PHYSICVS
DE MOTV LOCALI,
Motum naturalem, violentum, & mixtum pertinent,
explicantur, & ex principiis Phy&longs;icis
demon&longs;trantur.
CVNCTA EXCERPTA
Apud IOANNEM CHAMPION,
in foro Cambij.
D C. XLVI.
Cum Priuilegio Regis, & Approbatione Doctorum.
NOBILISSIMOQVE DOMINO,
DOMINO DE FLECHERES,
SANCTIORIS CONSILII REGIS
Con&longs;iliario, in Lugdunen&longs;i Curia Prætori prima
rio, & &longs;ecundùm Mercatorum Præpo&longs;ito, &c.
phiæ fœtum in&longs;cribo, cui iam
primum in&longs;crip&longs;i
AMPLISSIME)
e&longs;&longs;e debeo, quia tu &longs;emper idem
es: non muta&longs;ti merita, non mu
tabo officia: multos non expo&longs;cam Patronos, qui
iam omnium optimum, & meriti&longs;simum habeo; neo
enim &longs;acra Philo&longs;ophiæ anathemata rudi, & ru
&longs;tico muro appendam, quæ ex &longs;acro tholo templi
Themidos amœniter pendent: Nec leuem toti rei li
terariæ iniuriam inferrem, &longs;i alium illi, quàm li-& verò Tracta
tum hunc de Motu Locali, alteri quàm tibi in&longs;cri
bere non debui, cuius imperia Ludgunen&longs;is orbis, po
tiùs quàm vrbis, componunt: Tu prudens Intelli
gentia, huic orbi &longs;emper a&longs;si&longs;tis; ita motibus in
uigilas, vt quieti publicæ con&longs;ulas, remque ita pu
blicam admini&longs;tras, vt &longs;ingulis commoda procures:
Cæterùm dubitare non po&longs;&longs;um, quin hunc meu&mtail;
quantulumcumque conatum, fidemque meam ia&mtail;
tibi &longs;emel oppigneratam, & nunc altero voto peni
tus ob&longs;trictam, æqui bonique &longs;is con&longs;ulturus, Val&etail;.
NIHIL habeo præfari (Beneuole Lector)
in gratiam huius tractatus de Motu Locali,
cuius amœnitatem & vtilitatem, rerum co
piam & &longs;yluam, tuo gu&longs;tui & iudicio re
linquo: Multi &longs;anè hactenus in hac mate
ria feliciter de&longs;udarunt; & quidem præ cæteris magnus
ille Galileus, qui mirificâ, & ferè diuinâ ingenij acie,
motum localem eò perduxit, quò mortalium nemo per
duxerat; quia tamen multa omi&longs;it, quæ ad motum &longs;pe
ctant, vt nemo ne&longs;cit; nec ex principijs Phy&longs;icis mira
biles illos effectus demon&longs;trauit, &longs;ed tantùm certis qui
bu&longs;dam proportionibus ex geometricis addixit; vt Phy
&longs;icæ con&longs;ulamus, aliam inimus viam: Geometriam qui
dem adhibemus, ad explicandas, exponenda&longs;que præ
dictas illas proportiones, quæ motibus in&longs;unt; &longs;ed effe
ctus illos prædictis proportionibus affixos ad principia
Phy&longs;ica reducimus; id e&longs;t, cùm &longs;upponamus quòd &longs;int,
propter quid &longs;int demon&longs;tramus: in votis erat motus
omnes vno volumine complecti; id e&longs;t effectus omnes
cuiu&longs;uis potentiæ motricis; tres enim agno&longs;cimus hu
iu&longs;modi potentias: primam naturalem voco, quæ e&longs;t
grauium: alteram animalem, quæ e&longs;t animantium:
ter
tiam mediam, quæ ten&longs;orum e&longs;t vel compre&longs;&longs;orum: In
hoc tractatu tùm à motu progre&longs;&longs;iuo animantium, tùm
ab alijs motibus, qui in animato corpore, neruorum & cùm &longs;ci
licèt eas notiones &longs;upponant, quæ huius loci e&longs;&longs;e non
po&longs;&longs;unt, ab&longs;tinemus etiam à mirifica illa ten&longs;orum &
compre&longs;&longs;orum vi, quæ mediæ illius virtutis e&longs;t; neque
adhuc eò rem Phy&longs;icam adduximus; Sed hîc tantùm na
turam impetus con&longs;ideramus, motus naturalis affectio
nes, violenti, mixti ex rectis, reflexi, circularis, mixti
ex circularibus, illius qui fit in planis inclinatis &longs;ur&longs;um
& deor&longs;um, vibrationum funependuli, diuer&longs;arum im
pre&longs;&longs;ionum, centri percu&longs;&longs;ionis, &c. Fortè aliquis poten
tias mechanicas de&longs;ideraret, lineas, motus, & cæle&longs;tes
&longs;piras; &longs;ed hæ quidquid phy&longs;icum habent, &longs;ingulari tra
ctatui de corpore cæle&longs;ti, reliqua verò A&longs;tronomiæ con
cedunt: potentiæ mechanicæ ad Staticam pertinent, qua
re illarum tantùm phy&longs;icum principium in hoc tractatu
explicamus, lineæ motus nihil phy&longs;icum habent. Quare
ad vitandam confu&longs;ionem ad Mathe&longs;im illas remittimus,
cuius non modicam facient acce&longs;&longs;ionem; igitur &longs;ecun
dum Tomum de motu locali non expectabis, qui ne
cuncta quidem, quæ ad motum &longs;pectant comprehende
ret, &longs;ed huic &longs;tatim Metaphy&longs;icam demon&longs;tratiuam &longs;ub
necto. Cæterùm de &longs;ubtili&longs;&longs;imo i&longs;torum omnium inuen
torum auctore nihil dicam, qui cum ægrè tulerit paucula
illa quæ in prima tractatu præfatus &longs;um, os mihi peni
tus ob&longs;truxit: omitto etiam quæ in me quidam iniquè
certè rerum æ&longs;timatores iactarunt: reponere po&longs;&longs;em cum
fænore; &longs;ed nos talem con&longs;uetudinem non habemus;
de
dici hactenus pati iniurias, non inferre; quod non modò
moralis Philo&longs;ophia, &longs;ed præ&longs;ertim Chri&longs;tiana Religio me
docet.
Vnum e&longs;t, de quo te monitum velim (Amice Lector)
opu&longs;culum i&longs;tud non &longs;ine aliquot erratis edi potui&longs;&longs;e,
præ&longs;ertim cùm in a&longs;&longs;ignandis cuilibet figuræ &longs;uis chara
cteribus &longs;æpiùs peccatum &longs;it; operas excu&longs;abis in rebus
Geometricis minimè ver&longs;atos: auctor tibi &longs;um, vt errata,
quæ fideliter adnotaui ca&longs;tiges, vt deinde cum maiore
gu&longs;tu Librum hunc perlegere po&longs;&longs;is.
TABELLE WAR HIER
BREVISSIMAM huius operis Epitomem hîc
habes (Amice Lector) quam ex The&longs;ibus no&longs;tri
Philo&longs;ophi huc traduxi, quæ tibi ampli&longs;&longs;imi
indicis loco erit.
1. IMPETVS e&longs;t qualitas exigens motum &longs;ui &longs;ubiecti:
datur impetus; quia non pote&longs;t e&longs;&longs;e alia cau&longs;a exi
gitiua motus: adde quòd, potentia motrix e&longs;t acti
ua; igitur aliquid producit, &longs;ed non aliud quàm
impetum, vt con&longs;tat ex dictis de motu: e&longs;t aliquid di&longs;tinctum à
&longs;ub&longs;tantia mobilis, quæ pote&longs;t e&longs;&longs;e &longs;ine impetu: non e&longs;t modus,
quia di&longs;tinguitur ab effectu &longs;uo formali &longs;ecundario: impetus non
producitur in eo mobili, quod moueri non pote&longs;t à potentia mo
trice applicata: & produci tantùm pote&longs;t, vel in omni parte, vel
in nulla; alioquin e&longs;&longs;et fru&longs;trà; & gratis ponitur ne&longs;cio quis impe
tus inefficax.
2. Primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, ne &longs;imul im
petus &longs;it in duobus locis. Impetus productus ad extra non produci
tur à quantitate, nec virtute re&longs;i&longs;titiua, nec ab alio, quàm ab impe
tu, qui maximè e&longs;t cau&longs;a connaturalis alterius impetus: agit tan
tùm ad extra, vt tollat impedimentum: hinc, cùm pro diuer&longs;a
applicatione &longs;it diuer&longs;um impedimentum, modò plùs, modò minùs
agit; maximè verò, cum maximum e&longs;t impedimentum:
hinc ictus
per lineam perpendicularem forti&longs;&longs;imus e&longs;t: portò omnes partes
impetus agunt ad extra actione communi.
3. Impetus inten&longs;us producere pote&longs;t remi&longs;&longs;um, minoris mobi
lis in maiore; & remi&longs;&longs;us inten&longs;um, maioris mobilis in minore, vt
patet; æqualis æqualem, æqualis mobilis in æquali, modò &longs;it debi-
cùm linea directionis connectit centra grauitatis vtriu&longs;que. Datur
impetus alio impetu perfectior, & imperfectior, &longs;ine quo non po
te&longs;t explicari natura vectis: itaque dato quocunque dari pote&longs;t per
fectior, & imperfectior: quia dato quocunque motu pote&longs;t dari ve
locior, & tardior.
4. Propagatur impetus vniformiter tantùm, cùm omnes partes
corporis mouentur motu recto æquali: ibi enim e&longs;t æqualis cau&longs;a,
vbi e&longs;t æqualis effectus: in motu circulari applicata potentia cen
tro vectis, producitur æqualis perfectionis versùs circunferentiam,
& inæqualis numerus; applicata verò potentia circunferentiæ, pro
ducitur æqualis numerus, &longs;ed inæqualis perfectionis versùs cen
trum; quia potentia non pote&longs;t producere immediatè perfectiorem,
& imperfectiorem in infinitum: eadem potentia nece&longs;&longs;aria æquali
bus temporibus, & ii&longs;dem circun&longs;tantiis, producit æqualem impe
tum, & inæqualibus inæqualem: e&longs;t enim hæc ratio cau&longs;æ nece&longs;
&longs;ariæ.
5. Impetus innatus e&longs;t tantùm determinatus ad lineam perpen
dicularem deor&longs;um; alioquin &longs;i ad aliam determinari po&longs;&longs;et, primo
e&longs;&longs;et æqualis motus per inclinatam, & perpendicularem; corpus
graue mi&longs;&longs;um per lineam inclinatam ab eo non declinaret; imò im
petus &longs;emel productus (&longs;i liberum e&longs;&longs;et medium) non de&longs;trueretur:
quæ omnia phy&longs;icis hypothe&longs;ibus repugnant: omnis alius impetus,
etiam acqui&longs;itus motu naturali deor&longs;um, e&longs;t indifferens ad omnem
lineam, ad vitanda infinita ferè naturæ incommoda.
6. Impetus indifferens determinatur ad lineam multis modis:
primò, à potentia motrice: &longs;ecundò, ab impetu:
tertiò, ab alio impe
tu concurrente; quartò, ab obice occurrente: quintò, ab ip&longs;o appli
cationis diuer&longs;o modo: quæ omnia clara &longs;unt: hinc duo impetus ad
motum mixtum &longs;æpè concurrunt, quod &longs;emper fit, ni&longs;i determina
tiones &longs;int oppo&longs;itæ ex diametro. Impetus e&longs;t capax inten&longs;ionis;
quia aliquando de&longs;truitur ex parte: eius exten&longs;io commen&longs;uratur
exten&longs;ioni mobilis; quod etiam cæteris qualitatibus commune e&longs;t:
impetus productus non con&longs;eruatur à cau&longs;a primò productiua, à
qua etiam &longs;eparatus exi&longs;tit.
7. Impetus non e&longs;t contrarius alteri ratione entitatis;
quia qui
libet cum quolibet in eodem &longs;ubiecto coëxi&longs;tere pote&longs;t: pugnat
tamen vnus cum alio ratione determinationis: hinc vnus impetus
pugnat cum alio ratione lineæ motus: hinc vnus videtur de&longs;trui ab
quanquam impetus tantùm de&longs;truitur, cùm e&longs;t fru&longs;trà:
hinc, &longs;i
e&longs;&longs;et tantùm vnicus in eodem mobili, & liberum e&longs;&longs;et medium,
nunquam de&longs;trueretur nec vnquam dici po&longs;&longs;et functus &longs;uo mune
re; quod omninò gratis dicitur.
8. Hinc, &longs;i &longs;int tantùm duo impetus in eodem mobili æquales
verbi gratia, vel ad eandem lineam determinantur, vel ad diver&longs;as;
&longs;i ad eandem, nihil impetus de&longs;truitur, &longs;ed e&longs;t duplò velocior mo
tus; &longs;i ad diuer&longs;as, vel &longs;unt oppo&longs;itæ ex diametro, vel concurrentes
faciunt angulum; &longs;i primum, vterque de&longs;truitur impetus; &longs;i &longs;e
cundum, de&longs;truitur aliquid illius, quod determinabimus in
frà. Impetus innatus nunquam de&longs;truitur:
dici po&longs;&longs;et grauitas ab
&longs;oluta; &longs;altem nihil e&longs;t, quod di&longs;tingui ab illa probare po&longs;&longs;it. Porrò
nunquam de&longs;truitur; quia nunquam e&longs;t fru&longs;trà; quippe eius finis,
vel v&longs;us, non e&longs;t tantùm motus deor&longs;um, &longs;ed grauitatio, &longs;eu ni&longs;us
quidam deor&longs;um. Sed de grauitate aliàs.
