ integrorum &longs;patiorum quæ &longs;cilicet toto motu percurruntur in a&longs;cen&longs;u & de&longs;cen&longs;u, v. g. corpus graue cadens ex data altitudine 48 pedum æqualem ictum infli­git in fine de&longs;cen&longs;us, & in principio a&longs;cen&longs;us, quo &longs;cilicet ad eamdem altitudinem a&longs;cenderet; probatur, quia æqualis acquiritur impetus in de&longs;cen&longs;u alteri, qui de&longs;truitur in a&longs;cen&longs;u, a&longs;&longs;umptis dumtaxat &longs;patiis illis æqualibus; igitur æqualis e&longs;t in fine de&longs;cen&longs;us, in quo e&longs;t totus acqui&longs;i­tus, atque in principio a&longs;cen&longs;us, in quo nullus e&longs;t de&longs;tructus: ad id verò, quod dicebatur &longs;uprà de &longs;agitta, cuius ictus maior e&longs;t initio a&longs;cen&longs;us, quàm in fine de&longs;cen&longs;us non diffiteor; quia materia &longs;agittæ, tùm lignea tùm plumea motum &longs;atis &longs;uperque retardat, vt differentia ictuum &longs;en&longs;u ip&longs;o percipi po&longs;&longs;it; quæ tamen nulla perciperetur in a&longs;cen&longs;u de&longs;cen&longs;u­que globi ferrei.

Theorema 45.

Hinc reiicies Galileum, & alios eius &longs;ectatores qui volunt impetum corpori impre&longs;&longs;um de&longs;trui tantùm ab aëre; quod plu&longs;quàm fal&longs;um e&longs;&longs;e comper­tum e&longs;t, vt demon&longs;trauimus &longs;uprà Th. 20. qua&longs;i verò non ad&longs;it aliqua cau&longs;a nece&longs;&longs;aria de&longs;tructiua, &longs;cilicet impetus innatus; hinc etiam eum­dem reiicies, qui vult numquam fieri po&longs;&longs;e, vt motu naturaliter accelera­to tanta acquiratur velocitas, quanta imprimitur in motu violento; vult enim motum acceleratum tran&longs;ire in æquabilem, cuius contrarium de­mon&longs;trauimus &longs;uprà Th. 131, l. 2. igitur cum cre&longs;cat &longs;emper velocitas, nullus e&longs;t finitus gradus, quem tandem non a&longs;&longs;equatur; immò vt dictum e&longs;t in præcedenti Th. a&longs;&longs;umptis æqualibus &longs;patiis, impetus, qui e&longs;t in principio a&longs;cen&longs;us, æqualis e&longs;t cum eo, qui e&longs;t in fine de&longs;cen&longs;us.

Diceret fortè aliquis cadentem globum ex alti&longs;&longs;imæ turris apice de­clinare à perpendiculari antequam terram feriat, vt con&longs;tat ex multis experimentis; igitur præualet tandem re&longs;i&longs;tentia aëris: &longs;ed re&longs;pondeo id rantùm accidere propter currentem illac aëris tractum; alioquin non e&longs;&longs;et potiùs ratio, cur in vnam partem declinaret, quàm in aliam.

Theoroma 46.

Non e&longs;t eadem ratio ictuum, &longs;eu percu&longs;&longs;ionum, quæ e&longs;t &longs;egmentorum in­tegri &longs;patij; v.g. in &longs;ubduplo &longs;patij &longs;egmento non e&longs;t &longs;ubduplus ictus, &longs;it enim &longs;patium integrum motus vîolenti OH, & principium motus &longs;it in O, finis in H; accipiatur &longs;egmentum OM, quod e&longs;t qua&longs;i &longs;ubduplum O H, ictus in M non e&longs;t profectò &longs;ubduplus ictus in O, &longs;ed tantùm in L, vt con&longs;tat e dictis; igitur rationes ictuum non &longs;unt, vt rationes &longs;egmen­torum integri &longs;patij.

Theorema 47.

Vt in praxi determinentur rationes ictuum; a&longs;&longs;umatur progre&longs;&longs;io Gali­lei in AF, ita vt &longs;i prima parte temporis &longs;en&longs;ibili percurratur &longs;patium FE 9 partium æqualium; &longs;ecunda percurratur ED. 7. partium, tertia DC 5. quarta CB 3; quinta BA 1. hoc po&longs;ito facilè erit determinare rationes ictuum; nam in de&longs;cen&longs;u ictus &longs;unt vt velocitates, & hæ vt tem­pora; igitur &longs;i AB percurritur in dato tempore, & AC in duobus prio­ri æqualibus; certè ictus in de&longs;cen&longs;u AC e&longs;t duplus ictus in de&longs;cen&longs;u AB; in AD triplus, &c. Igitur in a&longs;cen&longs;u ictus in F erit quintuplus, ictus in E quadruplus in D triplus, &c. igitur ictus &longs;unt in ratione dupli­cata &longs;patiorum facto &longs;patij initio à &longs;ummo puncto A.

Theorema 48.

Hinc cognitis viribus, quibus corpus graue proijcitur ad datam altitudi­nem, cogno&longs;ci po&longs;&longs;unt vires, quibus ad aliam quamcumque proijciatur; v. g. proiiciatur corpus graue ad altitudinem 48. pedum; vires &longs;unt iis æqua­les, quas acquirit in de&longs;cen&longs;u ciu&longs;dem altitudinis 48. pedum; &longs;it alia di­&longs;tantia 100. pedum; haud dubiè vires nece&longs;&longs;ariæ ad motum hunc violen­tum &longs;unt æquales iis, quas acquireret in de&longs;cen&longs;u 100. pedum per Th. 40. atqui ita &longs;e habent vires acqui&longs;itæ in de&longs;cen&longs;u 48. pedum ad vires acqui&longs;itas in de&longs;cen&longs;u 100. vt v.g. 48. ad v.g. 100. id e&longs;t ferè vt 7. ad 10.

Theorema 49.

Cognitis etiam &longs;patiis cogno&longs;cetur tempus; &longs;it enim decur&longs;um idem &longs;pa­tium 48. pedum motu violento &longs;ur&longs;um; idque v. g. tempore 2. &longs;ecundo­rum, quod ferè cum experientia con&longs;entit; &longs;it aliud &longs;patium 100. tempus primi motus e&longs;t ad tempus &longs;ecundi vt v. g. 48. ad v. g. 100. quia &longs;patia &longs;unt vt quadrata temporum; igitur tempora vt radices 4. hinc vires &longs;unt in ratione temporum; quia vt temporibus æqualibus acquiruntur æqua­lia velocitatis momenta in motu naturali, ita & de&longs;truuntur æqualia in motu violento, quæ omnia con&longs;tant; igitur ictus &longs;unt vt vires, vires vt tempora, tempora denique, vt radices que &longs;patiorum.

Theorema 150.

In vltimo contactu motus violenti nullus e&longs;t ictus, v. g. mobile projectum &longs;ur&longs;um per lineam FA nullam percu&longs;&longs;ionem infligeret in A; probatur quia non tendit vlteriùs; igitur non impeditur eius motus à &longs;uperficie corporis terminati ad punctum A; igitur nullum impetum in eo produ­cit, qui tantùm producitur ad tollendum impedimentum per Th.44.l.1. igitur nullum ictum infligit, qui tantùm infligitur per impetum, vt con&longs;tat.

Theorema 51.

Ex his &longs;atis facilè comparari po&longs;&longs;unt rationes pereu&longs;&longs;ionis, quæ infliguntur tùm ex ca&longs;u corporis grauis cadentis, tùm vi mallei impacti, tùm ex impetu corporis projecti, tùm ex grauitatione corporis grauis incum­bentis, quæ omnia hîc fu&longs;iùs e&longs;&longs;ent tractanda, ni&longs;i locum proprium infrà &longs;ibi vendicarent.

Theorema 52.

Ad motum violentum non concurrit impetus innatus, probatur, quia im­petus ad lineas oppo&longs;itas ex diametro determinati ad communem li­neam determinari non po&longs;&longs;unt, cur enim potiùs dextror&longs;um quam &longs;ini­ctror&longs;um? igitur non concurrunt ad communem motum, ni&longs;i dicatur impetus innatus veleo nomine concurrere ad violentum, quod eius li­neam &longs;ingulis temporibus qua&longs;i ca&longs;tiget, vltróque, vel vlteriùs currentem contineat.

Theorema 53.

Hinc ad motum violentum impetus ab exteriore potentia mobili impre&longs;&longs;us tantùm concurrit; patet, cum enim in mobili projecto &longs;ur&longs;um &longs;it tantùm ille impetus præter innatum, nec innatus concurrat per Th. 52. illum tantùm concurrere nece&longs;&longs;e e&longs;t: excipe &longs;emper impetum acqui&longs;itum, de quo iam &longs;uprà.

Theorema 54.

Primo instanti quo producitur impetus ille à potentia motrice in mobili, me­diante &longs;cilicet impetu producto in organo proprio, non e&longs;t motus; probatur, quia primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, per Th.34.l.1.

Theorema 55.

Impetus productus in manu producit impetum in organo vel in mobili pri­mo in&longs;tanti, quo e&longs;t; probatur, quia &longs;ecundo in&longs;tanti exigit motum &longs;ui &longs;ub­jecti; igitur tolli etiam impedimentum; igitur per motum medij; igitur priori in&longs;tanti in eodem mobili debet e&longs;&longs;e impetus; igitur produci ab impetu organi; igitur & in organo ab impetu manus.

Theorema 56.

Primo in&longs;tanti, quo producitur impetus in motu violento, nullus eius gra­dus de&longs;truitur; probatur, quia alioquin &longs;imul eodem in&longs;tanti, quo e&longs;&longs;e in­ciperet, e&longs;&longs;e de&longs;ineret, quod dici non pote&longs;t.

Theorema 57.

Impetus innatus impedit ne producatur tantus impetus in motu violento,probatur, quia certè tàm impedit primam productionem, quàm con&longs;er­uationem, vt patet; e&longs;t enim par vtrobique ratio; præterea agit in ip&longs;am manum.

Theorema 58.

Impetus violentus producitur minor, quàm produceretur vno dunt ax at gra­du aquali ip&longs;i impetui innato; quippe &longs;icut de&longs;truit &longs;ingulis in&longs;tantibu: æqualibus vnum gradum; quia pugnat pro rata; ita pror&longs;us impedit, ne producatur vnus gradus &longs;ibi æqualis primo in&longs;tanti; cur enim duo po­tiùs, quàm tres?

Theorema 59.

Secundo &longs;tatim in&longs;tanti de&longs;truit alterum gradum: quippe e&longs;t cau&longs;a ne­ce&longs;&longs;aria; igitur &longs;tatim primo in&longs;tanti exigit de&longs;tructionem; non certè pro primo in&longs;tanti per Th.56.igitur pro &longs;ecundo, atque ita pro aliis dein­ceps; de&longs;truitur autem, ne &longs;it fru&longs;trà eo modo, quo diximus &longs;uprà.

Theorema 60.

Hinc optimaratio illius institi naturæ, quo factum e&longs;t, vt impetus innatus numquam destruatur; ne &longs;i aliquando de&longs;trueretur, nulla e&longs;&longs;et cau&longs;a de­&longs;tructiua impetus violenti; ac proinde æquabilis e&longs;&longs;et, &longs;emperque dura­ret, de&longs;tructiua inquam &longs;uo modo.

Theorema 61.

Hinc corpus quod non grauitat, facilè proijcitur, vel impellitur: &longs;ic na­uis aquis innatans, nubes in aëre libratæ; halitus, atque adeo ip&longs;æ partes aquæ, quas perexiguus lapillus in orbes penè innumeros agit, ne quid dicam de partibus aëris, quæ tam citò & procul mouentur, vt con&longs;tat in &longs;ono, motu &longs;cilicet ferè æquabili.

Theorema 62.

Hinc etiam è contrario corpus grauius difficiliùs &longs;ur&longs;um proijcitur: tùm quia plures partes impetus &longs;unt producendæ in &longs;ubjecto grauiore quod pluribus partibus con&longs;tat, tùm impetus innatus maior e&longs;t, non quidem in inten&longs;ione &longs;ed in exten&longs;ione, ac proinde impedit ne plures gradus pro­ducantur; quippe maius impedimentum plus impedit, quis hoc neget?

Theorema 63.

Omnes partes impetus productæ in mobili primo instanti concurrunt ad motum &longs;ecundi instantis; probatur, quia alioqui aliqua e&longs;&longs;et fru&longs;trà, quod dici non debet.

Theorema 64.

Concurrunt omnes illæ, quæ in&longs;unt eidem parti &longs;eu puncto mobilis commqua&longs;i actione vel exigentia; patet ex dictis de impetu, quia concurrunt ad velocitatem, quæ e&longs;t indiui&longs;ibilis actu.

Theorema 65.

Non ponitur tamen totus motus &longs;ecundo instanti, quem cgunt prquia impetus innatus aliquid detrahit, cum exigat motum deor&longs;um per lineam oppo&longs;itam, igitur imminuitur motus pro rata.

Theorema 66.

Hinc ille gradus motus quinon ponitur &longs;ecundo instanti respondet gradus im petus qui destruitur; cum vterque habeat eamdem men&longs;uram, &longs;cilicet im petum innatum.

Theorema 67.

Hinc effectus pete&longs;t e&longs;&longs;e eo instanti quo non existit eius cau&longs;a partialis; v.g. motus qui ponitur &longs;ecundo in&longs;tanti non minùs exigitur ab eo gradu im­petus qui de&longs;truitur &longs;ecundò in&longs;tanti, quàm ab aliis, non exigitur qui­dem &longs;ecundo &longs;ed primo pro &longs;ecundo; vnde dixi cau&longs;am partialem, quia etiam exigitur ab aliis gradibus impetus, qui non de&longs;truuntur exigentiâ communi; quippe impetus non exigit ni&longs;i pro &longs;ecundo in&longs;tanti; nec vl­lum ab&longs;urdum e&longs;t eo in&longs;tanti cau&longs;am exigentiæ non exi&longs;tere cum poni­tur eius effectus, &longs;cilicet id quod exigebat priori in&longs;tanti quo erat; nul­lus e&longs;t enim influxus huius cau&longs;æ; præ&longs;ertim cum non &longs;it cau&longs;a totalis.

Vnde cum effectus qui ponitur &longs;ecundo in&longs;tanti non re&longs;pondeat per­fectioni cau&longs;æ totius propter impedimentum, aliquis gradus cau&longs;æ e&longs;&longs;et fru&longs;trà; igitur eodem in&longs;tanti &longs;ecundo de&longs;trui debet, alioqui ni&longs;i de&longs;true­retur &longs;ingulis in&longs;tantibus poneretur effectus non re&longs;pondens per&longs;ectioni cau&longs;æ; immò numquam de&longs;trueretur totus motus violentus, vt con&longs;tat; itaque primo in&longs;tanti omnes gradus impetus qui &longs;unt exigunt motum pro &longs;ecundo ne aliquis eo in&longs;tanti &longs;it &longs;ru&longs;trà &longs;i non exigeret, & &longs;ecundo in&longs;tanti aliquis gradus impetus de&longs;truitur, ne &longs;it fru&longs;trà codem in&longs;tanti &longs;ecundo, cum &longs;cilicet non &longs;int tot gradus motus, quot &longs;unt gradus impe­tus; atque ita deinceps tertio in&longs;tanti de&longs;truitur vnus gradus, vt iam &longs;u­prà dictum e&longs;t.

Theorema 68.

Ideo de&longs;truitur potiùs vnus gradus impetus quàm alius &longs;ecundo in&longs;tanti, tertioque, &c. quia talis e&longs;t per&longs;ectionis; hoc iam &longs;uprà explicatum e&longs;t; quia cum motus initio &longs;it velocior, in&longs;tantia &longs;unt minora, igitur minùs im­petus in &longs;ingulis de&longs;truitur, pater ex dictis.

Theorema 69.

Ille gradus impetus qui de&longs;truitur &longs;ecundo in&longs;tanti non concurrit ad motuns tertij in&longs;tantis; quia non pote&longs;t concurrere ad motum ni&longs;i exigendo; at­qui exigere tantùm pote&longs;t, quando e&longs;t; quod enim non e&longs;t non exigit, &longs;ed motus tertij in&longs;tantis exigitur &longs;ecundo; &longs;ic enim tota res motus pro­cedit vt impetus primo in&longs;tanti exigat motum pro &longs;ecundo; & &longs;ecundo pro tertio; & tertio pro quarto, atque ita deinceps; igitur impetus ille qui de&longs;tru&longs;tur; &longs;ecundo in&longs;tanti non exigit motum pro tertio, & qui de­&longs;truitur tertio non exigit pro quarto, atque ita deinceps.

Theorema 70.

Hinc impetus innatus non concurrit ad motum violentum, vt dictum e&longs;t, &longs;ed tantùm impedit, immediatè quidem, quia cum exigat motum deor­sùm, facit vt non &longs;it tantus motus &longs;ur&longs;um; mediatè verò, quia cum non &longs;it tantus motus &longs;ursùm, quantus e&longs;&longs;et, haud dubiè non re&longs;pondet adæ­quatè cau&longs;æ; igitur aliquid cau&longs;æ fru&longs;trà e&longs;t; igitur de&longs;trui debet; hinc de&longs;truitur etiam hic impetus per principium commune, ne aliquid &longs;it fru&longs;trà.

Theorema 71.

Linea motus &longs;ur&longs;um determinatur à potentia motrice; probatur, quia hæc determinat impetum productum in manu vel in organo; hic verò im­petum, quem producit in mobili &longs;ursùm projecto; patet, quia nulla e&longs;t alia cau&longs;a applicata.

Theorema 72.

Tandem duo impetus violentus, &longs;cilicet, & innatus ad æqualitatem perue­nirent, &longs;i vel vnus gradus violenti e&longs;&longs;et æqualis perfectionis cum innato; cum enim detrahatur &longs;emper pars aliquota alicuius totius, tandem perueni­tur ad vltimam; igitur &longs;int 100. gradus impetus violenti, quorum quili­bet &longs;it æqualis impetui innato; certè cum temporibus æqualibus æqua­lis gradus impetus de&longs;truatur; accipiatur illud tempus, in quo de&longs;trui­tur vnus, haud dubiè 100. æqualibus temporibus de&longs;truentur omnes 100. igitur 99. in&longs;tantibus de&longs;truentur 99. gradus; igitur &longs;upere&longs;t vnus; igitur duo illi impetus perueniunt tandem ad æqualitatem.

Theorema 73.

Vbivterque perueni&longs;&longs;et ad æqualitatem, non e&longs;&longs;et potior ratiocur mobile mo­ueretur &longs;ursùm quàm deor&longs;um in&longs;tanti &longs;equenti; probatur, quia tàm gra­dus impetus innati exigit motum deor&longs;um quàm gradus impetus vio­lenti &longs;ursùm; igitur neuter habebit motum per Th.133.l. 1.

Theorema 74.

Hinc ip&longs;o in&longs;tanti, quo e&longs;&longs;et æqualitas, e&longs;&longs;et adhuc motus; quia in&longs;tanti immediatè antecedenti erant duo gradus impetus violenti, & vnus in­nati; igitur duo illi præualent pro in&longs;tanti &longs;equenti, in quo e&longs;t æqua­litas.

Theorema 75.

Itaque quie&longs;ceret mobile ip&longs;o &longs;tatim in&longs;tanti, quod in&longs;tanti æqualitatis &longs;uc­cedit; patet, quia neuter impetus pro illo in&longs;tanti præualere po&longs;&longs;et per Th. 73.

Theorema 76.

Igitur in&longs;tanti quietis nullus e&longs;&longs;et ampliùs impetus violantus; cum enim &longs;ingulis in&longs;tantibus de&longs;truatur vnus gradus, v. g in&longs;tanti illo, quod &longs;e­quitur po&longs;t in&longs;tans æqualitatis, de&longs;truitur ille gradus, qui &longs;upere&longs;t; nec pote&longs;t vel plùs, vel minùs de&longs;trui; pugnant enim pro rata; quod certè cuiquam fortè paradoxor videbitur, &longs;cilicet nullum tune e&longs;&longs;e motum propter pugnam, cum tamen nulla e&longs;t amplius pugna.

Theorema 77.

Quies illa duraret tantùm vno in&longs;t anti, probatur, quia cum in&longs;tanti quie­tis &longs;it tantùm impetus innatus per Th. 76. certè non impeditur quomi­nus habeat motum pro in&longs;tanti &longs;equenti, quem reuerà exigit; igitur pro in&longs;tanti &longs;equenti moueritur; &longs;ed pro alio antecedente mouebatur; igi­tur quies illa durat tantùm vno in&longs;tanti.

Theorema 78.

Quies illa non fit propter æliquam reflexionem, vt aliqui dicunt; quia nul­la pror&longs;us e&longs;t reflexio, vbi nullum e&longs;t reflectens; atqui nullum e&longs;t refle­ctens, vt patet, quia nullum e&longs;t corpus impediens motus propagationem; licèt enim medium impediat, non tamen per modum reflectentis pro­priè; immo vt dicemus infrà in puncto reflexionis nulla datur quies; &longs;ed motus reflexus &longs;ibi vendicat librum &longs;ingularem.

Theorema 79.

Hinc &longs;iue præce&longs;&longs;erit motus violentus, &longs;iue non, corpus graue eodem vel æ­quali motu deor&longs;um cadit, quia nullus amplius remanet impetus violen­tus in fine motus violenti, per Th.76. igitur &longs;olus impetus naturalis li­bero motu deorsùm fertur.

Theorema 80.

Hinc reiicies aliquos æpud Galileum, qui volunt ideo motum naturalem accelerari, quia &longs;en&longs;im de&longs;truitur impetus violentus antè impre&longs;&longs;us, quod pe­nitus ridiculum e&longs;t; quia lapis deci&longs;us è rupe etiam motu naturaliter accelerato deor&longs;um cadit, licèt eò nunquam motu violento euectus fuerit.

Ob&longs;eruabis hanc hypothe&longs;im gradus impetus violenti æqualis perfe­ctionis cum innato e&longs;&longs;e fal&longs;am. Primò, quia commodius e&longs;t potentiæ motrici producere imperfectiorem impetum, &longs;ic enim plures illius gra­dus producere pote&longs;t. Secundò, quia in reflexo &longs;ur&longs;um vltimus gradus qui de&longs;truitur e&longs;t imperfectior innato, e&longs;t enim acqui&longs;itus; igitur in omni alio motu &longs;ursùm. Tertiò, quia violentus e&longs;t cuminnato in eadem &longs;ubie­cti parte; &longs;ed idem &longs;ubiectum formas homogeneas non patitur, de quò aliàs, hinc dicendum &longs;upere&longs;t non quie&longs;cere mobile in fine motus

Theorema 81.

Corpus quod non grauitat proiicitur &longs;ur&longs;um motu æquabili per &longs;e; patet, quia nihil e&longs;t quod de&longs;truat ip&longs;um impetum; igitur &longs;emper moueretur, ni&longs;i per accideens ab ip&longs;o medio eius motus retardaretur; vnde dixi per &longs;e,cum ratione medij retardetur; immò quò leuius e&longs;t, faciliùs à medio re­tinetur, vide Th.61.

Theorema 82.

Noncre&longs;c it impetus naturalis in motu violento &longs;ur&longs;um; probatur primò, quia impetus naturalis aduentitius &longs;upponit motum deor&longs;um, ad cuius inten&longs;ionem à natura fuit in&longs;titutus per re&longs;p. ad quartam obiect. in di&longs;­&longs;ert.l.2. adde quod tardiùs a&longs;cenderet, quàm de&longs;cenderet; deinde velo­ciùs de&longs;cenderet po&longs;tmotum violentum corpus graue, quàm &longs;i nullo mo­tu violento præuio demitteretur deor&longs;um, quæ omnia experimentis etiam vulgaribus repugnant; immò & cunctis ferè præmi&longs;&longs;is Theorematis.

Theorema 83.

Motus violentus non tendit ad quietem per omnes tarditatis gradus, vt pa&longs;&longs;im a&longs;&longs;erit Galileus; Primò, quia non &longs;unt infinita in&longs;tantia, &longs;ed retarda­tur tantùm &longs;ingulis in&longs;tantibus; Secundò in medio den&longs;iore minùs du­rat; igitur non tran&longs;it per tot gradus tarditatis; præterea in plano incli­nato &longs;ur&longs;um în minore proportione retardatur motus, quod etiam in plano horizontali certi&longs;&longs;imum e&longs;t; quorum omnium rationes &longs;uo loco videbimus.

Nec e&longs;t quod aliqui dicant infinito tribui non po&longs;&longs;e hæc prædicata æqualitatis vel inæqualitatis, quod fal&longs;um e&longs;t, loquamur de infinito actu; &longs;i enim e&longs;&longs;et numerus infinitus hominum, nunquid verum e&longs;&longs;et dicere numerum oculorum e&longs;&longs;e maiorem numero hominum; nec e&longs;t quod ali­qui confugiant ad di&longs;iunctiones; nos rem i&longs;tam &longs;uo loco fusè tractabi­mus & demon&longs;trabimus, ni fallor, cum Ari&longs;totele, fieri non pò&longs;&longs;e vt &longs;it aliquod creatum infinitum actu; licèt vltrò concedamus plura e&longs;&longs;e infi­nita potentiâ; & verò certum e&longs;t infinito potentiâ non ine&longs;&longs;e huiu&longs;inodi prædicata æqualitatis, vel inæqualitatis.

Theorema 84.

Immò &longs;i tran&longs;iret mobile &longs;ursùm proiectum per omnes tarditatis gradus, nunquam profectò de&longs;cenderat; quia cum &longs;ingulis in&longs;tantibus &longs;inguli gra­dus re&longs;pondeant, & duo in&longs;tantia &longs;imul e&longs;&longs;e non po&longs;&longs;int; nunquam certè verum e&longs;&longs;et dicere fluxi&longs;&longs;e infinita; igitur nec mobile per infinitos tar­ditatis gradus ad quietem perueni&longs;&longs;e; hoc Theorema &longs;upponit e&longs;&longs;e tan­tùm finita in&longs;tantia.

Theorema 85.

Re&longs;i&longs;tentia aëris est maior initio, quàm in fine motus violenti, vt con&longs;tat ex dictis, quia initio motus e&longs;t velocior, igitur plures partes aëris æquali tempore re&longs;i&longs;tunt; in fine verò è contrario.

Theorema 86.

Hinc oppo&longs;ita e&longs;t omninò ratio re&longs;istentia, quæ &longs;equitur ex motu violento illi, quæ cum naturali e&longs;t coniuncta, hæc enim initio minor, in fine maior, illa verò initio maior, & in fine minor; hinc prima cre&longs;cit cam &longs;uo motu, &longs;ecunda cum &longs;uo decre&longs;cit.

Theorema 87.

Decre&longs;cit igitur impetus eadem proportione, qua decre&longs;cit re&longs;i&longs;tentia; vt pa­tet ex dictis; igitur in toto motu eadem e&longs;t re&longs;i&longs;tentiæ proportio.

Theorema 88.

Variæ &longs;unt potentiæ motrices, à quibus mobile &longs;ur&longs;um proiici potest motu violento, v.g. potentia motrix animantium, potentia motrix grauium mo­bili &longs;cilicet &longs;ur&longs;um repercu&longs;&longs;o; potentia motrix, quæ &longs;equitur ex com­pre&longs;&longs;ione & rarefactione corporum, &longs;ed de his omnibus aliàs.

Scholium.

Ob&longs;eruabis primò &longs;i aliquando accidat, vt aliqui volunt ictum, qui &longs;tatim initio motus violenti infligitur, non e&longs;&longs;e maximum, &longs;ed minorem eo, qui po&longs;t aliquod confectum &longs;patium infligitur; quod probant in pila ex fi&longs;tula ænea &longs;ur&longs;um emi&longs;&longs;a, quæ moiorem ictum infligit in data di&longs;tantia, 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&longs;um pila pellatur ab igne, qui ab ore fi&longs;tulæ erumpens per aliquod &longs;pa­tium à tergo vrget; igni enim innatum e&longs;t &longs;ur&longs;um euolare.

Ob&longs;eruabis &longs;ecundò, vix po&longs;&longs;e manu mobile &longs;ur&longs;um rectà proiici, quia &longs;cilicet manus extremitas motu mixto mouetur ex duobus vel pluribus circularibus, de quo infrà.

Ob&longs;erua tertiò, non tantùm propter grauitationem con&longs;eruari impe­rum naturalem innatum, &longs;ed etiam vt motui violento re&longs;i&longs;tat; at verò non re&longs;i&longs;teret, ni&longs;i grauitaret.

Ob&longs;erua quartò, reciprocas rationes motus naturalis & violenti; in quibus mirabile pror&longs;us fuit naturæ in&longs;titutum, cum idem in vtroque il­larum &longs;it principium.

Ob&longs;erua quintò, finem motus violenti e&longs;&longs;e multiplicem, nullum ta­men à natura in&longs;titutum; quippe potentia motrix, quæ agit ex appetitu elicito, (vt vulgò aiunt,) &longs;eu cum cognitione, finem &longs;ibi proponit ad libi­tùm; illa verò quæ vi compre&longs;&longs;ionis excitatur per accidens &longs;ur&longs;um agit mobile potiùs, quàm per aliam lineam; repercu&longs;&longs;a &longs;ursùm videntur e&longs;&longs;e magis iuxta in&longs;titutum naturæ.

LIBER QVARTVS,

DE MOTV MIXTO EX duobus, vel pluribus rectis.

MOTVM mixtum eum e&longs;&longs;e non dico, qui ex pluribus aliis motibus componatur; &longs;eu mi&longs;ceatur; nec enim plures motus &longs;imul e&longs;&longs;e po&longs;&longs;unt in eodem mobili; cùm tantùm e&longs;&longs;e po&longs;&longs;it vno dumtaxat in&longs;tan­ti vnica migratio ex loco in locum; nec plura loca naturali virtute &longs;imul acquiri po&longs;&longs;unt; Igitur nec &longs;i­mul e&longs;&longs;e duo motus; Itaque motus mixtus &longs;implex e&longs;t, &longs;i con&longs;ideretur ratio, & linea motus; mixtus verò dicitur, quod ex pluribus re&longs;ultet, qui reuerâ non &longs;unt, &longs;ed cùm e&longs;&longs;e po&longs;&longs;int, qua&longs;i confluunt in tertium motum communi &longs;umptu qua&longs;i de vtroque partici­pantem, quod totum fit propter diuer&longs;os impetus, vel eumdem ad diuer&longs;as lineas determinatum, vt fusè explicabimus infrà: Porrò in hoc Libro explicamus tantùm motum mixtum, qui re&longs;ultat ex pluribus re­ctis, vt titulus ip&longs;e præfert.

DEFINITIO 1.

MOtus mixtus e&longs;t, qui &longs;equitur ex multiplici impetu ad eamdem, vel di­uer&longs;as lineas determinato, vel eodem ad diuer&longs;as; hæc definitio cla­ra e&longs;t; ob&longs;eruabis tantùm ad motum mixtum &longs;ufficere duplicem impe-tum ad eamdem lineam determinatam, deor&longs;um, v.g. in mobili proiecto; nec enim e&longs;t motus purè naturalis, nec etiam violentus, vt con&longs;tat; igi­tur mixtus.

Hypothe&longs;is 1.

Cum proiicitur corpus per lineam horizont alem, vel inclinatum &longs;ur&longs;um, vel deor&longs;um mobile percurrit lineam curuam; quod etiam pueri &longs;ciunt, qui di&longs;co ludunt.

Hypothe&longs;is 2.

Globus etiam plumbeus è &longs;ummo malo malo mobilis nauis demi&longs;&longs;us per lineam perpendicularem deor&longs;um minimè cadit, &longs;ed per curuam inclinatam: hæc hypothe&longs;is mille &longs;altem nititur experimentis; modò &longs;ufficiat quod &longs;it; nam propter quid &longs;it, demon&longs;trabo.

Hypothe&longs;is 3.

Proiectum per horizontalem &longs;ub finem motus minùs ferit quàm initio, imò & proiectum per inclinatam deor&longs;um; hæc hypothe&longs;is centies probata fuit; nec in dubium reuocari pote&longs;t.

Axioma 1.

Omnis impetus qui mobili ine&longs;t dum ip&longs;um mouetur, præ&longs;tat aliquid ad mo­tum; vel enim retardat, vt impetus innatus retardat violentum, vt &longs;uprà diximus; vel ad motum vnà cum alio, vel &longs;olus concurrit. Ax.2.

Axioma 2.

