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version 1.19, 2002/08/08 23:25:30

version 1.27, 2002/08/18 16:43:32

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<?xml version="1.0"?>

<!DOCTYPE archimedes SYSTEM "../dtd/archimedes.dtd" ><archimedes> <info>

<author>Fabri, HonorŽ</author>

<author>Fabri, Honoré</author>

<title>Tractatus physicus de motu locali</title>

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<cvs_file>fabri_tract_01_la_1646</cvs_file>

<cvs_version></cvs_version>

</info> <text> <front> </front> <body> <chap> <pb/>

<s><emph type="center"/>TRACTATVS <lb/>PHYSICVS <lb/>DE MOTV LOCALI, <lb/><emph type="italics"/>IN QVO<emph.end type="italics"/><emph.end type="center"/></s>

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<s>quia vnus e&longs;t hic & <lb/>alius non e&longs;t hic, vnum tango, & alium non tango, vnus e&longs;t meus, & <lb/>alius non e&longs;t meus, &c. </s>

<s>vides prædicata contradictoria, quæ cum eidem <lb/>&longs;imul ine&longs;&longs;e non po&longs;&longs;int per Ax. </s>

<s>vides prædicata contradictoria, quæ cum eidem <lb/>&longs;imul ine&longs;&longs;e non po&longs;&longs;int per Ax. 1. diuer&longs;is, & di&longs;tinctis ine&longs;&longs;e nece&longs;&longs;e <lb/>e&longs;t. </s>

<s>1. diuer&longs;is, & di&longs;tinctis ine&longs;&longs;e nece&longs;&longs;e <lb/>e&longs;t. </s>

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

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<s><emph type="italics"/>Tunc ponenda e&longs;t forma distincta &longs;ub&longs;t antialis vel accidentalis, dum e&longs;t ali<lb/>qua proprietas &longs;en&longs;ibilis, quæ non pote&longs;t tribui ip&longs;i materiæ,<emph.end type="italics"/> hîc res tantùm <lb/>naturales con&longs;idero, nec &longs;uper naturales attingo, quæ &longs;uas regulas diui<lb/>næ fidei debent, non &longs;en&longs;ibus. </s>

<s>Hoc Axioma omninò certum e&longs;t, & per Ax. </s>

<s>Hoc Axioma omninò certum e&longs;t, & per Ax. 3. confirmatur, vt enim <lb/>dicas aliquid di&longs;tinctum ab omni alio exi&longs;tere, vel debet id &longs;en&longs;u percipi, <lb/>vel aliqua ratione probari quod &longs;it; atqui formam &longs;ub&longs;tantialem &longs;en&longs;u <lb/>non percipis immediatè; igitur aliquem eius effectum &longs;en&longs;ibilem vel me<lb/>diatè, vel immediatè; qui certè &longs;i tribui po&longs;&longs;it materiæ, haud dubiè per il<lb/>lum &longs;ormam non probabis, ni&longs;i formæ ip&longs;ius e&longs;&longs;e antè demon&longs;tres; &longs;i ve<lb/>to e&longs;t forma accidentalis, quam &longs;en&longs;u percipis; certè id tantùm accidit ex <pb pagenum="8"/>aliqua affectione, quâ&longs;en&longs;um ip&longs;um afficit hæc forma, igitur ex effectu il<lb/>lo illam percipis, quod clarum e&longs;t. </s>

<s>3. confirmatur, vt enim <lb/>dicas aliquid di&longs;tinctum ab omni alio exi&longs;tere, vel debet id &longs;en&longs;u percipi, <lb/>vel aliqua ratione probari quod &longs;it; atqui formam &longs;ub&longs;tantialem &longs;en&longs;u <lb/>non percipis immediatè; igitur aliquem eius effectum &longs;en&longs;ibilem vel me<lb/>diatè, vel immediatè; qui certè &longs;i tribui po&longs;&longs;it materiæ, haud dubiè per il<lb/>lum &longs;ormam non probabis, ni&longs;i formæ ip&longs;ius e&longs;&longs;e antè demon&longs;tres; &longs;i ve<lb/>to e&longs;t forma accidentalis, quam &longs;en&longs;u percipis; certè id tantùm accidit ex <pb pagenum="8"/>aliqua affectione, quâ&longs;en&longs;um ip&longs;um afficit hæc forma, igitur ex effectu il<lb/>lo illam percipis, quod clarum e&longs;t. </s>

<s>Huc reuoca vulgare illud principium, <emph type="italics"/>Frustrà fit per plura, quod po<lb/>test fieri per pauciora,<emph.end type="italics"/> quod ad Tertium etiam reuocatur; quod ita in<lb/>telligi non debet, vt &longs;ine gutta aquæ Oceanus, &longs;ine &longs;tella c&oelig;lum, &longs;ine gra<lb/>nulo arenæ terra, &longs;ine altero oculorum homo &longs;tare non po&longs;&longs;int; quæ <lb/>omnia fal&longs;i&longs;&longs;una e&longs;&longs;e con&longs;tat; &longs;ed tantùm quod illud dicatur exi&longs;tere &longs;iue <lb/>&longs;it &longs;ub&longs;tantia, &longs;iue accidens, quod vel experientia certa euincit, vel nece&longs;<lb/>&longs;itas, vel ratio, vel diuina fides (immò & humana in rebus humanis, non <lb/>tamen in &longs;cientiis.) </s>

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<s>atqui vnus tantùm ex æquo re&longs;pondet, &longs;cilicet 5. &longs;ed <lb/>quod vnum e&longs;t determinatum e&longs;t, per Axioma 5. igitur pugnant pro <lb/>rata. </s>

<s>Nec dicas A totum de&longs;trui à B, quòd e&longs;t contra hypothe&longs;im, nam <lb/>modicum caloris non de&longs;truit totum frigus: in impetu res e&longs;t clari&longs;&longs;ima; <lb/>adde quod minor cau&longs;a minùs agit per Ax. </s>

<s>3. igitur minùs exi<lb/>git; porrò cum dico vnum ab alio de&longs;trui, intelligo tantùm ex applica<lb/>tione vnius &longs;equi de&longs;tructionem alterius &longs;alrem ex parte. </s>

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<s><emph type="italics"/>Motus e&longs;t aliquid realiter di&longs;tinctum à mobili.<emph.end type="italics"/></s><s> Demon&longs;tratur; Motus <lb/>e&longs;t in mobili, in quo antè non erat per hypoth. </s>

<s>3. & de&longs;init e&longs;&longs;e in mobili, <lb/>in quo antè erat per hypoth.4. igitur mobile e&longs;t, & non e&longs;t motus; igi<lb/>tur à motu &longs;eparatum; igitur realiter di&longs;tinctum per Ax. 2. præterea <lb/>moueri, & non moueri &longs;unt prædicata contradictoria, vt con&longs;tat; igi<lb/>tur cidem &longs;imul ine&longs;&longs;e non po&longs;&longs;unt per Ax. 1. igitur cum co non &longs;unt <lb/>idem; alioquin &longs;imul e&longs;&longs;ent; igitur alterum illorum e&longs;t di&longs;tinctum à <lb/>mobili; non quies, vt con&longs;tat, quæ e&longs;t tantùm negatio motus, &longs;eu per<lb/>&longs;euerantia in codem loco; igitur nullam dicit mutationem; at verò <lb/>motus mutationem dicit, per Def. </s>

<s>2. præterea <lb/>moueri, & non moueri &longs;unt prædicata contradictoria, vt con&longs;tat; igi<lb/>tur cidem &longs;imul ine&longs;&longs;e non po&longs;&longs;unt per Ax. </s>

<s>1. igitur cum co non &longs;unt <lb/>idem; alioquin &longs;imul e&longs;&longs;ent; igitur alterum illorum e&longs;t di&longs;tinctum à <lb/>mobili; non quies, vt con&longs;tat, quæ e&longs;t tantùm negatio motus, &longs;eu per<lb/>&longs;euerantia in codem loco; igitur nullam dicit mutationem; at verò <lb/>motus mutationem dicit, per Def. </s>

<s>1. hoc Theorema fusè demon&longs;trabo <lb/>in Metaph. </s>

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<s>Dices potentia motrix e&longs;t actiua; igitur agit; igitur producit, &longs;ed ni<lb/>hil ni&longs;i motum. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. potentiam motricem e&longs;&longs;e actiuam vt dicemus, <lb/>& ab eâ produci impetum, qui deinde exigit motum, vt dicemus <lb/>infrà. </s>

<s>potentiam motricem e&longs;&longs;e actiuam vt dicemus, <lb/>& ab eâ produci impetum, qui deinde exigit motum, vt dicemus <lb/>infrà. </s>

<s>Nec e&longs;t quod aliqui ita mirentur hæc à me dici; cum certum &longs;it effe<lb/>ctus &longs;ormales &longs;ecundarios principum ferè qualitatum tales e&longs;&longs;e, vt mini<lb/>mè producantur; &longs;ed qua&longs;i re&longs;ultent ab exigentia; v. </s>

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<s><emph type="italics"/>Entitas &longs;eu &longs;ubstantia mobilis non e&longs;t cau&longs;a immediata motus,<emph.end type="italics"/> Sit enim <lb/>lapis proiectus per Po&longs;tul. </s>

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

<s>2. Adde quod motus &longs;ecundi generis habet tantùm caulam <lb/>immediatam exigentem, &longs;ed potentia motrix non exigit; quia primò <lb/>non defatigaretur exigendo; &longs;ecundò quia lapis &longs;eparatus à manu etiam <lb/>mouetur, &longs;ed non ad exigentiam potentiæ motricis, vt patet; quia &longs;tatim <lb/>po&longs;t &longs;eparationem pote&longs;t illa potentia de&longs;trui, licèt lapis longo pò&longs;t <lb/>temporemoueatur; &longs;ed quod non e&longs;t, nihil exigit. </s>

<s>Aliquis fortè diceret potentiam motri&oelig;m exigere primam partem <lb/>motus, quæ deinde &longs;ecundam exigit, & &longs;ecunda tertiam, tertia quar<lb/>tam, &c. </s>

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<s>Dices cum graue aliquod mouetur deor&longs;um, vel leue &longs;ur&longs;um, vel <lb/>corpus animatum &longs;e ip&longs;um mouet, dici pote&longs;t &longs;ub&longs;tantia corporis cau&longs;a <lb/>immediata motus. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. negando, tùm quia omnis potentia motrix <lb/>agit; igitur producit aliquid aliud, quod e&longs;t cau&longs;a motus: præterea po<lb/>tentia motrix corporis animati, agit v&longs;que ad defatigationem, &longs;udorem, <lb/>licèt non &longs;it motus, igitur aliud producit, de corpore graui probabi<lb/>mus infrà. </s>

<s>negando, tùm quia omnis potentia motrix <lb/>agit; igitur producit aliquid aliud, quod e&longs;t cau&longs;a motus: præterea po<lb/>tentia motrix corporis animati, agit v&longs;que ad defatigationem, &longs;udorem, <lb/>licèt non &longs;it motus, igitur aliud producit, de corpore graui probabi<lb/>mus infrà. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s>

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<s>Demon&longs;tro primò dari impetum: Quidquid e&longs;t, & antè non erat, non <lb/>e&longs;t à &longs;e, &longs;ed habet cau&longs;am per Ax.8. Motus de nouo e&longs;t per hypothe&longs;im <lb/>tertiam; igitur habet cau&longs;am, &longs;ed non aliam, quam impetum, quod pro<lb/>bo: Lapis cadens, vel impactus in alium lapidem mouet illum per hy<lb/>poth.7. &longs;ed &longs;ub&longs;tantia lapidis in alium impacti non e&longs;t cau&longs;a huius mo<lb/>tus, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt patet; igitur applicata eundem effe<lb/>ctum produceret per Ax.12. &longs;ed etiam applicata immediata non agit, vt <lb/>con&longs;tat experientia; igitur per idem Axioma non e&longs;t cau&longs;a. </s>

<s>Scio e&longs;&longs;e aliquas re&longs;pon&longs;iones, quas infrà refellemus; nunc &longs;ufficiat <lb/>dixi&longs;&longs;e lapidem impactum non producere motum, qui propriè non pro<lb/>ducitur per Th.2. nec exigere, vt con&longs;tat ex &longs;ecunda probatione Th. </s>

<s>5. <lb/>igitur &longs;i aliquid exigit, vel producit, voco impetum. </s>

<s>Secundò probatur; potentia motrix e&longs;t actiua, quia defatigatur, quis <lb/>hoc neget? </s>

<s>igitur aliquid producit; non motum, qui propriè non pro<lb/>ducitur per Th.2. igitur aliquid aliud; voco impetum; adde quod etiam <lb/>&longs;ine motu agit, & defatigatur vt iam dictum e&longs;t; igitur habet alium effe<lb/>ctum immediatum; denique mouere, pellere, trahere, proiicere, percu<lb/>tere, nihil ni&longs;i actionem &longs;onant. </s>

<s>Tertiò probatur; pila di&longs;iuncta à manu proiicientis diu adhuc mo<lb/>uetur per hypoth.6. igitur hic motus habet cau&longs;am per Ax. </s>

<s>Tertiò probatur; pila di&longs;iuncta à manu proiicientis diu adhuc mo<lb/>uetur per hypoth.6. igitur hic motus habet cau&longs;am per Ax. 8. quælibet <lb/>enim pars motus de nouo e&longs;t, neque duæ illius partes &longs;imul e&longs;&longs;e po&longs;&longs;unt. </s>

<s>8. quælibet <lb/>enim pars motus de nouo e&longs;t, neque duæ illius partes &longs;imul e&longs;&longs;e po&longs;&longs;unt. </s>

<s><lb/>atqui potentia motrix non e&longs;t can&longs;a per Ax.10. immò pote&longs;t e&longs;&longs;e de&longs;tru<lb/>cta; igitur non e&longs;t cau&longs;a per Ax. </s>

<s><lb/>atqui potentia motrix non e&longs;t can&longs;a per Ax.10. immò pote&longs;t e&longs;&longs;e de&longs;tru<lb/>cta; igitur non e&longs;t cau&longs;a per Ax. 9. Non e&longs;t etiam cau&longs;a &longs;ub&longs;tantia <gap/>pilæ <lb/>mobilis per Th.5.5. nec priores pattes motus per re&longs;p. </s>

<s>9. Non e&longs;t etiam cau&longs;a &longs;ub&longs;tantia <gap/>pilæ <lb/>mobilis per Th.5.5. nec priores pattes motus per re&longs;p. </s>

<s>ad primam in<lb/>&longs;tantiam Th 5. igitur aliquid aliud; voco impetum. </s>

<s>Quartò probatur; pila proiecta &longs;en&longs;im &longs;ine &longs;en&longs;u tardiore motu <lb/>mouetur; donec tandem moueri omnino de&longs;inat per hypoth. </s>

<s>5. igitur <lb/>non e&longs;t &longs;emper æqualis, & eadem cau&longs;a huius motus per Ax. </s>

<s>5. igitur <lb/>non e&longs;t &longs;emper æqualis, & eadem cau&longs;a huius motus per Ax. 12. & 13. <lb/>num.3. igitur cau&longs;a huius motus eodem modo debilitatur, &longs;eu remitti<lb/>tur, quo ip&longs;e motus; &longs;ed decre&longs;cit &longs;ub&longs;tantia mobilis, nec potentia mo-<pb pagenum="16"/>trix, vel corpus prius impactum; ergo e&longs;t alia cau&longs;a præ&longs;ens, quæ mi<lb/>nuitur; voco impetum. </s>

<s>12. & 13. <lb/>num.3. igitur cau&longs;a huius motus eodem modo debilitatur, &longs;eu remitti<lb/>tur, quo ip&longs;e motus; &longs;ed decre&longs;cit &longs;ub&longs;tantia mobilis, nec potentia mo-<pb pagenum="16"/>trix, vel corpus prius impactum; ergo e&longs;t alia cau&longs;a præ&longs;ens, quæ mi<lb/>nuitur; voco impetum. </s>

<s>Quintò corpus graue deor&longs;um cadens accelerat &longs;uum motum, vt patet <lb/>experientia; quæ maximè clara e&longs;t in funependulis, de qua in &longs;equen<lb/>tibus libris; igitur debet e&longs;&longs;e cau&longs;a huius motus velocioris; non e&longs;t au<lb/>tem &longs;ub&longs;tantia lapidis, nec grauitas per Ax. </s>

<s>12. nec aliud quidpiam ex<lb/>trin&longs;ecum, vt videbimus &longs;uo loco; igitur aliquid aliquid inttin&longs;ecum, <lb/>voco impetum. </s>

<s>Igitur certum e&longs;t dari impetum; qui certè tribui non <lb/>pote&longs;t, vel vlli connotationi, vel alteri exigentiæ, vt con&longs;tat ex <lb/>dictis. </s>

<s>Diceret fortè alius hæc omnia e&longs;&longs;e dubia; nam fieri pote&longs;t vt Deus <lb/>tantùm moueat; quod &longs;ine impetu fieri po&longs;&longs;e certum e&longs;t; Re&longs;p. </s>

<s>Diceret fortè alius hæc omnia e&longs;&longs;e dubia; nam fieri pote&longs;t vt Deus <lb/>tantùm moueat; quod &longs;ine impetu fieri po&longs;&longs;e certum e&longs;t; Re&longs;p. equi<lb/>dem per miraculum hoc fieri po&longs;&longs;e; &longs;ed quemadmadum certum e&longs;t phy<lb/>ficè ignem applicatum calefacere, niuem frigefacere, & modò calamum <lb/>à me hæc &longs;cribente moueri, ita certum o&longs;t phy&longs;icè &longs;agittam à &longs;agittario <lb/>emitti, & pilam à proiiciente, &c. </s>

<s>equi<lb/>dem per miraculum hoc fieri po&longs;&longs;e; &longs;ed quemadmadum certum e&longs;t phy<lb/>ficè ignem applicatum calefacere, niuem frigefacere, & modò calamum <lb/>à me hæc &longs;cribente moueri, ita certum o&longs;t phy&longs;icè &longs;agittam à &longs;agittario <lb/>emitti, & pilam à proiiciente, &c. </s>

<s>adde quod Deus, vt auctor naturæ <lb/>e&longs;t, agit tantùm; vel de&longs;init agere iuxta exigentiam cau&longs;arum &longs;ecunda<lb/>rum; denique cau&longs;am phy&longs;icè appello, ex cuius applicatione nunquam <lb/>non &longs;equitur effectus per Ax.11. num.1. </s>

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<s><emph type="italics"/>Impetus est aliquid distinctum à &longs;ubstantiâ mobilis.<emph.end type="italics"/></s><s> Demon&longs;tratur. </s>

<s><lb/>Quia &longs;ub&longs;tantia mobilis non e&longs;t cau&longs;a exigens motum per Th. </s>

<s><lb/>Quia &longs;ub&longs;tantia mobilis non e&longs;t cau&longs;a exigens motum per Th. 5. Impe<lb/>tus e&longs;t cau&longs;a exigens per Def. </s>

<s>5. Impe<lb/>tus e&longs;t cau&longs;a exigens per Def. </s>

<s>3. & Th. 6. de codem contradictoria dici <lb/>non po&longs;&longs;unt per Ax. 1. n. </s>

<s>6. de codem contradictoria dici <lb/>non po&longs;&longs;unt per Ax. </s>

<s>3. Igitur impetus non e&longs;t idem cum &longs;ub&longs;tantià <lb/>mobilis; igitur di&longs;tinctus; dcinde &longs;eparari pote&longs;t à &longs;ub&longs;tantia mobilis <lb/>per Hypoth. </s>

<s>4. igitur e&longs;t di&longs;tinctus per Ax. </s>

<s>4. igitur e&longs;t di&longs;tinctus per Ax. 2. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s>

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<s>& ex Hypoth. </s>

<s>3. & 4. igitur di&longs;tingui<lb/>tur à motu; igitur & ab impetu per Ax. </s>

<s>3. & 4. igitur di&longs;tingui<lb/>tur à motu; igitur & ab impetu per Ax. 2. igitur impetus non e&longs;t &longs;ub<lb/>&longs;tantia; igitur accidens. </s>

<s>2. igitur impetus non e&longs;t &longs;ub<lb/>&longs;tantia; igitur accidens. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus est qualitas Phy&longs;ica.<emph.end type="italics"/></s><s> Quia impetus e&longs;t di&longs;tinctus realiter à &longs;ue <lb/>&longs;ubiecto per Th. </s>

<s><emph type="italics"/>Impetus est qualitas Phy&longs;ica.<emph.end type="italics"/></s><s> Quia impetus e&longs;t di&longs;tinctus realiter à &longs;ue <lb/>&longs;ubiecto per Th. 7. E&longs;t enim &longs;eparabilis per Hypoth. </s>

<s>7. E&longs;t enim &longs;eparabilis per Hypoth. </s>

<s>3. & 4. igitur di<lb/>&longs;tinctus per Ax. </s>

<s>3. & 4. igitur di<lb/>&longs;tinctus per Ax. 2. &longs;ed qualitatem realiter di&longs;tinctam apello Phy&longs;icam; <lb/>præ&longs;ertim cum nec moralis &longs;it, nec Logica, &c. </s><pb pagenum="18"/>

<s>2. &longs;ed qualitatem realiter di&longs;tinctam apello Phy&longs;icam; <lb/>præ&longs;ertim cum nec moralis &longs;it, nec Logica, &c. </s><pb pagenum="18"/>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 12.<emph.end type="center"/></s>

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<s>7. Phy&longs;. </s>

<s>7. Phy&longs;. cap.

<s>cap.

2. Aliqui excu&longs;ant ip&longs;um Ari&longs;torelem, putantque <lb/>non e&longs;&longs;e locutum ex propriâ &longs;ententiâ: Alij dicunt Ari&longs;totelem quidem <lb/>tribui&longs;&longs;e aliquam vim extrin&longs;ecam aëri; non tamen nega&longs;&longs;e intrin&longs;ecam <lb/>impetus; quidquid &longs;it, ip&longs;a verba Ari&longs;totclis demon&longs;trant ip&longs;um agno<lb/>ui&longs;&longs;e vim motricem impre&longs;&longs;am aëri, hoc e&longs;t impetum (<emph type="italics"/>potentia enim<emph.end type="italics"/> (in<lb/>quit) &longs;cilicet motrix, <emph type="italics"/>quâ pollet proijciens qua&longs;i vim impre&longs;&longs;am tradit vtrique<emph.end type="italics"/>) <lb/>id e&longs;t aëri &longs;ur&longs;um, &deor&longs;um; quid porrò e&longs;t illa vis motrix, ni&longs;i impetus. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus non producit motum.<emph.end type="italics"/></s><s> Probatur, quia motus non dicitur pro<lb/>ductus per Th. </s>

<s><emph type="italics"/>Impetus non producit motum.<emph.end type="italics"/></s><s> Probatur, quia motus non dicitur pro<lb/>ductus per Th. 2. Adde &longs;i vis rationem metaphy&longs;icam; quia nihil cogit <lb/>dicere accidens aliquod, ex iis &longs;cilicet, quæ &longs;en&longs;u percipimus, agere ad <lb/>intra; quod videtur e&longs;&longs;e proprium &longs;ub&longs;tantiæ, &longs;altem naturaliter; vt <lb/>demon&longs;trabimus in Metaph. </s>

<s>2. Adde &longs;i vis rationem metaphy&longs;icam; quia nihil cogit <lb/>dicere accidens aliquod, ex iis &longs;cilicet, quæ &longs;en&longs;u percipimus, agere ad <lb/>intra; quod videtur e&longs;&longs;e proprium &longs;ub&longs;tantiæ, &longs;altem naturaliter; vt <lb/>demon&longs;trabimus in Metaph. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus exigit motum, id est fluxum mobilis in loco<emph.end type="italics"/>; quia cau&longs;a imme<lb/>diata motus e&longs;t tantum exigens, per Th. </s>

<s><emph type="italics"/>Impetus exigit motum, id est fluxum mobilis in loco<emph.end type="italics"/>; quia cau&longs;a imme<lb/>diata motus e&longs;t tantum exigens, per Th. 4. &longs;ed impetus e&longs;t cau&longs;a motus <lb/>immediata per Th. 5. & 6. igitur e&longs;t cau&longs;a exigens, adde quod id tantùm <lb/>accidens &longs;en&longs;ibile præ&longs;tare pote&longs;t in &longs;uo &longs;ubiecto, vt aliquam illius mu<lb/>tationem præ&longs;tet, vel exigat; quæ vel e&longs;t localis, hoc e&longs;t fluxus quidam: <pb pagenum="19"/>per &longs;patium loci; vel alteratiua, vt vulgò vocatur; quà &longs;cilicet vel re<lb/>&longs;oluuntur partes, vel rare&longs;iunt, vel lique&longs;cunt, vel concre&longs;cunt &c. </s>

<s>4. &longs;ed impetus e&longs;t cau&longs;a motus <lb/>immediata per Th. </s>

<s>5. & 6. igitur e&longs;t cau&longs;a exigens, adde quod id tantùm <lb/>accidens &longs;en&longs;ibile præ&longs;tare pote&longs;t in &longs;uo &longs;ubiecto, vt aliquam illius mu<lb/>tationem præ&longs;tet, vel exigat; quæ vel e&longs;t localis, hoc e&longs;t fluxus quidam: <pb pagenum="19"/>per &longs;patium loci; vel alteratiua, vt vulgò vocatur; quà &longs;cilicet vel re<lb/>&longs;oluuntur partes, vel rare&longs;iunt, vel lique&longs;cunt, vel concre&longs;cunt &c. </s>

<s>vel <lb/>demùm mutant &longs;en&longs;ibilem &longs;tatum; vel e&longs;t perfectiua aliquo modo, qua<lb/>tenus &longs;ubiectum nouam aliquam habitudinem acquirit ad &longs;en&longs;us; &longs;ic <lb/>lumen illuminando obiectum reddit illud vi&longs;ibile. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 15.<emph.end type="center"/></s>

<s><emph type="italics"/>Motus e&longs;t effectus formalis &longs;ecundarius impetus.<emph.end type="italics"/></s><s> Cum enim &longs;it cau&longs;a <lb/>exigens per Th. </s>

<s><emph type="italics"/>Motus e&longs;t effectus formalis &longs;ecundarius impetus.<emph.end type="italics"/></s><s> Cum enim &longs;it cau&longs;a <lb/>exigens per Th. 121. Voco effectum formalem &longs;ecundarium, quem in <lb/>mobili exigit impetus; quippe, vt iam dictum e&longs;t, cau&longs;a exigens redu<lb/>citur ad formalem; nec enim cau&longs;at aliquid producendo, quod &longs;pectat ad <lb/>efficientem; nec mouendo, quod &longs;pectat ad finalem; nec determinando, <lb/>quod &longs;pectat ad obiectiuam; nec recipiendo, quod &longs;pectat ad materia<lb/>lem; nec dirigendo, quod &longs;pectat ad idæalem, vel exemplarem; &longs;ed <lb/>exigendo; quatenus &longs;cilicet ad id à natura e&longs;t in&longs;tituta, vt ex eius in <lb/>&longs;ubiecto præ&longs;entia talis affectio, vel mutatio con&longs;equatur; vocatur au<lb/>tem effectus formalis &longs;ecundarius; non verò primarius, qui e&longs;t tantùm <lb/>concretum ex ip&longs;a formâ, & &longs;ubiecto. </s>

<s>121. Voco effectum formalem &longs;ecundarium, quem in <lb/>mobili exigit impetus; quippe, vt iam dictum e&longs;t, cau&longs;a exigens redu<lb/>citur ad formalem; nec enim cau&longs;at aliquid producendo, quod &longs;pectat ad <lb/>efficientem; nec mouendo, quod &longs;pectat ad finalem; nec determinando, <lb/>quod &longs;pectat ad obiectiuam; nec recipiendo, quod &longs;pectat ad materia<lb/>lem; nec dirigendo, quod &longs;pectat ad idæalem, vel exemplarem; &longs;ed <lb/>exigendo; quatenus &longs;cilicet ad id à natura e&longs;t in&longs;tituta, vt ex eius in <lb/>&longs;ubiecto præ&longs;entia talis affectio, vel mutatio con&longs;equatur; vocatur au<lb/>tem effectus formalis &longs;ecundarius; non verò primarius, qui e&longs;t tantùm <lb/>concretum ex ip&longs;a formâ, & &longs;ubiecto. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s>

<s><emph type="italics"/>Ni&longs;i e&longs;&longs;et motus non e&longs;&longs;et impetus.<emph.end type="italics"/></s><s> Probatur quia motus e&longs;t finis intrin<lb/>&longs;ecus impetus per Th. 16. igitur &longs;i nullus motus e&longs;&longs;e po&longs;&longs;et, &longs;uo finc ca<lb/>reret impetus; igitur non e&longs;&longs;et, vt patet, igitur non e&longs;&longs;et; quia quod <lb/>fru&longs;trà e&longs;t, non e&longs;t per Ax. 6. nec ob&longs;tat quod &longs;uprà indicatum e&longs;t de im <lb/>petu naturali primo vel innato (&longs;ic enim deinceps appellabimus vtrecti <lb/>di&longs;tinguamus ab acqui&longs;ito quem vocabimus impetum accelerationis) <lb/>qui &longs;ine motu con&longs;eruatur in corpore grauitante; quia ni&longs;i po&longs;&longs;ibilis e&longs;<lb/>&longs;et motus deor&longs;um nulla e&longs;&longs;et grauitatio; quippe grauitare e&longs;t dcor<lb/>&longs;um inclinari, motumque inclinationis impediri; hinc dicemus <pb pagenum="20"/>in &longs;ecundo libro impetum innatum &longs;æpiùs e&longs;&longs;e &longs;ine motu; cum &longs;cilicet <lb/>impeditur à corpore &longs;u&longs;tinente? </s>

<s>16. igitur &longs;i nullus motus e&longs;&longs;e po&longs;&longs;et, &longs;uo finc ca<lb/>reret impetus; igitur non e&longs;&longs;et, vt patet, igitur non e&longs;&longs;et; quia quod <lb/>fru&longs;trà e&longs;t, non e&longs;t per Ax. </s>

<s>6. nec ob&longs;tat quod &longs;uprà indicatum e&longs;t de im <lb/>petu naturali primo vel innato (&longs;ic enim deinceps appellabimus vtrecti <lb/>di&longs;tinguamus ab acqui&longs;ito quem vocabimus impetum accelerationis) <lb/>qui &longs;ine motu con&longs;eruatur in corpore grauitante; quia ni&longs;i po&longs;&longs;ibilis e&longs;<lb/>&longs;et motus deor&longs;um nulla e&longs;&longs;et grauitatio; quippe grauitare e&longs;t dcor<lb/>&longs;um inclinari, motumque inclinationis impediri; hinc dicemus <pb pagenum="20"/>in &longs;ecundo libro impetum innatum &longs;æpiùs e&longs;&longs;e &longs;ine motu; cum &longs;cilicet <lb/>impeditur à corpore &longs;u&longs;tinente? </s>

<s>immò dicemus infrà primo in&longs;tanti, <lb/>quo e&longs;t impetus, nondum e&longs;&longs;e motum. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s>

<s><emph type="italics"/>Ni&longs;i e&longs;&longs;et impetus, non e&longs;&longs;et naturaliter motus.<emph.end type="italics"/></s><s> Quia ni&longs;i e&longs;&longs;etcau&longs;a, non <lb/>e&longs;&longs;et naturaliter effectus per Ax. 8. Impetus enim e&longs;t cau&longs;a motus per <lb/>Th.15. Deinde omnis motus e&longs;t ab aliqua potentia motrice, vt patet ex <lb/>omni hypothe&longs;i; &longs;iue &longs;it naturalis in grauibus, & leuibus, &longs;iue &longs;evitalis <lb/>in viuentibus; &longs;iue &longs;it media in compre&longs;&longs;is, & dilatatis; &longs;iue alia quæli<lb/>bet: &longs;ed omnis potentia motrix e&longs;t actiua, quia mouet; ergo agit, &longs;ed <lb/>motum non producit per Th. 2. Igitur impetum, qui deinde exigit mo<lb/>tum per Th. 14. Dixi naturaliter; quia non e&longs;t dubium, quin Deus &longs;ine <lb/>impetu aliquo modo mouere po&longs;&longs;it; ide&longs;t, facere &longs;ine impetu, vt corpus <lb/>mutet locum: nec dicas Deum non po&longs;&longs;e &longs;upplere vices cau&longs;æ formalis; <lb/>nam concedo id quidem pro effectu formali primario; nec enim Deus <lb/>pote&longs;t facere, vt aliquid &longs;it calidum &longs;ine calore; cum e&longs;&longs;e calidum &longs;it <lb/>idem, ac e&longs;&longs;e habens calorem; id tamen nego pro effectu &longs;ecundario, <lb/>quem &longs;cilicet cau&longs;a formalis exigit: Etenim &longs;icut pote&longs;t &longs;ummo iure non <lb/>&longs;atisfacere exigentiæ; ita pote&longs;t id <expan abbr="cõferre">conferre</expan> &longs;ine exigentiâ, quòd cum exi<lb/>gentia conferre pote&longs;t; &longs;ic pote&longs;t corpus re&longs;oluere &longs;ine calore, mouere <lb/>fine impetu &c. </s>

<s>8. Impetus enim e&longs;t cau&longs;a motus per <lb/>Th.15. Deinde omnis motus e&longs;t ab aliqua potentia motrice, vt patet ex <lb/>omni hypothe&longs;i; &longs;iue &longs;it naturalis in grauibus, & leuibus, &longs;iue &longs;evitalis <lb/>in viuentibus; &longs;iue &longs;it media in compre&longs;&longs;is, & dilatatis; &longs;iue alia quæli<lb/>bet: &longs;ed omnis potentia motrix e&longs;t actiua, quia mouet; ergo agit, &longs;ed <lb/>motum non producit per Th. </s>

<s>2. Igitur impetum, qui deinde exigit mo<lb/>tum per Th. </s>

<s>14. Dixi naturaliter; quia non e&longs;t dubium, quin Deus &longs;ine <lb/>impetu aliquo modo mouere po&longs;&longs;it; ide&longs;t, facere &longs;ine impetu, vt corpus <lb/>mutet locum: nec dicas Deum non po&longs;&longs;e &longs;upplere vices cau&longs;æ formalis; <lb/>nam concedo id quidem pro effectu formali primario; nec enim Deus <lb/>pote&longs;t facere, vt aliquid &longs;it calidum &longs;ine calore; cum e&longs;&longs;e calidum &longs;it <lb/>idem, ac e&longs;&longs;e habens calorem; id tamen nego pro effectu &longs;ecundario, <lb/>quem &longs;cilicet cau&longs;a formalis exigit: Etenim &longs;icut pote&longs;t &longs;ummo iure non <lb/>&longs;atisfacere exigentiæ; ita pote&longs;t id <expan abbr="cõferre">conferre</expan> &longs;ine exigentiâ, quòd cum exi<lb/>gentia conferre pote&longs;t; &longs;ic pote&longs;t corpus re&longs;oluere &longs;ine calore, mouere <lb/>fine impetu &c. </s>

<s>quanquam vt verum fatear non e&longs;&longs;et propriè motus, &longs;ed <lb/>qua&longs;i continuæ reproductionis modus; nam motus dicit aliquam pa&longs;<lb/>&longs;ionem; &longs;cilicet actum entis in potentiâ, vt aiunt. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s>

<s><emph type="italics"/>Si e&longs;&longs;et aliquod corpus e&longs;&longs;entialiter immobile, e&longs;&longs;et incapax impetus.<emph.end type="italics"/></s><s> Pro<lb/>batur; quia, ni&longs;i e&longs;&longs;et motus, non e&longs;&longs;et impetus per Th. </s>

<s>17. igitur &longs;ubic<lb/>ctum incapax motus e&longs;t incapax impetus. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s>

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<s><emph type="italics"/>Omne corpus, quod e&longs;t capax motus, e&longs;t capax impetus, & vici&longs;&longs;im.<emph.end type="italics"/><lb/>Probatur 1. pars; quia impetus in eo non e&longs;&longs;et fru&longs;trà; haberet enim <lb/>fuum effectum formalem, & finem intrin&longs;ecum. </s>

<s>Probatur 2.pars; quia in <lb/>eo impetus non e&longs;&longs;et fru&longs;trà per Ax. </s>

<s>Probatur 2.pars; quia in <lb/>eo impetus non e&longs;&longs;et fru&longs;trà per Ax. 6. igitur haberet &longs;uum effectum; <lb/>igitur motum. </s>

<s>6. igitur haberet &longs;uum effectum; <lb/>igitur motum. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s>

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<s>Dices &longs;patio imaginario; apage i&longs;tas nugas: <lb/>de i&longs;to &longs;patio plura demon&longs;trabimus in Metaphy. </s>

<s>Probatur 2. pars; quia <lb/>&longs;i e&longs;t capax motus, e&longs;t capax impetus per Th. </s>

<s>Probatur 2. pars; quia <lb/>&longs;i e&longs;t capax motus, e&longs;t capax impetus per Th. 24. Quod dixi de corpo<lb/>re; dicendum e&longs;t de omni re creata finita permanente. </s>

<s>24. Quod dixi de corpo<lb/>re; dicendum e&longs;t de omni re creata finita permanente. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s>

<s><emph type="italics"/>Duratio moueri non pote&longs;t.<emph.end type="italics"/></s><s> Cum enim &longs;it &longs;ucce&longs;&longs;iua, fluit per partes, <lb/>igitur quælibet illius pars, &longs;eu quod durat vna in&longs;tanti tantùm e&longs;t inca<lb/>pax motus, per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc actio moueri non pote&longs;t<emph.end type="italics"/>; cum enim actio per quam res con&longs;erua<lb/>tur, &longs;it eius duratio; vt con&longs;tabit ex iis, quæ demon&longs;trabimus in Me<lb/>taphy&longs;ica, & cum duratio moucri non po&longs;&longs;it, per Th. </s>

<s>29. certè neque <lb/>actio moueri pote&longs;t. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s>

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<s>Ob&longs;eruabis tertiò aliud naturæ in&longs;titutum, quo &longs;cilicet in eo tan<lb/>tùm &longs;ubiecto recipi pote&longs;t cau&longs;a formalis, in quo recipi pote&longs;t cius effe<lb/>ctus formalis &longs;ecundarius: nec alia regula, præter eam excogitari pote&longs;t; <lb/>cum enim aliqua forma ad talem, vel talem finem à natura in&longs;tituta e&longs;t; <lb/>certè propter illum finem e&longs;t, igitur in co non e&longs;t, in quo &longs;uum finem <lb/>con&longs;equi non pote&longs;t; alioquin fru&longs;trà e&longs;&longs;et; & contra in eo e&longs;&longs;e pote&longs;t, <lb/>in quo fru&longs;trà non e&longs;t; cum &longs;cilicet in eo &longs;uum finem con&longs;equatur; ad<lb/>de quod finis ille intrin&longs;ecus phy&longs;icus &longs;cilicet, non moralis, aliquis no<lb/>uus effectus e&longs;t; atqui nouus effectus &longs;ine &longs;ua cau&longs;a e&longs;&longs;e non pote&longs;t, neque <lb/>cau&longs;a nece&longs;&longs;aria &longs;ine e&longs;&longs;ectu; igitur ibi, &longs;cilicet in hoc &longs;ubiecto, in quo <lb/>e&longs;t, vel e&longs;&longs;e pote&longs;t effectus formalis, cau&longs;a formalis e&longs;t, vel e&longs;&longs;e pote&longs;t, <lb/>e&longs;t inquam citra miraculum. </s>

<s>Ob&longs;eruabis quartò egregiam rationem; propter quam res eadem in <lb/>pluribus locis naturaliter e&longs;&longs;e non pote&longs;t; quippe cum res fuerit primo <lb/>producta in aliquo loco, illa certè nouum locum acquirere non pote&longs;t <lb/>naturaliter; ni&longs;i per motum, atqui motus dicit nece&longs;&longs;ario priorem lo<lb/>tum relictum, & nouum acqui&longs;itum; igitur cum tot acquirantur loca <lb/>per motum, quot relinquuntur; &longs;i ante motum vnus tantùm erat eiu&longs;<lb/>dem rci locus, po&longs;t motum etiam vnus e&longs;t: quod autem producatur tan-<pb pagenum="24"/>tùm res in vno loco patet; vel enim à cau&longs;a prima vel ab aliqua 2. pro<lb/>ducitur; &longs;i à 2. ergo ab aliqua aplicata; igitur ex &longs;uppo&longs;itione quòd il<lb/>la cau&longs;a 2. in vno tantùm loco producta &longs;it, vni tantum applicari po<lb/>te&longs;t; quod autem cau&longs;a 1. in pluribus locis naturaliter eundem effectum <lb/>non producat, certum e&longs;t, & demon&longs;trabimus in Metaphy&longs;. </s>

<s>quia &longs;in<lb/>gulis effectibus &longs;ingulæ &longs;ufficiunt actiones; &longs;ingulis terminis &longs;ingulæ <lb/>viæ; immò hoc requiri videtur, &longs;eu &longs;pectare ad huius vniuer&longs;itatis or<lb/>dinem; quippe &longs;i res eadem in pluribus locis e&longs;&longs;et; cur potius in duo<lb/>bus quam in tribus? </s>

<s>deinde multiplex iure po&longs;&longs;et exi&longs;timari; denique <lb/>quod vnum e&longs;t in entitate creata, &longs;eu dependente ab cadem cau&longs;a, vnum <lb/>e&longs;t etiam in dependentia; quæ e&longs;t actio, per quam dependet; &longs;ed de his <lb/>aliàs. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus non producitur in eo mobili, quod moueri non pote&longs;t à potentia <lb/>motrice applicata, licèt à fortiori moueri po&longs;&longs;it.<emph.end type="italics"/></s><s> Probatur, quia impetus <lb/>e&longs;t tantùm propter motum, qui eius effectus e&longs;t, & finis, per Th. 15. <lb/>& 16. Igitur vbi non e&longs;t motus, fru&longs;trà e&longs;t impetus; &longs;ed quod fru&longs;trà <lb/>e&longs;t, non e&longs;t; id e&longs;t non e&longs;t, quod fru&longs;trà e&longs;&longs;et, &longs;i e&longs;&longs;et per Ax. 6. Exci<lb/>pio tamen impetum naturalem innatum, qui nunquam e&longs;t fru&longs;trà, vt <lb/>dictum e&longs;t &longs;uprà in Theorem. </s>