1. DAtur motus naturalis grauium deor&longs;um ab intrin&longs;eco,
quippe non pote&longs;t e&longs;&longs;e, vel à vi tractrice terræ vel fila
mentis quibu&longs;dam, vel materia quadam tenui expultrice. Eius finis
e&longs;t globi terre&longs;tris compactio, &c. E&longs;t autem motus naturalis ab
impetu: primò, quia eius acceleratio &longs;ine impetu explicari non po
te&longs;t: &longs;ecundò, quia, cùm graue deor&longs;um cadens imprimat impetum
in corpore occurrente, certè debet habere impetum: nec alio ar
gumento mihi probabis, Solem e&longs;&longs;e lucidum, ignem calidum.
2. Motus hic e&longs;t naturaliter acceleratus, &longs;cilicet, ab intrin&longs;eco;
patet experientiâ. Ratio e&longs;t:
quia, cùm in libero medio non impe
diatur motus, & impetus productus primo in&longs;tanti non con&longs;erue
tur &longs;ecundo à cau&longs;a primò productiua, &longs;ed ab alia, &longs;itque ip&longs;a mo
bilis &longs;ub&longs;tantia cau&longs;a nece&longs;&longs;aria; certè &longs;ecundo in&longs;tanti producit
nouum impetum: idem dica de tertio, quarto, &c. igitur cre&longs;cit
cau&longs;a motus; igitur & motus: quæ ratio clari&longs;&longs;ima e&longs;t:
hinc æquali
bus temporibus æqualia acquiruntur velocitatis momenta; quia
cau&longs;a nece&longs;&longs;aria æqualibus temporibus, æqualem effectum produ
cit: quid clarius?
3. Hinc non pote&longs;t cre&longs;cere hic impetus &longs;ecundùm porportio-
temporum, etïam ex mente Galilei: cre&longs;cit autem velocitas, vt im
petus; effectus, &longs;cilicet, vt cau&longs;a:
idem dico de motu, ratione velo
citatis; quippe motus ip&longs;e e&longs;t &longs;ua velocitas: at verò ip&longs;a &longs;patia,
quæ decurruntur illo motu, &longs;i con&longs;ideretur crementum in in&longs;tan
tibus, cre&longs;cunt iuxta progre&longs;&longs;ionem arithmeticam &longs;implicem,
id e&longs;t, &longs;i primo in&longs;tanti, acquiritur vnum &longs;patium, &longs;ecundo acquiri
tur vnum &longs;patium, &longs;ecundo acquiruntur duo, tertio 3. quarto 4. at
que ita deinceps.
4. Hoc autem facilè pote&longs;t
quia, cùm velocitas cre&longs;
cat iuxta proportionem temporum, &longs;i primo in&longs;tanti &longs;it vnus gradus
velocitatis, &longs;ecundo erunt duo, tertio tres, at que ita deinceps: igitur,
&longs;i mobile cum vno gradu velocitatis acquirit vnum &longs;patium, certè
cum duobus acquiret duo &longs;patia, cum tribus tria, atque ita dein
ceps: debet autem vera progre&longs;&longs;io crementorum a&longs;&longs;umi in &longs;ingulis
in&longs;tantibus, quia reuerà &longs;ingulis in&longs;tantibus phy&longs;icis (nam de iis
loquor) noua fit huius crementi acce&longs;&longs;io.
5. Quia tamen in&longs;tantia non &longs;unt &longs;en&longs;ibilia, vt Phy&longs;icæ con&longs;u
latur, quæ res &longs;en&longs;ibiles con&longs;iderat, a&longs;&longs;umi debent partes temporis
&longs;en&longs;ibiles, in quibus reuerâ progre&longs;&longs;io &longs;patiorum non e&longs;t arithmeti
ca &longs;implex; &longs;ed tam propè accedit ad hanc numerorum imparium,
1. 3. 5. 7. &c. quam Galileus excogitauit, vt &longs;ine &longs;crupulo hæc a&longs;
&longs;umi po&longs;&longs;it: hinc &longs;patia &longs;unt ferè vt temporum quadrata: dixi, ferè:
nam e&longs;t paulò minor proportio, cùm tantùm finita &longs;int in&longs;tantia
phy&longs;ica, quæ reuerà &longs;i infinita e&longs;&longs;ent in qualibet temporis &longs;en&longs;ibilis
parte, haud dubiè &longs;patia e&longs;&longs;ent omninò in ratione duplicata tem
porum: &longs;ed, quia parum pro nihilo computatur, hanc progre&longs;&longs;io
nem Galilei deinceps v&longs;urpabimus in Phy&longs;ica.
6. Hinc ratio euidens maioris ictus inflicti à corpore graui,
cùm ex maiori altitudine cadit. Sunt autem ictus, vt impetus;
impetus, vt tempora; hæc demum, vt radices &longs;patiorum &longs;en&longs;ibi
liter quæ omnia con&longs;tant ex dictis. Impetus acqui&longs;itus in de&longs;cen&longs;u
e&longs;t &longs;emper imperfectior, &longs;i a&longs;&longs;umantur &longs;ingula in&longs;tantia, quæ reuerâ
&longs;unt &longs;emper minora; quia motus fit &longs;emper velocior: cùm graue
de&longs;cendit in medio, quod re&longs;i&longs;tit, minùs accuratè &longs;eruantur prædi
ctæ proportiones, quæ in vacuo modico accurati&longs;&longs;imè &longs;eruaren
tur.
7. Re&longs;i&longs;tentia medij non e&longs;t propter vllam formam improportio
natam, qua&longs;i verò impetus &longs;it forma improportionata aëri: &longs;ed in
primò, eò quòd medium detrahat ali
quid grauitationis corporis grauis; &longs;ecundò, eò quòd partes medij
aliquam implicationem habeant, quæ &longs;olui non pote&longs;t &longs;ine aliqua
compre&longs;&longs;ione, vel ten&longs;ione; vtraque autem re&longs;i&longs;tit impetui: quod
&longs;pectat ad primum, &longs;i medium &longs;it æqualis grauitatis cum ip&longs;o cor
pore, detrahitur tota grauitatio, &longs;i &longs;ubduplæ &longs;ubduplum, &c. de quo
aliàs.
8. Hinc corpus graue per medium rarius, cæteris paribus, fa
cilè de&longs;cendit; non tamen ex re&longs;i&longs;tentia medij cognita, pote&longs;t co
gno&longs;ci proportio grauitatis vtriu&longs;que, propter &longs;ecundum caput, ex
quo etiam petitur re&longs;i&longs;tentia. Idem corpus cum eodem medio
comparatum, habet tres coniugationes: nam, vel e&longs;t grauius, vel
e&longs;t grauius, vel æquè graue, vel minùs. Sunt etiam tres aliæ con
iugationes, &longs;cilicet, eiu&longs;dem mobilis cum diuer&longs;is mediis, duorum
mobilium cum eodem medio, duorum mobilium cum duobus
mediis.
9. Figura corporis grauis deor&longs;um cadentis motum vel retardat
vel accelerat; retardat quidem, &longs;i plures partes medij amouendæ
&longs;unt vel pauciores velociori motu; accelerat è contrario: hinc idem
corpus
&longs;o de&longs;cendere pote&longs;t: hinc ratio, cur acuminata tam facilè de&longs;cen
dant. Cubus, qui de&longs;cendit, imprimit aëri velociorem motum,
quàm ip&longs;e habeat; & quò maior e&longs;t eius &longs;uperficies, eò velociorem.
10. Duo globi, vel cubi eiu&longs;dem materiæ æquè velociter de&longs;
cendunt: ratio e&longs;t, quia, licèt maioris vires habeant maiorem pro
portionem ad molem aëris re&longs;i&longs;tentis, quàm vires minoris ad alte
ram aëris molem, quæ proprium illius motum retardat, cùm tamen
aër, qui re&longs;i&longs;tit maiori cubo, debeat amoueri velociori motu, quàm
aër, qui re&longs;i&longs;tit minori, &longs;itque eadem proportio re&longs;i&longs;tentiæ ratione
motus, minoris ad maiorem, quæ e&longs;t ratione molis, maioris ad mi
norem; certè ratio compo&longs;ita vtriu&longs;que erit eadem in vtroque cu
bo: igitur æqualiter de&longs;cendet vterque.
11. Si tamen &longs;int diuer&longs;æ materiæ, haud dubiè, qui con&longs;tat leuio
ri materia, tardiùs de&longs;cendet; quia eius vires habent minorem
proportionem ad re&longs;i&longs;tentiam. Corpu&longs;cula etiam ex graui&longs;&longs;ima ma
teria tardi&longs;&longs;imè de&longs;cendunt: tum, quia à filamentis illis, quibus par
tes aëris implicantur, facilè detinentur; analogiam habes in lapil
lo, qui ab araneæ tela intercipitur: tum, quia, cùm lati&longs;&longs;imam ali
quando habeant &longs;uperficiem pro modica mole, minimam habent
impetu agitari po&longs;&longs;int ab aëre mobili, vnus motus alium impedit.
12. Singulis in&longs;tantibus motus naturaliter accelerati cre&longs;cit
re&longs;i&longs;tentia; quia, cùm motus cre&longs;cat, æqualibus temporibus, plures
partes medij occurrunt; cre&longs;cunt tamen vires in eadem proportio
ne, &longs;cilicet, impetus: igitur non mutatur progre&longs;&longs;io motus. Hinc
colligo, contra Galilæum, motum rectum ex naturaliter accelerato
nunquam fieri æquabilem: dixi motum rectum; quia motus corpo
rum cœle&longs;tium ex accelerato factus e&longs;t æqualis.
1. MOtus violentus &longs;ur&longs;um vulgò dicitur e&longs;&longs;e à principio ex
trin&longs;eco. Triplici modo accidere pote&longs;t:
primò, &longs;i reuerà
imprimatur impetus ab extrin&longs;eco, vt, cùm mitto lapidem &longs;ur&longs;um:
&longs;ecundò, &longs;i corpus deor&longs;um cadens deinde reflectatur &longs;ur&longs;um; tunc
autem nihil e&longs;t ab extrin&longs;eco, ni&longs;i determinatio noua, quæ e&longs;t à cor
pore reflectente: tertiò, &longs;i terra vtrinque e&longs;&longs;et peruia;
nam lapis haud
dubiè non &longs;i&longs;teret in centro, &longs;altem po&longs;t primum de&longs;cen&longs;um; igitur
a&longs;cenderet per eandem lineam; nullum tamen e&longs;t principium ex
trin&longs;ecum; igitur motus violentus dicit tantùm motum &longs;ur&longs;um
corporis grauis.
2. Dari autem motum violentum, dubium e&longs;&longs;e non pote&longs;t, qui
&longs;upponit impetum, vel impre&longs;&longs;um ab extrin&longs;eco, vel in de&longs;cen&longs;u
acqui&longs;itum, qui reuerâ ine&longs;t ip&longs;i mobili, cùm ip&longs;um medium hunc
motum potiùs impediat, quàm iuuet: hinc, &longs;i nullus e&longs;&longs;et impetus
extrin&longs;ecus, vel acqui&longs;itus, nullus e&longs;&longs;et motus violentus; quia im
petus innatus illius cau&longs;a e&longs;&longs;e non pote&longs;t. Portò hic motus non e&longs;t
acceleratus, nec æqualis, alioquin
3. Hinc nece&longs;&longs;ariò e&longs;t retardatus:
igitur de&longs;truitur impetus, non
quidem ab ip&longs;a medij re&longs;i&longs;tentia; quippe idem medium non magis
re&longs;i&longs;tit motui &longs;ur&longs;um, quàm motui deor&longs;um, vt patet: igitur de&longs;trui
tur ille impetus motus violenti ab impetu innato aliquo modo; non
quidem vt à contrario ratione entitatis, &longs;ed ratione determinatio
nis: cùm enim impetus innatus exigat motum deor&longs;um, & alius &longs;ur
&longs;um: hic quidem præualet, attamen fru&longs;trà e&longs;t, ratione gradus
æqualis impetui innato: igitur de&longs;truitur ille gradus illo in&longs;tanti.
4. Hinc &longs;ingulis temporibus æqualibus de&longs;truitur gradus impe
tui innato; e&longs;t enim eadem ratio pro omnibus:
igitur temporibus
æqualibus de&longs;truitur æqualis impetus: igitur amittit ille motus
æqualia velocitatis momenta: igitur e&longs;t naturaliter retardatus:
igi
tur iuxta eam proportionem decre&longs;cit motus violentus, iuxtaquam
cre&longs;cit naturalis: igitur dici debent de hac progre&longs;&longs;ione retardatio
nis, quæ dicta &longs;unt de illa progre&longs;&longs;ione accelerationis.