Ille impetus qui alium retardat, haud dubiè retardat tantùm pro rata; hoc etiam &longs;uprà demon&longs;trauimus, & qui de&longs;truitur, de&longs;truitur quoque pro rata, ne &longs;it fru&longs;trà qui de&longs;truitur.

Axioma 3.

Ille impetus qui cum alio ad eumdem motum concurrit, concurrit etiam pro rata; hoc etiam &longs;uprà demon&longs;tratum e&longs;t, e&longs;t enim cau&longs;a nece&longs;&longs;aria, igitur quantum pote&longs;t concurrit, igitur pro rata &longs;uæ virtutis.

Axioma 4.

Licèt &longs;int plures impetus in eodem mobili, non &longs;unt tamen plures &longs;imul li­ueæ motus; ne mobile &longs;it &longs;imul in pluribus locis.

Po&longs;tulatum 1.

Licedt a&longs;&longs;ismere quamlibet coniugætionem motuum, v. g. vel duorum æ­quabilium, vel alterius æquabilis, & alterius retardati, vel alterius æqua­bilis, & alterius accelerati, vel alterius reterdati, & alterius accelera­ti, &c.

Po&longs;tulatum 2.

Illa linea vocetur curua quæ con&longs;tat infinitis prope lateribus polygoni.

Theorema 1.

Motus mixtus ex duobus æquabilibus æquælibus e&longs;t rectus; &longs;it enim mo-bile in A, &longs;itque impetus per AB, & alter æqualis per AD, motus mixtus fiet per AE, a&longs;&longs;umpra fcilicet DE æquali, & parallela AB, quod probatur per Th.137.l.1.

Theorema 2.

Linea AE e&longs;t diagonalis quadrati, quotie&longs;cumque vterque impetus e&longs;t æ­qualis, & liueæ determinationum decu&longs;&longs;antur ad angulos rectos; probatur per idem Th.137.

Theorema 3.

Hinc de&longs;truitur aliquid impetus; alioquin motus e&longs;&longs;et duplus cuiu&longs;li­bet &longs;eor&longs;im &longs;umpti, quod fal&longs;um e&longs;t; nam motus &longs;unt vt lineæ &longs;ed diago­nalis quadrati non e&longs;t dupla lateris; hoc etiam probatur per Th. 141. & 142.l.1.

Theorema 4.

Motus mis ex duobus æquabilibus inæqualibus est etiam rectus; &longs;it enim mobile in A cadem figura &longs;itque impetus per AC, & alter &longs;ubdu­plus prioris per AD, motus fiet per AF ducta DF æquali, & parallela AC, quod probatur per Th.137.l.1.

Theorema 5.

Linea AF e&longs;t diagonalis rectanguli, quotie&longs;cunqne lineæ deterninationum decu&longs;&longs;antur ad angulos rectos; probatur per idem Th.137.

Theorema 6.

Hinc de&longs;truitur aliquid impetus per Th.141. & 142.l.1. idque pro rata ne aliquid &longs;it fru&longs;trà per Ax.2. & &longs;æpè iam probatnm e&longs;t.

Theorema 7.

Hinc determinari pote&longs;t portio vtriu&longs;que impetus destructi, v.g. &longs;i &longs;int æ­quales, portio detracta vtrique æqualibus temporibus e&longs;t differentia diagonalis & compo&longs;itæ ex DA, AB, quod clarum e&longs;t; &longs;i vero impetus &longs;int inæquales, portio de&longs;tructa erit &longs;emper differentia diagonalis, v.g. AF & compo&longs;itæ ex AC.AD.

Theorema 8.

Aliquando impetus qui remanet in motu mixto est rationalis; id e&longs;t habet proportionem ad vtrumque, quæ appellari pote&longs;t, aliquando ad neutrum, aliquando ad alterutrum; ad vtrumque v.g. &longs;i alter impetuum &longs;it 8.alter 6. haud dubiè linea motus mixti erit 10. ad neuttum vt in diagonali qua­drati, & in multis aliis; ad alterum denique v. g. &longs;i alter &longs;it &longs;ubduplus la­teris æquilateri; alter verò eiu&longs;dem perpendicularis; nam diagonalis, &longs;eu linea motus mixti erit latus ip&longs;um æquilateri.

Theorema 9.

Si lineæ determinationum decu&longs;&longs;entur ad angulum obtu&longs;um, &longs;intque æqua­les impetus, linea motus mixti erit diagonalis Rhombi; vt patetper Th.140. l.1. pote&longs;t autem hæc diagonalis e&longs;&longs;e vel æqualis alteri laterum, vel ma-ior, vel minor; e&longs;t æqualis, quando angulus maior Rhombi e&longs;t 120. e&longs;t minor cùm angulus minor e&longs;t 60. denique e&longs;t maior, cùm maior angu­lus e&longs;t minor 120, quæ omnia con&longs;tant ex Geometria.

Theorema 10.

Si lineæ determinationum decu&longs;&longs;entur ad angulum acutum, & &longs;int æqua­les impetus, linea motus mixti erit diaganalis Rhombi; quæ certè eò longior erit, quò angulus erit acutior per Th. 139. l.1. porrò e&longs;t &longs;emper maior lateribus &longs;eor&longs;im &longs;umptis.

Scholium.

Ob&longs;erua in Rhombo e&longs;&longs;e duas diagonales inæquales, vt con &longs;tat; igi­tur cùm lineæ determinationum decu&longs;&longs;antur ad angulum obtu&longs;um, linea motus mixti &longs;emper e&longs;t diagonalis minor; cùm verò decu&longs;&longs;antur ad an­gulum acutum, &longs;emper e&longs;t diagonalis maior.

Corollarium 1.

Hinc quò acutior e&longs;t angulus diagonalis accedit propiùs ad duplum lateris, donec tandem vtraque linea coëat; tunc enim linea motus e&longs;t du­pla lateris.

Corollarium 2.

Hinc quoque quò angulus e&longs;t obtu&longs;ior diagonalis accedit propiùs ad nullam, vt &longs;ic loquar, donec tandem vtraque linea concurrat in rectam, tunc enim nulla e&longs;t diagonalis; igitur nulla linea motus.

Theorema 11.

Cum alter impetuum e&longs;t maior, linea motus e&longs;t diagonalis Rhomboidis, mi­nor quidem &longs;i lineæ decu&longs;&longs;entur ad angulum obtu&longs;um; maior verò &longs;i decu&longs;&longs;en­tur ad angulum acutum; vt patet ex dictis.

Theorema 12.

Cum alter impetus in motu mixto est maior, linea motus mixti accedit propiits ad lineam maioris; hoc est facit angulum acutiorem cum illa; v.g. in eadem figura &longs;it linea impetus maioris AC, & minoris AD, linea motus mixti e&longs;t diagonalis AF, quæ accedit propiùs ad AC, quàm ad AD, id e&longs;t facit angulum acutiorem cum AC, vt patet ex dictis.

Theorema 13.

Cum verò impetus &longs;unt æquales, linea motus mixti facit ængulum æqualem cum linea vtriu&longs;que; vt AE in eadem figura quod etiam dici debet, licèt lineæ determinationum decu&longs;&longs;entur ad angulum obtu&longs;um vel acutum, vt AC, EG. IM.

Theorema 14.

Non cre&longs;cit, vel decre&longs;cit in eadem rætione, in quæ vnus impetus &longs;uperat alium; cum enim impetus &longs;int vt lineæ, &longs;ub quibus fiunt rectangula vel Rhomboides; v.g. impetus AC e&longs;t duplus impetus AD, &longs;ed angulus D AF non e&longs;t duplus anguli FAC, vt con&longs;tat ex Geometria.

Scolium.

Ob&longs;eruabis dari de facto hunc motum mixtum ex duobus æquabilibus in rerum natura; talis e&longs;t motus nauis, quam geminus ventus impellit in mari, vel nubis, imò aëris pars in medio aëre, atque adeo ip&longs;ius venti, &longs;unt enim hi motus æquabiles per &longs;e; quippe retardantur &longs;olummodo propter re&longs;i&longs;tentiam medij, non verò propter vllam grauitationem.

Theorema 15.

Motus mixtus ex duobus retardatis e&longs;t rectus; &longs;it enim duplex impetus per AE & AH æqualis; ita vt in dato tempore percurrat &longs;eor&longs;im AE mo­tu retardato; item AH iuxta proportionem Galilei; certè eo tempore quo percurreret AD in AE, & AI in AH percurrit AG motu mîxto per Th. 5. Similiter eo tempore quo percurreret AE &longs;eor&longs;im, & AH, percurrit AF per Th.5. Igitur hic motus mixtus e&longs;t rectus, dum &longs;it vterque retar­datus iuxta eamdem progre&longs;&longs;ionem; &longs;imiliter &longs;i alter impetus impetus &longs;it inæqualis, vt patet in &longs;equenti figura, &longs;it enim impetus per AE, & alter minor per AH, certè ex AD, AI fit AG, & ex AE, AH fit AF, quam rectam e&longs;&longs;e con&longs;tat ex Geometria; nec vlla e&longs;t difficultas, quæ ex &longs;upe­rioribus Theorematis facilè &longs;olui non po&longs;&longs;it.

Corollarium. 1.

Hinc linea motus mixti ex duobus retardatis &longs;iue æqualibus, &longs;iue inæqualibus e&longs;t diagonalis parallelogrammatis &longs;ub lineis determina­tionum.

Scholium.

Ob&longs;eruabis dari de facto hunc motum in rerum natura, &longs;i v. g. in pla­no horizontali idem globus, vel &longs;imul gemino ictu impellatur, vel &longs;i iam impul&longs;um mobile per nouam lineam imppellatur.

Theorema 16.

Motus mixtus ex duobus acceleratis uniformiter e&longs;t etiam rectus; Proba­tur, quia debet tantùm inuerti linea prioris &longs;cilicet mixti ex duobus re­tardatis; &longs;i enim à puncto F pellatur per FE, FH, motu accelerato, ita primo, tempori re&longs;pondeat FM, FN, &longs;ecundo NH, ME; haud dubiè li­nea motus mixti erit FA; nam primò tempori re&longs;pondebit FG, & duo­bus FA, vt con&longs;tat ex dictis, &longs;iue vterque impetus &longs;it æqualis, &longs;iue alter maior altero.

Corollarium 1.

Hinc etiam linea motus mixti ex duobus acceleratis e&longs;t diagonalis, vt iam &longs;uprà dictum e&longs;t de omnibus aliis.

Scholium.

Ob&longs;eruabis hunc motum dari in rerum natura &longs;altem in corporibus &longs;ublunaribus; nec enim e&longs;t acceleratus ni&longs;i &longs;it motus naruralis, qui à duplici impetu e&longs;&longs;e non pote&longs;t.

Theorema 17.

Si motus mixtus con&longs;tet ex æquabili, & accelerato naturaliter &longs;it per li­neam curuam; &longs;it enim impetus per AF motu æquabili, & per AC motu accelerato naturaliter, ita vt eo tempore quo percurritur &longs;eor&longs;im &longs;pa­tium AB percurratur AD triplum; certè ex vtroque primo tempore ro­&longs;ultat linea motus mixti AE, &longs;ecundo tempore EG, &longs;ed AEG non e&longs;t recta; alioquin duo triangula ABE, ACG e&longs;&longs;ent proportionalia, quod e&longs;t ab&longs;urdum.

Theorema 18.

Hæc linea e&longs;t Parabola; quod ip&longs;e Galileus toties in&longs;inuauit, & quiuis etiam rudior Geometra intelliget; in quo diutiùs non hæreo, præ&longs;ertim cùm nullus &longs;it motus, qui con&longs;tet ex æquabili, & naturaliter accelerato, vt demon&longs;trabimus infrà.

Theorema 19.

Si motus mixtus con&longs;tet ex æquabili & naturaliter retardato, fit per lineam curuam; &longs;i enim eo tempore quo per NE &longs;ur&longs;um proiicitur corpus graue & con&longs;equenter motu naturaliter retardato impellatur per NI motu æquabili, diuidatur NI in 4. partes æquales v.g. ductis parallelis RD, NE, PC, &c. a&longs;&longs;umatur NS vel RM, cui affigatur quilibet numerus impar; putà 7. itaque RM &longs;int 7. ducatur HM parallelæ IN, a&longs;&longs;umatur QL 5. ducatur GL parallela, accipiatur VK 3. ducatur FK: denique a&longs;&longs;umatur FAI ducaturque AE parallela IN, & de&longs;cribatur per puncta AKLMN, linea curua; hæc e&longs;t Parabola, vt con&longs;tat ex Geometria; nam &longs;i BK e&longs;t 1. CL erit 4. DM 9. EV 16. &longs;ed æquales &longs;unt AF.AG.AH.AI. prioribus vt patet; igitur &longs;agittæ &longs;unt vt quadrata applicatarum; igitur hæc e&longs;t Parabola; igitur curua, atqui motus mixtus prædictus fieret per hanc lineam, nam eo tempore quo mobile e&longs;&longs;et in S, erit in M, concurrit enim vterque im­petus pro rata, & eo tempore, quo e&longs;&longs;et in K erit in L, atque ita deinceps.

Scholium.

Ob&longs;eruabis e&longs;&longs;e pror&longs;us inuer&longs;am prioris, quæ &longs;it ex motu æquabili, & naturaliter acceletato; &longs;i enim per AE &longs;it æquabilis & æqualis priori per NI, & per AI &longs;it acceleratus, &longs;i quo tempore peruenit in B motu æ­quabili perueniat in F motu accelerato; haud dubiè perueniet in K, mox in L, &c. quia eadem proportione, &longs;ed inuer&longs;a quâ retardatur, acceleratur; igitur &longs;i vltimo tempore retardati acquirit tantùm YE; primo tempore æquali &longs;cilicet accelerati acquiret AF, atque ita deinceps &longs;i per NE &longs;it retardatus, & per NI æquabilis linea motus mixti erit NLA; &longs;i verò &longs;it per AI acceleratus, & per AE æquabilis æqualis priori per NI, lineamosus mixti erit ALN eadem &longs;cilicet cum priori mutatis tantùm terminis à quo, & ad quem; vtrùm verò in rerum natu­tra &longs;it huiu&longs;modi motus videbimus infrà.

Theorema 20.

Si con&longs;tet ex retardato & accelerato, vt fit in perpendiculari &longs;ur&longs;um, & deor&longs;um motus mixtus, linea per quam fit e&longs;t curua, &longs;it enim retardatus per AD, &longs;it acceleratus per AG, a&longs;&longs;umatur AB cum numero impari, putà 5.BC.3. CD.1. accipiatur AE.1. EF.3. ducantur parallelæ BK. CL. DI. & aliæ EM. FH. GI. & per puncta AM. HI. ducatur linea curua, hæc e&longs;t linea motus mixti ex retardato & accelcrato; hæc porrò non e&longs;t Parabo­la, vt con&longs;tat, quia quadratum AE non e&longs;t ad ad quadratum AF, vt qua­dratum AB, vel EM ad quadratum FH, vel AC.

Scholium.

Ob&longs;eruabis in fine huius motus amplitudinem, &longs;eu &longs;inum rectum li­neæ &longs;cilicet GI, e&longs;&longs;e æqualem altitudini &longs;eu &longs;inui ver&longs;o, vel &longs;agittæ AG; cùm enim motus naturaliter acceleratus in eadem proportione cre&longs;cat, quod hic &longs;uppono, in qua retardatus decre&longs;cit; certè AG quæ e&longs;t linea accelerati e&longs;t æqualis GI, quæ e&longs;t linea retardati: non tamen dicendum e&longs;t lineam AI e&longs;&longs;e circulum, alioquin GH e&longs;&longs;et æqualis GI, &longs;ed GH e&longs;t, v. g. 89. cum GI &longs;it radix quadr.81. e&longs;t enim 9. licèt GM &longs;it æqualis GH. &longs;ed de his lineis infrà. Vtrùm verò &longs;it aliquis motus huiu&longs;modi, videbi­mus in &longs;equentibus Theorematis.

Theorema 21.

Quando corpus proiicitur per horizontalem in aëre libero, mouetur motu mixto; probatur, quia &longs;unt duo impetus in eo corpore, &longs;cilicet innatus deor&longs;um, & impre&longs;&longs;us per horizontalem, vt patet; igitur vterque aliquid præ&longs;tat ad illum motum per Ax. 1. igitur e&longs;t motus mixtus per def. 1.

Theorema 22.

Ille motus non e&longs;t mixtus ex vtroque æquabili. Demon&longs;tro; motus mixtus ex vtroque æquabili e&longs;t rectus per Th.1.& 4. &longs;ed hic motus proiecti per horizontalem non e&longs;t rectus per hyp.1.

Theorema 23.

Ille motus non e&longs;t mixtus ex naturali æquabili & alio accelerato; patet, quia nulla e&longs;t cau&longs;a, à qua violentus po&longs;&longs;it accelerari.

Theorema 24.

Non est mixtus ex naturali æquabili & violento retardato; Primò, quia cùm pro tata concurrant po&longs;t integrum quadrantem vix &longs;patium vnius palmi confeci&longs;&longs;et in perpendiculari deor&longs;um per Th.59.l.2.quod tamen e&longs;t contra experientiam.Secundò, quia ad aliquod tandem punctum per­ueniretur, in quo mobile haberet tantùm impetum innatun; igitur nul­lus e&longs;&longs;et ictus contra experientiam. Tertiò, quia naturalis impetus in­tenditur in plano inclinato; igitur in motu per inclinatam, e&longs;t enim motus deor&longs;um; igitur intenditur impetus naturalis, vt patet ex lib. 2. igitur non e&longs;t mixtus.

Theorema 25.

Motus ille non e&longs;t mixtus ex naturali retardator & violento æquabili, vel accelerato; quia numquam de&longs;truitur impetus innatus, vt &longs;æpiùs dictum e&longs;t &longs;uprà, tùm primo, tùm &longs;ecundo libro, nec in hoc e&longs;t vlla diffi­cultas.

Theorema 26.

Non est mixtus ex naturali accelerato & violento æquabili; demon&longs;tra­tur, primò, quia &longs;ub finem motus e&longs;&longs;et maior impetus; quippè nihil de­traheretur violento, &longs;ed multùm accederet naturali; igitur e&longs;&longs;et maior, igitur e&longs;&longs;et maior ictus contra hyp. 3. &longs;ecundò, quotie&longs;cunque &longs;unt duo impetus in codem mobili ad diuer&longs;as lineas determinati, aliquid illo­rum de&longs;truitur per Th.141.l.1.tertiò &longs;i e&longs;&longs;et vterque æquabilis, aliquid de&longs;trueretur per Theorema 6. igitur potiori iure, &longs;i impetus naturalis cre&longs;cat.

Diceret fortè aliquis impetum de&longs;trui ab aëre, &longs;ed iam &longs;uprà re&longs;pon­&longs;um e&longs;t modicum inde imminui; nec enim vnquam aër in corpore graui de&longs;truit tantùm impetus, quantùm producitur naturalis &longs;i &longs;it acceleratus; alioquin motus dcor&longs;um non cre&longs;ceret contra experientiam, & &longs;uprà in toto ferè 2.lib. demon&longs;trauimus.

Theorema 27.

Hinc linea huius motus non e&longs;t Parabola; quia vt &longs;it Parabola, debet ille motus con&longs;tare vel ex naturali æquabili, & violento retardato per Th. 19. vel ex naturali accelerato & violento æquabili per Th. 18. &longs;ed hic motus neuter e&longs;t, non primum per Th. 25. non &longs;ecundum per Theo­rema 26.

Theorema 28.

Hinc reiicies Galileum, qui in dialogis hæc &longs;emper &longs;uppo&longs;uit, &longs;ed nun­quam probauit, nec probare vnquam potuit; hoc etiam &longs;upponunt multi Galilei &longs;ectatores, qui cen&longs;ent impetum nunquam de&longs;trui ni&longs;i à re&longs;i&longs;tentia medij; &longs;ed quæro ab illis quodnam medium de&longs;truat partem impetus in motu mixto; nec enim linea motus mixti adæquat duas alias ex quibus qua&longs;i re&longs;ultat; certè hoc non potc&longs;t explicari cum infinitis fetè aliis, ni&longs;i dicatur impetum de&longs;trui ab alio impetu, eo modo quo &longs;æpè diximus, hoc e&longs;t ne &longs;it fru&longs;trà; igitur impetus violentus de&longs;truitur ab in­nato, non tamen innatus à violento, vt &longs;æpiùs inculcauimus.

Theorema 29.

Non e&longs;t mixtus ex naturali accelerato eo modo quo acceleratur deor&longs;um per lineam perpendicularem & ex violento retardato: Probatur, &longs;i ita e&longs;t, tantùm additur naturali, quantum detrahitur violento, imò plùs; igitur &longs;emper e&longs;t in eo mobili æqualis vel maior impetus; igitur æqualis e&longs;t &longs;emper, vel maior ictus contra hyp. 3. adde quod non minùs impeditur ab im­petu violento naturalis motus, quàm ab inclinato plano; &longs;ed in plano inclinato non acceleratur motus cum eadem acce&longs;&longs;ione, qua &longs;cilicet in­tenditur in perpendiculari deorsùm; nec enim tam citò de&longs;cendit mobi­le, quod certum e&longs;t, & in lib.de planis inclinatis demon&longs;trabo, cum tan­tùm hîc &longs;upponam ad in&longs;tar phy&longs;icæ hypothe&longs;eos; adde quod idem mo­bile proiectum per horizontalem in data di&longs;tantia minùs ferit, quàm pro­iectum per inclinatam deor&longs;um.

Theorema 30.

Itaque motus prædictus mixtus est ex violento retardato & naturali acce­lerato, non eo quidem modo quo acceleratur in perpendiculari, &longs;ed eo quo acce­leratur in plano inclinato, quod hic &longs;ingulis in&longs;tantibus mutatur; probatur pri­mo, quia inductione facta non conftat ex omnibus aliis; &longs;unt enim tantùm 9 combinationes, quia &longs;unt tres differentiæ, &longs;cilicet æquabilibus, retarda­tio, acceleratio; igitur &longs;i 3.ducantur in 3. &longs;unt 9. &longs;unt autem prima ex na­turali, quem deinceps voco primum, æquabili & violento (quem voca­bo &longs;ecundum) æquabili, &longs;ecunda ex prima æquabili & &longs;ecundo accelera­to, tertia ex primo æquabili & &longs;ecundo retardato, quarta ex primo acce­lerato & &longs;ecundo æquabili, quinta ex primo accelerato & &longs;ecundo acce­lerato, &longs;exta ex primo accelerato & &longs;ecundo retardato, &longs;eptima ex primo retardato & &longs;ecundo æquabili, octaua ex primo retardato & &longs;ecundo ac­celerato, nona ex primo rardato, & &longs;ecundo retardato: non e&longs;t prima per Th.22. non &longs;ecunda per Th. 21. non tertia per Th. 24. non quarta, per Th.26. non quinta per T.2h.23. non &longs;exta per Th.29. eo modo quo diximus, non &longs;eptia per Th. 25. non octaua per Th. 25. non denique nona per Th.25. igitur debet e&longs;&longs;e alius motus, &longs;ed alius excogitari non pote&longs;t præter ill quem adduxi. Probatur &longs;ecundò, quia non minùs impeditur ab impetu violento impetus naturalis acqui&longs;itus quàm à pla­no inclinato vt iam dictum e&longs;t; igitur acceleratur quidem &longs;ed minùs; nec enim vterque e&longs;t æquabilis, nam linea e&longs;&longs;et recta per Th.4. & naturalis cre&longs;cit quia de&longs;cendit deor&longs;um; præterea per Th.24. non pote&longs;t impetus naturalis e&longs;&longs;e æquabilis, igitur non pote&longs;t violentus e&longs;&longs;e vel æquabilis, vel acceleratus, igitur retardatus.

Theorema 31.

Motus naturalis acceleratus ex quo hic motus con&longs;tat acceleratur in alia proportione quàm fit ea, in qua acceleratur, dum per idem planum inclina­tum de&longs;cendit; probatur, quia &longs;ingulis in&longs;tantibus mutatur inclinatio pla­ni &longs;eu lineæ; igitur &longs;ingulis in&longs;tantibus mutatur proportio accelera­tionis.

Theorema 32.

Hinc perpetuò cre&longs;cit praportio accelerationis, quia &longs;emper cre&longs;cit inclina­tio plani, vt patec, cùm enîm &longs;it linea curua per hyp. 1. quo magis incur­uatur, accedit propiùs ad perpendicularem, igitur motus magis accele­ratur.

Theorema 33.

Hinc ratio hypothe&longs;eos primæ, cùm enim con&longs;tet hic motus ex accelera­to & retardato, eius linea e&longs;t curua per Th.20. non tamen e&longs;t Parabola, vt con&longs;tat ex eodem Th.20. Vnde reiicies Galileum, qui vult lineam mo­tus proiecti per horizontalem in aëre libero e&longs;&longs;e Parabolam.

Theorema 34.

In hoc motu retardatur in maiori proportione violentus quàm acceleretur natur alis; probatur, non in minore, quia plùs impetus adderetur quàm de­traheretur; igitur maior e&longs;&longs;et in fine motus quàm initio, igitur maior ictus contra hyp.;. non in æquali, quia &longs;emper e&longs;&longs;et æqualis ictus con­tra hyp.3.& contra Th.29.

Theorema 35.

Hinc plùs detrahitur impetus quàm addatur, quia &longs;cilicet detrahitur pro rata, vt dicemus infrà; at verò cùm acceleretur tantùm naturalis iuxta rationem motus, & motus &longs;it iuxta rationem plani, minùs accele­ratur quàm &longs;i caderet mobile perpendiculariter deor&longs;um.

Theorema 36.

Hinc ratio clara cur &longs;it minor ictus in &longs;ine huius motus; quia &longs;cilicet e&longs;t minùs impetus, quia plùs detractum e&longs;t quàm additum; nec e&longs;t quod tribuant hanc retardationem medio; quippe aër non plùs re&longs;i&longs;tit motui violento quàm naturali; &longs;ed id quod detrahitur ab aëre corpori graui, v. g. pilæ plumbeæ e&longs;t in&longs;en&longs;ibile, vt fatentur omnes; igitur idem dicen­dum e&longs;t de motu violento & mixto, hinc hoc ip&longs;um etiam fieret in vacuo.

Theorema 37.

Impetus naturalis concurrit ad hunc motum; probatur, quia alioquin e&longs;&longs;et rectus contra hyp. 3. prætereà pote&longs;t concurrere; nec enim &longs;unt li­neæ determinationum oppo&longs;itæ; igitur concurrit per Th.137.l.1.

Theorema 38.

Si impetus naturalis non concurreret ad hunc motum, proiectum moueretur per lineam horizontalem rectam, vt con&longs;tat, motu æquabili; po&longs;ito quod non retardaretur in horizontali, eodem modo moueretur quo in verticali &longs;ur&longs;um, quæ omnia con&longs;tant ex dictis &longs;uprà.

Theorema 39.

Patest vtrimquc de&longs;cribi linea curua huius motus; &longs;it enim mobile pro­jectum ex E per horizontalem EI eam &longs;cilicer velocitate, quam acqui&longs;iui&longs;­&longs;et motu naturaliter accelerato de&longs;cendendo ex A in E; &longs;itque AB &longs;pa­tium acqui&longs;itum primo in&longs;tanti de&longs;cen&longs;us; BC duplum, CD triplum, &c. iuxta progre&longs;&longs;ionem arithmeticam, &longs;it EI æqualis EA, diuidatur que eo­dem modo in 4. &longs;patia vt diui&longs;a e&longs;t EA; a&longs;&longs;umpta EO æqualis AB, ducan­tur FN. GM. HL. IK. parallelæ EV; a&longs;&longs;umatur OP æqualis OE, & PQquæ &longs;it ad OE, vt OE ad hypothenu&longs;im &longs;eu planum inclinm EN, a&longs;-&longs;iunatur QR æqualis OE, tum RS quæ &longs;it ad OE vt OQ ad planum incli­natum NM; denique a&longs;&longs;umatur ST æqualis OE, tum TV, quæ &longs;it ad OF, vt QS ad inclinatam ML; ducantur ON. QM. SL. VK. parallclæ EI, tùm per puncta E.N.M.L.X ducatur curua, hæc e&longs;t linea prædicti motus, demon&longs;tratur.

Impetus violentus percurrit EF eo tempore, quo naturalis percurtit EO; igitur linea motus mixti ex vtroque ducitur per punctum N, & licèt videatur e&longs;&longs;e recta EN, &longs;cilicet diagonalis rectanguli OF, e&longs;t tamen cur­ua, quia mobile non percurrit EF vno in&longs;tanti; igitur nec EO, igitur motu æqualiter accelerato percurrit EO; igitur EN non e&longs;t recta per Th.20. Præterea.Secundo tempore impetus innatus remanet; igitur per­curratur OP cui additut PQ, quia impetus naturalis minùs cre&longs;cit, vt di­ctum e&longs;t in Th.34. quippe cre&longs;cit iuxta rationem plani inclinati EN.ad EO permutando, quæ &longs;it v.g. &longs;ubquadrupla; igitur PQ e&longs;t &longs;ubquadrupla EO; & cùm de&longs;trui &longs;upponatur vnus gradus violenti, v.g. &longs;uper&longs;unt tan­tùm 3. quibus percurritur FG; igitur linea huius motus duci debet per punctum M, idem dico de punctis L & K, igitur hæc e&longs;t linea motus mixti, quàm &longs;cilicet corpus graue proiectum per horizontalem &longs;uo fluxu de&longs;cribit, & cuius alias proprietates demon&longs;trabimus.

Theorema 40.

Hinc impetus naturalis in motu mixto cre&longs;cit &longs;emper in maiori proportionev.g. Oq.e&longs;t maior EO, & QS maior OQ atque ita deinceps.

Theorema 41.

Impetus violentus hîc &longs;upponitur decre&longs;cere &longs;emper in eadem proportione; v.g. FG e&longs;t minor EF vno &longs;patio, GH minor EF vno &longs;patio; HI minor GH vno &longs;patio, quæ omnia con&longs;tant. Vtrùm verò id fiat, dicemus infià, & exempli gratia tantùm dictum e&longs;&longs;e volo.

Theorema 42.

Hinc quò maior e&longs;t impetus violentus in hoc motu, amplitudo huius linea e&longs;t maior v.g. VK, quæ &longs;emper maior e&longs;t altitudine VE, vt enim e&longs;&longs;et æ­qualis, impetus naturalis deberet cre&longs;cere in eadem proportione, in qua decre&longs;cit violentus, vt dictum e&longs;t &longs;uprà.

Theorema 43.

Determinari po&longs;&longs;et hæc amplitudo, &longs;i decre&longs;cat violentus in EI, vt decre­&longs;cit in verticali EA; nam EI & EA &longs;unt æquales, &longs;ed EI & VK &longs;unt æqua­les, AE verò e&longs;t linea, vel quam conficit mobile proiectum &longs;ur&longs;um cum eodem, vel æquali impetu alteri quo proiicitur per horizontalem; &longs;eu e&longs;t linea quam percurrit corpus graue deor&longs;um, dum acquirit æqualem impetum alteri impre&longs;&longs;o eidem mobili per horizontalem EI.

Theorema 44.

Hinc non pote&longs;t proijci in liero medio mobile graue per rectam horizonta­lem; quippe moueri non pote&longs;t ni&longs;i motu mixto ex naturali accelerato eo modo quo diximus, & violento retardato; igitur linea e&longs;t curua; dixi in medio libero, cùm in plano duro horizontali per lineam rectam pro­iici po&longs;&longs;it.

Theorema 45.

Hinc funis ten&longs;us, cuius &longs;cilicet vtraque extremitas immobiliter affixa e&longs;t, nunquam e&longs;t rectus, &longs;ed inflectitur; ratio e&longs;t, quia haud dubiè grauitat, igi­tur incuruatur; vtrùm verò faciat Parabolam hæc linea curua, vt vult Galileus, examinabimus in libro de lineis motus.

Scholium.

Ob&longs;eruabis funem ten&longs;um &longs;emper incuruari, ni&longs;i fortè ex maxima tra­ctione &longs;uam flexibilitatem amittat, cuius ope tantùm curuatur, imò ita tendi pote&longs;t, vt ten&longs;ioni cedens frangatur: Equidem po&longs;ito quod vel in­flecti po&longs;&longs;it, vel reduci, nece&longs;&longs;ariò inflectetur in medio, vt benè demon­&longs;trat Galileus in dialogis, no&longs;que infrà ad potentiam vectis reducemus, ne multiplicemus figuras.

Theorema 46.