<s>15. <lb/>& 16. Igitur vbi non e&longs;t motus, fru&longs;trà e&longs;t impetus; &longs;ed quod fru&longs;trà <lb/>e&longs;t, non e&longs;t; id e&longs;t non e&longs;t, quod fru&longs;trà e&longs;&longs;et, &longs;i e&longs;&longs;et per Ax. </s>

<s>6. Exci<lb/>pio tamen impetum naturalem innatum, qui nunquam e&longs;t fru&longs;trà, vt <lb/>dictum e&longs;t &longs;uprà in Theorem. </s>

<s>3. id manife&longs;tè probari; nam qui diceret in rupe im<lb/>mobili impetum imprimi; certè po&longs;itiuo argumento probare tenere<lb/>tur, quod tantùm duci pote&longs;t, vel ab experimento; atqui hîc nullum e&longs;t; <lb/>vel à nece&longs;&longs;itate, quæ nulla e&longs;t; vel ex alio quocumque capite, quod <lb/>nullum excogitari pote&longs;t; &longs;ed maiorem lucem huic Th. </s>

<s>3. ex proximè <lb/>&longs;equentibus accer&longs;emus; nec e&longs;t quòd aliqui dicant produci impetum <lb/>inefficacem; qui cum fru&longs;trà &longs;it, &longs;i e&longs;t, ex nullo capite probari pote&longs;t: ad<lb/>de quòd de&longs;truitur impetus, ne &longs;it fru&longs;trà; Igitur non producitur, ne &longs;it <lb/>fru&longs;trà; nam con&longs;eruatio e&longs;t vera actio, vt dicemus &longs;uo loco; Igitur &longs;i <lb/>hæc non ponitur, ne aliquid &longs;it fru&longs;trà; etiam 1. productio poni non <lb/>debet; vnde commentum illud impetus inefficacis pror&longs;us inefficax e&longs;t. </s>

<s>17. adde quod non pote&longs;t cogno&longs;oi <lb/>impetus, ni&longs;i vel ex motu, vel ex ictu, vel ex contrario ni&longs;u, vel <lb/>impul&longs;u; &longs;ed nihil horum cernitur in rupe quam ferio; Igitur non <lb/>e&longs;t dicendum in ea produci impetum, cuius rationem afferemus infrà; <lb/>nunc&longs;atis e&longs;t Ax. 3. id manife&longs;tè probari; nam qui diceret in rupe im<lb/>mobili impetum imprimi; certè po&longs;itiuo argumento probare tenere<lb/>tur, quod tantùm duci pote&longs;t, vel ab experimento; atqui hîc nullum e&longs;t; <lb/>vel à nece&longs;&longs;itate, quæ nulla e&longs;t; vel ex alio quocumque capite, quod <lb/>nullum excogitari pote&longs;t; &longs;ed maiorem lucem huic Th. 3. ex proximè <lb/>&longs;equentibus accer&longs;emus; nec e&longs;t quòd aliqui dicant produci impetum <lb/>inefficacem; qui cum fru&longs;trà &longs;it, &longs;i e&longs;t, ex nullo capite probari pote&longs;t: ad<lb/>de quòd de&longs;truitur impetus, ne &longs;it fru&longs;trà; Igitur non producitur, ne &longs;it <lb/>fru&longs;trà; nam con&longs;eruatio e&longs;t vera actio, vt dicemus &longs;uo loco; Igitur &longs;i <lb/>hæc non ponitur, ne aliquid &longs;it fru&longs;trà; etiam 1. productio poni non <lb/>debet; vnde commentum illud impetus inefficacis pror&longs;us inefficax e&longs;t. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s>

<s><emph type="italics"/>Ideo potentia motrix non producit impetum in pradicta rupe.<emph.end type="italics"/> v.g. <emph type="italics"/>quia de<lb/>bilior e&longs;t.<emph.end type="italics"/></s><s> Probatur, & explicatur; quippedebilior potentia minorem ef<lb/>fectum producit per. </s>

<s>Ax. 13. <emph type="italics"/>num.<emph.end type="italics"/> 2. igitur pauciores partes impetus <lb/>æquales vni certæper idem <emph type="italics"/>num.<emph.end type="italics"/> 1. igitur &longs;i &longs;int plures partes &longs;ubiecti <lb/>mobilis, &longs;eu rupis, quàm impetus; cum vna pars impetus duobus parti<lb/>bus &longs;ubiecti ine&longs;&longs;e non po&longs;&longs;it; licet plures vni &longs;imul in e&longs;&longs;e po&longs;&longs;int; <lb/>non e&longs;t mirum &longs;i nullus impetus producatur; cum non po&longs;&longs;int tot partes <lb/>illius produci, quot e&longs;&longs;ent nece&longs;&longs;ariæ; vt &longs;altem &longs;ingulæ &longs;ingulis &longs;ubie<lb/>cti, &longs;eu rupis partibus di&longs;tribuerentur. </s><pb pagenum="25"/>

<s>13. <emph type="italics"/>num.<emph.end type="italics"/> 2. igitur pauciores partes impetus <lb/>æquales vni certæper idem <emph type="italics"/>num.<emph.end type="italics"/> 1. igitur &longs;i &longs;int plures partes &longs;ubiecti <lb/>mobilis, &longs;eu rupis, quàm impetus; cum vna pars impetus duobus parti<lb/>bus &longs;ubiecti ine&longs;&longs;e non po&longs;&longs;it; licet plures vni &longs;imul in e&longs;&longs;e po&longs;&longs;int; <lb/>non e&longs;t mirum &longs;i nullus impetus producatur; cum non po&longs;&longs;int tot partes <lb/>illius produci, quot e&longs;&longs;ent nece&longs;&longs;ariæ; vt &longs;altem &longs;ingulæ &longs;ingulis &longs;ubie<lb/>cti, &longs;eu rupis partibus di&longs;tribuerentur. </s><pb pagenum="25"/>

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

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<s><emph type="italics"/>Vel producitur impetus in omnibus &longs;ubiecti partibus vnitis, vel in nulla; <lb/>modò nulla fiat &longs;eparatio, neque compre&longs;&longs;io<emph.end type="italics"/>: Certum e&longs;t enim in ijs omni<lb/>bus partibus, quæ auolant ab ictu, produci impetum. </s>

<s>Probatur igitur <lb/>1. quia &longs;i non producatur in omnibus partibus, & nulla &longs;eparetur ab <lb/>alijs; certè nulla mouetur, vt certum e&longs;t; igitur nulla habet impetum; <lb/>quia ibi non e&longs;t cau&longs;a formalis, vbi non e&longs;t effectus formalis; alicquin <lb/>e&longs;&longs;et fru&longs;trà, contra Ax. </s>

<s>6.2. Tu dicis produci impetum in aliquot parti<lb/>hus; hoc dicis, hoc proba? </s>

<s>an potes digno&longs;cere impetum ni&longs;i ex motu? </s>

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<s>Igitur certum e&longs;tvel produci in omnibus, vel <lb/>in nulla, ni&longs;i forte aliquæ auolent, &longs;ed tunc &longs;eparantur. </s>

<s>Obiiciet aliquis 1. e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam applicatam &longs;ubiecto ap<lb/>to: igitur agit per Ax. </s>

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

<s>12. Re&longs;pondeo e&longs;&longs;e impeditam; nam re&longs;i&longs;tentia <lb/>&longs;ubiecti &longs;uperat vires potentiæ vt dictum e&longs;t; immò in ip&longs;o motu re<lb/>torqueo argumentum; licèt enim &longs;it applicata cau&longs;a nec&longs;&longs;aria mouens, <lb/>non tamen mouet. </s>

<s>Obiiciet 2. Ignis applicatus agit in nonnullas partes fubiecti, licèt <lb/>non agat in omnes; igitur & potentia motrix. </s>

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<s>Re&longs;pondeo &longs;ingulos producere 100. <lb/>actione &longs;cilicet communi indiui&longs;ibiliter; &longs;int enim duo trahentes A. & <lb/>B. A. producit 100. &longs;ed non &longs;olus; B. producit ea&longs;dem 100. &longs;ed non &longs;o<lb/>lus; &longs;ed explicabimus hunc modum actionis communis in Metaphys. </s>

<s><lb/>quod autem agant actione communi patet per Ax. </s>

<s><lb/>quod autem agant actione communi patet per Ax. 13. </s>

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

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s>

<s><emph type="italics"/>Primo in&longs;tanti, quo est impetus, non est ille motus, cuius hic impetus e&longs;t <lb/>cau&longs;a.<emph.end type="italics"/></s><s> Probatur; quia non pote&longs;t e&longs;&longs;e motus, ni&longs;i &longs;it locus prior reli<lb/>ctus, & nouus acqui&longs;itus, igitur &longs;i eodem in&longs;tanti, quo e&longs;t impetus, <lb/>haberet motum, codem in&longs;tanti e&longs;&longs;et in duobus locis, quod dici non <lb/>pote&longs;t; & iam diximus in Th. </s>

<s>26. igitur impetus primo in&longs;tanti quo <lb/>e&longs;t non habet &longs;uum motum. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s>

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<s><lb/>2. &longs;icut vna pars caloris non re&longs;oluit alias partes &longs;ubiecti; ita nec im<lb/>petus. </s>

<s>3. Ratio à priori e&longs;t; quia impetus non e&longs;t cau&longs;a efficiens motus <lb/>per Th. </s>

<s>3. Ratio à priori e&longs;t; quia impetus non e&longs;t cau&longs;a efficiens motus <lb/>per Th. 13. &longs;ed tantùm cau&longs;a formalis per Th. 15. Igitur præ&longs;tat tantùm <lb/>&longs;uum effectum formalem in eo &longs;ubiecto, in quo e&longs;t. </s>

<s>13. &longs;ed tantùm cau&longs;a formalis per Th. </s>

<s>15. Igitur præ&longs;tat tantùm <lb/>&longs;uum effectum formalem in eo &longs;ubiecto, in quo e&longs;t. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s>

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

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s>

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<s>&longs;i <lb/>mobile &longs;it ferrum calidum, recipitur in ip&longs;a &longs;ub&longs;tantia ferri; non verò <lb/>in ip&longs;o calore (ex &longs;uppo&longs;itione quod calor &longs;it accidens, vt aliàs demon<lb/>&longs;trabimus; nec in alijs accidentibus, &longs;i quæ &longs;unt, in eodem &longs;ubiecto; pro<lb/>batur 1. quia &longs;i produceretur etiam impetus in accidentibus, quo plu<lb/>ra e&longs;&longs;ent accidentia in aliquo &longs;ubiecto; plures quoque partes impetus <lb/>producendæ e&longs;&longs;ent; igitur maiori potentiâ opus e&longs;&longs;et per Ax. </s>

<s>4. <lb/>Igitur difficiliùs mouerentur, quod e&longs;t ab&longs;urdum. </s>

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<s>Dices, igitur &longs;imiliter dici pote&longs;t non recipi impetum in omni<lb/>bus partibus &longs;ubiecti mobilis, &longs;ed in vnâ dumtaxat; cui cum <lb/>aliæ &longs;int vnitæ, exigunt moueri &longs;ine impetu ad illius motum? </s>

<s>cum <lb/>hoc ip&longs;um ad omnem vnionem &longs;pectare videatur; Re&longs;pondeo vnam <pb pagenum="30"/>partem plumbi ita coniungi cum alia, vt etiam &longs;eparata naturaliter <lb/>exi&longs;tere po&longs;&longs;it; igitur non e&longs;t par ratio; præterea vna pars plumbi non <lb/>e&longs;t in loco alterius; nec enim inuicem penetrantur cum &longs;it compene<lb/>tratio accidentium cum &longs;ubiecto; deinde, quò plures &longs;unt partes vnitæ, <lb/>maior e&longs;t re&longs;i&longs;tentia, quæ ip&longs;o etiam &longs;en&longs;u percipitur; denique non vide<lb/>rur cur potius produceretur in vna parte, quam in alia; quæ omnia <lb/>iam &longs;uprà Th. </s>

<s>33. demon&longs;trauimus. </s>

<s>Adde quod &longs;i impetus produceretur in ip&longs;is accidentibus, etiam in <lb/>ip&longs;o impetu prius producto alius impetus produceretur; cum &longs;cilicet <lb/>noua fit impetus acce&longs;&longs;io; quod &longs;atis ridiculum e&longs;t; qua&longs;i verò impetus <lb/>indigeat impetu &c. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s>

<s><emph type="italics"/>Quando corpus pellitur ab alio corpore per impetum impre&longs;&longs;um; band da<lb/>biè impetus ille impre&longs;&longs;us ab aliqua cau&longs;a efficiente producijur<emph.end type="italics"/>; patet <lb/>per Ax. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s>

<s><emph type="italics"/>Ille impetus non producitur à &longs;ub&longs;tantia corporis in aliud impacti.<emph.end type="italics"/></s><s> Proba<lb/>tur; quia &longs;i produceretur, e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt <expan abbr="clar&utilde;">clarum</expan> e&longs;t; igitur appli<lb/>cata, & non impedita ageret per Ax. </s>

<s>32. quod e&longs;t contra experientiam. </s>

<s><lb/>Dicunt aliqui requiri <expan abbr="mot&utilde;">motum</expan> præuium, vt agat; &longs;ed contra; nam motus <lb/>præuius non requiritur vt cau&longs;a, vt patet; quia cau&longs;a vt agat debet exi<lb/>&longs;tere per Ax. 9. Igitur requiritur, vt conditio, quod dici non pote&longs;t; <lb/>quia primo etiam conditio debet e&longs;&longs;e præ&longs;ens; &longs;ed motus præuius de <lb/>nihil pre&longs;enti e&longs;t &longs;ecundo quia non pote&longs;t excogitari aliud munus con<lb/>ditionis; ni&longs;i vel vt tollat impedimentum, vel vt applicet cau&longs;am &longs;ubie<lb/>cto apto; præterea motus præuius non e&longs;t; igitur codem modo &longs;e <lb/>habet, ac &longs;i nunquam extiti&longs;&longs;et; & &longs;i co in&longs;tanti quo corpus impa<lb/>ctum primo tangit, amitteret totum impetum, ita vt expræterito motu <lb/>nihil reliquum e&longs;&longs;et, haud dubiè corpus aliud non pelleret. </s>

<s>9. Igitur requiritur, vt conditio, quod dici non pote&longs;t; <lb/>quia primo etiam conditio debet e&longs;&longs;e præ&longs;ens; &longs;ed motus præuius de <lb/>nihil pre&longs;enti e&longs;t &longs;ecundo quia non pote&longs;t excogitari aliud munus con<lb/>ditionis; ni&longs;i vel vt tollat impedimentum, vel vt applicet cau&longs;am &longs;ubie<lb/>cto apto; præterea motus præuius non e&longs;t; igitur codem modo &longs;e <lb/>habet, ac &longs;i nunquam extiti&longs;&longs;et; & &longs;i co in&longs;tanti quo corpus impa<lb/>ctum primo tangit, amitteret totum impetum, ita vt expræterito motu <lb/>nihil reliquum e&longs;&longs;et, haud dubiè corpus aliud non pelleret. </s>

<s>Diceret alius impetum e&longs;&longs;e tantùm conditionem, quæ &longs;emper e&longs;t <lb/>de præ&longs;enti: ad hanc in&longs;tantiam non valet &longs;uperior re&longs;pon&longs;io; & certè <lb/>&longs;i eo ip&longs;o in&longs;tanti contactus noua fieret impetus acce&longs;&longs;io; haud dubiè <lb/>maior e&longs;&longs;et ictus; licèt cum codem motu præuio, & tamen idem e&longs;&longs;et <lb/>corpus <expan abbr="impact&utilde;">impactum</expan>, Igitnr ad hanc <expan abbr="in&longs;tantiã">in&longs;tantiam</expan> alio modo re&longs;pondeo, ex appli<lb/>catione impetus &longs;emper &longs;equitur productio alterius impetus; dum &longs;cili<lb/>cet &longs;ubiectum, cui applicatur &longs;it capax motus; ex applicatione corporis <lb/>&longs;eu &longs;ubiecti ip&longs;ius non &longs;emper &longs;equitur; igitur dicendum e&longs;t impetum <lb/>ip&longs;um e&longs;&longs;e cau&longs;am alterius per Ax. 11. n. </s>

<s>1. voco enim illud cau&longs;am, <lb/>ex cuius applicatione &longs;emper &longs;equitur &longs;imilis effectus; alioquin &longs;i hoc <lb/>neges; proba mihi aliter ignem accendi ab alio igne; dicam enim tibi <lb/>ignem applicatum e&longs;&longs;e tantùm conditionem, & produci à c&oelig;lo; proba <lb/>mihi aliter calorem produci à calore? </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 42.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus producitur &longs;emper ad extra ab alio impetu.<emph.end type="italics"/></s><s> Quia cum &longs;emper <lb/>ad illius productionem requiratur applicatio alterius impetus; certè <lb/>non e&longs;t ponenda alia cau&longs;a per Ax. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s>

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

<s>17.ce rtè totus e&longs;t impetus propter motum <lb/>qui e&longs;t eius finis; igitur non agit ni&longs;i propter motum: &longs;ed non pote&longs;t <lb/>excogitari, quid faciat propter motum, dum agit, ni&longs;i dicamus ideo <lb/>tantùm agere, vt tollatur impedimentum; cum certum &longs;it corpus im<lb/>mobile, in quod impingitur aliud mobile, impedire eius motum. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 46.<emph.end type="center"/></s>

<s><emph type="italics"/>Omnes partes impetus mobilis agunt ad extra actione communi.<emph.end type="italics"/></s><s> Probatur <lb/>per Ax. </s>

<s><emph type="italics"/>Omnes partes impetus mobilis agunt ad extra actione communi.<emph.end type="italics"/></s><s> Probatur <lb/>per Ax. 13. n. </s>

<s>1. ni&longs;i enim agerent actione communi &longs;ed quælibet &longs;uam <lb/>produceret; cur potius in hac parte &longs;ubiecti, quam in alia, demde ap<lb/>pìicatur tantùm vna immediatè; Igitur agunt omnes actione commu<lb/>ni; omnes inquam illæ, quæ impediuntur; cum enim impetus agat <lb/>tantùm ad extrà vt tollat impedimentum &longs;ui motus; ille pro&longs;ectò age<lb/>re non debet, cuius motus vel effectus non impeditur. </s><pb pagenum="34"/>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc maiora corpora putà onerariæ naues, licèt tardi&longs;&longs;imo motu ferantur, <lb/>cum in aliud corpus impinguntur maxima vi illud impellunt.<emph.end type="italics"/></s><s> Ratio e&longs;t; <lb/>quia cum &longs;int plures partes impetus in pluribus partibus &longs;ubiecti, & <lb/>omnes agant actione communi, non mirum e&longs;t &longs;i maiorem effectum <lb/>producant, per Ax. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 48.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus, cuius motus non impeditur, non agit ad extrà.<emph.end type="italics"/></s><s> Probatur per <lb/>Th. </s>

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

<s>44. hinc &longs;i aliud corpus affigas mobili à tergo, nullum impetum in <lb/>eo producet, cuius effectus, qui certè impetui &longs;ingularis e&longs;t, alia ratio <lb/>e&longs;&longs;e non pote&longs;t; tam enim corpus e&longs;t applicatum à tergo, quam in <lb/>ip&longs;a fronte; & nihil e&longs;t in vno, quod non &longs;it in alio, ni&longs;i quod in fronte <lb/>impedit motum, à tergo verò non impedit. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s>

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

<s>12. vnde vt agat cau&longs;a nece&longs;&longs;aria, debet <lb/>applicari debito modo; &longs;i agat in orbem, omnis applicatio &longs;ufficiens <lb/>e&longs;t: &longs;i verò agat tantùm per vnam lineam; certè applicari debet in ca <lb/>linea; alioquin non aget defectu debitæ applicationis. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s>

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

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s>

<s><emph type="italics"/>Si linea motus &longs;it inter vtramque; est minus impedimentum.<emph.end type="italics"/> &longs;it globus <lb/>immobilis BA; &longs;it linea motus GC cum impedimento, de qua in Th. 50. <lb/>&longs;it alia KB cum nullo impedimento, de qua in Th. 51. &longs;int aliæ HD, <lb/>IE; certè minus e&longs;t impedimentum in contactu D, quàm in C; quia ca<lb/>dit obliquè in D, perinde atque &longs;i caderet in tangentem NO; Igitur <lb/>minus impeditur; in qua vero proportione, dicemus aliàs, cum de re<lb/>flexione, & de motu mixto. </s>

<s>50. <lb/>&longs;it alia KB cum nullo impedimento, de qua in Th. </s>

<s>51. &longs;int aliæ HD, <lb/>IE; certè minus e&longs;t impedimentum in contactu D, quàm in C; quia ca<lb/>dit obliquè in D, perinde atque &longs;i caderet in tangentem NO; Igitur <lb/>minus impeditur; in qua vero proportione, dicemus aliàs, cum de re<lb/>flexione, & de motu mixto. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc producitur in contactu<emph.end type="italics"/> C, <emph type="italics"/>totus impetus; in contactu<emph.end type="italics"/> D, <emph type="italics"/>minùs; <gap/><lb/>contactu<emph.end type="italics"/> E <emph type="italics"/>adhuc minùs; in<emph.end type="italics"/> B <emph type="italics"/>nihil<emph.end type="italics"/>; quia in ca proportione producitui <lb/>plùs vel minùs impetus, quo plùs e&longs;t, vel minùs impedimenti per <lb/>Th. </s>

<s>49. &longs;ed minùs e&longs;t impedimentum in E, quàm in C; & in E, quàm <lb/>in D, per Th. </s>

<s>52; Igitur in D producitur minùs impetus, quàm in C, <lb/>& minùs in E, quàm in D. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s>

<s><emph type="italics"/>Si linea directionis ducatur per centrum vtriu&longs;que globi, mobilis &longs;cilicet <lb/>& immobilis, impetus producit totum impetum quem pote&longs;t producere &longs;iue in <lb/>maiori globo, &longs;iue in minori, &longs;iue in æquali<emph.end type="italics"/>; patet experientia; cuius ratio <lb/>e&longs;t; quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria; Igitur idem impetus codem mo<lb/>do applicatus æquali tempore, æqualem &longs;emper effectum producit, per <lb/>Ax. 12. igitur cum impetus agat tantùm, vt tollat impedimentum per <lb/>Th. 44. & cum in prædicta linea agat quantum pote&longs;t per Th. 50. cer<lb/>tè æqualem effectum producat nece&longs;&longs;e e&longs;t; &longs;iue in maiori &longs;iue in mino<lb/>ri, &longs;iue in æquali globo immobili. </s>

<s>12. igitur cum impetus agat tantùm, vt tollat impedimentum per <lb/>Th. </s>

<s>44. & cum in prædicta linea agat quantum pote&longs;t per Th. </s>

<s>50. cer<lb/>tè æqualem effectum producat nece&longs;&longs;e e&longs;t; &longs;iue in maiori &longs;iue in mino<lb/>ri, &longs;iue in æquali globo immobili. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc impetus remi&longs;&longs;us potest producere inten&longs;um; & hæc e&longs;t altera diffcul<lb/>tas; cum &longs;cilicet maior globus in minorem impingitur<emph.end type="italics"/>; cum enim omnes <lb/>partes impetus maioris globi agant actione communi per Th. 46. & <lb/>cum agant quantùm maximè po&longs;&longs;unt; in minore globo, tot partes pro<lb/>ducunt impetus, quot in maiore, vt patet; igitur in minore globo pau<lb/>cioribus partibus &longs;ubiecti di&longs;tribuuntur plures partes impetus; crgo in <lb/>qualibet parte &longs;ubiecti &longs;unt plures; &longs;ed hoc e&longs;t e&longs;&longs;e inten&longs;um, vt con&longs;tat, <lb/>igitur impetus remi&longs;&longs;us producit inten&longs;um; quod e&longs;t paradoxon egre<lb/>gium. </s>

<s>46. & <lb/>cum agant quantùm maximè po&longs;&longs;unt; in minore globo, tot partes pro<lb/>ducunt impetus, quot in maiore, vt patet; igitur in minore globo pau<lb/>cioribus partibus &longs;ubiecti di&longs;tribuuntur plures partes impetus; crgo in <lb/>qualibet parte &longs;ubiecti &longs;unt plures; &longs;ed hoc e&longs;t e&longs;&longs;e inten&longs;um, vt con&longs;tat, <lb/>igitur impetus remi&longs;&longs;us producit inten&longs;um; quod e&longs;t paradoxon egre<lb/>gium. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus globi impacti in alium globum eo modo, quo diximus, id est, linea<lb/>directionis ducta per centra grauitatis vtriu&longs;que producit in eo æqualem<emph.end type="italics"/>; Pro<lb/>batur, quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria, quæ tunc agit quantum pote&longs;t <lb/>per Th. </s>

<s>57. &longs;ed æqualis pote&longs;t producere æqualemi. </s>

<s>Probatur primò, <lb/>exemplo aliarum qualitatum; &longs;ecundò, quia ideo agit vt tollat impedi<lb/>mentum, hoc e&longs;t vt corpus illud amoueat loco; igitur æquali motu per <lb/>&longs;e; alioquin ni&longs;i æquali motu amoueret, non tolleret impedimentum, <lb/>vt pater; tertiò &longs;int 30. partes impetus, certèvel producent plures vel <lb/>pauciores, vel totidem, non plures; cur enim potius 31. quam 32. <lb/>nec etiam pauciores; cur enim potius 20. quam 18, &c. </s>

<s>Igitur totidem; <lb/>quia cum &longs;int plures numeri plurium partium &longs;upra 30. & pauciorum <lb/>infra vt patet; &longs;itque tantùm vnicus numerus æqualium; certè quod <lb/>vnum e&longs;t, determinatum e&longs;t, per Ax. 5. hæc ratio licèt videatur negati<lb/>ua e&longs;t tamen potenri&longs;&longs;ima: quartò, quia actus &longs;ecundus, re&longs;pondet actui <lb/>primo, id e&longs;t, effectus productus virtuti cau&longs;æ producentis; itaque cum <lb/>virtus agendi impetus &longs;it eius entitas, vt patet, certè impetus productus <lb/>e&longs;t per &longs;e æqualis impetui producenti per &longs;e; id e&longs;t remoto omni <lb/>impedimento, & facto co contactu iuxta modum prædictum, ca quo-<pb pagenum="38"/>que lege, vt impetu<gap/> agat quantum pote&longs;t, & omnes partes mobilis <lb/>moueantur æquali motu. </s>

<s>5. hæc ratio licèt videatur negati<lb/>ua e&longs;t tamen potenri&longs;&longs;ima: quartò, quia actus &longs;ecundus, re&longs;pondet actui <lb/>primo, id e&longs;t, effectus productus virtuti cau&longs;æ producentis; itaque cum <lb/>virtus agendi impetus &longs;it eius entitas, vt patet, certè impetus productus <lb/>e&longs;t per &longs;e æqualis impetui producenti per &longs;e; id e&longs;t remoto omni <lb/>impedimento, & facto co contactu iuxta modum prædictum, ca quo-<pb pagenum="38"/>que lege, vt impetu<gap/> agat quantum pote&longs;t, & omnes partes mobilis <lb/>moueantur æquali motu. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s>

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

<s>47. Se<lb/>cundò, ratio propo&longs;ita nulla e&longs;t; quia quando globus impactus impellit <lb/>alium, e&longs;t veluti potentiâ, quæ cum tota&longs;ua vi, & cum impetu agit, <lb/>cuius nulla pars transfertur in alium globum; nec enim migrat de <lb/>de &longs;ubiecto in &longs;ubiectum, &longs;ed producit &longs;ibi æqualem: equidem &longs;i duo <lb/>globi æquales e&longs;&longs;ent vel coniuncti, vel contigui in linea directionis, <lb/>quilibet pro rata acciperet impetus producti partem à potentia applica<lb/>ta; &longs;i e&longs;&longs;ent æquales, qui&longs;que &longs;ubduplum: &longs;i alter &longs;ubduplus &longs;ubtri<lb/>plum, &c. </s>

<s>&longs;ed hæc &longs;unt &longs;atis facilia. </s>

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<s><lb/>quis non vidit maiores illas onerarias naues etiam pigro, & tardo motu <lb/>labentes maximum impetum minori occurrenti cymbæ ctiam impri<lb/>mere? </s>

<s>Rationem habes in Th. </s>

<s>Rationem habes in Th. 47. &longs;ed dices; igitur aliquis velocitatis <lb/>gradus nullam habet cau&longs;am; igitur e&longs;t à nihilo, quod dici non pote&longs;t. </s>

<s>47. &longs;ed dices; igitur aliquis velocitatis <lb/>gradus nullam habet cau&longs;am; igitur e&longs;t à nihilo, quod dici non pote&longs;t. </s>

<s>37. at verò cum partes impetus producti in minore globo <lb/>di&longs;tribuantur paucioribus partibus &longs;ubiecti, faciunt inten&longs;iorem im<lb/>petum; igitur velociorem motum, quippe omnes producuntur ab <lb/>omnibus illis actione communi per Ax. </s>

<s><lb/>Re&longs;pondeo, plures partes impetus non produci in minore globo, quàm <lb/>&longs;int in maiore; igitur nulla pars e&longs;t impetus minoris globi, quæ &longs;ui <lb/>cau&longs;am &longs;ufficientem non habeat; &longs;ed cum partes impetus maioris globi <lb/>di&longs;tribuantur pluribus partibus &longs;ubiecti, faciunt remi&longs;&longs;um impetum, igi<lb/>tur & tardum; cum &longs;cilicct impetus vnius partis non iuuet motum alte<lb/>rius per Th. 37. at verò cum partes impetus producti in minore globo <lb/>di&longs;tribuantur paucioribus partibus &longs;ubiecti, faciunt inten&longs;iorem im<lb/>petum; igitur velociorem motum, quippe omnes producuntur ab <lb/>omnibus illis actione communi per Ax. 17. num. </s>

<s>1. quid clarius. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 62.<emph.end type="center"/></s>

<s><emph type="italics"/>Globus minor imprimit maiori remi&longs;&longs;iorem impetum & tardiorem motum <lb/>& æqualis, aqitali æqualem<emph.end type="italics"/>; hæc omnia probantur per Th. </s>

<s><emph type="italics"/>Globus minor imprimit maiori remi&longs;&longs;iorem impetum & tardiorem motum <lb/>& æqualis, aqitali æqualem<emph.end type="italics"/>; hæc omnia probantur per Th. 60. & præ-, <lb/>cedentia. </s><pb pagenum="40"/>

<s>60. & præ-, <lb/>cedentia. </s><pb pagenum="40"/>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s>

<s><emph type="italics"/>Ex hac hyp<gap/>e&longs;i globus impactus producit in alie mouas partes impetus<emph.end type="italics"/>; <lb/>quia impeditur cius motus, igitur vt tollat impedimentum, agit ad <lb/>cxtra per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 65.<emph.end type="center"/></s>

<s><emph type="italics"/>Hie impetus neuus preductus miner e&longs;t eo qus preduceretur in codem glob<lb/>immobili<emph.end type="italics"/>: ratio c&longs;t; quia &longs;i &longs;i&longs;tcret, maius e&longs;&longs;et impedimentum, quia <lb/>totum motum impediret, cuius tantùm partem impedit, dum mouctur, <lb/>licèt paulò tardius; igitur minus agit ad cxtra per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s>

<s><emph type="italics"/>Mobile adbarens alseri mobili à terge; dum vtr<gap/>que æqu<gap/> velociter <lb/>feratur nullum preducis in ce i<gap/>pesum.<emph.end type="italics"/></s><s> Probatur, quia mobile quod præit, <lb/>non impedit motum &longs;ub&longs;equentis; igitur nullum impetum ab co acci <lb/>pit per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 67.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/></s>

<s>Ex dictis ctiam colliges diuer&longs;as percu&longs;&longs;ionum rationes &longs;uppo&longs;ita di<lb/>uer&longs;a ratione ponderum globi percutientis, & percu&longs;&longs;i; cum enim impe<lb/>tus productus &longs;it æqualis per &longs;e impetui producenti, per Th.60. modò <lb/>debita fiat applicatio, de qua in Th.50. &longs;i percutiens &longs;it duplus percu&longs;&longs;i, <lb/>&longs;uppo&longs;ita eadem materia, motus percu&longs;&longs;i erit duplò velocior; quia im<lb/>petus erit duplò inten&longs;ior, vt con&longs;tat ex Th. </s>

<s>61. &longs;i verò &longs;it quadruplus, <lb/>quadruplo, &c. </s>

<s>Igitur velocitates motuum &longs;unt in ratiòne ponderum <lb/>permutando. </s>

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<s>&longs;it parallelipedum EB; quod <lb/>moueatur motu recto parallclo, lineis CD, HG, &c. </s>

<s>&longs;itque globus in <lb/>D; haud dubiè agit quantùm pote&longs;t, quia &longs;cilicet e&longs;t maximum impedi<lb/>mentum per Th.68. Tam enim globus in D impedit motum parallcli<lb/>pedi, quàm parallclip edum motum globi impacti per lineam ID; impedit <lb/>inquam ratione oppo&longs;itionis; quia centra grauitatis vtriu&longs;que con<lb/>currunt in cadem linea; igitur &longs;i maximum e&longs;t impedimentum, agit <lb/>quantùm pote&longs;t Th. </s>

<s>50. hinc producitur impetus æqualis per Th.60. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 37.<emph.end type="center"/></s>

<s><emph type="italics"/>Si circa centrum immobile rotetur corpus parallelipedum<emph.end type="italics"/> CA, <emph type="italics"/>diuer&longs;a e&longs;t <lb/>ratio percu&longs;&longs;ionum ab ea, quàm &longs;uprà propo&longs;uimus<emph.end type="italics"/>; moucatur enim circa <lb/>centrum C, fitque CA diui&longs;a bifariam in B, haud dubiè punctum A <lb/>faciet arcum AE eo tempore, quò punctum B faciet BD &longs;ubduplum <lb/>AE; igitur punctum A duplò velociùs mouetur quàm B, vt con&longs;tat; igi<lb/>tur habet duplò maiorem impetum; cum effectum habeat duplò maio<lb/>rem per Ax. </s>

<s>4. igitur cum totus motus &longs;egmenti AB &longs;it ad to<lb/>tum motum &longs;egmenti BC, vt &longs;patia acqui&longs;ita; certè &longs;patia acqui&longs;ita <lb/>&longs;unt vt arcus; igitur & trapezus BAED, continet 3/4 totius CAE, vt <lb/>con&longs;tat; &longs;unt enim &longs;ectores &longs;imilis in ratione duplicata radiorum; igi<lb/>tur totus motus &longs;egmenti BC &longs;ubquadruplus motus totius CA;, gitur <lb/>& impetus; vt autem habeatur centrum impre&longs;&longs;ionis, vel impetus; &longs;it &longs;e<lb/>ctor CHI, &longs;ubduplus totius CAE quod quomodo fiat, patet ex Geo<lb/>metria; accipiatur tantùm &longs;ubdupla diagonalis quadrati lateris CA, igi<lb/>tur in puncto H e&longs;t centrum impre&longs;&longs;ionis, &longs;eu media proportionalis in<lb/>ter totam CA, & &longs;ubduplam CB: vt autem habeatur percu&longs;&longs;ionis, a&longs;<lb/>&longs;umatur CY dupla YA; Dico punctum Y e&longs;&longs;c centrum percu&longs;&longs;ionis; <lb/>quia perinde &longs;e habet, atque &longs;i e&longs;&longs;et trianguli cadentis ictus, vt demon<lb/>&longs;trabimus aliàs nunc tantùm indica&longs;&longs;e &longs;ufficiat. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus &longs;imilis, id e&longs;t, ad eamdem lineam determinatus, & æqualis in in<lb/>ten&longs;ione, non pote&longs;t intendere alium &longs;imilem<emph.end type="italics"/>; Probatur, quia agit tantùm ad <lb/>extra, vt tollat impedimentum per Th. 44. &longs;ed eorum mobilium, quæ <lb/>ver&longs;us eamdem partem pari velocitate mouentur, neutrum impedit al<lb/>terius motum, vt con&longs;tat; igitur impetus &longs;imilis, &c. </s>

<s>44. &longs;ed eorum mobilium, quæ <lb/>ver&longs;us eamdem partem pari velocitate mouentur, neutrum impedit al<lb/>terius motum, vt con&longs;tat; igitur impetus &longs;imilis, &c. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s>

<s><emph type="italics"/>Datur impetus altero impetu perfectior &longs;ecundum entitatem<emph.end type="italics"/>; dixi &longs;ecun<lb/>dum entitatem; quia iam dictum e&longs;t &longs;uprà dari perfectiorem &longs;ecundum <lb/>inten&longs;ionem; huius Theorematis veritas mihi maximè demon&longs;tranda <lb/>e&longs;t, ex quo tàm multa infrà deducemus; &longs;ic autem probamus; Quotie&longs;<lb/>cunque mouetur corpus, producuntur &longs;altem tot partes impetus quot <lb/>&longs;unt partes mobilis per Th. 33. Quotie&longs;cunque producuntur in mobili <lb/>tot partes impetus quot &longs;unt in mobili partes &longs;ubjecti, mouetur mobile, <lb/>modó non impediatur; quia po&longs;ita cau&longs;a nece&longs;&longs;aria, & non impedita per <lb/>Ax. 11. ponitur effectus, quod de omni cau&longs;a, &longs;ed de fotmali poti&longs;&longs;imum <lb/>dicidebet; præterea-datur aliquod pondus, quod data potentia &longs;ine me<lb/>chanico organo moucre non pote&longs;t, licèt cum organo facilè moueat; hæc <lb/>hypothe&longs;is certa e&longs;t; igitur cum mouet, producit tot partes impetus quot <lb/>&longs;unt nece&longs;&longs;ariæ, vt omnibus partibus mobilis di&longs;tribuantur per idem Th. </s>

<s>33. Quotie&longs;cunque producuntur in mobili <lb/>tot partes impetus quot &longs;unt in mobili partes &longs;ubjecti, mouetur mobile, <lb/>modó non impediatur; quia po&longs;ita cau&longs;a nece&longs;&longs;aria, & non impedita per <lb/>Ax. </s>

<s>11. ponitur effectus, quod de omni cau&longs;a, &longs;ed de fotmali poti&longs;&longs;imum <lb/>dicidebet; præterea-datur aliquod pondus, quod data potentia &longs;ine me<lb/>chanico organo moucre non pote&longs;t, licèt cum organo facilè moueat; hæc <lb/>hypothe&longs;is certa e&longs;t; igitur cum mouet, producit tot partes impetus quot <lb/>&longs;unt nece&longs;&longs;ariæ, vt omnibus partibus mobilis di&longs;tribuantur per idem Th. </s>

<s><lb/>33. cum verò non mouet, non producit tot partes impetus vt con&longs;tat ex <lb/>dictis; igitur producit plures cnm organo in mobili, quàm &longs;ine organo; <lb/>igitur imperfectiores, quod demon&longs;tro: &longs;it enim vectis BF, cuius cen<lb/>trum &longs;eu fulcrum &longs;it in A, potentia in B, pondus G, quod attollitur in F; <lb/>plures partes impetus produci po&longs;&longs;unt in F, vel in E, quàm in B, &longs;cilicet <lb/>in ip&longs;o pondere; quia pondus quod non pote&longs;t attolli in B, attollitur in <lb/>E, vel in F, vt patet ex dictis; præterea punctum F mouetur tardius, quàm <lb/>B; quia motus &longs;unt vt arcus, atcus vt &longs;emidiametri, hæ demum vt AF, <lb/>ad AB; igitur motus puncti F, e&longs;t tardior, vel imperfectior; igitur im<lb/>petus puncti F, e&longs;t imperfectior impetu puncti B, per Ax. </s>

<s>13 num.4. atqui <lb/>non e&longs;t imperfectior ratione numeri partium, igitur ratione entiratis, <lb/>quæ imperfectior e&longs;t; igitur datur impetus altero impetu imperfectior. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="italics"/>Impetus perfectus nunquam producitur ab imperfecto, per Ax.<emph.end type="italics"/> 3. <emph type="italics"/>num.<emph.end type="italics"/> 2. <lb/>adde quod nunquam effectus perfectio &longs;uperat perfectionem cau&longs;æ; dixi <lb/>perfectum ab imperfecto; &longs;cilicet &longs;i con&longs;ideretur perfectio ratione en-<pb pagenum="49"/>titatis; cum reuerâ, vt dictum e&longs;t &longs;uprà, remi&longs;&longs;us producat inten&longs;um, <lb/>quod in vecte clari&longs;&longs;unum e&longs;t; quippe momentum applicatum in F, quod <lb/>tardiùs mouetur deor&longs;um, quàm B, &longs;ur&longs;um, vt patet, habet impetum re<lb/>mi&longs;&longs;iorem, qui tamen producit in B, inten&longs;iorem: Pro quo, ob&longs;eruabis <lb/>impetum imperfectum cum alio perfecto actione communi agentem <lb/>po&longs;&longs;e concurrere ad producendum perfectum, vt patet; non tamen in <lb/>ratione cau&longs;æ totalis: &longs;imiliter plures imperfecti &longs;imul concurrentes <lb/>po&longs;&longs;unt producere perfectum; quia plures imperfecti conjunctim adæ<lb/>quant perfectionem alterius perfectioris &longs;inguli &longs;eor&longs;im. </s>