5. Hinc impetus imperfectior initio de&longs;truitur:
quia, cùm motus
ille &longs;it velocior initio, in&longs;tantia &longs;unt minora: atqui minori tempore
minùs retardatur: igitur inperfectior impetus de&longs;truitur;
cùm è
contrario in motu acceleratio initio acquiratur imperfectior, quia
in&longs;tantia &longs;unt maiora: vnde vides, gradus impetus e&longs;&longs;e heteroge
neos, & principium illud etiam in impetu valere, &longs;cilicet, &longs;ubiectum
ita compleri ab vna forma, vt alterius homogeneæ non &longs;it ampliùs
capax, &longs;altem naturaliter.
6. Hinc vltimus gradus impetus violenti e&longs;t omnium perfecti&longs;
&longs;imus, vt con&longs;tat. Quie&longs;ceret vno in&longs;tanti mobile iactum &longs;ur&longs;um, &longs;i
gradus vltimus violenti e&longs;&longs;et æqualis perfectionis, cum impetu in
nato: vbi enim ventum e&longs;&longs;et ad in&longs;tans æqualitatis, neutrum præ
ualere po&longs;&longs;et: igitur in&longs;tanti &longs;equenti e&longs;&longs;et quies:
cùm tamen &longs;int
diuer&longs;æ perfectionis, perfectior præualet: vter autem &longs;it perfectior,
dicemus infrà.
7. Cum mobile &longs;ur&longs;um reflectitur, vel terra perforata &longs;uam lineam
motus &longs;ur&longs;um versus oppo&longs;itam cœli plagam promouet, vel aliud
æqualis ponderis, vel maioris, &longs;ur&longs;um mouet, tunc certum e&longs;t, inna
tum e&longs;&longs;e perfectiorem: &longs;i verò imprimitur ab alia potentia motrice,
tunc etiam imperfectior e&longs;t impetu innato; nam inæqualis e&longs;t;
alio
quin, &longs;i e&longs;&longs;et æqualis, &longs;imul e&longs;&longs;ent in eodem &longs;ubiecto duo gradus
homogenei: præ&longs;tat autem e&longs;&longs;e imperfectiorem, quàm perfectio
rem, vt plura impetus puncta à potentia imprimantur; quòd mul
tum facit ad mouenda maiora pondera: hinc nullo in&longs;tanti quie&longs;
cunt proiecta &longs;ur&longs;um.
8. Tandiu durat &longs;en&longs;ibiliter de&longs;cen&longs;us globi proiecti &longs;ur&longs;um,
quandiu durauit a&longs;cen&longs;us; e&longs;t enim eadem ratio: &longs;agittæ verò mi
nùs durat a&longs;cen&longs;us, quàm de&longs;cen&longs;us propter mixtionem materiæ.
Si motus violentus e&longs;&longs;et æquabilis, percurreret proiectum &longs;patium
ferè duplum eo tempore, quo retardato percurrit &longs;ubduplum: hinc
&longs;onus tam citò auditur; quia propagatur cum particulis aëris æqua
bili ferè motu: e&longs;&longs;e autem &longs;patium ferè duplum, probatur ex eo,
de
cur&longs;um verò motu retardato, re&longs;pondet triangulo, &longs;ubduplo rectan
guli: a&longs;&longs;umpto &longs;cilicet, æquali tempore.
9. Vites potentiæ proiicientis toto ni&longs;u re&longs;pondent velocitati
acqui&longs;itæ in toto de&longs;cen&longs;u corporis proiecti;
impetus in de&longs;cen&longs;u acquiritur, quantùm in a&longs;cen&longs;u deperditur.
Impetus primo in&longs;tanti, quo e&longs;t, agit, &longs;i e&longs;t aliquod impedimen
tum; e&longs;t enim cau&longs;a nece&longs;&longs;aria:
primo in&longs;tanti motus aliquid im
petus de&longs;truitur: &longs;iue præce&longs;&longs;erit motus violentus, &longs;iue non præce&longs;
&longs;erit, corpus graue æquali motu deor&longs;um cadit: re&longs;i&longs;tentia aëris e&longs;t
quidem maior initio; &longs;ed etiam &longs;unt maiores vires.
1. PLanum inclinatum e&longs;t &longs;ur&longs;um, vel deor&longs;um:
in hoc de&longs;cen
dit corpus graue, ni&longs;i fortè retineatur ab a&longs;peritate, vel pro
pria, vel ip&longs;ius plani: impeditur autem motus naturalis in plano
prædicto, quia impeditur eius linea: ideò e&longs;t tardior hic motus in
plano inclinato, quàm in perpendiculari: in ea porrò proportione
e&longs;t tardior, in qua perpendiculum e&longs;t minus linea inclinata, eiu&longs;dem
&longs;cilicet, altitudinis; quippe eò tardior e&longs;t, quò magis impeditur, &
magis impeditur, quò maius &longs;patium decurrendum e&longs;t, ad acqui
rendam eandem altitudinem: igitur eadem e&longs;t proportio impe
dimenti, quæ &longs;patij, &c.
2. Hinc motus &longs;unt vt lineæ permutando:
hinc mobile de&longs;cendit
per &longs;e in prædicto plano: licet enim motus impediatur, non tamen
qui&longs;ito in perpendiculari in eadem proportione; nam impetus &longs;unt
vt motus: hinc pote&longs;t perfectio impetus imminui in infinitum, cùm
po&longs;&longs;it e&longs;&longs;e in infinitum linea magis, ac magis inclinata: igitur mo
tum imminui po&longs;&longs;e in infinitum, non tantùm ex vecte, &longs;ed etiam
ex planis inclinatis haberi pote&longs;t.
3. Hinc producit impetum imperfectiorem impetus acqui&longs;itus
in hoc eodem plano, quàm acqui&longs;itus in perpendiculari, æqualibus
&longs;cilicet temporibus, quia cau&longs;a imperfectior imperfectiorem pro
ducit effectum: motus in plano inclinato deor&longs;um e&longs;t acceleratus
iuxta eandem proportionem, iuxta quam acceleratur in perpendi-tempora, quibus percurruntur perpendiculum, & linea plani
inclinati, &longs;unt vt lineæ; &longs;patia autem, quæ in prædictis lineis acqui
runtur æqualibus temporibus, &longs;unt vt motus, id e&longs;t, vt lineæ per
mutando, vt patet ex dictis.
4. Ex his concludo, nece&longs;&longs;ariò per plana omnia eiu&longs;dem altitu
dinis acquiri eandem velocitatem, quantumuis a&longs;&longs;umantur longi&longs;
&longs;ima, modò &longs;cilicet perpendicula &longs;int &longs;emper parallela. Hinc habes
apud Galileum, per omnes chordas circuli erecti de&longs;cen&longs;um fieri
æqualibus temporibus. Vires, quæ &longs;u&longs;tinent pondus in plano in
clinato per lineam plano
pendiculo, vt lineæ permutando; quia debent adæquare impetum,
qui producitur, tùm in plano inclinato, tùm in perpendiculo.
5. Porrò minùs grauitat in ip&longs;um planum inclinatum corpus gra
ue, quàm in planum horizontale: e&longs;t autem grauitatio in horizonta
li, &longs;eu Tangente, ad grauitationem in inclinata, &longs;eu &longs;ecante, vt ip&longs;æ
lineæ permutando: quod facilè demon&longs;tramus. Proiicitur mobile
faciliùs per inclinatum planum &longs;ur&longs;um, quàm per ip&longs;am perpendi
cularem: patet experientia: cuius ratio e&longs;t, quia minùs re&longs;i&longs;tit im
petus innatus, cuius minor e&longs;t ni&longs;us per inclinatam, vt con&longs;tat ex
dictis.
6. Illæ vires, quæ &longs;ufficiunt ad eum motum &longs;ur&longs;um in perpendi
culo, &longs;ufficiunt ad motum &longs;ur&longs;um in plano inclinato eiu&longs;dem alti
tudinis: quia illæ vires &longs;ufficiunt ad a&longs;cen&longs;um, quæ acquiruntur in
toto de&longs;cen&longs;u: &longs;ed in de&longs;cen&longs;u inclinatæ, & perpendiculi acquirun
tur vires æquales, id e&longs;t, velocitas æqualis, vt dictum e&longs;t &longs;uprà. Om
nia puncta plani inclinati rectilinei, imò & horizontalis, &longs;unt di
uer&longs;æ inclinationis: in iis tamen planis inclinatis quæ vulgò a&longs;&longs;u
muntur, non mutatur &longs;en&longs;ibiliter inclinatio.
7. Hinc minùs de&longs;truitur impetus in plano inclinato &longs;ur&longs;um,
quàm in perpendiculo; quia diutiùs durat:
cùm enim minùs ac
quiratur in de&longs;cen&longs;u, vt dictum e&longs;t, minùs etiam de&longs;truitur in a&longs;
cen&longs;u: hinc accedit propriùs hic motus ad æquabilem:
in eodem
plano rectilineo pote&longs;t e&longs;&longs;e a&longs;cen&longs;us, & de&longs;cen&longs;us, versùs eandem
partem: tale e&longs;&longs;et planum horizontale, in cuius vnico tantùm pun
cto nulla e&longs;t inclinatio: in quolibet puncto huius plani e&longs;t &longs;ingu
laris inclinatio, vt patet, quæ e&longs;t ad perpendiculum, vt Tangens ad
&longs;ecantem é&longs;tque eadem proportio motuum.
8. Corpus graue in &longs;uperficie quadrantis caua, deor&longs;um cadit
motu naturaliter accelerato; quia &longs;ingulis in&longs;tantibus accedit nouus
non tamen æqualibus temporibus, acquiruntur æqualia
velocitatis momenta; quia in &longs;ingulis punctis quadrantis, e&longs;t diuer
&longs;a tangens; igitur mutatur progre&longs;&longs;io accelerationis, quæ certè ma
jor e&longs;t initio, & &longs;ub finem minor; quia initio tangentes acce
dunt propriùs ad perpendiculum, & &longs;ub finem ad horizonta
lem.
9. De&longs;cendit etiam in &longs;uperficie conuexa globi erecti motu ac
celerato; initio quidem, in minore proportione;
&longs;ub finem, in maio
re; vnde e&longs;t inuer&longs;a prioris:
pote&longs;t etiam de&longs;cendere corpus graue
v&longs;que ad centrum terræ motu accelerato, in &longs;uperficie conuexa &longs;e
micirculi: &longs;i &longs;uperficies terræ e&longs;&longs;et læuigati&longs;&longs;ima, corpus proje
ctum moueretur in ea motu æquabili, nec de&longs;trueretur impetus im
pre&longs;&longs;us, vt con&longs;tat; pote&longs;t quoque de&longs;cendere per &longs;piralem: &longs;unt in
finita plana curua, in quibus faciliùs moueri pote&longs;t, quam in ho
rizontali recta.
1. DAri motum mixtum ille non dubitat, qui di&longs;cum proiicit.
Mixtus ex duobus rectis æquabilibus e&longs;t rectus, e&longs;t que
diagonalis vtriu&longs;que: hinc de&longs;truitur aliquid impetus, iuxta pro
portionem differentiæ diagonalis, & vtriu&longs;que lateris &longs;imul &longs;ump
ti; quia, &longs;cilicet, e&longs;t fru&longs;trà:
quò maior e&longs;t angulus, quem faciunt li
neæ determinationum, minor e&longs;t diagonalis; igitur plùs impetus
de&longs;truitur, donec tandem concurrant in oppo&longs;itas lineas, tunc enim
totius impetus de&longs;truitur.
2.
de&longs;truitur; quia diagonalis maior e&longs;t;
donec tandem conueniant in
eandem lineam, tunc enim nihil de&longs;truitur: datur de facto hic mo
tus in rerum natura; talis e&longs;t motus nauis à duobus ventis impre&longs;
&longs;us; vel eiu&longs;dem partis aëris; imò & ip&longs;ius venti: motus mixtus ex
duobus retardatis iuxta eandem progre&longs;&longs;ionem e&longs;t rectus; quia fit
per hypothenu&longs;im triangulorum proportionalium: idem dico de
duobus acceleratis.
3. Si mixtus &longs;it ex æquali, & accelerato, vel ex duobus accelera
tis in diuer&longs;a progre&longs;&longs;ione, vel ex duobus retardatis &longs;imiliter, fit per
lineam curuam, vt patet: dum proiicitur corpus graue per horizon-
retardato violento: e&longs;t enim acceleratus naturalis, cùm deor&longs;um
deor&longs;um tendat qua&longs;i per gradus, &longs;eu diuer&longs;a plana inclinata.
4. Non tamen impetus acqui&longs;itus in eo motu e&longs;t eiu&longs;dem perfe
ctionis cum illo, qui acquireretur in perpendiculari eiu&longs;dem longi
tudinis; &longs;ed tantùm eiu&longs;dem altitudinis:
nam perinde cre&longs;cit ille
impetus, atque cre&longs;ceret in diuer&longs;is planis inclinaris: impetus verò
violentus in hoc motu retardatur; tùm, quia, &longs;i maneret idem, maior
e&longs;&longs;et ictus &longs;ub finem iactus, quod e&longs;t ridiculum; nec e&longs;t, quòd aliqui
dicant, ab aëre de&longs;trui, qui non minùs re&longs;i&longs;tit naturali, quàm vio
lento.