Hinc ducitur optima ratio, cur proiectum per lineam horizontalem, v.g.pi­la è tormento explo&longs;a, vel &longs;agitta arcu emi&longs;&longs;a per plura &longs;ecunda minuta mo­ueatur in medio aëre antequam terram attingat; quod plu&longs;-quàm mille ex­perimentis comprobatum e&longs;t; plura leges apud Mer&longs;ennum, v. g. &longs;it tor­mentum horizonti parallelum extans &longs;upra horizontem tribus pedibus; certum e&longs;t &longs;patium illud trium pedum confici à globo perpendiculariter demi&longs;&longs;o tempore 30. tertiorum; cùm tamen explo&longs;us per lineam hori­zontalem terram tantùm attingat po&longs;t 4. &longs;ecunda, ide&longs;t 240. tertia; ita Mer&longs;ennus l.2. de motu Prop. vltima, imò l. 5. &longs;uæ ver&longs;ionis art.5. con­tra Galileum o&longs;tendit glandem emi&longs;&longs;am è tormento minori conficere 75. exapedas, tempore vnius &longs;ecundi minuti in linea, quæ parùm decli­nat ab horizontali; atqui tempore vnius &longs;ecundi minuti conficit 2.exa­pedas in perpendiculari deor&longs;um; igitur deberet glans infrà &longs;copum de­&longs;cendere notabiliter, id e&longs;t, toto 12. pedum interuallo, cùm tamen vix tantillùm aberret à &longs;copo 1.Idem Mer&longs;ennus habet in Bali&longs;tica Prop.25. globum è maiore tormento horizonti parallelo emi&longs;&longs;um in aëre tractu continuo vola&longs;&longs;e toto tempore 8. &longs;ecundorum, antequam planum hori­zontale attigi&longs;&longs;et, cùm tamen &longs;ex tantùm exapedis tormentum extaret &longs;upra horizontem; alter globus ex alio tormento explo&longs;us 6. tantum &longs;e­cunda in aëre con&longs;ump&longs;it; imò bombardarum globi aliquando tota 14. &longs;ecunda po&longs;uerunt; habet idem Mer&longs;ennus alia plura, quorum fides &longs;it penes authores à quibus accepit; nam vt dicam quod res e&longs;t vix accu­ratè minima illa tempora metiri po&longs;&longs;umus; quidquid &longs;it, ex illis &longs;altem euinco mobile projectum per horizontalem plùs temporis in&longs;umere in &longs;uo fluxu, quam &longs;i ex eadem altitudine perpendiculariter demittatur; vt vult Galileus; cuius ratio alia non e&longs;t ab ea, quàm &longs;uprà indicauimus, quòd &longs;cilicet motus naturalis minùs cre&longs;cat in motu mixto quàm in na-turali, vt &longs;uprà demon&longs;trauimus; imò &longs;i cre&longs;ceret vt vult Galileus, ictus; haud dubiè e&longs;&longs;et maior in fine motus quàm initio, quod omninò expe­rientiæ repugnat.

Nec e&longs;t quod aliquis dicat glandem emi&longs;&longs;am per horizontalem tan­tillùm a&longs;cendere; vnde plus temporis in a&longs;cen&longs;u &longs;imul & de&longs;cen&longs;u col­locatur, quàm in &longs;olo de&longs;cen&longs;u; nam primò vix hoc aliquis &longs;ibi per&longs;ua­&longs;erit, cùm experimento percipi non po&longs;&longs;it; Secundò licèt verum e&longs;&longs;et, non tamen e&longs;t tantus a&longs;cen&longs;us, quin adhuc plùs temporis ponat in a&longs;­cen&longs;u, atqué in de&longs;cen&longs;u, quàm in alti&longs;&longs;ima perpendiculari quadruplæ al­titudinis, vt con&longs;tat; &longs;it enim horizontalis AF, di&longs;tans à plano hori­zontali altitudine BA; &longs;it tormentum directum per lineam AF, & glo­bus percurrat lineam curuam AEF, idque &longs;patio 8.&longs;ecundorum minu­torum; &longs;itque DE 3. pedum; certè eo tempore quo conficit AE, &longs;i in perpendiculari conficiat ED, cum ED conficiat tempore 30‴; haud dubiè AE eodem tempore conficere deberet; &longs;ed concit AE tempore 4. &longs;ecundorum, vt con&longs;tat ex ip&longs;is multorum ob&longs;eruationibus; igitur to­tam AEF deberet percurrere tempore 1″, id e&longs;t eo tempore quo in per­pendiculari deor&longs;um percurruntur 12. pedes; denique &longs;i verum &longs;it glo­bum a&longs;cendere tantillùm dum emittitur è tormento horizonti paralle­lo; crediderim id e&longs;&longs;e tùm ex aliqua repercu&longs;&longs;ione aëris, tùm eo quod à flamma &longs;ur&longs;um a&longs;cendente &longs;ur&longs;um etiam aliquantulum inclinetur; quod verò &longs;pectat ad &longs;agittam, alia cau&longs;a non e&longs;t ni&longs;i modica aëris repercu&longs;&longs;io; e&longs;t enim leuior &longs;agittæ materia; &longs;ed de repercu&longs;&longs;ione fusè agemus infrà.

Theorema 47.

Motus projecti &longs;ur&longs;um per inclinatam e&longs;t mixtus; probatur, quia con&longs;tat ex naturali, & violenti; qui cùm non &longs;int in oppo&longs;itis lineis, ad commu­nem motum concurrunt, vt patet.

Theorema 48.

Non e&longs;t mixtus ex vtroque æquabili; quia linea e&longs;&longs;et recta per Th.1.&longs;ed linea huius motus e&longs;t curua per hyp. non pertinet etiam hic motus ad &longs;ecundam combinationem de qua Th. 30. nec ad quintam, nec ad octauam, nec ad nonam, de aliis videbimus infrà.

Theorema 49.

Non e&longs;t mixtus ex naturali accelerato, & violento æquabili; probatur, quia in fine motus e&longs;&longs;et maior impetus, igitur e&longs;&longs;et maior ictus contra ex­perientiam; imò longè maior quàm &longs;i mobile proiiceretur per horizon­talem, quia diutiùs durat ille motus; igitur plures gradus impetus na­turalis acquiruntur; igitur longè maior e&longs;t ictus; prætereà &longs;i impetus naturalis de&longs;truit impetum &longs;ur&longs;um in verticali, cur non in inclinata? nam e&longs;t eadem omninò ratio; quippe ideò de&longs;truitur in verticali, quia cor­pus graue &longs;ur&longs;um attollitur; cùm tamen &longs;ua &longs;ponte deor&longs;um ferri debe­ret; &longs;ed non minùs, cùm per inclinatam &longs;ur&longs;um proiicitur, remouetur à &longs;uo centro, & &longs;ur&longs;um rapitur; nec ob&longs;tat oppo&longs;itio lineæ vertcalis &longs;ur­&longs;um cum perpendiculari deor&longs;um; quia etiam per inclinatam deor&longs;um fertur in plano inclinato, quæ opponitur ex diametro alteri inclinatæ &longs;ur&longs;um.

Theorema 50.

Non e&longs;t mixtus in a&longs;cen&longs;u ex primo accelerato & &longs;ecundo retardato, acce­lerato inquam eo modo quo acceleratur in perpendiculari deor&longs;um; probatur primò, quia motus ille e&longs;&longs;et &longs;emper æqualis, quia tantùm adderetur im­petus quantùm detraheretur, igitur e&longs;&longs;et idem ictus in fine qui in princi­pio; Secundò, quia tempora motuum e&longs;&longs;ent breuiora quàm par &longs;it con­tra experientiam, vt patet ex Th.46.

Theorema 51.

Non e&longs;t mixtus in a&longs;cen&longs;u ex violento retardato, & naturali accelerato, eo modo quo diximus in Th. 30. probatur, quia cùm acceleretur iuxta ratio­nem plani inclinati deor&longs;um, vt dictum e&longs;t, &longs;upra horizontalem; nullum e&longs;t amphùs planum inclinatum deor&longs;um; igitur nulla acceleratio, imò impetus naturalis, vt iam &longs;uprà dictum e&longs;t cre&longs;cit tantùm vt motus deor­&longs;um acceleretur; &longs;ed nullus e&longs;t hîc motus deor&longs;um; modicùm figuræ rem ob oculos ponit; motus in plano AB e&longs;t ad motum in AC vt AC ad AB, & in AD, vt AD ad AB, & in AE, vt AE ad AB; igitur immi­nuitur in infinitum; &longs;ed acceleratur in inclinata deor&longs;um iuxta hanc ra­tionem, igitur nulla &longs;upere&longs;t ampliùs proportio, &longs;ecundum quam acce­lerari po&longs;&longs;et in inclinata &longs;ur&longs;um.

Theorema 52.

Hic motus e&longs;t mixtus ex naturali æquabili, & violento retardato in a&longs;cen­&longs;u; probatur, quia nulla alia combinatio præter hanc &longs;upere&longs;t, quam tertio loco &longs;uprà collocauimus in Th. 30. ratio à priori e&longs;t, quia natura­lis innatus non retardatur; quia nunquam de&longs;truitur, nec acceleratur; quia &longs;ur&longs;um tendit mobile; igitur &longs;upere&longs;t tantùm quod &longs;it æquabilis, violentus verò non acceleratur, vt patet, quia nulla e&longs;t cau&longs;a: non e&longs;t æquabilis, quia coniunctus e&longs;t cum cau&longs;a de&longs;tructiua; igitur e&longs;t re­tardatus.

Theorema 53.

Hic motus e&longs;t mixtus in arcu de&longs;cen&longs;us ex naturali accelerato eo modo, quo diximus &longs;uprà in Th. 30. & violento retardato; probatur per idem Th.e&longs;t enim par vtrique motui ratio; quippe hic perinde &longs;e habet, atque &longs;i mo­bile per horizontalem proiiceretur, nam præuius motus nequidquam facit.

Theorema 54.

Arcus vterque constat linea curua; probatur per Th.19. non e&longs;t tamen Parabola linea arcus de&longs;cen&longs;us per Th.20.& 27.

Theorema 55.

Pote&longs;t hac linea vtcumque de&longs;cribi, &longs;uppo&longs;ita retardatione violenti in pro-portione arithmetica &longs;implici; &longs;it enim verticalis, AG horizontalis AN, linea projectionis AD; &longs;itque primum &longs;egmentum AD, quod &longs;cilicet percurritur eo tempore quo in perpendiculari deor&longs;um percurritur DF, id e&longs;t, v.g. &longs;exta eius pars, ducatur AFG, &longs;itque FG 5. partium, quarum &longs;cilicet AD e&longs;t 6. a&longs;&longs;umatur GH æqualis DF, ducaturque FHI; &longs;itque HI 4. partium, a&longs;&longs;umatur IP æqualis GH, ducaturque HP; accipiatur PK 3. partium; iam motus naturalis acceleratur eo modo quo &longs;uprà di­ctum e&longs;t iuxta rationem inclinationis deor&longs;um; itaque a&longs;&longs;umatur KL paulo maior IP; &longs;imiliter ducatur PLM, &longs;itque LM duarum partium, & MN paulò maior KL, tum &longs;it LNO, &longs;itque NO 1. partis, & OB ma­ior MN, & ducatur curua per puncta A.F.H.P.L.N.B. & habebis intentum.

Porrò hæc linea non e&longs;t parabolica, vt con&longs;tat ex Geometria & plura puncta habebis &longs;i minora &longs;patiola a&longs;&longs;umas; &longs;uppono enim DF e&longs;&longs;e tan­tùm id &longs;patij quod primo in&longs;tanti in perpendiculari deor&longs;um à corpore graui percurritur.

Theorema 56.

Aliter hæc linea pote&longs;t de&longs;cribi &longs;uppo&longs;ita retardatione per numeros impa­res; vt habes in fig. 46.T.1. in qua AC e&longs;t verticalis, AB horizontalis, AD inclinata 9. partium, FG 7. HI 5. reliqua vt &longs;uprà dictum e&longs;t.

Si verò linea inclinata recedat longiùs ab horizontali, & accedat pro­piùs ad verticalem; vt habeantur puncta, transferantur cadem &longs;patia, & habebis puncta, per quæ de&longs;cribes prædictam lineam.

Denique &longs;i inclinata accedat propiùs ad horizontalem, transferantur &longs;imiliter &longs;patia vnius in alteram.

Ob&longs;eruabis autem crementa de&longs;cen&longs;us in GH. IB e&longs;&longs;e iuxta nume­ros impares 1.3.5.7.&c. quandoquidem a&longs;&longs;umitur &longs;patium quod confi­citur in tempore &longs;en&longs;ibili, habita tamen &longs;emper ratione accelerationis, quæ fit in plano inclinato, vnde cre&longs;cit &longs;emper proportio acceleratio­nis, vt &longs;uprà demon&longs;trauimus; quæ certè proportionum inæqualitas ef­ficit, ne po&longs;&longs;int accuratè de&longs;cribi prædictæ lineæ, &longs;ed tantùm rudi Miner­uâ, cum &longs;ingulis in&longs;tantibus mutetur proportio accelerationis.

Scholium.

Ob&longs;eruabis nondum e&longs;&longs;e à nobis determinatam proportionem illam, in qua de&longs;truitur impetus violentus in motu mixto, quæ tamen ex dictis &longs;uprà pote&longs;t colligi; quippe de&longs;truitur pro rata, ide&longs;t qua proportione linea motus mixti e&longs;t minor linea compo&longs;ita ex vtroque, &longs;it ergo.

Theorema 57.

Impetus violentus &longs;olus de&longs;truitur in arcu a&longs;cen&longs;us; probatur, quia natu­ralis non cre&longs;cit, vt patet; con&longs;tat enim arcus a&longs;cen&longs;us ex naturali æqua­bili, &longs;ed aliquis impetus decre&longs;cit, vt con&longs;tat ex dictis, igitur &longs;olus violentus.

Theorema 58.

Impetus naturalis non decre&longs;eit etiam in arcu de&longs;cen&longs;us; probatur quia cre&longs;cit, vt dictum e&longs;t &longs;uprà, igitur non decre&longs;cit.

Theorema 59.

De&longs;truitur impetus violentus pro rata. id e&longs;t, qua proportione e&longs;t frustràsv.g. &longs;it impetus per AD inclinatam &longs;ur&longs;um, & alius per AB perpendi­cularem deor&longs;um; haud dubiè motus erit per AC; igitur concurrunt ad motum AC motus AB & AD, vel potiùs impetus; igitur debet de­&longs;trui impetus in ea proportione, in qua AC e&longs;t minor AG, id e&longs;t com­po&longs;ita ex AD, DC, quod impetus AB non po&longs;&longs;it de&longs;trui; totum id quod de&longs;truetur detrahetur impetui AD; igitur a&longs;&longs;umatur DF &longs;cilicet differentia AC, & AG; impetus de&longs;tructus ita &longs;e habet ad impetum AD, vt DF ad AD, & ad re&longs;iduum impetum ex AD, vt DF ad FA, quæ omnia con&longs;tant ex Th.7. &longs;it ergo AC fig. 49. perpendicularis &longs;ur­&longs;um, AD inclinata, AB horizontalis; &longs;it impetus violentus re&longs;pondens AD, & naturalis DG, ducatur AGK, ex AD detrahatur DF, id e&longs;t differentia AG & compo&longs;itæ ex AD. DG, &longs;upere&longs;t AF, cui a&longs;&longs;umitur æqualis GK, ex qua detrahitur KH, id e&longs;t differentia GL, & compo&longs;itæ ex GK, KL, &longs;upere&longs;t GH, cui LO accipitur æqualis, cui detrahitur OM, id e&longs;t differentia LP & compo&longs;itæ ex LO, OP, &longs;upere&longs;t ML, cui æqualis accipitur PR, atque ita deinceps. Porrò demon&longs;tratur de&longs;trui impetum violentum iuxta hanc proportionem; quia de&longs;truitur, qua proportione e&longs;t fru&longs;trà, pro rata per Ax.2.& Th.7.&longs;ed totus impetus qui concurrit ad &longs;ecundam lineam AG, e&longs;t compo&longs;itus ex AD, GD; quia &longs;i naturalis &longs;olus e&longs;&longs;et, percurreret &longs;patium æquale DG; &longs;i verò &longs;olus e&longs;&longs;et violentus percurreret &longs;patium æquale AD; igitur vterque &longs;imul &longs;umptus e&longs;t vt compo&longs;ita, ex AG. DG. igitur &longs;i ea proportione e&longs;t fru&longs;trà, qua motus deficit, cùm AG &longs;it motus; certè motus e&longs;t ad impetum, vt AG ad compo­&longs;itam ex AD. DG; igitur deficit motus tota DF quæ e&longs;t differentia AG & compo&longs;itæ ex AD. DG; igitur impetus e&longs;t fru&longs;trà in ratione DF; igitur de­bet de&longs;trui in ratione DF; &longs;ed impetus DG &longs;eu naturalis nihil de&longs;trui­tur per Th.57. & 58. igitur ex violento AD de&longs;truitur DF; igitur &longs;u­pere&longs;t tantum AF vel æqualis GK; &longs;imiliter impetui GK & KL re­&longs;pondet motus GL, &longs;ed GL e&longs;t minor compo&longs;ita ex GK & KL &longs;eg­mento KH; igitur e&longs;t fru&longs;trà impetus in ratione KH; igitur de&longs;truitur in eadem ratione KH, non ex naturali KL; igitur ex violento GK; igitur &longs;upere&longs;t tantum GH, vel æqualis LO, in qua &longs;imiliter procedi­tur. & &longs;upere&longs;t LM vel æqualis PR, atque ita deinceps.

Corollarium 1.

Hinc de&longs;truitur impetus initio motus in maiori quantitate, quia DF. v. g. e&longs;t maxima omnium differentiarum.

Corollarium 2.

Hinc &longs;ub finem differentia lineæ motus v. g. TB &longs;emper e&longs;t maius latus trianguli TXB; idem dico de aliis; igitur differentia lineæ motus & compo&longs;itæ ex duplici impetu e&longs;t &longs;emper minor & minor in in­finitum.

Corollarium 3.

Po&longs;&longs;unt determinari à Geometria omnes anguli triangulorum ADG. GKL. OLP. nam ADG e&longs;t æqualis CAD, at verò GKL æqualis KGD, & hic duobus &longs;imul ADG & DAG, igitur determinari facilè poterunt ex doctrina triangulorum.

Corollarium. 4.

Hinc etiam &longs;ciri poterit in quo puncto linea motus v.g. LP cum per­pendiculari OP faciat angulum rectum, quod &longs;atis e&longs;t indica&longs;&longs;e, nam hic Geometram non ago.

Corollarium 5.

Hinc quoque &longs;ciri pote&longs;t maxima altitudo huius projectionis, quæ &longs;cilicet in eo puncto e&longs;t, in quo linea motus cum perpendiculari deor­&longs;um facit angulum rectum, v.g. in puncto P, &longs;i angulus LPO e&longs;t rectus.

Corollarium 6.

Hinc pote&longs;t etiam &longs;ciri altitudo operâ triangulorum productorum AG 2. GK 3. OLP. quod quiuis Geometra facilè intelliget; hîc quo­que obiter ob&longs;erua vnum, quod &longs;æpè aliàs indicauimus, quanti videlicet momenti &longs;it Geometria in rebus phy&longs;icis.

Corollarium 7.

Hinc etiam colligo arcum a&longs;cen&longs;us maiorem e&longs;&longs;e arcu de&longs;cen&longs;us &longs;u­pra idem planum horizontale AB; quia in arcu de&longs;cen&longs;us acceleratur pro ratione diucr&longs;æ inclinationis impetus naturalis; igitur lineam mo­tus addunt propiùs ad perpendicularem, vt vides in TB; igitur minùs acquirit in horizontali; igitur minor amplitudo horizontalis &longs;ube&longs;t ar­cui de&longs;cen&longs;us projectorum quàm arcui a&longs;cen&longs;us; dixi &longs;uprà idem pla­num, quia arcus de&longs;cen&longs;us infra planum AB propagatur ferè in infi­nitum.

Corollarium 8.

Hinc reiicio Galileum qui nulla pror&longs;us fultus ratione phy&longs;ica vult vtrumque e&longs;&longs;e æqualem, quod tamen omnibus experimentis repugnat, & ip&longs;i etiam pueri, qui di&longs;co ludunt ob&longs;eruare po&longs;&longs;unt arcum de&longs;cen&longs;us &longs;ui di&longs;ci e&longs;&longs;e longè minorem, nec e&longs;t quod ad &longs;uam Parabolam confugiat, quæ duo fal&longs;a &longs;upponit principia, &longs;cilicet æquabilitatem motus violen­ti, & accelerationem naturalis co &longs;cilicet modo quo fieret in perpendi­culari; at vtrumque fal&longs;um e&longs;&longs;e &longs;uprà demon&longs;trauimus, adde quod vt iam dixi in &longs;agitta emi&longs;&longs;a, projecto di&longs;co, &c. omnes ob&longs;eruare po&longs;&longs;unt ar­cum a&longs;cen&longs;us maiorem e&longs;&longs;e arcu de&longs;cen&longs;us, quod etiam &longs;upponunt om­nes, qui de re tormentaria &longs;crip&longs;erunt; præ&longs;ertim Vfanus tract. 3. c. 13.

Corollarium 9.

Hinc etiam colliges contra Vfanum globum è tormento emi&longs;&longs;um per inclinatam &longs;ur&longs;um non ferri primò per lineam rectam, quia mouetur motu mixto, qui rectus e&longs;&longs;e non pote&longs;t in hoc ca&longs;u per Th.54.

Corollarium 10.

Motus mixtus arcus de&longs;cen&longs;us v&longs;que ad centrum terræ durare pof&longs;et &longs;i producerentur tot partes impetus quot &longs;unt in&longs;tantia illius motus; quia cùm &longs;emper de&longs;truatur minor impetus, & minor in infinitum, po&longs;t ali­quod &longs;patium de&longs;cen&longs;us tam parùm de&longs;truitur v&longs;que ad centrum terræ vt non adæquet totus ille impetus primam partem primo in&longs;tanti de&longs;tru­ctam, at tunc linea motus à perpendiculari deor&longs;um di&longs;tingui non pote&longs;t.

Corollarium 11.

Sed ne Geometriam omninò de&longs;picere videar, in circulo demon&longs;tro proportiones omnes in quibus decre&longs;cit motus violentus per quamlibet lineam inclinatam &longs;ur&longs;um, vel deor&longs;um; &longs;it ergo circulus ADGQ cen­tro B; &longs;it motus violentus &longs;ur&longs;um BD coniunctus cum naturali BR, &longs;int­que ex gr. BR. RQ æquales; hand dubiè linea motus erit BC, quia na­turalis BR pugnat pro rata per Th.134.l.1. eritque BC &longs;ubdupla BD; igitur centro R. &longs;emidiametro RC de&longs;cribatur circulus CLPS, erit æqualis priori, ducanturque ex centro B infinitæ lineæ BE. BF. BK. BN, & vt res fit clarior, &longs;int omnes anguli DBE. EBF. FBG, &c. æquales &longs;cilicer grad. 30. & ex punctis E.F.G.K.N.q. ducantur lineæ ad circunferentiam circuli CLPS. parallelæ DP.Dico omnes e&longs;&longs;e æqua­les DC; nam primò FH. GL. KM. QP &longs;unt æquales, vt patet: deinde CE & QO &longs;unt æquales; igitur EV. OX, quod etiam certum e&longs;t; igi­tur &longs;i &longs;upponatur idem motus violentus æqualis BD per omnes inclina­tas BE. BF, &c. coniunctus naturali æquali BR; primum &longs;patium erit BC, &longs;ecundum BV, tertium BH, quartum BL, quintum BM, &longs;extum BO2 &longs;eptimum BP. quod certè mirabile e&longs;t; nam ex BE. EV. fit BV per Th.5. funiliter ex BF. FH. fit BH, ex BG. GL. fit BL; denique ex Bque QP fit BP; iam verò proportiones i&longs;tarum linearum ex Trigo­nometria facilè intelligi po&longs;&longs;unt.

Theorema 60.

Iactus per horizontalem, & per verticalem nihil acquirit per &longs;e eodem plane horizontali, vnde incipit iactus; probatur, quia verticatis iactus per eamdem lineam redit; horizontalis verò &longs;tatim de&longs;cendit; quia motus mixtus e&longs;t per Th.44. dixi per &longs;e, nam fortè per accidens fieri pote&longs;t, vt iactus horizontalis habeat arcum a&longs;cen&longs;us, & de&longs;cen&longs;us.

Theorema 61.

Hinc quò iactus propiùs accedit ad horizontalem &longs;eu verticalem, minùs acquirit in eodem plano horizontali, &longs;cilicet in eo à cuius extremitate inci­pit iactus; probatur, quia cùm iactus verticalis nihil pror&longs;us acqui­rat in hotizontali plano per Theorema 60. certè quò propiùs ad illum iactus inclinatus accedet, minùs acquiret; idem dico de iactu hori­zontali.

Theorema 62.

Hinc quò iactus longiùs recedit ab vtroque &longs;cilicet à verticali, & hori­zontali, plùs acquiret in eodem plano horizontali; &longs;i enim quò plùs ac­cedit ad vtrumque, minùs acquitit, igitur plùs acquirit, quò plùs re­cedit.

Theorema 63.

Hinc iactus medius &longs;eu per inclinatam qua cum verticali, vel horizontali facit angulum 45.&longs;eu &longs;omirectum, e&longs;t omnium maximus, id e&longs;t plùs acqui­rit in eodem plano horizontali, quàm reliqui omnes; experientia certi&longs;&longs;ima e&longs;t, ratio e&longs;t quia ab horizontali & verticali maximè omnium di&longs;tat; igitur maximus e&longs;t per Theorema 62. nec e&longs;t vlla alia ratio geome­trica.

Theorema 64.

Iactus qui æqualiter ab horizontali & verticali di&longs;tant, &longs;unt æquales; probatur, quia qua proportione ad horizontalem &longs;eu verticalem acce­dit iactus, in ea proportione minor e&longs;t; igitur qui æqualiter acce­dunt in proportione æquali, minores &longs;unt; igitur æquales, quod mo­dica figura ob oculos ponet; &longs;it enim quadrans ABF, iactus verti­calis AB, horizontalis AF, medius AD, hic maximus omnium erit; at verò AC, & AE, qui ab AD æqualiter di&longs;tant, erunt æ­quales.

Scholium.

Ob&longs;eruabis primò, omitti à me multa quæ &longs;uis Parabolis aliqui af­fingunt, quæ nec experimentis, nec vllis rationibus con&longs;en­tiunt.

Secundò rationem i&longs;torum omnium Theorematum; quia quo iactus d verticalem propiùs accedit, maior quantitas impetus de&longs;truitur v.g. in AD plùs quàm in GK; igiturcitò deficiunt vires huic iactui; adde quod acquirit in verticali, quod alius acquirit in horizontali; at verò qui propiùs accedit ad horizontalem citò de&longs;cendit infra planum horizontale, tùm quia propior e&longs;t, tum quia citò naturalis impetus acceleratur; igitur plùs acquirit in perpendiculari deor&longs;um, quàm in horizontali; quæ omnia ex certis principiis, non fictitiis dedu­cuntur.

Tertiò, ob&longs;eruabis talem e&longs;&longs;e hypothe&longs;im illam Paraboli&longs;tarum, de qua &longs;uprà; &longs;it enim iactus verticalis EA; medius EB; certè ex corum etiam principio eo tempore, quo motu æquabili percurreret mobile &longs;pa­tium EA, motu naturalirer retardato percurreret &longs;patium EG &longs;ubdu­plum; atqui percurrit EG eo tempore, quo idem percurreret GE motu naturaliter accelerato; &longs;ed percurret inclinatam EC eo tempore quo percurret EA, &longs;cilicet motu æquabili; &longs;unt enim æquales: Volunt autem FE diuidi in 16. partes, & ED in 8. ducique parallelas HQ IP, &c. & ac­cipi VR (1/16) FE, ita vt RQ &longs;it ad RH vt 9.ad 7. & PS (4/16) & NT (9/16), vel O T (1/16) PS (4/16) PR (9/16); igitur eo tempore, quo mobile e&longs;&longs;et in IX, erit in M; igitur motus naturalis acqui&longs;iuit XM, id e&longs;t 1/4 AE; igitur eo tempore quo e&longs;&longs;et in B erit in D; igitur motus naturalis acqui&longs;iuit BD quadruplum X M; nam &longs;i vno tempore motu æquabili conficit EX, duobus conficit E D & &longs;i motu naturaliter accelerato conficit vno tempore XM, duobus conficit BD iuxta proportionem Galilei, in qua &longs;patia &longs;unt vt temporum quadrata; & quo tempore motu æquabili conficeret EA, vel EB naturali conficeret GE vel CZ æqualem GE; ducatur igitur linea per puncta E. RS, OM, hæc e&longs;t &longs;emiparabola cui &longs;i addas MZD, habebis totam ampli­tudinem Parabolæ ED, hoc e&longs;t totum &longs;patium, quod acquirit in plano horizontali ED iactus medius EB.

Si verò &longs;it inclinata EY; vt habeatur iuxta hanc hypothe&longs;im amplitu­do horizontalis; fiat &longs;emicirculus centro G, &longs;emidiametro GE; &longs;it per­pendicularis YK, erit &longs;ubdupla amplitudo; &longs;icut perpendicularis XL de­finit &longs;ubduplam amplitudinem LE iactus EB; &longs;imiliter YK definit &longs;ubdu­plam amplitudinem iactus E4.3.nam arcus YX e&longs;t æqualis arcui X 4. igitur anguli YEC, CE. 3. &longs;unt æquales; hinc iactus &longs;unt æquales &longs;upra, & infra grad.45. vt autem habeatur altitudo Parabolæ &longs;ubdupla XL e&longs;t al­titudo Parabolæ iactus EC, &longs;ubdupla YX e&longs;t altitudo iactus EY, &longs;ubdu­pla 4.K e&longs;t altitudo iactus E 3.

Ex his facilè iuxta hypethe&longs;im tabulæ omnium iactuum, cuiu&longs;libet eleuationis con&longs;trui po&longs;&longs;unt; de quibus habes plura apud Galileum in dialogis, & plurima apud Mer&longs;ennum in Bali&longs;tica; quare ab illis ab&longs;ti­neo: præ&longs;ertim cum &longs;it fal&longs;a illa hypothe&longs;is, eiu&longs;que &longs;ectatores vltrò fa­teantur tabulas illas non parum à vero abe&longs;&longs;e, de quo vide Mer&longs;ennum prop. 30. Bali&longs;t.

Quartò, po&longs;&longs;unt iuxta no&longs;tram hypothe&longs;im tabulæ nouæ con&longs;trui, quod & ego præ&longs;tarem, ni&longs;i pror&longs;us inutiles e&longs;&longs;ent; quare prudenter omi&longs;&longs;as e&longs;&longs;e prudentes omnes cen&longs;ebunt, cum hîc calculatorem non agam, &longs;ed phi­lo&longs;ophu; id certè tolerari potuit in analyticis, quæ &longs;ine calculationibus intelligi non po&longs;&longs;unt; &longs;ed minimè ferendum in Phy&longs;ica, quæ &longs;ucculen-tior e&longs;t, quàm vt numeris tantùm, &longs;icci&longs;que calculis nutriatur; adde quod Praxis Theoricæ in his omninò præferenda e&longs;t; quamquam huic etiam parti dce&longs;&longs;e nolumus, &longs;ed in &longs;ingularem libellum omnes i&longs;tas tabulas & alias huiu&longs;modi remittimus; cum hic tantùm rerum phy&longs;icarum cau&longs;as explicemus.

Theorema 65.

Si accipiatur planum horizontale intra illud vnde incipit iactus haud du­biè iactus omnium maximus erit horizontalis in vtraque hypothe&longs;i. Primo in hypothe&longs;i Galilci, in qua Parabola GD figurâ &longs;uperiore habet maximum omnium amplitudinem; licèt iactus per GX; ex quo &longs;equitur, non ha­beat impetum maiorem, quâm iactus per EY, vel EX; in no&longs;tra verò, ia­ctus per BG primo tempore plùs acquirit in horizontali BG, quàm ia­ctus per BF; igitur plùs etiam &longs;ecundo tempore; nam BF acquirit tantùm primo tempore BH, at verò BG acquirit RL; adde quod minùs perit ex iactu BG; quippe a&longs;&longs;umatur BL in B 2. & GL in 2. 3. detrahitur tantùm G. 3.ex BG; at verò a&longs;&longs;umatur BH in B 4. & FH in 4.5. detrahitur F 5.ex BF; igitur plùs ex BF quàm ex BG; quæ omnia ex &longs;uperioribus regulis iu&longs;ta no&longs;tram hypothe&longs;un præ&longs;criptis con&longs;equuntur.