<s>Ob&longs;eruabis &longs;ecundò præclarum naturæ in&longs;titutum, quo factum e&longs;t; <lb/>vt cum vires hominum maiora pondera leuare non po&longs;&longs;int, &longs;i &longs;eor&longs;un <lb/>con&longs;iderentur; cum organis tamen mechanicis conjunctæ nullum pon<lb/>dus quantumuis immane leuare non po&longs;&longs;int; quod certè nullo modo ac<lb/>cideret, ni&longs;i plures partes impetus producerent neque plures producere <lb/>po&longs;&longs;ent, ni&longs;i minoris perfectionis e&longs;&longs;ent; quia faciliùs producitur effe<lb/>ctus imperfectus, quam perfectus per Ax. </s>

<s>Tertiò hinc optimè à natura proui&longs;um e&longs;t, vt motus tardior in infi<lb/>nitum e&longs;&longs;e po&longs;&longs;it; quod reuerâ fieri non po&longs;&longs;et, ni&longs;i dari po&longs;&longs;et impetus <lb/>alio imperfectior. </s>

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<s><emph type="italics"/>Potentia naturalis grauium producit tantùm vno in&longs;tanti ad intra vnicum <lb/>punctum impetus in quolibet puncto &longs;ubiecti; &longs;i tamen impetum producit, quod <lb/>definiam lib.<emph.end type="italics"/> 20. <emph type="italics"/>& &longs;i dentur puncta &longs;ubiecti, quod ad præ&longs;ens in&longs;titutum non <lb/>pertinet<emph.end type="italics"/>; Probatur, quia fru&longs;trà e&longs;&longs;ent plura puncta impetus; nec enim <lb/>&longs;unt multiplicandæ formæ &longs;ine nece&longs;&longs;itate, ratione &c. </s>

<s>1. Præterea non e&longs;t, cur potius produceret 2. quàm 3. 4. &c. </s>

<s>atqui <lb/>quod vnum e&longs;t, determinatum e&longs;t per Ax. </s>

<s>atqui <lb/>quod vnum e&longs;t, determinatum e&longs;t per Ax. 5. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 88.<emph.end type="center"/></s>

<s><emph type="italics"/>Eadem potentia inæqualibus temporibus impetum inæqualem in perfectio<lb/>ne producit<emph.end type="italics"/>; accipiatur enim totum illud tempus, quo vnicum tantùm <lb/>punctum impetus producit (vocetur in&longs;tans) de quo in Th. </s>

<s>86; certè <lb/>&longs;i in minori tempore agat, minùs aget, per Ax. </s>

<s>4. &longs;ed non <lb/>pote&longs;t minùs agere ratione numeri, vt patet; igitur ratione perfectio<lb/>nis. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 89.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus violenti, qui &longs;en&longs;im de&longs;truitur in proiectis, po&longs;itis ij&longs;dem circum<lb/>&longs;tantiis medij, & re&longs;i&longs;tentiæ, minori tempore minùs de&longs;truitur; plus verò ma<lb/>jori:<emph.end type="italics"/> Quia hæc de&longs;tructio habet cau&longs;am; nam quidquid de&longs;truitur, ad <lb/>exigentiam alicuius de&longs;truitur, per Ax. </s>

<s>2. igitur minori <lb/>tempore minùs de&longs;truitur per Ax. </s>

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<s><emph type="italics"/>Impetus propagatur nece&longs;&longs;ariò per totum corpus impul&longs;um, &longs;eu proiectum.<emph.end type="italics"/></s>

<s>Probatur; quia cum omnes eius partes moueantur, nec vlla &longs;ine im<lb/>petu moueri po&longs;&longs;it per Th. </s>

<s>Probatur; quia cum omnes eius partes moueantur, nec vlla &longs;ine im<lb/>petu moueri po&longs;&longs;it per Th. 18. & 33. cum etiam potentia motrix non <lb/>&longs;it omnibus immediatè applicata, vt con&longs;tat; certè &longs;ine propagatione, <lb/>vel diffu&longs;ione non pote&longs;t explicari productio huius motus. </s><pb pagenum="51"/>

<s>18. & 33. cum etiam potentia motrix non <lb/>&longs;it omnibus immediatè applicata, vt con&longs;tat; certè &longs;ine propagatione, <lb/>vel diffu&longs;ione non pote&longs;t explicari productio huius motus. </s><pb pagenum="51"/>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 92.<emph.end type="center"/></s>

<s><emph type="italics"/>Illa progatio non fit per motum localem, ita vt pars impetus producta in <lb/>prima parte &longs;ubiecti tran&longs;eat ad &longs;ecundam,<emph.end type="italics"/> patet; quia cum impetus &longs;it ac<lb/>cidens per Th. </s>

<s>8. de &longs;ubiecto in &longs;ubiectum tran&longs;ire non pote&longs;t per deff. </s>

<s><lb/>accidentis; de qua in Metaphy&longs;icâ; nec e&longs;t quod aliqui dicant &longs;e <expan abbr="nõ">non</expan> po&longs;&longs;e <lb/>concipere, quomodo id fiat &longs;ine motu locali; cum ip&longs;is etiam oculis <lb/>qua&longs;i ocrnatur; cum enim percutis corpus oblongum AE, & cadit ictus <lb/>in extremitatem A, corpus ip&longs;um totum &longs;unul moues; igitur pars impe<lb/>tus, quæ recipitur in A, non migrat in E, &longs;ed hæc producitur in A, & <lb/>alia in B, alia in C, atque ita deinceps. </s>

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<s>globi A & B, æquales &longs;ibi inuicem contigui in C, &longs;it applicata po<lb/>tentia in D, non modò producet impetum in globo A, &longs;ed etiam in B: <lb/>probatur primò, quia &longs;e habent per modum vnius, vt patet ex re&longs;i&longs;ten<lb/>&longs;tia, nec enim A moueri pote&longs;t &longs;ine B per lineam DE, quod certè cla<lb/>ri&longs;&longs;imum e&longs;t; probatur &longs;ecundò quia &longs;i A produceret impetum in B, duo <lb/>globi, vel 3. vel 5. vel infiniti tantùm re&longs;i&longs;terent, quantùm vnicus glo<lb/>bus, quod fal&longs;um & ab&longs;urdum e&longs;t. </s>

<s>Tertiò, Ratio à priori e&longs;t; quia idco <lb/>producitur, & propagatur impetus in toto A; quia vna pars non pote&longs;t <lb/>moueri &longs;ine alia per Th. </s>

<s>Tertiò, Ratio à priori e&longs;t; quia idco <lb/>producitur, & propagatur impetus in toto A; quia vna pars non pote&longs;t <lb/>moueri &longs;ine alia per Th. 33. &longs;ed non pote&longs;t A moueri ni&longs;i moucatur B; <lb/>igitur in vtroque &longs;imul, & æqualiter propagatur impetus. </s>

<s>33. &longs;ed non pote&longs;t A moueri ni&longs;i moucatur B; <lb/>igitur in vtroque &longs;imul, & æqualiter propagatur impetus. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s>

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<s><emph type="italics"/>Cum applicatur potentia centro motus circularis, ita propagatur impetus, vt <lb/>plures partes impetus continuò producantur ver&longs;us <expan abbr="circumferentiã">circumferentiam</expan><emph.end type="italics"/>; &longs;it enim <lb/>cylindrus CA, fig. </s>

<s>Th. 73. &longs;it centrum motus C; haud dubiè plures <lb/>partes impetus producuntur in B, quàm in C, & plures in A, quam in B; <lb/>quia, cum pars B moueatur velociùs, quàm C, & A quàm B; certè, vbi e&longs;t <lb/>maior motus, vel effectus, ibi debet e&longs;&longs;e maior impetus, vel cau&longs;a per <lb/>Ax. 13. n. </s>

<s>73. &longs;it centrum motus C; haud dubiè plures <lb/>partes impetus producuntur in B, quàm in C, & plures in A, quam in B; <lb/>quia, cum pars B moueatur velociùs, quàm C, & A quàm B; certè, vbi e&longs;t <lb/>maior motus, vel effectus, ibi debet e&longs;&longs;e maior impetus, vel cau&longs;a per <lb/>Ax. </s>

<s>4. quod autem &longs;it maior motus, con&longs;tat ex maioribus &longs;patiis, <lb/>vel arcubus æquali tempore confectis; quod verò &longs;it impetus inten&longs;ior <pb pagenum="54"/>versùs circumferentiam, non perfectior, patet per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 98.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s>

<s>Hinc &longs;i di&longs;tantia CA e&longs;t dupla di&longs;tantiæ CB, impetus in A e&longs;t du<lb/>plus impetus in B: at verò impetus &longs;egmenti e&longs;t ad impetum alterius, <lb/>vt diximus in Th. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 2.<emph.end type="center"/></s>

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<s>Hinc etiam manife&longs;ta ratio &longs;equitur illius experimenti, quod propo<lb/>&longs;uimus corol. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 5.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 100.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus, qui producitur ver&longs;us centrum vectis in pondere, licèt cre&longs;cat nu<lb/>mero, decre&longs;cit tamen in perfectione.<emph.end type="italics"/></s><s> Probatur per Th.81. ex motu imper<lb/>ctiore, cui re&longs;pondet impetus imperfectior per Ax. 17.num.4. non ratio<lb/>ne numeri, qui maior e&longs;t per Th.99. igitur ratione entitatis, &longs;eu perfe<lb/>ctionis entitatiuæ. </s>

<s>17.num.4. non ratio<lb/>ne numeri, qui maior e&longs;t per Th.99. igitur ratione entitatis, &longs;eu perfe<lb/>ctionis entitatiuæ. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 101.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 107.<emph.end type="center"/></s>

<s><emph type="italics"/>Ex his constat produci impetum æqualem numero in omnibus punctis vectis <lb/>a circumferentia ad centrum, cum &longs;cilicet applicatur potentia circumferentiæ<emph.end type="italics"/>; <lb/>probatur, quia non producitur numerus minor per Th.105. neque maior <lb/>per Th. 106. igitur æqualis; adde quod res explicari non pote&longs;t per ma<lb/>iorem, neque per minorem; ita vt &longs;cilicet pondera, quæ à data potentia <lb/>leuantur, &longs;int vt di&longs;tantiæ, de quo &longs;uprà. </s><pb pagenum="58"/>

<s>106. igitur æqualis; adde quod res explicari non pote&longs;t per ma<lb/>iorem, neque per minorem; ita vt &longs;cilicet pondera, quæ à data potentia <lb/>leuantur, &longs;int vt di&longs;tantiæ, de quo &longs;uprà. </s><pb pagenum="58"/>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="italics"/>Iam facilè explicatur ex dictis, quomodo, & cuius rationis pondera attol<lb/>lantur ex diuer&longs;is punctis vectis<emph.end type="italics"/>; &longs;it enim idem vectis AC, & producan<lb/>tur.v.g. </s>

<s>in &longs;ingulis punctis vectis &longs;ingula puncta impetus, &longs;ed diuer&longs;æ <lb/>perfectionis; haud dubiè plures partes impetus imperfecti po&longs;&longs;unt face<lb/>re impetum æqualem in perfectione alteri, qui con&longs;tat paucioribus, &longs;ed <lb/>perfectioribus; igitur cum impetus B &longs;it imperfectior duplò quàm im<lb/>petus in A, duplò plures partes impetus producentur in B, quàm in A, er<lb/>go duplò maius pondus moucbitur; atque ita deinceps; eum enim ap<lb/>ponitur pondus in B, producuntur in eo partes impetus omnes eiu&longs;dem <lb/>perfectionis; quæ &longs;cilicet re&longs;pondet B, id e&longs;t, quæ e&longs;t &longs;ubdupla perfectio<lb/>nis impetus A; igitur plures partes producuntur, quàm &longs;i e&longs;&longs;ent perfe<lb/>ctionis A; &longs;ed pauciores quàm &longs;i e&longs;&longs;ent perfectionis O, quæ minor e&longs;t; <lb/>quippe eadem potentia, &longs;eu cau&longs;a, quæ agit quantum pote&longs;t (quod &longs;up<lb/>pono modò) producit æqualem effectum in perfectione, per Ax. </s>

<s><lb/>4. &longs;ed æqualis perfectio pote&longs;t con&longs;tare pluribus, vel paucioribus parti<lb/>bus perfectionis, nam 4. pattes perfectionis vt 4. faciunt æqualem effe<lb/>ctum alteri qui con&longs;tat 8. partibus perfectionis vt 2. quod certum e&longs;t; &longs;ed <lb/>de his plura aliàs. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 111.<emph.end type="center"/></s>

<s><emph type="italics"/>Si &longs;int tantum duo puncta vel duæ partes vectis, illa potentia ad illum mo<lb/>uendum &longs;ufficiens motu circulari est ad aliam &longs;ufficientem ad illum mouen<lb/>dum motu recto, vt<emph.end type="italics"/> 1/2 <emph type="italics"/>ad<emph.end type="italics"/> 2. &longs;i &longs;int tria puncta vt 2. ad 3. &longs;i 4. vt 2. 1/2 ad 4. <lb/>&longs;i 5. vt 3. ad 5. &longs;i 6. vt 3. 1/2 ad 6. atque ita deinceps iuxta hanc propor<lb/>tionem in quo non e&longs;t difficultas, cum hoc totum &longs;equatur ex Th. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s>illa, quæ applicatur <lb/>vecti. </s>

<s>2. punctorum e&longs;t ad eam, quæ applicatur vecti trium punctorum, <lb/>&longs;cu partium, vt 1. 1/2 ad 2. & &longs;i vectis &longs;it 4. punctorum ad 2. 1/2; &longs;i 5. ad 3. <lb/>&longs;i 6. ad 3. 1/2; &longs;i 7. ad 4. &longs;i 8. ad 4. 1/2. Vides egregiam progre&longs;&longs;ionem; &longs;it <lb/>enim vectis 2. punctorum AB, in puncto A, quod e&longs;t extremitas, produ<lb/>catur punctum impetus datæ perfectionis, in B producetur aliud, cuius <lb/>perfectio e&longs;t &longs;ubdupla prioris per Th. 109. igitur caracter, &longs;eu momen<lb/>tum totius impetus e&longs;t 1. 1/2. &longs;it porrò vectis 4. punctorum CDEF, in <lb/>C, quod e&longs;t extremitas; producatur vnum punctum impetus ciu&longs;dem <lb/>perfectionis cum eo, quod productum e&longs;t in A; certè in D producetur <lb/>aliud cuius perfectio erit ad priorem vt 3.ad 4. per idem Th. &longs;ic autem <lb/>notetur 1/4, in E 2/4, in F 3/4, in C vero 4/4; perfectiones enim &longs;unt vt lon-<pb pagenum="60"/>gitudines; quæ &longs;i colligantur, habebis characterem totius impetus, 2 1/2: <lb/>igitur totus impetus productus in minore vecte, qui con&longs;tat 2. punctis, <lb/>e&longs;t ad impetum, qui producitur in maiore con&longs;tante 4.punctis, vt 1. 1/2 ad <lb/>2. 1/2; igitur vectis maior maiorem potentiam ad mouendum ip&longs;um ve<lb/>ctem requirit; non certè in de&longs;cen&longs;u; quippe &longs;uo pondere de&longs;cendit, &longs;ed <lb/>in plano horizontali; ni&longs;i enim potentia po&longs;&longs;it mouere vectem; haud <lb/>dubiè nullum pondus vecte mouebit. </s>

<s>109. igitur caracter, &longs;eu momen<lb/>tum totius impetus e&longs;t 1. 1/2. &longs;it porrò vectis 4. punctorum CDEF, in <lb/>C, quod e&longs;t extremitas; producatur vnum punctum impetus ciu&longs;dem <lb/>perfectionis cum eo, quod productum e&longs;t in A; certè in D producetur <lb/>aliud cuius perfectio erit ad priorem vt 3.ad 4. per idem Th. </s>

<s>&longs;ic autem <lb/>notetur 1/4, in E 2/4, in F 3/4, in C vero 4/4; perfectiones enim &longs;unt vt lon-<pb pagenum="60"/>gitudines; quæ &longs;i colligantur, habebis characterem totius impetus, 2 1/2: <lb/>igitur totus impetus productus in minore vecte, qui con&longs;tat 2. punctis, <lb/>e&longs;t ad impetum, qui producitur in maiore con&longs;tante 4.punctis, vt 1. 1/2 ad <lb/>2. 1/2; igitur vectis maior maiorem potentiam ad mouendum ip&longs;um ve<lb/>ctem requirit; non certè in de&longs;cen&longs;u; quippe &longs;uo pondere de&longs;cendit, &longs;ed <lb/>in plano horizontali; ni&longs;i enim potentia po&longs;&longs;it mouere vectem; haud <lb/>dubiè nullum pondus vecte mouebit. </s>

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

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<s>quàm tribus? </s>

<s>Præterca, cum re&longs;i&longs;tens, vel im<lb/>pediens e&longs;t æquale agenti; cer<gap/>e &longs;icut agens refundit in pa&longs;&longs;um totum <lb/>id, quod habet, id e&longs;t æqualem impetum in inten&longs;ione, & æquè velo<lb/>cem motum per Th. 60. Ita re&longs;i&longs;tens, vel impediens refundit æquale <lb/>impedimentum, quod tantùm &longs;umi pote&longs;t ex æqualitate mobilium; &longs;ed <lb/>ex æquali impedimento duci tantùm pote&longs;t æqualis determinatio priori; <lb/>denique pote&longs;t dari determinatio noua æqualis priori, vt con&longs;tat, &longs;ed <lb/>aliunde duci non pote&longs;t quàm ex ip&longs;a mobilium æqualitate, modò fiat <lb/>contactus per lineam connectentem centra. </s>

<s>60. Ita re&longs;i&longs;tens, vel impediens refundit æquale <lb/>impedimentum, quod tantùm &longs;umi pote&longs;t ex æqualitate mobilium; &longs;ed <lb/>ex æquali impedimento duci tantùm pote&longs;t æqualis determinatio priori; <lb/>denique pote&longs;t dari determinatio noua æqualis priori, vt con&longs;tat, &longs;ed <lb/>aliunde duci non pote&longs;t quàm ex ip&longs;a mobilium æqualitate, modò fiat <lb/>contactus per lineam connectentem centra. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 128.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 134.<emph.end type="center"/></s>

<s><emph type="italics"/>Si verò alter impetus &longs;it inten&longs;ior, po&longs;ito eodem ca&longs;u, haud dubiè eius de<lb/>terminatio præualebit pro rata<emph.end type="italics"/>; patet etiam experientià; ratio e&longs;t, quia im<lb/>petus fortior debiliorem vincit; pugnant enim pro rata per Ax. 15. <lb/>hinc &longs;i &longs;it duplò inten&longs;ior, &longs;ubduplum &longs;uæ velocitatis amittet, &longs;i triplè <lb/>&longs;ubtriplum, &c. </s>

<s>15. <lb/>hinc &longs;i &longs;it duplò inten&longs;ior, &longs;ubduplum &longs;uæ velocitatis amittet, &longs;i triplè <lb/>&longs;ubtriplum, &c. </s>

<s>de quo aliàs. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 135.<emph.end type="center"/></s>

<s><emph type="italics"/>Siduo globi projecti &longs;ibi inuicem occurrant in lineæ directionis connectente <lb/>centra, reflectitur vterque æquali motu, quo antè.<emph.end type="italics"/></s><s> Probatur; &longs;unt enim globi <pb pagenum="66"/>A & B, & A feratur per lineam DE, & B per lineam ED, punctum con<lb/>tactus &longs;it C, haud dubiè globus A impactus in B amittit totum &longs;uum im<lb/>petum per Th.127. & 128. B, item impactus in A amittit totum &longs;uum per <lb/>camdem rationem; globus A producit impetum in B æqualem &longs;uo per <lb/>Th.60. item B producit in A æqualem per idem Th. </s>

<s>igitur tantùm perit <lb/>impetus quantùm accedit; igitur in vtroque globo remanet æqualis im<lb/>petus priori; igitur æquali motu vterque mouetur, quod erat dem. </s>

<s>& hæc <lb/>e&longs;t ratio veri&longs;&longs;ima toties probatæ experientiæ. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 139.<emph.end type="center"/></s>

<s><emph type="italics"/>Si lineæ duplicis impetus, faciunt angulum acutiorem, longius erit &longs;patium<emph.end type="italics"/><pb pagenum="67"/><emph type="italics"/>acqui&longs;itum<emph.end type="italics"/>: &longs;int duæ lineæ IK IL, mobili &longs;eilicet &longs;tatuto in I; <lb/>haud dubiè noua linea erit IM; & quo angulus KIL, erit acutior (&longs;up<lb/>po&longs;itis æqualibus &longs;emper lateribus IK IL) Diagonalis IM, erit ma<lb/>ior; donec tandem IL & IK coeant in eandem lineam; tunc enim li<lb/>nea erit dupla IK per Th. </s>

<s>&longs;uperius: quandiu verò e&longs;t aliquis angulus in <lb/>I quantumuis acutus, linea motus erit minor dupla IK, ad quam tamen <lb/>propiùs &longs;emper accedit; quæ omnia con&longs;tant ex elementis. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 140.<emph.end type="center"/></s>

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

<s>137. & &longs;i accipia<lb/>tur angulus obtu&longs;ior HEF; noua linea erit EG, eo rectè breuior, <lb/>quò angulus e&longs;t obtu&longs;ior, non tamen iuxta rationem angulorum; donec <lb/>tandem de&longs;inat angulus, & ED EF coëant in vnam lineam; tunc enim <lb/>nullum erit &longs;patium, quia &longs;i&longs;ter omninò mobile per Th.133.quæ omnia <lb/>ip&longs;a luce clariora e&longs;&longs;e con&longs;tat; quippe quæ cum certis experimentis, & <lb/>clari&longs;&longs;imis principiis con&longs;entiant; &longs;ed de his plura infrà. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 141.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 143.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus &longs;emel productus, quamdiu durat motus, con&longs;eruatur.<emph.end type="italics"/></s><s> Probatur, <lb/>quia non pote&longs;t e&longs;&longs;e effectus, ni&longs;i &longs;it eius cau&longs;a per Ax. </s>

<s>8. igitur &longs;i e&longs;t mo<lb/>tus, e&longs;t impetus. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 144.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 145.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc ab alia causâ con&longs;eruari nece&longs;&longs;e e&longs;t impetum<emph.end type="italics"/>: Probatur, quia impe<lb/>tus non e&longs;t à &longs;e, quia de&longs;truitur aliquando per Ax. </s>

<s>14. igitur con&longs;eruatur <lb/>ab alio per Ax.14. num. </s>

<s>1. non à cau&longs;a primò productiua per Th.144.igi<lb/>tur ab alia, eaque applicata per Ax. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 149.<emph.end type="center"/></s>

<s><emph type="italics"/>In lineis oppo&longs;itis impetus de&longs;truitur ab impetu &longs;uo modo<emph.end type="italics"/>; &longs;it enim globus <lb/>proiectus ver&longs;us au&longs;trum; cui deinde imprimatur nouus impetus ver<lb/>&longs;us Boream; de&longs;truitur prior vt con&longs;tat, igitur ad exigentiam alicuius, <lb/>&longs;ed nihil e&longs;t quod po&longs;&longs;it exigere, ni&longs;i nouus impetus, &longs;cilicet mediatè; <lb/>nihil enim aliud e&longs;t applicatum, igitar nihil aliud exigit per Ax. </s>

<s>10. <lb/>hæc porrò exigentia non e&longs;t immediata, &longs;ed mediata, vt dixi. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 150.<emph.end type="center"/></s>

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

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

<s>14. num.2. cum verò determinatio altera maior e&longs;t, certè præ<lb/>ualet tantùm pro rata; igitur minor e&longs;t motus; igitur, ne aliqui gradus <lb/>impetus &longs;int fru&longs;trà, de&longs;truuntur, cum verò &longs;unt duo impetus in eodem <lb/>mobili, vt in naui mobili ad lineas oppo&longs;itas determinati; haud dubiè <lb/>maior impetus præualet pro rata per Ax. </s>

<s>15. Igitur non modò totus <lb/>impetus minor perit, ne &longs;it fru&longs;trà; &longs;ed etiam aliquot gradus maioris, ne <lb/>&longs;int etiam fru&longs;trà; nec enim in communem lineam coïre po&longs;&longs;unt. </s>

<s>Denique quando &longs;unt duo impetus ad lineas diuer&longs;as determinati, <lb/>&longs;ed non oppo&longs;itas ex diametro, pugnant pro diuer&longs;o oppo&longs;itionis gradu, <lb/>vt &longs;uprà fusè dictum e&longs;t. </s>

<s>Igitur cum totus impetus non habeat totum <lb/>motum, quod duplex illa determinatio impedit, ne aliqui gradus <lb/>&longs;int fru&longs;trà, de&longs;truuntur; igitur vides impetum impre&longs;&longs;um ab ex<lb/>trin&longs;eco de&longs;trui tantùm ne &longs;it fru&longs;trà; faceret enim vt e&longs;&longs;et fru&longs;trà vel <lb/>nouus impetus, vel determinato noua, & in hoc &longs;en&longs;u dicitur impetus <lb/>de&longs;trui ab impetu. </s>

<s>Quintò, &longs;i de&longs;trueretur mobile, etiam de&longs;trueretur impetus per idem <lb/>Ax. </s>

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

<s>6. quia e&longs;&longs;et fru&longs;trà &longs;eparatum; immò ex hoc vno principio demon<lb/>&longs;tramus accidentia & formas &longs;ub&longs;tantiales materiales non po&longs;&longs;e natura<lb/>liter con&longs;eruari extra &longs;uum &longs;ubiectum, quia &longs;cilicet e&longs;&longs;ent fru&longs;trà; quip<lb/>pe finem &longs;uum habent in &longs;ubiecto. </s>

<s>Sextò, Impetus naturalis innatus nunquam de&longs;truitur; quia nunquam <lb/>e&longs;t fru&longs;trà; quippe &longs;emper habet alterum &longs;uorum effectuum formalium, <lb/>id e&longs;t vel motum deor&longs;um, vel grauitationem, adde quod fru&longs;trà de<lb/>&longs;trueretur, cum &longs;it &longs;emper applicata potentia, id e&longs;t ip&longs;a grauitas, &longs;ed de <lb/>his infrâ fusè. </s><pb pagenum="71"/>

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

<s>Denique &longs;atis e&longs;t mihi, vt dicatur aliquid cau&longs;a, Phy&longs;icè loquendo, &longs;i <lb/>ex illius applicatione &longs;emper &longs;equatur effectus; nam non nego po&longs;&longs;e fie<lb/>ri effectus omnes, qui no&longs;tris &longs;en&longs;ibus &longs;ubiiciuntur, &longs;altem extrin&longs;ecos, <lb/>e&longs;&longs;e à cau&longs;a prima, quippe &longs;i &longs;emper ex ignis applicatione Deus diffun<lb/>deret lucem, & calorem, quem &longs;olus ip&longs;e produceret, igne ip&longs;o inerte re<lb/>licto, nullam pror&longs;us mutationem perciperemus; & nemo e&longs;&longs;et, qui non <lb/>exi&longs;timaret lucom hanc & calorem ltunc e&longs;&longs;e ab igne; igitur Phy&longs;icè lo<lb/>quendo cau&longs;am appellamus id, ex cuius applicatione &longs;emper &longs;equitur <lb/>effectus, vt iam diximus in Ax. 11.l.1. n.1. Igitur cum ex corpore graui <lb/>po&longs;ito in aëre libero &longs;equatur motus deor&longs;um; dicendum e&longs;t, Phy&longs;icè lo<lb/>quendò, e&longs;&longs;e huius motus cau&longs;am, id e&longs;t in ordine ad Phy&longs;icam, perinde <lb/>omninò &longs;e habere, atque &longs;i e&longs;&longs;et cau&longs;a, licèt cau&longs;a non e&longs;&longs;et. </s><pb pagenum="77"/>

<s>11.l.1. n.1. Igitur cum ex corpore graui <lb/>po&longs;ito in aëre libero &longs;equatur motus deor&longs;um; dicendum e&longs;t, Phy&longs;icè lo<lb/>quendò, e&longs;&longs;e huius motus cau&longs;am, id e&longs;t in ordine ad Phy&longs;icam, perinde <lb/>omninò &longs;e habere, atque &longs;i e&longs;&longs;et cau&longs;a, licèt cau&longs;a non e&longs;&longs;et. </s><pb pagenum="77"/>

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

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

<s>Sextò, &longs;unt denique multi, <expan abbr="ii&qacute;ue">iique</expan> ex &longs;euerioribus Peripateticis, qui <lb/>exi&longs;timant grauia moueri deor&longs;um à generante, quod expre&longs;&longs;rs verbis <lb/>traditum e&longs;t ab <emph type="italics"/>Ari&longs;totele l.<emph.end type="italics"/>8. <emph type="italics"/>phy&longs;. cap.<emph.end type="italics"/>4. <emph type="italics"/>iuxta<emph.end type="italics"/> principium illud vniuer<lb/>fali&longs;&longs;imum; <emph type="italics"/>Quidquid mouetur; ab alio mouetur<emph.end type="italics"/>; &longs;ed profectò ij ip&longs;i, qui <lb/>motum grauium generanti tribuunt, tanquam principi cau&longs;æ, non ne<lb/>gant ine&longs;&longs;e grauibus grauitatem, quæ &longs;it principium actiuum minus <lb/>principale motus; ad quem etiam, vt ip&longs;i exi&longs;timant, forma &longs;ub&longs;tantialis <lb/>concurrit; In hoc quippe conueniunt omnes tùm &longs;ectarum Principes, <lb/>tùm recentiores: quidquid &longs;it etiam ex iis ip&longs;is datur motus naturalis, <lb/>qui e&longs;t à virtute proxima intrin&longs;eca; hoc ip&longs;um etiam &longs;en&longs;it Ari&longs;toteles <lb/>lib.4. de cælo cap.

<s>cap.<emph.end type="italics"/>4. <emph type="italics"/>iuxta<emph.end type="italics"/> principium illud vniuer<lb/>fali&longs;&longs;imum; <emph type="italics"/>Quidquid mouetur; ab alio mouetur<emph.end type="italics"/>; &longs;ed profectò ij ip&longs;i, qui <lb/>motum grauium generanti tribuunt, tanquam principi cau&longs;æ, non ne<lb/>gant ine&longs;&longs;e grauibus grauitatem, quæ &longs;it principium actiuum minus <lb/>principale motus; ad quem etiam, vt ip&longs;i exi&longs;timant, forma &longs;ub&longs;tantialis <lb/>concurrit; In hoc quippe conueniunt omnes tùm &longs;ectarum Principes, <lb/>tùm recentiores: quidquid &longs;it etiam ex iis ip&longs;is datur motus naturalis, <lb/>qui e&longs;t à virtute proxima intrin&longs;eca; hoc ip&longs;um etiam &longs;en&longs;it Ari&longs;toteles <lb/>lib.4. de cælo cap.

3. t. </s>

<s>25. vbi ait grauibus & leuibus ine&longs;&longs;e principium <lb/>actiuum &longs;uorum motuum; immò &longs;i totum cap.4. l.8. phy&longs;. </s>

<s>25. vbi ait grauibus & leuibus ine&longs;&longs;e principium <lb/>actiuum &longs;uorum motuum; immò &longs;i totum cap.4. l.8. phy&longs;. attentè lega<lb/>tur, vbi dicit moueri à generante, haud dubiè intelligetur nihil aliud in<lb/>tendi&longs;&longs;e Ari&longs;totelem quàm grauia à generante, in&longs;tanti, quo generan<lb/>tur, accipere actum primum huius motus; id e&longs;t virtutem, à qua po&longs;<lb/>&longs;int reduci ad actum &longs;ecundum, id e&longs;t ad ip&longs;um motum, de cuius rei ve<lb/>ritate iam mihi non e&longs;t laborandum. </s>

<s>attentè lega<lb/>tur, vbi dicit moueri à generante, haud dubiè intelligetur nihil aliud in<lb/>tendi&longs;&longs;e Ari&longs;totelem quàm grauia à generante, in&longs;tanti, quo generan<lb/>tur, accipere actum primum huius motus; id e&longs;t virtutem, à qua po&longs;<lb/>&longs;int reduci ad actum &longs;ecundum, id e&longs;t ad ip&longs;um motum, de cuius rei ve<lb/>ritate iam mihi non e&longs;t laborandum. </s>

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

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/></s>

<s><emph type="italics"/>Motus naturalis est aliquid distinctum realiter à mobili:<emph.end type="italics"/> Probatur; <lb/>mobile ip&longs;um aliquando quie&longs;cit per hypoth.4.lib.1. igitur e&longs;t &longs;ine mo<lb/>tu; igitur &longs;eparatum à motu; igitur realiter di&longs;tinctum per Ax.2. lib.1. <lb/>hoc etiam probatus per Th. </s>

<s>1.lib.

1. Et certè mirari &longs;atis non po&longs;&longs;um <lb/>aliquos recentiores non po&longs;&longs;e concipere, vt ip&longs;i aiunt, motum e&longs;&longs;e ali<lb/>quid ab ip&longs;o mobili di&longs;tinctum; nam quotie&longs;cunque duo prædicata, vel <pb pagenum="80"/>attributa contradictoria, quorum &longs;cilicet vnum negat aliud, eidem &longs;ub<lb/>jecto diuer&longs;is temporibus ine&longs;&longs;e dicuntur, haud dubiè alterum &longs;altem ab <lb/>eo di&longs;tingui realiter nece&longs;&longs;e e&longs;t; alioqui &longs;i vtrumque idem e&longs;&longs;e cum vno <lb/>tertio vere dicitur; <emph type="italics"/>mouetur, non monetur,<emph.end type="italics"/> quæ &longs;unt prædicata contradi<lb/>ctoria; igitur vel moueri, vel non moueri dicit di&longs;tinctum realiter à mo<lb/>bili; Secundum e&longs;t mera negatio; nam eo ip&longs;o, quod mobile e&longs;t &longs;ine vllo <lb/>addito, non mouetur; igitur &longs;uprà ip&longs;um mobile dicit puram putam ne<lb/>gationem motus; igitur moueri, dicit aliquid di&longs;tinctum. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 5.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc motus naturalis e&longs;t ab impetu.<emph.end type="italics"/></s><s> Probatur; e&longs;t ab aliqua cau&longs;a per <lb/>Ax.8. lib.1. ab aliqua intrin&longs;eca per Th. 1. non à &longs;ub&longs;tantia corporis <lb/>grauis per Th. 3. non à grauitace per Th. 4. igitur ab impetu, quia <lb/>nihil aliud e&longs;&longs;e pote&longs;t intrin&longs;ecum, à quo &longs;it motus pet definitionem <lb/>3. lib.

<s>1. non à &longs;ub&longs;tantia corporis <lb/>grauis per Th. </s>

<s>3. non à grauitace per Th. </s>

<s>4. igitur ab impetu, quia <lb/>nihil aliud e&longs;&longs;e pote&longs;t intrin&longs;ecum, à quo &longs;it motus pet definitionem <lb/>3. lib.

1. </s><pb pagenum="81"/>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus productus primo instanti durat proximè &longs;equenti.<emph.end type="italics"/></s><s> Probatur pri<lb/>mò; quia &longs;emper habet &longs;uum effectum formalem; vel grauitationis, &longs;i <lb/>impeditur; vel motus in medio libero; igitur non e&longs;t fru&longs;trà; igitur <lb/>non de&longs;truitur per Th.162.lib.1. nihil enim exigit de&longs;tructionem; non <lb/>tota natura, quia non e&longs;t fru&longs;trà per Ax. </s>

<s>6. non à contrario impetu, qui <lb/>&longs;æpè abe&longs;t, vt cum liberè mouetur corpus graue in aëre, vel &longs;u&longs;tinetur, <lb/>v.g. </s>

<s>glans plumbea ab ingenti rupe: adde quod, licèt producatur in cor<lb/>pore graui impetus violentus &longs;ur&longs;um, non de&longs;truitur, tamen innatus; alio<lb/>quin nihil e&longs;&longs;et, quod de&longs;trueret violentum per Th.150. & Schol. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus ille innatus, qui durat &longs;ecundo instanti, con&longs;eruatur ab aiiqua cau<lb/>&longs;a<emph.end type="italics"/>; e&longs;t certum per Ax. </s>

<s><emph type="italics"/>Impetus ille innatus, qui durat &longs;ecundo instanti, con&longs;eruatur ab aiiqua cau<lb/>&longs;a<emph.end type="italics"/>; e&longs;t certum per Ax. 14.lib.1.num.1. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 11.<emph.end type="center"/></s>

<s><emph type="italics"/>Non con&longs;eruatur à cau&longs;a primò productiua.<emph.end type="italics"/></s><s> Probatur per Th.144. lib.1. <lb/>alioquin non po&longs;&longs;et intendi ab eadem cau&longs;a per Th. 146. lib 1. quippè <lb/>con&longs;eruatio nihil e&longs;t aliud, quàm repetita productio, vt con&longs;tat; nam <lb/>cau&longs;a con&longs;eruans verè influit; igitur &longs;i e&longs;t can&longs;a nece&longs;&longs;aria primo, & &longs;e<lb/>cundo in&longs;tanti æquali ni&longs;u influit; influit enim quantum pote&longs;t per Ax. </s>

<s>146. lib 1. quippè <lb/>con&longs;eruatio nihil e&longs;t aliud, quàm repetita productio, vt con&longs;tat; nam <lb/>cau&longs;a con&longs;eruans verè influit; igitur &longs;i e&longs;t can&longs;a nece&longs;&longs;aria primo, & &longs;e<lb/>cundo in&longs;tanti æquali ni&longs;u influit; influit enim quantum pote&longs;t per Ax. </s>

<s><lb/>12.lib.1.quòd autem impetus intendatur, demon&longs;trabimus infrà; con&longs;ule <lb/>Schol.Th.146.lib.1.in quo habes rationem præclari natura in&longs;tituti; quo <lb/>&longs;cilicet factum e&longs;t, vt qualitates, quæ contrario carent à causâ primò pro<lb/>ductiua; aliæ verò, quæ contrarium habent, ab alia causà con&longs;er<lb/>uentur. </s>

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<s><emph type="italics"/>Quando graue e&longs;t in medio libero, per quod &longs;cilicet de&longs;cendere pote&longs;t, &longs;ecun<lb/>do instanti producitur nouus impetus, itemque tertio, quarto, quinto. </s>

<s>&c.<emph.end type="italics"/></s><s> Pro<lb/>batur primò; quia &longs;ecundo in&longs;tanti e&longs;t eadem cau&longs;a quæ primo non ma<lb/>gis impedita, eáque nece&longs;&longs;aria; igitur nece&longs;&longs;ariò agit per Ax. 12. lib.1. <lb/>igitur aliquem effectum producit; &longs;ed hic effectus non e&longs;t impetus pro<lb/>ductus primo in&longs;tanti, quia non con&longs;eruatur à cau&longs;a primò productiua <lb/>per Th.11. igitur e&longs;t nouus. </s>

<s>12. lib.1. <lb/>igitur aliquem effectum producit; &longs;ed hic effectus non e&longs;t impetus pro<lb/>ductus primo in&longs;tanti, quia non con&longs;eruatur à cau&longs;a primò productiua <lb/>per Th.11. igitur e&longs;t nouus. </s>

<s>Probatur &longs;ecundò; cre&longs;cit motus grauium in <lb/>libero medio per hypoth. </s>

<s>1.2.3. igitur cre&longs;cit impetus; quia cum motus <lb/>naturalis &longs;it ab impetu per Th.5. quâ proportionc cre&longs;cit effectus, &longs;cilicet <lb/>&longs;ormalis, & exigentiæ; &longs;ic enim motus e&longs;t effectus impetus per Th. 15. <lb/>lib.1.eàdem cre&longs;cit cau&longs;a per Ax.2. Probatur tertiò, quia corpus graue ex <lb/>maiore altitudine cadens maiorem quoque ictum infligit per hypoth.1. <lb/>igitur maior impetus imprimitur in corpore percu&longs;&longs;o; &longs;ed impetus ad ex<lb/>tra producitur ab alio impetu per Th.42.lib.1. igitur &longs;i cre&longs;cit productus <lb/>inpetus, cre&longs;cit impetus producens. </s>

<s>15. <lb/>lib.1.eàdem cre&longs;cit cau&longs;a per Ax.2. Probatur tertiò, quia corpus graue ex <lb/>maiore altitudine cadens maiorem quoque ictum infligit per hypoth.1. <lb/>igitur maior impetus imprimitur in corpore percu&longs;&longs;o; &longs;ed impetus ad ex<lb/>tra producitur ab alio impetu per Th.42.lib.1. igitur &longs;i cre&longs;cit productus <lb/>inpetus, cre&longs;cit impetus producens. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hinc reiicies quorumdam placitum, qui volunt cau&longs;am velociotis <lb/>motus e&longs;&longs;e grauitatem ip&longs;am, quatenus connotat motum præuium; quia <lb/>&longs;cilicet grauitas non producit illum maiorem impetum ad extra, vt con<lb/>&longs;tat; nec &longs;ub&longs;tantia ip&longs;ius corporis grauis per Th.40.lib.1.igitur ip&longs;e im<lb/>petus: præterea &longs;i hoc e&longs;&longs;et, fru&longs;trà requireretur impetus contra Th. </s>

<s>5. <lb/>Denique motus præuius nihil e&longs;t amplius, cum alius &longs;uccedit: Vide Th. </s>