5. Adde, quòd e&longs;t duplex determinatio:
igitur aliquid de&longs;trui de
bet, non acqui&longs;iti; igitur impre&longs;&longs;i: de&longs;trui autem non dicitur acqui
&longs;itus, quòd, &longs;cilicet, plùs de nouo accedat, quàm pereat; e&longs;t enim ac
celeratus: adde, quòd non infligitur tantus ictus &longs;ub finem;
igitur
de&longs;truitur aliquid impetus, non acqui&longs;iti, eo modo, quo diximus;
igitur impre&longs;&longs;i: ita tamen &longs;en&longs;im de&longs;truitur, vt pro æquabili per ali
quod &longs;patium qua&longs;i haberi po&longs;&longs;it.
6. Hinc mobile proiectum per horizontalem, ne primo quidem
in&longs;tanti per horizontalem mouetur, alioqui non e&longs;&longs;et motus mix
tus: tardiùs cadit mobile ita proiectum in planùm horizontale &longs;ub
iectum, quàm cum &longs;ua &longs;ponte, ex eadem altitudine de&longs;cendit: cuius
rei clari&longs;&longs;ima e&longs;t experientia: ratio e&longs;t; quia impetus acqui&longs;itus in
hoc iactu non e&longs;t eiu&longs;dem perfectionis, cùm acqui&longs;ito in perpendi
culo: cùm proiicitur mobile per inclinatam &longs;ur&longs;um, mouetur motu
mixto ex naturali æquabili, & violento retardato: patet prima pars;
quia acceleratur tantùm naturalis deor&longs;um, &longs;altem in inclinata: &longs;e
cunda pars etiam patet; quia &longs;ub finem minor e&longs;t ictus.
7. Hinc linea motus e&longs;t curua:
iuxta diuer&longs;am progre&longs;&longs;ionem de
&longs;truitur hic impetus impre&longs;&longs;us: tùm pro diuer&longs;a inclinatione plani,
cuius etiam hîc habetur ratio; nam &longs;ingulis in&longs;tantibus mutatur:
tùm, quia modò plùs impetus e&longs;t fru&longs;trà, modò minùs; plùs
certè, cùm linea determinationis impetus impre&longs;&longs;i facit obtu
&longs;iorem: atqui initio e&longs;t obtu&longs;ior; &longs;ub finem verò a&longs;cen&longs;us acu
tior.
8. A&longs;cen&longs;us proiecti per inclinatam diutiùs durat, quàm de&longs;
cen&longs;us, ratione eiu&longs;dem plani horizontalis; quia, &longs;cilicet, a&longs;
cen&longs;us longior e&longs;t, quàm de&longs;cen&longs;us: e&longs;t autem longior;
quia, vt
e&longs;&longs;et æqualis, nihil impetus impre&longs;&longs;i deberet de&longs;trui in a&longs;cen&longs;u
& retardato violento, vt con&longs;tat ex dictis: iactus per incli
natam ad angulum 45. e&longs;t omnium maximus, ratione eiu&longs;dem
plani horizontalis: clara e&longs;t experientia. Ratio e&longs;t:
quia per verti
calem &longs;ur&longs;um, nihil acquiritur in plano horizontali, ex quo fit ia
ctus; nihil etiam per ip&longs;am horizontalem; igitur plùs acquiritur per
illam, quæ maximè ab vtraque &longs;imul recedit.
9. Hæc ratio e&longs;t verè phy&longs;ica, geometrica nulla e&longs;t: hinc illi
iactus æquale &longs;patium acquirunt in prædicto plano horizontali,
qui fiunt per inclinatas æqualiter à prædicta inclinata ad ang. 45.
di&longs;tantes. Cùm emittitur mobile per inclinatum deor&longs;um, in libero
medio, mouetur motu mixto ex naturali accelerato, & impre&longs;
&longs;o retardato, vt con&longs;tat ex dictis; ille autem primus accelera
tur per acce&longs;&longs;ionem impetus perfectionis quàm in iactu per ho
rizontalem; &longs;ed imperfectionis, quàm in perpendiculo:
retarda
tur verò impetus minùs, quàm in iactu per horizontalem; plùs ve
rò, quàm in iactu per ip&longs;um perpendiculum, in quo nihil impetus
de&longs;truitur.
10. Cùm è naui mobili &longs;ur&longs;um mittitur corpus graue, e&longs;t motus
mixtus ex tribus, in a&longs;cen&longs;u, &longs;cilicet, ex naturali æquabili, ex verti
cali retardato, & horizontali æquabili: mouetur &longs;ur&longs;um per cur
uam, &longs;empérque capiti iaculatoris imminet; quippe tantùm acqui
rit in horizontali, quantùm nauis: in de&longs;cen&longs;u verò e&longs;t motus
ex horizontali retardato, & naturali accelerato: quia tamen bre
ui&longs;&longs;imo illo tempore, retardatio illa horizontalis non e&longs;t &longs;en&longs;ibilis,
ferè in ip&longs;ius iaculatoris caput de&longs;cendit; quod certè phænomenon
ex no&longs;tris principiis euincitur.
11. Parum cautè Vfanus vniuer&longs;im a&longs;&longs;erit, iaculationem pilæ ex
tormento, maiorem e&longs;&longs;e ex naui in continentem, & minorem vi
ci&longs;&longs;im, cùm vtriu&longs;que differentia peti po&longs;&longs;it, vel à puluere tormen
tario, vel ab eius compre&longs;&longs;ione, vel humiditate, vel tormenti fabri
ca, vel ip&longs;ius demum nauigij motu, qui pilæ motum, vel accelerat, &longs;i
versùs eandem partem e&longs;t, vel retardat è contrario: in plano ho
rizontali duro pote&longs;t e&longs;&longs;e motus mixtus ex duobus, tribus, qua
tuor, & pluribus aliis.
12. Cùm è naui mobili emittitur &longs;agitta per horizontalem, quæ fa
cit angelum rectum cum linea directionis nauis, fertur qua&longs;i per dia
gonalem vtriu&longs;que, &longs;altem per aliquod &longs;patium: cùm verò emitti-
iactus e&longs;t longior toto illo &longs;patio, quod nauis decurrit, dum iactus
durat; breuior tamen, &longs;i in partem oppo&longs;itam fiat iactus in hoc ca
&longs;u, &longs;i nauis æqualem impetum imprimeret, deor&longs;um rectà ferretur
mobile motu naturali; imò &longs;agitta po&longs;&longs;et retorqueri in iaculatorem:
&longs;i terra e&longs;&longs;et vtrimque peruia, lapis demi&longs;&longs;us per multa annorum
millia libraretur; non tamen e&longs;&longs;et motuus perpetuus.
1. MOtus reflexi vera cau&longs;a e&longs;t impetus prior, ad nouam li
neam determinatus ab occurrente obice; planum refle
ctens e&longs;t cau&longs;a nouæ determinationis &longs;uo modo; cau&longs;am enim di
co eam, ex qua aliquid &longs;equitur: ex gemina determinatione, noua,
&longs;cilicet, per ip&longs;am perpendicularem erectam in puncto contactus,
& priore per lineam incidentiæ, ab eodem puncto contactus pro
pagatam, fit determinatio mixta per lineam reflexionis; quæ omnia
patent ex terminis: hinc nullus impetus producitur à plano refle
ctente; quippe prior pote&longs;t determinari ad nouam lineam: adde,
quòd planum, quod caret impetu, impetum producere non pote&longs;t.
2. Imò nihil impetus de&longs;truitur in reflexione pura per &longs;e;
quia ni
hil impetus e&longs;t fru&longs;trà per &longs;e in pura reflexione; multus tamen im
petus de&longs;truitur per accidens, tùm ab ip&longs;o attritu tùm mollitie
& ce&longs;&longs;ione, tùm pre&longs;&longs;ione: hinc &longs;uppo&longs;ito eodem iactu, perpendi
cularis reflexa e&longs;t omnium reflexarum minima; quia per eam li
neam maximus ictus infligitur; igitur maxima e&longs;t partium colli&longs;io,
& pre&longs;&longs;io: hinc etiam corpora duriora longiùs reflectuntur, per ip&longs;am
quoque
3. Determinatio noua dupla e&longs;t prioris, po&longs;ita linea incidentiæ
perpendiculari, & po&longs;ito etiam plano reflectente immobili; quia
alioquin anguli reflexionis non e&longs;&longs;ent æquales angulis incidentiæ:
&longs;i globus reflectens &longs;it æqualis impacto, æqualis e&longs;t ce&longs;&longs;io re&longs;i&longs;tenciæ
cùm &longs;it æquale agens re&longs;i&longs;tenti, perid enim reflectens re&longs;i&longs;tit, per
quod e&longs;t: igitur, &longs;i æqualis re&longs;i&longs;tit, & cedit, certè æqualiter ce
dit, & re&longs;i&longs;tit: hinc noua determinatio æqualis e&longs;t priori:
hinc glo
bus impactis &longs;i&longs;tit immobilis; quia ex duabus determinationibus
oppo&longs;itis neutra præualet.
4. Tantum e&longs;t ab æqualitate prædicta ce&longs;&longs;ionis, & re&longs;i&longs;tentiæ, ad
nullam ce&longs;&longs;ionem, & notam re&longs;i&longs;tentiam, quantum e&longs;t ad nullam hinc, cùm à tota ce&longs;&longs;ione ad æqua
litatem prædictam acquiratur tantùm noua determinato æqualis
priori; igitur ab eadem æqualitate ad nullam ce&longs;&longs;ionem tantun
dem acquiritur; igitur dupla prioris, vt iam &longs;uprà dictum e&longs;t;
nulla
e&longs;&longs;et re&longs;i&longs;tentia in vacuo; nulla e&longs;t ce&longs;&longs;io, cùm ip&longs;um corpus refle
ctens nullo modo mouetur ab ictu.
5. Determinatio noua per lineam obliquam, e&longs;t ad nouam per
lineam perpendicularem, vt &longs;inus rectus anguli incidentiæ, ad &longs;i
num totum, in qualibet hypothe&longs;i; quia &longs;unt hæ, vt ictus, per vtran
que lineam; ictus verò vt grauitationes in horizontale planum, &
in planum inclinatum, &longs;ub angulo complementi anguli incidentiæ:
hinc noua determinatio per lineam obliquam, e&longs;t vt dupla &longs;inus re
cti anguli incidentiæ, ad &longs;inum totum: hinc &longs;upra angulum inci
dentiæ 30, noua e&longs;t maior priore, infrà minor; in ip&longs;o angulo 30.
æqualis, &longs;uppo&longs;ita hypothe&longs;i plani reflectentis immobilis.
6. Ex hoc po&longs;itiuo principio demon&longs;tratur accurati&longs;&longs;imè æqua
litas anguli reflexionis, & incidentiæ, quod certè demon&longs;tratum
non fuit ab Ari&longs;t. in problematis, &longs;ect. 17. problem. 4. & 13. quibus
in locis fusè &longs;atis explicatur hoc Theorema, ducta comparatione,
tùm à grauibus, quæ cadunt, tùm ab orbibus, quæ rotantur, rùm à
&longs;peculis: &longs;ed minimè demon&longs;tratur ex certis principiis &longs;ine petitio
ne principij. In puncto reflexionis, po&longs;ita hypothe&longs;i plani immo
bilis reflectentis, nulla datur quies; quia vnum tantùm e&longs;t conta
ctus in&longs;tans; &longs;ed eo in&longs;tanti e&longs;t motus, quo primo acquiritur locus.
7. Omnes lineæ reflexæ per &longs;e &longs;unt æqualis longitudinis, & ab
eodem puncto contactus, ad communem peripheriam terminan
tur: &longs;i globus impactus &longs;it æqualis reflectenti, &longs;itque linea inciden
tiæ obliqua quælibet terminata ad idem punctum contactus, re
flectitur prædictus globus per lineam tangentem globum refle
ctentem in eodem puncto; quia hæc tangens e&longs;t diagonalis com
munis, & determinatio mixta communis omnibus lineis inciden
tiæ: e&longs;t tamen modò longior, modò breuior linea reflexa, é&longs;tque vt
vt &longs;inus complementi anguli incidentiæ, ad &longs;inum totum, qui &longs;it
determinatio prior, vt facilè demon&longs;tramus.
8. Si globus impactus &longs;it minor corpore reflectente, reflectitur
etiam per ip&longs;am perpendicularem, & determinatio noua e&longs;t dupla
prioris, minùs ratione globorum v. g. &longs;i globus impactus &longs;it &longs;ubdu-
&c. ratio e&longs;t, quia in ea proportione globus reflectens cedit, in
qua mouetur, igitur tantùm detrahitur determinationis impacto
globo, quantùm additur motus reflectenti: at verò noua determina
tio per lineam incidentiæ obliquam, e&longs;t ad nouam per ip&longs;am per
pendicularem, vt &longs;inus rectus anguli incidentiæ ad &longs;inum totum.
9. In hac hypothe&longs;i lineæ reflexæ omnes &longs;unt &longs;upra prædictam
tangentem, &longs;eu &longs;ectionem plani, maiores, vel minores, pro diuer&longs;a
men&longs;ura diagonalis: in &longs;uperiori verò hypothe&longs;i æqualium globo
rum, &longs;unt omnes in ip&longs;a &longs;ectione plani: &longs;i denique globus impactus
&longs;it maior alio, omnes &longs;unt infra prædictam &longs;ectionem. Porrò in hac
hypothe&longs;i vltima, determinatio noua per ip&longs;am perpendicularem
e&longs;t minor priore: hinc non modò nulla fit reflexio in perpendicula
ri, &longs;ed linea directa vlteriùs propagatur; quia prior determinatio
præualet.