Theorema 66.

Immò probabile e&longs;t æquales fore iactus per inclinatas &longs;ur&longs;um, & deor&longs;um æqualiter ab horizontali, vnde incipit iactus, distantes; æquales inquam in ali­quo plano horizontali, inferiore; &longs;i enim iactus fiat per BD eadem figura & BP nihil acquiritur in horizontali, vt con&longs;tat; &longs;i verò iactus &longs;it per BG maximum &longs;patium acquirunt in horizontali plano inferiore; igitur qua proportione propiùs accedent lineæ &longs;eu iactus ad BD, PP minùs acqui­rent; qua verò proportione propiùs accedent ad RG plùs acquirent; igi­tur æqualiter plùs, & minùs hinc inde, &longs;i æqualiter hinc inde di&longs;tent; im­mò hoc ip&longs;um præ&longs;entibus oculis intueri licèt; &longs;i enim iactus BF compa­retur cum iactu BK; certè BK acquirit RK, BF acquirit BH æqualem B K; &longs;ed BF & BK æqualiter di&longs;tant ab horizontali BG; nam arcus GF, & GK &longs;unt æquales, vt con&longs;tat: idem dico de iactu BE, & BX, qui acquirunt æquale &longs;patium in horizontali æquale &longs;cilicet BZ.

Scholium.

Ob&longs;eruabis hoc omninò licèt mirum cuiquam fortè videatur, certè in&longs;titutum e&longs;&longs;e à natura; &longs;i enim comparentur omnes iactus &longs;uprà hori­zontalem BG, haud dubiè cum duo extremi &longs;cilicet BD, & BG nihil pror&longs;us acquirant, vt con&longs;tat ex dictis, iactus medius &longs;cilicet ad gradum 45.erit omnium maximus, quia æqualiter ab vtraque extremitate di&longs;tat, vt demon&longs;trauimus &longs;uprà; &longs;i verò comparentur omnes iactus, qui po&longs;­&longs;unt fieri à centro B per totum &longs;emicirculum DGque certè cum duo ex­tremi BD, BQ nihil pror&longs;us acquirant, vt con&longs;tat, iactus medius, &longs;cilicet ad gradum 90.qui e&longs;t BG erit omnium maximus, quia æqualiter ab vtra-que di&longs;tat extremitate; &longs;imiliter quemadmodum iactus æqualiter à me­dio iactu 45. di&longs;tantes æqualem amplitudinem acquirunt in horizontali BG, ita qui æqualiter di&longs;tant à medio iactu 90.vel horizontali BG æqua­lem amplitudinem acquirunt in aliquo plano hoaizontali, &longs;cilicet in eo vnde vterque iactus de&longs;init in perpendicularem deor&longs;um.

Ob&longs;eruabis &longs;ecundo, omnes perpendiculares deor&longs;um perinde accipi, atque &longs;i e&longs;&longs;ent parallelæ propter in&longs;en&longs;ibilem differentium; quod certè ab omnibus admittitur; quomodo verò per diuer&longs;a plana deor&longs;um cor­pus rendere po&longs;&longs;it, v&longs;que ad cuntrum terræ, Libro &longs;equenti explica­bimus.

Theorema 67.

In iactu per inclinatam deor&longs;um dato tempore minùs detrahitur de impetu violento, quàm in iactu per inclinatam &longs;ur&longs;um &longs;it enim circulus centro A &longs;emidiametro AG; &longs;itque AG horizontalis, & AO perpendiculatis deor­&longs;um; &longs;it iactus per inclinatam &longs;ur&longs;um AD, &longs;itque impetus violentus vt A D, & naturalis deor&longs;um vt DE; linea motus erit DAE; igitur a&longs;&longs;umatur A E in AC, & DE in CB, ex impetu AD detrahitur DB, vt con&longs;tat ex dictis quia totius ille fru&longs;trà e&longs;t; &longs;it autem inclinata deor&longs;um cum impetu vio­lento æquali AI æqualis AD, &longs;itque naturalis deor&longs;um acceleratus prrata plani inclinati vt IL, linea motus erit AL; a&longs;&longs;umatur AK, vt AL, & KH vt IL, detrahitur tantùm IH, &longs;ed IH e&longs;t minor DB; igitur tempore &longs;equenti æquali impetus violentus inclinatæ &longs;ur&longs;um erit vt EF æqualis AB inclinatæ deor&longs;um, vt LM, quæ maior e&longs;t EF, quia e&longs;t æqua­lis AH.

Ratio à priori e&longs;t, quia cum inclinata deor&longs;um faciat acutum angu­lum cum perpendiculari deor&longs;um, cum quo obtu&longs;um facit inclinata &longs;ur­&longs;um, maior e&longs;t in illa linea motus; e&longs;t enim maior diagonalis, in hac ve­rò minor, igitur in illa minùs impetus e&longs;t fru&longs;trà, in i&longs;ta verò plùs, igitur minùs impetus in illa de&longs;truitur, plùs in i&longs;ta; quæ omnia con&longs;tant ex Th. 110. & 139. & 140. l.1. habes etiam in qua proportione decre&longs;cat impetus.

Theorema 68.

Hinc in iactu qui fit per inclinatam deor&longs;um minùs detrahitur, & in eo qui fit per inclinationem &longs;ur&longs;um plùs detrahitur, in perpendiculari deor­&longs;um nihil detrahitur, in perpendiculari &longs;ur&longs;um totus detrahitur qui po­te&longs;t extrahi, id e&longs;t ex collectione vtriu&longs;que naturalis, & violenti dupli naturalis in prima linea motus; hæc omnia &longs;equuntur ex dictis.

Obiici pote&longs;t vnum &longs;atis difficile; quia &longs;i in perpendiculari deor&longs;um purà in AP nihil detrahitur impetus violenti, igitur cre&longs;cit &longs;emper vis ictus, quod videtur e&longs;&longs;e contra experientiam.

Re&longs;p, me aliquando fui&longs;&longs;e in ea &longs;ententiâ, vt reuerâ exi&longs;timarem de­cre&longs;cere impetum violentum in iactu perpendiculari deor&longs;um; cum etiam exi&longs;timarem decre&longs;cere vim ictus; &longs;ed re melius con&longs;iderata, cum nunquam id experiri potuerim; nam &longs;emper fentio vim ictus maiorem, cum deorfum mobile proiicitur, quàm cum &longs;ua &longs;ponte ex eadem altitu­dine de&longs;cendit; certè ni fallor cum ratio demon&longs;tratiua pro hac &longs;en­tentia faciat, non dubitaui ampliùs priorem &longs;ententiam immutare.

Porrò ratio, quæ pro hac &longs;ententia facit, remque ip&longs;am euincit, talis e&longs;t; certum e&longs;t impetum violentum de&longs;trui à naturali aliquando in ma­iori, aliquando in minori proportione, vt con&longs;tat ex dictis; illa autem, &longs;eu maior, &longs;eu minor proportio aliam regulam non habet præter illam quam toties inculcauimus, id e&longs;t impetum de&longs;trui pro rata, id e&longs;t qua propo rtione e&longs;t fru&longs;trà, id e&longs;t qua proportione e&longs;t minor motus co, qui e&longs;&longs;et ab vtroque impetu &longs;i ad eamdem lineam vterque determinatus e&longs;&longs;et atqui cum proiicitur mobile deor&longs;um, vterque impetus ad eamdem li­ne am e&longs;t determinatus; igitur nihil motus dec&longs;t per Th.138.l.1. igitur nihil impetus e&longs;t fru&longs;trà; igitur nihil impetus illius de&longs;truitur.

Quod dictum e&longs;&longs;e velim non con&longs;iderata medij re&longs;i&longs;tentiâ, quæ certè aliquid impetus de&longs;truit, quod tamen in&longs;en&longs;ibile e&longs;t in medio libero, pu­tà in aëre; &longs;i enim in&longs;en&longs;ibilis e&longs;t hæc re&longs;i&longs;tentia in motu naturali; dum mobile &longs;it eius &longs;oliditatis, quæ &longs;uperet facilè vim aëris; certè etiam in­&longs;en&longs;ibilis e&longs;t in motu proiectorum, præ&longs;ertim in mediocri &longs;patio, e&longs;t enim par vtrobique ratio.

Equidem fateor in longi&longs;&longs;imo &longs;patio po&longs;&longs;e tandem de&longs;trui totum im­petum violentum; nam &longs;i aliquid in dato &longs;patio de&longs;truitur; igitur in ma­iore piùs de&longs;truitur; atque ita deinceps, donec tandem totus de&longs;tructus &longs;it; at verò in iis altitudinibus, ex quibus corpus deor&longs;um proiicere po&longs;­&longs;umus, vix quidquam facit prædicta re&longs;i&longs;tentia.

Nec e&longs;t quod aliquis dicat ab hac re&longs;i&longs;tentia non de&longs;trui impetum naturalem in motu naturaliter accelerato, vt dictum e&longs;t in &longs;ecundo lib. Igitur nec de&longs;trui violentum; nam qua proportione cre&longs;cit medij re&longs;i­ftentia, cre&longs;cunt vires impetus, qui perpetuò augetur; nde cum remaneat &longs;emper eadem re&longs;i&longs;tentiæ proportio &longs;icut primo tempore mo­tus impedit hæc re&longs;i&longs;tentia, ne tantillùm impetus producatur; ita &longs;ecun­do tempore impedit ne tantillùm æquale producatur; igitur nihil pro­ducti impetus ab illa de&longs;truitur propter augmentum continuum: at ve­rò cum impetus violentus non intendatur; certè &longs;i tantillùm illus perit, primo vel &longs;ecundo in&longs;tanti motus, propter medij re&longs;i&longs;tentis, tantillùm æquale &longs;ingulis temporibus æqualibus de&longs;truitur; igitur cum infinitus non &longs;it po&longs;t longi&longs;&longs;imum &longs;patij tractum totus tandem de&longs;truetur vio­lentus &longs;olo &longs;uper&longs;tite naturali.

Hinc fortè &longs;agitta ex notabili altitudine minùs ferit; quia materia illa lignea, & plumea, ex qua con&longs;tat, multùm ab aëre re&longs;i&longs;tente accipit de­trimenti: adde quod licèt initio deor&longs;um rectà emittatur; attamen mini­mo aëris flatu declinat tantillùm obliqua; hæc verò obliquitas maximam ictus vim infringit, & conflictus impetuum qua&longs;i ip&longs;um ictum diftrahit, quod facilè probabis, &longs;i modico ferè tactu cadentem perpendiculariter fagiam à &longs;uo tramite deturbes.

Dices, etiam in glande è tormento explo&longs;a hoc ip&longs;um cernitur

Re&longs;p. e&longs;t minor vis ictus inflicti à glande deor&longs;um, quàm &longs;ur&longs;um vt aliqui putant; id autem ex duplici capite procedere; primum e&longs;t, cum fe­ratur glans ab igne per aliquod tempus, non e&longs;t dubium, quin vis ignis &longs;ur&longs;um maior &longs;it quàm deor&longs;um; cum &longs;ur&longs;um gemino qua&longs;i impetu fera­tur, deor&longs;um verò impetu tantùm explo&longs;ionis; &longs;ecundum e&longs;t, quia cum glans iam deor&longs;um &longs;ua &longs;ponte de&longs;cendat, haud dubiè ab igne minus eò impelli pote&longs;t, vt &longs;æpè diximus &longs;uprà; quidquid &longs;it, &longs;i proiiciatur dcor&longs;um globus plumbeus vel arcu, vel manu, ob&longs;eruabitur maiorem ab co ictum infligi, quàm &longs;i &longs;ua &longs;ponte de&longs;cenderet.

Theorema 69.

Si corpus moueatur deor&longs;um perpendicul ariter motu mixto, eo tempore que motu naturali acquireret illum impetum quem habet motu violento, acquirit triplum illius &longs;patium v.g. in figura &longs;uperiore &longs;it linea perpendiculatis deor&longs;um A E, in qua motu naturali dato tempore acquiratur AB, & &longs;e­cundo tempore æquali BC; &longs;itque impetus violentus vt AC: Dico quod æquali tempore prioribus acquireret AE triplum AC, quia motu ve­loci vt AC acquirit CE eo tempore, quo motu veloci vt AB acquirit A B, & veloci vt BC acquirit BC; nam eo tempore, quo acquirit AB acqui­rit CD, & eo tempore, quo acquirit BC acquirit DE; ergo eo tempore, quo acquirit AC acquirit CE; ergo &longs;i iungatur motus naturalis violeto, co tempore, quo motu naturali acquiretur tantùm AC, motu mixto ex naturali & tali violento acquiretur AE, id e&longs;t triplum: &longs;i verò moueatur duobus temporibus, ita vt primò acquirat AC, & altero triplum AC, &longs;itque coniunctus impetus violentus vt AC; certè duobus temporibus acquiretur motu mixto octuplum AC, &longs;ed hæc &longs;unt facilia.

Theorema 70.

Si corpus graue proiiciatur deor&longs;um per medium aëra, quire&longs;i&longs;tat, cum tandem de&longs;truatur impetus violentus, vbitotus de&longs;tructus e&longs;t, minor e&longs;t ictus quàm e&longs;&longs;et. &longs;i corpus graue &longs;olo impetu natur ali eò de&longs;cendi&longs;&longs;et; quod demon­&longs;tro, &longs;it enim &longs;patium AD, quod percurrit motu mixto eo tempore, quo motu naturali puro &longs;patium BC idem mobile percurreret, &longs;itque de&longs;tru­ctus in puncto D totus impetus violentus; certè remanet tantùm natu­ralis acqui&longs;itus co tempore, quo mobile percurrit BC; &longs;ed temporibus æ­qualibus acquiruntur æqualia velocitatis momenta; igitur æqualis im­petus; igitur in C tantùm ille impetus, qui e&longs;&longs;et in E vel in D; &longs;ed dum percurreret ED motu puro naturali, augetur impetus; igitur maior e&longs;&longs;et impetus in D &longs;ub finem motus naturalis per AD, quam motus mixti per camdem AD; igitur maior ictus &longs;ub finem naturalis; igitur minus &longs;ub fi­nem violenti.

Theorema 71.

Hinc paradoxon egregium; mobile proiectum in data di&longs;t antia minùs ferit quàm &longs;ua &longs;ponte demi&longs;&longs;um; quod nece&longs;&longs;ariò &longs;equitur ex dictis.

Ob&longs;eruabis &longs;crupulum adhuc fortè hærere, cur &longs;cilicet impetus violentus non de&longs;truatur à naturali, cuius &longs;cilicet iu&longs;tam impedit ptopa­gationem; &longs;ed profectò nullo modo impetus ille violentus impedit effe­ctum impetus naturalis innati vel addititij; quia vterque totum &longs;uum ef­fectum &longs;ortitur; quod autem &longs;pectat ad propagationem; certè ita propa­gatur, vt temporibus æqualibus æqualis impetus accedat.

Dices, debes quidem nouus impetus accedere, &longs;ed non tali modo.

Re&longs;p. non e&longs;&longs;e alium modum à natura in&longs;titutum, ni&longs;i vt temporibus æqualibus æqualia velocitatis momenta acquirantur.

Dices præterea, fru&longs;trà accedit nouus impetus naturalis, cum iam ad­&longs;it violentus, qui eius munere defungi pote&longs;t.

Re&longs;p. cau&longs;am nece&longs;&longs;ariam nece&longs;&longs;ariò agere; igitur corpus graue perpe­tuò in medio libero &longs;uum motum intendit.

Theorema 72.

Pote&longs;t vtcumque delineari linea motus mixti per inclinat am deor&longs;um &longs;it enim perpendicularis door&longs;um AB &longs;it iactus per inclinatam AF; &longs;itque impetus violentus vt AE naturalis vt EC, linea motus erit AC; a&longs;&longs;umatur AF æqualis AC, & DF æqualis EC, &longs;itque CH vt AD, & impetus natu­ralis auctus vt HK, linea motus crit CK; &longs;it CI æqualis DK, & IG æqua­lis HK, & KL æqualis CG; &longs;it que impetus naturalis &longs;ecundò auctus vt L M; linea motus erit KM; igitur connectantur puncta AC, KM per lineam curuam, hæc e&longs;t linea quæ&longs;ita, vt con&longs;tat ex dictis &longs;uprà.

Theorema 73.

Hinc pote&longs;t aliquo tempore tantùm impetus violenti de&longs;trui quantùm pro­ducitur naturalis; igitur &longs;i non con&longs;ideres re&longs;i&longs;tentiam medij, tunc æqua­ is e&longs;&longs;et ictus, & æquabilis motus.

Theorema 74.

Quando mobile peruenit in M, & acqui&longs;iuit in perpendiculari deor&longs;um to­tam altitudinem AR, non habet totum impetum naturalem, quem acquireret motu naturali per totam AR, &longs;ed tantùm illum, quem acquireret in compo&longs;ita ex &longs;egmentis NO, PB, QR; quia ad motum i&longs;tum deor&longs;um non tantùm concurrit impetus naturalis, &longs;ed etiam violentus vt con&longs;tat.

Theorema 75.

Hinc reiicies Galileum, & alios, qui volunt in linea motus AC ac­quiri tantumdem impetus naturalis quuntum in perpendiculari AB ac­quireretur.

Theorema 76.

In naui mobili &longs;iè &longs;ummo malo remittatur corpus graue, de&longs;cendit motmixto; probatur, quia duplex impetus concurrit ad illum motum, &longs;cilicet naturalis deor&longs;um, & horizontalis impre&longs;&longs;us à naui, vt con&longs;tat ex defini­tione 1.hyp.2. & Ax.1.

Theorema 77.

Ille motus e&longs;t mixtus ex naturali accelerato, & violento per horizontalem retardato; quod eodem modo probatur, quo &longs;uprà probatum e&longs;t in mobi­li proiecto per horizontalem Th.30. e&longs;t enim pror&longs;us eadem, cum à na­ui reuera imprimatur impetus iis omnibus, quæ motu nauis fe­runtur.

Theorema 78.

Hinc reiicio omnes alias combinationes recepta &longs;exta; immò &longs;extam ip&longs;am ex parte; nec enim naturalis acceleratur in hoc motu in ea proportione, in qua acceleratur per lineam perpendicularem deor­&longs;um per Th. 29.&longs;ed iuxta rationem planorum inclinatorum per Theo­rema 31. nec etiam violentus de&longs;truitur vniformiter, &longs;ed pro rata per Th. 39.

Theorema 79.

Hinc initio plùs detrahitur violenti, & minùs additur naturalis, in fine plùs additur naturalis & minùs detrahitur violenti; hinc minor e&longs;t ictus in fine ni&longs;i malus nauis ad eam altitudinem a&longs;cenderet, ad quam profectò nullus a&longs;cendit, quæ omnia con&longs;tant per Theorema 34. 35. 36.

Theorema 80.

Hinc ratio curuitatis huius lineæ, vel hypothe&longs;is &longs;ecundæ; quæ tamen non e&longs;t Parabola vt volunt aliqui; hinc non eo tempore de&longs;cendit in nauim prædictus globus, quo de&longs;cenderet per ip&longs;am perpendicularem motu purè naturali ex eadem altitudine, &longs;ed maiore tempore; quia motu mix­to non acceleratur iuxta proportionem motus naturalis puri per Th. 77. quod confirmatur illis omnibus experimentis, quæ &longs;uprà adduxi Th. 46.

Theorema 81.

Hinc &longs;i nauis moueretur eadem velocitate, qua funis arcus cum re­dit, e&longs;&longs;etque aptata &longs;agitta, & directa horizontaliter in naui; haud dubiè &longs;i po&longs;t aliquod tempus &longs;taret illicò immota nauis: emitteretur &longs;a­gita, non minore certè vi quàm ab ip&longs;o rcu; hinc ctiam cum nauis appellitur ad littus, &longs;i &longs;tatim &longs;ub&longs;i&longs;tat; omnia quæ &longs;unt in naui &longs;uccutiuntur & plerique cadunt incauti in partem aduer&longs;am propter impetum à naui acceptum; ex quo certè experimento maximè confir­matur hic impetus à naui impre&longs;&longs;us, per quem Galileus ex hypothe&longs;i mo­tus æ&longs;tum maris explicat exemplo appul&longs;arum nauium ad littus, quæ aquam vehunt.

Theorema 82.

Hinc demi&longs;&longs;us globus plumbeus, vel alterius materiæ, quæ facilè vim aëris infringat è &longs;ummo malo nauis ad imum ferè malum de&longs;cendit, hæc e&longs;t ex­perientia à Galileo producta, non tamen adinuenta, à Ga&longs;&longs;endo do­cti&longs;&longs;imè & eleganti&longs;&longs;imè explicata, ab omnibus Copernici &longs;ectatoribus toties decantata, quæ vulgus ignobile ad admirationem adducit; imò plures è Philo&longs;ophis fuere, qui eam in dubium adducerent, cum cam &longs;uis principiis, ne dicam fortè &longs;omniis aduer&longs;ari putarent; certi&longs;&longs;imum tamen e&longs;t illud experimentum centies, imò millies comprobatum, totis etiam vrbibus &longs;pectantibus. Nec ratio huius experimenti adco ab&longs;tru&longs;a e&longs;t, vel recondita, quin à vulgari, ne dicam triobolari Philo&longs;opho &longs;tatim ex­plicari po&longs;&longs;it; cum enim imprimatur à naui mobili impetus pendulo globo per horizontalem, & alius ab ip&longs;a grauitate deor&longs;um per Th. 71. certè mouetur globus demi&longs;&longs;us re&longs;ecto funiculo motu mixto ex hori­zontali nauis, naturali corporis grauis; igitur per lineam curuam, quæ ferè ad imum malum terminatur &longs;ed modicum figuræ adhibendum e&longs;t; &longs;it planum aquæ horizontale, cui innatat nauis IH; &longs;it malus IA perpen­dicularis altus 48. pedes; diuidatur in 4. partes æquales; corpus graue conficiat &longs;patium illud duobus &longs;ecundis, v.g.igitur AK vno &longs;ecundo; e&longs;t autem VK 12. pedum; iam verò moueatur nauis per horizontalem IH, vel AL maxima qua&longs;i velocitate qua triremis moueri pote&longs;t; ita vt vna hora faciat 16. milliaria Germanica, & 15′.4. milliaria, 3′ 800. pa&longs;&longs;us, 1′ 266. 1″ 4. pa&longs;&longs;us & (13/30); &longs;upponamus 1″ conficere 18. pedes, &longs;itque AC 18. & AK vel CE 12. haud dubiè motu mixto faciet lineam AE, & &longs;e­cundo tempore lineam EH, donec tandem cadat in punctum H nauis, quò ferè peruenit punctum I; nam eodem modo retardatur motus nauis; immò plùs quàm motus globi; quod &longs;cilicet partes aquæ, quæ à naui diuiduntur multum re&longs;i&longs;tant; vnde fit compen&longs;atio; nam initio motus violentus, qua&longs;i &longs;ecum rapit motum naturalem initio tardi&longs;&longs;i­mum; præ&longs;ertim cum non acceleretur, ni&longs;i iuxta rationem plani incli­nati, vt &longs;uprà dictum e&longs;t, & in fine naturalis rapit violentum.

Dixi ad imum ferè malum; nam reuera aliquid dee&longs;t quod tamen in­&longs;en&longs;ibile e&longs;t; &longs;ed quia modico tempore globus de&longs;cendit; &longs;it enim malus 108. pedum altitudinis, de&longs;cendit globus tempore 3″; &longs;it 192.4; &longs;it &longs;i fieri pote&longs;t 432. de&longs;cendet 6″, &longs;ed nunquam accedit ad tantam altitudi­nem, igitur duobus vel tribus &longs;ecundis de&longs;cendit; igitur modico tem­pore; igitur violentus motus cen&longs;eri debet eo tempore æquabilis &longs;en&longs;i­biliter; & cum motus nauis nunquam &longs;it eiu&longs;dem velocitatis cum illa quæ acquiritur tempore 2″ in de&longs;cen&longs;u, quia cum in de&longs;cen&longs;u acquiran­tur, hoc dato tempore ferè 48. pedes &longs;patij; certè motu æquabili cuius e&longs;&longs;et eadem velocitas acquirerentur 96. &longs;ed vix acquirerentur 24.vt di­ctum e&longs;t &longs;uprà; igitur vix nauis percurrit in horizontali æqualem lineam longitutidini mali eo tempore, quo globus nauim attingit &longs;it enim altitudo mali FA 48. pedum; &longs;it amplitudo &longs;patij horizontalis æqualis FA; haud dubiè 1″ percurret AD, id e&longs;t 12.pedes ferè, quo tempore per­currat FG. 24. pedes & 20″ percurret DF, & GI. &longs;i motus &longs;umatur vt æquabilis, vel GH, &longs;i retardatur, igitur 1°″ mobile percurrit &longs;egmentum curuæ AE & 2° EH.

Et licèt videatur tantùm acquirere MI, quæ e&longs;t minor DF 15. per­pendiculari deor&longs;um, acquirit totam EH, quæ non modo e&longs;t à motuna­turali, verùm etiam à motu violento; nec enim motu naturali dum mi­&longs;cetur cum alio, tantùm acquiritur deor&longs;um, quantùm reuerâ acquiritur motu naturali puro, vt &longs;uprà monuimus; quia tamen etiam deor&longs;um mo­tus violentus dflectitur, etiam aliquid &longs;patij ratione violenti deor&longs;um acquiritur; &longs;i enim vbi peruenit in E vterque impetus intactus remane­ret &longs;ine acce&longs;&longs;ione, &longs;ine imminutione; haud dubiè per eamdem EM, quæ &longs;it tangens huius curuæ AEH &longs;uum cur&longs;um pro&longs;cqueretur; igitur ac­quireret deor&longs;um totam DN, vel EO propter impetum naturalem præ­uium; &longs;i verò aliquid naturalis accedat, quid mirum &longs;i ratione illius ac­quiratur MI, vel NF?

Dices non de&longs;cendit tam citò motu naturali accelerato, mixto cum violento, quàm motu puro naturali.

Re&longs;pondeo concedo; vnde nunquam ex A in H 2″ de&longs;cendit; &longs;ed tardiùs, licèt FA &longs;it 48. ped. &longs;ed parùm abe&longs;t tùm propter minorem re&longs;i­&longs;tentiam huius impetus violenti, qui facilè detorquetur, & con&longs;equen­tur minùs illius perit, tùm quia etiam de&longs;truitur aliquid violenti; igitur paulò plùs temporis collocat in GI, quàm in FG.

Scholium.

Ob&longs;eruabis primò, &longs;i nouus impetus accedat, non e&longs;&longs;e expectandum hunc effectum; quippe nihil accipit à naui globus deinceps, vbi &longs;emel re&longs;ecto fune ab ea qua&longs;i &longs;eparatur.

Secundò, &longs;i &longs;tatim &longs;i&longs;tat nauis demi&longs;&longs;o globo ad vnum malum nullo modo de&longs;cendet, vt patet, &longs;ed antè.

Tertiò, &longs;i demittatur globus dum &longs;i&longs;tit nauis, tùm deinde, vbi demi&longs;&longs;us e&longs;t, impellatur nauis; non de&longs;cendet etiam ad radicem, &longs;ed retrò.

Quartò, motus nauis non e&longs;t æquabilis, quidquid dicat Galileus; alio­quin vna remorum impul&longs;ione opus e&longs;&longs;et, vt &longs;emper eodem motu mouo­retur, aut certè &longs;i continua remigatione impellatur; cre&longs;ceret in infini­tum velocitas motus, &longs;i nihil de priori, velocitate detraheretur; retarda­tur igitur ille nauis motus propter re&longs;i&longs;tentiam aquæ, cuius partes & im­pellendæ & &longs;ulcandæ, &longs;eu diuidendæ &longs;unt; hinc fiunt ro&longs;tratæ naues vel cu&longs;pidatæ vt faciliùs aquam findere po&longs;&longs;int; igitur ille motus nauis non e&longs;t æquabilis; Idem pror&longs;us dicendum e&longs;t de impetu impre&longs;&longs;o in globo, cuius aliquæ partes de&longs;truuntur, ne &longs;int fru&longs;trà, quod &longs;uprà de pro­jecto per horizontalem vel incliatam luculenter demon&longs;trauimus.

Quintò &longs;i demittatur ex ali naui proxima immobili perpendiculari­ter omninò de&longs;cendet; Vnde valde hallucinantur ij, qui exi&longs;timant hunc motum e&longs;&longs;e ab aëre quem nauis commouet, quod fal&longs;i&longs;&longs;imum e&longs;t, quia pertica ad in&longs;tar mali parùm aëris commouet; adde quod aër retrò agi­tur, vt patet in aqua; præterea &longs;i è curru immobili demittatur globus co tempore, quo alius currus præteruolat, de&longs;cendit perpendiculariter; &longs;i ve­rò è curru mobili etiam in maiori di&longs;tantia porrecta &longs;cilicet maximè extra currum demittente dextera; globus ab ip&longs;o curru capietur; hîc etiam ob&longs;eruabis idem pror&longs;us accidere in curru mobili, quod in naui; &longs;i enim è fene&longs;tra currus mobilis demittas pilam, &longs;emper cadet ex aduer&longs;o; idem dico de currente equo, cui in&longs;idens demittat globum, imò &longs;i locus &longs;it planus & politus, pila per aliquod tempus currum, vel equitem in&longs;e­quetur, quod qui&longs;que probare poterit, vt reuerâ centies probatum fuit.

Sextò ad rationem Galilei, qui contendit motum circularem circa centrum terræ e&longs;&longs;e æquabilem, quia &longs;cilicet mobile non recedit à centro: leuis e&longs;t omninò ratio; quia globus in medio aëre motu mixto mouetur, id e&longs;t habet impetum partim deor&longs;um, partim per tangentem, & nullo modo per circularem, vt certum e&longs;t; nec enim rotata alium impetum im­primunt, igitur violentus e&longs;t; igitur de&longs;trui debet etiam iuxta commu­nia principia: adde quod motus mixtus fit per Diagonalem quod etiam ip&longs;e admittit; igitur totus impetus æqualem motum non habet; nec enim Diagonalis æqualis e&longs;t vnquam duobus lateribus; igitur aliquid illius fru&longs;trà e&longs;t; igitur de&longs;trui debet; præterea licèt motus circularis &longs;it peren­nis circa centrum mundi; nam de illo tantùm e&longs;t quæ&longs;tio, hoc ip&longs;um &longs;upponit primò motum illum e&longs;&longs;e &longs;implicem; &longs;ecundò, nullam pror&longs;us e&longs;&longs;e re&longs;i&longs;tentiam; atqui in hoc ca&longs;u vtrumque deficit; nam motus ille circularis non e&longs;t &longs;implex &longs;ed mixtus, & obe&longs;t re&longs;i&longs;tentia aquæ, vt &longs;uprà dictum e&longs;t; ni&longs;i verò con&longs;ideres de&longs;cendentem globum è &longs;ummo malo, quis dicat e&longs;&longs;e circularem? adde quod nauis imprimit tantùm rectum per tangentem, vt iam &longs;uprà dictum e&longs;t; porrò ad illud, quod dicit non de­&longs;trui motum circularem à naturali, cui non e&longs;t contrarius, cum non re­moueat longiùs à centro; videtur omninò di&longs;&longs;unulare cau&longs;am impetus de&longs;tructiuam, quæ cettè in contrarietate tantùm determinationis po&longs;ita e&longs;t, vt &longs;uprà dictum e&longs;t; ex qua &longs;equitur aliquid impetus fru&longs;trà e&longs;&longs;e; ac pro­inde de&longs;trui per Axioma illud toties decantatum, Quod frustr à e&longs;t, non e&longs;t: Præterea non video quomodo hanc rationem proponat magnus Gali­leus, qui nullum alium impetum violentum de&longs;trui putat, nî&longs;i tantùm il­lum, qui e&longs;t per lineam verticalem &longs;ur&longs;um; nam ex motu illo impre&longs;&longs;o æquabili, & naturali accelerato &longs;uas Parabolas ad&longs;truit.