<s><lb/>40.lib.1. vbi hæc fusè di&longs;cu&longs;&longs;imus. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus productus &longs;ecundo instanti in medio libero con&longs;eruatur tertio, & <lb/>productus tertio con&longs;eruatur, quarto, atque ita deinceps<emph.end type="italics"/>; quia &longs;cilicet nec con<lb/><gap/>antur à cau&longs;a primo productiua per Th.144.libri: nec aliquid exigit <lb/>de&longs;tructionem; non contrarius impetus, quia nullus e&longs;t applicatus, vt <lb/>con&longs;tat; non re&longs;i&longs;tentia medij, quæ quidem alicuius momenti e&longs;t; &longs;ed <lb/>non tanti, vt impedire po&longs;&longs;it motum omninò, vt con&longs;tat; nam &longs;uppono <lb/>liberum medium, igitur nec de&longs;truere impetum; cum tamdiu duret cau<lb/>&longs;a quamdiu durat effectus, vt patet; igitur nihil e&longs;t quod exigat impe<lb/>ms huius de&longs;tructionem; igitur non de&longs;truitur per Ax. </s>

<s>14. lib.1. <lb/><expan abbr="q&umacr;anta">qunanta</expan> verò &longs;it, & quid &longs;it cuiu&longs;libet medij re&longs;i&longs;tentia, dicemus <lb/>infrà. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 14.<emph.end type="center"/></s>

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<s>Secunda naturalis acqui&longs;iti. </s>

<s><lb/>Tertia violenti; innatus e&longs;t qui vel à generante &longs;imul cum corpore <lb/>graui productus e&longs;t; qui&longs;quis tandem &longs;it generans, de quo aliàs; vel ab <lb/>ip&longs;o graui qua&longs;i profunditur, id e&longs;t, producitur in &longs;e ip&longs;o &longs;tatim initio, <lb/>quo e&longs;t; porrò cum in corpore graui duplex qua&longs;i proprietas &longs;en&longs;ibilis <lb/>e&longs;&longs;e videatur, &longs;cilicet grauitas, &longs;eu pondus & motus deor&longs;um; certè de<lb/>bet e&longs;&longs;e in co aliquid per quod tùm cogno&longs;ci po&longs;&longs;it eins pondus, tùm in<lb/>cipiat moueri deor&longs;um; quippe maximè corpora ex pondere cogno&longs;ci<lb/>mus, vnumque ab alio di&longs;tinguimus; igitur debet e&longs;&longs;e aliquid, quod &longs;en<lb/>&longs;um afficiat, vt cogno&longs;ci po&longs;&longs;it; atqui illud ip&longs;um non e&longs;t &longs;ub&longs;tantia cor<lb/>poris; nam corpus graue meæ manui &longs;u&longs;tinenti impetum imptimit; <lb/>immò vim alterius impetus infringit; igitur operâ alterius per Th. 40. <lb/>& 42.lib.1. Præterea illud ip&longs;um, quod agit, &longs;eu deor&longs;um pellit &longs;u&longs;tinen<lb/>tem manum, e&longs;t illud ip&longs;um quod inclinat corpus graue deor&longs;um imme<lb/>diatè, &longs;eu quod exigit motum naturalem deor&longs;um; illud autem quod <lb/>immediatè præ&longs;tat hunc motum, nec e&longs;t &longs;ub&longs;tantia corporis grauis per <lb/>Th.3. igitur ip&longs;e impetus per Th.5. adde quod primo in&longs;tanti, quo e&longs;t im<lb/>petus, non e&longs;t motus ille, quem exigit per Th.34. lib.1. igitur præexi&longs;tit <lb/>&longs;emper impetus, qui ne &longs;it fru&longs;trà, habet primum effectum &longs;uum forma<lb/>lem, id e&longs;t grauitationem: Ex his dicendum e&longs;t hunc impetum natiuum <lb/>nunquam de&longs;trui, quia nunquam e&longs;t fru&longs;trà, habet enim &longs;emper aliquem <lb/>effectum, primum quidem &longs;i caret &longs;ecundo; &longs;ecundum verò &longs;i caret pri<lb/>mo; quippe vtrumque &longs;imul habere non pote&longs;t; nam corporis pondus <lb/>cogno&longs;ci non pote&longs;t, dum fertur deor&longs;um accelerato motu, quot verò, <lb/>& quanta commoda ex cognitione ponderis cuiu&longs;libet materiæ proce<lb/>dant, vix explicari pote&longs;t. </s>

<s>40. <lb/>& 42.lib.1. Præterea illud ip&longs;um, quod agit, &longs;eu deor&longs;um pellit &longs;u&longs;tinen<lb/>tem manum, e&longs;t illud ip&longs;um quod inclinat corpus graue deor&longs;um imme<lb/>diatè, &longs;eu quod exigit motum naturalem deor&longs;um; illud autem quod <lb/>immediatè præ&longs;tat hunc motum, nec e&longs;t &longs;ub&longs;tantia corporis grauis per <lb/>Th.3. igitur ip&longs;e impetus per Th.5. adde quod primo in&longs;tanti, quo e&longs;t im<lb/>petus, non e&longs;t motus ille, quem exigit per Th.34. lib.1. igitur præexi&longs;tit <lb/>&longs;emper impetus, qui ne &longs;it fru&longs;trà, habet primum effectum &longs;uum forma<lb/>lem, id e&longs;t grauitationem: Ex his dicendum e&longs;t hunc impetum natiuum <lb/>nunquam de&longs;trui, quia nunquam e&longs;t fru&longs;trà, habet enim &longs;emper aliquem <lb/>effectum, primum quidem &longs;i caret &longs;ecundo; &longs;ecundum verò &longs;i caret pri<lb/>mo; quippe vtrumque &longs;imul habere non pote&longs;t; nam corporis pondus <lb/>cogno&longs;ci non pote&longs;t, dum fertur deor&longs;um accelerato motu, quot verò, <lb/>& quanta commoda ex cognitione ponderis cuiu&longs;libet materiæ proce<lb/>dant, vix explicari pote&longs;t. </s>

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

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 16.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc motus naturalis deor&longs;um acceleratur<emph.end type="italics"/>; hoc ip&longs;um &longs;uppo&longs;ui &longs;uprà <lb/>Quod e&longs;&longs;et in hyp.1.2.3. iam verò demon&longs;tro propter quid e&longs;t; &longs;ie cnim <lb/>hypothe&longs;is in Theorema conuerti pote&longs;t, vt fæpè monuimus in metho<lb/>do; igitur probatur hoc Theorema facilè; cre&longs;cit impetus in corpore gra<lb/>ui, quod tendit deor&longs;um in libero medio per T. 15. igitar cre&longs;cit cau&longs;a <lb/>motus; nam impetus e&longs;t can&longs;a immediata motus naturalis per Th. 51. <lb/>&longs;ed quâ proportione cre&longs;cit cau&longs;a, debet cre&longs;cere effectus per Ax.2. igi<lb/>tur motus naturalis deor&longs;um cre&longs;cit, id e&longs;t acceleratur, id e&longs;t fit velo<lb/>cior, quod erat dem: nec e&longs;t quod aliquis exi&longs;timet hic à me committi <lb/>vitio&longs;um argumentationis circulum; quippe probaui &longs;uprà cre&longs;cere im<lb/>petum, quia cre&longs;cit motus; iam verò probo cre&longs;cere motum, quia cre&longs;<lb/>cit impetus; nam primò probaui produci nouum impetum in Th.12. co <lb/>quod &longs;ecundo in&longs;tanti. </s>

<s>51. <lb/>&longs;ed quâ proportione cre&longs;cit cau&longs;a, debet cre&longs;cere effectus per Ax.2. igi<lb/>tur motus naturalis deor&longs;um cre&longs;cit, id e&longs;t acceleratur, id e&longs;t fit velo<lb/>cior, quod erat dem: nec e&longs;t quod aliquis exi&longs;timet hic à me committi <lb/>vitio&longs;um argumentationis circulum; quippe probaui &longs;uprà cre&longs;cere im<lb/>petum, quia cre&longs;cit motus; iam verò probo cre&longs;cere motum, quia cre&longs;<lb/>cit impetus; nam primò probaui produci nouum impetum in Th.12. co <lb/>quod &longs;ecundo in&longs;tanti. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s>

<s><emph type="italics"/>Duo motus æquabiles, qui durant æqualibus temporibus, &longs;unt vt &longs;patia<emph.end type="italics"/>; <lb/>patet; cùm enim impetus &longs;int vt motus per Ax. </s>

<s><emph type="italics"/>Duo motus æquabiles, qui durant æqualibus temporibus, &longs;unt vt &longs;patia<emph.end type="italics"/>; <lb/>patet; cùm enim impetus &longs;int vt motus per Ax. 2. motus &longs;unt vt &longs;patia; <lb/>quippe vt ex impetu &longs;equitur motus, ita ex motu confectum &longs;pa<lb/>tium. </s>

<s>2. motus &longs;unt vt &longs;patia; <lb/>quippe vt ex impetu &longs;equitur motus, ita ex motu confectum &longs;pa<lb/>tium. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 22.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s>

<s><emph type="italics"/>Si duo mobilia ferantur motu æquabili per diuer&longs;a &longs;patia, & diuer&longs;a velo<lb/>eitate, tempora erunt in ratione compo&longs;ita ex ratione &longs;paliorum & ratioue <lb/>velocitatum permutata<emph.end type="italics"/>; probatur eodem modo quo &longs;uperius Th. 30. &longs;it <lb/>ratio &longs;patiorum 4/1, velocitatum 4/2; permutetur hæc 1/4; componetur ex <lb/>vtraque 4/1, ide&longs;t 1/2, quæ e&longs;t ratio temporum. </s>

<s>30. &longs;it <lb/>ratio &longs;patiorum 4/1, velocitatum 4/2; permutetur hæc 1/4; componetur ex <lb/>vtraque 4/1, ide&longs;t 1/2, quæ e&longs;t ratio temporum. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s>

<s><emph type="italics"/>In motu naturaliter accelerato impetus nouus acquiritur &longs;ingulis in&longs;tanti<lb/>bus<emph.end type="italics"/>; Probatur quia &longs;ingulis in&longs;tantibus e&longs;t eadem cau&longs;a nece&longs;&longs;aria, igi<lb/>tur &longs;ingulis in&longs;tantibus aliquem effectum producit, per Ax. </s>

<s>12. l.1. &longs;ed <lb/>priorem non con&longs;eruat, vt dictum e&longs;t &longs;uprà, igitur nouum producit. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 34.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc &longs;ingulis in&longs;tantibus intenditur impetus in hoc motu<emph.end type="italics"/>; cum &longs;ingulis <lb/>in&longs;tantibus producatur nouus, & prior con&longs;eruetur, cui cum addatur, <lb/>intenditur per Ax. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 43.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc &longs;patia cre&longs;cunt &longs;ingulis in&longs;tantibus æqualibus &longs;ecundùm pregre&longs;&longs;io<lb/>nem arithmeticam<emph.end type="italics"/>; quia cre&longs;cit vt velocitas per Th.40. hæc vt impetus <lb/>per Th.38. hic demum iuxta progre&longs;&longs;ionem arithmeticam per Th. 37. <lb/>igitur &longs;i &longs;patium acqui&longs;itum primo in&longs;tanti &longs;it 1. acqui&longs;itum &longs;ecundo crit <lb/>2. tertio 3. quarto 4. &c. </s>

<s>37. <lb/>igitur &longs;i &longs;patium acqui&longs;itum primo in&longs;tanti &longs;it 1. acqui&longs;itum &longs;ecundo crit <lb/>2. tertio 3. quarto 4. &c. </s>

<s>hinc &longs;patia acqui&longs;ita &longs;ingulis in&longs;tantibus &longs;unt <lb/>vt &longs;eries numerorm, qui componunt progre&longs;&longs;ionem &longs;implicem, &longs;cilicet <lb/>1.2.3.4.5.6. &c. </s>

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<s><emph type="italics"/>Ex dictis hactenus facilè redditur ratio maioris ictus eiu&longs;dem corporis im<lb/>pacti quod cadit ex maiori altitudine<emph.end type="italics"/>; fuit hyp. </s>

<s>1. &longs;ed ideò e&longs;t maior ictus, <lb/>quia maior imprimitur impetus, vt patet, at ideò maior impetus impri<lb/>mitur, quia maior e&longs;t imprimens per Ax. </s>

<s>1. &longs;ed ideò e&longs;t maior ictus, <lb/>quia maior imprimitur impetus, vt patet, at ideò maior impetus impri<lb/>mitur, quia maior e&longs;t imprimens per Ax. 2. cre&longs;cit enim impetus, vt <lb/>con&longs;tat ex dictis. </s>

<s>2. cre&longs;cit enim impetus, vt <lb/>con&longs;tat ex dictis. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 51.<emph.end type="center"/></s>

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<s><emph type="italics"/>Hinc po&longs;&longs;unt comparari duæ percu&longs;&longs;iones duorum grauium inæqualium <lb/>dum cadunt deor&longs;um<emph.end type="italics"/>; &longs;i enim cadunt æqualibus temporibus, percu&longs;&longs;io<lb/>nes erunt vt corpora &longs;eu grauitates, vt patet v.g. </s>

<s>corpus 2. librarum po&longs;t <lb/>2. in&longs;tantia motus infligit duplam percu&longs;&longs;ionem illius, quam infligit cor<lb/>pus vnius libræ po&longs;t 2. in&longs;tantia motus; &longs;i verò tempora motus &longs;unt inæ<lb/>qualia, & grauitates æquales, percu&longs;&longs;iones erunt vt tempora; &longs;i demum <lb/>grauitates inæquales, & tempora motus inæqualia, percu&longs;&longs;iones erunt <lb/>in ratione compo&longs;ita ex ratione grauitatum & temporum, quæ omnia <lb/>patent ex dictis in Th. </s>

<s>&longs;uperioribus, v. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 59.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc &longs;i corpus graue de&longs;cenderet motu æquabili eoque æquali motui primi <lb/>m&longs;tantis; certè vix modicum &longs;patium post multos annos decurreret<emph.end type="italics"/>; &longs;uppo<lb/>namus enim quod plures habent, licèt accuratè experimento &longs;ubii<lb/>ci non po&longs;&longs;it, &longs;cilicet vno &longs;ecundo minuto temporis decurri à corpore <lb/>graui deor&longs;um 12. pedes &longs;patij; in &longs;ecundo minuto &longs;upponamus e&longs;&longs;e <lb/>mille in&longs;tantia, quamuis infinita penè contineat; &longs;itque in primo in<lb/>&longs;tanti motus vnus gradus impetus; &longs;ic enim vocetur illa pars impetus, que <lb/>producitur primo in&longs;tanti; certè po&longs;t mille in&longs;tantia motus, erunt mille <lb/>gradus impetus; iam vcrò &longs;i accipiatur &longs;ubduplum maximæ & minimæ <lb/>velocitatis; id e&longs;t vnius gradus, & mille graduum, &longs;cilicet 500. 1/2 tri<lb/>buaturque motui æquabili; haud dubiè vno fecundo minuto percur<lb/>rentur 12. pedes &longs;patij per Th. </s>

<s>46. Igitur &longs;i cum velocitate vt 500, 1/2 <lb/>percurrentur 12. pedes 1.minuto, cum velocitate vt 1. percurrentur <lb/>12. pedes 500.&longs;ecundis minutis, &; 30. tertiis; &longs;i verò accipiantur plura <lb/>in&longs;tantia, v.g. </s>

<s>1000000.in&longs;tantia, percurrentur 12. pedes 500000. &longs;e<lb/>cundis minutis; &longs;i verò 1000000000000. percurremur 500000000000. <lb/>&longs;ecundis, id e&longs;t 8333333333. minutis, id e&longs;t 138888888. horis <pb pagenum="96"/>id e&longs;t 5787037. diebus id e&longs;t 89031. annis, omitto minutias; atqui lon<lb/>gè adhuc plura in vno minuto continentur in&longs;tantia. </s>

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<s>Prima ratio, quam affert Galileus e&longs;t; quia cum natura in &longs;uis opera<lb/>tionibus adhibeat &longs;implici&longs;&longs;ima media; & cum acceleratio motus natu<lb/>ralis non po&longs;&longs;it fieri iuxta faciliorem, vel &longs;impliciorem progre&longs;&longs;ionem, <lb/>quàm &longs;it-ea quæ fit per quadrata; non e&longs;t dubium, quin iuxta illam pro<lb/>gre&longs;&longs;io motus naturaliter accelerati fieri debeat; præ&longs;ertim cùm omni<lb/>bus experimentis con&longs;entiat, & in ea omnia phænomena explicari <lb/>po&longs;&longs;int. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. Primò progre&longs;&longs;ionem arithmeticam &longs;implicem iuxta hos nu<lb/>meros 1.2.3.4. longè &longs;impliciorem e&longs;&longs;e alia quæ fit iuxta illos 1.3.5.7.vt <lb/>nemo non iudicabit. </s>

<s>Primò progre&longs;&longs;ionem arithmeticam &longs;implicem iuxta hos nu<lb/>meros 1.2.3.4. longè &longs;impliciorem e&longs;&longs;e alia quæ fit iuxta illos 1.3.5.7.vt <lb/>nemo non iudicabit. </s>

<s>Secundò <expan abbr="c&utilde;">cum</expan> accidit duas hypothe&longs;es conuenire cum <lb/>omnibus experimentis &longs;eu phænomonis, debet e&longs;&longs;e aliqua ratio, cur ad<lb/>hibeatur vna potiùs quàm alia; &longs;ed nulla e&longs;t ratio, cur Galileus adhibeat <lb/>&longs;uam, vti videbimus; nos verò ratione demon&longs;tratiuâ probamus no&longs;tram; <lb/>igitur no&longs;tra e&longs;t præferenda pro theorica rei veritate; quia verò alia in <lb/>temporibus &longs;en&longs;ibilibus proximè ad verum accedit eam adhibendam e&longs;&longs;e <lb/>decernemus infrà ad praxim, & communem i&longs;torum motuum men<lb/>&longs;uram. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc qua preportione decre&longs;cunt instantia, decre&longs;cit etiam per&longs;ectio <lb/>impetus<emph.end type="italics"/>; quia temporibus æqualibus cadem cau&longs;a nece&longs;&longs;aria æqualem ef<lb/>fectum producit per Ax. </s>

<s>tertium igitur inæqualem inæqualibus, per Ax. </s>

<s><lb/>13. num.4. igitur minorem minore tempore; igitur minorem in eadem <lb/>proportione, in qua tempus e&longs;t; igitur qua proportione, &c. </s>

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<s><emph type="italics"/>Hinc vides quâm &longs;it nece&longs;&longs;aria illa diuer &longs;a perfestio impetus, quam indi<lb/>cauimus lib.<emph.end type="italics"/>1. hinc impetus productus &longs;ecundo, & tertio in&longs;tanti adæ<lb/>quat impetum productum primo, quem etiam adæquat productus quar<lb/>to, quinto, &longs;exto, item productus &longs;eptimo, octauo, nono; decimo, atque ita <lb/>deinceps; hinc e&longs;t eadem differentia impetuum, quæ <expan abbr="in&longs;tãtium">in&longs;tantium</expan>; hinc &longs;in<lb/>gulis &longs;patiis æqualibus primo &longs;patio, quod percurritur primo in&longs;tanti; <lb/>re&longs;pondent &longs;ingula in&longs;tantia, & &longs;ingulis in&longs;tantibus &longs;inguli, & &longs;ingulares <lb/>impetus; hinc non e&longs;t quod primo in&longs;tanti dicantur produci plura pun<lb/>cta impetus in eodem puncto corporis grauis; &longs;ed vnicum tantùm pun<lb/>ctum talis perfectionis &longs;cilicet phy&longs;icum; cur enim potius duo puncta, <lb/>quam tria? </s>

<s>&longs;ed quod vnum e&longs;t determinatum e&longs;t per Ax. </s>

<s>&longs;ed quod vnum e&longs;t determinatum e&longs;t per Ax. 5. lib.

<s>5. lib.

1. hinc <lb/>optima ratio cur potius tali in&longs;tanti producatur impetus talis perfectie<lb/>nis quàm alterius? </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus naturalis addititius de&longs;truitur<emph.end type="italics"/>; patet experientiâ; quippe pila <lb/>deor&longs;um cadens tandem quie&longs;cit, licèt à terra reflectatur ratione impe<lb/>dimenti, ex quo re&longs;ultat duplex determinatio, ratione cuius idem im<lb/>petus &longs;ibi aliquo modo redditur <expan abbr="cõtrarius">contrarius</expan>; &longs;ed de his fusè in primo libro <lb/>à Th.148. ad finem v&longs;que libri: nam reuerâ duæ determinationes op<lb/>po&longs;itæ pugnant pro rata per Ax. 15.l.1. & quotie&longs;cunque idem impetus <lb/>e&longs;t ad lineas oppo&longs;itas determinatus eodem modo &longs;e habet, ac &longs;i duplex <lb/>e&longs;&longs;et, & quilibet &longs;uæ &longs;ube&longs;&longs;et determinationi; atqui &longs;i duplex e&longs;&longs;et oppo<lb/>&longs;itus, pugnarent pro rata; igitur tàm pugnant duæ determinationes op<lb/>po&longs;itæ in codem impetu, quàm duo impetus ad oppo&longs;itas lincas deter<lb/>minati; igitur impetus naturalis aduentitius de&longs;truitur, &c. </s>

<s>15.l.1. & quotie&longs;cunque idem impetus <lb/>e&longs;t ad lineas oppo&longs;itas determinatus eodem modo &longs;e habet, ac &longs;i duplex <lb/>e&longs;&longs;et, & quilibet &longs;uæ &longs;ube&longs;&longs;et determinationi; atqui &longs;i duplex e&longs;&longs;et oppo<lb/>&longs;itus, pugnarent pro rata; igitur tàm pugnant duæ determinationes op<lb/>po&longs;itæ in codem impetu, quàm duo impetus ad oppo&longs;itas lincas deter<lb/>minati; igitur impetus naturalis aduentitius de&longs;truitur, &c. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 73.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc etiam impetus producitur imperfectior<emph.end type="italics"/>; quia ex imperfectione ef<lb/>fectus requiritur imperfectic cau&longs;æ per Ax. </s>

<s><emph type="italics"/>Hinc etiam impetus producitur imperfectior<emph.end type="italics"/>; quia ex imperfectione ef<lb/>fectus requiritur imperfectic cau&longs;æ per Ax. 13.l. </s>

<s>1. & quâ proportione <lb/>e&longs;t tardior motus eâdem impetus e&longs;t imperfectior, per Ax. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 85.<emph.end type="center"/></s>

<s><emph type="italics"/>Medium graue detrahit aliquid de &longs;ingulari grauit atione corporis grauio<lb/>ris<emph.end type="italics"/>; certa e&longs;t hypothe&longs;is; nec enim plumbum e&longs;t eius ponderis &longs;ingula<lb/>ris in aqua, cuius e&longs;t in aëre; dixi &longs;ingularis; nam &longs;i plumbum & ip&longs;a <lb/>aqua &longs;imul appendantur, haud dubiè totum habebis pondus plumbi, & <lb/>totum pondus aquæ; ratio verò huius effectus non e&longs;t huius loci; quid<lb/>quid &longs;it, &longs;i æqualis grauitas medij tollit totam æqualem alterius corpo<lb/>ris; certè maiorem alterius corporis totam non tollit per Th. </s>

<s>80. &longs;ed <lb/>tantùm aliquid illius, quod quomodo fiat, dicemus Tomo quinto cum de <lb/>graui, & leui. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 86.<emph.end type="center"/></s>

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<s><emph type="italics"/>Si mobilia duo eiu&longs;dem materiæ, figuræ, & grauitatis in eodem medio de<lb/>&longs;<gap/>endant, æquali motu feruntur<emph.end type="italics"/> dem. </s>

<s>vbi e&longs;t eadem proportio cau&longs;æ & re&longs;i<lb/>&longs;bentiæ ibi e&longs;t idem effectus, per Ax. </s>

<s>vbi e&longs;t eadem proportio cau&longs;æ & re&longs;i<lb/>&longs;bentiæ ibi e&longs;t idem effectus, per Ax. 5. &longs;ed in hoc ca&longs;u eadem e&longs;t illa pro<lb/>portio; nam e&longs;t æqualis cau&longs;a, &longs;cilicet grauitas; idem medium æqualiter <lb/>vtrique re&longs;i&longs;tens, cum non plures medij partes re&longs;i&longs;tant vni, quam alteri; <lb/>igitur æqualis proportio. </s>

<s>5. &longs;ed in hoc ca&longs;u eadem e&longs;t illa pro<lb/>portio; nam e&longs;t æqualis cau&longs;a, &longs;cilicet grauitas; idem medium æqualiter <lb/>vtrique re&longs;i&longs;tens, cum non plures medij partes re&longs;i&longs;tant vni, quam alteri; <lb/>igitur æqualis proportio. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 106.<emph.end type="center"/></s>

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<s><emph type="italics"/>Hinc plùs minuitur grauitas, quàm re&longs;i&longs;tentia minoris cubi<emph.end type="italics"/>; quia grauitas <lb/>re&longs;pondet &longs;olido, & re&longs;i&longs;tentia primç facici; re&longs;i&longs;tentia <expan abbr="inqu&amacr;">inquam</expan> ratione par<lb/>tium medij; &longs;ed &longs;olidum plus miuuitur quàm facies, vt dictum e&longs;t; igitur <lb/>plus minuitur grauitas, quæ e&longs;t cau&longs;a virium quàm hæc re&longs;i&longs;tentia; crgo <lb/>decre&longs;cunt vires in maiore proportione quàm hæc re&longs;i&longs;tentia, quod be<lb/>nè ob&longs;eruauit Galileus in dìalogis. </s>

<s>Hinc concludit Galileus duos cubos eiu&longs;dem materiæ, &longs;ed inæquales <lb/>de&longs;cendere inæquali motu; maiorem &longs;cilicet velociùs minori; demon<lb/>ftrare videtur, quia maior habet maiorem proportionem virium ad re<lb/>&longs;i&longs;tentiam, quàm minor; igitur maiorem habet effectum per Ax. </s>

<s>5. igi<lb/>tur maiorem, & velociorem motum. </s>

<s>Scio non dee&longs;&longs;e multos viros doctos qui acriter in hanc &longs;ententiam <pb pagenum="126"/>in&longs;urgant: Obiicient fortè primò, experientiam e&longs;&longs;e contrariam; &longs;i enim <lb/>accipiantur duo cubi maior, & minor eiu&longs;dem materiæ, & dimittantur <lb/>ex eadem altitudine codem pror&longs;us momento terram ferient; Re&longs;ponde<lb/>ri pote&longs;t momentum illud &longs;en&longs;u percipi non po&longs;&longs;e; &longs;i enim dicam ma<lb/>iorem tangere terram 1000. in&longs;tantibus ante minorem, an fortè &longs;en&longs;u <lb/>hoc percipies, vi&longs;u &longs;cilicet vel auditu? </s>

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<s>Secundò obiicient, &longs;i &longs;uperponatur cubus minor maiori in &longs;uo motu <lb/>nunquam &longs;eparantur; igitur æquali motu de&longs;cendunt. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. videri po<lb/>te&longs;t equidem æquali motu de&longs;cendere quia &longs;unt veluti partes ciu&longs;dem <lb/>corporis, & grauitant grauitatione communi, neque minor habet &longs;ingu<lb/>larem re&longs;i&longs;tentiam &longs;uperandam; immò &longs;i &longs;uperponatur minor maiori, <lb/>vel maior minori, motus e&longs;t velocior quàm e&longs;&longs;et &longs;olius maioris; quia <lb/>cum non &longs;it maior re&longs;i&longs;tentia, maiores illi vires opponuntur; igitur fa<lb/>ciliùs &longs;uperatur. </s>

<s>videri po<lb/>te&longs;t equidem æquali motu de&longs;cendere quia &longs;unt veluti partes ciu&longs;dem <lb/>corporis, & grauitant grauitatione communi, neque minor habet &longs;ingu<lb/>larem re&longs;i&longs;tentiam &longs;uperandam; immò &longs;i &longs;uperponatur minor maiori, <lb/>vel maior minori, motus e&longs;t velocior quàm e&longs;&longs;et &longs;olius maioris; quia <lb/>cum non &longs;it maior re&longs;i&longs;tentia, maiores illi vires opponuntur; igitur fa<lb/>ciliùs &longs;uperatur. </s>

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

<s>Quartò obij:igitur motus po&longs;&longs;et e&longs;&longs;e velocior, & velocior in infini<lb/>tum; &longs;i enim maior cubus de&longs;cenderet velociùs; igitur &longs;i detur maior ad<lb/>huc velociùs, atque ita deinceps: Re&longs;p. inanem pror&longs;us e&longs;&longs;e difficulta<lb/>tem; quia cubus ille quantumuis maximus in vacuo de&longs;cendit velociùs <lb/>quàm in aliquo medio v.g.in aëre, igitur nunquam augmentum veloci<lb/>tatis infinitum e&longs;t; quippe inter duos gradus velocitatis infiniti &longs;unt <lb/>po&longs;&longs;ibiles. </s>

<s>inanem pror&longs;us e&longs;&longs;e difficulta<lb/>tem; quia cubus ille quantumuis maximus in vacuo de&longs;cendit velociùs <lb/>quàm in aliquo medio v.g.in aëre, igitur nunquam augmentum veloci<lb/>tatis infinitum e&longs;t; quippe inter duos gradus velocitatis infiniti &longs;unt <lb/>po&longs;&longs;ibiles. </s>

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<s>Vt &longs;olui po&longs;&longs;it præ&longs;ens difficultas, quæ cettè maxima e&longs;t, totam rem <lb/>i&longs;tam paulò fu&longs;iùs e&longs;&longs;e explicandam iudico. </s>

<s>Primò itaque certum e&longs;t <lb/>partes medij, quæ prius in fronte erant, retroire; hoc ip&longs;um videmus in <lb/>naui quæ &longs;ulcat aquas, hoc ip&longs;um accidit in omni corpore natante etiam <lb/>immobili, quippe partes aquæ retinentur ab illa membranula, de qua &longs;u<lb/>prà; &longs;ic enim &longs;æpè a&longs;&longs;urgunt, & intume&longs;cunt &longs;upra labra va&longs;is; cur verò <lb/>continui penè circulares limbi dilatentur: Re&longs;p. nullo flante vento <lb/>vix aliquem circulum huiu&longs;inodi in &longs;uperficie aquæ apparere à fronte, <lb/>&longs;ed tantùm à tergo, & lateribus, qua&longs;i ad in&longs;tar pyramidis; &longs;ed de his aliàs <lb/>fusè. </s>

<s>nullo flante vento <lb/>vix aliquem circulum huiu&longs;inodi in &longs;uperficie aquæ apparere à fronte, <lb/>&longs;ed tantùm à tergo, & lateribus, qua&longs;i ad in&longs;tar pyramidis; &longs;ed de his aliàs <lb/>fusè. </s>

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

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 120.<emph.end type="center"/></s>

<s><emph type="italics"/>Globus materiæ leuioris pote&longs;t de&longs;cendere velociori motu quam parallelipe<lb/>dum grauioris<emph.end type="italics"/>; con&longs;tat experientia; ratio e&longs;t, quia cum globus ferreus de&longs;<lb/>cendat velociùs, quàm ligneus per Th. 118. in data ratione, putà (1/100) <lb/>haud dubiè bractea ferri non modo (1/100) tardiùs de&longs;cendet, verùm etiam <lb/>(20/100) in quo non e&longs;t difficultas. </s>

<s>118. in data ratione, putà (1/100) <lb/>haud dubiè bractea ferri non modo (1/100) tardiùs de&longs;cendet, verùm etiam <lb/>(20/100) in quo non e&longs;t difficultas. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 121.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 124.<emph.end type="center"/></s>

<s><emph type="italics"/>In modico vacuo omnia æquè velociter de&longs;cenderent<emph.end type="italics"/>: Probatur, quia tota <lb/>diuer&longs;itas vel inæqualitas mediorum petitur à diuer&longs;a proportione acti<lb/>uitatis cum re&longs;i&longs;tentia medij per Ax. 5. &longs;ed in vacuo nulla e&longs;t re&longs;i&longs;ten<lb/>tia; igitur nulla proportio; igitur nulla ratio motus inæqualis. </s>

<s>5. &longs;ed in vacuo nulla e&longs;t re&longs;i&longs;ten<lb/>tia; igitur nulla proportio; igitur nulla ratio motus inæqualis. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 125.<emph.end type="center"/></s>

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<s><emph type="italics"/>Hinc cre&longs;cit re&longs;istentia medij in eademratione, in qua cre&longs;cunt vires mobi<lb/>lis<emph.end type="italics"/>; demon&longs;tr. </s>

<s>quia cre&longs;cunt vires, vt cre&longs;cit impetus; nam impetus e&longs;t <lb/>vis illa, quâ mobile &longs;uperat re&longs;i&longs;tentiam medij vt con&longs;tat, &longs;ed re&longs;i&longs;ten<lb/>tia cre&longs;cit vt impetus per Th. </s>

<s>128. igitur cre&longs;cit in ratione virium. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 130.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 132.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc certè concludo contra Galileum, & alios quo&longs;dam motum grauium <lb/>po&longs;t aliquod &longs;patium decur&longs;um ex naturaliter accelerato non fieri æquabilem,<emph.end type="italics"/><lb/>quia in tantum fieret æquabilis in quantum tanta e&longs;&longs;et re&longs;i&longs;tentia, vt no<lb/>uam accelerationem impediret; &longs;ed hæc ratio nulla e&longs;t; quia in eadem <lb/>ratione cre&longs;cit re&longs;i&longs;tentia, in qua cre&longs;cunt vires per Th. 129. igitur non <lb/>mutatur progre&longs;&longs;io motuum per Th. 130. igitur nec acceleratio; igitur <lb/>motus naturalis ex accelerato non fit æquabilis: Equidem, vt iam &longs;uprà <lb/>dictum e&longs;t, in minori &longs;emper ratione cre&longs;cit velocitas, itémque ip&longs;a re&longs;i<lb/>&longs;tentia quod in omni progre&longs;&longs;ione arithmetica iuxta numeros 1.2.3.4.5. </s>

<s>129. igitur non <lb/>mutatur progre&longs;&longs;io motuum per Th. </s>

<s>130. igitur nec acceleratio; igitur <lb/>motus naturalis ex accelerato non fit æquabilis: Equidem, vt iam &longs;uprà <lb/>dictum e&longs;t, in minori &longs;emper ratione cre&longs;cit velocitas, itémque ip&longs;a re&longs;i<lb/>&longs;tentia quod in omni progre&longs;&longs;ione arithmetica iuxta numeros 1.2.3.4.5. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s>Diceret aliquis ab aëre extrin&longs;ecùs ambiente mobile ip&longs;um propelli; <lb/>&longs;ed contra, nam aër, & omne aliud medium re&longs;i&longs;tit potiùs quàm iuuet, vt <lb/>demon&longs;trauimus l. </s>

<s>&longs;ecundo Th. </s>

<s>&longs;ecundo Th. 1. Nec dicas fui&longs;&longs;e mentem Ari&longs;totelis, <lb/>cum nobiles Peripatetici contrâ &longs;entiant; Albertus Magnus, Toletus, <lb/>Scaliger, Suarius, & recentiores; neque hoc negauit vnquam Ari&longs;tote-<pb pagenum="136"/>les, &longs;ed in hoc non multùm laboramus; nec dicas hinc &longs;equi motum <lb/>violentum e&longs;&longs;e à principio intrin&longs;eco contra def. </s>

<s>1. Nec dicas fui&longs;&longs;e mentem Ari&longs;totelis, <lb/>cum nobiles Peripatetici contrâ &longs;entiant; Albertus Magnus, Toletus, <lb/>Scaliger, Suarius, & recentiores; neque hoc negauit vnquam Ari&longs;tote-<pb pagenum="136"/>les, &longs;ed in hoc non multùm laboramus; nec dicas hinc &longs;equi motum <lb/>violentum e&longs;&longs;e à principio intrin&longs;eco contra def. </s>

<s>1. nam e&longs;t quidem à <lb/>principio intrin&longs;eco formali, non tamen à principio intrin&longs;eco mouen<lb/>te vel agente; nec enim impetus e&longs;t cau&longs;a efficiens motus &longs;ui &longs;ubjecti; <lb/>&longs;ed cau&longs;a formalis vt &longs;æpè explicuimus. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s>

<s><emph type="italics"/>I&longs;te impetus producitur ab aliqua cau&longs;a<emph.end type="italics"/>; Probatur, quia e&longs;t de nouo; igi<lb/>tur non e&longs;t à &longs;e per Ax. </s>

<s><emph type="italics"/>I&longs;te impetus producitur ab aliqua cau&longs;a<emph.end type="italics"/>; Probatur, quia e&longs;t de nouo; igi<lb/>tur non e&longs;t à &longs;e per Ax. 8. l. </s>

<s>1. igitur e&longs;t ab alio; igitur ab aliqua <lb/>cau&longs;a. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 7.<emph.end type="center"/></s>

<s><emph type="italics"/>Producitur ab aliqua cau&longs;a extrin&longs;eca<emph.end type="italics"/>; Probatur primò, quia aliquis <lb/>motus violentus e&longs;t à cau&longs;a extrin&longs;eca per def.1. Secundò, e&longs;t ab aliqua <lb/>cau&longs;a applicata, &longs;ed e&longs;t tantùm applicata potentia motrix; igitur e&longs;t cau<lb/>&longs;a, per Ax. </s>

<s>1. nec enim producitur hic impetus ab entitate corpo<lb/>ris projecti, quod plu&longs;quàm certum e&longs;t ex dictis; hîc enim tantùm <lb/>e&longs;t quæ&longs;tio de illo motu, qui extrin&longs;ecùs aduenit, non vero de reflexo <lb/>&longs;ursùm, &c. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 8.<emph.end type="center"/></s>

<s><emph type="italics"/>Producitur ab alio impetu<emph.end type="italics"/>; quia potentia motrix non agit ad extra ni&longs;i <lb/>per impetum productum in organo, vt patet; præterea &longs;i e&longs;t cau&longs;a vni<lb/>uoca &longs;ufficiens applicata, non e&longs;t ponenda æquiuoca per Ax.11.l.1. adde <lb/>quod impetus producitur &longs;emper ad extra ab alio impetu per Th. </s>

<s>42. <lb/>l.1.nec in his hactenus propo&longs;itis vlla e&longs;t difficultas. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s>

<s><emph type="italics"/>Impetus impre&longs;&longs;us mobili &longs;ur&longs;um con&longs;eruatur per aliquod tempus<emph.end type="italics"/>; Probatur, <pb pagenum="137"/>quia mobile &longs;eparatum à potentia motrice adhuc mouetur per hyp.6.l.1, <lb/>igitur ille motus habet cau&longs;am, vt &longs;æpè dictum e&longs;t; non aliam, quàm im<lb/>petum per Th.4. non productum de nouo, quippe nulla e&longs;t cau&longs;a mobili <lb/>applicata per Th. </s>

<s>7. & 8. igitur iam antè productam; igitur con&longs;er<lb/>uatur. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s>

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<s>Tertiò, globus &longs;ursùm projectus <lb/>a&longs;cendit, & deinde de&longs;cendit æquali tempore; igitur &longs;altem &longs;ingulis in<lb/>&longs;tantibus de&longs;truitur vnus gradus impetus violenti æqualis primo gradui <lb/>innato; atqui aër non pote&longs;t vno in&longs;tanti de&longs;truere impetum æqualem <lb/>primo innato; alioqui non intenderetur motus naturalis. </s>

<s>Quartò, & hæc <lb/>e&longs;t ratio à priori, quotie&longs;cumque &longs;unt in eodem mobili duo impetus ad <lb/>oppo&longs;itas lineas determinati, pugnant pro rata, vt demon&longs;trauimus l.1. <lb/>Th. </s>

<s>149. 150. 152. & in toto Schol. </s>

<s>& multis aliis pa&longs;&longs;im; atqui con&longs;er<lb/>natur &longs;emper impetus naturalis innatus per Sch. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc vera ratio cur &longs;ingulis in&longs;tantibus aliquid de&longs;truatur,<emph.end type="italics"/> quia &longs;ingulis <lb/>in&longs;tantibus e&longs;t cau&longs;a de&longs;tructiua applicata, igitur &longs;ingulis in&longs;tantibus de<lb/>&longs;truit per Ax. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 26.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc illa inuer&longs;a communis dicti, æqualibus temporibus æqualia de&longs;truun<lb/>tur velocitatis momenta in motu violento<emph.end type="italics"/>; quippe eadem cau&longs;a eidem &longs;ub<lb/>jecto applicata æqualibus temporibus æqualem effectum producit per <lb/>Ax.3.l.2. &longs;ed impetus innatus e&longs;t cau&longs;a de&longs;tructiua impetus violenti per <lb/>Th. </s>

<s>22. igitur æqualibus temporibus, &c. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s>

<s><emph type="italics"/>In cadem proportione retardatur motus violentus, in qua naturaiis accele<lb/>ratur<emph.end type="italics"/>: probatur quia &longs;ingulis in&longs;tantibus æqualibus acquiritur æqualis <lb/>gradus impetus, vt &longs;æpè dictum e&longs;t &longs;uprà; atqui &longs;ingulis in&longs;tantibus de<lb/>&longs;truitur vnus gradus impetus violenti per Th.24. &longs;ed ille gradus re&longs;pon<lb/>det impetui innato per Th. 25. igitur æqualibus temporibus tantùm de<lb/>&longs;truitur violenti, quantùm acquiritur naturalis; cum enim primo in<lb/>&longs;tanti &longs;it impetus naturalis, & &longs;ecundo tempore æquali acquiratur æqua<lb/>lis, item tertio, quarto, &c. </s>

<s>25. igitur æqualibus temporibus tantùm de<lb/>&longs;truitur violenti, quantùm acquiritur naturalis; cum enim primo in<lb/>&longs;tanti &longs;it impetus naturalis, & &longs;ecundo tempore æquali acquiratur æqua<lb/>lis, item tertio, quarto, &c. </s>