10. Detrahitur priori portio æqualis rationi globorum;
v. g. glo
bus reflectens e&longs;t &longs;ubduplus impacto de trahitur priori determina
tioni vna &longs;ecunda; e&longs;t &longs;ubquadruplus, vna quarta; atque ita dein
ceps: ratio patet ex dictis:
in linea verò incidentiæ obliqua, deter
minatio e&longs;t ad determinationem in perpendiculari, vt &longs;inus rectus
anguli incidentiæ ad &longs;inum totum: linea demum reflexa e&longs;t modò
maior, modò minor pro diuer&longs;a diagonali.
11. Si duo globi æquales in &longs;e inuicem impingantur æquali mo
tu, per lineam connectentem centra, vterque æquali motu priori re
troagitur; quia æqualis in æqualis æqualem impetum imprimit:
non
e&longs;t tamen motus reflexus; quia totus prior impetus de&longs;truitur, vt
patet ex dictis: &longs;i autem inæquali motu concurrant, retroaguntur
ii&longs;dem motibus, permutando; quod etiam clarum e&longs;t: hinc egre
gium paradoxum, &longs;i quod aliud con&longs;equitur, &longs;cilicet, globum A, v.
g. æqualem motum imprimere globo B, &longs;iue hic moueatur, &longs;iue
quie&longs;cat.
12. Si verò linea incidentiæ &longs;it obliqua, vterque globus reflecte
tur pror&longs;us vt à plano immobili: hinc reflexio &longs;it ad angulos æqua
les, & lineæ omnes reflexionis &longs;unt æquales: ratio e&longs;t; quia, quantùm
detrahit globus reflectens re&longs;i&longs;tendo, tantùm addit in partem op
po&longs;itam repellendo, po&longs;itiuo ni&longs;u, vel impetu: quòd &longs;i alter globus
maiore, vel minore motu moueatur, vel &longs;i globi &longs;int inæquales,
cum æquali motu, vel inæquali, res etiam determinari pote&longs;t ex
præmi&longs;&longs;is.
13. Cum duo globi in &longs;e&longs;e inuicem impinguntur æquali motu,
minor retroagitur velociore motu, quàm ante moueretur, vt clarum
e&longs;t: maior verò, &longs;i duplus e&longs;t alterius, &longs;i&longs;tit immobilis in puncto
contactus; &longs;i maior duplo &longs;uum iter pro&longs;equitur, &longs;ed tardiore mo
tu; &longs;i minor duplo, retroagitur: quæ omnia facilè ex dictis demon
&longs;trantur. Pote&longs;t impetus e&longs;&longs;e æqualis alteri, & præualere; pote&longs;t
æqualem impetum producere hoc in&longs;tanti, & &longs;tatim in&longs;tanti, quod
&longs;equitur, totus de&longs;trui.
14. Pote&longs;t globus retroagi in plano horizontali, licèt in aliud cor
pus non incidat, ita vt initio tendat in ortum, verbi gratia: tùm
deinde, licèt nihil pror&longs;us addatur, versùs occa&longs;um; quod accidit,
cum globus vtroque motu, centri, &longs;cilicet, & orbis, mouetur, &longs;ed
contrario; primùm enim motus centri præualet, &longs;ed facilè cedit
propter attritum maiorem partium. Nullus datur propriè motus
refractus: licèt enim incuruetur linea motus, dum per aquam &longs;u
bit mobile; hæc tamen e&longs;t reflexionis &longs;pecies.
15. Globus reflectens, qui ab ictu alterius mouetur, non mouetur
in&longs;tanti contactus; quia impetus primo in&longs;tanti, quo e&longs;t, non mo
uetur; producitur enim impetus primo in&longs;tanti contactus:
&longs;i impe
tus e&longs;&longs;et tantùm determinatus ad vnam lineam, nulla fieri po&longs;&longs;et
reflexio, &longs;ed tantùm repercu&longs;&longs;io; quia veri&longs;&longs;ima cau&longs;a reflexionis
con&longs;i&longs;tit in noua determinatione: per reflexionem po&longs;&longs;unt colligi
plures partes aëris &longs;onori ad Echometriam: &longs;agitta emi&longs;&longs;a per ho
rizontalem &longs;ursùm, tantillùm a&longs;cendit per arcum; quia tantillùm
reflectitur ab aëre.
1. DAri motum circularem, probatur infinitis ferè experimen
tis: cuius ratio à priori e&longs;t, quòd po&longs;&longs;int extremitates eiu&longs;
dem cylindri in partes oppo&longs;itas pelli; vnde &longs;equitur nece&longs;&longs;ariò
motus circularis; quem ij negare coguntur, qui ex punctis mathe
maticis quantitatem componunt. Motus circularis in &longs;ublunaribus
oritur ex recto impedito; quia, &longs;cilicet, determinatur tantùm im
petus ad lineam rectam: hinc quidam motus circularis e&longs;t merè
per accidens, vt cùm retinetur extremitas funependuli, &longs;eu quidam tamen
non e&longs;t merè peraccidens, vt cùm pellitur extremitas cylindri in
plano horizontali; e&longs;t enim, iuxta in&longs;titutionem naturæ, ad facili
tatem motus.
2. Quippe tale e&longs;t naturæ in&longs;titutum, vt eo motu corpora mo
ueantur, quo faciliùs moueri po&longs;&longs;unt: atqui cùm pellitur altera cy
lindri extremitas, in plano horizontali putà innatantis, faciliùs
mouetur, quàm recto, & qua&longs;i minore &longs;umptu, cùm minùs &longs;patij
acquirat: æquali tempore: pote&longs;t dari motus circularis mixtus ex
duobus rectis, quorum vnus &longs;it, vt &longs;inus recti, alius vt ver&longs;i; vix
tamen hoc accidit vnquàm, &longs;ed tantùm oritur hic motus ex
determinatione per tangentem impedita, ratione alicuius puncti
immobilis.
3. Hinc, &longs;i tollatur impedimentum, &longs;tatim per tangentem or
bis fit motus, vt patet in funda: inæqualiter partes radij prædicti
orbis mouentur, iuxta proportionem di&longs;tantiæ maioris, & minoris
à centro: hinc propagatio impetus inæqualis, de qua iam &longs;uprà,
&longs;ingulis in&longs;tantibus & punctis e&longs;t noua determinatio; quia, &longs;cilicet,
&longs;ingulis punctis &longs;ua tangens re&longs;pondet: hinc, &longs;i imponatur rotæ
aliud corpus, &longs;tatim abigitur, &longs;ine &longs;it in &longs;itu verticali, &longs;iue in &longs;itu ho
rizontali; hinc dum turbo rotatur, &longs;i vel aquæ guttula eius &longs;uper
ficies a&longs;pergitur, & &longs;tatim di&longs;pergitur.
4 Dari impetum in motu circulari certi&longs;&longs;imum e&longs;t:
punctum phy
&longs;icum e&longs;t capax huius motus; cuius finis multiplex e&longs;t; corpus mo
uetur motu circulari circa centrum immobile cum motus centri
impeditur non tamen motus orbis, ad quem impetus facilè deter
minatur, cùm &longs;it ad omnes lineas indifferens: adde v&longs;um vectis,
trochleæ, aliorúmque organorum, qui &longs;ine motu circulari e&longs;&longs;e non
pote&longs;t: omitto motum progre&longs;&longs;iuum, ipsúmque brachiorum, & ti
biarum v&longs;um, qui motu circulari carere non pote&longs;t.
5. Motus circularis rotæ in plano verticali e&longs;t æquabilis per &longs;e;
quia nihil e&longs;t, quod impetum &longs;emel impre&longs;&longs;um de&longs;truat: licèt enim
&longs;ingulis in&longs;tantibus &longs;it noua determinatio, nullus tamen impetus
e&longs;t fru&longs;trà; quippe illud &longs;patium acquiritur in linea curua, quod in
recta, &longs;i nullum e&longs;&longs;et impedimentum, percurreret: quemadmodum
enim in reflexione, quæ fit à plano immobili, nullus de&longs;truitur im
petus; ita nullus hîc de&longs;truitur;
tam enim centrum illud immobile
ad &longs;e qua&longs;i trahit mobile, quàm planum immobile à &longs;e repellit; in
quo e&longs;t perfectè analogia.
6. Hinc per &longs;e motus circularis integri orbis e&longs;t perpetuus;
de
&longs;truitur tamen per accidens, &longs;cilicet, propter attritum axis: hinc
tam diu durat hic motus: clari&longs;&longs;imum experimentum habes in tur
bine, cuius cu&longs;pis læuigati&longs;&longs;ima in plano læuigati&longs;&longs;imo rotatur; nec
vnquam ce&longs;&longs;aret hic motus &longs;ine prædicto attritu, & partium a&longs;peri
tate: nec quidquam ob&longs;tat, quòd aliquæ partes rotæ, quæ in circu
lo verticali voluitur, a&longs;cendant; quia etiam aliquæ de&longs;cendunt: qua
re &longs;emper remanet perfectum æquilibrium, & harum de&longs;cen&longs;us, il
larum a&longs;cen&longs;um compen&longs;at. Quò diutiùs potentia motrix manet
applicata manubrio axis rotæ, ita vt nouum &longs;emper producat im
petum, rotæ motus velocior e&longs;t, atque diutiùs durat: idem pror&longs;us
dico de rota circulo horizontali parallela.
7. Cùm mouetur æquali ni&longs;u acus circa immobile centrum, tùm
in plano
alia, per &longs;e plures gyros non de&longs;cribit vna, quàm alia; quia per &longs;e
mouetur motu æquabili: per accidens tamen &longs;ecus accidit;
quippe
maior e&longs;t maioris attritus: dixi, cùm mouetur æquali ni&longs;u;
nam &longs;æpè
contingit, maiore ni&longs;u potentiam motricem agere circa maiorem;
æquali tamen tempore numerus circuitionum minoris, e&longs;t ad nu
merum circuitionum maioris per &longs;e vt acuum quadrata permu
tando; &longs;unt enim motus vt &longs;patia, &longs;pacia vt quadrata.
8. Verbi gratia, &longs;it acus maior 2. minor 1. certè cùm tota area or
bis maioris &longs;it quadrupla minoris, &longs;itque area maioris, &longs;patium ma
ioris, & area minoris &longs;patium minoris, haud dubiè de&longs;cribet minor
quatuor circuitiones, eo tempore, quo maior decurret vnicam: li
cèt enim extremitas minoris, quæ impellitur, habeat tantùm du
plum impetum extremitatis maioris, &longs;itque impetus inten&longs;io in
minore, dupla inten&longs;ionis impetus in maiore; e&longs;t tamen quadrupla
illius, quæ e&longs;t in &longs;egmento maioris versùs centrum æquali minori
acui: porrò motus circulares æquabiles in vtraque cum eodem
impetu, &longs;unt vt motus recti.
9. Rota in plano verticali faciliùs mouetur, quàm in horizonta
li; quia in illo mouetur per minimam impetus, vel potentiæ acce&longs;
&longs;ionem; &longs;ecùs in i&longs;to;
quippe per minimam acce&longs;&longs;ionem tollitur
æquilibrium; imò moueri pote&longs;t in plano verticali, licèt nullus im
primatur impetus rotæ, v. g. per additionem minimi ponderis, vel
momenti, vt patet; cùm tamen in plano horizontali moueri non
po&longs;&longs;it, ni&longs;i impetus imprimatur.
10. Si cylindrus in plano horizontali læuigato in altera extremi
tate per tangentem impellatur, mouebitur motu circulati, &longs;cilicet,
quarta totius cylindri: ratio e&longs;t: quia faciliùs mouetur circa illud
centrum, quàm circa alia puncta, quòd, &longs;cilicet, minùs &longs;patij decur
ratur, po&longs;ito eodem &longs;emper motu alterius extremitatis, cui appli
catur immediatè potentia motrix.
11. Cùm rota mouetur in verticali, atque præponderat alter &longs;emi
circulus, haud dubiè hic præponderans producit impetum in alio
&longs;emicirculo: hinc fortè e&longs;t, quòd mirere, impetus determinatus
deor&longs;um producit alium &longs;ur&longs;um: hinc impetus vnius partis mobi
lis pote&longs;t producere &longs;imilem in alia parte continua; quod tantùm in
hoc ca&longs;u locum habet: quando corpus incumbit plano, quod mo
uetur motu recto æquabili, ab eo non &longs;eparatur; &longs;ecùs verò, &longs;i in
cumbat plano, quod mouetur motu circulari.
1. FVnependulum de&longs;cendit per arcum motu naturaliter acce
lerato: experientia clari&longs;&longs;ima e&longs;t: cùm enim ex maiori &longs;ubli
mitate de&longs;cendit, maiorem ictum infligit. Ratio à priori e&longs;t quia
priori impetui acqui&longs;ito nouus accedit: non acceleratur in eadem
proportione, in qua &longs;uprà dictum e&longs;t accelerari in linea recta; quia
in hac acceleratur vniformiter, id e&longs;t, æqualibus temporibus,
æqualia acquiruntur velocitatis momenta; quia vel e&longs;t &longs;emper ea
dem inclinatio plani, vel idem perpendiculum: at verò in fune
pendulo in &longs;ingulis punctis e&longs;t noua tangens; igitur noua inclina
tio plani; igitur noua ratio motus.