Septimò, non e&longs;t tamen quod diffitear ingeniosè excogitatum ab co fui&longs;&longs;e, ideo globum è &longs;ummo malo demi&longs;&longs;um ad imum de&longs;cendere, quod &longs;cilicet de&longs;cendat motu mixto ex naturali accelerato, & violento æqua-bili, quod vt breuiter ob oculos ponatur &longs;it malus nauis mobilis IA, quæ eo tempore, quo corpus graue de&longs;cendit ab A in D motu naturali, percurrit FG æquabili motu, & con&longs;equenter GI æqualem FG eo tem­pore, quo idem corpus graue percurrit DF triplam AD; igitur globus demi&longs;&longs;us ex A &longs;uo motu de&longs;cribit Parabolam AEH; quod etiam accidet a&longs;&longs;umpta quacunque altitudine mali vel quocunque &longs;patio confecto à naui mobili eo tempore, quo corpus graue motu naturali accelerato conficit &longs;patium æquale altitudini mali.

Octauò, non e&longs;t tamen di&longs;&longs;imulandum, quod etiam non di&longs;&longs;imulauit Mer&longs;ennus, talem non fore de&longs;cen&longs;um, &longs;i nauis v. g. eadem cum emi&longs;&longs;a &longs;agitta, vel explo&longs;a è tormento glande velocitate moueretur; non quod aër vel medium ob&longs;i&longs;tat, vt ip&longs;i dicunt; hoc enim iam &longs;uprà rejecimus; &longs;ed quod major impetus violentus efficiat, vt iam &longs;uprà dictum e&longs;t, ne in tanta proportione naturalis acceleretur; quod etiam &longs;uo boatu intonant tormenta maiora, è quibus horizontaliter directis explo&longs;æ pilæ per plu­ra &longs;ecunda in libero aëre moueantur, licèt os tormenti à plano horizon­tis vix tribus pedibus ab&longs;it; igitur non de&longs;cribunt &longs;uo motu Parabolas; hinc &longs;ub finem minor e&longs;t ictus; hinc etiam fatetur idem Mer&longs;ennus &longs;e­cundum &longs;patium horizontale confici tardiore motu quàm primum & tertium quàm &longs;ecundum, atque ita deinceps.

Theorema 83.

Si corpus graue proiiciatur &longs;ur&longs;um perpendicul ariter è naui mobili, &longs;unt tres impetus qui concurrunt ad illum motum &longs;it enim nauis mobilis per hori­zontalem LF, è qua &longs;ur&longs;um rectâ per lineam perpendicularem LA pro­iiciatur corpus graue; huic certè ine&longs;t triplus impetus, &longs;cilicet duo vio­lenti, alter per verticalem LA impre&longs;&longs;us à proiiciente; alter per horizon­talem LF impre&longs;&longs;us à naui; tertius denique naturalis per ip&longs;am perpen­dicularem deor&longs;um LP; igitur tres i&longs;ti impetus &longs;uo modo concurrunt ad motum per Ax.1.certè &longs;i ine&longs;&longs;ent tantùm duo impetus &longs;cilicet LA, & LF, motus &longs;ieret per inclinatam rectam LC; vel &longs;i tantùm duo LP, & LA fieret per ip&longs;am LA motus retardatus; vel &longs;i LF & LP fieret per curuam deor&longs;um, vt con&longs;tat ex dictis; igitur per aliam lineam fieri de­bet ad quam tres illi impetus concurrunt.

Theorema 84.

Tam pugnat impetus naturalis per LP cum verticali LA quando e&longs;t con­junctus cum horizontali LF, quàm cum nullus e&longs;t horizontalis, probatur, quia &longs;emper mobile deor&longs;um trahit, vt patet.

Theorema 85.

Hinc naturalis e&longs;t æquabilis, & violentus &longs;ur&longs;um e&longs;t retardatus; horizon­talis verò e&longs;t æquabilis &longs;altem æquiualenter; quia cum illo non pugnat ho­rizontalis, in a&longs;cen&longs;u &longs;altem perinde &longs;e habet; immò cum illo conuenit ad de&longs;truendum violentum &longs;ur&longs;um, id e&longs;t ad deflectendum deor&longs;um mobile vt con&longs;tat; igitur hic motus con&longs;tat ex naturali & horizontali æquabilibus, & violento retardato &longs;int enim tres impetus ab eodem puncto E &longs;cilicet EF, ED, EA; ex EA ED fit mixtus EG, ex EA, EF, violentus EB; denique ex mixto EG à naturali EF fit EC, quæ omnia &longs;unt clara.

Theorema 86.

A&longs;cendit mobile ad eamdem altitudinem hoc motu, ad quem a&longs;cenderet &longs;ine horizontali v. g. &longs;ine hozontali a&longs;cendit in B, cum horizontali a&longs;cendit in C, &longs;ed DC, & EB &longs;unt eiu&longs;dem altitudinis.

Scholium.

Ob&longs;eruabis, licèt i&longs;te motus non fiat per lineam parabolicam, vt &longs;uprà demon&longs;trauimus Th. 54. & reliquis; quia tamen &longs;en&longs;ibiliter proximè accedit, deinceps vtemur Parabola vt in fig. Th. 83. & horizontalem motum accipiemus pro æquabili; licèt omninò æquabilis non &longs;it; ni&longs;i tantùm æquiualenter; dixi æquiualenter, quia eodem modo &longs;e habet hic motus, ac &longs;i per inclinatam &longs;ur&longs;um LC impetu &longs;cilicet LC mobile pro­iiceretur; &longs;ed in hoc ca&longs;u de&longs;trueretur impetus ille per inclinatam &longs;im­plex; igitur & mixtus; quia tamen ille qui remanet partim ex LA, par­tim ex LF eodem modo ferè &longs;e habet ac &longs;i totus LF intactus maneret; hinc dictum e&longs;t &longs;uprà æquiualenter e&longs;&longs;e æquabilem.

Theorema 87.

A&longs;cendit hoc motu ad &longs;ubduplam altitudinem illius, ad quam motu mixto tantum ex verticali & horizontali &longs;ine naturali a&longs;cenderet; quippe a&longs;cende­ret in C fig. Th.83. &longs;inc impetu naturali, &longs;ed FC & LA æquales &longs;unt; atqui motu violento puro, ni&longs;i naturalis obe&longs;&longs;et, a&longs;cenderet in A; at ve­rò &longs;i obe&longs;t naturalis; a&longs;cendit tantùm motu violento in K, & mixto in in D; quia ex K in L motu naturali tot acquireret mobile gradus impe­tus naturalis quot amittit in motu violento ab L in K; &longs;ed cum in impe­tu acqui&longs;ito à K in L motu æquabili a&longs;cenderet ab L in A, quæ e&longs;t dupla LK vt o&longs;tendimus in &longs;ecundo libro; &longs;ed motu mixto, & verticali, & ho­rizontali a&longs;cenderet in C; &longs;ed FD e&longs;t &longs;ubdupla FE; igitur motu mixto a&longs;cendit ad &longs;ubduplam altitudinem, &c.

Theorema 88.

Mobile projectum è nai mobili, vbi ad &longs;ummam altitudinem peruenit mo­tumixto ex verticali retardato, horizontali æquabili, & naturali item æqua­bili, de&longs;cendit etiam motu mixto ex horizontali retardato &longs;altem æquinenter, & naturali accelerato; dixi æquiualenter, quia vt dixi in Sch. Th.86.licèt remaneat aliquid impetus veiticalis qui in communem lineam abit cum horizontali; res tamen perinde &longs;e habet atque &longs;i totus verticalis de&longs;true­retur, & totus horizontalis intactus permaneret; igitur de&longs;cen&longs;us fit mo­tu mixto ex naturali accelerato & horizontali retardato per Th.30.quia tamen modico illo tempore parùm retardatur, vt &longs;uprà monui, &longs;en&longs;ibili­ter accipi pote&longs;t pro æquabili.

Theorema 89.

Hinc &longs;en&longs;ibiliter ex a&longs;cen&longs;u & de&longs;cen&longs;u fit integra Parabola; nam pro­iiciatur ex L in A, co tempere, quo nauis mouetur ex L in F, certè &longs;i tempus illud diuidatur bifariam prima parte mobile percurret LI tri­plam IK in verticali, & LM &longs;ubduplam LF in horizontali; igitur erit in G; &longs;ecunda verò perte temporis in verticali percurrit IK, & MF in horizontali; igitur erit in D; præterea &longs;i accipiantur duæ aliæ partes tem­poris æquales; prima in perpendiculari deor&longs;um percurret DE æqua­lem LK, & in horizontali DO; igitur crit in N; &longs;ecunda vero in per­pendiculari percurret NQ triplam NO, & NR in horizontali; igitur erit in S; &longs;ed hæc e&longs;t Parabola; nam vt &longs;e habent quadrata applicatarum v.g. EG, FL, ita &longs;agittæ DE, DF; dixi &longs;en&longs;ibiliter, nam vt &longs;uprà mo­nui e&longs;t alia linea, quæ tamen proximè accedit ad Parabolam.

Theorema 90.

Hinc ferè recedit mobile in idem punctum nauis, è quo &longs;ur&longs;um proiectum e&longs;t; dixi ferè, quia non e&longs;t omninò Parabola; immò &longs;upponitur motus horizontalis tùm nauis tùm mobilis omninò æquabilis, à quo tamen tantillùm deficit, &longs;ed in tam breui tempore non e&longs;t &longs;en&longs;ibile.

Theorema 91.

Hinc quantùm initio detrahit horizontali verticalis inten&longs;ior, & &longs;ub finem remittit, tantùm initio remittit horizontali naturalis tardior, & &longs;ub finem ve­locior detrahit; &longs;ic in a&longs;cen&longs;u linea curua LD, initio parùm recedit à ver­ticali LK, & multùm &longs;ub finem; in de&longs;cen&longs;u verò curua DS accedit propiùs ad horizontalem DT, à qua multùm recedit &longs;ub finem.

Theorema 92.

Hinc eadem, quâ mobilis proijcitur &longs;ur&longs;um è naui mobili, recipitur manu; probata centies experientia; idem dico de &longs;agitta, arcu emi&longs;&longs;a, glande tormento explo&longs;a, &c. &longs;ic dum demittis manu in eadem naui aliquod graue deor&longs;um, eadem &longs;emper à te di&longs;tantia cadit; &longs;ic in rhodis currenti­bus poma odorifera, &longs;ur&longs;um modica vi projecta eadem &longs;emper excipiun­tur manu, perinde atque &longs;i currus ip&longs;e &longs;taret. Ita pror&longs;us &longs;e res habet dum in&longs;idens equo etiam pernici&longs;&longs;imè currenti ludis huiu&longs;modi moti­bus; quorum nullum pror&longs;us di&longs;crimen ob&longs;eruabis in naui, &longs;iue &longs;tet &longs;iue moueatur &longs;olito cur&longs;u; &longs;i enim eadem velocitate, qua vel emi&longs;&longs;a &longs;agitta, vel glans explo&longs;a moueretur; haud dubiè maximum di&longs;crimen inter­cederet.

Theorema 93.

Hinc &longs;i pilam projectam è naui mobili continuo intuitu pro&longs;equaris &longs;ur&longs;um rectà ferri iudicabis; quippe cum perpetuò mutes perpendicularem pro­pter motum nauis, in eadem &longs;emper e&longs;&longs;e putas, in qua pila &longs;emper occurrat; licèt reuerâ qui &longs;unt in naui immobili rem aliter e&longs;&longs;e iudicent; quippe vident pilam &longs;uo moiu de&longs;cribere curuam non &longs;imi­iem illi, quam di&longs;cus per lineam inclinatam &longs;ur&longs;um proiectus &longs;uo mo­tu de&longs;criberet; neque mirum e&longs;t, cum &longs;int eædem vtriu&longs;que rationes, cum hac tantum differentia, quòd inclinata di&longs;ci &longs;it motus &longs;umplicis, inclina­ta verò pilæ a&longs;cendentis &longs;it motus mixti ex horizontali & verticali, æ­quabili quidem in a&longs;cen&longs;us accelerato in de&longs;cenfu.

Theorema 94.

Ex his vides non valere vulgarem rationem, quæ vulgò a&longs;fertur contra mo­tum terræ, &longs;equi &longs;cilicet ex eo lapidem proiectum &longs;ur&longs;um per verticalem longo interuallo ver&longs;us occa&longs;um retrò de&longs;cen&longs;urum, quod tamen etiam ex motu terræ &longs;uppo&longs;ito non &longs;equeretur, cum non &longs;equatur ex motu nauis.

Igitur alia ratione impugnari debet hypothe&longs;is illa, quæ terræ motude&longs;truit; quod certè &longs;i à me fieri po&longs;&longs;it, in tractatu de corporibus cœle&longs;ti­bus, vel de nouo &longs;y&longs;temate aliquando præ&longs;tabimus; non tamen e&longs;t quod hîc di&longs;&longs;imulem aliquorum agendi methodum, qui ex hoc phœnome­no con&longs;tanter a&longs;ferunt terram moueri; nam primò, &longs;equeretur tantùm moueri circa centrum id e&longs;t motu orbis, non verò motu centri; quæ e&longs;t hypothe&longs;is Origani. Secundò ex quiete terræ hoc idem phœnomenon &longs;equitur; quippe, &longs;i terra quie&longs;cit, cadem manu cadentem excipio lapi­dem, quæ &longs;ur&longs;um rectà proiicit; igitur quemadmodum ex hoc non infero terræ quietem, &longs;ed aliunde; ita neque ex hoc inferri pote&longs;t terræ motus; cum enim duplex hypothe&longs;is codem phœnomeno &longs;tare pote&longs;t, neutra ex eo euincitur; igitur &longs;icuti fateor ex hoc phœnomeno minimè demon­&longs;trari terræ quietem ita & tu fateri debes ex eo minimè ad&longs;trui po&longs;&longs;e terræ motum.

Adde quod, haud dubiè &longs;i terra quie&longs;cit citiùs proiectus lapis &longs;ur&longs;um de&longs;cendit, quàm &longs;i mouetur; nec enim vt dictum e&longs;t &longs;uprà proiecta velo­ci&longs;&longs;imo motu per horizontalem de&longs;cendunt eo tempore, quo ex eadem altitudine motu purè naturali de&longs;cenderent; quod multis euincitur ex­perimentis, vt vidimus in Th.46. atqui punctum terræ &longs;ub æquatore ve­loci&longs;&longs;imè moueretur, quod vno temporis &longs;ecundo conficeret 1250.pedes geometricos &longs;i 5. pedes geometrici tribuantur pa&longs;&longs;ui, 4000. pa&longs;&longs;us leucæ germanicæ, 15. leucæ germanicæ gradui Æquatoris, toti demum Æqua­tori 360. gradus; cum autem iactus medius tormenti validi&longs;&longs;imi &longs;it 15000. pedum, duretque 30″ temporis; certè 30″ temporis con&longs;icit pun­ctum æquatoris 37500. pedes; igitur mouetur velociùs explo&longs;a glande; igitur &longs;i hæc velocitas glandis impedit, ne tàm citò deor&longs;um cadat, ma­jor velocitas motus terræ potiori iure illud ip&longs;um impediet; igitur &longs;i terra quie&longs;cit, globus &longs;ur&longs;um proiectus velociùs recidet in terram, et&longs;i terra moueatur tardiùs.

Scholium.

Ob&longs;eruabis duos tantùm motus in naui mobili fui&longs;&longs;e hactenus explica­tos; primus e&longs;t, quo demittitur plumbea pila è &longs;ummo mali; &longs;ecundus e&longs;t, quo ex summo malo, vel ex alio nauis mobilis puncto proiicitur &longs;ursum cor-pus graue per lineam verticalem; &longs;unt autem plures alij motus, tot &longs;cili­cet, quot po&longs;&longs;unt duci lineæ è &longs;ummo malo in orbem quoquo ver&longs;um; quarum hæ &longs;unt præcipuæ. &longs;it apex mali B; circa quem de&longs;cribatur cir­culus ACDE, &longs;itque primò circulus ille verticalis parallelus &longs;cilicet li­neæ directionis nauis BA, quæ &longs;it v. g. ver&longs;us Boream; primò habes li­neam verticalem &longs;ur&longs;um BE; &longs;ecundò perpendicularem deor&longs;um BC; tertiò lineam directionis ver&longs;us Boream BA; quartò illi oppo&longs;itum ver&longs;us Au&longs;trum BD; tùm voluatur circulus circa axem immobilem AD per quadrantem integrum, dum &longs;cilicet BE &longs;it ad Ortum, quæ e&longs;t quinta linea, & BC ip&longs;i oppo&longs;ita ad Occa&longs;um, quæ e&longs;t &longs;exta. Igitur habes 6. li­neas; &longs;cilicet &longs;ur&longs;um, deor&longs;um, ver&longs;us Boream & Au&longs;trum, ver&longs;us Ortum, & Occa&longs;um; linea quæ tendit deor&longs;um pote&longs;t dupliciter con&longs;iderari, vel enim demittitur &longs;ua &longs;ponte, vel proiicitur.

Iam verò inter Boream, & Occa&longs;um habes lineas triplicis generis, primò horizonti parallelas, quæ vt con&longs;iderentur; cen&longs;eatur prædictus circulus parallelus horizonti, ita vt ex centro B ducantur ad circumferentiam tot lineæ, quot &longs;unt puncta in circumferentia; &longs;ecundò inclinatas &longs;ur&longs;um & inclinatas deor&longs;um; &longs;imiliter inter Occa&longs;um & Au&longs;trum, inter Au&longs;trum & Ortum, inter Ortum & Boream; porrò exprimes omnes lineas, &longs;i api­cem mali fingas centrum globi, &longs;eu &longs;i in circulo prædicto verticali à centro B ad circumferentiam ducantur tot lineæ quot po&longs;&longs;unt duci, tuncque circa axem EC immobilem voluatur circulus, &c. his po&longs;i­tis &longs;it.

Theorema 95.

Si proijciatur globus deor&longs;um à &longs;ummo malo, de&longs;cendet ferè ad imum ma­lum; probatur, quia de&longs;cendet quidem velociùs quàm &longs;i motu naturali de&longs;cenderet vt con&longs;tat per Th. 69. &longs;ed profectò nihil acquiret in hori­zontali globus, quod non acquirat nauis; igitur imùm ferè malum attin­git &longs;ed opus e&longs;t aliqua figurâ; &longs;it enim apex mali A, de&longs;cendatque pri­mò ex A &longs;ua &longs;ponte in H; haud dubiè &longs;i eo tempore, quo motu na­turali conficit AD, mixto deor&longs;um conficit AF, eo tempore cadet in G ex A &longs;i hic impetus deor&longs;um adueniat; &longs;ed res e&longs;t clara; hæc porrò figura non e&longs;t Parabola, licèt &longs;it curua; con&longs;tat autem hîc motus ex naturali accelerato, ex impre&longs;&longs;o deor&longs;um æquabili per &longs;e, & horizontali &longs;en&longs;i­biliter æquabili; pote&longs;t autem de&longs;ignari hæc linea motus ex &longs;uprà dictis.

Theorema 96.

Si in circulo verticali prædicto proijciatur per lineam horizontalem ver­&longs;us Boream, mouebitur globus motu mixto ex duplici horizontali per eamdem lineam ferè æquabili; id e&longs;t &longs;en&longs;ibiliter, licèt geometricè loquendo retardetur, & naturali accelerato; &longs;it perpendicularis deor&longs;um AH, horizontalis AC, quam conficiat eo tempore, quo conficit AH motu naturali, motu mixto perueniet in K; &longs;i verò duplicetur horizontalis, ita vt eo tempore quo conficit AH, conficiat AD, motu mixto perueniet in L; hæc autem curua HL accedit ad Parabolam licèt non &longs;it vera Parabola; quia quando ia­ctus horizontalis e&longs;t veloci&longs;&longs;imus, qualis in arce, vel in tormentis belli­cis, eodem tempore mobile non decidit in terram, quo de&longs;cenderet mo­tu purè naturali ex eadem altitudine.

Theorema 97.

Hinc, &longs;i motus nauis e&longs;&longs;et æqualis motui &longs;agittæ, motus ex vtroque mixtus duplam amplitudinem in plano hòrizontali acquireret, v.g. &longs;i tantùm &longs;agitta emi&longs;&longs;a arcu extra nauim ex A perueniret in K, in naui mebili perueniret in L; &longs;i verò nauis, vt reuerâ fit, tardiùs moueatur, &longs;agitta è naui emi&longs;&longs;a ver&longs;us Boream &longs;cilicet acquiret pro rata, id e&longs;t &longs;i nauis motus &longs;it tantùm &longs;ubduplus perueniret in M; &longs;i &longs;ubquadruplus in N &c.

Theorema 98.

Hinc tormentum bellicum quod e&longs;t in prora directum ad eamdem lineam, quam &longs;uo motu conficit nauis maiorem iactum habebit, non tamen &longs;en&longs;ibiliter; quia motus nauis parum addit; ob&longs;eruabis tamen non videri maiorem quàm &longs;i nauis quie&longs;ceret, quia eo tempore, quo &longs;agitta ex A peruenit in L, nauis ex H peruenit in K; igitur videtur &longs;emper e&longs;&longs;e idem iactus, &longs;iue moueatur nauis &longs;iue non, quia e&longs;t &longs;emper eadem di&longs;tantia nauis, & ter­mini iactus; cum nauis id totum acquirat &longs;patij, quod motui &longs;agittæ accedit.

Theorema 99.

Hinc vt quis maiore ni&longs;u lapidem v. g. proijciat, tùm longiore tempore brachium rotat, tm præuio cur&longs;u impetum auget, quia non tantùm impe­tus brachij imprimitur mobili, &longs;ed etiam impetus totius corporis; hinc etiam &longs;i præmittatur cur&longs;us longiore &longs;altu plano horizontali maius &longs;patium traiicitur; quæ omnia ex ii&longs;dem principiis manife&longs;tè &longs;e­quuntur.

Theorema 100.

Si verò per oppo&longs;itam lineam ver&longs;us Au&longs;trum proijcitur mobile, moucbitur motu mixto ex duobus horizontalibus ad oppo&longs;itas lineas, & ex naturali ac­celerato; &longs;it proiectio per AB, ita vt mobilè perueniat in L ni&longs;i impedia­tur; certè &longs;i nauis motu &longs;ubduplo in oppo&longs;itam partem feratur, peruenit tantùm in K, quæ omnia con&longs;tant ex dictis; nam impetus oppo&longs;iti pu­gnant pro rata, vt &longs;æpè diximus; videbitur tamen e&longs;&longs;e æqualis iactus; &longs;i enim eo tempore, quo &longs;agitta peruenit in K, nauis fertur in oppo&longs;itam partem &longs;patio æquali KL, haud dubiè di&longs;tantia &longs;emper crit æqualis; tan­tùm enim recedit ver&longs;us Boream nauis, quantùm &longs;agitta à puncto L ad punctum K reducitur.

Theorema 101.

Si motus nauis e&longs;&longs;et æqualis motui &longs;agittæ v. g. &longs;i nauis ferretur per ineam GC &longs;eu TA ver&longs;us Boream, & &longs;agitta è &longs;ummo malo emitteretur per lineam TO ver&longs;us Au&longs;trum, de&longs;cenderet per lineam T.G. nec quidquamacquireret in horizontali; quod probatur per Th. 133. l.1. fic globus tor­menti etiam ne latum quidem vnguem pertran&longs;iret in horizontali, vide­tur tamen &longs;emper e&longs;&longs;e idem iactus; nam eo tempore, quo &longs;agitta caderet à T in G, nauis e&longs;&longs;et in C, atqui CG & GM &longs;unt a&longs;&longs;umptæ æquales; hinc potiùs arcus e&longs;&longs;et emi&longs;&longs;us quàm &longs;agitta, & tormentum explo&longs;um quàm globus.

Scholium.

Ob&longs;eruabis, &longs;i nauis motus &longs;it ad motum &longs;agittæ v. g. in ratione &longs;ub­dupla, &longs;cilicet vt FG, vel LM ad GM peruenit in L per Parabolam TL; &longs;t vt EG vel KM ad GL peruenit in K per Parabolam TK; &longs;i vt DG vel I M ad GM peruenitin I per Parabolam TI, &c. vnde vides Parabolas i&longs;tas &longs;emper in infinitum contrahi, donec tandem in rectam TG de&longs;i­nant vbi motus nauis e&longs;t æqualis motui &longs;agittæ: Parabolas dixi &longs;en&longs;ibi­liter, &longs;cilicet eo modo, quo &longs;uprà.

Theorema 102.

Si verò motus nauis e&longs;&longs;et maior motu &longs;agittæ, &longs;agitta fèrretur in eamdem partem in quam fertur nauis per &longs;patium æquale differentia illorum motuum,v.g. &longs;i nauis moueatur per GM & &longs;agitta per TA, &longs;itque motus nauis ad motum &longs;agittæ, vt GM, ad IM; eo tempore quo nauis attinget M, &longs;agitta cadet in I, & &longs;i motus &longs;it vt GM ad KM cadet in K vel vt GM ad GL cadet in L. per Parabolas, quæ omnia con&longs;tant ex dictis, & ex Theore­mate per 134. l.1.

Corollarium 1.

Ex illa hypothe&longs;i &longs;equitur egregium paradoxon &longs;cilicet &longs;agittam retor queri in &longs;agittarium; &longs;it enim motus nauis ad motum &longs;agittæ vt GM ad LM; haud dubiè per Th. &longs;uperius eo tempore, quo nauis peruenit ad M &longs;a­gitta attinget punctum L, & eo tempore quo nauis e&longs;&longs;et in L &longs;agitta e&longs;­&longs;et in puncto Y, &longs;i cum nauis peruenit in L illicò &longs;i&longs;tat &longs;agitta, cadet in ip&longs;am nauim; nam cadet in L quod clarum e&longs;t: dixi &longs;i nauis &longs;i&longs;tat po&longs;t emi&longs;&longs;am &longs;agittam, &longs;i enim nauis &longs;emper moueatur, æquabilis &longs;emper e&longs;&longs;e videbitur &longs;agittæ iactus, &longs;i enim è naui immobili emi&longs;&longs;a fui&longs;&longs;et prædicta &longs;agitta per horizontalem TO, acqui&longs;iui&longs;&longs;et &longs;patium vel amplitudinem G L; &longs;ed videtur confeci&longs;&longs;e ML, cum nauis mouetur; atqui ML e&longs;t æqualis LG, quid clarius?

Hinc &longs;i quis in naui currat per lineam directionis id e&longs;t ver&longs;us eain partem, in quam mouetur nauis, curret velociùs; immò &longs;i ambulet, ingen­tes faciet pa&longs;&longs;us &longs;eu &longs;altus v.g.&longs;i nauis conficit &longs;patium GM eo tempore quo aliquis &longs;altat ex G in H; haud dubiè amplitudo eius &longs;altus erit com­po&longs;ita ex tota GM & GH; &longs;i verò in partem oppo&longs;itam ver&longs;us C currat: vel currit velociùs, vel tardiùs, vel æquali motu: &longs;i primum, aliquid &longs;patij acquiret ver&longs;us C æqualis &longs;cilicet differentiæ motuum; &longs;i &longs;ecundum, recedet ver&longs;us M &longs;patio æquali eidem differentiæ; &longs;i tertium, nec acceder, nec re­cedet, &longs;ed totis viribus currens &longs;eu tentans currere in eodem &longs;emper lo-co &longs;tabit, vel &longs;i &longs;it rotatus globus in tabulato nauis mouebitur motu or­bis circa centrum immobile.

Theorema 103.

Si proiiciatur mobile per lineam inclinatam deor&longs;um, quæ &longs;it hypothenu&longs;is trianguli orthoganij, is ba&longs;is &longs;it horizontalis & perpendiculum &longs;patium,quod percurritur motu naturali æquali tempore, idque in naui mobili in eam partem, ver&longs;us quam mouetur nauis, erit motus mixtus ex naturali accelerato & inclinato mixto ex horizontali & alio inclinato &longs;it enim horizontalis AD, perpendicularis AMK, &longs;it AM &longs;patium quod percurri­tur in perpendiculari motu purè naturali, eo tempore, quo percurritur AC &longs;ubdupla AD, &longs;itque AM &longs;ubdupla AC, & &longs;ecundo tempore æquali percurratur in horizontali CD, & in perpendiculari MK tripla AM; erit motus mixtus per lineam parabolicam ANH; nam &longs;uppono hori­zontalem æquabilem, cùm parùm ab eo ab&longs;it, vt &longs;upradictum e&longs;t; præ&longs;er­tim cum &longs;en&longs;ibiliter hæc linea &longs;it parabolica.

Iam verò in eadem naui proiiciatur mobile per inclinatam AP, quæ &longs;it diagonalis quadrati AP, & impetus perinclinatam AP &longs;it ad impetum per horizontalem AC, vt AP ad AC; ducatur LPF parallela MN, & CF parallela AP; denique diagonalis AF: haud dubiè ML e&longs;t æqualis AM, vt patet; & &longs;i motus e&longs;&longs;et tantum mixtus ex AC & AP fieret per diagona­lem AF, quam mobile eodem tempore percurreret quo vel AC vel AP; igitur &longs;i dum percurrit AF percurrit AM, motu naturali, certè dum per­currit AN &longs;ubdupla AF, percurret tantùm &longs;ubquadruplam AM; a&longs;&longs;uma­tur ergo NO æqualis AS, & FG æqualis AM; ducaturque curua AOG, hæc e&longs;t linea quç&longs;ita.

Itaque idem dicendum e&longs;t de his inclinatis, quod de aliis &longs;uprà di­ctum e&longs;t Th.72. ni&longs;i quod accipitur inclinata mixta ex horizontali & da­ta inclinata, v.g. ANF ex AC & AP; hæc autem linea non e&longs;t Parabolica, quia quadratum MN, vel VO e&longs;t ad quadratum RG vt 1.ad 4.at verò &longs;a­gitta AV e&longs;t ad &longs;agittam AP, vt 5.ad 12.porrò hæc linea &longs;ecat Parabolam vt patet; &longs;i verò accipiatur inclinatata AI, mixta inclinata erit AH igitur a&longs;&longs;umatur HX æqualis AM, & PZ æqualis AS ducetur linea huius mo­tus per AZX. quænam verò &longs;int hç lineæ, dicemus aliàs Tomo &longs;equenti.

Theorema 104.

Si proiiciatur per inclinatam &longs;ur&longs;um in eam partem, in quam mouetur nauis, erit etiam mixtus ex naturali, & inclinato ex horizontali, & data inclinata; vnde idem pror&longs;us dicenduin e&longs;t de mixta inclinata, quod de &longs;implici in­clinata, de qua multa &longs;uprà dicta &longs;unt à Th.47. &longs;uppo&longs;ito tamen motu na­turali accelerato, ad quem proximè accedit propter mutationem perpe­tuam lineæ. &longs;it enim inclinata &longs;ur&longs;um AB, quæ percurratur motu æquabili eo tempore, quo horizontalis AE, vel quo motu naturali LA; diuidatur AE bifariam in D; ducatur DG, tùm DC, AC, hæc e&longs;t linea mo­tus mixti ex inclinata AG, & horizontali AD; &longs;equitur deinde Parabola; nam &longs;ico tempore quo percurritur AD, percurritur AG, & LM vel FA; certè eodem percurritur AC, igitur &longs;ubduplo tempore percurrentur AN; igitur FO, quæ e&longs;t &longs;ubquadrupla FA; igitur a&longs;&longs;umatur NH æqualis FO, & CK æqualis FA, & ducatur curua per puncta AHK; hæc e&longs;t &longs;emiparabo­la, nam KI e&longs;t ad KE vt quadratum IH ad quadratum EA.

Vnde vides omnes inclinatas &longs;ur&longs;um v&longs;que ab horizontali DB ad verticalem DA inclu&longs;iuè e&longs;&longs;e Parabolas; omnes verò inclinatas ab ca­dem horizontali DB ad perpendicularem DC inclu&longs;iuè non e&longs;&longs;e Para­bolas, &longs;ed propiùs accedere ad rectam, vnde aliquis &longs;u&longs;picari po&longs;&longs;et e&longs;&longs;e Hyperbolas.

Theorema 105.

Si proijciatur mobile per inclinatam &longs;ur&longs;um vel deor&longs;um in partem oppo&longs;i­tam directionis nauis, &longs;cilicet per diagonales de&longs;cendit & a&longs;cendit per li­neam rectam, &longs;ur&longs;um vel deor&longs;um, v.g. &longs;it horizontalis KL, inclinata deor&longs;um KB, mixta crit KL; &longs;it etiam inclinata KL, & horizontalis CH; mixta erit KH, cui addatur in eadem KF portio &longs;patij, quod motu naturali percurritur; idem dico de aliis inclinatis.