<s>certè cum impetus innatus pugnet cum vio<lb/>lento pro rata; nec &longs;it potior ratio cur maiorem portionem quàm mino<lb/>rem de&longs;truat, æqualem certè de&longs;truit, itemque &longs;ecundo in&longs;tanti æqua<lb/>lem, item tertio, quarto; igitur in eadem proportione decre&longs;cit violentus, <lb/>&longs;eu retardatur, in qua naturalis acceleratur. </s><pb pagenum="140"/>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 31.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc primo in&longs;tanti motus violenti de&longs;truitur minor gradus impetus quàm <lb/>&longs;ecundo,<emph.end type="italics"/> quod demon&longs;tro; quia cadem cau&longs;a breuiore tempore minùs agit <lb/>per Ax.3.l.2. & Ax. 13.l.1. num.4. igitur minùs impetus de&longs;truitur pri<lb/>mo, quàm &longs;ecundo, & minùs &longs;ecundo quàm tertio, atque ita deinceps; <lb/>idem enim dici debet de cau&longs;a de&longs;tructiua, quod de productiua. </s>

<s>13.l.1. num.4. igitur minùs impetus de&longs;truitur pri<lb/>mo, quàm &longs;ecundo, & minùs &longs;ecundo quàm tertio, atque ita deinceps; <lb/>idem enim dici debet de cau&longs;a de&longs;tructiua, quod de productiua. </s>

<s>Dices, igitur idem impetus de&longs;truitur primo in&longs;tanti, quo e&longs;t, &longs;i de&longs;trui<lb/>tur primo in&longs;tanti motus. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc reiicies Galileum, & alios eius &longs;ectatores qui volunt impetum corpori <lb/>impre&longs;&longs;um de&longs;trui tantùm ab aëre<emph.end type="italics"/>; quod plu&longs;quàm fal&longs;um e&longs;&longs;e comper<lb/>tum e&longs;t, vt demon&longs;trauimus &longs;uprà Th. 20. qua&longs;i verò non ad&longs;it aliqua <lb/>cau&longs;a nece&longs;&longs;aria de&longs;tructiua, &longs;cilicet impetus innatus; hinc etiam eum<lb/>dem reiicies, qui vult numquam fieri po&longs;&longs;e, vt motu naturaliter accelera<lb/>to tanta acquiratur velocitas, quanta imprimitur in motu violento; vult <lb/>enim motum acceleratum tran&longs;ire in æquabilem, cuius contrarium de<lb/>mon&longs;trauimus &longs;uprà Th. 131, l. </s>

<s>20. qua&longs;i verò non ad&longs;it aliqua <lb/>cau&longs;a nece&longs;&longs;aria de&longs;tructiua, &longs;cilicet impetus innatus; hinc etiam eum<lb/>dem reiicies, qui vult numquam fieri po&longs;&longs;e, vt motu naturaliter accelera<lb/>to tanta acquiratur velocitas, quanta imprimitur in motu violento; vult <lb/>enim motum acceleratum tran&longs;ire in æquabilem, cuius contrarium de<lb/>mon&longs;trauimus &longs;uprà Th. </s>

<s>2. igitur cum cre&longs;cat &longs;emper velocitas, <lb/>nullus e&longs;t finitus gradus, quem tandem non a&longs;&longs;equatur; immò vt dictum <lb/>e&longs;t in præcedenti Th. </s>

<s>a&longs;&longs;umptis æqualibus &longs;patiis, impetus, qui e&longs;t in <lb/>principio a&longs;cen&longs;us, æqualis e&longs;t cum eo, qui e&longs;t in fine de&longs;cen&longs;us. </s>

<s>2. igitur cum cre&longs;cat &longs;emper velocitas, <lb/>nullus e&longs;t finitus gradus, quem tandem non a&longs;&longs;equatur; immò vt dictum <lb/>e&longs;t in præcedenti Th. a&longs;&longs;umptis æqualibus &longs;patiis, impetus, qui e&longs;t in <lb/>principio a&longs;cen&longs;us, æqualis e&longs;t cum eo, qui e&longs;t in fine de&longs;cen&longs;us. </s>

<s>Diceret fortè aliquis cadentem globum ex alti&longs;&longs;imæ turris apice de<lb/>clinare à perpendiculari antequam terram feriat, vt con&longs;tat ex multis <lb/>experimentis; igitur præualet tandem re&longs;i&longs;tentia aëris: &longs;ed re&longs;pondeo id <lb/>rantùm accidere propter currentem illac aëris tractum; alioquin non <lb/>e&longs;&longs;et potiùs ratio, cur in vnam partem declinaret, quàm in aliam. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc ad motum violentum impetus ab exteriore potentia mobili impre&longs;&longs;us <lb/>tantùm concurrit<emph.end type="italics"/>; patet, cum enim in mobili projecto &longs;ur&longs;um &longs;it tantùm <lb/>ille impetus præter innatum, nec innatus concurrat per Th. 52. illum <lb/>tantùm concurrere nece&longs;&longs;e e&longs;t: excipe &longs;emper impetum acqui&longs;itum, de <lb/>quo iam &longs;uprà. </s>

<s>52. illum <lb/>tantùm concurrere nece&longs;&longs;e e&longs;t: excipe &longs;emper impetum acqui&longs;itum, de <lb/>quo iam &longs;uprà. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 54.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 75.<emph.end type="center"/></s>

<s><emph type="italics"/>Itaque quie&longs;ceret mobile ip&longs;o &longs;tatim in&longs;tanti, quod in&longs;tanti æqualitatis &longs;uc<lb/>cedit<emph.end type="italics"/>; patet, quia neuter impetus pro illo in&longs;tanti præualere po&longs;&longs;et per <lb/>Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 77.<emph.end type="center"/></s>

<s><emph type="italics"/>Quies illa duraret tantùm vno in&longs;t anti,<emph.end type="italics"/> probatur, quia cum in&longs;tanti quie<lb/>tis &longs;it tantùm impetus innatus per Th. </s>

<s><emph type="italics"/>Quies illa duraret tantùm vno in&longs;t anti,<emph.end type="italics"/> probatur, quia cum in&longs;tanti quie<lb/>tis &longs;it tantùm impetus innatus per Th. 76. certè non impeditur quomi<lb/>nus habeat motum pro in&longs;tanti &longs;equenti, quem reuerà exigit; igitur pro <pb pagenum="150"/>in&longs;tanti &longs;equenti moueritur; &longs;ed pro alio antecedente mouebatur; igi<lb/>tur quies illa durat tantùm vno in&longs;tanti. </s>

<s>76. certè non impeditur quomi<lb/>nus habeat motum pro in&longs;tanti &longs;equenti, quem reuerà exigit; igitur pro <pb pagenum="150"/>in&longs;tanti &longs;equenti moueritur; &longs;ed pro alio antecedente mouebatur; igi<lb/>tur quies illa durat tantùm vno in&longs;tanti. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 3.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc de&longs;truitur aliquid impetus<emph.end type="italics"/>; alioquin motus e&longs;&longs;et duplus cuiu&longs;li<lb/>bet &longs;eor&longs;im &longs;umpti, quod fal&longs;um e&longs;t; nam motus &longs;unt vt lineæ &longs;ed diago<lb/>nalis quadrati non e&longs;t dupla lateris; hoc etiam probatur per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 4.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s>

<s><emph type="italics"/>Si lineæ determinationum decu&longs;&longs;entur ad angulum acutum, & &longs;int æqua<lb/>les impetus, linea motus mixti erit diaganalis Rhombi<emph.end type="italics"/>; quæ certè eò longior <lb/>erit, quò angulus erit acutior per Th. </s>

<s>139. l.1. porrò e&longs;t &longs;emper maior <lb/>lateribus &longs;eor&longs;im &longs;umptis. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 21.<emph.end type="center"/></s>

<s><emph type="italics"/>Quando corpus proiicitur per horizontalem in aëre libero, mouetur motu <lb/>mixto<emph.end type="italics"/>; probatur, quia &longs;unt duo impetus in eo corpore, &longs;cilicet innatus <lb/>deor&longs;um, & impre&longs;&longs;us per horizontalem, vt patet; igitur vterque aliquid <lb/>præ&longs;tat ad illum motum per Ax. </s>

<s>1. igitur e&longs;t motus mixtus per def. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s>

<s><emph type="italics"/>Itaque motus prædictus mixtus est ex violento retardato & naturali acce<lb/>lerato, non eo quidem modo quo acceleratur in perpendiculari, &longs;ed eo quo acce<lb/>leratur in plano inclinato, quod hic &longs;ingulis <expan abbr="in&longs;tãtibus">in&longs;tantibus</expan> mutatur<emph.end type="italics"/>; probatur pri<lb/>mo, quia inductione facta non <expan abbr="cõftat">conftat</expan> ex omnibus aliis; &longs;unt enim tantùm <lb/>9 combinationes, quia &longs;unt tres differentiæ, &longs;cilicet æquabilibus, retarda<lb/>tio, acceleratio; igitur &longs;i 3.ducantur in 3. &longs;unt 9. &longs;unt autem prima ex na<lb/>turali, quem deinceps voco primum, æquabili & violento (quem voca<lb/>bo &longs;ecundum) æquabili, &longs;ecunda ex prima æquabili & &longs;ecundo accelera<lb/>to, tertia ex primo æquabili & &longs;ecundo retardato, quarta ex primo acce<lb/>lerato & &longs;ecundo æquabili, quinta ex primo accelerato & &longs;ecundo acce<lb/>lerato, &longs;exta ex primo accelerato & &longs;ecundo retardato, &longs;eptima ex primo <lb/>retardato & &longs;ecundo æquabili, octaua ex primo retardato & &longs;ecundo ac<lb/>celerato, nona ex primo r<gap/>ardato, & &longs;ecundo retardato: non e&longs;t prima <lb/>per Th.22. non &longs;ecunda per Th. </s>

<s>21. non tertia per Th. </s>

<s>24. non quarta, <lb/>per Th.26. non quinta per T.2h.23. non &longs;exta per Th.29. eo modo quo <lb/>diximus, non &longs;epti<gap/>a per Th. </s>

<s>25. non octaua per Th. </s>

<s>25. non denique <lb/>nona per Th.25. igitur debet e&longs;&longs;e alius motus, &longs;ed alius excogitari non <lb/>pote&longs;t præter ill<gap/> quem adduxi. </s>

<s>Probatur &longs;ecundò, quia non minùs <lb/>impeditur ab impetu violento impetus naturalis acqui&longs;itus quàm à pla<lb/>no inclinato vt iam dictum e&longs;t; igitur acceleratur quidem &longs;ed minùs; nec <lb/>enim vterque e&longs;t æquabilis, nam linea e&longs;&longs;et recta per Th.4. & naturalis <lb/>cre&longs;cit quia de&longs;cendit deor&longs;um; præterea per Th.24. non pote&longs;t impetus <lb/>naturalis e&longs;&longs;e æquabilis, igitur non pote&longs;t violentus e&longs;&longs;e vel æquabilis, <lb/>vel acceleratus, igitur retardatus. </s>

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<s>&longs;it tor<lb/>mentum horizonti parallelum extans &longs;upra horizontem tribus pedibus; <lb/>certum e&longs;t &longs;patium illud trium pedum confici à globo perpendiculariter <lb/>demi&longs;&longs;o tempore 30. tertiorum; cùm tamen explo&longs;us per lineam hori<lb/>zontalem terram tantùm attingat po&longs;t 4. &longs;ecunda, ide&longs;t 240. tertia; ita <lb/>Mer&longs;ennus l.2. de motu Prop. </s>

<s>vltima, imò l. </s>

<s>5. &longs;uæ ver&longs;ionis art.5. con<lb/>tra Galileum o&longs;tendit glandem emi&longs;&longs;am è tormento minori conficere <lb/>75. exapedas, tempore vnius &longs;ecundi minuti in linea, quæ parùm decli<lb/>nat ab horizontali; atqui tempore vnius &longs;ecundi minuti conficit 2.exa<lb/>pedas in perpendiculari deor&longs;um; igitur deberet glans infrà &longs;copum de<lb/>&longs;cendere notabiliter, id e&longs;t, toto 12. pedum interuallo, cùm tamen vix <lb/>tantillùm aberret à &longs;copo 1.Idem Mer&longs;ennus habet in Bali&longs;tica Prop.25. <lb/>globum è maiore tormento horizonti parallelo emi&longs;&longs;um in aëre tractu <lb/>continuo vola&longs;&longs;e toto tempore 8. &longs;ecundorum, antequam planum hori<lb/>zontale attigi&longs;&longs;et, cùm tamen &longs;ex tantùm exapedis tormentum extaret <lb/>&longs;upra horizontem; alter globus ex alio tormento explo&longs;us 6. tantum &longs;e<lb/>cunda in aëre con&longs;ump&longs;it; imò bombardarum globi aliquando tota 14. <lb/>&longs;ecunda po&longs;uerunt; habet idem Mer&longs;ennus alia plura, quorum fides &longs;it <lb/>penes authores à quibus accepit; nam vt dicam quod res e&longs;t vix accu<lb/>ratè minima illa tempora metiri po&longs;&longs;umus; quidquid &longs;it, ex illis &longs;altem <lb/>euinco mobile projectum per horizontalem plùs temporis in&longs;umere in <lb/>&longs;uo fluxu, quam &longs;i ex eadem altitudine perpendiculariter demittatur; vt <lb/>vult Galileus; cuius ratio alia non e&longs;t ab ea, quàm &longs;uprà indicauimus, <lb/>quòd &longs;cilicet motus naturalis minùs cre&longs;cat in motu mixto quàm in na-<pb pagenum="165"/>turali, vt &longs;uprà demon&longs;trauimus; imò &longs;i cre&longs;ceret vt vult Galileus, ictus; <lb/>haud dubiè e&longs;&longs;et maior in fine motus quàm initio, quod omninò expe<lb/>rientiæ repugnat. </s>

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<s><emph type="italics"/>Non e&longs;t mixtus ex vtroque æquabili<emph.end type="italics"/>; quia linea e&longs;&longs;et recta per Th.1.&longs;ed <lb/>linea huius motus e&longs;t curua per hyp. </s>

<s>non pertinet etiam hic motus ad <lb/>&longs;ecundam combinationem de qua Th. </s>

<s>non pertinet etiam hic motus ad <lb/>&longs;ecundam combinationem de qua Th. 30. nec ad quintam, nec ad <lb/>octauam, nec ad nonam, de aliis videbimus infrà. </s>

<s>30. nec ad quintam, nec ad <lb/>octauam, nec ad nonam, de aliis videbimus infrà. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 49.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 52.<emph.end type="center"/></s>

<s><emph type="italics"/>Hic motus e&longs;t mixtus ex naturali æquabili, & violento retardato in a&longs;cen<lb/>&longs;u<emph.end type="italics"/>; probatur, quia nulla alia combinatio præter hanc &longs;upere&longs;t, quam <lb/>tertio loco &longs;uprà collocauimus in Th. 30. ratio à priori e&longs;t, quia natura<lb/>lis innatus non retardatur; quia nunquam de&longs;truitur, nec acceleratur; <lb/>quia &longs;ur&longs;um tendit mobile; igitur &longs;upere&longs;t tantùm quod &longs;it æquabilis, <lb/>violentus verò non acceleratur, vt patet, quia nulla e&longs;t cau&longs;a: non e&longs;t <lb/>æquabilis, quia coniunctus e&longs;t cum cau&longs;a de&longs;tructiua; igitur e&longs;t re<lb/>tardatus. </s>

<s>30. ratio à priori e&longs;t, quia natura<lb/>lis innatus non retardatur; quia nunquam de&longs;truitur, nec acceleratur; <lb/>quia &longs;ur&longs;um tendit mobile; igitur &longs;upere&longs;t tantùm quod &longs;it æquabilis, <lb/>violentus verò non acceleratur, vt patet, quia nulla e&longs;t cau&longs;a: non e&longs;t <lb/>æquabilis, quia coniunctus e&longs;t cum cau&longs;a de&longs;tructiua; igitur e&longs;t re<lb/>tardatus. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 53.<emph.end type="center"/></s>

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<s><emph type="italics"/>In iactu per inclinatam deor&longs;um dato tempore minùs detrahitur de impetu <lb/>violento, quàm in iactu per inclinatam &longs;ur&longs;um<emph.end type="italics"/> &longs;it enim circulus centro A <lb/>&longs;emidiametro AG; &longs;itque AG horizontalis, & AO perpendiculatis deor<lb/>&longs;um; &longs;it iactus per inclinatam &longs;ur&longs;um AD, &longs;itque impetus violentus vt A <lb/>D, & naturalis deor&longs;um vt DE; linea motus erit DAE; igitur a&longs;&longs;umatur A <lb/>E in AC, & DE in CB, ex impetu AD detrahitur DB, vt con&longs;tat ex dicti<gap/>s <lb/>quia totius ille fru&longs;trà e&longs;t; &longs;it autem inclinata deor&longs;um cum impetu vio<lb/>lento æquali AI æqualis AD, &longs;itque naturalis deor&longs;um acceleratus pr<gap/><lb/>rata plani inclinati vt IL, linea motus erit AL; a&longs;&longs;umatur AK, vt AL, & <lb/>KH vt IL, detrahitur tantùm IH, &longs;ed IH e&longs;t minor DB; igitur tempore <lb/>&longs;equenti æquali impetus violentus inclinatæ &longs;ur&longs;um erit vt EF æqualis <lb/>AB inclinatæ deor&longs;um, vt LM, quæ maior e&longs;t EF, quia e&longs;t æqua<lb/>lis AH. </s>

<s>Ratio à priori e&longs;t, quia cum inclinata deor&longs;um faciat acutum angu<lb/>lum cum perpendiculari deor&longs;um, cum quo obtu&longs;um facit inclinata &longs;ur<lb/>&longs;um, maior e&longs;t in illa linea motus; e&longs;t enim maior diagonalis, in hac ve<lb/>rò minor, igitur in illa minùs impetus e&longs;t fru&longs;trà, in i&longs;ta verò plùs, igitur <lb/>minùs impetus in illa de&longs;truitur, plùs in i&longs;ta; quæ omnia con&longs;tant ex <lb/>Th. </s>

<s>110. & 139. & 140. l.1. habes etiam in qua proportione decre&longs;cat <lb/>impetus. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s>

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<s>Dices, etiam in glande è tormento explo&longs;a hoc ip&longs;um cernitur </s><pb pagenum="176"/>

<s>Re&longs;p. </s>

<s>Re&longs;p. e&longs;t minor vis ictus inflicti à glande deor&longs;um, quàm &longs;ur&longs;um vt <lb/>aliqui putant; id autem ex duplici capite procedere; primum e&longs;t, cum fe<lb/>ratur glans ab igne per aliquod tempus, non e&longs;t dubium, quin vis ignis <lb/>&longs;ur&longs;um maior &longs;it quàm deor&longs;um; cum &longs;ur&longs;um gemino qua&longs;i impetu fera<lb/>tur, deor&longs;um verò impetu tantùm explo&longs;ionis; &longs;ecundum e&longs;t, quia cum <lb/>glans iam deor&longs;um &longs;ua &longs;ponte de&longs;cendat, haud dubiè ab igne minus eò <lb/>impelli pote&longs;t, vt &longs;æpè diximus &longs;uprà; quidquid &longs;it, &longs;i proiiciatur dcor&longs;um <lb/>globus plumbeus vel arcu, vel manu, ob&longs;eruabitur maiorem ab co ictum <lb/>infligi, quàm &longs;i &longs;ua &longs;ponte de&longs;cenderet. </s>

<s>e&longs;t minor vis ictus inflicti à glande deor&longs;um, quàm &longs;ur&longs;um vt <lb/>aliqui putant; id autem ex duplici capite procedere; primum e&longs;t, cum fe<lb/>ratur glans ab igne per aliquod tempus, non e&longs;t dubium, quin vis ignis <lb/>&longs;ur&longs;um maior &longs;it quàm deor&longs;um; cum &longs;ur&longs;um gemino qua&longs;i impetu fera<lb/>tur, deor&longs;um verò impetu tantùm explo&longs;ionis; &longs;ecundum e&longs;t, quia cum <lb/>glans iam deor&longs;um &longs;ua &longs;ponte de&longs;cendat, haud dubiè ab igne minus eò <lb/>impelli pote&longs;t, vt &longs;æpè diximus &longs;uprà; quidquid &longs;it, &longs;i proiiciatur dcor&longs;um <lb/>globus plumbeus vel arcu, vel manu, ob&longs;eruabitur maiorem ab co ictum <lb/>infligi, quàm &longs;i &longs;ua &longs;ponte de&longs;cenderet. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 69.<emph.end type="center"/></s>

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<s>Dices, debes quidem nouus impetus accedere, &longs;ed non tali <lb/>modo. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. non e&longs;&longs;e alium modum à natura in&longs;titutum, ni&longs;i vt temporibus <lb/>æqualibus æqualia velocitatis momenta acquirantur. </s>

<s>non e&longs;&longs;e alium modum à natura in&longs;titutum, ni&longs;i vt temporibus <lb/>æqualibus æqualia velocitatis momenta acquirantur. </s>

<s>Dices præterea, fru&longs;trà accedit nouus impetus naturalis, cum iam ad<lb/>&longs;it violentus, qui eius munere defungi pote&longs;t. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. cau&longs;am nece&longs;&longs;ariam nece&longs;&longs;ariò agere; igitur corpus graue perpe<lb/>tuò in medio libero &longs;uum motum intendit. </s>

<s>cau&longs;am nece&longs;&longs;ariam nece&longs;&longs;ariò agere; igitur corpus graue perpe<lb/>tuò in medio libero &longs;uum motum intendit. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 78.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc reiicio omnes alias combinationes recepta &longs;exta; immò &longs;extam <lb/>ip&longs;am ex parte<emph.end type="italics"/>; nec enim naturalis acceleratur in hoc motu in ea <lb/>proportione, in qua acceleratur per lineam perpendicularem deor<lb/>&longs;um per Th. 29.&longs;ed iuxta rationem planorum inclinatorum per Theo<lb/>rema 31. nec etiam violentus de&longs;truitur vniformiter, &longs;ed pro rata per <lb/>Th. 39. </s>

<s>29.&longs;ed iuxta rationem planorum inclinatorum per Theo<lb/>rema 31. nec etiam violentus de&longs;truitur vniformiter, &longs;ed pro rata per <lb/>Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 79.<emph.end type="center"/></s>

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<s><emph type="italics"/>Hinc demi&longs;&longs;us globus plumbeus, vel alterius materiæ, quæ facilè vim aëris <lb/>infringat è &longs;ummo malo nauis ad imum ferè malum de&longs;cendit,<emph.end type="italics"/> hæc e&longs;t ex<lb/>perientia à Galileo producta, non tamen adinuenta, à Ga&longs;&longs;endo do<lb/>cti&longs;&longs;imè & eleganti&longs;&longs;imè explicata, ab omnibus Copernici &longs;ectatoribus <lb/>toties decantata, quæ vulgus ignobile ad admirationem adducit; imò <lb/>plures è Philo&longs;ophis fuere, qui eam in dubium adducerent, cum cam &longs;uis <lb/>principiis, ne dicam fortè &longs;omniis aduer&longs;ari putarent; certi&longs;&longs;imum tamen <lb/>e&longs;t illud experimentum centies, imò millies comprobatum, totis etiam <lb/>vrbibus &longs;pectantibus. </s>

<s>Nec ratio huius experimenti adco ab&longs;tru&longs;a e&longs;t, <lb/>vel recondita, quin à vulgari, ne dicam triobolari Philo&longs;opho &longs;tatim ex<lb/>plicari po&longs;&longs;it; cum enim imprimatur à naui mobili impetus pendulo <lb/>globo per horizontalem, & alius ab ip&longs;a grauitate deor&longs;um per Th. 71. <lb/>certè mouetur globus demi&longs;&longs;us re&longs;ecto funiculo motu mixto ex hori<lb/>zontali nauis, naturali corporis grauis; igitur per lineam curuam, quæ <lb/>ferè ad imum malum terminatur &longs;ed modicum figuræ adhibendum e&longs;t; <lb/>&longs;it planum aquæ <expan abbr="horizõtale">horizontale</expan>, cui innatat nauis IH; &longs;it malus IA perpen<lb/>dicularis altus 48. pedes; diuidatur in 4. partes æquales; corpus graue <lb/>conficiat &longs;patium illud duobus &longs;ecundis, v.g.igitur AK vno &longs;ecundo; e&longs;t <lb/>autem VK 12. pedum; iam verò moueatur nauis per horizontalem IH, <lb/>vel AL maxima qua&longs;i velocitate qua triremis moueri pote&longs;t; ita vt vna <lb/>hora faciat 16. milliaria Germanica, & 15′.4. milliaria, 3′ 800. pa&longs;&longs;us, <lb/>1′ 266. 1″ 4. pa&longs;&longs;us & (13/30); &longs;upponamus 1″ conficere 18. pedes, &longs;itque AC <lb/>18. & AK vel CE 12. haud dubiè motu mixto faciet lineam AE, & &longs;e<lb/>cundo tempore lineam EH, donec tandem cadat in punctum H nauis, <lb/>quò ferè peruenit punctum I; nam eodem modo retardatur motus <lb/>nauis; immò plùs quàm motus globi; quod &longs;cilicet partes aquæ, quæ à <lb/>naui diuiduntur multum re&longs;i&longs;tant; vnde fit compen&longs;atio; nam initio <lb/>motus violentus, qua&longs;i &longs;ecum rapit motum naturalem initio tardi&longs;&longs;i<lb/>mum; præ&longs;ertim cum non acceleretur, ni&longs;i iuxta rationem plani incli<lb/>nati, vt &longs;uprà dictum e&longs;t, & in fine naturalis rapit violentum. </s>

<s>71. <lb/>certè mouetur globus demi&longs;&longs;us re&longs;ecto funiculo motu mixto ex hori<lb/>zontali nauis, naturali corporis grauis; igitur per lineam curuam, quæ <lb/>ferè ad imum malum terminatur &longs;ed modicum figuræ adhibendum e&longs;t; <lb/>&longs;it planum aquæ <expan abbr="horizõtale">horizontale</expan>, cui innatat nauis IH; &longs;it malus IA perpen<lb/>dicularis altus 48. pedes; diuidatur in 4. partes æquales; corpus graue <lb/>conficiat &longs;patium illud duobus &longs;ecundis, v.g.igitur AK vno &longs;ecundo; e&longs;t <lb/>autem VK 12. pedum; iam verò moueatur nauis per horizontalem IH, <lb/>vel AL maxima qua&longs;i velocitate qua triremis moueri pote&longs;t; ita vt vna <lb/>hora faciat 16. milliaria Germanica, & 15′.4. milliaria, 3′ 800. pa&longs;&longs;us, <lb/>1′ 266. 1″ 4. pa&longs;&longs;us & (13/30); &longs;upponamus 1″ conficere 18. pedes, &longs;itque AC <lb/>18. & AK vel CE 12. haud dubiè motu mixto faciet lineam AE, & &longs;e<lb/>cundo tempore lineam EH, donec tandem cadat in punctum H nauis, <lb/>quò ferè peruenit punctum I; nam eodem modo retardatur motus <lb/>nauis; immò plùs quàm motus globi; quod &longs;cilicet partes aquæ, quæ à <lb/>naui diuiduntur multum re&longs;i&longs;tant; vnde fit compen&longs;atio; nam initio <lb/>motus violentus, qua&longs;i &longs;ecum rapit motum naturalem initio tardi&longs;&longs;i<lb/>mum; præ&longs;ertim cum non acceleretur, ni&longs;i iuxta rationem plani incli<lb/>nati, vt &longs;uprà dictum e&longs;t, & in fine naturalis rapit violentum. </s>

<s>Dixi ad imum ferè malum; nam reuera aliquid dee&longs;t quod tamen in<lb/>&longs;en&longs;ibile e&longs;t; &longs;ed quia modico tempore globus de&longs;cendit; &longs;it enim malus <lb/>108. pedum altitudinis, de&longs;cendit globus tempore 3″; &longs;it 192.4; &longs;it &longs;i <lb/>fieri pote&longs;t 432. de&longs;cendet 6″, &longs;ed nunquam accedit ad tantam altitudi<lb/>nem, igitur duobus vel tribus &longs;ecundis de&longs;cendit; igitur modico tem<lb/>pore; igitur violentus motus cen&longs;eri debet eo tempore æquabilis &longs;en&longs;i<lb/>biliter; & cum motus nauis nunquam &longs;it eiu&longs;dem velocitatis cum illa <lb/>quæ acquiritur tempore 2″ in de&longs;cen&longs;u, quia cum in de&longs;cen&longs;u acquiran<lb/>tur, hoc dato tempore ferè 48. pedes &longs;patij; certè motu æquabili cuius <pb pagenum="180"/>e&longs;&longs;et eadem velocitas acquirerentur 96. &longs;ed vix acquirerentur 24.vt di<lb/>ctum e&longs;t &longs;uprà; igitur vix nauis percurrit in horizontali æqualem lineam <lb/>longitutidini mali eo tempore, quo globus nauim attingit &longs;it enim <lb/>altitudo mali FA 48. pedum; &longs;it amplitudo &longs;patij horizontalis æqualis <lb/>FA; haud dubiè 1″ percurret AD, id e&longs;t 12.pedes ferè, quo tempore per<lb/>currat FG. 24. pedes & 20″ percurret DF, & GI. &longs;i motus &longs;umatur vt <lb/>æquabilis, vel GH, &longs;i retardatur, igitur 1°″ mobile percurrit &longs;egmentum <lb/>curuæ AE & 2° EH. </s>

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<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Ob&longs;eruabis, licèt i&longs;te motus non fiat per lineam parabolicam, vt &longs;uprà <lb/>demon&longs;trauimus Th. </s>

<s>Ob&longs;eruabis, licèt i&longs;te motus non fiat per lineam parabolicam, vt &longs;uprà <lb/>demon&longs;trauimus Th. 54. & reliquis; quia tamen &longs;en&longs;ibiliter proximè <lb/>accedit, deinceps vtemur Parabola vt in fig. </s>

<s>54. & reliquis; quia tamen &longs;en&longs;ibiliter proximè <lb/>accedit, deinceps vtemur Parabola vt in fig. </s>

<s>Th. 83. & horizontalem <lb/>motum accipiemus pro æquabili; licèt omninò æquabilis non &longs;it; ni&longs;i <lb/>tantùm æquiualenter; dixi æquiualenter, quia eodem modo &longs;e habet hic <lb/>motus, ac &longs;i per inclinatam &longs;ur&longs;um LC impetu &longs;cilicet LC mobile pro<lb/>iiceretur; &longs;ed in hoc ca&longs;u de&longs;trueretur impetus ille per inclinatam &longs;im<lb/>plex; igitur & mixtus; quia tamen ille qui remanet partim ex LA, par<lb/>tim ex LF eodem modo ferè &longs;e habet ac &longs;i totus LF intactus maneret; <lb/>hinc dictum e&longs;t &longs;uprà æquiualenter e&longs;&longs;e æquabilem. </s>

<s>83. & horizontalem <lb/>motum accipiemus pro æquabili; licèt omninò æquabilis non &longs;it; ni&longs;i <lb/>tantùm æquiualenter; dixi æquiualenter, quia eodem modo &longs;e habet hic <lb/>motus, ac &longs;i per inclinatam &longs;ur&longs;um LC impetu &longs;cilicet LC mobile pro<lb/>iiceretur; &longs;ed in hoc ca&longs;u de&longs;trueretur impetus ille per inclinatam &longs;im<lb/>plex; igitur & mixtus; quia tamen ille qui remanet partim ex LA, par<lb/>tim ex LF eodem modo ferè &longs;e habet ac &longs;i totus LF intactus maneret; <lb/>hinc dictum e&longs;t &longs;uprà æquiualenter e&longs;&longs;e æquabilem. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 87.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 95.<emph.end type="center"/></s>

<s><emph type="italics"/>Si proijciatur globus deor&longs;um à &longs;ummo malo, de&longs;cendet ferè ad imum ma<lb/>lum<emph.end type="italics"/>; probatur, quia de&longs;cendet quidem velociùs quàm &longs;i motu naturali <lb/>de&longs;cenderet vt con&longs;tat per Th. 69. &longs;ed profectò nihil acquiret in hori<lb/>zontali globus, quod non acquirat nauis; igitur imùm ferè malum attin<lb/>git &longs;ed opus e&longs;t aliqua figurâ; &longs;it enim apex mali A, de&longs;cendatque pri<lb/>mò ex A &longs;ua &longs;ponte in H; haud dubiè &longs;i eo tempore, quo motu na<lb/>turali conficit AD, mixto deor&longs;um conficit AF, eo tempore cadet in G <lb/>ex A &longs;i hic impetus deor&longs;um adueniat; &longs;ed res e&longs;t clara; hæc porrò figura <lb/>non e&longs;t Parabola, licèt &longs;it curua; con&longs;tat autem hîc motus ex naturali <lb/>accelerato, ex impre&longs;&longs;o deor&longs;um æquabili per &longs;e, & horizontali &longs;en&longs;i<lb/>biliter æquabili; pote&longs;t autem de&longs;ignari hæc linea motus ex &longs;uprà <lb/>dictis. </s>

<s>69. &longs;ed profectò nihil acquiret in hori<lb/>zontali globus, quod non acquirat nauis; igitur imùm ferè malum attin<lb/>git &longs;ed opus e&longs;t aliqua figurâ; &longs;it enim apex mali A, de&longs;cendatque pri<lb/>mò ex A &longs;ua &longs;ponte in H; haud dubiè &longs;i eo tempore, quo motu na<lb/>turali conficit AD, mixto deor&longs;um conficit AF, eo tempore cadet in G <lb/>ex A &longs;i hic impetus deor&longs;um adueniat; &longs;ed res e&longs;t clara; hæc porrò figura <lb/>non e&longs;t Parabola, licèt &longs;it curua; con&longs;tat autem hîc motus ex naturali <lb/>accelerato, ex impre&longs;&longs;o deor&longs;um æquabili per &longs;e, & horizontali &longs;en&longs;i<lb/>biliter æquabili; pote&longs;t autem de&longs;ignari hæc linea motus ex &longs;uprà <lb/>dictis. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 96.<emph.end type="center"/></s>

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<s><emph type="italics"/>Si motus nauis e&longs;&longs;et æqualis motui &longs;agittæ v. </s>

<s>g.<emph.end type="italics"/> <emph type="italics"/>&longs;i nauis ferretur per <lb/><gap/> ineam GC &longs;eu TA ver&longs;us Boream, & &longs;agitta è &longs;ummo malo emitteretur <lb/>per lineam TO ver&longs;us Au&longs;trum, de&longs;cenderet per lineam T.G. nec quidquam<emph.end type="italics"/><pb pagenum="188"/><emph type="italics"/>acquireret in horizontali<emph.end type="italics"/>; quod probatur per Th. 133. l.1. fic globus tor<lb/>menti etiam ne latum quidem vnguem pertran&longs;iret in horizontali, vide<lb/>tur tamen &longs;emper e&longs;&longs;e idem iactus; nam eo tempore, quo &longs;agitta caderet <lb/>à T in G, nauis e&longs;&longs;et in C, atqui CG & GM &longs;unt a&longs;&longs;umptæ æquales; hinc <lb/>potiùs arcus e&longs;&longs;et emi&longs;&longs;us quàm &longs;agitta, & tormentum explo&longs;um quàm <lb/>globus. </s>

<s>133. l.1. fic globus tor<lb/>menti etiam ne latum quidem vnguem pertran&longs;iret in horizontali, vide<lb/>tur tamen &longs;emper e&longs;&longs;e idem iactus; nam eo tempore, quo &longs;agitta caderet <lb/>à T in G, nauis e&longs;&longs;et in C, atqui CG & GM &longs;unt a&longs;&longs;umptæ æquales; hinc <lb/>potiùs arcus e&longs;&longs;et emi&longs;&longs;us quàm &longs;agitta, & tormentum explo&longs;um quàm <lb/>globus. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Corollarium<emph.end type="italics"/> 1.<emph.end type="center"/></s>

<s><emph type="italics"/>Ex illa hypothe&longs;i &longs;equitur egregium paradoxon &longs;cilicet &longs;agittam retor queri <lb/>in &longs;agittarium<emph.end type="italics"/>; &longs;it enim motus nauis ad motum &longs;agittæ vt GM ad LM; <lb/>haud dubiè per Th. &longs;uperius eo tempore, quo nauis peruenit ad M &longs;a<lb/>gitta attinget punctum L, & eo tempore quo nauis e&longs;&longs;et in L &longs;agitta e&longs;<lb/>&longs;et in puncto Y, &longs;i cum nauis peruenit in L illicò &longs;i&longs;tat &longs;agitta, cadet in <lb/>ip&longs;am nauim; nam cadet in L quod clarum e&longs;t: dixi &longs;i nauis &longs;i&longs;tat po&longs;t <lb/>emi&longs;&longs;am &longs;agittam, &longs;i enim nauis &longs;emper moueatur, æquabilis &longs;emper e&longs;&longs;e <lb/>videbitur &longs;agittæ iactus, &longs;i enim è naui immobili emi&longs;&longs;a fui&longs;&longs;et prædicta <lb/>&longs;agitta per horizontalem TO, acqui&longs;iui&longs;&longs;et &longs;patium vel amplitudinem G <lb/>L; &longs;ed videtur confeci&longs;&longs;e ML, cum nauis mouetur; atqui ML e&longs;t æqualis <lb/>LG, quid clarius? </s>

<s>&longs;uperius eo tempore, quo nauis peruenit ad M &longs;a<lb/>gitta attinget punctum L, & eo tempore quo nauis e&longs;&longs;et in L &longs;agitta e&longs;<lb/>&longs;et in puncto Y, &longs;i cum nauis peruenit in L illicò &longs;i&longs;tat &longs;agitta, cadet in <lb/>ip&longs;am nauim; nam cadet in L quod clarum e&longs;t: dixi &longs;i nauis &longs;i&longs;tat po&longs;t <lb/>emi&longs;&longs;am &longs;agittam, &longs;i enim nauis &longs;emper moueatur, æquabilis &longs;emper e&longs;&longs;e <lb/>videbitur &longs;agittæ iactus, &longs;i enim è naui immobili emi&longs;&longs;a fui&longs;&longs;et prædicta <lb/>&longs;agitta per horizontalem TO, acqui&longs;iui&longs;&longs;et &longs;patium vel amplitudinem G <lb/>L; &longs;ed videtur confeci&longs;&longs;e ML, cum nauis mouetur; atqui ML e&longs;t æqualis <lb/>LG, quid clarius? </s>

<s>Hinc &longs;i quis in naui currat per lineam directionis id e&longs;t ver&longs;us eain <lb/>partem, in quam mouetur nauis, curret velociùs; immò &longs;i ambulet, ingen<lb/>tes faciet pa&longs;&longs;us &longs;eu &longs;altus v.g.&longs;i nauis conficit &longs;patium GM eo tempore <lb/>quo aliquis &longs;altat ex G in H; haud dubiè amplitudo eius &longs;altus erit com<lb/>po&longs;ita ex tota GM & GH; &longs;i verò in partem oppo&longs;itam ver&longs;us C currat: <lb/>vel currit velociùs, vel tardiùs, vel æquali motu: &longs;i primum, aliquid &longs;patij <lb/>acquiret ver&longs;us C æqualis &longs;cilicet <expan abbr="differ&etilde;tiæ">differentiæ</expan> motuum; &longs;i <expan abbr="&longs;ecund&utilde;">&longs;ecundum</expan>, recedet <lb/>ver&longs;us M &longs;patio æquali eidem differentiæ; &longs;i tertium, nec acceder, nec re<lb/>cedet, &longs;ed totis viribus currens &longs;eu tentans currere in eodem &longs;emper lo-<pb pagenum="189"/>co &longs;tabit, vel &longs;i &longs;it rotatus globus in tabulato nauis mouebitur motu or<lb/>bis circa centrum immobile. </s>

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<s><emph type="italics"/>Grauitatio ponderis in planum inclinatum e&longs;t ad grauit at tonem eiu&longs;dem <lb/>in planum horizontale, vt Tangens, vel herizontalis ad &longs;ecantem, vel incli<lb/>natam,<emph.end type="italics"/> quod demon&longs;tro. </s>

<s>Primò &longs;ir planum inclinatum GD, pondus in-<pb pagenum="203"/>eubans F; dico grauitationem ponderis F in inclinatam GD e&longs;&longs;e ad gra<lb/>uitationem in horizontalem CD vt CD ad GD; quia pondus F pellit <lb/>planum per lineam FE &longs;eu GB Tangentem; quia determinari non po<lb/>te&longs;t &longs;eu percu&longs;&longs;io, &longs;eu impre&longs;&longs;io ex alio capite quàm ex linea ducta à <lb/>centro grauitatis perpendiculariter in planum, vt demon&longs;trauimus <lb/>in Th. </s>

<s>1. atqui libræ extremitas G initio de&longs;cendit per Tangen<lb/>tem GB, id e&longs;t per minimum arcum, qui ferè concurrit cum Tangente<gap/><lb/>&longs;ed ideò de&longs;cendit in AB, quia pellitur deor&longs;um à pondere; igitur men<lb/>&longs;ura grauitationis e&longs;t de&longs;cen&longs;us libræ, &longs;ed libra faciliùs de&longs;cendit ex A <lb/>deor&longs;um quàm ex G in proportione AD ad CD vel GD ad CD; igitur <lb/>grauitatio ponderis in A e&longs;t ad grauitationem ciu&longs;dem in G, vt GD ad <lb/>CD; quia rationes cau&longs;arum &longs;unt eædem cum rationibus effectuum. </s>