2. Initio acceleratur motus per maiora crementa, &longs;ub finem per mi
nora; v.g. &longs;i dato tempore acqui&longs;iuit vnum gradum impetus initio,
æquali deinde tempore acquiret minùs: ratio clara e&longs;t: quia, vt ac
quireret æqualem, deberet e&longs;&longs;e eadem plani inclinatio; &longs;ed &longs;emper
cre&longs;cit Inclinatio; igitur &longs;emper imminuitur impetus æquali
acqui&longs;itus: acquiritur tamen æqualis velocitas in arcu, & in chor
da, &longs;eu plano inclinato, eiu&longs;dem altitudinis; igitur &longs;emper cre&longs;cit
motus funependuli in de&longs;cen&longs;u, &longs;ed minoribus incrementis.
3. Hinc breuiore tempore de&longs;cendit per radium perpendicula
rem, quàm per quadrantis arcum eiu&longs;dem radij; tùm quia breuior
e&longs;t linea; tùm, quia in perpendiculari acceleratur motus per maiora
crementa. Vibratio maior eiu&longs;dem funependuli æquali ferè tem-
quia, cùm ferè decurrantur
arcus iuxta &longs;ubten&longs;arum proportionem, certè cùm &longs;ubten&longs;æ om
nes æquali tempore decurrantur, idem ferè fit in ip&longs;is arcubus: dixi
ferè: nam reuerà minor vibratio citiùs, maior tardiùs perficitur, vt
4. Non a&longs;cendit funependulum ad eam altitudinem, ex qua priùs
de&longs;cenderat: clara e&longs;t experientia:
neque ratio tantùm petitur ab
aëris re&longs;i&longs;tentia; tam enim re&longs;i&longs;tit de&longs;cen&longs;ui, quàm a&longs;cen&longs;ui;
&longs;ed ex
eo, quòd &longs;ingulis in&longs;tantibus &longs;it quædam pugna, inter impetum in
natum, & alium determinatum ad arcum &longs;ur&longs;um: quippe impetus
innatus ad totum de&longs;cen&longs;um, &longs;ed nullo modo ad a&longs;cen&longs;um con
currit: hinc in maiori vibratione imminuitur motus, & &longs;patium in
maiori proportione, quàm in minori; quia in hac lineæ &longs;ingulæ a&longs;
cen&longs;us qua&longs;i
5. Hinc diu vibratur funependulum per minores arcus, quippe
facilis e&longs;t a&longs;cen&longs;us per planum proximè ad horizontale accedens:
hinc etiam in funependulo maiori diutiùs durant huiu&longs;modi vi
brationes, idque in arcubus paulò maioribus; quia &longs;ubten&longs;æ his
arcubus &longs;unt inclinatiores: hinc refutabis eos, qui dicunt, vibra
tiones funependuli in vacuo fore perpetuas: arcus vibratio
nis a&longs;cen&longs;us fit motu naturaliter retardato, &longs;ed per imminu
tiones inæquales; quia pro diuer&longs;a inclinatione plani diuer&longs;imodè
retardatur.
6. Vltimum punctum impetus acqui&longs;itus acqui&longs;itum in de&longs;cen&longs;u,
nullo modo ad de&longs;cen&longs;um concurrit, &longs;ed ad a&longs;cen&longs;um, vnico tan
tùm in&longs;tanti; quippe e&longs;t omnium imperfecti&longs;&longs;imum;
quod reuerà &longs;i
e&longs;&longs;et eiu&longs;dem perfectionis cum innato, a&longs;cen&longs;us æqualis e&longs;t de&longs;cen
&longs;ui: &longs;i &longs;int funependula inæqualia, vibrationes non &longs;unt æquè diu
turnæ: ratio e&longs;t: quia, &longs;i a&longs;&longs;umantur, v.g. duo quadrantes inæquales,
&longs;unt eju&longs;dem inclinationis; igitur minor citiùs percurritur.
7. Porrò tempora vibrationum &longs;unt in ratione &longs;ubduplicata ar
cuum &longs;imilium, vel chordarum &longs;imilium, vel radiorum; id e&longs;t, vt
radices &longs;patiorum &longs;imilium: verbi gratia, &longs;it quadruplus alterius,
tempus vibrationis maioris e&longs;t duplum temporis vibrationis mino
ris; quod ita intelligendum e&longs;t, vt hæc proportio con&longs;ideretur in
partibus temporis &longs;en&longs;ibilibus, vt iam dictum e&longs;t de motu natura
liter accelerato deor&longs;um in perpendiculo, & in planis inclinatis;
nam progre&longs;&longs;io arithmetica; a&longs;&longs;umpta in &longs;ingulis in&longs;tantibus, tran
&longs;it in hanc, &longs;i a&longs;&longs;umantur partes temporis &longs;en&longs;ibiles, quarum &longs;ingu
læ infinitis ferè con&longs;tent in&longs;tantibus.
8. In maiori quadrante, circa &longs;upremam extremitatem, e&longs;t minor
inclinatio, quàm in minore; hic enim &longs;tatim detorquetur à perpen
diculo, cum quo facit angulum maiorem: at verò circa infirmam
extremitatem, e&longs;t maior inclinatio in maiore, quàm in minore: hinc,
&longs;i comparetur vibratio maioris, cum vibratione minoris in modico
arcu, tempus illius e&longs;t paulò maius duplo, temporis huius; in maxi
mo arcu paulò minùs duplo, dum, &longs;cilicet, longitudinum ratio
&longs;it quadrupla.
9. In de&longs;cen&longs;u funependuli velocitas acqui&longs;ita e&longs;t eadem cum ea,
quæ in &longs;ubten&longs;a eiu&longs;dem arcus acquiritur: hinc &longs;unt ijdem ictus:
numerus, vibrationum non e&longs;t infinitus, licèt in vacuo vibraretur
funependulum; quia, cùm &longs;ingulæ imminuantur, & infinitis pun
ctis non con&longs;tent; tandem ad vltimam peruenitur:
illa autem e&longs;t vl
tima, in cuius de&longs;cen&longs;u acquiritur tantùm vnum punctum impetus
&longs;upra innatum; in ea tamen &longs;ententia, quæ vel infinitas partes actu,
vel infinita puncta cogno&longs;cit, certè nunquam quie&longs;ceret funepen
dulum in vacuo vibratum.
10. Funependulum in fine a&longs;cen&longs;us non quie&longs;cit vno in&longs;tanti;
quia impetui innato po&longs;ita ta
men illa æqualitate, in&longs;tanti &longs;equenti e&longs;&longs;et quies: funependulum
grauius citiùs de&longs;cendit; e&longs;t enim eadem ratio, quæ fuit pro mo
tu naturali; corpus oblongum &longs;olidum circa punctum immobile
in circulo verticali rotatum vibratur adin&longs;tat funependuli; de&longs;
cendit tamen citiùs, quàm funependulum eiu&longs;dem longitudinis.
11. Ratio facilis e&longs;t;
quia partes &longs;olidæ, quæ accedunt propiùs
ad extremitatem immobilem, accelerant motum aliarum, quæ
ad mobilem extremitatem accedunt; faciunt enim arcum mino
rem: hinc a&longs;cen&longs;us non peruenit ad tantam &longs;ublimitatem;
quia, vt
prædictæ partes accelerant motum aliarum in de&longs;cen&longs;u, ita retar
dant in de&longs;cen&longs;u: hinc citiùs quie&longs;cit hoc penduli genus, quàm
aliud: ex hoc colligo paradoxon, &longs;cilicet, corpus moueri po&longs;&longs;e &longs;ua
&longs;ponte velociùs in arcu deor&longs;um, quàm in perpendiculo; v.g. &longs;i iuxta
extremitatem immobilem &longs;it nodus plumbeus, cuius vi, altera ex
tremitas longiùs di&longs;tans deor&longs;um rapiatur.
1. ROta, quæ mouetur in &longs;uperficie plana, mouetur motu mixto
ex recto centri, & circulari orbis: axis tantùm rotæ mouetur
motu recto: punctum contactus rotæ mouetur motu tardi&longs;&longs;imo,
runtur; punctum verò oppo&longs;itum veloci&longs;&longs;imo, quia in motu huius
rotus motus orbis additur motui centri; in motu verò illius, to
tus motus orbis, motui centri detrahitur: quod autem detrahit mo
tus orbis, nunquam æquale e&longs;t toti motui centri.
2. Hinc omnia puncta eiu&longs;dem circuli rotæ mobilis in plano
hoc motu mixto mouentur in æquali motu: hoc etiam motu mo
uetur globus de&longs;cendens in plano inclinato, in quo reuerâ motu
hæc habes: primò, non modò accelerari
motum orbis;
lis partibus eiu&longs;dem circuli, & plani in æqualiter di&longs;tribuatur, tertiò
hoc motu motum rectum non impediri à circulari, & &longs;ed iuuari.
3. Cùm rota voluitur in &longs;uperficie connexa, mouetur motu mix
to ex duobus circularibus: &longs;imilis e&longs;t hic motus motui epicycli. Ca
lamus volatilis, cuius mi&longs;&longs;io frequens, & repercu&longs;&longs;io, ludi non in
grati copiam facit: mouetur motu mixto ex recto, & circulari:
in
hoc porrò motu præit calami caput, & &longs;equuntur pennæ; quia aër
fortiùs re&longs;i&longs;tit pennis, quàm thecæ: hinc pennarum motum theca
grauior accelerat, cuius motum pennæ retardant.
4. Hinc, &longs;i quando accidat, penas educi ex theca in libero medio;
&longs;tatim theca velociori motu mouetur, cùm tamen pennæ ip&longs;æ &longs;i
&longs;tant: ex hac inæqualitate, ne impetus &longs;it fru&longs;trà, propter detortas
in alteram partem pennas ab aëre re&longs;i&longs;tente totum iaculum defle
ctitur, agitúr que in orbem; hinc motus orbis traducitur ex theca in
pennas, non contrà, vt aliquis fortè exi&longs;timaret, licèt pennarum tar
ditas, & obliqua deflexio, ratione cuius ab aëre re&longs;tante, in alteram
partem qua&longs;i reflectentur, &longs;int nece&longs;&longs;aria conditio huius traductio
nis.
5. Hinc motu recto prædictum iaculum in vacuo tantùm mo
ueretur, vt patet: hinc: cùm pennæ &longs;unt explicatiores, tardiùs;
cùm
verò contractiores, velociùs mouetur, etiam motu orbis; cui non
minùs aër re&longs;i&longs;tit, in pennis, &longs;cilicet, quàm motui axis: hinc, &longs;i theca
&longs;it grauior, velociùs; &longs;i leuior, tardiùs iaculum fertur;
etiam tenera
plumarum lanugo tarditatem conciliat: porrò, &longs;i axis mouetur mo
tu recto, quod reuerà fit, cùm iaculum deor&longs;um demittitur in per
pendiculo, hic motus e&longs;t &longs;piralis cylindricus: ex his infinita ferè
phænomena explicari po&longs;&longs;unt.
6. Sunt infiniti propemodum motus mixti;
v. g. cylindri ab alte
ra extremitate rotata emi&longs;&longs;i; longioris ha&longs;tæ, quæ &longs;ur&longs;um facta cir
cuitione emittitur; brachij, gladij, &c. &longs;ed poti&longs;&longs;imùm turbinis, qui
mè apparet motus centri, & orbis: ratio motus orbis e&longs;t impe
tus impre&longs;&longs;us vtrique extremitati diametri va&longs;is in partes contra
rias; ratio verò motus centri e&longs;t, quia adducitur funiculo vel ex
ploditur, &longs;eu expellitur &longs;cutica: huius motus phænomena &longs;unt ferè
infinita: &longs;ingula ex no&longs;tris principiis facilè explicantur.
1. CVm &longs;u&longs;tinetur manus, &longs;eu brachium, in &longs;itu horizontali im
mobile, producitur nece&longs;&longs;ariò impetus æqualis impetui gra
uitationis; alioquin, &longs;i maior e&longs;&longs;et, &longs;ur&longs;um ferretur brachium; &longs;i verò
minor, deor&longs;um: quia præualeret grauitatio, porrò hic impetus pro
ducitur tantùm à potentia motrice animantis, in &longs;ingulari organo;
non verò in aliis partibus, etiam animatis, ni&longs;i quando mouentur;
nec in ip&longs;o pondere, &longs;i aliquod &longs;u&longs;tinetur: &longs;ic men&longs;a in pondere &longs;u
per po&longs;ito impetum nullum producit. Si anima immediatè in toto
corpore po&longs;&longs;et producere impetum, homo facilè volare po&longs;&longs;et.
2. Cùm &longs;u&longs;tinetur funependulum, nullus impetus producitur à
&longs;u&longs;tinente in ip&longs;o globo, ne &longs;cilicet, &longs;it fru&longs;trà; &longs;ecùs verò, &longs;i attolla
tur: &longs;ic per quamlibet lineam corpus retineri pote&longs;t &longs;ine impetu in
eo corpore producto per &longs;e: hinc, cùm duo &longs;e&longs;e inuicem trahunt ad
uer&longs;o ni&longs;u, neuter in altero producit impetum per &longs;e; &longs;ed per acci
dens, propter mollitiem, & ten&longs;ionem partium: cùm verò defertur
aliquid coniunctum, producitur haud dubiè æqualis impetus; hinc
&longs;eparari non pote&longs;t; quia æqualis e&longs;t motus latoris, & delati: exem
plum habes in naui.