Præterea &longs;it horizontalis VX, inclinata &longs;ursum VN; mixta erit VY; &longs;ic ex VOVX fiet VS detracta &longs;cilicet portioni &longs;patij, quod detrahitur à motu naturali; &longs;i verò &longs;it vel major motus horizontalis, vel minor eo, quem a&longs;&longs;ump&longs;unus, non percurrit mobile lineam rectam &longs;ed vel Para­bolam &longs;i &longs;ur&longs;um proiiciatur, vel &longs;i deor&longs;um aliam nouam, quam ad Hy­perbolam accedere &longs;uprà diximus.

Hinc certè, quod mirabile dictu e&longs;t, &longs;i è puncto nauis V &longs;ur&longs;um per inclinatam VO proiiciatur, &longs;tatimque po&longs;t proiectionem &longs;i&longs;tat nauis, in ip&longs;am nauim de&longs;cendet mobile; atque ita ex his habeo omnes motus cir­culi verticalis paralleli lineæ directionis; quare &longs;upere&longs;t vt explicemus alios motus; ac primò quidem per circulum horizontalem, cuius habeo quoque duas lineas, &longs;eilicet communes &longs;ectiones horizontalis & prio­ris verticalis, id e&longs;t lineam directionis ver&longs;us Boream, & oppo&longs;itam ver­&longs;us Au&longs;trum.

Theorema 106.

Si proijciatur mobile per horizontalem ver&longs;us Ortum è naui mobili, monebitur motu mixto ex duplici horizontali, & naturali deor&longs;um, &longs;it enim horizontalis ver&longs;us Boream AC, & alia horizontalis AH ver&longs;us ortum in eodem plano horizontali; certè ex vtraque fit mixta AK, quæ &longs;i percurratur æquali tempore cum AC, & cius &longs;ubdupla cum AB, AC verò æquali tempore cum AF; quamquàm &longs;uppono iam e&longs;&longs;e perpendi­cularem deor&longs;um AB; denique cum AG &longs;ubquadrupla AF a&longs;&longs;umatur ED æqualis AG perpendiculariter ducta in AD, & KL æqualis AF parallela ED, & per puncta AEL ducatur curua, hæc e&longs;t linea motus quæ&longs;ita; voluatur autom triangulum AKL, donec &longs;it parallelum circulo verticali vel alteri, ACO erit in proprio &longs;itu; vnde eo tempore, quo e&longs;­&longs;et in DE punctum nauis A e&longs;&longs;et in B, & co, quo e&longs;&longs;et in KL, punctum A e&longs;&longs;et in C; hoc e&longs;t &longs;ingula puncta AK, è regione AC ductis parallelis BD, CK, ac proinde nauis & mobile &longs;emper e&longs;&longs;ent è regione in linea ver&longs;us ortum.

Hinc &longs;i ex A dirigas &longs;agittam in H feris punctum K, quam artem probè no&longs;&longs;e debent rei tormentariæ præfecti; quippe &longs;agitta aberrabit à &longs;copo ver&longs;us Boream declinans toto eo &longs;patio, quod conficit nauis codem tem­pore, quo mouetur &longs;agitta; ita pror&longs;us &longs;i moueatur H ver&longs;us K, vt attin­gas ex puncto immobili A debes dirigere ictum in K, &longs;i quo tempore &longs;agitta conficit AK &longs;copus H percurrit HK.Idem pror&longs;us dicendum e&longs;t de iaculatione per lineam oppo&longs;itam ver&longs;us occa&longs;um.

Si verò proiiciatur mobile per lineam inter Boream, & Ortum, linea motus erit Parabola cuius Tangens erit mixta ex horizontali ver&longs;us Boream, & declinante ver&longs;us Ortum, v. g. &longs;it horizontalis vei&longs;us Boream AF, quam hactenus a&longs;&longs;ump&longs;i pro linea directionis; &longs;it linea ver&longs;us Ortum AC; &longs;it declinans ver&longs;us Boream AL; &longs;itque impetus AL, ad AE vt AL ad AE, quod hactenus &longs;uppo&longs;ui; &longs;it LG æqualis AE, AG e&longs;t mixta ex AE, AL; a&longs;&longs;umatur KI, & GH vt iam diximus; fiatque Parabola AIH, quæ circa axem AE ita voluatur, vt &longs;it perpendicularis plano horizontali LF.

Idem dico de omni alia declinante vel à Borea ad Ortum, vel ad Oc­ea&longs;um.

Theorema 107.

Si mobile proiiciatur per declinantem ab Austro ad Ortum, cuius impetus &longs;it vt linea; conficit lineam parabolicam, cuius tangens vel amplitudo e&longs;t re­sta ad Ortum; &longs;it enim NF ad Boream, NA ad Au&longs;trum, NI ad Or­tum, ND ad Occa&longs;um; &longs;it NL declinans ab au&longs;tro ad Ortum, &longs;itque im­petus per NL ad impetum per NF, vt NL ad NF; mixta ex NF NL e&longs;t HK; &longs;it autem KH æqualis &longs;patio, quod conficitur motu naturali eo tempore, quo percurritur NF, &longs;it KI æqualis NK, & IG quadrupla KH; Parabola NHG e&longs;t linea motus quæ&longs;ita dum voluatur NIG circa axem NI, dum IG pendeat perpendicularitur ex plano horizontali ON.

Idem fiet, &longs;i proiiciatur per declinantem NB ab Au&longs;tro &longs;cilicet ad Occa&longs;um.

Theorema 108.

Si mobile proiiciatur per inclinantem &longs;ur&longs;um in circulo verticali, cuius &longs;e­ctio cum horizontali tendit ad Ortum, conficit lineam parabolicam, cuius am­plitudo e&longs;t mixta ex horizontali ver&longs;us Boream, & horizontali ver&longs;us Ortum, &longs;it linea ver&longs;us Boream AB, ver&longs;us Ortum AK, mixta ex vtraque AF, linea inclinata &longs;ur&longs;um AP, Parabola AMN, quæ vertatur circa A do­nec incubet AFG, denique AFG circa FA voluatur, donec incubet perpendiculariter plano; porrò perinde e&longs;t, &longs;iue proiiciatur per inclina­tam &longs;ur&longs;um ver&longs;us Ortum, &longs;iue ver&longs;us Occa&longs;um.

Si verò proiiciatur per inclinatam deor&longs;um ver&longs;us Ortum, de&longs;cribit lineam, quæ non e&longs;t Parabola, &longs;ed propiùs accedit ad Hyperbolam, cuius tangens e&longs;t mixta ex inclinata deor&longs;um ex horizontali ver&longs;us Boream, &longs;it enim AC ver&longs;us Boream, AB ver&longs;us Ortum, AD inclinata deor­&longs;um &longs;ub horizontali AB, AG quæ e&longs;t in eodem plano cum AD DG, mixta ex AD, & AC; a&longs;&longs;umatur EF æqualis &longs;patio, quod conficitur motu naturali eo tempore, quo conficitur AE, & GH æqualis &longs;patio, quod conficitur motu naturali eo tempore, quo percurritur AG; duca­tur curua AFH, cuius &longs;itus vt habeatur &longs;it AB ver&longs;us Ortum, ex qua pendeat perpendiculariter deor&longs;um triangulum ABH, tùm circa axem AD voluatur triangulum ADH, donec HD &longs;it parallela horizonti; tùm circa axem AG voluatur triangulum AGH, dum GH &longs;it perpendicu­laris deor&longs;um, tunc enim linea motus AFH habebit proprium &longs;itum; idem fiet &longs;i proiiciatur per inclinatam deor&longs;um ver&longs;us Occa&longs;um.

Theorema 109.

Si proijciatur per inclinatam &longs;ur&longs;um, & declinantem ad Ortum, linea mo­tus erit Parabola, cuius amplitudo erit mixta ex declinante horizontali, & horizontali ver&longs;us Boream, &longs;it enim horizontalis ver&longs;us Boream AK, horizontalis ver&longs;us Ortum AR, declinans à Borea in Ortum AD, mixta ex AD, AK &longs;it AI, &longs;itque Rhomboides AE parallelus horizonti; &longs;it EG perpendicularis &longs;ur&longs;um, &longs;it HD parallela GE; differentia &longs;patij, quod acquiritur motu naturali eo tempore, quo percurritur AI, & FC, quæ &longs;it &longs;ubdupla EG. Dico lineam motus AHF e&longs;&longs;e parabolicam, quæ omnia con&longs;tant ex dictis; idemque dictum e&longs;to de omni alia inclinata &longs;ur&longs;um &longs;imul, & declinante, &longs;eu ver&longs;us Ortum &longs;eu ver&longs;us Occa&longs;um; porrò triangulum AEG incubat perpendiculariter plano horizontali ADEK; &longs;i verò proiiciatur per inclinatam deor&longs;um voluatur AKE, dum KO &longs;it perpendicularis deor&longs;um; &longs;it planum RK horizontale, voluatur AKE circa A, ita vt KO &longs;it &longs;emper perpendicularis deor&longs;um, donec AE &longs;ecet planum RK in AD &longs;int IO. & EA vt EF, GH in &longs;uperio­re figura, & per puncta AOM ducatur curua; hæc e&longs;t linea motus quæ&longs;ita.

Theorema 110.

Si proiiciatur per declinantem ab Austro ad Ortum & inclinatam &longs;ur&longs;um, de&longs;cribet Parabolam, cuius amplitudo erit mixta ex horizontali ver&longs;us Bo­ream & declinante horizontali ab Au&longs;tro ad Ortum &longs;it AF horizontalis ver&longs;us Boream, AG ver&longs;us Ortum, AI declinans ab Au&longs;tro ad Ortum, AG mixta ex AF AI AL inclinata, ANK Parabola; &longs;it enim planum FI horizontale cui triangulum ALI incubet perpendiculariter in &longs;e­ctione AG, reliqua &longs;unt facilia; idem dico de inclinata &longs;ur&longs;um &longs;imul, & declinante ab Au&longs;tro ad Occa&longs;um; &longs;i verò &longs;it inclinata deor&longs;um, &longs;it pla­num ACB horizontale, AB &longs;it declinans, AC &longs;it mixta ex AB & ho­rizontali ver&longs;us Boream AF; &longs;it AD inclinata deor&longs;um, fiatque cur­ua AQE more &longs;olito, ita vt triangulum ACE perpendiculariter deor&longs;um pendeat ex plano horizontali ACB, reliqua &longs;unt facilia.

Scholium.

Ob&longs;eruabis a&longs;&longs;umptam e&longs;&longs;e à me hactenus Parabolam, licèt accurate non &longs;int parabolicæ lineæ, quia proximè ad Parabolas accedunt; certè Phy&longs;icè loquendo & &longs;en&longs;ibiliter pro Parabolis a&longs;&longs;umi po&longs;&longs;e ni­nil vetat.

Corollaria.

Ex his colligis mirabilium motuum rationem. Primò mobile proje­ctum per lineam declinantem ab Ortu ferri po&longs;&longs;e rectà ad Ortum.

Secundò projectum per inclinatam deor&longs;um, ferri po&longs;&longs;e per ip&longs;am perpendicularem deor&longs;um.

Tertiò projectum per inclinatam fur&longs;um, ferri po&longs;&longs;e per verti­calem.

Quartò, rationem à priori habes, cur &longs;i ex equo vel &longs;puas, vel ali­quid demittas deor&longs;um, rectà perpendiculariter non cadat, &longs;ed &longs;emper è regione, quod maximè videre e&longs;t cum purgatur nauis mobilis, ciecta &longs;cilicet aquâ, quæ &longs;emper nauim in&longs;equi videtur, imò & cum quis pe­dem effert in naui hunc motum quoque ob&longs;eruat.

Quintò non erit etiam iniucundum inde elicere quomodo in maiore naui, di&longs;co ludere vel pila quis po&longs;&longs;it, licèt nauis motus nullo modo lu­dum impediat; quæ omnia ex iis, quæ diximus nece&longs;&longs;ariò con&longs;equuntur, & quæ manife&longs;tum probat experimentum.

Sextò, inde etiam eruuntur rationes motuum mixtorum ex pluribus motibus v.g.4.5.6.7.&c.in infinitum &longs;iue in eodem plano, &longs;iue in diuer­&longs;is; In diuer&longs;is vt hactenus explicuimus; in eodem vero &longs;iv.g.per BC, BE, BA &longs;imul imprimantur impetus eidem mobili qui &longs;int vt ip&longs;æ li­neæ; primò fiat ex BA BC mixta BD, & ex BD BE, mixta BF, vel ex BE BC mixta BG, & ex BG BA mixta BF, vel ex BE BA mixta BH, & ex BH BC mixta BF; vides &longs;emper e&longs;&longs;e camdem vltimam mixtam in diuer&longs;is planis; iam o&longs;tendimus e&longs;&longs;e plures &longs;uprà in naui mobili v.g. per planum verticale, horizontale, & inclinatum.

Septimò, &longs;i in naui mobili curreret equus, vel currus, e&longs;&longs;et motus mix­tus ex quatuor aliis, & &longs;i terra moueretur in naui mobili e&longs;&longs;ent quatuor motus, &longs;i ex ea aliquod mobile proiiceretur; inuenitur autem linea mix­ta in diuer&longs;is planis per quamdam planorum circuitionem, de qua &longs;uprà.

Octauò, po&longs;&longs;et facilè in eodem plano motus mixtus conflari ex qua­tuor aliis vel etiam pluribus, &longs;int enim quatuor in eodem plano AD AE. AF. AH. ex AD AE fit AB, ex AB, A fi fit AC, ex AC AH fit AG, quæ e&longs;t longior AC, & AC longior AB: po&longs;&longs;es etiam compo­nere ex AH AF, atque ita deinceps eodem ordine, & &longs;emper vltima linea erit AG, quod certè mirabile e&longs;t, & à Geometris demon&longs;trari pote&longs;t.

Nonò, ex his motibus mixtis educi po&longs;&longs;unt rationes multorum effe-ctuum naturalium, qui ob&longs;eruantur in rebus naturalibus, quales &longs;unt v.g. nubium, vaporum, ventorumque motus, qui &longs;æpè turbinatim procellas agunt, quorum turbinum ratio referri non debet, vt videbimus &longs;uo loco, in repercu&longs;&longs;ionem aliquam, quæ fiat à concauis montibus, qui longi&longs;&longs;i­mo interuallo &longs;æpiùs ab&longs;unt; &longs;ed potiùs petenda e&longs;t ab ip&longs;a mixti motus naturâ; quippè rara materies venti facilè recipit omnem impetum; ita­que ex prægnantibus &longs;æpè nubibus conferta tenui&longs;&longs;imorum halituum examina fractis qua&longs;i carceribus quacumque linea erumpunt; hinc infiniti propemodum motus, hinc turbines illi, &c. atque hæc de motu mixto ex pluribus rectis &longs;int &longs;atis.

LIBER QVINTVS,

DE MOTV IN DIVERSIS Planis.

HACTENVS con&longs;iderauimus motum in libe­ro medio; iam verò con&longs;iderabimus in planis durioribus, in quibus mobilè feratur vel &longs;ua &longs;ponte vel ab extrin&longs;eco impul&longs;um.

DEFINITIO 1.

PLanum inclinatum e&longs;t corpus durum læuigati&longs;&longs;imum, in quo mobile quod­piam moueri po&longs;&longs;it, quod nec &longs;it verticale &longs;ur&longs;um, nec perpendiculare deor­&longs;um, non addo, nec horizonti parallelum; quia planum rectilineum hori­zontale e&longs;t etiam decliue, vt &longs;uo loco videbimus.

Hypothe&longs;is 1.

Corpus graue per planum inclinatum de&longs;cendit, & quidem velociùs per illud planum, quod minùs recedit à perpendiculari, tardiùs verò per illud, quod plùs recedit.

Hypothe&longs;is 2.

Corpus graue in plano inclinato minùs grauitat, id e&longs;t faciliùs &longs;ustinetur, & tardiore motu de&longs;cendit, quàm in perpendiculari deor&longs;um.

Vtraque hypothe&longs;is certa e&longs;t, & de vtraque &longs;upponimus tantùm, quòd &longs;it, nam demon&longs;trabimus infrà propter quid &longs;it.

Axioma 1.

Corpus graue ideò tantùm mouetur &longs;ua &longs;ponte, vt deor&longs;um tendat: hoc Axioma con&longs;tat ex iis, quæ fusè demon&longs;traui &longs;ecundò lib. adde quoddeor&longs;um tendere, & corpus graue &longs;ua &longs;ponte moueri idem pror&longs;us &longs;onare videntur; nec enim loquor de potentiâ motrice animantium, vel de alia quacumque magneticâ, &longs;ed de potentiâ motrice grauium; graue autem illud appello, quod in medio rariore po&longs;itum deor&longs;um tendit, ni&longs;i impe­diatur, denique hîc fuppono dari motum naturalem grauium deor&longs;um quod demon&longs;tratum e&longs;t &longs;ecundo lib. & verò &longs;i tibi adhuc non fiat &longs;atis, probetur hoc Axioma per hypothe&longs;im primam; nam reuerâ &longs;uppono quòd omnibus experimentis comprobatur, &longs;cilicet corpus graue per pla­num Inclinatum deor&longs;um &longs;ua &longs;ponte de&longs;cendere, non verò a&longs;cendere ni&longs;i propter aliquam reflexionem.

Axioma 2.

Motus, qui impeditur, imminuitur, idque pro rata, & vici&longs;&longs;im impeditur qui imminuitur; cur enim imminueretur &longs;eu retardaretur, &longs;i nullum &longs;it impedimentum?

Axioma 3.

Omne quod impedit motum, debet e&longs;&longs;e applicatum mobili vei per &longs;e, vel per &longs;uam virtutem; hoc Axioma etiam certum e&longs;t.

Po&longs;tulatum.

Liceat accipere in perpendiculari deor&longs;um, parailelas, cum &longs;cilicet a&longs;&longs;umi­tr modica altitudo; licèt enim non &longs;int parallelç, quia tamen in&longs;en&longs;ibili interuallo ad &longs;e&longs;e inuicem accedunt, pro parallelis accipiuntur.

Theorema 1.

Impeditur motus corporis in plano inclinato; certum e&longs;t quod impedia­tur, quia tardiore motu de&longs;cendit mobile per hyp. 2. igitur impeditur per Axio.2.

Theorema 2.

Ideo impeditur, quia impeditur linea ad quam determinatus e&longs;t impetus innatus; cum &longs;it determinatus ad lineam perpendicularem deor&longs;um per Ax.1. cur enim potiùs ad vnam lineam quàm ad aliam? atqui id tan­tùm planum inclinatum efficit, vel impedit, ne deor&longs;um rectà tendere po&longs;&longs;it; igitur ex eo tantùm capite impedit.

Theorema 3.

Non totus impeditur motus in plano inclinato; quia &longs;i totus impediretur, nullus e&longs;&longs;et omninò motus &longs;uper eodem plano, &longs;ed per planum inclina­tum mobile deor&longs;um mouetur per hyp.1.igitur totus motus non impedi­tur; hinc ratio à priori primæ hypothe&longs;eos.

Theorema 4.

In ea proportione minùs mouetur, in quæ plùs impeditur; probatur per Axioma 2.cum enim motus imminuatur, quia impeditur per idem Axio­ma; certè quò plùs impeditur, plùs imminuitur; &longs;ed quò plùs imminui­tur, minor e&longs;t, ergo quò plùs impeditur, minor e&longs;t.

Theorema 5.

Eò plùs impeditur motus, quò maius &longs;patium conficiendum e&longs;t ad ac­quirendam eamdem altitudinem, &longs;eu di&longs;tantiam à centro, ill &longs;patio, quod conficitur in perpendiculari deor&longs;um; hoc Theor. vt clariùs demon&longs;tretur, aliquid figuræ tribuendum e&longs;t. &longs;it perpendicularis deor-&longs;um, AB, &longs;it planum inclinatum AE duplum AB; certè vbi mobile ex A peruenit in E per planum AE, di&longs;tat æquè à centro, ac &longs;i e&longs;&longs;et in B; &longs;up­pono enim perpendiculares omnes deor&longs;um e&longs;&longs;e parallelas per po&longs;tula­tum; igitur non acce&longs;&longs;it propiùs ad centrum confecto &longs;patio AE, quàm confecto AB; igitur impeditur in plano AE in ea proportione, in qua AB e&longs;t minor AE, nam haud dubiè AE e&longs;t maior AB, &longs;it autem dupla v.g. igitur impeditur non quidem totus motus &longs;ed &longs;ubduplus; in plano verò AD impeditur iuxta cam proportionem in qua AB e&longs;t minor AD, nec enim aliunde pote&longs;t impediri, cum &longs;cilicet impediatur tantùm, quia im­peditur lineaad quam ab ip&longs;a natura determinatus e&longs;t per Th.2. v. g.li­nea deor&longs;um AB; quippè lineæ comparantur inter &longs;e v.g. AE cum AB, nam impedimentum lineæ AE in eo tantùm po&longs;itum e&longs;t, quòd difficiliùs per illam quàm per AB ad centrum feratur mobile, quod certum e&longs;t, cum imperimentum petatur a difficultate; atqui difficultas motus, qui fit per lineam AE in eo tantùm e&longs;t, quòd &longs;it maius &longs;patium conficiendum, igi­tur quò maius &longs;patium e&longs;t, maior difficuitas e&longs;t; igitur quò maior linea e&longs;t, maius impedimentum e&longs;t.

Adde quod vel impedimenti proportio petitur ab angulis vel à Tan­gentibus, vel à &longs;ecantibus; nihil enim aliud ade&longs;&longs;e pote&longs;t; igitur per Ax. 3. pote&longs;t tantùm impediri ab his; &longs;ed proportio impedimenti non pote&longs;t e&longs;&longs;e ab angulis; quod probatur primò, quia &longs;i ego quæram à te in qua proportione motus per AE e&longs;t tardior motu per AB; dices in ea, in qua angulus EAB e&longs;t maior nullo angulo, quod e&longs;t ridiculum: Equidem di­ceres motum per AD e&longs;&longs;e velociorem motu per AE in ea proportione, in qua angulus EAB e&longs;t maior angulo BAD, quod tamen fal&longs;um e&longs;t; e&longs;&longs;et enim ferè duplò maior, quod repugnat experimentis omnibus; at &longs;i accipiam angulum BA, qui &longs;it tantùm vnius gradus &longs;eu minuti, &longs;itque EAB angu­lus 60. grad. &longs;i velocitas motus per AI e&longs;&longs;et ad velocitatem motus per AE vt angulus EAB ad angulum BAI, motus per AI e&longs;&longs;et &longs;exagecuplò velocior, quàm per AE, quod e&longs;t ab&longs;urdum: Diceret fortè aliquis in to­to angulo 90. GAB di&longs;tribui huius impedimenti motum v.g. &longs;i angulus BAI &longs;it 1.grad. motus per AI amittit tantùm (1/90) &longs;ui motus; &longs;i angulus D AB circiter 40.grad. motus per AD amittit tantùm (40/90), & per AE (60/90); cum &longs;it angulus BAE 60. grad. igitur motus per AB e&longs;t ad motum per AE vt 3.ad 1. quod omnibus experimentis repugnat.

Secundò probatur, quia &longs;i fiat inclinata proximè accedens ad AG v. g.4′.& a&longs;&longs;umatur alia accedens 3′. differentia anguli erit tantùm 2′. cum tamen differentia longitudinis plani &longs;eu &longs;ecantis huius, & illius, &longs;it ma­xima, vt con&longs;tat ex canone &longs;inuum, igitur non imminueretur motus in plano inclinato ratione impedimenti contra Th.4. quis enim neget e&longs;&longs;e maximum impedimentum motus tantum &longs;patium, quod conficiend e&longs;t.

Tertiò, omnia experimenta con&longs;entiunt huic Theoremati, & repu­gnant huic propo&longs;itioni quæ petitur ab angulis; adde quod angulus ni­hil pror&longs;us facit ad motum, &longs;ed linea feu &longs;patium; denique hoc ip&longs;um e&longs;t quod ab omnibus Mechanicis vulgò &longs;upponitur perinde qua&longs;i prima notio, quæ tamen aliquâ demon&longs;tratione indig

Equidem explicari pote&longs;t hæc demon&longs;tratio operâ libræ; &longs;it enim libra CG cuius centrum immobile e&longs;t A; &longs;it autem diameter libræ CG, pondus in C &longs;e habet ad pondus in D, tran&longs;lata &longs;cilicet diametro in DH vt CA, ad BA; igitur pondus in D grauitaret minùs in planum inclina­tum DA, quàm in horizontali CAI; nam pondus in D idem præ&longs;tat, quod præ&longs;taret appen&longs;um in D fune DE; igitur grauitatio in C e&longs;t ad grauita­tionem in D, vt CA, vel DA ad BA; &longs;ed quâ proportione decre&longs;cit graui­tatio in planum, cre&longs;cit motus in plano inclinato, quia minùs impeditur per Th.4. igitur in perpendiculari ea nulla e&longs;t gtauitatio in planum; nec impeditur vllo modo motus, igitur ab E ver&longs;us C ita impeditur motus, vt AC ver&longs;us C impeditur grauitatio in planum, &longs;ed impeditur grauitatio in D v.g. in ratione totius CA ad EA, vel DA ad DI; igitur impeditur motus in eadem proportione v.g. in plano DA ad DB vel AI, igitur in ratione plani inclinati ad perpendicularem.

Hæc omnia veri&longs;&longs;ima &longs;unt; &longs;upere&longs;t tamen vt &longs;ciatur ratio phy&longs;ica cur pondus in D æquiualeat ponderi in B quod &longs;upponunt quidem omnes Mechanici, & omnibus experimentis congruit: Equidem pondus pendu­lum ex D fune DB, vel longiore, e&longs;t eiu&longs;dem momenti, cuius e&longs;t affixum in D, ita vt linea directionis, quæ ducitur ab eius centro re&longs;pondeat fu­ni DB; vnde rectè concluditur ab Archimede idem pondus affixum bra­chio BA eiu&longs;dem e&longs;&longs;e momenti cum pendulo DB, vel affixo puncto D, quod certè veri&longs;&longs;unum e&longs;t, nondum tamen rationem phy&longs;icam video; verum quidem e&longs;t idem pondus pendulum fune DB minoris e&longs;&longs;e momenti, quàm &longs;i e&longs;&longs;et affixum puncto C; nam &longs;uppono CG e&longs;&longs;e libram in &longs;itu horizontali; tum quia pondus illud DB trahit deor&longs;um extremum libræ D per arcum DC longo circuitu, maximè declinante à &longs;ua linea directionis DB; tùm quia ex hoc &longs;equitur nece&longs;&longs;ariò pondus B deflecti à &longs;ua perpendiculari curua linea; tùm quia linea DA, quæ rigida &longs;uppo­nitur, re&longs;i&longs;tit motui DB & patet; in qua verò proportione, dictum e&longs;t certè hactenus, &longs;ed phy&longs;icè non demon&longs;tratum.

Pater Mer&longs;ennus multis locis ex docti&longs;&longs;imo Roberuallo demon&longs;trat rem i&longs;tam ingenio&longs;i&longs;&longs;imè; &longs;it enim circulus centro R; &longs;int vectes æqua­les BF horizonti, DN perpendiculari paralleli; tùm CL, FO, æqualiter inclinati, ducantur CO EL; haud dubiè &longs;i pondera C & L &longs;int æqualia erit æquilibrium; quod certum e&longs;t, & demon&longs;trabimus cum de libra; e&longs;t enim quarta propo&longs;itio Vbaldi de libra; &longs;ed pondus in O pendulum &longs;ci­licet filo CO e&longs;t eiu&longs;dem momenti, cuius e&longs;t pondus in P; igitur pon­dus in P æquale ponderi O &longs;u&longs;tineret pondus ML, &longs;ed pondus in P e&longs;t ad pondus in B vel in F, ad hoc, vt &longs;it æquiblirium, RF ad R P; igitur pondus in A vel in R, quod erit ad pondus in L, vt P ad R L, &longs;u&longs;tinebit pondus in L; &longs;ed &longs;i applicetur potentia in C quæ trahat per tangentem CT, faciet idem momentum quod faceret in B trahens per tangentem BA; at vicem illius potentiæ gerit pondus B vel A, quod gra­uitat per BA; igitur potentia applicata C per CT, æqualis ponderi A retineret pondus in L; ducatur autem KLG Tangens parallela CT; certè eadem potentia in L per LG retinebit pondus in L; quæ idem retine­ret applicata in C per CT; cum enim RC & RL &longs;int æquales &longs;i &longs;int ap­plicatæ duæ potentiæ æquales in C quidem per CT, & in L per LG; haud dubiè erit perfectum æquilibrium; igitur &longs;i pondus A pendeat in H fune LGH, retinebit pondus L in plano inclinato GLK; e&longs;t autem pondus H ad pondus LN SR ad RL; &longs;ed triangula RSL, & GKI &longs;unt proportionalia; igitur pondus in H e&longs;t ad pondus L, vt GI ad G K; igitur &longs;i vires, quæ retinent pondus in plano inclinato GK &longs;unt ad vi­res, quæ retinent pondus in perpendiculari GI, vt GI ad GK; igitur im­petus &longs;eu motus mobilis in plano GK e&longs;t ad impetum, &longs;eu motum eiu&longs;­dem in perpendiculo GI, vt GI ad GK.

Hæc omnia veri&longs;&longs;ima &longs;unt, &longs;emper tamen de&longs;iderari videtur ratio phy­&longs;ica, cur idem pondus pendulum ex C in O, &longs;it ciu&longs;dem momenti cum pondere affixo puncto P, &longs;eu brachio libræ horizontalis PS. quod certè Mechanica Axiomatis, vel hypothe&longs;eos loco iure a&longs;&longs;umere pote&longs;t; at ve­rò phy&longs;ica non &longs;atis habet de re cogno&longs;cere quod &longs;it, ni&longs;i &longs;ciat propter quid &longs;it; igitur nos aliquam afferre conabimur. Suppono tantùm tunc e&longs;&longs;e æquilibrium perfectum duorum ponderum æqualium cum vtrimque æqualia illa pondera ita &longs;unt appen&longs;a, vt linea directionis vnius æqua­lis &longs;it lineæ directionis alterius, cur enim alterum præualeret &longs;i &longs;int æ­qualia? hoc po&longs;ito.

Dico pondus affixum P æquale ponderi L facere aquilibrium; cum enim linea directionis &longs;it PO, &longs;i de&longs;cenderet liberè per PO. L eodem tempore attolleretur per LS, quod certè applicatis planis SL PO facilè fieri po&longs;&longs;et; &longs;ed eodem modo P grauitat, quo &longs;i de&longs;cenderet per PO; e&longs;t enim eius linea directionis; atqui tunc faceret æquilibrium, quod o&longs;ten­do; æquale &longs;patium conficeret L, per LS a&longs;cendendo, quod P per PO de&longs;cendendo; igitur &longs;i attolleret L in S, &longs;imiliter pondus L æquale P in S attolleret pondus P ex O in P, igitur neutrum præualere pote&longs;t; &longs;ed quia hæc fu&longs;iùs explicabimus cum de libra, nunc tantùm indica&longs;&longs;e &longs;ufficiat.

Supere&longs;t vt breuiter o&longs;tendamus accipi non po&longs;&longs;e hanc proportio­nem imminutionis motus in plano inclinato à Tangente BE tùm quia; iam à &longs;ecante accipi o&longs;tendimus, tùm quia &longs;it Tangens BD æqualis &longs;umi toti &longs;eu perpendiculari AB; &longs;equeretur motum per AD æqualem e&longs;&longs;e motui per AB; Equidem in maxima di&longs;tantia accedit Tangens ad &longs;ecantem; igitur eò plùs impeditur motus, quò maius &longs;patium conficien­dum e&longs;t, &c.

Theorema 6.

Ex hoc &longs;equitur nece&longs;&longs;ariò motum in plano inclinato e&longs;&longs;e ad motum in per­pendiculari, vt ip&longs;a perpendicularis ad ip&longs;um planum inclinatum, v.g. velo­citas motus per AE e&longs;t ad velocitatem motus per AB, vt ip&longs;a AB e&longs;t ad ip&longs;am AE, &longs;it enim AE dupla AB, velocitas per AB e&longs;t dupla veloci­atis per AE.

Ob&longs;erua quæ&longs;o, cum dico motum in plano inclinato e&longs;&longs;e ad motum in perpendiculo, vt ip&longs;æ lineæ permutando, ita intelligendum e&longs;&longs;e, vt vel a&longs;&longs;umatur motus in &longs;ingulis in&longs;tantibus, ita vt eo inftanti, quo datum &longs;patium in inclinata acquiritur, acquiratur duplum in perpendiculo; quo po&longs;ito valet certè tantùm illa proportio ratione motus æquabilis, &longs;i &longs;er­uari debet; nam perinde &longs;e habet phy&longs;icè, atque &longs;i e&longs;&longs;et, vt iam fusè ex­plicatum e&longs;t lib.2. in re &longs;imili.