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<s>&longs;it enim cylindrus AE horizontalis, &longs;u&longs;tineaturque in A immo<lb/>biliter, itemque in E; certè qui &longs;u&longs;tinet in E æqualiter &longs;u&longs;tinet; at verò <lb/>&longs;i attollatur in AD; certè potentia quæ in D &longs;u&longs;tinet, e&longs;t ad potentiam <lb/>quæ &longs;u&longs;tinet in E, vt AF ad AE, quia pondus grauitaret in D & in E in <lb/>cadem ratione per Th. 16. &longs;ed potentia &longs;u&longs;tinens adæquat ponderis ra<lb/>tionem, &longs;u&longs;tinens inquam, per DH; nam reuerà &longs;u&longs;tinens per DF æqua<lb/>lis e&longs;&longs;e debet potentiæ in E: idem dico &longs;i attollatur in AP, nam potentia <lb/>trahens in P, per CP, e&longs;t ad potentiam in E, vt QA ad AP, vel AE; <lb/>igitur pondus in D e&longs;t ad pondus in P vt FA ad QA. </s>

<s>16. &longs;ed potentia &longs;u&longs;tinens adæquat ponderis ra<lb/>tionem, &longs;u&longs;tinens inquam, per DH; nam reuerà &longs;u&longs;tinens per DF æqua<lb/>lis e&longs;&longs;e debet potentiæ in E: idem dico &longs;i attollatur in AP, nam potentia <lb/>trahens in P, per CP, e&longs;t ad potentiam in E, vt QA ad AP, vel AE; <lb/>igitur pondus in D e&longs;t ad pondus in P vt FA ad QA. </s>

<s>Quintò, hinc &longs;i duo ferant parallelipedum in &longs;itu inclinato v.g.vt AD, <lb/>ferunt inæqualiter, &longs;cilicet in ratione AD FA, itemque &longs;i ferant in &longs;itu <lb/>inclinato AP, vel AC, donec tandem AE attollatur in B, nihil amplius <lb/>&longs;u&longs;tinet potentia in B, & potentia in A totum &longs;u&longs;tinet. </s>

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<s>motus in AE, e&longs;t enim <lb/>vt AE ad AC per Th.6. igitur cum &longs;ubduplo motu æquali tempore ac<lb/>quiritur &longs;ubduplus impetus; igitur tempore duplo æqualis impetus; at<lb/>qui tempus motus per AC e&longs;t ad tempus motus per AE vt AC ad AE, <lb/>ide&longs;t duplum; adde quod &longs;i æqualis impetus e&longs;t in C & in E; igitur æqua<lb/>lis in D & in B, &longs;ed AB e&longs;t ad BC vt AD ad DE; igitur &longs;i cre&longs;cit impe<lb/>tus per partes &longs;ubduplas in AC, nece&longs;&longs;ariò cre&longs;cit per partes duplas in <lb/>&longs;patio, atque in tempore; cùm enim motus &longs;it &longs;ubduplus, tarditas e&longs;t &longs;ub<lb/>dupla; igitur acquiritur in AC &longs;patium AB &longs;ubduplum AE co tempore, <lb/>quo percurtitur AE, &longs;i enim accipiantur æqualia tempora, &longs;patia &longs;unt vt <lb/>motus; &longs;ed motus per AC e&longs;t &longs;ubduplus; igitur &longs;patium AB e&longs;t &longs;ubdu<lb/>plum AE; &longs;ed tempore æquali conficit BC triplum AB, igitur tota AC <lb/>e&longs;t dupla AE; &longs;ed percurritur tempore duplo; igitur tempora &longs;unt vt <lb/><expan abbr="lõgitudines">longitudines</expan> planorum; &longs;ed clariùs, & brcuiùs illud demon&longs;tro; In ea pro<lb/>portione erit maius tempus per AC quàm per AE, in qua minor e&longs;t <lb/>motus per AC quàm per AE; &longs;i enim motus per AF e&longs;&longs;et ad motum per <lb/>AE vt AF ad AE, certè æquali tempore AF & AE percurrerentur; igitur <lb/>qua proportione motus per AF e&longs;t minor, tempus e&longs;t maius; tantumdem <lb/>enim additur tempori, quantum detrahitur motui; igitur tempora &longs;unt <pb pagenum="209"/>vt lineæ. </s>

<s>Hinc acquiritur velocitas æqualis, vt dictum e&longs;t Th. </s>

<s>Hinc acquiritur velocitas æqualis, vt dictum e&longs;t Th. 20. quia <lb/>&longs;i tantùm addit tempus per AF &longs;upra tempus per AE, quantum addit <lb/>motus per AE &longs;upra motum per AF, haud dubiè e&longs;t æqualitas. </s>

<s>20. quia <lb/>&longs;i tantùm addit tempus per AF &longs;upra tempus per AE, quantum addit <lb/>motus per AE &longs;upra motum per AF, haud dubiè e&longs;t æqualitas. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc datis duabus inclinatis æqualibus pote&longs;t determinari ratio tempo<lb/>rum, in quibus percurruntur<emph.end type="italics"/>; &longs;int enim AG.AH æquales, &longs;ed diuer&longs;æ incil<lb/>nationis; haud dubiè cum æquali tempore AG. AF percurrantur per <lb/>Th. </s>

<s>27. tempora quibus percurruntur AGAH erunt vt tempora quibus <lb/>percurruntur AF AH, & hæc vt tempora quibus percurruntur AE. </s>

<s>A <lb/>K, & hæc vt radices quadratæ illorum &longs;patiorum AE. AK, cum autem <lb/>&longs;patia &longs;int vt quadrata temporum, vel in duplicata ratione, &longs;i inter AE <lb/>& AK &longs;it media propprtionalis AN. v. </s>

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<s>cogno&longs;co tem<lb/>pus quo percurritur AK, & volo cogno&longs;cere tempus quo percurritur K <lb/>E, con&longs;equenti motu ex AK, &longs;cio tempus quo percurritur &longs;ola AE, quod <lb/>e&longs;t ad tempus quo percurritur AK vt AE ad AN per Th. 28. igitur <lb/>tempus quo percurritur KE con&longs;equenti motu ex AK e&longs;t ad tempus, <lb/>quo percurritur AK vt EN ad NA, vel vt NK, ad NA. </s>

<s>28. igitur <lb/>tempus quo percurritur KE con&longs;equenti motu ex AK e&longs;t ad tempus, <lb/>quo percurritur AK vt EN ad NA, vel vt NK, ad NA. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc in planis inæqualibus tùm in longitudine, tùns in inclinatione, <lb/>pote&longs;t &longs;<gap/>iri ratio temporum, quibus percurruntur<emph.end type="italics"/>; &longs;int enim AC AR duo pla<lb/>na; &longs;it autem AE perpendicularis indefinita; diuidatur AC bifariam <lb/>in V ducta perpendiculari VB; ex B fiat circulus, &longs;ecabit puncta <lb/>ACE; &longs;ecat etiam AR; in D igitur AC, & AD percurruntur æquali <lb/>tempore per Th. 27. &longs;imiliter fiat circulus ART eodem modos certè A <lb/>R & AT percurruntur æqualibus temporibus per Th. 27. igitur tempus, <lb/>quopercurritur AR, vel AD e&longs;t ad tempus, quo percurritur AR vt <lb/>tempus, quo percurritur AE ad tempus, quo percurritur AT; &longs;ed hæc <pb pagenum="211"/>&longs;unt vt radices AEAT, id e&longs;t tempus quo percurritur AE e&longs;t ad tem<lb/>pus, quo percurritur AT, vt AE ad mediam proportionalem inter AE <lb/>AT, vel vt AD ad mediam proportionalem inter AD AR; quippe AD <lb/>e&longs;t ad AR vt AE ad AT. </s>

<s>27. &longs;imiliter fiat circulus ART eodem modos certè A <lb/>R & AT percurruntur æqualibus temporibus per Th. </s>

<s>27. igitur tempus, <lb/>quopercurritur AR, vel AD e&longs;t ad tempus, quo percurritur AR vt <lb/>tempus, quo percurritur AE ad tempus, quo percurritur AT; &longs;ed hæc <pb pagenum="211"/>&longs;unt vt radices AEAT, id e&longs;t tempus quo percurritur AE e&longs;t ad tem<lb/>pus, quo percurritur AT, vt AE ad mediam proportionalem inter AE <lb/>AT, vel vt AD ad mediam proportionalem inter AD AR; quippe AD <lb/>e&longs;t ad AR vt AE ad AT. </s>

<s>Galileus verò demon&longs;trat rationem i&longs;torum temporum e&longs;&longs;e compo&longs;i<lb/>tam ex ratione longitudinem planorum & ex ratione &longs;ubduplicata al<lb/>titudinum eorumdem permutatim accepta: pro quo ob&longs;erua à Galileo <lb/>rationem duplicatam appellari duplam, & &longs;ubduplicatam appellari &longs;ub<lb/>duplam. </s>

<s>Ob&longs;eruabis denique plurima ex his colligi po&longs;&longs;e præ&longs;ertim ex Th. </s>

<s>Ob&longs;eruabis denique plurima ex his colligi po&longs;&longs;e præ&longs;ertim ex Th. 27. <lb/>quæ quia &longs;unt purè geometrica, certè phy&longs;icç minimè competunt; aliqua <lb/>tamen omittere non po&longs;&longs;um. </s>

<s>27. <lb/>quæ quia &longs;unt purè geometrica, certè phy&longs;icç minimè competunt; aliqua <lb/>tamen omittere non po&longs;&longs;um. </s>

<s>Primò, &longs;i &longs;int duo plana inæqualia ad angulum rectum, qui &longs;u&longs;tinea<lb/>tur ab horizontali, determinari po&longs;&longs;unt tempora de&longs;cen&longs;uum &longs;it enim <lb/>triangulum orthogonium ABE, ita vt AE &longs;it horizontalis; ducatur B <lb/>G indefinita perpendicularis in ba&longs;im AE; tùm FA perpendicularis in <lb/>AB; tùm FC perpendicularis in BE; tùm denique GE in BE; dico BA <lb/>BFBC percurri temporibus æqualibus, item BE, BG, EG, etiam æqua<lb/>libus; igitur tempus, quo percurritur BA e&longs;t ad tempus quo percurrri<lb/>tur BE, vt tempus, quo percurritur BF ad tempus quo percurritur BG; <lb/>hæc porrò &longs;unt in &longs;ubduplicata ratione BFBG vel BC, & BE. </s>

<s>Secundò, &longs;i planum &longs;u&longs;tinens angulum rectum non &longs;it parallelum <lb/>horizonti 6. res &longs;imiliter determinari poterit; &longs;it enim triangulum or<lb/>thogonium ABC ex B, ducatur perpendicularis deor&longs;um indefinitè BF, <lb/>tùm EA in AB, tùm DC in CB, tùm EH parallela DC, tùm GC in A <lb/>C; denique AG parallela BF; dico quod BABEHE AE percurren<lb/>tur æqualibus temporibus item BCCDBD. </s>

<s>Tertiò, &longs;iue de&longs;cendat ex B in C per lineam perpendicularem BC, <lb/>&longs;iue ex A per inclinatam AC, eodem modo de&longs;cendet &longs;iue per CD, &longs;iue <lb/>per CE; ratio e&longs;t clara, quia acquirit æqualem velocitatem &longs;iue ex A &longs;i<lb/>ue ex B de&longs;cendat pet Th. 20. erit autem tempus per CE ad tempus per <lb/>CD, vt CE ad CD per Th.23.& motus per CE ad motum per CD, vt <lb/>CD ad CE per Th.6. po&longs;ito initio motus in C. </s>

<s>20. erit autem tempus per CE ad tempus per <lb/>CD, vt CE ad CD per Th.23.& motus per CE ad motum per CD, vt <lb/>CD ad CE per Th.6. po&longs;ito initio motus in C. </s>

<s>Quartò, præuio motu ex A vel ex B ad C pote&longs;t inueniri inclinata, <lb/>per quam mobile pergat moueri motu &longs;cilicet naturaliter accelerato, ita <lb/>vt æquali tempore illam conficiat; &longs;i enim BC conficiet dato tempore; <lb/>igitur CF triplum CB conficiet tempore æquali; &longs;it autem planum ho<lb/>rizontale EDK ad quod ex C ducendum &longs;it planum inclinatum, quod <lb/>eodem tempore percurratur, quo CF, diuidatur CF bifariam in H, & ex <lb/>puncto H fiat arcus CK, ducaturque CK: Dico CF & CK æquali tem<lb/>pore confici per Th. 27. modò ex quiete C procedat motus: &longs;imiliter a&longs;<lb/>&longs;umi pote&longs;t alia horizontalis LM ducto arcu LF ex centro H; nam CL <lb/>& CF æquali tempore percurruntur; &longs;i verò præ&longs;upponatur motus præ<lb/>uius ex A vel ex B, haud dubiè CK breuiori tempore percurretur, quàm <lb/>CF, idem dico de CL; alioqui CE & CI eodem præuio motu &longs;uppo <pb pagenum="212"/>&longs;ito æquali tempore percurrerentur, quod fal&longs;um e&longs;t; nam &longs;it AC ad A <lb/>N vt AN ad AE; &longs;itque BC ad BO vt BO ad BI; certè tempus, quo <lb/>percurritur BC e&longs;t ad tempus, quo percurritur CI vt CB ad CO, & <lb/>tempus quo percurritur BC e&longs;t ad tempus quo percurritur CE vt BC ad <lb/>CN; &longs;ed CN e&longs;t minot quàm CO, vt con&longs;tat ex Geometria, quod bre<lb/>niter in tironum <expan abbr="gratiã">gratiam</expan> in terminis rationabilibus o&longs;tendo, &longs;it planum <lb/>inclinatum AE 9. &longs;itque AE id e&longs;t 9. ad AD. 6. vt AD ad AC 4. ex <lb/>centro C a&longs;&longs;umpta CH 3. ducatur arcus HB & ex A ad prædictum ar<lb/>cum Tangens AB, tùm ex BC G indefinitè & ex E, EG perpendicularis <lb/>in EA; haud dubiè triangula CGE, CAB &longs;unt proportionalia; igitur vt <lb/>CB;.ad CA. 4.ita CE 5. ad CG 6. 2/3; igitur tota BG e&longs;t 9. 2/3; &longs;itque B <lb/>G ad BF, vt BF ad DC, quod vt fiat BG 9. 2/3 in BC 3. productum erit <lb/>29. igitur BF e&longs;t Rad. </s>

<s>27. modò ex quiete C procedat motus: &longs;imiliter a&longs;<lb/>&longs;umi pote&longs;t alia horizontalis LM ducto arcu LF ex centro H; nam CL <lb/>& CF æquali tempore percurruntur; &longs;i verò præ&longs;upponatur motus præ<lb/>uius ex A vel ex B, haud dubiè CK breuiori tempore percurretur, quàm <lb/>CF, idem dico de CL; alioqui CE & CI eodem præuio motu &longs;uppo <pb pagenum="212"/>&longs;ito æquali tempore percurrerentur, quod fal&longs;um e&longs;t; nam &longs;it AC ad A <lb/>N vt AN ad AE; &longs;itque BC ad BO vt BO ad BI; certè tempus, quo <lb/>percurritur BC e&longs;t ad tempus, quo percurritur CI vt CB ad CO, & <lb/>tempus quo percurritur BC e&longs;t ad tempus quo percurritur CE vt BC ad <lb/>CN; &longs;ed CN e&longs;t minot quàm CO, vt con&longs;tat ex Geometria, quod bre<lb/>niter in tironum <expan abbr="gratiã">gratiam</expan> in terminis rationabilibus o&longs;tendo, &longs;it planum <lb/>inclinatum AE 9. &longs;itque AE id e&longs;t 9. ad AD. 6. vt AD ad AC 4. ex <lb/>centro C a&longs;&longs;umpta CH 3. ducatur arcus HB & ex A ad prædictum ar<lb/>cum Tangens AB, tùm ex BC G indefinitè & ex E, EG perpendicularis <lb/>in EA; haud dubiè triangula CGE, CAB &longs;unt proportionalia; igitur vt <lb/>CB;.ad CA. 4.ita CE 5. ad CG 6. 2/3; igitur tota BG e&longs;t 9. 2/3; &longs;itque B <lb/>G ad BF, vt BF ad DC, quod vt fiat BG 9. 2/3 in BC 3. productum erit <lb/>29. igitur BF e&longs;t Rad. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s>

<s><emph type="italics"/>Ex duobus ferentibus idem parallelipedum in &longs;itu inclinato pote&longs;t alter fer<lb/>re tantùm vnam libram, licèt pendat centum libras<emph.end type="italics"/>; &longs;it enim ita inclina-<pb pagenum="215"/>tum, vt linea inclinationis &longs;it centupla horizontalis oppo&longs;itæ; certè qui <lb/>&longs;u&longs;tinet in altera extremitate eleuata (1/100) tantùm &longs;u&longs;tinet ponderis par<lb/>tem per Th. </s>

<s>18. alius verò &longs;u&longs;tinet in altera extremitate, quæ deor&longs;um <lb/>e&longs;t (93/100). </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s>

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<s>&longs;i proii<lb/>citur per BA in verticali, illa eadem <expan abbr="pot&etilde;tia">potentia</expan> quæ proiicit in A ex B, pro<lb/>iiciet <expan abbr="quoq;">quoque</expan> ex F in A, ex M in A, atque ita deinceps ex &longs;ingulis punctis <lb/>horizontalis BM; ratio e&longs;t, quia in ea proportione de&longs;truitur impetus <lb/>per BA, in qua motus per AB de&longs;cendit; nam impetus innatus deor<lb/>&longs;um qua&longs;i trahit mobile graue; impetus verò impre&longs;&longs;us &longs;ur&longs;um attollit; <lb/>igitur pugnant pro rata, vt &longs;æpè diximus in tertio libro, & alibi: &longs;imiliter <lb/>in inclinata FA impetus innatus qua&longs;i reducit mobile deor&longs;um dum <lb/>impre&longs;&longs;us violentus &longs;ur&longs;um promouet; igitur &longs;i impetus innatus per AB, <lb/>& per AT æqualem vim haberet, haud dubiè æquale &longs;patium contine<lb/>ret mobile projectum per BA & FA; nam eadem potentia cum æquali <lb/>re&longs;i&longs;tentia idem præ&longs;tat & inæqualiter de&longs;cendit per AB AF, & motus <lb/>per AF e&longs;t ad motum per AB, vt AB ad AF. v.g. </s>

<s>&longs;ubduplus; igitur re<lb/>&longs;i&longs;tentia per BA erit dupla re&longs;i&longs;tentiæ per FA; igitur &longs;patium per FA <lb/>erit duplum; igitur ex F a&longs;cendet in A, quo cum eo impetu ex B a&longs;cendet <lb/>in A, &longs;uppo&longs;ita eadem potentia; idem etiam dicendum de aliis punctis <lb/>horizontalis BM: præterea ille impetus &longs;ufficit ad motum &longs;ur&longs;um per <lb/>FA, qui accipitur in de&longs;cen&longs;u AF, vt con&longs;tat ex dictis; itemque &longs;ufficit <lb/>ad motum &longs;ur&longs;um per BA qui acquiritur in de&longs;cen&longs;u AB; &longs;ed æqualis ve<lb/>locitas, vel impetus acquiritur in vtroque de&longs;cen&longs;u AB AF per Th. </s>

<s>20. <lb/>igitur idem impetus &longs;ufficit ad de&longs;cen&longs;um BA FA. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 39.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 44.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc erecta perpendiculari<emph.end type="italics"/> FC, <emph type="italics"/>ductaque horizontali<emph.end type="italics"/> FL, <emph type="italics"/>productaque <lb/>in infinitum, &longs;i ex quolibet illius puncto eleuetur planum inclinatum termina<lb/>tum ad<emph.end type="italics"/> C, <emph type="italics"/>eadem potentia que ex<emph.end type="italics"/> F <emph type="italics"/>in<emph.end type="italics"/> C <emph type="italics"/>mobile proiiciet, etiam ex quolibet <lb/>puncto de&longs;ignato in horizontali proiiciet in<emph.end type="italics"/> C <emph type="italics"/>per planum inclinatum<emph.end type="italics"/>; quod <lb/>probatur per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s>

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<s><emph type="italics"/>Pote&longs;t determinari motus proportio cuiu&longs;libet puncti a&longs;&longs;ignati in plano EN<emph.end type="italics"/>; <pb pagenum="220"/>&longs;it enim punctum G; ducatur à centro A recta AGH; haud dubiè e&longs;t per<lb/>pendicularis; ducatur IGK &longs;ecans GH; ad angulos rectos; hæc e&longs;t ho<lb/>rizontalis, quæ ad hanc perpendicularem pertinet; ducatur HI parallela <lb/>EG; hæc e&longs;t inclinata, vt patet ex dictis; immò per ip&longs;am deff. </s>

<s>1. &longs;ed mo<lb/>tus in inclinata e&longs;t vt ip&longs;um perpendiculum ad inclinatam per Th. </s>

<s>1. &longs;ed mo<lb/>tus in inclinata e&longs;t vt ip&longs;um perpendiculum ad inclinatam per Th. 6. <lb/>igitur motus per HI in ip&longs;o puncto H, vel per GE in ip&longs;o Buncto G e&longs;t <lb/>ad motum per HG, vt HG ad HI. </s>

<s>6. <lb/>igitur motus per HI in ip&longs;o puncto H, vel per GE in ip&longs;o Buncto G e&longs;t <lb/>ad motum per HG, vt HG ad HI. </s>

<s>Aliter ducatur HZ perpendicularis IH; dico motum in G vel ex G <lb/>initio e&longs;&longs;e ad motum per VE vel GL vt GH ad GZ; &longs;unt enim duo <lb/>triangula IGH, ZGH proportionalia. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 56.<emph.end type="center"/></s>

<s><emph type="italics"/>In &longs;ingulis punctis plani EN e&longs;t diuer&longs;us motus<emph.end type="italics"/>; nam in puncto E nullus <lb/>e&longs;t motus per Th. </s>

<s><emph type="italics"/>In &longs;ingulis punctis plani EN e&longs;t diuer&longs;us motus<emph.end type="italics"/>; nam in puncto E nullus <lb/>e&longs;t motus per Th. 50.atqui in puncto G e&longs;t motus; idem dico de puncto <lb/>O, atqui in puncto O e&longs;t maior motus, quàm in G, &longs;cilicet initio, id e&longs;t <lb/>velocior incipit motus in O, quàm in G; probatur quia in G e&longs;t ad mo<lb/>tum maximum qui fit in perpendiculari vt QL ad LA, & in puncto O <lb/>vt YP ad PA, &longs;ed YP e&longs;t maior QL, vt con&longs;tat; igitur initio e&longs;t maior <lb/>motus in O quàm in G; igitur quâ proportione horizontalis EN erit <lb/>longior, puncta, quæ longiùs di&longs;tabunt, habebunt rationem plani ma<lb/>gis inclinati. </s>

<s>50.atqui in puncto G e&longs;t motus; idem dico de puncto <lb/>O, atqui in puncto O e&longs;t maior motus, quàm in G, &longs;cilicet initio, id e&longs;t <lb/>velocior incipit motus in O, quàm in G; probatur quia in G e&longs;t ad mo<lb/>tum maximum qui fit in perpendiculari vt QL ad LA, & in puncto O <lb/>vt YP ad PA, &longs;ed YP e&longs;t maior QL, vt con&longs;tat; igitur initio e&longs;t maior <lb/>motus in O quàm in G; igitur quâ proportione horizontalis EN erit <lb/>longior, puncta, quæ longiùs di&longs;tabunt, habebunt rationem plani ma<lb/>gis inclinati. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 57.<emph.end type="center"/></s>

<s><emph type="italics"/>Pote&longs;t determinari grauitatio in &longs;ingulis punctis plani EN<emph.end type="italics"/>; cum enim <lb/>grauitatio in plano inclinato &longs;it ad grauitationem in horizontali vt <lb/>Tangens ad &longs;ecantem, vel vt horizontalis, in quam &longs;cilicet cadit perpen<lb/>lum ad inclinatam per Th. 16. &longs;it punctum, G grauitatio in eo puncto <lb/>e&longs;t ad grauitationem in puncto E, vt QA ad AL, & in puncto O ve YA <lb/>ad AP: idem dico de aliis punctis. </s>

<s>16. &longs;it punctum, G grauitatio in eo puncto <lb/>e&longs;t ad grauitationem in puncto E, vt QA ad AL, & in puncto O ve YA <lb/>ad AP: idem dico de aliis punctis. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 58.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 60.<emph.end type="center"/></s>

<s><emph type="italics"/>Ille motus acceleratur per partes inæquales<emph.end type="italics"/>; quia &longs;cilicet motus additus <lb/>in O minor e&longs;&longs;et quàm in N, & in G quàm in O per Th. 56. igitur per <lb/>partes inæquales acceleraretur, immò pote&longs;t determinari proportio cre<lb/>menti motus in &longs;ingulis; cum enim in O &longs;it vt YP, in QL. in Yvt T <foreign lang="greek">d</foreign><lb/>ad AC; certè cre&longs;cit in proporrione &longs;inuum rectorum ad &longs;inum totum. </s>

<s>56. igitur per <lb/>partes inæquales acceleraretur, immò pote&longs;t determinari proportio cre<lb/>menti motus in &longs;ingulis; cum enim in O &longs;it vt YP, in QL. in Yvt T <foreign lang="greek">d</foreign><lb/>ad AC; certè cre&longs;cit in proporrione &longs;inuum rectorum ad &longs;inum totum. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 61.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 71.<emph.end type="center"/></s>

<s><emph type="italics"/>Omnes gradus acqui&longs;iti in de&longs;cen&longs;u concurrunt ad de&longs;cen&longs;um præter vnum <lb/>&longs;cilicet præter acqui&longs;itum vltimo instanti de&longs;cen&longs;us<emph.end type="italics"/>; quia impetus non con<lb/>currit ad motum primo in&longs;tanti quo e&longs;t, per Th. 34. lib.1. de omnibus <lb/>aliis certum e&longs;t quod concurrant, quia non impediuntur, igitur concur<lb/>runt per Ax.12. lib.1. </s>

<s>34. lib.1. de omnibus <lb/>aliis certum e&longs;t quod concurrant, quia non impediuntur, igitur concur<lb/>runt per Ax.12. lib.1. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 72.<emph.end type="center"/></s>

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<s><emph type="italics"/>Hinc in ea proportione cre&longs;cit impetus in de&longs;cen&longs;u, qua decre&longs;cit in a&longs;cen&longs;u, <lb/>& in eadem cre&longs;cit, & decre&longs;cit motus in eadem cre&longs;cunt, & decre&longs;cunt &longs;pa<lb/>tia,<emph.end type="italics"/> v.g. </s>

<s>&longs;int &longs;ex in&longs;tantia de&longs;cen&longs;us iuxta proportionem &longs;cilicet in&longs;tan<lb/>tium, in qua res i&longs;ta faciliùs explicatur: primo in&longs;tanti motus &longs;unt duo <lb/>gradus impetus, quorum alter tantùm concurrit, &longs;cilicet qui præextitit; <lb/>qui enim producitur primo illo in&longs;tanti, non concurrit ad illum motum <lb/>per Th. </s>

<s>34. lib.

1. igitur primo in&longs;tanti &longs;unt duo gradus impetus, vnus <lb/>gradus motus, & vnum &longs;patium; &longs;ecundo verò in&longs;tanti &longs;unt tres gradus <lb/>impetus quorum vnus non concurrit, 2. gradus motus, 2.&longs;patia, atque ita <lb/>deinceps; donec tandem &longs;exto eo vltimo in&longs;tanti de&longs;cen&longs;us &longs;int 7. gra<lb/>dus impetus, quorum vnus non concurrit, 6. gradus motus, & 6. <lb/>&longs;patia. </s>

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<s><emph type="italics"/>Hinc æqualia ferè vtrimque &longs;unt &longs;patia de&longs;cen&longs;us &longs;cilicet, & a&longs;cen&longs;us<emph.end type="italics"/>; v.g. </s>

<s><lb/>MF æquale FN, quia e&longs;t &longs;umma eorumdem terminorum per Th. </s>

<s><lb/>MF æquale FN, quia e&longs;t &longs;umma eorumdem terminorum per Th. 74. <lb/>igitur ex F mobile a&longs;cendit ad altitudinem FN æqualem altitudini FM, <pb pagenum="225"/>ex qua priùs de&longs;cenderat dixi ferè, quia cum innatus &longs;it perfectior vlti<lb/>mo acqui&longs;ito paulò plùs &longs;patij acquiritur in de&longs;cen&longs;u, quàm in a&longs;cen&longs;u, <lb/>&longs;ed minimum e&longs;t &longs;en&longs;ibile. </s>

<s>74. <lb/>igitur ex F mobile a&longs;cendit ad altitudinem FN æqualem altitudini FM, <pb pagenum="225"/>ex qua priùs de&longs;cenderat dixi ferè, quia cum innatus &longs;it perfectior vlti<lb/>mo acqui&longs;ito paulò plùs &longs;patij acquiritur in de&longs;cen&longs;u, quàm in a&longs;cen&longs;u, <lb/>&longs;ed minimum e&longs;t &longs;en&longs;ibile. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 76.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 81.<emph.end type="center"/></s>

<s><emph type="italics"/>Initio a&longs;cen&longs;us per FN de&longs;truuntur gradus impetus producti &longs;ub finem de<lb/>&longs;ien&longs;us, & &longs;ub finem a&longs;cen&longs;us destruuntur producti initio de&longs;cen&longs;us:<emph.end type="italics"/> ratio e&longs;t <lb/>clara, quia producti &longs;ub finem de&longs;cen&longs;us &longs;unt imperfectiores, cùm plùs <lb/>recedant à perpendiculari, per Th. 55. &longs;imiliter initio a&longs;cen&longs;us longiùs <lb/>recedit linea à verticali; igitur minùs de&longs;truetur impetus, vt &longs;æpè incul-<pb pagenum="226"/>cauimus; nam idem de&longs;truitur in dato puncto a&longs;cen&longs;us, qui producere<lb/>tur in eodem puncto de&longs;cen&longs;us. </s>

<s>55. &longs;imiliter initio a&longs;cen&longs;us longiùs <lb/>recedit linea à verticali; igitur minùs de&longs;truetur impetus, vt &longs;æpè incul-<pb pagenum="226"/>cauimus; nam idem de&longs;truitur in dato puncto a&longs;cen&longs;us, qui producere<lb/>tur in eodem puncto de&longs;cen&longs;us. </s>

<s>Dices, gradus productus vltimo in&longs;tanti de&longs;cen&longs;us non de&longs;truitur pri<lb/>mo a&longs;cen&longs;us. </s>

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<s>Sed ne hoc fortè excidat &longs;i Globus CGLH de&longs;cendat ex A ad cen<lb/>trum mundi &longs;eu grauium E, quæri pote&longs;t vtrum omnes partes mouean<lb/>tur &longs;ua &longs;ponte ver&longs;us L etiam illæ quæ vltra centrum E proce&longs;&longs;erunt, &longs;eu <lb/>quod idem e&longs;t, vtrum globus CGLH, cuius centrum E e&longs;t coniun<lb/>ctum cum centro grauium E tran&longs;latus in IFKB eiu&longs;dem &longs;it ponderis, <lb/>cuius e&longs;&longs;et in A. v.g. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. primò globum prædictum, cuius centrum e&longs;t in E, nullius e&longs;&longs;e <lb/>ponderis, vt con&longs;tat; nec enim potiùs in vnam partem, quàm in aliam <lb/>inclinat. </s>

<s>primò globum prædictum, cuius centrum e&longs;t in E, nullius e&longs;&longs;e <lb/>ponderis, vt con&longs;tat; nec enim potiùs in vnam partem, quàm in aliam <lb/>inclinat. </s>

<s>Re&longs;pondeo &longs;ecundò globum eumdem, cuius centrum e&longs;t D ex<lb/>tra centrum grauium E grauitare, quia inclinat ver&longs;us E.R e&longs;pondeo ter<lb/>tiò non æqualiter grauitare, &longs;iue &longs;it in D, &longs;iue &longs;it in A; quia grauitat per <lb/>&longs;uam entitatem quatenus coniuncta e&longs;t cum inclinatione; &longs;ed non e&longs;t ea<lb/>dem entitas in A quæ in D cum cadem inclinatione, igitur nec cadem <lb/>grauitas; non enim grauitat inde &longs;ecundum totam &longs;uam entitatem; <lb/>quia &longs;cilicet &longs;ectio MFNE non pote&longs;t ampliùs grauitare infrà E, quan<lb/>doquidem E e&longs;t locus infimus. </s>

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<s>versùs P; certè de&longs;cenderet v&longs;que ad A per conuexum &longs;emicir<lb/>culi QLA; per conuexum, inquam, non per concauum, vt dictum e&longs;t <lb/>de quadrante LVA. </s>

<s>Ratio e&longs;t, quia accederet &longs;emper propiùs ad cen<lb/>trum A; igitur e&longs;&longs;et planum inclinatum per Th. </s>

<s>Ratio e&longs;t, quia accederet &longs;emper propiùs ad cen<lb/>trum A; igitur e&longs;&longs;et planum inclinatum per Th. 2. igitur per illud de<lb/>&longs;cenderet, nec vlla e&longs;&longs;et difficultas; quod autem accedat &longs;emper propiùs <lb/>ad A per &longs;emicirculum QLA, certum e&longs;t; quia PA minor e&longs;t QA; nam <lb/>diamcter e&longs;t maxima &longs;ubten&longs;arum in circulo. </s>

<s>2. igitur per illud de<lb/>&longs;cenderet, nec vlla e&longs;&longs;et difficultas; quod autem accedat &longs;emper propiùs <lb/>ad A per &longs;emicirculum QLA, certum e&longs;t; quia PA minor e&longs;t QA; nam <lb/>diamcter e&longs;t maxima &longs;ubten&longs;arum in circulo. </s>

<s>Immò per alium &longs;emi<lb/>circulum ASQ a&longs;cenderet denuóque de&longs;cenderet repetitis pluribus vi<lb/>brationibus; nunquam tamen a&longs;cenderet v&longs;que ad punctum Q propter <lb/>tamdem rationem, quam in Theoremate 92. adduximus. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 1.<emph.end type="center"/></s>

<s><emph type="italics"/>Datur motus re&longs;lexus<emph.end type="italics"/>; nemo dubitat: quippe aliquod corpus in aliud <lb/>impactum reflectitur per Ax. </s>

<s><emph type="italics"/>Datur motus re&longs;lexus<emph.end type="italics"/>; nemo dubitat: quippe aliquod corpus in aliud <lb/>impactum reflectitur per Ax. primum &longs;ed &longs;i corpus reflectitur e&longs;t motus <lb/>reflexus; igitur certum e&longs;t de motu reflexo quod &longs;it; infrà verò videbi<lb/>mus propter quid &longs;it. </s>

<s>primum &longs;ed &longs;i corpus reflectitur e&longs;t motus <lb/>reflexus; igitur certum e&longs;t de motu reflexo quod &longs;it; infrà verò videbi<lb/>mus propter quid &longs;it. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 2.<emph.end type="center"/></s>

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<s>Igitur &longs;i tantùm agit, quo <lb/>maius e&longs;t plùs agit; quæ omnia &longs;unt perab&longs;urda; Igitur non producitur <lb/>ille impetus à corpore reflectente. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 6.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 9.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc impetus ille, qui e&longs;t cau&longs;a motus re&longs;lexi, e&longs;t idem cum præuio con&longs;er<emph.end type="italics"/>-<pb pagenum="239"/><emph type="italics"/>uato<emph.end type="italics"/>; quia vel e&longs;t productus de nouo, vel præuius, per Th. </s>

<s>4. non pri<lb/>mum per Th.8.igitur e&longs;t præuius. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc non destruitur totus impetus in puncto reflexionis.<emph.end type="italics"/></s><s> Probatur primò, <lb/>quia motus reflexus e&longs;t ab impetu per Th. </s>

<s><emph type="italics"/>Hinc non destruitur totus impetus in puncto reflexionis.<emph.end type="italics"/></s><s> Probatur primò, <lb/>quia motus reflexus e&longs;t ab impetu per Th. 3. &longs;ed non producitur nouus <lb/>impetus per Thcorema 8. igitur e&longs;t impetus, qui erat ante reflexionem <lb/>per Th.9. igitur non de&longs;truitur totus, &longs;altem per &longs;e, in puncto reflexio<lb/>nis. </s>

<s>3. &longs;ed non producitur nouus <lb/>impetus per Thcorema 8. igitur e&longs;t impetus, qui erat ante reflexionem <lb/>per Th.9. igitur non de&longs;truitur totus, &longs;altem per &longs;e, in puncto reflexio<lb/>nis. </s>

<s>Probatur &longs;ecundò à priori; quia nunquam de&longs;truitur impetus, ni&longs;i <lb/>quando e&longs;t fru&longs;tra per Ax.3.&longs;ed corpus reflectens non facit, vt &longs;it fru&longs;trà, <lb/>quia non impedit omnem lineam motus; igitur &longs;i ad aliquam determi<lb/>nari pote&longs;t, impetus non erit fru&longs;trà: ad quam autem determinari de<lb/>beat, dicemus infrà. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 27.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc, &longs;i linea incidentiæ e&longs;t perpendicularis GD, linea quoque re&longs;lexionis <lb/>e&longs;t eadem DG<emph.end type="italics"/>; quia huic e&longs;t maximum impedimentum, quia &longs;cilicet e&longs;t <lb/>maximus ictus; igitur maxima determinatio per Th. 25. &longs;ed maxima e&longs;t <lb/>illa, quâ mobile per ip&longs;am perpendicularem DG à puncto contactus D <lb/>retorquetur per Th.26. Igitur &longs;i linea incidentiæ, &c. </s>

<s>25. &longs;ed maxima e&longs;t <lb/>illa, quâ mobile per ip&longs;am perpendicularem DG à puncto contactus D <lb/>retorquetur per Th.26. Igitur &longs;i linea incidentiæ, &c. </s>

<s>quod erat proban<lb/>dum. </s>

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<s>&longs;it linea incidentia AD, linea reflexionis non e&longs;t per<lb/>pendicularis DG; quia tunc non e&longs;t maximus ictus, nec maximum im<lb/>pedimentum per Th.23.igitur nec maxima determinatio per Theor.24. <lb/>igitur nonfit per ip&longs;am perpendicularem DG per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 29.<emph.end type="center"/></s>

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<s>&longs;it linea incidentiæ AD, linea reflexionis <lb/>DH; non tantùm determinatur hæc linea à plano FB, alioqui e&longs;&longs;et DG, <lb/>nec e&longs;t eadem cum prima; alioqui e&longs;&longs;et DE, &longs;ed partim determinatur à <lb/>plano FB per DG partimque reti netaliquid primæ determinationis, & <lb/>ex vtraque fit DH, vt con&longs;tat, quia quò linea incidentiæ e&longs;t obliquior, <lb/>planum minùs determin at per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 30.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 40.<emph.end type="center"/></s>

<s><emph type="italics"/>Determinatio per DG à plano e&longs;t dupla determinationis prioris per lineam <lb/>incidentiæ GD<emph.end type="italics"/>; quod &longs;ic demon&longs;tro; &longs;it linea incidentiæ ID, linea re<lb/>flexionis erit DN, &longs;cilicet ad angulos æquales, per Th. 33. &longs;it autem an<lb/>gulus NDM 30. graduum, & NDG 60. ducatur NO parallela GD; <pb pagenum="246"/>tùm ID producatur in O, denique ducatur NG: prima determinatio <lb/>lineæ incidentiæ ID, e&longs;t per DO, determinatio plani e&longs;t per DG; &longs;ed <lb/>DO e&longs;t æqualis DG; nam DON, DNG &longs;unt æquilatera æqualia; <lb/>hinc determinatio mixta e&longs;t per DN, diuidens angulum GDO bifa<lb/>riam; igitur &longs;i &longs;it linea incidentiæ ID & angulus ID B. 30. graduum, <lb/>æqualis e&longs;t determinatio plani determinationi prioris lineæ; hinc angu<lb/>lus diuiditur æqualiter bifariam; &longs;it verò linea incidentiæ AD produ<lb/>cta v&longs;que ad E, linea reflexionis DH; ducatur HE; a&longs;&longs;umatur DT <lb/>æqualis EH: dico determinationem plani e&longs;&longs;e ad determinationem <lb/>prioris lineæ AD vel DE, vt DT ad DE; cum enim determinatio mix<lb/>ta &longs;it per DH; certè DH accedit propiùs ADDG, quàm ad DE; igi<lb/>tur determinatio per DG e&longs;t ad determinationem, per DE vt DT <lb/>æqualis HE ad DE; nam perinde &longs;e habent, atque &longs;i e&longs;&longs;ent duo impe<lb/>tus determinati ad duas lineas, de quibus hoc ip&longs;um demon&longs;trauimus <lb/>tùm libro 1. Th.137. 138. 139. &c. </s>

<s>33. &longs;it autem an<lb/>gulus NDM 30. graduum, & NDG 60. ducatur NO parallela GD; <pb pagenum="246"/>tùm ID producatur in O, denique ducatur NG: prima determinatio <lb/>lineæ incidentiæ ID, e&longs;t per DO, determinatio plani e&longs;t per DG; &longs;ed <lb/>DO e&longs;t æqualis DG; nam DON, DNG &longs;unt æquilatera æqualia; <lb/>hinc determinatio mixta e&longs;t per DN, diuidens angulum GDO bifa<lb/>riam; igitur &longs;i &longs;it linea incidentiæ ID & angulus ID B. 30. graduum, <lb/>æqualis e&longs;t determinatio plani determinationi prioris lineæ; hinc angu<lb/>lus diuiditur æqualiter bifariam; &longs;it verò linea incidentiæ AD produ<lb/>cta v&longs;que ad E, linea reflexionis DH; ducatur HE; a&longs;&longs;umatur DT <lb/>æqualis EH: dico determinationem plani e&longs;&longs;e ad determinationem <lb/>prioris lineæ AD vel DE, vt DT ad DE; cum enim determinatio mix<lb/>ta &longs;it per DH; certè DH accedit propiùs ADDG, quàm ad DE; igi<lb/>tur determinatio per DG e&longs;t ad determinationem, per DE vt DT <lb/>æqualis HE ad DE; nam perinde &longs;e habent, atque &longs;i e&longs;&longs;ent duo impe<lb/>tus determinati ad duas lineas, de quibus hoc ip&longs;um demon&longs;trauimus <lb/>tùm libro 1. Th.137. 138. 139. &c. </s>