3. Si verò nauis illicò &longs;i&longs;tat, vel tardiùs moueri pergat, tunc fit &longs;e
paratio: hinc liquida effunduntur, &longs;i dum feruntur, breuior quietis
in va&longs;e intercedat morula. Vt feratur cylindrus humeris
debet &longs;u&longs;tineri in ideò in circulo
voluitur &longs;cyphus aqua plenus &longs;ine effu&longs;ione; quia impetus determi
natus per tangentem circuli aquam ip&longs;am à centro circuli remouet.
4. Cùm trahitur aliquod corpus impetus impre&longs;&longs;us in vna parte
non producit impetum in alia, alioquin daretur proce&longs;&longs;us in infi
nitum; &longs;i chorda vtrinque trahatur, rumpetur in medio:
&longs;i affixa
extremitati immobili, trahatur à potentia applicata alteri extremi-
ticulo &longs;uppo&longs;ito tendatur, vel pondere deprimente, in eo puncto
rumpetur. Ratio communis i&longs;torum omnium e&longs;t:
quia inter illas
duas partes fieri debet diui&longs;io per &longs;e, quarum vna mouetur, &longs;ecùs
alia; vel quarum vtraque in partes oppo&longs;itas mouetur.
5. Vt quodlibet pondus faciliùs trahatur, &longs;inguli equi trahere
debent fune communi, potiùs quàm bigati; quia tunc nihil ferè pe
rit impetus: cùm plures idem pondus trahunt, agunt actione com
muni, alioqui &longs;inguli in toto pondere &longs;uum impetum producerent;
igitur &longs;inguli &longs;eor&longs;um trahere? e&longs;&longs;ent, quod fal&longs;um e&longs;t:
ideò currus
paulò po&longs;t initium motus faciliùs mouetur; quia aliquid impetus
priùs producti remanet: hinc etiam rupto fune, quo trahitur currus,
currus ip&longs;e modicum tempus adhuc mouetur.
6. Si, dum quis trahit toto ni&longs;u magnum aliquod pondus, funis
rumpatur, pronùs corruit: quia maiorem impetum in &longs;e producit,
totum, &longs;cilicet, illum, quem in toto pondere produxi&longs;&longs;et eo in&longs;tan
ti, quo rumpitur finis, qui reuerà maior e&longs;t, propter impedimen
tum, ex præmi&longs;&longs;is principiis, maiorique applicatione potentiæ, ner
uorum ten&longs;ione, &c. dum trahitur vnco an nullus immobilis ver
sùs nauim, nauis fertur versùs littus; dum pellitur aduersùm littus,
recedit à littore, quia pede, vel genu, imprimitur naui impetus in
contrariam pattem.
7. Cùm trahitur cylindrus vtrinque æqualiter, qui neque flecti,
neque tendi pote&longs;t, nullum impetum accipit; imò in tractione nul
lus impetus e&longs;t inutilis: brachium infligit maiorem ictum, cùm ma
iorem quia, &longs;cilicet, mouetur motu natu
raliter accelerato: hinc auer&longs;a manu validior impingitur colaphus,
quàm aduer&longs;a; quia illa maiorem arcum de&longs;cribit:
hinc longius bra
chium cæteris paribus grauiùs ferit: hinc diu qua&longs;i rotatur bra
chium, vt longiùs mittatur lapis.
8. Maiore fu&longs;te maior ictus infligitur;
quia potentia toto ni&longs;u
agens, diutiùs manet applicata maiori, quàm minori; &longs;untque ictus
in ratione &longs;ubduplicata vtriu&longs;que fu&longs;tis; v. g. fu&longs;tis pendens vnam
libram per maximum arcum impactus, infligit &longs;ubduplum ictum
alterius, quem infligit fu&longs;tis quatuor pendens libras per eundem
arcum impactus: idem dicatur de mi&longs;&longs;o lapide: principium huius
veritatis pendet ex iis, quæ diximus lib. 2. de motu naturali
ter accelerate, iuxta progre&longs;&longs;ionem numerorum imparium,
1. 3. 5. &c.
9. Fu&longs;tis circa centrum immobile vibratus, maximum ictum in-
lindrus, vel parallelipedum; nec in extremitate mobili;
&longs;ed in eo
puncto, in quo e&longs;t centrum impetus impre&longs;&longs;i, id e&longs;t, quod æqualem
vtrinque dirimit impetum: ratio e&longs;t; quia tunc totus impetus agit,
quantùm pote&longs;t; illud autem punctum Geometria demon&longs;trat e&longs;&longs;e
terminum mediæ proportionalis, inter totum cylindrum, & &longs;ub
duplum; modò nulla ratio vectis habeatur alioquin centrum pro
cu&longs;&longs;ionis di&longs;tat 2/3 ab extremitate immobili.
10. Cùm fu&longs;tis inflectitur, reditque ad pri&longs;tinum &longs;tatum, vt
videre e&longs;t in tudicula maiore, maior ictus imprimitur: quia non
tantùm agit impetus extrin&longs;ecùs adueniens; verùm etiam potentia
quædam media, quæ corpora compre&longs;&longs;a, vel ten&longs;a, ad pri&longs;tinum
&longs;tatum reducit: hinc maximus e&longs;t ictus tudiculæ, cùm eo in&longs;tanti,
quo reductum e&longs;t omninò manubrium priori rectitudini, infligitur
ictus, quia tunc vis potentiæ mediæ e&longs;t maxima.
11. Rotato flagello ideò maxima vis ine&longs;t, quia diutiùs potentia
manet applicata: hinc vides hoc principium e&longs;&longs;e vniuer&longs;ali&longs;&longs;imum,
quod iactis, pul&longs;is, & impactis competit; de malleorum ictu idem
pror&longs;us dicendum e&longs;t, quod de fu&longs;te; &longs;i autem mallei cadant
ex eadem altitudine, motu naturali accelerato, ictus &longs;unt vt
mallei, quia duplus malleus, v. g. duplum impetum acquirit: nam
&longs;ingulæ partes &longs;eor&longs;im æqualem impetum acquirunt.
12. Si verò ex diuer&longs;a altitudine cadant, vel &longs;unt æquales, vel
inæquales: &longs;i primum, ictus &longs;unt vt tempora, quibus cadunt:
&longs;i
&longs;ecundum, ictus &longs;unt in ratione compo&longs;ita temporum, & mal
leorum: &longs;i &longs;unt infinitæ, partes actu, nulla e&longs;t proportio percu&longs;&longs;ionis
granuli cadentis, & rupis ingentis grauitantis; &longs;ed hoc vltimum fal
&longs;um e&longs;&longs;e con&longs;tat; non pote&longs;t tamen determinari proportio vitium
grauitationis, & percu&longs;&longs;ionis, ni&longs;i numerus in&longs;tantium: quibus durat
motus deor&longs;um cogno&longs;catur.
13. Leui&longs;&longs;imi lapides vix emittuntur ad modicam di&longs;tantiam;
quia &longs;tatim &longs;eparantur à potentia: parallelipedum cadens de or
&longs;um in &longs;itu horizontali maximum ictum infligit in centro grauita
tis, id e&longs;t, in medio; quia tunc totus impetus agit, totus enim impe
ditur: in aliis punctis minor e&longs;t ictus, iuxta proportionem maioris
di&longs;tantiæ à prædicto centro: &longs;i verò percutiatur cylindrus innatans,
maxima erit vis, vel effectus ictus in centro grauitatis propter ean
dem rationem.
TRACTATVM hunc de motu locali
ab ip&longs;o impetu au&longs;picamur, ex cuius
profectò cognitione tota res i&longs;ta de
pendet; cum enim impetus &longs;it cau&longs;a
immediata motus, vt fusè demon&longs;tra
bimus infrà; & cum propter quid &longs;it res cogno&longs;ci
non po&longs;&longs;it, ni&longs;i eius cau&longs;a cogno&longs;catur; dubium e&longs;&longs;e
non pote&longs;t, quin præmittenda &longs;it tractatio illa, quæ
e&longs;t de impetu, vt deinde affectiones ip&longs;ius motus
per cau&longs;am eiu&longs;dem demon&longs;trentur; immò au&longs;im
dicere ex vnius impetus cognitione, non modò mo
tum ip&longs;um, verùm etiam totam rem Phy&longs;icam pen
dere.
MOTVS
Huius definitionis explicationem habebis in Metaphy&longs;icâ,
quæ &longs;anè explicatio ad rem præ&longs;entem non facit.
æquale &longs;patium minori tempore; contrà verò motus tardus.
qua est cau&longs;a proxima motus illius mobilis, cui ine&longs;t, eo &longs;cilicet modo, quo
pote&longs;t e&longs;&longs;e cau&longs;a motus.
Dico e&longs;&longs;e qualitatem &longs;iue di&longs;tincta &longs;it, &longs;iue non di&longs;tincta;
quod hîc
certè non di&longs;cutio; nec enim affirmo in hac definitione dari impetum;
&longs;ed definio tantùm quid &longs;it impetus; qui reuera aliud non e&longs;t, &longs;i e&longs;t:
quippe id tantùm concipio, cum impetum appello; &longs;iue &longs;it, &longs;iue non &longs;it,
ne quis fortè initio &longs;tatim mihi litem intendat; quemadmodum definit
circulum Geometra; licèt non a&longs;&longs;erat dari perfectum circulum;
ita Phy
&longs;icus definit impetum, quamuis non affirmet dari impetum; quod tamen
in &longs;exto Theoremate demon&longs;trabimus; itaque &longs;i e&longs;t impetus, haud dubiè
nihil omninò præ&longs;tat in &longs;uo &longs;ubiecto ni&longs;i motum; quod quomodò fiat,
explicabimus intrà in Theorematis.
quis enim non videt volantem auem, natantem
pi&longs;cem; currentem equum, rotatum globum; denique vnum corpus mi
grans è loco in locum? &longs;ed hoc e&longs;t moueri per Def. 1. igitur infinitis fe
rè experimentis nititur hæc hypothe&longs;is, quam veram e&longs;&longs;e nece&longs;&longs;e e&longs;t, &longs;i
illa vera &longs;unt; &longs;ed illa certa &longs;unt phy&longs;icè, neque citra miraculum fallere
po&longs;&longs;unt.
Diceret fortè aliquis etiam motum &longs;ube&longs;&longs;e oculorum fallaciæ; cùm è
naui mobili littus ip&longs;um moueri, ip&longs;umque nauigium non moueri iudi
cemus. Quis enim oculos in Solem intendens, primo intuitu Solem &longs;ta
re non iudicet? cum tamen deinde pernici&longs;&longs;imo cur&longs;u rotari demon&longs;tre
mus; adde alias oculorum fallacias circa motum;
&longs;ic rotata &longs;cintilla, vel
carbo accen&longs;us immotum orbem de&longs;cribere videtur; &longs;ic nota inu&longs;ta
trocho, dum celerrimè rotatur, orbem etiam immobilem de&longs;cribere iu
dicatur; &longs;ic &longs;tella cadens, vel exhalatio continenti &longs;ucce&longs;&longs;ione accen&longs;a
moueri videtur; licet minimè moueatur;
idem dicendum de puluere
tormentario, vel alia qualibet materia; quæ continuata con&longs;ecutione
accenditur; immò trochus ip&longs;e in orbem celerrimè agitatus, quie&longs;cere
videtur; &longs;ic qui vertigine laborant, ea moueri exi&longs;timant, quæ quie&longs;cunt;
idem exemplum habemus in ebrio&longs;is, iracundis, in iis qui ex graui febris
ardore delirant, & in pueris qui diu in gyros eunt, vbi verti de&longs;ierint;
&longs;ic eorum quæ motu æquali feruntur, remotiora tardiùs moueri viden
tur; immò &longs;i per eandem lineam oculus, & mobile pari velocitate ince
dant, ip&longs;um mobile quie&longs;cere videtur, plura leges apud Opticos, de
quibus agemus &longs;uo loco: Igitur ex his omnibus con&longs;tat minimè con&longs;ta
re dari motum, ex eo quòd oculis aliquid moueri videatur.
Re&longs;pondeo equidem fateri me, vi&longs;um ip&longs;um plurimis &longs;ube&longs;&longs;e fraudi
bus; attamen &longs;i rectè oculus admoueatur, iu&longs;ta di&longs;tantià, nec vllum &longs;it
impedimentum exterius nec interius; fieri non pote&longs;t, quin oculus mo
tum ob&longs;eruet; an fortè currentis calami motus oculum meum fallere po-quidquid &longs;it, fateor vltrò hanc hypothe&longs;im in eo tantùm certitudi
nis gradu e&longs;&longs;e reponendam, in quo reponitur hæc cognitio, quâ modo
cogno&longs;co me &longs;cribere, manu&longs;que, & calami motum ob&longs;eruo; &longs;iue id tan
tùm oculis fiat, &longs;iue intellectu ex oculis; quod aliàs di&longs;cutiemus; &longs;i quis
fortè in Phy&longs;ica maiorem certitudinem po&longs;tularet, cum eo certè conue
nire non po&longs;&longs;um.