Theorema 7.

Hinc de&longs;cendit mobile per &longs;e in plano inclinato; ratio e&longs;t, quia totus mo­tus non impeditur, cum &longs;it eadem proportio, quæ e&longs;t perpendicularis ad inclinatam; dixi per &longs;e, nam per accidens in plano &longs;cabro tantillùm inclinato mobile de&longs;cendit, adde quod corpus graue tamdiu mouetur quandiu accedere pote&longs;t ad centrum terræ.

Theorema 8.

Motus in infinitum imminui pote&longs;t, probatur, quia proportio perpen­dicularis ad inclinatam pote&longs;t e&longs;&longs;e minor in infinitum, quia inclinata pote&longs;t e&longs;&longs;e longior, & in infinitum.

Theorema 9.

Ex his ver a redditur ratio cur in plano inclinato ad angulum BG motus &longs;it &longs;ubduplus illius qui fit in perpendiculari; v.g. &longs;it angulus BAE 60. certè AE e&longs;t dupla AB, &longs;ed motus in AB e&longs;t ad motum in AE vt AE ad AB per Th.6. igitur e&longs;t duplus.

Ex his reiicies quoque Cardanum, & alios quo&longs;dam, qui diuer&longs;am proportionem motuum in planis inclinatis deducunt ex diuerfis angu­lis inclinationis; iuxta quam proportionem motus in AE e&longs;&longs;et &longs;ubtri­plus in AB contra experimentum.

Theorema 10.

Motus acceleratur in plano inclinato; experientia clari&longs;&longs;ima e&longs;t, ratio eadem cum illa, quam adduximus lib.3. cum de motu naturali, quia &longs;ci­licet prior impetus con&longs;eruatur, & acquiritur nouus, Imò acceleratur iuxta eamdem proportionem, vel no&longs;tram &longs;ingulis in&longs;tantibus, vel Gali­lei in partibus temporum &longs;en&longs;ibilibus; vnde a&longs;&longs;umemus deinceps i&longs;tam Galilei proportionem, quia &longs;cilicet partes temporis &longs;en&longs;ibiles tantùm a&longs;&longs;umere po&longs;&longs;umus.

Theorema 11.

In plano inclinato e&longs;t idem impetus innatus qui est in perpendiculari, &longs;ed in hac habet totum &longs;uum motum, non verò in illa, quia impeditur, ni&longs;i enim totus e&longs;&longs;et, non grauitaret corpus illud in planum inclinatum; quippe &longs;uas omnes vires impetus ille exereret circa motum; igitur ali­quid illarum exerit circa motum aliquid circa planum, in quod ex parte grauitat; igitur idem e&longs;t impetus innatus, adde quod ille e&longs;t in&longs;epa­tabilis.

Theorema 12.

Impetus naturalis aduentitius productus à corpore graui in plano inclinato e&longs;t minor eo, qui producitur in perpendiculari; probatur, quia e&longs;t minor motus, igitur minor impetus, vt &longs;æpè diximus; &longs;ecundò (hæc e&longs;t ratio à priori;) quia cum ideo producatur impetus i&longs;te aduentitius, vt motus acceleretur; certè debet re&longs;pondere motui, qui competit impetui innati; &longs;i enim nullum habet motum, nullus accedit de nouo impetus, è con­tra verò &longs;i e&longs;t motus, &longs;ed maior, &longs;i maior e&longs;t motus, & minor &longs;i e&longs;t minor; quia hic impetus tantùm e&longs;t propter motum.

Theorema 13.

Impetus qui producitur in acceleratio motus habet totum motum quem exigit (præ&longs;cindendo à re&longs;i&longs;tentia medij); nec enim per illum mobile graui­tat in planum; alioquin cre&longs;ceret &longs;emper grauitatio; igitur totus exerce­tur circa motum; ratio e&longs;t quia hic impetus addititius non e&longs;t in&longs;titutus propter grauitationem, &longs;ed tantùm propter motum: adde quod ad om­nem lineam determinari pote&longs;t, &longs;ecùs verò naturalis &longs;altem om­ninò.

Theorema 14.

Imminuitur motu illo granitatio corporis in planum; ratio e&longs;t primò; quia quò velociùs mouetur in plano, breuiori tempore &longs;ingulis partibus in­cumbit: &longs;ecundò quia motu illo accelerato qua&longs;i di&longs;trahitur mobile ab illa linea grauitationis in planum; hinc mobile celeri motu moueretur in plano illo inclinato, quod eiu&longs;dem &longs;ub&longs;i&longs;tentis grauitationi & ponde­ri vltrò cederet.

Theorema 15.

Impetus innatus ex &longs;e e&longs;t &longs;emper determinatus ad lineam perpendicularem deor&longs;um; quia grauitas tendit ad commune centrum, vt videbimus tra­ctatu &longs;equenti; tamen ratione plani qua&longs;i detorquetur ad lineam plani ad quam tamen omninò non determinatur, alioquin non grauitaret in planum: vnde dixi, detorquetur &longs;eu qua&longs;i diuiditur, perinde qua&longs;i e&longs;&longs;et duplex impetus, quorum alter per lineam perpendicularem deor&longs;um e&longs;&longs;et determinatus, in quo non e&longs;t difficultas; impetus tamen aduenti­tius determinatur omninò ad lineam plani.

Scholium.

Dubitari pote&longs;t an grauitatio in planum inclinatum &longs;it vt re&longs;iduum plani, cui detrahitur perpendiculum v.g. &longs;it planum inclinatum CD ad angulum ACD 60. potentia quæ &longs;u&longs;tinet pondus B per EB e&longs;t ad præ­dictum pondus vt CA ad CD; detrahitur CA ex CD, &longs;upere&longs;t FD æqua­lis &longs;cilicet CA; an fortè grauitatio ponderis B in planum inclinatum C D e&longs;t ad grauitationem eiu&longs;dem in planum horizontale; quæ e&longs;t graui­tatio tota, id e&longs;t nihil imminuta vt DF ad DC; attollatur enim totum triangulum CAD in eadem &longs;itu altera manu, & altera filo EB paralle-lo CF, retineatur pondus B ne &longs;cilicet deor&longs;um cadat; tùm &longs;ubtrahatur pondus trianguli CAD; nunquid fortè altera manus &longs;u&longs;tinebit tantùm &longs;ubduplum ponderis B? & altera &longs;ubduplum? igitur vt habeatur quod &longs;u&longs;tinet &longs;uppo&longs;ita dextra v.g. debet &longs;ub&longs;trahi, quod &longs;u&longs;tinet &longs;ini&longs;tra, &longs;ed quod &longs;u&longs;tinet &longs;ini&longs;tra, e&longs;t vt ip&longs;a potentia, id e&longs;t vt CA ad CD; igitur tota CD repræ&longs;entat totum pondus, &longs;egmentum CF partem ponderis quæ competit potentiæ E, FD verò partem quæ fu&longs;tinetur à pla­no CF.

Hinc facilè po&longs;&longs;et determinari quota pars ponderis incubet plano,&longs;it enim planum inclinatum AC, perpendiculum AB, accipiatur AB æqualis AB, &longs;itque AC tripla AB, duæ tertiæ ponderis incubant plano &longs;i verò &longs;it horizontale planum, totum pondus grauitat in illud; nulla e&longs;t enim perpendicularis, &longs;i &longs;it perpendiculare planum, nihil pror&longs;us gra­uitat; quia nulla e&longs;t inclinata, & quò propiùs accedit planum inclina­tum ad horizontalem plùs grauitat pondus in illud, minùs verò; quò propiùs accedit ad perpendicularem.

Hinc e&longs;&longs;et oppo&longs;ita ratio grauitationis, & motus, in plano inclinato; nam quò plùs e&longs;t grauitationis minùs e&longs;t motus, quò plùs motus, minùs grauitationis; quando verò planum inclinatum e&longs;t duplum perpendicu­culi vt planum CFD, tunc tantumdem detrahitur de grauitatione in planum quantùm de motu in eodem plano; ide&longs;t vtrique &longs;ubduplum, &longs;i verò vt in plano ADC perpendiculum e&longs;t &longs;ubtriplum plani, detrahun­tur de motu 2/3 & de grauitatione 1/3, idem dico de aliis, quæ certè omnia ex veris principiis phy&longs;icis con&longs;equi videntur, quò enim plus grauitat mobile in planum, plùs &longs;u&longs;tinetur; quò plùs &longs;u&longs;tinetur, plùs impeditur il­lius motus; &longs;ed hoc repugnat communi Mechanicorum &longs;ententiæ, qui cen&longs;ent grauitationem in planum inclinatum e&longs;&longs;e ad grauitationem in horizontale, vt Tangens e&longs;t ad &longs;ecantem, quæ &longs;it linea plani inclinati, v.g. vt AB ad CD, quod certè omnes &longs;upponunt, &longs;ed minimè demon­&longs;trant, &longs;i quid video &longs;altem phy&longs;icè; nec enim illud nemon&longs;trant propriè ex eo quòd pondus in extremitate libræ affixum habeat diuer&longs;a momenta iuxta rationem Tangentium ad &longs;ecantes, v.g. in &longs;ecunda figura Th.5. pondus in D e&longs;t ad pondus in C vt BA ad DA, quod veri&longs;&longs;imum e&longs;t, & &longs;uprà demon&longs;trauimus; quippe hoc pertinet ad rationem momenti, non verò grauitationis in planum; adde quod affixum e&longs;t pondus vecti; igi­tur vectis &longs;u&longs;tinet totum illius pondus; vtrùm verò &longs;i pondus in plano inclinato veluti in vecte moueatur pondus quo grauitat in planum &longs;it ad pondus quo grauita in horizontali vt Tangens ad &longs;ecantem, certè non demon&longs;trant; attamen ita res pror&longs;us &longs;e haber; quare fit.

Theorema 16.

Grauitatio ponderis in planum inclinatum e&longs;t ad grauit at tonem eiu&longs;dem in planum horizontale, vt Tangens, vel herizontalis ad &longs;ecantem, vel incli­natam, quod demon&longs;tro. Primò &longs;ir planum inclinatum GD, pondus in-eubans F; dico grauitationem ponderis F in inclinatam GD e&longs;&longs;e ad gra­uitationem in horizontalem CD vt CD ad GD; quia pondus F pellit planum per lineam FE &longs;eu GB Tangentem; quia determinari non po­te&longs;t &longs;eu percu&longs;&longs;io, &longs;eu impre&longs;&longs;io ex alio capite quàm ex linea ducta à centro grauitatis perpendiculariter in planum, vt demon&longs;trauimus in Th. 120. l. 1. atqui libræ extremitas G initio de&longs;cendit per Tangen­tem GB, id e&longs;t per minimum arcum, qui ferè concurrit cum Tangente&longs;ed ideò de&longs;cendit in AB, quia pellitur deor&longs;um à pondere; igitur men­&longs;ura grauitationis e&longs;t de&longs;cen&longs;us libræ, &longs;ed libra faciliùs de&longs;cendit ex A deor&longs;um quàm ex G in proportione AD ad CD vel GD ad CD; igitur grauitatio ponderis in A e&longs;t ad grauitationem ciu&longs;dem in G, vt GD ad CD; quia rationes cau&longs;arum &longs;unt eædem cum rationibus effectuum.

Præterea &longs;it planum inclinatum GD, &longs;it IF parallela GD; &longs;int IK, I M & quadrans KFR; punctum I &longs;it centrum libræ immobile; certè &longs;i &longs;it alterum brachium libræ æquale IF in&longs;tructum æquali pondere F, erit æ­quilibrium; &longs;ed pondus illud in F e&longs;t ad idem in R, vt IM ad IF, &longs;eu vt CD ad GD, quod erat dem.

Scholium.

Ob&longs;eruabis po&longs;&longs;e facilè ex dictis explicari diuer&longs;as potentias applica­tas ponderi F in eodem plano GD, primò &longs;i accipiatur IHF parallela GH cum centro immobili I pondus retinebitur, &longs;i potentia in I &longs;it ad globum vt GC ad GD, vt demon&longs;tratum e&longs;t; &longs;i verò pellat potentia per lineam IF, globus de&longs;cendet, vt patet.

Hinc &longs;ecundò &longs;u&longs;tinens MF totum pondus F &longs;u&longs;tinet, patet, quia &longs;i­ue planum inclinatum pondus ip&longs;um tangat, &longs;iue perpendiculare, totum &longs;u&longs;tinet pondus; &longs;ub&longs;tracto enim plano pondus immobile manet, adde quod non pote&longs;t pondus F &longs;u&longs;tineri in brachio IM, ni&longs;i æquale pondus ex æquali brachio oppo&longs;ito pendeat.

Tertiò ex puncto T lineâ TFE non pote&longs;t &longs;u&longs;tineri pondus licèt po­tentia in T e&longs;&longs;et infinita, quia ex TE de&longs;cendet in TV, patet; idem dico de omnibus aliis lineis ductis ab F ad aliquod punctum inter TM.

Quartò ex puncto X linea XF &longs;u&longs;tinebitur pondus dum potentia ap­plicetur in X, maior quidem potentia applicata in I, &longs;ed minor applica­ta in M; nam potentia M e&longs;t ad potentiam I vt IF ad MF; igitur poten­tia X e&longs;t ad potentiam M vt MF ad XF; ad potentiam verò I vt IF ad XF.

Quintò, cùm triangula IF M.HF 4. &longs;int proportionalia, potentia M e&longs;t ad potentiam I vt HF ad 4. F.

Sextò, &longs;i applicetur potentia, vel in T pellendo per lineam TFE, quæ cadit perpendiculariter in planum GD, vel &longs;i applicetur in A per lineam AE trahendo, non poterit retineri globus, quæcunque tandem poten­tia applicetur; quia &longs;emper per GD globus rotari poterit nullo cor­pore impediente; &longs;uppono enim tùm planum tùm globum e&longs;&longs;e perfectè politum, quo tamen nobis ce&longs;&longs;e certum e&longs;t ad experimentum, &longs;uppo­no nullam e&longs;&longs;e partium compre&longs;&longs;ionem, qua vna pars in aliam qua&longs;i pe­netret; &longs;i enim totus locus datur ad de&longs;cen&longs;um; certè non e&longs;t vlla ratio propter quam non de&longs;cendat; nec dicas affigi plano GD ab ip&longs;a vi ex­teriùs affigente; quia nullo modo impeditur motus, per datam lineam, ni&longs;i vel aliquod corpus opponatur, vel alius impetus detrahat ab eadem linea; atqui nihil horum prorsùs e&longs;t in hoc ca&longs;u.

Si potentia applicetur in N per lineam NF, maior e&longs;&longs;e debet quàm in I, &longs;ed minor quàm in A; e&longs;t autem ad potentiam in I vt IF ad NF; quippe re&longs;i&longs;tit planum GD huic potentiæ in N, non tamen re&longs;i&longs;tit in I; igitur illa maior e&longs;&longs;e debet, quod autem potentia in N &longs;it ad potentiam in I, vt IF ad NF (po&longs;ito &longs;cilicet quod vtraque pondus E &longs;u&longs;tineat) plùs quàm certum e&longs;t; quia cùm pondus po&longs;&longs;it tantùm moueri per EG &longs;eu per lineam FI potentia NF trahit per FN; igitur potentia in N &longs;u&longs;tinens pondus F e&longs;t ad potentiam in I &longs;u&longs;tinentem idem pondus, vt IF ad NF; &longs;imiliter potentia in K &longs;u&longs;tinens idem pondus F e&longs;t ad potentiam in I vt IF ad ZF, nam IZ e&longs;t perpendicularis in KF, donec tandem potentia &longs;it in A applicata per AF in quam IF cadit perpendiculariter, igitur po­tentia in A debet e&longs;&longs;e infinita.

Octauò, &longs;i pellatur pondus F per omnes lineas contentas &longs;ini&longs;tror&longs;um inter FT & FA deor&longs;um faciliùs cadet; &longs;i verò trahatur per lineas con­tentas inter TF & FA dextror&longs;um, etiam deor&longs;um cadit; quia perinde e&longs;t &longs;iue trahatur per lineam IF, &longs;iue pellatur æquali ni&longs;u per lineam VF quæ concurrit cum FI; & perinde e&longs;t &longs;iue pellatur per IF, &longs;iue trahatur per FV; idem dictum &longs;it de omnibus aliis lineis, quæ per centrum F hinc inde ducuntur.

Vnum e&longs;t, quod de&longs;iderari videtur ex quo reliqua ferè omnia depen­dent, quomodo &longs;cilicet potentia in N trahens per FN &longs;it ad potentiam in I trahentem per FI vt FI e&longs;t ad FN, quod &longs;ic breuiter demon&longs;tro: &longs;it horizontalis BD, & triangulum ECD; ex centro D ducatur arcus BE, qui &longs;it v.g. 30.grad. vt CE &longs;it &longs;ubdupla ED; certè potentia in B e&longs;t ad potentiam in E per EC vt BD, vel ED ad CD; &longs;ed potentia in E per EA Tangentem e&longs;t æqualis potentiæ in B; &longs;it autem planum EA, & connectatur AC; triangula AEC & ECD &longs;unt proportionalia; igitur &longs;it AC verticalis, EC horizontalis, & AE inclinata; &longs;it potentia in A per AE trahens pondus E; &longs;it potentia C trahens per CE; dico quod impeditur tractio toto angulo AEC, &longs;icut ante impediebatur grauitatio toto angulo AEC; igitur vtrobique e&longs;t æquale impedimentum; &longs;ed in primo ca&longs;u ratione impedimenti ita &longs;e habet potentia in E per EA ad potentiam in E per EC, vt ED ad CD, vel vt EA ad EC; igitur in &longs;e­cundo in quo e&longs;t idem impedimentum potentia in A per EA e&longs;t ad po­tentiam in C per EC, vt ip&longs;a inclina AE ad EC.

Nonò denique ob&longs;eruabis, egregium e&longs;&longs;e apud Mer&longs;ennum tractatum authore docti&longs;&longs;imo Roberuallo &longs;uper hac tota re, in quo certè Geome-tria nihil de&longs;iderare pote&longs;t; licèt phy&longs;ica fortè aliquid de&longs;iderare po&longs;&longs;it; adde quod implicatior illa figura infinitis ferè contexta lineis, quam ha­bet, equidem erudito Geometræ faciet &longs;atis, non tamen rudiori Tyroni, qui vix in hoc labyrintho tutum &longs;e e&longs;&longs;e putabit.

Theorema 17.

Si globus incumbat plano inclinato rotatur nece&longs;&longs;ariò deor&longs;um; &longs;it enim globus F in plano ED; ducatur FH perpendicularis deor&longs;um; hæc e&longs;t linea directionis centri grauitatis, vt con&longs;tat; igitur cùm non &longs;u&longs;tinea­tur in prædicta linea, nec enim terminatur ad punctum contactus G, cer­tè debet rotari; adde quod non e&longs;t in æquilibrio, vt patet, ratio autem inæqualitatis e&longs;t vt GF ad FN, nec vlla e&longs;t difficultas; igitur duplici qua&longs;i motu de&longs;cendet in prædicto plano ille globus, &longs;cilicet motu centri propter inclinationem plani, & motu orbis, tùm quia non e&longs;t in æqui­ibrio, tùm quia in linea directionis FH non &longs;u&longs;tinetur à plano.

Theorema 18.

Si corpus aliquod incumbat plano inclinato, &longs;ique linea directionis centri grauitatis &longs;ecet ip&longs;um planum intra ba&longs;im corpus repit quidem in prædicto plano &longs;ed non rotatur, &longs;i verò cadat extra ba&longs;im rotatur, non repit; &longs;it enim planum inclinatum BC, cui incubet cubus DL, cuius cen­trum grauitatis &longs;it I; ducatur RG perpendicularis dium per cen­trum grauitatis I cadit in punctum G intra ba&longs;im BG; igitur non ro­tabitur, &longs;ed repet; quia &longs;i &longs;u&longs;tinetur in G remoto &longs;en&longs;im plano BC; haud dubiè portio GD non præponderat portioni GL, vt patet ex libra.

Sit quoque parallelipedum EK, centrum grauitatis N, perpendicu­laris ducta per centrum HNM cadit intra ba&longs;im; igitur non rotabi­tur, quia &longs;ubmoto plano BC non &longs;u&longs;tinetur quidem in M, &longs;ed minimè inclinabitur dextror&longs;um; igitur non rotabitur. Si verò cadat extra ba­&longs;im haud dubiè rotabitur, &longs;it enim planum inclinatum AC, cui in­cumbat parallelipedum FN, cuius centrum grauitatis &longs;it L; ducatur L perpendicularis, cadit in E extra ba&longs;im FD; certè latus DN inclinabi­tur deor&longs;um; igitur rotabitur, quia eodem modo &longs;e habet, quo &longs;e ha­beret, &longs;i &longs;ubmoto plano &longs;u&longs;tineretur in linea DX, &longs;ed trapezus DX PN triangulo FXD præponderat per regulas libræ, de quibus &longs;uo loco.

Ob&longs;eruabis autem primò &longs;ciri po&longs;&longs;e data plani inclinatione & ba&longs;i parallelipedi maximam illius altitudinem, qua po&longs;ita non rotetur; &longs;ecus verò po&longs;ita cunque alia maiore; &longs;it enim planum AC, ba­&longs;is parallelipedi FD; erigantur FO, DN perpendiculares in AC; tùm erigatur perpendicularis DX parallela AB; connectantur R M: dico FX e&longs;&longs;e maximam altitudinem, vt con&longs;tat ex dictis.

Secundò, quotie&longs;cunque rectangulum, ita e&longs;t &longs;itum, vt eius diagonalis &longs;it perpendicularis; dico e&longs;&longs;e in perfecto æquilibrio; &longs;it enim rectangulum BE, cuius diagonalis BE perpendicula­riter cadit in horizontalem AC; certè erit in æqualibrio; &longs;it enim diui&longs;um per lineam BE ita vt FH vel KI &longs;it libra quæ &longs;u&longs;tineatur in ful­cro BG; &longs;itque totum pondus trianguli BED appen&longs;um brachio GH, & aliud BET appen&longs;um brachio æquali GF, erit perfectum æquili­brium per regulas libræ, &longs;ed duo triangula eodem modo &longs;e habent conjuncta, quo &longs;e haberent &longs;eparata & appen&longs;a, vt patet.

Tertiò, omnia rectangula proportionalia in eodem æquilibrio rema­nerent v.g. rectangulum BG cum rectangulo BE, idem dico de Rhom­bo, Rhomboide, &c.

Quartò, inde etiam cogno&longs;citur in qua proportione minuatur pondus. v. g. &longs;it enim cylindrus AE horizontalis, &longs;u&longs;tineaturque in A immo­biliter, itemque in E; certè qui &longs;u&longs;tinet in E æqualiter &longs;u&longs;tinet; at verò &longs;i attollatur in AD; certè potentia quæ in D &longs;u&longs;tinet, e&longs;t ad potentiam quæ &longs;u&longs;tinet in E, vt AF ad AE, quia pondus grauitaret in D & in E in cadem ratione per Th. 16. &longs;ed potentia &longs;u&longs;tinens adæquat ponderis ra­tionem, &longs;u&longs;tinens inquam, per DH; nam reuerà &longs;u&longs;tinens per DF æqua­lis e&longs;&longs;e debet potentiæ in E: idem dico &longs;i attollatur in AP, nam potentia trahens in P, per CP, e&longs;t ad potentiam in E, vt QA ad AP, vel AE; igitur pondus in D e&longs;t ad pondus in P vt FA ad QA.

Quintò, hinc &longs;i duo ferant parallelipedum in &longs;itu inclinato v.g.vt AD, ferunt inæqualiter, &longs;cilicet in ratione AD FA, itemque &longs;i ferant in &longs;itu inclinato AP, vel AC, donec tandem AE attollatur in B, nihil amplius &longs;u&longs;tinet potentia in B, & potentia in A totum &longs;u&longs;tinet.

Sextò, hinc cùm attollitur cylindrus continuò minùs &longs;entitur pondus & faciliùs attollitur; &longs;ic qui attollunt pontes illos ver&longs;atiles, initio maxi­mo ni&longs;u, & modico &longs;ub finem trahunt.

Septimò ob&longs;eruabis, &longs;i circa centrum immobile A attollatur cylindrus AE fune BE, potentia po&longs;ita in B, vel fune EO, potentia po&longs;ita in O; hæc deber e&longs;&longs;e minor quàm po&longs;ita in B, vt autem cogno&longs;catur propor­tio, fiat angulus PAE æqualis angulo OEB; ducatur Pque dico poten­tiam in O e&longs;&longs;e ad potentiam B, vt AQ ad AP, quia &longs;i anguli OEB & PAQ &longs;unt æquales etiam anguli APQ & AEB &longs;unt æquales; igitur perinde e&longs;t &longs;iue trahatur PA ci A per lineam PQ, &longs;iue trahatur EA circa A per lineam EB. Idem dictum &longs;it de aliis lincis.

Octauò &longs;i attollendum &longs;it rectangulum non quidem circa axem; &longs;ed circa angulum immobilem, etiam dec&longs;cit proportio ponderis, &longs;it enim v.g.quadratũ ACFD, &longs;itque AD horizontalis, AI perpendicularis, duca­tur diagonalis AF, attollatur circa punctum A, ita vt trans&longs;eraturin AG, ducatur GB perpendicularis: dico potentiam in G e&longs;&longs;e ad potentiam in in A, vt AB ad AD; quippe res codem modo &longs;e habet, ac &longs;i AF a&longs;cenderet per arcum FM, donec vbi AF traducta &longs;it in AM, tunc enim nulla crit potentia in M propter æquilibrium.

Nonò, hinc initio decre&longs;cit in maiori proportione ratione præpon­derantiæ; quia po&longs;ita ba&longs;i KN, angulus KAN e&longs;t omnium maximus; at verò decre&longs;cit in minori proportione initio ratione &longs;egmenti horizon­talis AD, in quam cadit perpendicularis.

Decimò, &longs;i &longs;it rectangulum oblongum horizontale vt AE diffici­liùs attolletur; quia quadratum AF figuræ prioris debet tantùm attolli per arcum FM, vt &longs;tatuatur in æquilibro; at verò rectangulum AE fi­guræ huius attolli debet per arcum EC longè maiorem; igitur difficiliùs: porrò potentia in D e&longs;t ad potentiam in F vt AG ad AF, vt con&longs;tat ex dictis.

Vndecimò, denique, &longs;i &longs;it rectangulum oblongum, &longs;ed verticale vt HK longè faciliùs attolletur, quia diagonalis HK debet tantùm percur­rere arcum KM vt &longs;tatuatur in æquilibrio; igitur minorem, igitur longè faciliùs; porrò hæc omnia omnibus experimentis con&longs;entiunt, & ex principiis facillimis demon&longs;trantur. Hæc paulò fu&longs;iùs pro&longs;equutus &longs;um, quia pertinent ad rationem plani inclinati.

Theorema 19.

In plano inclinato acceleratur motus in eadem proptione qua acceleratur in perpendiculari; &longs;it enim planum inclinatum AC, perpendicularis A E, in qua primo tempore &longs;en&longs;ibili percurrat AD; &longs;ecundò DE; certè dato etiam tempore licèt maiore percurret AB; igitur alio æquali percurret CB; nam vt &longs;e habet AE ad AG; ita &longs;e habet AD ad AB, & DE ad BC; quæ omnia &longs;unt certa.

Theorema 20.

Hinc æqualis ine&longs;t velocitas mobili decur&longs;a AC, inclinata & decur&longs;a AE perpendiculari, probatur, motus per AC e&longs;t ad motum per AE, vt AE, ad AC per Th.6.igitur motus per AC e&longs;t tardior; &longs;ed motu tardiore minùs &longs;patium conficitur æquali tempore in ca proportione, in qua motus e&longs;t tardior; &longs;ed proportio velocitatis e&longs;t vt AC ad AE: atqui quâ propor­tione motus e&longs;t tardior alio, maius &longs;patium decurri debet, vt motu acce­lerato per minora crcmenta acquiratur velocitas alteri æqualis; igitur eò &longs;patium debet e&longs;&longs;e maius, quò motus erit tardior; igitur debet percur­ri AC in inclinata, & AE in perpendiculari, vt &longs;it æqualis velocitas; &longs;it autem v.g. AC dupla AE, certè motus per AC e&longs;t &longs;ubduplus motus pes AE; ducatur EB perpendicularis, certè AB e&longs;t &longs;ubdupla AE; igitur eo tempore, quo percurret AE, percurret tantùm AB &longs;ubduplum &longs;cili­cet motu &longs;ubduplo; igitur tempore æquali BC triplam AB; &longs;ed tem­poribus æqualibus acquiruntur æqualia velocitatis momenta; igitur ve­locitas in C e&longs;t dupla illius, quæ erat in B; &longs;ed quæ e&longs;t in E e&longs;t dupla il­lius, quæ e&longs;t in B; igitur quæ e&longs;t in E e&longs;t æ&longs;iualis illi, quæ e&longs;t in C. Adde quod in ea proportione in qua motus e&longs;t tardior, &longs;patium e&longs;t maius, vt æqualis velocitas acquiratur; igitur &longs;i quælibet pais &longs;patij motum augec minùs quidem qua proportione motus e&longs;t tardior, & &longs;i &longs;patium AC ma­jus e&longs;t &longs;patio AE in ca proportione in qua motus per AE e&longs;t velocior; pauciores partes &longs;patij AE augent motum, &longs;ed plùs &longs;ingulæ, & plures &longs;patij AC augent motum, &longs;ed minùs &longs;ingulæ; &longs;ed cum &longs;int plures in ea­dem proportione, in qua minùs augent; certè plures quarum &longs;ingulæ mi­nùs augent, &longs;imul &longs;umptæ æqualiter augent, v.g. &longs;int AC 4. partes, & AE 2. &longs;ingulæ AE augeant motum vt 4. & &longs;ingulæ AC vt 2. quia in ca pro­portione minùs augent in qua 2. &longs;unt ad 4. certè 2. &longs;imul &longs;umptæ augent motum vt 8. & 4. &longs;imul &longs;umptæ etiam vt 8. quæ dicta &longs;unt in gratiam Geometrarum, &longs;ed meliùs adhuc ex dictis patebit.

Theorema 21.

Hinc aqualis e&longs;&longs;et ictus ab eodem mobili po&longs;t motum per AE. AF. AC. AG. quia e&longs;&longs;et acqui&longs;itus æqualis impetus; igitur e&longs;&longs;et æqualis ictus, quodcertè mirabile e&longs;t.

Theorema 22.

Hinc pote&longs;t determinari &longs;patij quæcunque petita proportio ad &longs;patium da­tum; v. g. &longs;it ictus inflictus à mobili decur&longs;a perpendiculari AE: vis æ­qualem ictum &longs;ed confecto &longs;patio duplo; accipe AC duplam AE: vis æ­qualem ictum &longs;ed confecto &longs;patio triplo, accipe AG triplam AE.

Theorema 23.

Tempora quibus percurruntur &longs;patia planorum &longs;unt vt planorum longitu­dines, v.g.tempus quo percurritur planum inclinatum AC e&longs;t ad tempus quo percurritur perpendicularis AE, vt AC ad AE; probatur, cùm enim mobile in C & in E habeat æqualem impetum &longs;eu velocitatem per Th. 20. certè cùm motus in AC &longs;it &longs;ubduplus v.g. motus in AE, e&longs;t enim vt AE ad AC per Th.6. igitur cum &longs;ubduplo motu æquali tempore ac­quiritur &longs;ubduplus impetus; igitur tempore duplo æqualis impetus; at­qui tempus motus per AC e&longs;t ad tempus motus per AE vt AC ad AE, ide&longs;t duplum; adde quod &longs;i æqualis impetus e&longs;t in C & in E; igitur æqua­lis in D & in B, &longs;ed AB e&longs;t ad BC vt AD ad DE; igitur &longs;i cre&longs;cit impe­tus per partes &longs;ubduplas in AC, nece&longs;&longs;ariò cre&longs;cit per partes duplas in &longs;patio, atque in tempore; cùm enim motus &longs;it &longs;ubduplus, tarditas e&longs;t &longs;ub­dupla; igitur acquiritur in AC &longs;patium AB &longs;ubduplum AE co tempore, quo percurtitur AE, &longs;i enim accipiantur æqualia tempora, &longs;patia &longs;unt vt motus; &longs;ed motus per AC e&longs;t &longs;ubduplus; igitur &longs;patium AB e&longs;t &longs;ubdu­plum AE; &longs;ed tempore æquali conficit BC triplum AB, igitur tota AC e&longs;t dupla AE; &longs;ed percurritur tempore duplo; igitur tempora &longs;unt vt longitudines planorum; &longs;ed clariùs, & brcuiùs illud demon&longs;tro; In ea pro­portione erit maius tempus per AC quàm per AE, in qua minor e&longs;t motus per AC quàm per AE; &longs;i enim motus per AF e&longs;&longs;et ad motum per AE vt AF ad AE, certè æquali tempore AF & AE percurrerentur; igitur qua proportione motus per AF e&longs;t minor, tempus e&longs;t maius; tantumdem enim additur tempori, quantum detrahitur motui; igitur tempora &longs;unt vt lineæ. Hinc acquiritur velocitas æqualis, vt dictum e&longs;t Th. 20. quia &longs;i tantùm addit tempus per AF &longs;upra tempus per AE, quantum addit motus per AE &longs;upra motum per AF, haud dubiè e&longs;t æqualitas.