<s>tùm lib.4. à Th. 1. ad Th.14.quippe <lb/>linea determinationis mixtæ e&longs;t diagonalis, vt &longs;æpè probauimus: deinde <lb/>&longs;it linea incidentiæ per KD; &longs;it DX linea reflexionis; &longs;it XQ, ip&longs;ique <lb/>æqualis DZ, dico determinationem per DG e&longs;&longs;e ad determinationem <lb/>per DQ vt DZ ad DQ, &longs;ed XQ e&longs;t minor GS, vt con&longs;tat; igitur quò <lb/>linea incidentiæ accedit propiùs ad perpendicularem GD, determinatio <lb/>plani e&longs;t maior, e&longs;tque vt chordæ NO, HE, <expan abbr="Xq;">Xque</expan> igitur &longs;i tandem li<lb/>nea incidentiæ &longs;it perpendicularis GD, determinatio plani e&longs;t ad deter<lb/>minationem lineæ incidentiæ, vt DY æqualis GS ad DG: &longs;ed cum ex <lb/>Th.4. multa lux reliquis con&longs;equentibus immò & antecedentibus afful<lb/>gere po&longs;&longs;it, paulò fu&longs;iùs explicandum, & demon&longs;trandum e&longs;&longs;e videtur: <lb/>itaque duobus modis, primò ex hypothe&longs;i anguli reflexionis æqualis an<lb/>gulo incidentiæ, quod iam reuerâ præ&longs;titum e&longs;t; &longs;ed cum ex hoc Theo<lb/>remate prædicta æqualitas angulorum reflexionis tanquam per princi<lb/>pium immediatum po&longs;itiuum demon&longs;trari po&longs;&longs;it, ne &longs;it aliqua circuli <lb/>&longs;pecies, quo determinatio noua dupla prioris po&longs;ita linea incidentiæ <lb/>perpendiculari per æqualitatem anguli reflexionis, & hæc æqualitas per <lb/>illam eandem determinationem duplam demon&longs;tretur, aliam viam inire <lb/>oporter, vnde intima totius reflexionis principia eruantur, quod vt <lb/>fiat. </s>

<s>1. ad Th.14.quippe <lb/>linea determinationis mixtæ e&longs;t diagonalis, vt &longs;æpè probauimus: deinde <lb/>&longs;it linea incidentiæ per KD; &longs;it DX linea reflexionis; &longs;it XQ, ip&longs;ique <lb/>æqualis DZ, dico determinationem per DG e&longs;&longs;e ad determinationem <lb/>per DQ vt DZ ad DQ, &longs;ed XQ e&longs;t minor GS, vt con&longs;tat; igitur quò <lb/>linea incidentiæ accedit propiùs ad perpendicularem GD, determinatio <lb/>plani e&longs;t maior, e&longs;tque vt chordæ NO, HE, <expan abbr="Xq;">Xque</expan> igitur &longs;i tandem li<lb/>nea incidentiæ &longs;it perpendicularis GD, determinatio plani e&longs;t ad deter<lb/>minationem lineæ incidentiæ, vt DY æqualis GS ad DG: &longs;ed cum ex <lb/>Th.4. multa lux reliquis con&longs;equentibus immò & antecedentibus afful<lb/>gere po&longs;&longs;it, paulò fu&longs;iùs explicandum, & demon&longs;trandum e&longs;&longs;e videtur: <lb/>itaque duobus modis, primò ex hypothe&longs;i anguli reflexionis æqualis an<lb/>gulo incidentiæ, quod iam reuerâ præ&longs;titum e&longs;t; &longs;ed cum ex hoc Theo<lb/>remate prædicta æqualitas angulorum reflexionis tanquam per princi<lb/>pium immediatum po&longs;itiuum demon&longs;trari po&longs;&longs;it, ne &longs;it aliqua circuli <lb/>&longs;pecies, quo determinatio noua dupla prioris po&longs;ita linea incidentiæ <lb/>perpendiculari per æqualitatem anguli reflexionis, & hæc æqualitas per <lb/>illam eandem determinationem duplam demon&longs;tretur, aliam viam inire <lb/>oporter, vnde intima totius reflexionis principia eruantur, quod vt <lb/>fiat. </s>

<s>Primò certum e&longs;t, corpus reflectens in perpendiculari, (quæ e&longs;t cum <lb/>linea incidentiæ terminata ad punctum contactus ducitur per centrum <lb/>grauitatis globi reflexi) certum e&longs;t inquam corpus reflectens in prædi<lb/>cta linea aliquando cedere, aliquando non cedere; cedere autem dici<lb/>tur cùm vel amouetur à corpore impacto, vel &longs;altem concutitur: <lb/>tunc autem nullo modo cedere dicitur, cum ab ictu nullo modo mo<lb/>uetur. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 45.<emph.end type="center"/></s>

<s><emph type="italics"/>Nullus impetus de&longs;truitur per &longs;e in pura reflexione<emph.end type="italics"/>; nam per accidens vt <lb/>plurimùm de&longs;truitur, vt dicemus infrà: dixi in pura reflexione; quia cum <lb/>fit aliqua compre&longs;&longs;io, vel repellitur corpus impactus ni&longs;u po&longs;itiuo, etiam <lb/>de&longs;truitur impetus; demon&longs;tratur Th. quia nihil impetus e&longs;t fru&longs;trà; <lb/>igitur nihil de&longs;truitur: con&longs;equentia patet ex dictis; probatur antece<lb/>dens, quia linea determinationis mixtæ e&longs;t &longs;emper æqualis lineæ prioris <lb/>determinationis, &longs;i remoto obice fui&longs;&longs;et propagata. </s>

<s>quia nihil impetus e&longs;t fru&longs;trà; <lb/>igitur nihil de&longs;truitur: con&longs;equentia patet ex dictis; probatur antece<lb/>dens, quia linea determinationis mixtæ e&longs;t &longs;emper æqualis lineæ prioris <lb/>determinationis, &longs;i remoto obice fui&longs;&longs;et propagata. </s>

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<s>Obiiceret fortè aliquis <expan abbr="pilã">pilam</expan> reflexam nunquam ad eam a&longs;cendere <expan abbr="&longs;ubli-mitat&etilde;">&longs;ubli<lb/>mitatem</expan> ex qua priùs demi&longs;&longs;a fuerat. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. hoc ve<gap/><expan abbr="i&longs;&longs;im&utilde;">i&longs;&longs;imum</expan> e&longs;&longs;e &longs;ed per acci<lb/>dens hoc ita fieri certum e&longs;t propter diui&longs;ionem, attritum, compre&longs;&longs;io<lb/>nem, ce&longs;&longs;ionemque partium; vnde pila eò altiùs a&longs;cendit, quò durior, & <lb/>leuigatior e&longs;t illa materia, ex qua con&longs;tat, planumque ip&longs;um leuigatius, <lb/>durius & ad libellam acuratius ita compo&longs;itum, vt &longs;it omninò horizonti <lb/>parallelum: adde quod planum debet e&longs;&longs;e pror&longs;us immobile; &longs;i enim mo<lb/>bile &longs;it, multus impetus de&longs;t <gap/>itur. </s>

<s>hoc ve<gap/><expan abbr="i&longs;&longs;im&utilde;">i&longs;&longs;imum</expan> e&longs;&longs;e &longs;ed per acci<lb/>dens hoc ita fieri certum e&longs;t propter diui&longs;ionem, attritum, compre&longs;&longs;io<lb/>nem, ce&longs;&longs;ionemque partium; vnde pila eò altiùs a&longs;cendit, quò durior, & <lb/>leuigatior e&longs;t illa materia, ex qua con&longs;tat, planumque ip&longs;um leuigatius, <lb/>durius & ad libellam acuratius ita compo&longs;itum, vt &longs;it omninò horizonti <lb/>parallelum: adde quod planum debet e&longs;&longs;e pror&longs;us immobile; &longs;i enim mo<lb/>bile &longs;it, multus impetus de&longs;t <gap/>itur. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 47.<emph.end type="center"/></s>

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<s><emph type="italics"/>Si globus in alium æqualem impingitur, ita vt punctum contactus, & cen<lb/>trum vtriu&longs;que &longs;int in eadem linea, multa <expan abbr="&longs;equ&utilde;tur">&longs;equuntur</expan> phænomena, quæ iam atti<lb/>gimus lib.<emph.end type="italics"/>1.<emph type="italics"/>à Th.<emph.end type="italics"/>60.Primò, æqualis impetus in globo, in quem impactus <lb/>e&longs;t, producitur per Th.60.lib.1. Secundò, æqualis e&longs;t determinatio noua <lb/>priori; probatur per Th.127.lib.1. Tertiò, de&longs;truitur totus impetus prior <lb/>per Th.128. hinc quie&longs;cit globus impactus; cuius rei non pote&longs;t e&longs;&longs;e alia <lb/>cau&longs;a; nec enim dicas de&longs;trui totum impetum illum (vt reuerâ totus de<lb/>&longs;truitur) ratione re&longs;i&longs;tentiæ, quæ minor e&longs;t, quàm e&longs;&longs;et, &longs;i in parietem il<lb/>lideretur; igitur tota ratio, cur de&longs;truatur totus impetus, duci tantùm <lb/>pote&longs;t ex eo, quod &longs;it fru&longs;trà; e&longs;t autem fru&longs;trà, quia cum prior deter<lb/>minatio ferat globum impactùm per eandem lineam, & noua per oppo<lb/>&longs;itam; vtraque certè æqualis e&longs;t; igitur neutra præualet; igitur globus <lb/>con&longs;i&longs;tit; &longs;i quis enim diceret non e&longs;&longs;e æquales; igitur altera maior e&longs;t; <lb/>igitur debet præualere; igitur &longs;i prior e&longs;t, debet vlteriùs propagari motus <pb pagenum="255"/>in cadem linea; &longs;i noua, igitur debet tantillùm reflecti; igitur cum nec <lb/>vlteriùs producatur motus, nec retrò agatur mobile, vtraque determi<lb/>natio nece&longs;&longs;ariò æqualis e&longs;t. </s>

<s>Quænam verò &longs;it huius æqualitatis ratio à <lb/>priori, difficilè dictu e&longs;t; dico tamen petendam e&longs;&longs;e ab æqualitate glo<lb/>borum; cum enim determinatio noua &longs;it duplò maior à plano immobili <lb/>& duro; certè à plano mobili minor e&longs;t, vt con&longs;tat, quia cedit; igitur <lb/>quâ proportione plùs, vel minùs cedit, e&longs;t minor dupla; &longs;ed maior glo<lb/>bus minùs cedit, quàm æqualis; quia ce&longs;&longs;io e&longs;t minor impul&longs;ione; igitur <lb/>quando ce&longs;&longs;io e&longs;t æqualis impul&longs;ioni, æquales &longs;unt determinationes; at<lb/>qui cum producitur æqualis impetus, & imprimitur æqualis motus, <lb/>æqualis e&longs;t ce&longs;&longs;iò impul&longs;ioni, id e&longs;t æquè cedit, ac impellitur; cum tamen, <lb/>&longs;i maior &longs;it globus, non æquè citò cedat, quia tardior motus imprimitur, <lb/>& hæc e&longs;t, ni fallor, vera ratio huius æqualitatis determinationum, & <lb/>hæc vera cau&longs;a quietis globi impacti, de qua iam &longs;uprà Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 63.<emph.end type="center"/></s>

<s><emph type="italics"/>Cum verò globus impellitur in globum æqualem per lineam obliquam, num<lb/>quam quie&longs;cit<emph.end type="italics"/>; quod demon&longs;tratur, quia &longs;emper e&longs;t determinatio mixta; <lb/>quod vt meliùs intelligatur, opus e&longs;t nouâ figurâ &longs;it ergo punctum con<lb/>tactus duorum globorum B, & ip&longs;a CBN &longs;it Tangens communis, &longs;eu <lb/>&longs;ectio plani, quæ gerit vicem plani reflectentis; fit autem primò linea <lb/>incidentiæ connectens centra FBA; nulla fit in ea reflexio per Th. 61. <lb/>quia &longs;cilicet determinatio noua per lineam BF e&longs;t æqualis priori per <lb/>FB; &longs;it EB linea incidentiæ faciens angulum EBC cum Tangente <lb/>NC; determinatio noua e&longs;t ad determinationem priorem vt BG vel <lb/>ER ad BE, & &longs;i &longs;it linea incidentiæ DB vt BH, vel SD ad BD; deni<lb/>que &longs;i &longs;it BV vt TV ad BV, donec tandem linea incidentiæ &longs;it CB, quâ <lb/>po&longs;itâ nulla e&longs;t determinatio noua; vides e&longs;&longs;e eandem viam proportio<lb/>num quæ fuit &longs;uprà; licèt non &longs;it futura eadem angulorum reflexionis <lb/>proportio, quia determinationum nouarum rationes non &longs;unt eædem; <lb/>producatur enim EBL DBM &c. </s>

<s>61. <lb/>quia &longs;cilicet determinatio noua per lineam BF e&longs;t æqualis priori per <lb/>FB; &longs;it EB linea incidentiæ faciens angulum EBC cum Tangente <lb/>NC; determinatio noua e&longs;t ad determinationem priorem vt BG vel <lb/>ER ad BE, & &longs;i &longs;it linea incidentiæ DB vt BH, vel SD ad BD; deni<lb/>que &longs;i &longs;it BV vt TV ad BV, donec tandem linea incidentiæ &longs;it CB, quâ <lb/>po&longs;itâ nulla e&longs;t determinatio noua; vides e&longs;&longs;e eandem viam proportio<lb/>num quæ fuit &longs;uprà; licèt non &longs;it futura eadem angulorum reflexionis <lb/>proportio, quia determinationum nouarum rationes non &longs;unt eædem; <lb/>producatur enim EBL DBM &c. </s>

<s>determinatio prior per EB e&longs;t ad <lb/>nouam per BF, vt BE ad BG; igitur ducantur EP PL; a&longs;&longs;umatur LI <lb/>æqualis BG, & GI, BL æqualis BE; denique ducatur BI: dico BI e&longs;&longs;e <lb/>lineam reflexionis &longs;eu determinationem mixtam ex BG BL per Th. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 64.<emph.end type="center"/></s>

<s><emph type="italics"/>Si globus minor in maiorem impingatur, qui ab eo tamen moueatur per li<lb/>neam connectentem centra vtriu&longs;que impactus, reflectitur<emph.end type="italics"/>; ratio e&longs;t, quiama<lb/>ior globus e&longs;t maius impedimentum, vt iam diximus Th. 131.lib.1.id <lb/>e&longs;t, vt clariùs hic explicetur, quæ ibidem tantùm obiter indicauimus, <lb/>noua determinatio maior e&longs;t priore, quia ce&longs;sio e&longs;t minor impul&longs;ione; &longs;it <lb/>autem. </s>

<s>131.lib.1.id <lb/>e&longs;t, vt clariùs hic explicetur, quæ ibidem tantùm obiter indicauimus, <lb/>noua determinatio maior e&longs;t priore, quia ce&longs;sio e&longs;t minor impul&longs;ione; &longs;it <lb/>autem. </s>

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<s><emph type="italics"/>Si globus minor in maiorem impingatur per lineam obliquam incidentiæ, <lb/>&longs;emper reflectitur<emph.end type="italics"/>; quippè &longs;it determinatio mixta ex priore, & noua, quæ <lb/>determinari pote&longs;t, &longs;i aliquid à nouæ figuræ de&longs;cribatur; &longs;it circulus <lb/>FQCD; &longs;int diametri QD, FC; &longs;it AI dupla AF, &longs;itque determi<lb/>natio prior vt FA, &longs;i &longs;ecunda &longs;it vt AI, erit dupla prioris; igitur corpus <lb/>reflectens erit immobile; igitur &longs;i linea incidentiæ &longs;it EA, reflexa erit <lb/>AT, ita vt anguli TAF, EAF &longs;int æquales; &longs;i autem determinatio no<lb/>ua &longs;it ad priorem vt AH ad AF, id e&longs;t, v.g. </s>

<s>vt 3. ad 2. po&longs;itâ &longs;cilicet li<lb/>neâ incidentiæ perpendiculari FA in planum reflectens QD, quod certè <lb/>mouebitur per Th. </s>

<s>vt 3. ad 2. po&longs;itâ &longs;cilicet li<lb/>neâ incidentiæ perpendiculari FA in planum reflectens QD, quod certè <lb/>mouebitur per Th. 64. aliter procedendum e&longs;t vt inueniatur linea re<lb/>flexa re&longs;pondens lineæ incidentiæ obliquæ; diuidatur FAMK ita vt <lb/>KN &longs;it ad AF vt 3.ad 2. ac proinde AH &longs;it diui&longs;a bifariam in K; de<lb/>&longs;cribatur circulus KMNR, &longs;it linea quælibet incidentiæ obliqua EA; <lb/>producatur in B; ducantur OX BT parallelæ AH; a&longs;&longs;umatur AG æqua<lb/>lis OX, & GS æqualis AB; certè BS erit æqualis OX vel AG; duca<lb/>tur AS, hæc erit reflexa quæ&longs;ita: idem dico de omnibus aliis lineis in<lb/>cidentiæ; demon&longs;tratur eodem modo quo &longs;uprà in Th. 30. 31. 32. quæ <lb/>con&longs;ule, ne hic repetere cogar. </s>

<s>64. aliter procedendum e&longs;t vt inueniatur linea re<lb/>flexa re&longs;pondens lineæ incidentiæ obliquæ; diuidatur FAMK ita vt <lb/>KN &longs;it ad AF vt 3.ad 2. ac proinde AH &longs;it diui&longs;a bifariam in K; de<lb/>&longs;cribatur circulus KMNR, &longs;it linea quælibet incidentiæ obliqua EA; <lb/>producatur in B; ducantur OX BT parallelæ AH; a&longs;&longs;umatur AG æqua<lb/>lis OX, & GS æqualis AB; certè BS erit æqualis OX vel AG; duca<lb/>tur AS, hæc erit reflexa quæ&longs;ita: idem dico de omnibus aliis lineis in<lb/>cidentiæ; demon&longs;tratur eodem modo quo &longs;uprà in Th. </s>

<s>30. 31. 32. quæ <lb/>con&longs;ule, ne hic repetere cogar. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 66.<emph.end type="center"/></s>

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<s><lb/>63. &longs;i enim globi &longs;unt æquales, ce&longs;&longs;io æqualis e&longs;t impul&longs;ioni; &longs;i globus <lb/>impactus &longs;it maior, ce&longs;&longs;io e&longs;t maior impul&longs;ione, vt con&longs;tat; igitur, &longs;i globus <lb/>e&longs;t ad globum vt FB ad FB; determinatio noua erit ad priorem vt FB <lb/>ad FB; igitur quie&longs;cet globus impactus per Th. 62. &longs;i verò globus impa<lb/>ctus &longs;it ad alium vt EB ad ER; determinatio noua erit ad priorem, vt <lb/>BG ad BF; igitur motus retardatus globi impacti e&longs;t ad non retardatum <lb/>vt FG ad FB; quod &longs;i globus impactus e&longs;t ad alium vt DB ad DS, deter<lb/>minatio noua e&longs;t ad priorem vt BH ad BF; &longs;i &longs;it vt TV, ad VB, deter<lb/>minatio noua erit ad priorem vt BX ad BF, donec tandem nullus &longs;it <lb/>globus re&longs;i&longs;tens; neque res aliter e&longs;&longs;e pote&longs;t. </s>

<s>62. &longs;i verò globus impa<lb/>ctus &longs;it ad alium vt EB ad ER; determinatio noua erit ad priorem, vt <lb/>BG ad BF; igitur motus retardatus globi impacti e&longs;t ad non retardatum <lb/>vt FG ad FB; quod &longs;i globus impactus e&longs;t ad alium vt DB ad DS, deter<lb/>minatio noua e&longs;t ad priorem vt BH ad BF; &longs;i &longs;it vt TV, ad VB, deter<lb/>minatio noua erit ad priorem vt BX ad BF, donec tandem nullus &longs;it <lb/>globus re&longs;i&longs;tens; neque res aliter e&longs;&longs;e pote&longs;t. </s>

<s>Hinc vides duos terminos oppo&longs;itos, qui &longs;unt, nulla re&longs;i&longs;tentia, & infi<lb/>nita re&longs;i&longs;tentia; nulla e&longs;t re&longs;i&longs;tentia, cum globus impactus in nullum in<lb/>cidit, &longs;ed e&longs;t veluti infinita ce&longs;&longs;io; cum verò globus in corpus immobile <lb/>impingitur, e&longs;t veluti infinita re&longs;i&longs;tentia ratione huius motus; cum verò <lb/>globus in alium globum, quem mouet, impingitur, &longs;i vterque æqualis e&longs;t; <lb/>e&longs;t etiam æqualis ce&longs;&longs;io re&longs;i&longs;tentiæ; igitur globus impactus quie&longs;cit, & <lb/>hoc e&longs;t iu&longs;tum medium extremorum prædictorum, id e&longs;t, inter nullam <lb/>ce&longs;&longs;ionem, & infinitam ce&longs;&longs;ionem; media e&longs;t æqualis ce&longs;&longs;io; & inter nul<lb/>lam re&longs;i&longs;tentiam & infinitam re&longs;i&longs;tentiam media e&longs;t æqualis re&longs;i&longs;tentia; <pb pagenum="258"/>re&longs;i&longs;tentia autem con&longs;ideratur in globo impacto, cuius re&longs;i&longs;titur motui; <lb/>ce&longs;&longs;io verò in alio, qui motui cedit; appello autem infinitam re&longs;i&longs;ten<lb/>tiam cui nulla re&longs;pondet ce&longs;&longs;io; nihil enim aliud præ&longs;taret infinita; por<lb/>rò cum nulla e&longs;t ce&longs;&longs;io, determinatio noua e&longs;t dupla prioris, vt demon<lb/>&longs;tratum e&longs;t &longs;uprà; igitur nihil prioris remanet; cum verò nulla e&longs;t re&longs;i<lb/>&longs;tentia, tota prior remanet, & nulla e&longs;t noua: denique cum ce&longs;&longs;io æqua<lb/>lis e&longs;t re&longs;i&longs;tentiæ, tantùm remanet prioris quantùm e&longs;t nouæ; igitur <lb/>vtraque æqualis e&longs;t: Vnde vides, ni fallor, perfectam analogiam, &c. </s>

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<s><emph type="italics"/>Si globus maior impingatur in minorem per lineam obliquam &longs;emper re<lb/>flectitur, licèt aliquando iu&longs;en&longs;ibiliter, quia fit determinatio mixta ex noua & <lb/>priore, cuius proportio determinari pote&longs;t<emph.end type="italics"/>; &longs;it enim determinatio noua ad <lb/>priorem in linea incidentiæ perpendiculari vt C<foreign lang="greek">d</foreign> ad CA fig. </s>

<s>Th. 65. <lb/> vel vt AZ ad AF, &longs;it linea incidentiæ obliqua EA producta in B; <lb/>certè &longs;i determinatio noua per lineam incidentiæ obliquam EA e&longs;t ad <lb/>priorem, vt AZ ad AF; &longs;umatur B<foreign lang="greek">u</foreign> æqualis AY; ducantur Y<foreign lang="greek">u</foreign> A<foreign lang="greek">u</foreign><lb/>dico A<foreign lang="greek">u</foreign> e&longs;&longs;e lineam reflexionis, quia e&longs;t mixta ex AY & AB, vt con<lb/>&longs;tat ex dictis; Idem dico de aliis incidentiæ. </s>

<s>65. <lb/> vel vt AZ ad AF, &longs;it linea incidentiæ obliqua EA producta in B; <lb/>certè &longs;i determinatio noua per lineam incidentiæ obliquam EA e&longs;t ad <lb/>priorem, vt AZ ad AF; &longs;umatur B<foreign lang="greek">u</foreign> æqualis AY; ducantur Y<foreign lang="greek">u</foreign> A<foreign lang="greek">u</foreign><lb/>dico A<foreign lang="greek">u</foreign> e&longs;&longs;e lineam reflexionis, quia e&longs;t mixta ex AY & AB, vt con<lb/>&longs;tat ex dictis; Idem dico de aliis incidentiæ. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 68.<emph.end type="center"/></s>

<s><emph type="italics"/>Si globus in æqualem globum impingatur, qui æquali impetu in eum etiam <lb/>impingitur per lineam connectentem centra<emph.end type="italics"/>; vterque retro agitur æquali <lb/>p&oelig;nitus motu, quo &longs;uam lineam vlteriùs propaga&longs;&longs;et, &longs;i in alterum glo<lb/>bum non incidi&longs;&longs;et per Th.137.lib.1.&longs;i autem inæquali impetu mouean<lb/>tur, non e&longs;t determinatum &longs;uprà; pote&longs;t autem &longs;it determinari, fig. </s>

<s>1. <lb/>Tab.1.&longs;it globus A impactus in alium B motu vt 4. codem tempore, quo <lb/>globus B impingitur in A motu vt 2. certè globus B retrò agetur motu vt <lb/>4. quippè &longs;iue moueatur æquali motu, &longs;iue minori, &longs;iue etiam quie&longs;cat, <lb/>&longs;emper æquali motu à globo A impelletur; quod certè mirabile e&longs;t; pri<lb/>mum con&longs;tat per Th. </s>

<s>135.lib.

tertium con&longs;tat per Theor.128.lib.1.Igi<lb/>tur &longs;ecundum con&longs;tat, &longs;i enim impellitur motu vt 4.dum in contrariam <lb/>partem mouetur vt 4. multò magis &longs;i tantùm mouetur vt 2. & &longs;i tantùm <lb/>impellitur motu vt 4. dum quie&longs;cit multò magis motu vt 4. dum in <pb pagenum="259"/>contrariam partem mouetur motu vt 2. at verò globus A non retroage<lb/>tur: motu vt 4. &longs;ed tantùm motu vt 2. vt patet; quippe omninò con&longs;i&longs;teret, <lb/>&longs;iglobus B nullum præuium impetum habui&longs;&longs;et; &longs;i verò habui&longs;&longs;et mo<lb/>tum vt 4. tùm etiam A retroageretur motu vt 4. igitur motu vt duo, &longs;i <lb/>B impre&longs;&longs;it impetum vt duo. </s>

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<s>Th.65. res ferè eodem modo determinari pote&longs;t; quippè mo<lb/>tus impre&longs;&longs;us à globo B per lineam perpendicularem e&longs;t ad motum im<lb/>pre&longs;&longs;um A per inclinatam EA vt AZ ad AY; &longs;it autem linea inci<lb/>dentiæ DB fig. </s>

<s>Th. 63. eiu&longs;dem incidentiæ cum EA fig. </s>

<s>63. eiu&longs;dem incidentiæ cum EA fig. </s>

<s>Th. 65. igitur <lb/>globus A incidat per DB, & globus B per MB, ita vt punctum conta<lb/>ctus &longs;it B, & linea connectens centra FA; determinatio noua ratione in<lb/>cidentiæ e&longs;t vt BH, cui addatur HF æqualis AY fig. </s>

<s>65. igitur <lb/>globus A incidat per DB, & globus B per MB, ita vt punctum conta<lb/>ctus &longs;it B, & linea connectens centra FA; determinatio noua ratione in<lb/>cidentiæ e&longs;t vt BH, cui addatur HF æqualis AY fig. </s>

<s>alterius ratione <lb/>motus impre&longs;&longs;i à globo B; tota determinatio erit BF; a&longs;&longs;umatur MT <lb/>æqualis BF: dico nouam lineam quæ&longs;itam e&longs;&longs;e B<foreign lang="greek">q</foreign> mixtam &longs;cilicet ex <lb/>BF BM, quod probatur vt &longs;uprà. </s>

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<s>&longs;i globus plumbeus ex <lb/>aëre perpendiculariter cadat in &longs;uperficiem aquæ, haud dubiè ip&longs;am <lb/>aquam &longs;ubit, &longs;ed minore motu; quippe frangitur ab ip&longs;a den&longs;itate aquæ <lb/>vis primi impetus, quo &longs;cilicet per liberiorem aëra priùs ferebatur: vnde <lb/>&longs;i habeatur proportio re&longs;i&longs;tentiæ aquæ po&longs;ita linea incidentiæ perpendi<lb/>culari, non e&longs;t dubium, quin habeatur etiam re&longs;i&longs;tentia po&longs;ita linea in<lb/>cidentiæ obliqua; nam eodem modo hoc determinandum e&longs;t, quo &longs;uprà <lb/>determinatum fuit Th. </s>

<s>Th. 65. determinatio noua <lb/>po&longs;ita perpendiculari &longs;it ad priorem vt AZ ad AF, ita vt per mediam <lb/>aquam conficiat tantùm &longs;patium A<foreign lang="greek">d</foreign> v. </s>

<s>65. determinatio noua <lb/>po&longs;ita perpendiculari &longs;it ad priorem vt AZ ad AF, ita vt per mediam <lb/>aquam conficiat tantùm &longs;patium A<foreign lang="greek">d</foreign> v. </s>

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<s>Sed dicunt ab eodem plano e&longs;&longs;e non po&longs;&longs;e determinationem inæqua<lb/>lem; quia idem principium eundem effectum habet. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. negando ante<lb/>cedens; cùm enim pro diuer&longs;a re&longs;i&longs;tentia diuer&longs;a &longs;it determinatio, & <lb/>cùm planum prædictum modò plùs, modò minùs re&longs;i&longs;tat; quid mirum &longs;i <lb/>diuer&longs;a &longs;it etiam determinatio? </s>

<s>negando ante<lb/>cedens; cùm enim pro diuer&longs;a re&longs;i&longs;tentia diuer&longs;a &longs;it determinatio, & <lb/>cùm planum prædictum modò plùs, modò minùs re&longs;i&longs;tat; quid mirum &longs;i <lb/>diuer&longs;a &longs;it etiam determinatio? </s>

<s>In&longs;tant, lineam determinationis eiu&longs;dem impetus e&longs;&longs;e &longs;emper æqua<lb/>lem. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. negando; quia idem impetus ad duas lineas pote&longs;t determi<lb/>nari &longs;imul, quæ faciant determinationem mixtam; vnde licèt idem im<lb/>petus habeat eandem lineam &longs;patij, non tamen eandem lineam determi<lb/>nationis. </s>

<s>negando; quia idem impetus ad duas lineas pote&longs;t determi<lb/>nari &longs;imul, quæ faciant determinationem mixtam; vnde licèt idem im<lb/>petus habeat eandem lineam &longs;patij, non tamen eandem lineam determi<lb/>nationis. </s>

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<s>Quæres, quid &longs;it illa determinatio: facilis quæ&longs;tio. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. e&longs;&longs;e ip&longs;um <pb pagenum="271"/>impetum cum habitudine actuali ad talem vel talem lineam; quod au<lb/>tem po&longs;&longs;it e&longs;&longs;e plùs vel minùs determinatus ad vnam, quàm ad aliam, du<lb/>bium e&longs;&longs;e non pote&longs;t, nec in dubium reuocari, & benè di&longs;tinguitur li<lb/>nea quanta in ratione determinationis, & quanta in ratione &longs;patij: immò <lb/>hoc ip&longs;i &longs;upponunt; nam &longs;i KD e&longs;t mixta ex K <foreign lang="greek">b</foreign> & K <foreign lang="greek">q</foreign>, quis non vi<lb/>det e&longs;&longs;e eundem impetum cum determinatione duplici inæquali? </s>

<s>e&longs;&longs;e ip&longs;um <pb pagenum="271"/>impetum cum habitudine actuali ad talem vel talem lineam; quod au<lb/>tem po&longs;&longs;it e&longs;&longs;e plùs vel minùs determinatus ad vnam, quàm ad aliam, du<lb/>bium e&longs;&longs;e non pote&longs;t, nec in dubium reuocari, & benè di&longs;tinguitur li<lb/>nea quanta in ratione determinationis, & quanta in ratione &longs;patij: immò <lb/>hoc ip&longs;i &longs;upponunt; nam &longs;i KD e&longs;t mixta ex K <foreign lang="greek">b</foreign> & K <foreign lang="greek">q</foreign>, quis non vi<lb/>det e&longs;&longs;e eundem impetum cum determinatione duplici inæquali? </s>

<s>præ<lb/>terea, quis neget globum impactum perpendiculariter in alium æqua<lb/>lem quie&longs;cere? </s>

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<s>impe<lb/>tus in Q e&longs;t ad impetum in C, vt longitudo KQ ad KC, vt con&longs;tat ex <lb/>dictis; accipio autem omnes partes impetus, quæ &longs;unt in Q, & compa<lb/>ro omnes illas cum omnibus illis, quæ in&longs;unt poncto C; nam certum e&longs;t <lb/>ex his quæ fusè diximus lib.1.non produci plures partes impetus in C, <expan abbr="quã">quam</expan> <lb/>in <expan abbr="q;">que</expan> &longs;ed perfectiorem impetum produci in C, quàm in Q: recole quæ <lb/>diximus lib.1. à Th. 99. ad Th.112. in quibus habes totam propagatio<lb/>nem impetus determinati ad motum circularem; &longs;iue applicetur po<lb/>tentia centro, id e&longs;t iuxta centrum; &longs;iue circumferentiæ. </s>

<s>99. ad Th.112. in quibus habes totam propagatio<lb/>nem impetus determinati ad motum circularem; &longs;iue applicetur po<lb/>tentia centro, id e&longs;t iuxta centrum; &longs;iue circumferentiæ. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 18.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorèma<emph.end type="italics"/> 22.<emph.end type="center"/></s>

<s><emph type="italics"/>Rota minor in eodem &longs;itu de quo &longs;uprà æquè facilè moueri pote&longs;t, ac maior<emph.end type="italics"/><pb pagenum="282"/><emph type="italics"/>per &longs;e.<emph.end type="italics"/></s><s> Probatur primò, quia vtraque minimo impetu moueti pote&longs;t per <lb/>Th. 21. Secundò, quia addita minima vi impetus in F, & minima in A <lb/>tàm facilè maior rota de&longs;cendit, quàm minor, quia æqualiter tollitur <lb/>æquilibrium vtriu&longs;que: dixi per &longs;e, quia maior rota propter maius pon<lb/>dus maiore affrictu motum impedit. </s>

<s>21. Secundò, quia addita minima vi impetus in F, & minima in A <lb/>tàm facilè maior rota de&longs;cendit, quàm minor, quia æqualiter tollitur <lb/>æquilibrium vtriu&longs;que: dixi per &longs;e, quia maior rota propter maius pon<lb/>dus maiore affrictu motum impedit. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc &longs;i nullus &longs;it partium affrictus, e&longs;&longs;et motus ille perpetuus<emph.end type="italics"/>; quia nul<lb/>lus de&longs;truitur impetus per Th. </s>

<s><emph type="italics"/>Hinc &longs;i nullus &longs;it partium affrictus, e&longs;&longs;et motus ille perpetuus<emph.end type="italics"/>; quia nul<lb/>lus de&longs;truitur impetus per Th. 34. igitur ille motus e&longs;&longs;et perpetuus. </s>

<s>34. igitur ille motus e&longs;&longs;et perpetuus. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 36.<emph.end type="center"/></s>

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<s><emph type="italics"/>Si vectis BH ita pellatur in B in plano horizontali, in quo liberè moueri <lb/>po&longs;&longs;it<emph.end type="italics"/> <emph type="italics"/>v.g. </s>

<s>dum aquæ &longs;upernatat, nulli centro immobili affixus, &longs;it que aqualis <lb/>den&longs;itatis in omnibus &longs;uis partibus; mouebitur circa aliquod centrum, etiam&longs;i <lb/>nulli centro affigatur.<emph.end type="italics"/></s><s> Probatur, quia punctum B velociùs mouebitur, quàm <lb/>A vel H, vt patet experientiâ: ratio e&longs;t, quia minùs impetus producitur <lb/>in toto cylindro BH, applicata potentia in B, quàm in A, quod e&longs;t cen<lb/>trum grauitatis cylindri BA, vt iam o&longs;tendimus Th. </s>

<s>68. 69. BB; porrò <lb/>ratio à priori e&longs;t, quia cùm impetus producatur tantùm ad extra, vt tol<lb/>latur impedimentum motus, vt fusè o&longs;tendimus lib.