Theorema 24.

Hinc pote&longs;t determinari longitudo plani, quæ dato tempore percurratur, v. g. perpendicularis 3. pedum percurritur 30‴. igitur &longs;i a&longs;&longs;umas planum inclinatum 6. pedum, percurretur 1″. &longs;i 12. 2′. &longs;i 24. 4″. atque ita dein­ceps; hinc po&longs;&longs;et dari planum inclinatum quod tantùm 100. annis per­curretur, &longs;cilicet &longs;i longitudo plani a&longs;&longs;umpti &longs;it æquemultiplex longitu­dinis 12. pedum atque 100. anni vnius &longs;ecundi; quod facilè e&longs;t, imò da­to plano cuiu&longs;cunque longitudinis, pote&longs;t dari tempus quodcunque quo porcurratur, de quo infrà.

Theorema 25.

Determinari pote&longs;t quantum &longs;patium conficiat mobile in plano inclinato; dum conficit perpendicularem; &longs;it enim perpendiculum AE, inclinata AC; ducatus, EB perpendicularis in AC; dico quod codem tempore percur­ret AE & AB, quod demon&longs;tro; quia triangula EAB, EAC &longs;unt pro­portionalia: igitur AB e&longs;t ad AE vt AE ad AC; igitur motus in AB e&longs;t ad motum in DE vt AB ad AE; igitur &longs;i tempora a&longs;&longs;umantur æqua­lia &longs;patia erunt vt motus, vt patet, id e&longs;t motu &longs;ubduplo acquiritur &longs;pa­tium &longs;ubduplum: nec alia e&longs;&longs;e pote&longs;t regula tarditatis, igitur &longs;patia erunt vt AB ad AE, id e&longs;t in ratione motuum; licèt enim motus veloci­tas cre&longs;cat, attamen &longs;i accipiatur velocitas compo&longs;ita ex &longs;ubdupla maxi­mæ & minimæ, percurretur AE motu æquabili æquali tempore; &longs;ed compo&longs;ita ex &longs;ubdupla maximæ & minimæ per AB habet eamdem ra­tionem ad priorem compo&longs;itam, quàm motus per AB ad motum per AE. & hic quam habet AB ad AE. Sed hæc &longs;unt clara.

Theorema 26.

Hinc æqualitempore de&longs;cendit per inclinatam BE, &longs;it enim inclinata AG, perpendicularis AE; &longs;it quoque FC perpendicularis in AG, & FD, in CF. Dico quòd eo tempore, quo conficit CD perpendicularem conficit CF inclinatam per Th.24. e&longs;t enim DF perpendicularis in IC. &longs;icut FC in AG, &longs;ed CD e&longs;t æqualis AF, vt patet.

Theorema 27.

Hinc cognito &longs;patio quod percurritur in plano inclinato, cogno&longs;citur &longs;pa­tium quod conficeretur tempore æquali in perpendiculari, &longs;it enim tempus quo percurritur AC; ducatur ex C perpendicularis CF. Dico confici AF in perpendiculari eo tempore, quo percurritur AC: vel &longs;it inclinata C F, ducatur ex F perpendicularis FD; percurretur CD eo tempore, quo percurritur CF, quæ probantur per Th.24.& 25.

Theorema 28.

Hinc per omnes chordas in&longs;criptas circulo ad alteram extremitatem, diametri perpendicularis terminatas de&longs;cendit mobile æquali tempore; a &longs;it enim circulus centro B; &longs;it diameter AE perpendicularis deor&longs;um; du­catur AC inclinata, tùm CE; de&longs;cendat haud dubiè æquali tempore per AC.CE.AE. per Th.24.25.26. idem dico de omnibus aliis AD.D E. AG.GE.AF.FE; e&longs;t enim eadem omnibus ratio; hinc non pote&longs;t da­ri planum tam paruæ longitudinis, quo non pof&longs;it dari minus, quod dato tempore percurratur. Hæc e&longs;t illa propo&longs;itio toties à Galileo enuncia­ta; cum enim motus per BE &longs;it ad motum per GE vt GE ad BE, & tem­pus per BE ad tempus per GE vt BE ad GE; cumque &longs;it vt BE ad GE rita GE ad AE; certè motus per AE e&longs;t ad motum per GE vt AE ad G E; igitur tantùm addit AE &longs;upra GE ratione &longs;patij, quantum ratione motus: igitur tempore æquali per AE. & GE &longs;iet motus, idem dico de aliis chordis.

Theorema 29.

Hinc datis duabus inclinatis æqualibus pote&longs;t determinari ratio tempo­rum, in quibus percurruntur; &longs;int enim AG.AH æquales, &longs;ed diuer&longs;æ incil­nationis; haud dubiè cum æquali tempore AG. AF percurrantur per Th. 27. tempora quibus percurruntur AGAH erunt vt tempora quibus percurruntur AF AH, & hæc vt tempora quibus percurruntur AE. A K, & hæc vt radices quadratæ illorum &longs;patiorum AE. AK, cum autem &longs;patia &longs;int vt quadrata temporum, vel in duplicata ratione, &longs;i inter AE & AK &longs;it media propprtionalis AN. v. g. tempus quo percurretur AE erit ad tempus, quo percurretur AK vt AE ad AN, vel A Nad AK.

Theorema 30.

Hinc cognito tempore quo percurriitur data portio linea cogno&longs;ci potest tempus, quo percurritur aliud &longs;patium vel alia portio, v. g. cogno&longs;co tem­pus quo percurritur AK, & volo cogno&longs;cere tempus quo percurritur K E, con&longs;equenti motu ex AK, &longs;cio tempus quo percurritur &longs;ola AE, quod e&longs;t ad tempus quo percurritur AK vt AE ad AN per Th. 28. igitur tempus quo percurritur KE con&longs;equenti motu ex AK e&longs;t ad tempus, quo percurritur AK vt EN ad NA, vel vt NK, ad NA.

Theorema 30.

Hinc in planis inæqualibus tùm in longitudine, tùns in inclinatione, pote&longs;t &longs;iri ratio temporum, quibus percurruntur; &longs;int enim AC AR duo pla­na; &longs;it autem AE perpendicularis indefinita; diuidatur AC bifariam in V ducta perpendiculari VB; ex B fiat circulus, &longs;ecabit puncta ACE; &longs;ecat etiam AR; in D igitur AC, & AD percurruntur æquali tempore per Th. 27. &longs;imiliter fiat circulus ART eodem modos certè A R & AT percurruntur æqualibus temporibus per Th. 27. igitur tempus, quopercurritur AR, vel AD e&longs;t ad tempus, quo percurritur AR vt tempus, quo percurritur AE ad tempus, quo percurritur AT; &longs;ed hæc &longs;unt vt radices AEAT, id e&longs;t tempus quo percurritur AE e&longs;t ad tem­pus, quo percurritur AT, vt AE ad mediam proportionalem inter AE AT, vel vt AD ad mediam proportionalem inter AD AR; quippe AD e&longs;t ad AR vt AE ad AT.

Galileus verò demon&longs;trat rationem i&longs;torum temporum e&longs;&longs;e compo&longs;i­tam ex ratione longitudinem planorum & ex ratione &longs;ubduplicata al­titudinum eorumdem permutatim accepta: pro quo ob&longs;erua à Galileo rationem duplicatam appellari duplam, & &longs;ubduplicatam appellari &longs;ub­duplam.

Ob&longs;eruabis denique plurima ex his colligi po&longs;&longs;e præ&longs;ertim ex Th. 27. quæ quia &longs;unt purè geometrica, certè phy&longs;icç minimè competunt; aliqua tamen omittere non po&longs;&longs;um.

Primò, &longs;i &longs;int duo plana inæqualia ad angulum rectum, qui &longs;u&longs;tinea­tur ab horizontali, determinari po&longs;&longs;unt tempora de&longs;cen&longs;uum &longs;it enim triangulum orthogonium ABE, ita vt AE &longs;it horizontalis; ducatur B G indefinita perpendicularis in ba&longs;im AE; tùm FA perpendicularis in AB; tùm FC perpendicularis in BE; tùm denique GE in BE; dico BA BFBC percurri temporibus æqualibus, item BE, BG, EG, etiam æqua­libus; igitur tempus, quo percurritur BA e&longs;t ad tempus quo percurrri­tur BE, vt tempus, quo percurritur BF ad tempus quo percurritur BG; hæc porrò &longs;unt in &longs;ubduplicata ratione BFBG vel BC, & BE.

Secundò, &longs;i planum &longs;u&longs;tinens angulum rectum non &longs;it parallelum horizonti 6. res &longs;imiliter determinari poterit; &longs;it enim triangulum or­thogonium ABC ex B, ducatur perpendicularis deor&longs;um indefinitè BF, tùm EA in AB, tùm DC in CB, tùm EH parallela DC, tùm GC in A C; denique AG parallela BF; dico quod BABEHE AE percurren­tur æqualibus temporibus item BCCDBD.

Tertiò, &longs;iue de&longs;cendat ex B in C per lineam perpendicularem BC, &longs;iue ex A per inclinatam AC, eodem modo de&longs;cendet &longs;iue per CD, &longs;iue per CE; ratio e&longs;t clara, quia acquirit æqualem velocitatem &longs;iue ex A &longs;i­ue ex B de&longs;cendat pet Th. 20. erit autem tempus per CE ad tempus per CD, vt CE ad CD per Th.23.& motus per CE ad motum per CD, vt CD ad CE per Th.6. po&longs;ito initio motus in C.

Quartò, præuio motu ex A vel ex B ad C pote&longs;t inueniri inclinata, per quam mobile pergat moueri motu &longs;cilicet naturaliter accelerato, ita vt æquali tempore illam conficiat; &longs;i enim BC conficiet dato tempore; igitur CF triplum CB conficiet tempore æquali; &longs;it autem planum ho­rizontale EDK ad quod ex C ducendum &longs;it planum inclinatum, quod eodem tempore percurratur, quo CF, diuidatur CF bifariam in H, & ex puncto H fiat arcus CK, ducaturque CK: Dico CF & CK æquali tem­pore confici per Th. 27. modò ex quiete C procedat motus: &longs;imiliter a&longs;­&longs;umi pote&longs;t alia horizontalis LM ducto arcu LF ex centro H; nam CL & CF æquali tempore percurruntur; &longs;i verò præ&longs;upponatur motus præ­uius ex A vel ex B, haud dubiè CK breuiori tempore percurretur, quàm CF, idem dico de CL; alioqui CE & CI eodem præuio motu &longs;uppo &longs;ito æquali tempore percurrerentur, quod fal&longs;um e&longs;t; nam &longs;it AC ad A N vt AN ad AE; &longs;itque BC ad BO vt BO ad BI; certè tempus, quo percurritur BC e&longs;t ad tempus, quo percurritur CI vt CB ad CO, & tempus quo percurritur BC e&longs;t ad tempus quo percurritur CE vt BC ad CN; &longs;ed CN e&longs;t minot quàm CO, vt con&longs;tat ex Geometria, quod bre­niter in tironum gratiam in terminis rationabilibus o&longs;tendo, &longs;it planum inclinatum AE 9. &longs;itque AE id e&longs;t 9. ad AD. 6. vt AD ad AC 4. ex centro C a&longs;&longs;umpta CH 3. ducatur arcus HB & ex A ad prædictum ar­cum Tangens AB, tùm ex BC G indefinitè & ex E, EG perpendicularis in EA; haud dubiè triangula CGE, CAB &longs;unt proportionalia; igitur vt CB;.ad CA. 4.ita CE 5. ad CG 6. 2/3; igitur tota BG e&longs;t 9. 2/3; &longs;itque B G ad BF, vt BF ad DC, quod vt fiat BG 9. 2/3 in BC 3. productum erit 29. igitur BF e&longs;t Rad. quad. 29.igitur e&longs;t maior 5. &longs;ed &longs;i e&longs;&longs;et maior 5. C M & CD e&longs;&longs;ent æquales; igitur CF e&longs;t maior CD; e&longs;t enim BF ferè 3. 1/2 paulò minùs: vt autem reperiatur linea inclinata, quæ percurratur æ­quali tempore cum BC &longs;uppo&longs;ito præuio motu per BC, a&longs;&longs;umatur CK æqualis CB id e&longs;t 3.partium, fiatque vt AC ad AK, ita AK ad AN; haud dubiè percurret CN æquali tempore, quo BC; vt verò habeatur pun­ctum in horizontali, &longs;it AF perpendicularis bifariam diui&longs;a in K, &longs;it K F diui&longs;a in 4. partes æquales, quibus addatur FP 1/4 KFEK V dupla FA, & producatur in X; ita vt EX &longs;it 1/4 EK: dico quod præuio motu ex A in K, & deinde deflexo per KX conficietur KX æquali tempore cum AK; &longs;i enim caderet mobile ex V primo tempore percurreret VL, id e&longs;t 1/4 V K eo tempore, quo percurreret AK per Th.6. igitur &longs;ecundo tempore æquali LK, id e&longs;t 3/4 VK; igitur tertio tempore æquali KX 5/4 VK; nam eo­dem modo &longs;e habet in k &longs;iue de&longs;cendat ex V, &longs;iue ex A per Th.20.

Porrò vt habeatur in horizontali FS; &longs;it FR æqualis KF; &longs;it FT æ­qualis KR; &longs;it arcus TS ex k: Dico quod ks e&longs;t linea quæ&longs;ita; nam &longs;i &longs;it vt BS ad BZ, ita BZ ad BK, kz erit æqualis KF, vel AK; &longs;ed tempus quo percurritur AK e&longs;t ad tempus quo percurritur Dk vt BK ad AK per Th.23.& ad tempus, quo percurritur BS, vt Bk ad BZ, & ad tem­pus quo percurritur ks vt Bk ad kz; ergo Ak & ks percurruntur æ­quali tempore, &longs;i kz &longs;it æqualis KF, quod &longs;ic breuiter demon&longs;tro, cùm figura apud Galileum de&longs;ideretur. &longs;int AFFE æquales; ducatur AE quæ transferatur iu FG, &longs;itque GI æqualis AG, &longs;ictota AG mihi repræ­&longs;entat totam BS &longs;uperioris figuræ, vt con&longs;tat; &longs;it autem AG ad AH vt A H ad AI: Dico GH e&longs;&longs;e æqualem AF; &longs;it enim quadratum HD mediæ proportionalis: Dico e&longs;&longs;e æquale rectangulo IC, dùm AC &longs;it æqualis A G; igitur quadratum PR cuius latus e&longs;t æquale FG, &longs;eu AE continet duo quadrata RDSN; ergo GH e&longs;t æqualis VN; igitur GH quod erat demon&longs;trandum.

Quintò, hinc nunquam ks vel kx pote&longs;t e&longs;&longs;e tripla Ak donec tan­dem perueniatur ad perpendiculum kH; nam &longs;ecundo tempore percur­ritur kH triplum Ak, &longs;i primo percurritur Ak; nunquam etiam ks vel vlla alia inclinata pote&longs;t e&longs;&longs;e dupla tantùm Ak; &longs;ed &longs;emper e&longs;t maior, do-nec tandem perneniat ad horizontalem KY, quæ e&longs;t dupla AK, quia in horizontali non acceleratur motus; igitur cum impetu acqui&longs;ito in de&longs;­cen&longs;u AK, conficiet motu æquabili KY duplum AK per Th.42.l.3. po&longs;ito quòd non de&longs;truatur; atque ex his &longs;atis facilè intelligentur, quæcumque habes apud Galileum in dialog.3.à propo&longs;itione 3.ad 23.

Sextò non probat Galileus, &longs;ed tantùm &longs;upponit mobile ad eandem alti­tudinem a&longs;cendere po&longs;&longs;e motu reflexo ex qua de&longs;cendit, quod examinabi­mus lib. &longs;equenti, hinc non laborabimus in examinandis prop. 24.25.26.27.

Septimò, cognito tempore, quo percurrit mobile perpendiculum EC quod &longs;it diameter circuli; &longs;ciri pote&longs;t quo tempore percurrat duas chor­das &longs;imul EGGC; &longs;it enim Tangens EF, &longs;itque vt FG ad FD, ita FD ad FC; cum EG & EC de&longs;cendat æquali tempore per Th.27. cum in G &longs;it idem motus, &longs;iue ex E, &longs;iue ex F de&longs;cendat per Th.20. certè &longs;i de&longs;cendit per EG dato tempore, quod &longs;it vt EG, de&longs;cendit per GC tempore, quod e&longs;t vt GD; igitur tempus, quo de&longs;cendit per EC e&longs;t ad tempus, quo de&longs;­cendit per EGC, vt EG ad EGD.

Ob&longs;eruabis autem GF e&longs;&longs;e ad EF vt EF ad FC; igitur FD e&longs;t media inter FC GF, & e&longs;t æqualis FE, igitur anguli FDE.FED æquales; &longs;ed FD E e&longs;t æqualis duobus DCE.DEC, & FEG, e&longs;t æqualis DCE; igitur duo G DE DEC &longs;unt æquales.

Octauò, &longs;i accipiantur æquales horizontalis, & perpendicularis, v.g. BA AC, ducaturque BC: Dico nullum duci po&longs;&longs;e planum incliuatnm à puncto B ad perpendiculum AEM, quod breuiori tempore percurratur, quàm BC, nec intra angulum vt BR, nec extra vt BM; &longs;it enim vt BC ad BI ita BI ad BH, e&longs;t autem BI æqualis BA, igitur &longs;i BA, &longs;it 4.BC e&longs;t v.g. 32. & BH radix q.8.igitur HI e&longs;t ferè I paulò plùs; igitur cum BH percur­ratur æquali tempore cum AC, e&longs;t tempus, quo percurritur BH ad tem­pus quo percurritur HC vt BH ad HI.

Sit autem BR dupla AR, &longs;itque perpendicularis AK in BR; certè KR e&longs;t &longs;ubquadrupla BR; igitur percurritur BL æqualis KR eo tempore quo percurritur AR; igitur BL &longs;it ad BV vt BV ad BR; igitur temporibus æ­qualibus percurruntur BL LR; igitur &longs;i tempus quo percurritur BL &longs;it vt BH, tempus quo percurretur LR erit etiam vt BH; igitur totum tempus quo percurritur tota BR erit vt tota BE, &longs;ed tempus quo percurritut tota BC e&longs;t tantum vt BI qu&ecedil; e&longs;t minor BC; igitur BC breuiori tempore per­curritur quàm BR; &longs;it etiam vt BP ad BX ita BX ad BM, &longs;i BO e&longs;t 4. OP 2. certè BP e&longs;t rad.q. 12.id e&longs;t ferè 3.1/2 paulò minùs, BM verò e&longs;t dupla BA vel BO; igitur e&longs;t 8. ducatur ergo 8. in 4. 1/3 productum erit 28. cuius radix e&longs;t ferè 5.1/3 paulò minùs; igitur BX e&longs;t 5.1/3 paulò minùs; cum autem BH &longs;it 2.q.8.e&longs;t ferè 2.5/6, paulò minùs; igitur &longs;it vt BP 3.1/2 ad BX 5.1/3, ita BH 2.5/6 ad aliam; certè erit 144. id e&longs;t 4.(26/63), licèt minùs acceptum &longs;it; igitur 126.e&longs;t maior BI, quæ e&longs;t tantùm 4; igitur BE breuiori tempore percur­ritur, quàm BM.

Nonò, per duas chordas quadrantis de&longs;cendit breuiori tempore mo­bile, quàm per alteram tantùm inferiorem &longs;cilicet &longs;it enim tantùm quadrans ABG in quo &longs;int duæ chordæ GC, CB: Dico quòd per vtram­que ex G breuiori tempore de&longs;cendit, quàm per inferiorem CB; quia per CB, & GB æquali tempore de&longs;cendit per Th.27.&longs;ed per GCB bre­uiori tempore de&longs;cendit, quàm per GB; &longs;it enim GD perpendicularis parallela AB; &longs;it ED perpendicularis in CG, & per 3. puncta GCD ducatur circulus: his po&longs;itis, GH & GC eodem tempore percurrentur, & in C idem erit motus, &longs;iue ex G per GE, &longs;iue ex E per EC de&longs;cen­dat mobile per Th.27.& 20. &longs;it autem EB ad EK vt EK ad EC, &longs;itque BE v.g, dupla BE vel BA: dico EK e&longs;&longs;e æqualem BG; e&longs;t autem BH maior BC vel AB, vel HG minor CK; &longs;it etiam GH ad GI, ita GI ad GB: dico tempus, quo de&longs;cendit per GCB e&longs;&longs;e ad tempus quo de­&longs;cendit per GB vt GCK ad compo&longs;itam ex GC, HI; &longs;ed hæc e&longs;t ma­ior illa, vt patet ex Geometria, & analytica; igitur breuiori tempore de­&longs;cendit per GCB, quàm per GB; &longs;ed de hocaliàs.

Sit enim EB 8. dupla &longs;cilicet AB; &longs;it autem EE &longs;ubdupla EB ad EK vt EK ad EB; a&longs;&longs;umatur GE, &longs;itque tempus, quo continetur GC. vt GC, & quo conficitur BC vt CK; igitur quo conficitur GCB vt GCK: &longs;imiliter &longs;it &longs;ecunda linea GB, &longs;itque tempus, quo percurritur GH vt GC, vel NO æqualis GC, &longs;itque vt GH ad GN, ita GN ad GB certè &longs;i GH decurratur tempore GH, AB decurretur tempore HN; &longs;ed HN maior e&longs;t MB, vel CG, vt con&longs;tat ex analytica; adde quod in figura prima &longs;it GI ad GM vt GM ad GB; certè &longs;i tempore GI percurratur GI, percurretur GB tempore GM; e&longs;t autem GM æqua­lis AB, vel EC; &longs;imiliter &longs;it EC ad EK vt EK ad EB, &longs;i percurratur EC tempore EC, percurretur EB tempore EK; &longs;ed GC percurretur tempore GC &longs;ed GCK minor e&longs;t GIM; &longs;it enim GM. 4. EK R. que 32. id e&longs;t, 5 7/8 paulò minùs, quibus &longs;i &longs;ubtrahas CE 4. & &longs;ub&longs;tituas CG 2. paulò plùs habebis 3 7/8; igitur GCK minor e&longs;t GIM. Ex his habes omnes Galilei propo&longs;itiones de motu in planis inclinatis numero 38. in quo &longs;tudio, vt verum fatear, maximam &longs;ibi laudem peperit; in quo ta­men opere duo de&longs;iderari videntur, alterum à Philo&longs;ophis, quod ita phy&longs;i­cæ partes omnes neglexerit, vt ferè vni Geometriæ &longs;atisfaceret; alterum ab Geometris quod Geometriam equidem accuratè tractarit. Sed minùs ad captum Tyronum: atque hæc de his &longs;int &longs;atis, vt tandem no&longs;trorum Theorematum &longs;eriem interruptam repetamus.

Theorema 31.

Ex dictis &longs;equitur pondus centum librarum po&longs;&longs;e habere tantùm grauitatio­nem vnius libræ; &longs;it enim planum inclinatum centuplum horizontalis, id e&longs;t, &longs;ecans centupla Tangentis; haud dubiè grauitatio in prædictum pla­num erit tantùm &longs;ubcentupla per Th.16.

Theorema 32.

Ex duobus ferentibus idem parallelipedum in &longs;itu inclinato pote&longs;t alter fer­re tantùm vnam libram, licèt pendat centum libras; &longs;it enim ita inclina-tum, vt linea inclinationis &longs;it centupla horizontalis oppo&longs;itæ; certè qui &longs;u&longs;tinet in altera extremitate eleuata (1/100) tantùm &longs;u&longs;tinet ponderis par­tem per Th. 18. alius verò &longs;u&longs;tinet in altera extremitate, quæ deor&longs;um e&longs;t (93/100).

Theorema 33.

Qui pote&longs;t tantùm datum pondus &longs;ur&longs;um attollere per lineam verticalem, centuplum per inclinatum planum ad eamdem altitudinem attollet; &longs;i enim &longs;it inclinata ad perpendiculum in ratione centupla; haud dubiè qui attollit datum pondus per ip&longs;um derpendiculum &longs;ine viribus auctis per inclina­tum planum, pondus centuplò maius attollet, quia potentia per inclina­tam e&longs;t ad potentiam per ip&longs;um perpendiculum vel altitudo ad inclina­tam per Theor. 6. igitur &longs;i æqualis vtrobique applicetur potentia, pon­dus centuplò maius attollet per inclinatam, &longs;eu pellendo, &longs;eu tra­hendo.

Theorema 34.

Hinc ratio plani inclinati demon&longs;trat cochleæ vires. v.g. pellitur &longs;ur&longs;um per DE inclinatam faciliùs quàm verticalem DH in ratione DE ad DH, quæ &longs;i e&longs;t tripla, eadem potentia quæ datum pondus attollit per DH, triplò maius attollet per DE, vel &longs;i attollat per DA verticalem, triplò maius attollet per &longs;piras vel Helices DE EC, CF, &c. v&longs;que ad A; hinc quò Helix erit inclinatior, potentia maius pondus illius operâ attollet.

Theorema 35.

Hinc clarè vides compen&longs;ari longitudinem motus, &longs;patij vel temporis, pon­deris acce&longs;&longs;ione, v.g. triplò maius pondus attollitur per DE quàm per DH; quia &longs;patium DE e&longs;t triplum DH; igitur motus triplus, &longs;cilicet in duratione, (loquor enim de motu æquabili quo &longs;ur&longs;um corpus, vel tra­hitur, vel continuò pellitur.)

Theorema 36.

Hinc nullus mons e&longs;&longs;e pote&longs;t quantumuis arduus, ad cuius apicem via faci­li in modum cochleæ &longs;trata pertingi non po&longs;&longs;it; & quò plures erunt &longs;piræ, eo facilior erit & minùs decliuis via.

Theorema 37.

Quando de&longs;cendit mobile per multas &longs;piras, &longs;eu volutas, pote&longs;t determinari altitudo perpendicularis, ex qua eodem tempore de&longs;cenderet; &longs;it enim &longs;pira &longs;eu cochlea AFCHD, & perpendiculum AD; certè eodem tempore de&longs;cendit per AFC, quo de&longs;cenderet per AG duplam AF; &longs;ed co tem­pore, quo de&longs;cendit per AF inclinatam, conficit AD per Th.27. quæ e&longs;t ad AF vt AF ad BA; &longs;it autem dupla: &longs;imiliter codem tempore conficit AFG vel AFG, quo conficit AE duplam AG; denique eo tempore, quo conficit AF CHD, vel AGD, conficit duplam AE.

Sic etiam eo tempore, quo in perpendiculo conficit AD conficit &longs;ub­duplam &longs;cilicet AF, &longs;ed hæc &longs;unt clara.

Theorema 38.

Quando proiicitur mobile per planum inclinatum &longs;ur&longs;um in ea propertione proiicitur longiùs, quò inclinata ip&longs;a longior e&longs;t perpendiculari. v.g. &longs;i proii­citur per BA in verticali, illa eadem potentia quæ proiicit in A ex B, pro­iiciet quoque ex F in A, ex M in A, atque ita deinceps ex &longs;ingulis punctis horizontalis BM; ratio e&longs;t, quia in ea proportione de&longs;truitur impetus per BA, in qua motus per AB de&longs;cendit; nam impetus innatus deor­&longs;um qua&longs;i trahit mobile graue; impetus verò impre&longs;&longs;us &longs;ur&longs;um attollit; igitur pugnant pro rata, vt &longs;æpè diximus in tertio libro, & alibi: &longs;imiliter in inclinata FA impetus innatus qua&longs;i reducit mobile deor&longs;um dum impre&longs;&longs;us violentus &longs;ur&longs;um promouet; igitur &longs;i impetus innatus per AB, & per AT æqualem vim haberet, haud dubiè æquale &longs;patium contine­ret mobile projectum per BA & FA; nam eadem potentia cum æquali re&longs;i&longs;tentia idem præ&longs;tat & inæqualiter de&longs;cendit per AB AF, & motus per AF e&longs;t ad motum per AB, vt AB ad AF. v.g. &longs;ubduplus; igitur re­&longs;i&longs;tentia per BA erit dupla re&longs;i&longs;tentiæ per FA; igitur &longs;patium per FA erit duplum; igitur ex F a&longs;cendet in A, quo cum eo impetu ex B a&longs;cendet in A, &longs;uppo&longs;ita eadem potentia; idem etiam dicendum de aliis punctis horizontalis BM: præterea ille impetus &longs;ufficit ad motum &longs;ur&longs;um per FA, qui accipitur in de&longs;cen&longs;u AF, vt con&longs;tat ex dictis; itemque &longs;ufficit ad motum &longs;ur&longs;um per BA qui acquiritur in de&longs;cen&longs;u AB; &longs;ed æqualis ve­locitas, vel impetus acquiritur in vtroque de&longs;cen&longs;u AB AF per Th. 20. igitur idem impetus &longs;ufficit ad de&longs;cen&longs;um BA FA.

Theorema 39.

Hinc dicendum e&longs;t impetum naturalem per inclinatam FA vel MA n&longs;ur&longs;um intendi, &longs;eu cre&longs;cere; alioqui ex A mobile de&longs;cenderet citiùs in F, po&longs;tquàm ex F proiectum e&longs;&longs;et in A, quàm &longs;i tantùm ex A in F demit­teretur, quod e&longs;t contra experientiam; adde quòd impetus naturalis &longs;ur­&longs;um non cre&longs;cit, vt iam &longs;æpè dictum e&longs;t.

Theorema 40.

Destruitur aliquid impetus impre&longs;&longs;i in mobili per planum inclinatum.Probatur, quia tandem quie&longs;cit mobile; igitur ce&longs;&longs;at motus; igitur & im­petus: nec dicas id fieri ab aëre, vel plani &longs;cabritie; nam, &longs;i hoc e&longs;&longs;et, æquale &longs;patium conficeret in FA & LA; quippe æqualis portio plani æqualiter re&longs;i&longs;tit; Idem dico de aëre; igitur de&longs;truitur impetus impre&longs;­&longs;us ab impetu naturali.

Theorema 41.

Destruitur tantùm pro rata, hoc e&longs;t in ratione, quam habet perpendiculum ad inclinatam. v.g. &longs;it perpendiculum FCA; haud dubiè &longs;i non de&longs;true­retur motus &longs;ur&longs;um cum eo gradu impetus, quo ex F a&longs;cendit in C motu retardato, a&longs;cenderet in A motu æquabili, & eodem tempore; igitur eo tempore de&longs;truitur totus impetus; &longs;i verò proiiciatur per LC; certè im­petus totus non de&longs;truitur per LC, eo tempore, quo ex F a&longs;cenderet in C, &longs;ed pro rata, id e&longs;t in ratione FC ad LC, quæ &longs;it &longs;ubdupla v.g. igitur impetus de&longs;truitur tantùm &longs;ubduplus; igitur eo tempore, quo ex F a&longs;cen­dit in C, ex L a&longs;cendet in K, ita vt LM æquali FC addatur MK æqua­lis EB; e&longs;t autem EB &longs;ubdupla CA vel EF. Similiter &longs;it perpendicu­lum FG, & inclinata HF tripla FG; a&longs;&longs;umatur FC æqualis FG, item­que HO æqualis GF; certè eo tempore, quo perpendiculari detrahitur totus impetus, detrahitur tantùm &longs;ubtriplum per inclinatam HF; igitur a&longs;&longs;umatur ER &longs;ubtripla EF; & addatur OP æqualis FR: dico quod