1. certè in tantùm <lb/>amouetur impedimentum, in quantum amouetur corpus impediens mo<lb/>tum alterius; atqui amoucri tantùm pote&longs;t per motum; igitur eo motu <lb/>amouetur, quo faciliùs amoueri pote&longs;t, & minore &longs;umptu, vt ita dicam, <lb/>id e&longs;t minore impetu: porrò cum potentia &longs;it determinata ad producen<lb/>dum tabem impetum, immediatè &longs;cilicet, id e&longs;t, in ea parte, cui immedia<lb/>tè admouetur; alicqui &longs;i po&longs;&longs;et minorem, & minorem in infinitum pro<lb/>ducere po&longs;&longs;et etiam immediatè &longs;ine operâ organi mechanici quodlibet <lb/>pondus mouere, quod e&longs;t ab&longs;urdum, de quo iam &longs;uprà; &longs;it igitur potentia <lb/>applicata in A, &longs;cilicet in centro grauitatis cylindri BH; certè producit <lb/>maximum impetum, quem pote&longs;t producere in cylindro BH (&longs;uppono <lb/>enim e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam, & producere perfecti&longs;&longs;imum impetum, <lb/>quem producere po&longs;&longs;it) producit inquam maximum ratione numeri; <lb/>cùm in toto cylindro BH producat impetum eiu&longs;dem perfectionis; igi<lb/>tur mouetur motu recto; igitur æquali in omnibus partibus; igitur æqua<lb/>lis e&longs;t impetus in omnibus partibus, id e&longs;t, æquè inten&longs;us; &longs;it autem po-<pb pagenum="291"/>tentia applicata in B, ita vt in puncto B producatur impetus eiu&longs;dens <lb/>perfectionis, de quo &longs;uprà: &longs;i mouetur motu circulari circa aliquod cen<lb/>trum v. </s>

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<s><emph type="italics"/>Velocitates acqui&longs;itæ in funependulis inæqualibus &longs;unt vt altitudines<emph.end type="italics"/>; &longs;it <lb/>enim in figura. </s>

<s>Th. 10. Funependulum maius AH, minus GH; &longs;it vi<lb/>bratio minoris FYH; &longs;it vibratio maioris DKH: dico velocitatem <lb/>acqui&longs;itam in prima vibratione e&longs;&longs;e ad acqui&longs;itam in &longs;ecunda, vt AH ad <lb/>GH; &longs;i verò vibratio maioris &longs;it tantùm LKH; dico e&longs;&longs;e æqualem ve<lb/>locitatem vtriu&longs;que, quæ omnia patent ex dictis: hinc &longs;eruari po&longs;&longs;unt <lb/>quæ cumque proportiones ictuum inflictorum à malleis, vel &longs;imul, vel <lb/>&longs;ucce&longs;&longs;iue, &c. </s>

<s>10. Funependulum maius AH, minus GH; &longs;it vi<lb/>bratio minoris FYH; &longs;it vibratio maioris DKH: dico velocitatem <lb/>acqui&longs;itam in prima vibratione e&longs;&longs;e ad acqui&longs;itam in &longs;ecunda, vt AH ad <lb/>GH; &longs;i verò vibratio maioris &longs;it tantùm LKH; dico e&longs;&longs;e æqualem ve<lb/>locitatem vtriu&longs;que, quæ omnia patent ex dictis: hinc &longs;eruari po&longs;&longs;unt <lb/>quæ cumque proportiones ictuum inflictorum à malleis, vel &longs;imul, vel <lb/>&longs;ucce&longs;&longs;iue, &c. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 13.<emph.end type="center"/></s>

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<s>in figura Lem.3.percurruntur DT <lb/>dupla radij ID, eo tempore, quo percurritur DP; &longs;ed DP percurritur <lb/>tardiùs, quàm arcus DKP; igitur DKP citiùs quàm DT; igitur non <lb/>percurritur &longs;patium 6. pedum in perpendiculo eo tempore, quo percur<lb/>ritur arcus quadrantis DKP, cuius radius ID &longs;it tripedalis; præterea <lb/>non percurruntur tantùm in perpendiculo eodem tempore pedes &longs;patij <lb/>4 5/7, vel vndecim, &longs;i radius con&longs;tat 7. pedibus, vt voluit idem auctor l. </s>

<s>2. <lb/>de cau&longs;is &longs;onorum Prop. </s>

<s>2. <lb/>de cau&longs;is &longs;onorum Prop. 27. Cor. </s>

<s>3. quia &longs;i radius habet 3. arcus <lb/>quadrantis habet 4 5/7. &longs;i radius habet 7. arcus quadrantis habet 11. <lb/>&longs;ed eodem tempore conficitur maius &longs;patium in perpendiculo, quàm in <pb pagenum="317"/>arcu, cuius ratio con&longs;tat clari&longs;&longs;imè ex dictis, quia dum mobile mouea<lb/>tur in perpendiculo &longs;ingulis in&longs;tantibus nouum impetum æqualem pri<lb/>mo producit, in arcu verò minorem; igitur minor e&longs;t motus; igitur mi<lb/>nus &longs;patium eodem tempore percurritur in arcu, & maius in perpendi<lb/>culo; igitur non percurruntur 11. tantùm in perpendiculo eo tempore <lb/>quo 11. percurruntur in arcu; quantum verò &longs;patium in perpendiculo <lb/>percurratur eo tempore, quo arcus quadrantis dati conficitur, determi<lb/>nabimus infrà. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s>

<s><emph type="italics"/>Hinc non de&longs;truitur ille impetus ab impetu innato, vt fit in funependulis<emph.end type="italics"/>; <lb/>quia &longs;cilicet de&longs;truitur tantùm ab innato in a&longs;cen&longs;u; &longs;ed nullum pun<lb/>ctum globi a&longs;cendit, vt dictum e&longs;t, quod vt meliùs intelligatur, &longs;it in fi<lb/>gura Th. 1. globus centro O; &longs;itque OF perpendicularis deor&longs;um, quæ <lb/>percurritur ab eodem centro O motu centri; &longs;itque motus orbis ab L <lb/>in <expan abbr="q;">que</expan> intelligatur autem planium AI 6; certè punctum A, quod perinde <lb/>&longs;e habet, atque &longs;i e&longs;&longs;et punctum contactus, de&longs;cribit lineam ARP ergo <lb/>non a&longs;cendit; igitur non de&longs;truitur impetus productus ab impetu in<lb/>nato. </s>

<s>1. globus centro O; &longs;itque OF perpendicularis deor&longs;um, quæ <lb/>percurritur ab eodem centro O motu centri; &longs;itque motus orbis ab L <lb/>in <expan abbr="q;">que</expan> intelligatur autem planium AI 6; certè punctum A, quod perinde <lb/>&longs;e habet, atque &longs;i e&longs;&longs;et punctum contactus, de&longs;cribit lineam ARP ergo <lb/>non a&longs;cendit; igitur non de&longs;truitur impetus productus ab impetu in<lb/>nato. </s>

<s><emph type="center"/><emph type="italics"/>Scholium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s>Quintò, &longs;i vera e&longs;&longs;et hypothe&longs;is Copernici, terra moueretur hoc vlti<lb/>mo motu mixto ex motu centri, & motu orbis; vnde omnia puncta <lb/>ciu&longs;dem circuli parallcli mouerentur inæquali motui tardi&longs;&longs;imo qui<lb/>dem punctum contactus hoc e&longs;t meridiano re&longs;pondens, veloci&longs;&longs;imo ve<lb/>rò ip&longs;i oppo&longs;itum, &longs;cilicet de media nocte: porrò in hoc motu motus <lb/>centri e&longs;&longs;et ferè maior motu orbis iuxta communem de diametro ma<lb/>gni orbis &longs;ententiam. </s>

<s>Sextò, &longs;i motus maioris rotæ diragatur à minore res eodem modo <lb/>explicanda e&longs;t, quo explicuimus illam per <expan abbr="cõtactus">contactus</expan> diuer&longs;os inadæquatos <lb/>tùm Th. </s>

<s>11. tùm in digre&longs;&longs;ione multis locis: porrò po&longs;&longs;unt e&longs;&longs;e <lb/>diuer&longs;æ proportiones circuli mobilis, & immobilis; qui &longs;i maximus e&longs;t, <lb/>minimus illius arcus accipi pote&longs;t pro linea recta. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s>

<s><emph type="italics"/>Quando voluitur funis circa cylindrum, vel axem<gap/> mouetur motu <lb/>&longs;pirali, &longs;ed diuer&longs;o à prioribus<emph.end type="italics"/>; funt enim veræ &longs;piræ ad in&longs;tar &longs;apien<lb/>tia in diuer&longs;a volumina contorti; &longs;ic funis circa digitum fæpè <lb/>rotatur.; e&longs;t enim motus mixtus ex diuer&longs;is circul<gap/>ilius: quippè <pb pagenum="363"/>in &longs;ingulis punctis e&longs;t diuer&longs;a determinatio ad nouum circulum, quia <lb/>e&longs;t nouus radius, quia continuò radius huius vertiginis imminuitur; <lb/>porrò duobus modis pote&longs;t funis circa axem vel cylindrum conuolui. </s>

<s><emph type="italics"/>Quando voluitur funis circa cylindrum, vel axem<gap/> mouetur motu <lb/>&longs;pirali, &longs;ed diuer&longs;o à prioribus<emph.end type="italics"/>; &longs;unt enim veræ &longs;piræ ad in&longs;tar &longs;apien<lb/>tia in diuer&longs;a volumina contorti; &longs;ic funis circa digitum fæpè <lb/>rotatur.; e&longs;t enim motus mixtus ex diuer&longs;is circul<gap/>ilius: quippè <pb pagenum="363"/>in &longs;ingulis punctis e&longs;t diuer&longs;a determinatio ad nouum circulum, quia <lb/>e&longs;t nouus radius, quia continuò radius huius vertiginis imminuitur; <lb/>porrò duobus modis pote&longs;t funis circa axem vel cylindrum conuolui. </s>

<s><lb/>Primò, &longs;i &longs;emper circa eumdem cylindri circulum voluatur; tunc autem <lb/>facit veras &longs;piras, vt vides in A. Secundò, &longs;i circa diuer&longs;os eiu&longs;dem axis <lb/>circulos, vel potius diuer&longs;a eiu&longs;dem axis puncta voluatur, & hic e&longs;t mo<lb/>tus &longs;piralis conicus, vt vides in cono FDE; idem e&longs;&longs;et motus &longs;i conus <lb/>circa axem volueretur &longs;imulque aliquod punctum peripheriæ ba&longs;is coni <lb/>rectà ab ip&longs;a peripheria ad verticem coni tenderet; &longs;i enim totus conus <lb/>moueatur motu axis recto, quodlibet punctum &longs;uperficiei coni mouetur <lb/>motu &longs;pirali cylindrico, excepto dumtaxat ip&longs;o vertice; hoc denique <lb/>motu mouerentur &longs;ingula puncta baculi ED, qui in conum rotaretur à <lb/>vertice E eo tempore, quo rotans ip&longs;e per rectam EG moueretur. </s>

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<s>Secundò, &longs;i voluatur circulus radio AE circa centrum E, nec &longs;it vllus <lb/>motus circa centrum B; haud dubiè omnes partes excentrici ADOC <lb/>mouebuntur motu circulari &longs;ed inæquali, vt patet. </s>

<s>Tertiò, &longs;i &longs;it motus circularis circa vtrumque centrum; certè centrum <lb/>B circumagetur per circellum BGHF, punctum verò A excentrici <lb/>de&longs;cribet hanc lineam APIQBSIRA, vt con&longs;tat ex dictis Th. </s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s>

<s><emph type="italics"/>Si rota moueatur in circulo parallelo illi plane, cui incubat perpendicula<lb/>riter eodem ferè motu moneri videtur, quo turbo, de quo &longs;uprà<emph.end type="italics"/>; a&longs;&longs;umatur <lb/>enim figura prima Th. 15. in qua &longs;it circulus immobilis in plano hori<lb/>zontali BTXD, & erigatur rota BEDF, ita vt &longs;it parallela circulo <lb/>verticali, tangatque priorem circulum in B, cuius deinde periphæriam <lb/>&longs;en&longs;im percurrat; haud dubiè punctum B de&longs;cribet &longs;uo motu lineam, quæ <pb pagenum="375"/>pote&longs;t declinari; &longs;it enim circulus immobilis BDFC, mobilis FEG, <lb/>punctum F po&longs;t decur&longs;um quadrantem FD extat &longs;upra planum hori<lb/>zontis tota ID erecta; po&longs;t decur&longs;um verò &longs;emicirculum tota BK <lb/>erecta æquali BF, vt con&longs;tat; igitur vertatur FBK, circa FB, donec incu<lb/>bet perpendiculariter plano horizontali in BF; tùm circa FK, ita ere<lb/>ctam vertatur planum, donec incubet DI, erecta in I, fiet planum, in quo <lb/>de&longs;cribetur linea huius motus; a&longs;&longs;umatur autem DH æqualis AI; dico <lb/>quod ducetur per FHK: &longs;imiliter inuenientur alia puncta, quod &longs;uffi<lb/>ciat indica&longs;&longs;e; e&longs;t autem hic motus maximè inæqualis propter ratio<lb/>nem, de qua &longs;uprà: &longs;ed de his &longs;atis; immò certnm e&longs;t punctum F &longs;uo <lb/>motu prædicto de&longs;cribere perfectum circulum duplum circuli rota<lb/>ti, cuius centrum e&longs;t D crectum in A, nam DH, DF, DK &longs;unt æqua<lb/>les; &longs;i enim circulus tangat in M, punctum F erectum toto arcu FM, <lb/>re&longs;pondebit perpendiculariter puncto O, ita vt OM &longs;it æqualis PB, vel <lb/>HS, vel AN; erigatur autem OR, donec incubet perpendiculariter, <lb/>extat &longs;uper AD erecta in A tota QR, ita OQ &longs;it æqualis AD. </s>

<s>15. in qua &longs;it circulus immobilis in plano hori<lb/>zontali BTXD, & erigatur rota BEDF, ita vt &longs;it parallela circulo <lb/>verticali, tangatque priorem circulum in B, cuius deinde periphæriam <lb/>&longs;en&longs;im percurrat; haud dubiè punctum B de&longs;cribet &longs;uo motu lineam, quæ <pb pagenum="375"/>pote&longs;t declinari; &longs;it enim circulus immobilis BDFC, mobilis FEG, <lb/>punctum F po&longs;t decur&longs;um quadrantem FD extat &longs;upra planum hori<lb/>zontis tota ID erecta; po&longs;t decur&longs;um verò &longs;emicirculum tota BK <lb/>erecta æquali BF, vt con&longs;tat; igitur vertatur FBK, circa FB, donec incu<lb/>bet perpendiculariter plano horizontali in BF; tùm circa FK, ita ere<lb/>ctam vertatur planum, donec incubet DI, erecta in I, fiet planum, in quo <lb/>de&longs;cribetur linea huius motus; a&longs;&longs;umatur autem DH æqualis AI; dico <lb/>quod ducetur per FHK: &longs;imiliter inuenientur alia puncta, quod &longs;uffi<lb/>ciat indica&longs;&longs;e; e&longs;t autem hic motus maximè inæqualis propter ratio<lb/>nem, de qua &longs;uprà: &longs;ed de his &longs;atis; immò certnm e&longs;t punctum F &longs;uo <lb/>motu prædicto de&longs;cribere perfectum circulum duplum circuli rota<lb/>ti, cuius centrum e&longs;t D crectum in A, nam DH, DF, DK &longs;unt æqua<lb/>les; &longs;i enim circulus tangat in M, punctum F erectum toto arcu FM, <lb/>re&longs;pondebit perpendiculariter puncto O, ita vt OM &longs;it æqualis PB, vel <lb/>HS, vel AN; erigatur autem OR, donec incubet perpendiculariter, <lb/>extat &longs;uper AD erecta in A tota QR, ita OQ &longs;it æqualis AD. </s>

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<s>à men&longs;a; &longs;ed hæc non producit impetum in <lb/>pondere, quod &longs;u&longs;tinet, vt dicam paulò pò&longs;t; Tertiò, quia fru&longs;trà pro<lb/>duceretur; quia modò manus &longs;u&longs;tinens &longs;tet immobilis; haud dubiè etiam <lb/>&longs;ublato omni extrin&longs;eco impetu à pondere adhuc &longs;u&longs;tinebitur. </s>

<s>Dices; igitur fru&longs;trà produceretur impetus in manu; Re&longs;p. </s>

<s>Dices; igitur fru&longs;trà produceretur impetus in manu; Re&longs;p. negando <lb/>quia ni&longs;i potentia motrix produceret impetum in manu, ab ip&longs;o pon<lb/>dere deprimeretur; igitur non e&longs;t fru&longs;trà omninò ille impetus. </s>

<s>negando <lb/>quia ni&longs;i potentia motrix produceret impetum in manu, ab ip&longs;o pon<lb/>dere deprimeretur; igitur non e&longs;t fru&longs;trà omninò ille impetus. </s>

<s>Dices, non habet motum; igitur e&longs;t fru&longs;trà; Re&longs;p. </s>

<s>Dices, non habet motum; igitur e&longs;t fru&longs;trà; Re&longs;p. omnem impetum <lb/>non e&longs;&longs;e fru&longs;trà, licèt careat motu, vt patet in ip&longs;o impetu innato, <pb pagenum="379"/>cuius duplex e&longs;t effectum; &longs;cilicet grauitatio, & motus, vt aliàs iam in<lb/>dicauimus; &longs;imiliter impetus productus à potentia motrice, in &longs;uo or<lb/>gano habere pote&longs;t duplicem effectum; primus e&longs;t motus; &longs;ecundus e&longs;t <lb/>ni&longs;us &longs;eu conatus oppo&longs;itus extrin&longs;eco motui; quemadmodum enim in<lb/>natus &longs;emper habet motum, ni&longs;i impediatur ab alio corpore, ita & im<lb/>petus organi potentiæ motricis, nec e&longs;t magna difficultas; immò cla<lb/>ri&longs;&longs;ima vtriu&longs;que potentiæ analogia. </s>

<s>omnem impetum <lb/>non e&longs;&longs;e fru&longs;trà, licèt careat motu, vt patet in ip&longs;o impetu innato, <pb pagenum="379"/>cuius duplex e&longs;t effectum; &longs;cilicet grauitatio, & motus, vt aliàs iam in<lb/>dicauimus; &longs;imiliter impetus productus à potentia motrice, in &longs;uo or<lb/>gano habere pote&longs;t duplicem effectum; primus e&longs;t motus; &longs;ecundus e&longs;t <lb/>ni&longs;us &longs;eu conatus oppo&longs;itus extrin&longs;eco motui; quemadmodum enim in<lb/>natus &longs;emper habet motum, ni&longs;i impediatur ab alio corpore, ita & im<lb/>petus organi potentiæ motricis, nec e&longs;t magna difficultas; immò cla<lb/>ri&longs;&longs;ima vtriu&longs;que potentiæ analogia. </s>

<s>Vndecimò, hinc benè explicatur, quomodo defatigetur ten&longs;um bra<lb/>&longs;iue coniunctum &longs;iue coniunctum; &longs;it cum extrin&longs;eco <expan abbr="põdere">pondere</expan>, &longs;iue <expan abbr="c&utilde;">cum</expan> pro<lb/>pria tantùm grauitate; quia partes aliquæ tendunt deor&longs;um, aliæ verò &longs;ur<lb/>&longs;um; hinc &longs;emper fit aliqua ten&longs;io; igitur aliqua diui&longs;io; igitur dolor, &longs;ic <lb/>enim tenditur funis à <expan abbr="põdere">pondere</expan> pendulo, pondus verò <expan abbr="incubãs">incubans</expan> tùm aliquas <lb/>partes premit, tùm alias maximè di&longs;trahit, in quo non e&longs;t difficultas; &longs;i <lb/>autem manus incubet men&longs;æ, v. </s>

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<s>14. &longs;it globulus libram <lb/>pendens incubans men&longs;æ 99. librarum; haud dubiè qui men&longs;am pon<lb/>derat, centum librarum pondus &longs;u&longs;tinet; igitur globulus producit in <lb/>men&longs;a impetum. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. neg. </s>

<s><expan abbr="con&longs;eq.">con&longs;eque</expan> nam ideò &longs;entitur pondus 100. li<lb/>brarum; quia vtrumque pondus grauitatione communi in &longs;uppo&longs;itam <lb/>grauitat manum. </s>

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<s>Dices, alias partes re&longs;i&longs;tere. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. igitur vt moueantur, &longs;uperari debet <lb/>illarum re&longs;i&longs;tentia; igitur per aliquid de nouo proctum; igitur per <lb/>impetum: immò non producitur in vna, ni&longs;i producatur in aliis; <lb/>alioquin fru&longs;trà e&longs;&longs;et ille impetus, cui nullus effectus re&longs;pon<lb/>deret; igitur &longs;i de&longs;truitur, quando fru&longs;trà e&longs;&longs;et, &longs;i con&longs;eruaretur; ita <lb/>ctiam non producitur quando fru&longs;trà e&longs;&longs;et, &longs;i produceretur; e&longs;t enim <lb/>par vtrimque ratio. </s>

<s>igitur vt moueantur, &longs;uperari debet <lb/>illarum re&longs;i&longs;tentia; igitur per aliquid de nouo proctum; igitur per <lb/>impetum: immò non producitur in vna, ni&longs;i producatur in aliis; <lb/>alioquin fru&longs;trà e&longs;&longs;et ille impetus, cui nullus effectus re&longs;pon<lb/>deret; igitur &longs;i de&longs;truitur, quando fru&longs;trà e&longs;&longs;et, &longs;i con&longs;eruaretur; ita <lb/>ctiam non producitur quando fru&longs;trà e&longs;&longs;et, &longs;i produceretur; e&longs;t enim <lb/>par vtrimque ratio. </s>

<s>Quartò, hinc licèt trahatur ingens rupes, non propterea mouetur, <lb/>quia non pote&longs;t impetus produci in omnibus illius partibus ab applica<lb/>ta potentia; igitut in nulla per Th.33.l.1. </s>

<s>Dices, e&longs;t cau&longs;a nece&longs;&longs;aria applicata. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. e&longs;&longs;e quidem applicatam, &longs;ed <lb/>e&longs;&longs;e impeditam propter maximam rupis re&longs;i&longs;tentiam, quam debiliores <lb/>potentiæ vires &longs;uperare non po&longs;&longs;unt. </s>

<s>e&longs;&longs;e quidem applicatam, &longs;ed <lb/>e&longs;&longs;e impeditam propter maximam rupis re&longs;i&longs;tentiam, quam debiliores <lb/>potentiæ vires &longs;uperare non po&longs;&longs;unt. </s>

<s>Quintò, hinc vna pars tracta non &longs;equitur aliam vltrò; &longs;i enim vltrò <lb/>&longs;equeretur minima potentia, &longs;ufficeret ad trahendum maximum pondus; <lb/>prærerea &longs;ingulæ partes mouentur per impetum. </s>

<s>Diceret aliquis, impetus productus in vna parte producit impetum <lb/>in alia. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. negando; alioquin minima potentia quodlibet pondus <lb/>moueret contra experientiam. </s>

<s>negando; alioquin minima potentia quodlibet pondus <lb/>moueret contra experientiam. </s>

<s>Dices, impetus vnius corporis producit impetum in alio, à quo eius <lb/>motus impeditur; igitur impetus vnius partis producit impetum in <lb/>alia, à qua eius motus impeditur. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. impetum, qui reuerâ alicui <lb/>corpori ine&longs;t, hoc ip&longs;um præ&longs;tare; at impetus non producitur in vna <lb/>parte mobilis, ni&longs;i fimul in aliis producatur; vel enim producitur in <lb/>omnibus, vel in nulla; hinc colliges quantum ab&longs;urdum &longs;equerctur, <lb/>ni&longs;i hoc e&longs;&longs;et; quia perpetua e&longs;&longs;et impetus productio, & minimus im<lb/>petus totam ip&longs;am terram moueret; vide quæ diximus &longs;uper ea re toto <lb/>lib.1. nec enim totus impetus motoris producit totum &longs;uum effectum <lb/>in vnico puncto mobilis, quod ridiculum dictu e&longs;t; alioquin produ<lb/>ceretur impetus inten&longs;i&longs;&longs;imus; igitur in pluribus; igitur in omnibus, <lb/>quæ &longs;imul moueri debent, vel in multa. </s>

<s>impetum, qui reuerâ alicui <lb/>corpori ine&longs;t, hoc ip&longs;um præ&longs;tare; at impetus non producitur in vna <lb/>parte mobilis, ni&longs;i fimul in aliis producatur; vel enim producitur in <lb/>omnibus, vel in nulla; hinc colliges quantum ab&longs;urdum &longs;equerctur, <lb/>ni&longs;i hoc e&longs;&longs;et; quia perpetua e&longs;&longs;et impetus productio, & minimus im<lb/>petus totam ip&longs;am terram moueret; vide quæ diximus &longs;uper ea re toto <lb/>lib.1. nec enim totus impetus motoris producit totum &longs;uum effectum <lb/>in vnico puncto mobilis, quod ridiculum dictu e&longs;t; alioquin produ<lb/>ceretur impetus inten&longs;i&longs;&longs;imus; igitur in pluribus; igitur in omnibus, <lb/>quæ &longs;imul moueri debent, vel in multa. </s>

<s>Diceret aliquis; quando mouetur corpus equi, mouetur ctiam ani<lb/>ma; igitur &longs;ine impetu; igitur per impetum corporis; igitur nomine <lb/>tantùm vnionis; igitur pars corporis alteri vnita etiam &longs;ine impetu, <lb/>&longs;cilicet per impetum alterius moueri pote&longs;t: hanc difficultatem iam <lb/>&longs;oluimus &longs;uprà l.1.Th.38.Cor.12. </s>

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<s>Diceret aliquis hæc repugnare omnibus experimentis, quibus &longs;cili<lb/>cet clari&longs;&longs;imè con&longs;tat minorem e&longs;&longs;e breuiorum &longs;ari&longs;&longs;arum vim. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. hoc ip&longs;um accidere; quia brcuiores &longs;ari&longs;&longs;æ, quas habemus, vel <lb/>exiliores &longs;unt longioribus, vel &longs;altem non cra&longs;&longs;iores, cùm tamen cra&longs;&longs;io<lb/>res e&longs;&longs;e oporteat in eadem ratione, in qua illæ longiores &longs;unt vt æqualis <lb/>&longs;it ictus. </s>

<s>hoc ip&longs;um accidere; quia brcuiores &longs;ari&longs;&longs;æ, quas habemus, vel <lb/>exiliores &longs;unt longioribus, vel &longs;altem non cra&longs;&longs;iores, cùm tamen cra&longs;&longs;io<lb/>res e&longs;&longs;e oporteat in eadem ratione, in qua illæ longiores &longs;unt vt æqualis <lb/>&longs;it ictus. </s>

<s>Quintò, cur verò maior fu&longs;tis maiorem impetum à brachiorum vi <lb/>recipiat; ratio e&longs;t, primò quia maiori vtrumque brachium admouetur: <lb/>&longs;ecundò, quia vibratur antequam intendatur; atqui ex ea vibratione <lb/>multus impetus accedit, vt patet ex vibrato ariete: tertiò, quia maior <lb/>fu&longs;tis tardiùs mouetur, vt con&longs;tat; igitur plùs impetus in eo producit <lb/>potentia motrix, quæ &longs;ingulis in&longs;tantibus toto ni&longs;u fu&longs;tem impellit; & <lb/>hæc e&longs;t vera ratio à priori: quartò, adde quod pondus maioris fu&longs;tis <lb/>qua&longs;i neruos extendit; atqui ten&longs;i nerui fortiores &longs;unt; in qua verò <lb/>proportione &longs;it maior ictus, dicemus numero &longs;equenti; e&longs;t enim res <lb/>&longs;citu digni&longs;&longs;ima. </s>

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<s>Secundò, hinc ex hac hypothe&longs;i ictus &longs;unt in ratione &longs;ubduplicata <lb/>ponderum malleorum; con&longs;tat etiam, po&longs;ita &longs;cilicet cadem longitudine <lb/>manubrij. </s>

<s>Tertiò, maior incutitur ictus non quidem circa extremitatem <lb/>ba&longs;is mallei, nec circa medium, &longs;ed circa mediam proportionalem <lb/>inter diametrum ba&longs;is, & &longs;ubduplum, patet per Th. </s>

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<s>Septimò, diceret aliquis velocitatem C decur&longs;o CD, e&longs;&longs;e &longs;ubduplam <lb/>vclocitatis B decur&longs;o BF; &longs;ed velocitas C, decur&longs;o CG, e&longs;t dupla velo<lb/>citatis eiu&longs;dem C decur&longs;o CD; igitur velocitas C, decur&longs;o CG, e&longs;t <lb/>æqualis velocitati B, decur&longs;o BF; igitur æqualis ictus. </s>

<s>Re&longs;p. </s>

<s>Re&longs;p. conce&longs;&longs;a <lb/>primâ con&longs;equentiâ, vltimâ verò negatâ; quia non tantùm impetus <lb/>puncti C incutit ictum &longs;ed totius CA, qui cen&longs;etur e&longs;&longs;e collectus in <lb/>malleo in quo e&longs;t qua&longs;i centrum huius impetus, vt iam explicuimus <lb/>aliàs; &longs;ed velocitas totius CA confecto CAD e&longs;t æqualis velocitati <lb/>totius BA confecto BAF, cuius velocitas CA confecto CAG e&longs;t dupla, <lb/>vt iam probatum e&longs;t. </s>

<s>conce&longs;&longs;a <lb/>primâ con&longs;equentiâ, vltimâ verò negatâ; quia non tantùm impetus <lb/>puncti C incutit ictum &longs;ed totius CA, qui cen&longs;etur e&longs;&longs;e collectus in <lb/>malleo in quo e&longs;t qua&longs;i centrum huius impetus, vt iam explicuimus <lb/>aliàs; &longs;ed velocitas totius CA confecto CAD e&longs;t æqualis velocitati <lb/>totius BA confecto BAF, cuius velocitas CA confecto CAG e&longs;t dupla, <lb/>vt iam probatum e&longs;t. </s>

<s>Octanò, hinc ictus CA confecto CAD e&longs;t æqualis ictui AB con<lb/>fecto BAF, & ictus CA confecto CI duplo CD e&longs;t ad ictum CA con<lb/>fecto CD, vt radix CA ad radicem CI: hinc vides hunc motum con<lb/>uenire in co cum recto, quòd &longs;cilicet ictus inflictus motu recto à mi<lb/>nori mole, &longs;it ad ictum maioris, &longs;uppo&longs;ita linea motus æquali in ratio<lb/>ne &longs;ubduplicata ponderum; quòd dicitur etiam de motu circulari duo<lb/>rum fu&longs;tium inæqualium, quorum ictus &longs;unt in ratione &longs;ubduplicata <lb/>longitudinum, a&longs;&longs;umptis duntaxat arcubus æqualibus ab extremitate <lb/>vtriu&longs;que decur&longs;is. </s>

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<s>Ob&longs;eruabis e&longs;&longs;e plura alia ph&oelig;nomena in ludo minoris Tudiculæ <lb/>v.g. </s>

<s>1°.quod &longs;pectat ad proportionem ictuum ratione puncti contactus, <lb/>de qua idem dicendum e&longs;t, quod &longs;uprà dictum e&longs;t Th. </s>

<s>1°.quod &longs;pectat ad proportionem ictuum ratione puncti contactus, <lb/>de qua idem dicendum e&longs;t, quod &longs;uprà dictum e&longs;t Th. 15. num. </s>

<s>25. <lb/>2°.quod &longs;pectat ad lineam motus, per quam pila impacta impellit aliam, <lb/>de qua lib.1. Th.50. 51. 52.& alibi pa&longs;&longs;im. </s>

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<s>Quintò, ventilatio e&longs;t motio, quâ frumentum excernitur vanno; van<lb/>nus circuli e&longs;t vulgare &longs;atis frumentarium organum duabus an&longs;is in&longs;tru<lb/>ctum, quibus vibratur tùm in aduer&longs;am partem, vt ip&longs;o &longs;uccu&longs;&longs;u paleæ, <lb/>ari&longs;tæ, & aliæ fe&longs;tucæ auolent; tùm dextror&longs;um &longs;ini&longs;tror&longs;umque libratur <lb/>vt leuior materia extet; triticum enim grauius e&longs;t; igitur deor&longs;um ten<lb/>dit; palea verò &longs;ur&longs;um; ideo verò attollitur, &longs;ub&longs;ultatque triticum in van<lb/>no, quia po&longs;t impre&longs;&longs;um impetum per vibrationem &longs;ur&longs;um, manus ip&longs;a <lb/>deor&longs;um cum aliquo impetu truditur, in quo non e&longs;t difficultas, alio <lb/>verò motu qua&longs;i recto repit frumentum in vanni aluo, quia per addu<lb/>ctionem vanni impul&longs;æ priùs &longs;ini&longs;tror&longs;um frumentum in eam partem <lb/>adhuc propter priorem impetum fertur; &longs;ic cum nauis illicò &longs;i&longs;tit in <lb/>potu, qui &longs;unt in ea & portum a&longs;piciunt, proni cadunt, de quo iam <lb/>&longs;uprà. </s>

<s>Sextò, remigatio fit pellendo, trahendoque, de qua iam &longs;uprà Th. </s>

<s>Sextò, remigatio fit pellendo, trahendoque, de qua iam &longs;uprà Th. 6. <lb/>16.longior & latior remus maiorem vim aquæ impellit; difficiliùs taman <lb/>mouetur, quò maior e&longs;t illius portio à centro motus ver&longs;us manum re<lb/>migantis, faciliùs mouetur propter rationem vectis; faciliùs mouetur, &longs;i <lb/>aduer&longs;o flumine feratur nauis: ratio e&longs;t, quia aqua pul&longs;a ver&longs;us eam <lb/>partem, in quam fluir minùs re&longs;i&longs;tit, quando eundem remum tractant, <pb pagenum="417"/>ille plus confert, qui ad extremiiatem propiùs accedit; ratio clara e&longs;t: <lb/>&longs;ed de re nautica aliàs; vide interim locum citatum. </s>

<s>6. <lb/>16.longior & latior remus maiorem vim aquæ impellit; difficiliùs taman <lb/>mouetur, quò maior e&longs;t illius portio à centro motus ver&longs;us manum re<lb/>migantis, faciliùs mouetur propter rationem vectis; faciliùs mouetur, &longs;i <lb/>aduer&longs;o flumine feratur nauis: ratio e&longs;t, quia aqua pul&longs;a ver&longs;us eam <lb/>partem, in quam fluir minùs re&longs;i&longs;tit, quando eundem remum tractant, <pb pagenum="417"/>ille plus confert, qui ad extremiiatem propiùs accedit; ratio clara e&longs;t: <lb/>&longs;ed de re nautica aliàs; vide interim locum citatum. </s>

<s>Septimò, tritus fit, cum ab impacto aliquo duriore corpore malleo, <lb/>v.g. </s>

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<s>igi<lb/>tur perinde &longs;e habet impetus, qui ine&longs;t puncto D, atque &longs;i incubaret ip&longs;i <lb/>D.DC, & I, IH, & G, GF, &c. </s>

<s>atqui &longs;i hoc e&longs;&longs;et, centrum grauitatis <lb/>e&longs;&longs;et in I, vt patet ex dictis; ibique e&longs;&longs;et percu&longs;&longs;ionis, per Th. </s>

<s>atqui &longs;i hoc e&longs;&longs;et, centrum grauitatis <lb/>e&longs;&longs;et in I, vt patet ex dictis; ibique e&longs;&longs;et percu&longs;&longs;ionis, per Th. 3. igitur <lb/>I e&longs;t centrum percu&longs;&longs;ionis. </s>

<s>3. igitur <lb/>I e&longs;t centrum percu&longs;&longs;ionis. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 10.<emph.end type="center"/></s>

<s><emph type="italics"/>Si BG diuidatur in tres partes æquales B, D, I, G, rotetur que circa CA, <lb/>vt dictum e&longs;t &longs;uprà, centrum percu&longs;&longs;ionis e&longs;t in I<emph.end type="italics"/>; quia &longs;i volueretur &longs;ola <lb/>AF, e&longs;&longs;et in E, &longs;i &longs;ola CH, e&longs;&longs;et in K, &longs;i &longs;ola BG, e&longs;&longs;et in I, per Th. 8. <lb/>igitur centra percu&longs;&longs;ionis omnium &longs;unt in linea EK; &longs;ed lineæ EK, cuius <lb/>&longs;ingula puncta mouentur æquali motu, centrum percu&longs;&longs;ionis e&longs;t in I, per <lb/>Th.1. igitur centrum percu&longs;&longs;ionis totius CF acti circum CA, e&longs;t in I, <lb/>quod erat demon&longs;tr. </s>

<s>8. <lb/>igitur centra percu&longs;&longs;ionis omnium &longs;unt in linea EK; &longs;ed lineæ EK, cuius <lb/>&longs;ingula puncta mouentur æquali motu, centrum percu&longs;&longs;ionis e&longs;t in I, per <lb/>Th.1. igitur centrum percu&longs;&longs;ionis totius CF acti circum CA, e&longs;t in I, <lb/>quod erat demon&longs;tr. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s>DBE, circa quam voluatur triangulum, du<lb/>cantur AE, CD perpendiculares AD; aliæ duæ ip&longs;is æquales AFCG, <lb/>perpendicularis in AC; tùm FG connectantur; eleueturque Trapezus <lb/>AG, donec AF, CG incubent perpendiculariter plano ABC; denique <lb/>à B ducantur rectæ ad omnia puncta Trapezi erecti, habebitur pyramis, <lb/>cuius centrum grauitatis, dabit centrum percu&longs;&longs;ionis quæ&longs;itum, per Th. </s>

<s><lb/>11. quod vt &longs;iat, inueniatur centrum grauitatis Trapezi AG, modo di<lb/>cto, ducta &longs;cilicet FC, a&longs;&longs;umptoque I centro grauitatis trianguli FGC <lb/>& L centro grauitatis trianguli FAC; &longs;i enim ducatur LI, &longs;itque LI <lb/>ad LP, vt Trapezium AG, ad triangulum FGC; certè P e&longs;t centrum <lb/>grauitatis Trapezij per p.7. tùm ex P erecto ducatur recta ad B, hæc erit <lb/>axis pyramidis; porrò &longs;i ducatur perpendicularis PO; tùm BO habebi-<pb pagenum="427"/>tur orthogonium POB; denique a&longs;&longs;umatur OR 1/4 totius OB, R erit <lb/>centrum percu&longs;&longs;ionis trianguli ACB per Th. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

<s>Hinc colligo quid dicendum &longs;it de rectangulo ita rotato, vt diagona<lb/>lis cadat perpendiculariter in axem, circa quem rotatur; &longs;it enim re<lb/>ctangulum CF, cuius diagonalis AIA, axis circa quem voluitur BR, in<lb/>neniantur centra percu&longs;&longs;ionis vtriu&longs;que trianguli &longs;eor&longs;im AFH, ACH, <lb/>rotati circa axem BR per Th. 16. connectantur rectâ, in hac erit cen<lb/>trum percu&longs;&longs;ionis totius rectanguli; cù di&longs;tantiæ à centro communi <lb/>&longs;int vt pyramides permutando per p.7. vt con&longs;tat ex dictis; ex quibus <lb/>etiam &longs;atis intelligetur quid de alijs planis, tùm regularibus, tùm irre<lb/>gularibus dicendum &longs;it, cù &longs;cilicet po&longs;&longs;int in triangula diuidi. </s>

<s>16. connectantur rectâ, in hac erit cen<lb/>trum percu&longs;&longs;ionis totius rectanguli; cù di&longs;tantiæ à centro communi <lb/>&longs;int vt pyramides permutando per p.7. vt con&longs;tat ex dictis; ex quibus <lb/>etiam &longs;atis intelligetur quid de alijs planis, tùm regularibus, tùm irre<lb/>gularibus dicendum &longs;it, cù &longs;cilicet po&longs;&longs;int in triangula diuidi. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 17.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 23.<emph.end type="center"/></s>

<s><emph type="italics"/>Si voluatur rectangulum parallelum orbi in quo voluitur determinari<emph.end type="italics"/> <emph type="italics"/>po<lb/>test centrum percu&longs;&longs;ionis<emph.end type="italics"/>; &longs;it enim rectangulum AD, quod voluatur circa <lb/>centrum A, co modo, quo dictum e&longs;t &longs;it ducta AD, inueniatur centrum <lb/>I, trianguli ABD; itemque centrum H, trianguli ADF, per Th. 17. <lb/>tùm ducta IH, diuidatur bifariam in K; ducatur AK, tùm GK perpen<lb/>dicularis in AK: dico G e&longs;&longs;e centrum percu&longs;&longs;ionis, per po&longs;.7.& Theo<lb/>rema 17. </s>

<s>17. <lb/>tùm ducta IH, diuidatur bifariam in K; ducatur AK, tùm GK perpen<lb/>dicularis in AK: dico G e&longs;&longs;e centrum percu&longs;&longs;ionis, per po&longs;.7.& Theo<lb/>rema 17. </s>

<s><emph type="center"/><emph type="italics"/>Corollarium.<emph.end type="italics"/><emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 24.<emph.end type="center"/></s>

<s><emph type="italics"/>Pote&longs;t determinari centrum percu&longs;&longs;ionis &longs;olidi<emph.end type="italics"/> <emph type="italics"/>trium facierum ABDE<emph.end type="italics"/>; <lb/>vt demon&longs;trctur centrum percu&longs;&longs;ionis pyramidis, & pri&longs;matis, præmitti <lb/>debuit hoc &longs;olidum; &longs;it enim &longs;olidum priori &longs;imile, A.M. G.C. motus <lb/>puncti M, e&longs;t ad motum puncti G, vt recta BM ad rectam BG; igitur &longs;it <lb/>NK ad OH, vt BM ad BG; certè perinde &longs;e habet punctum M, atque <lb/>&longs;i NMK incubaret, non quidem per MG, &longs;ed per lineam perpendicu<lb/>larem ductam in BM, vt patet ex dictis: idem dico de puncto G, quod <lb/>perinde &longs;e habet, atque &longs;i incubaret OGH; itaque inuenire oporter <lb/>centrum grauitatis &longs;olidi ACHKNOA, quod vt fiat, a&longs;&longs;umatur IP <pb pagenum="432"/>æqualis AC; ducantur AP, CI centrum grauitatis &longs;olidi ACIKNP <lb/>re&longs;pondet per lineam directionis puncto E, ita vt EG &longs;it 1/3 GB per Co<lb/>roll.1. Th.3.&longs;i autem a&longs;&longs;umatur FG 1/4 totius BG, &longs;itque linea QFX, <lb/>& ex puncto F &longs;u&longs;tineatur vtraque pyramis AOPN, & CIHK, erit <lb/>perfectum æquilibrium per Th. 4. igitur &longs;it FE ad ED, vt &longs;olidum <lb/>ACHKNO ad vtramque pyramidem AOPN, CIHK, certè pun<lb/>ctum D erit centrum grauitatis &longs;olidi ACHKNO, per p.7. a&longs;&longs;umatur <lb/>GL æqualis GD; ducatur BL, hæc e&longs;t axis vt patet, modò GM &longs;it æqua<lb/>lis GB; &longs;i enim inæqualis e&longs;t, &longs;it GL ad GM, vt GD ad GB: præterea <lb/>ducatur DR parallela GM; denique ducatur perpendicularis FR in B <lb/>L; dico F e&longs;&longs;e centrum percu&longs;&longs;ionis, vt patet ex dictis &longs;uprà, præ&longs;ertim in <lb/>Th. 17. & alibi pa&longs;&longs;im, ne toties cadem repetere cogar ad nau&longs;eam; <lb/>quamquam enim hæc &longs;atis noua &longs;unt, illa tamen indicanda potiùs, quàm <lb/>fusè tractanda e&longs;&longs;e putaui. </s>

<s>4. igitur &longs;it FE ad ED, vt &longs;olidum <lb/>ACHKNO ad vtramque pyramidem AOPN, CIHK, certè pun<lb/>ctum D erit centrum grauitatis &longs;olidi ACHKNO, per p.7. a&longs;&longs;umatur <lb/>GL æqualis GD; ducatur BL, hæc e&longs;t axis vt patet, modò GM &longs;it æqua<lb/>lis GB; &longs;i enim inæqualis e&longs;t, &longs;it GL ad GM, vt GD ad GB: præterea <lb/>ducatur DR parallela GM; denique ducatur perpendicularis FR in B <lb/>L; dico F e&longs;&longs;e centrum percu&longs;&longs;ionis, vt patet ex dictis &longs;uprà, præ&longs;ertim in <lb/>Th. </s>

<s>17. & alibi pa&longs;&longs;im, ne toties cadem repetere cogar ad nau&longs;eam; <lb/>quamquam enim hæc &longs;atis noua &longs;unt, illa tamen indicanda potiùs, quàm <lb/>fusè tractanda e&longs;&longs;e putaui. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 25.<emph.end type="center"/></s>

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<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 32.<emph.end type="center"/></s>

<s><emph type="italics"/>Si voluatur planum triangulare circa angulum, eo modo quo diximus i<gap/><lb/>Th.<emph.end type="italics"/>11. <emph type="italics"/>&longs;unependulum i&longs;ochronum coniinet<emph.end type="italics"/> 3/4 <emph type="italics"/>axis prædicli irimguli<emph.end type="italics"/>; quia in <lb/>1/4 e&longs;t centrum percu&longs;&longs;ionis per Th. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 33.<emph.end type="center"/></s>

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<s>Colligo primò, cuilibet &longs;ectori funependulum i&longs;ochronum po&longs;&longs;e a&longs;&longs;i<lb/>gnari, quia cuiu&longs;libet &longs;ectoris, qui voluitur circa angulum, eo modo <lb/>quo diximus Th.13. centrum percu&longs;&longs;ionis determinatum e&longs;t. </s>

<s>Colligo &longs;ecundò, &longs;i rotetur planum circulare, eo modo quo diximus <lb/>Th.21. funependuli i&longs;ochroni longitudinem continere 2/3 diametri eiu&longs;<lb/>dem circuli, quia ibi e&longs;t centrum percu&longs;&longs;ionis eiu&longs;dem circuli, per <lb/>Th. </s>

<s>Colligo tertiò, &longs;i rotetur planum circulare circa diametrum, etiam <lb/>po&longs;&longs;e determinari ex centro percu&longs;&longs;ionis inuento, longitudinem fune<lb/>penduli i&longs;ochroni, vt patet ex dictis. </s>

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<s>quod certè ad veritatem tam pro<lb/>pè accedit ex geometrica calculatione, vt nullum pior&longs;us di&longs;crimen <pb pagenum="437"/>e&longs;&longs;e videatur, methodus huius calculationis facilis e&longs;t, & à mediocri <lb/>Logi&longs;ta haberi pote&longs;t. </s>

<s>Sextò, hinc etiam habetur longitudo funependuli i&longs;ochroni, &longs;i vol<lb/>uatur planum circulare parallelum plano, in quo voluitur, continet <lb/>enim 2/3 diametri circuli, qui voluitur; vt patet ex Th. </s>

<s>22. idem dico de <lb/>quolibet &longs;ectore, qui eodem modo voluatur. </s>

<s><emph type="center"/><emph type="italics"/>Theorema<emph.end type="italics"/> 35.<emph.end type="center"/></s>

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