Porrò quod &longs;pectat ad fallacias illas quæ &longs;upra adductæ &longs;unt;
certum
e&longs;t vel obiectum e&longs;&longs;e remotius, quam par &longs;it; vel moueri celeriùs, vel
e&longs;&longs;e aliquod impedimentum interius; præ&longs;ertim in iis, qui &longs;eu vertigine,
vel alio capitis morbo laborant; &longs;ed ne hîc opticum agere videar, harum
fallaciarum certi&longs;&longs;imas cau&longs;as in &longs;uum locum remittimus.
Cæterùm licèt ad &longs;tatuendam, firmandamque hanc hypote&longs;im, Phy
&longs;ica experimenta rectè applicato &longs;en&longs;u comprobata &longs;ufficere po&longs;&longs;int;
non de&longs;unt tamen rationes multæ à priori, vt vulgò aiunt, quibus euin
citur, non modò quid &longs;it motus, verùm etiam propter quid &longs;it.
Prima duci pote&longs;t à fine motus;
cum enim res creatæ vbique &longs;imul
e&longs;&longs;e non po&longs;&longs;int, certè, vt illo bono gaudeant, quo fortè carent, & vt
coniungantur &longs;uo fini, motu locali opus e&longs;t; &longs;itit equus, abe&longs;t aqua,
certè, ni&longs;i vel hæc propinetur, vel ille accedat, &longs;itim leuare non pote
rit; at neutrum &longs;ine motu haberi pote&longs;t: Lapis remouetur à &longs;uo centro,
à &longs;uo globo, à &longs;uo fine, vt &longs;e&longs;e illi re&longs;tituat, deor&longs;um cadat nece&longs;&longs;e e&longs;t.
Itaque ad cum finem res omnes creatæ in&longs;titutæ &longs;unt, quem &longs;ine motu
a&longs;&longs;equi non po&longs;&longs;unt; igitur dari motum nece&longs;&longs;e e&longs;t, vt res creatæ cum lo
cum acquirant, in quo &longs;uo bono, &longs;uo fini, &longs;uæ perfectioni coniungan
tur; vel &longs;altem id muneris obeant, cui ab ipsâ naturâ de&longs;tinantur.
Secunda ratio ducitur à cau&longs;a efficiente; ni&longs;i enim daretur motus,
fru&longs;trà daretur potentia motrix, tùm in animantibus, tùm in grauibus,
de quâ aliàs.
Tertia petitur à cau&longs;a formali; cum enim detur impetus, vt demon
&longs;trabimus infrà, nece&longs;&longs;e e&longs;t dari motum.
Quarta petitur à termino motus;
cum enim globus proiectus &longs;it in
nouo loco in quo ante non erat; certè nouus locus qui &longs;uccedit alteri
relicto, e&longs;t terminus motus citra miraculum; igitur &longs;i e&longs;t nouus locus,
e&longs;t quoque motus.
Quinta ab v&longs;u; nec enim &longs;ine motu flueret aqua, caderet lapis, gyros
agerent a&longs;tra, flaret ventus, volarent nubes, &c.
Sexta ab ip&longs;a Mechanica, quæ organa motui mini&longs;trat:
quis enim ne
garet maius momentum e&longs;&longs;e cum maiori di&longs;tantiâ coniunctum; &longs;i verò
maius momentum e&longs;t, nunquid præualebit; igitur deor&longs;um cadet, immò
&longs;euerior Geometria, vt omittam A&longs;tronomiam, motum &longs;upponit, cum ex
fluxu &longs;eu motu puncti infinitas fere lineas de&longs;cribat. Igitur certum e&longs;t
dari motum localem.
con&longs;ule &longs;en&longs;us rectè applicatos; tam enim certus &longs;um me iam in cathe
dra quie&longs;cere, quam &longs;um certus Solem lucere; igitur ex certis experi
mentis certa hypothe&longs;is con&longs;equitur. Non de&longs;unt rationes à priori; nam
primò res aliqua &longs;uo bono, &longs;eu fini coniuncta ab eo &longs;eparari non po&longs;tu
lat, igitur nec moueri. Secundò maximum incommodum e&longs;&longs;et, &longs;i res &longs;e
mel mota perpetuò moueretur. Tertiò, finis, &longs;eu terminus motus recti,
e&longs;t quies; nam ideo lapis deor&longs;um cadit, vt in &longs;uo centro &longs;eu globo
quie&longs;cat, id e&longs;t vt cum aliis partibus totum illud, &longs;eu globum componat,
vt dicemus aliàs.
Diceret fortè aliquis &longs;ententias prædictas non valere in &longs;ententiâ
Copernici, quæ terræ motum ad&longs;truit; præterea non modò falli &longs;en&longs;us
circa motum, verùm etiam circa quietem.
Re&longs;pondeo primò illam Copernici &longs;ententiam e&longs;&longs;e fal&longs;i&longs;&longs;imam, vt &longs;uo
loco o&longs;tendemus: &longs;ecundò, licèt terra moueretur &longs;ecundum Coperni
cum, Sol, & &longs;tellæ quie&longs;cerent.
Dices iuxta hypothe&longs;im Heraclidis Pontici, terra ip&longs;a, Sol etiam, &
&longs;tellæ mouentur. Re&longs;pondeo primò hypothe&longs;im illam e&longs;&longs;e fal&longs;am, vt &longs;uo
loco videbimus; &longs;ecundò etiam data illa hypothe&longs;i po&longs;&longs;et dari quies;
&longs;i
enim globus eodem ver&longs;us occa&longs;um impetu proiiceretur, quò ver&longs;us or
tum à terra ip&longs;a rapitur, haùd dubiè quie&longs;ceret: præterea iuxta hanc hy
pothe&longs;im, quietem appellarem vnius partis cum alia connexionem in ip
&longs;o toto &longs;eu globo, & quie&longs;cere dicerem lapidem, qui tantùm totius glo
bi motu mouetur, ex quo profectò tota &longs;oluitur difficultas.
Quod verò &longs;pectat ad fallacias oculi circa quietem;
eodem pror&longs;us
modo &longs;oluendæ &longs;unt, quo iam &longs;upra &longs;olutæ &longs;unt aliæ circa motum:
vtrùm verò motus, & quies dicant aliquid di&longs;tinctum à mobili, dice
mus infrà.
Video lapidem quie&longs;centem,
qui deinde proiectus mouetur; igitur ante non mouebatur, igitur cum
deinde mouetur, cœpit moueri; mille aliis experimentis hæc hypothe
&longs;is confirmari pote&longs;t.
Vi
deo rotatam pilam, quæ tandem quie&longs;cit, cadentem lapidem, qui tan
dem &longs;i&longs;tit, &c. igitur certa e&longs;t hæc hypothe&longs;is.
Video rotatum globum,
qui &longs;en&longs;im quie&longs;cit: &longs;entio ab eodem globo modò maiorem, modò mi
norem ictum infligi, &c. igitur e&longs;t certa hypothe&longs;is.
Oculos omnium te&longs;tes appello.
cit, nix frigefacit, Sol illuminat, corpus in aliud impactum illud ip&longs;um im
pellit. Prædictæ omnes Hypothe&longs;es certi&longs;&longs;imis nixæ experimentis certi
tudinem phy&longs;icam habent, & citra miraculum fallere non po&longs;&longs;unt.
Hoc ip&longs;um iam præ
mi&longs;imus Logicæ no&longs;træ demon&longs;tratiuæ, complectiturque prima illa
principia Metaphy&longs;icæ.
1.
2.
3.
negatur, non &longs;imul vtrumque.
po&longs;itio. Nihil enim a &longs;e ip&longs;o &longs;eparari po&longs;t;
quippe, vbi e&longs;t &longs;eparatio, &longs;eu
diui&longs;io, e&longs;t pluralitas; cur enim nummus A & nummus B eiu&longs;dem ma
teriæ, formæ, ponderis, realiter di&longs;tinguuntur? quia &longs;cilicet vnus
non e&longs;t alius inquies; & quare vnus non e&longs;t alius? quia vnus e&longs;t hic &
alius non e&longs;t hic, vnum tango, & alium non tango, vnus e&longs;t meus, &
alius non e&longs;t meus, &c. vides prædicata contradictoria, quæ cum eidem
&longs;imul ine&longs;&longs;e non po&longs;&longs;int per Ax. 1. diuer&longs;is, & di&longs;tinctis ine&longs;&longs;e nece&longs;&longs;e
e&longs;t.
Diceret fortè aliquis hominem reproductum in duobus locis e&longs;&longs;e po&longs;
&longs;e, & dum Romæ e&longs;t à &longs;e ip&longs;o Lugduni exi&longs;tente &longs;eiunctum e&longs;&longs;e; hoc
ip&longs;um aliàs examinabimus, dum con&longs;tet modò id totum, &longs;i fiat, mira
culo tribuendum e&longs;&longs;e, cum tamen res phy&longs;icas citra miraculum con&longs;ide
remus.
probari. Qui enim a&longs;&longs;erit rem aliquam po&longs;itiuam exi&longs;tere, certè po&longs;i
tiuo argumento demon&longs;trare debet quod &longs;it; illud porrò argumentum
duci pote&longs;t vel ab experimento certo; &longs;ic probo exi&longs;tere rem aliquam,
quam video; vel ab aliqua ratione; &longs;ic ex eo quòd cau&longs;a &longs;it nece&longs;&longs;aria
applicata &longs;ubiecto apto, probo effectum ip&longs;um produci; vel eo quòd &longs;it
effectus probo cau&longs;am e&longs;&longs;e vel ex nece&longs;&longs;itate, quâ aliquid e&longs;t nece&longs;&longs;a
rium ad aliquem finem à natura in&longs;titutum, quo natura ip&longs;a &longs;ine ab&longs;ur-
e&longs;&longs;e; vel demùm ex aliqua reuelatione certa in rebus fidei;
igitur hoc
Axioma certum e&longs;t phy&longs;icè; quod ni&longs;i recipiatur à Philo&longs;ophis;
cuique
licebit impunè mentiri; &longs;i enim dicam extra mundi huius fines e&longs;&longs;e
alios orbes, intra tuum mu&longs;æum, in quo &longs;olus fortè degis, e&longs;&longs;e quin
quaginta homines, e&longs;&longs;e mille Soles, & totidem Lunas in cœlo, &c.
numquid &longs;tatim oppones Axioma i&longs;tud,
nece&longs;&longs;itas, qua reuelatio?
reuoca principium illud commune.
1.
addas, vel &longs;ine ratione, vel &longs;ine experientia.
2.
terminatum e&longs;t.
Hoc Axioma explicatione modicâ indiget:
Determinatum illud
apello, quod illud ip&longs;um e&longs;t, quod e&longs;t, & nihil aliud; quod e&longs;t hoc, id
e&longs;t ab omni alio di&longs;tinctum; atqui quidquid productum e&longs;t, &longs;ingulare
e&longs;t, id e&longs;t, e&longs;t hoc; &longs;i enim producitur, alicubi producitur, & ali
quando, ergo dici pote&longs;t, e&longs;t hîc, e&longs;t nunc; igitur determinatum e&longs;t.
Aliquis fortè &longs;tatim opponet mihi partes indeterminatas quantitatis: &longs;ed
pro&longs;ectò nulla pars actu e&longs;t quæ non &longs;it hæc, & non alia; igitur quæ
non &longs;it determinata, de quo aliàs; quidquid &longs;it, &longs;altem partes illæ fa
ciunt aliquod totum quod e&longs;t determinatum, quod mihi &longs;atis e&longs;t modò
ad veritatem huius Axiomatis. Dices aliquid po&longs;&longs;e e&longs;&longs;e nullibi;
has
nugas refutabimus in Metaphy&longs;ica, quæ in mentem &longs;apientis viri ca
dere non po&longs;&longs;unt; nunc &longs;altem con&longs;tat id naturali modo fieri non
po&longs;&longs;e.
Quia quod vnum e&longs;t, e&longs;t hoc, &
nihil aliud; nihil enim aliud e&longs;t vnum, ni&longs;i indiui&longs;um in &longs;e, & diui
&longs;um à quolibet alio: quippè indifferentia, vel indeterminatio ibi tan
tum e&longs;t, vbi &longs;unt plura; &longs;i enim tantum vnum e&longs;t, certè non datur op
tio, &longs;i aliqua cau&longs;a e&longs;t indifferens ad effectum A & B, id e&longs;t &longs;i non e&longs;t,
cur vnum potius quàm alium producat? plures e&longs;&longs;e nece&longs;&longs;e e&longs;t; &longs;i enim
tantùm vnus e&longs;t, certè indifferens non e&longs;t.
Quidquid e&longs;t, id e&longs;t exi&longs;tit naturaliter
&longs;cilicet, & citra miraculum, fru&longs;trà non e&longs;t, id e&longs;t propter aliquem fi
nem e&longs;t ab ip&longs;a natura in&longs;titutum; finem autem rei ex ip&longs;o v&longs;u cogno
&longs;cimus; v&longs;um verò ip&longs;o ferè &longs;en&longs;u:
quod vt breui inductione confirme
mus, quidquid exi&longs;tit vel e&longs;t &longs;ub&longs;tantia, vel accidens; &longs;i &longs;ub&longs;tantia, vel
incorporea, vel corporea; &longs;i incorporea, vel e&longs;t Deus, vel Angelus, vel
atqui nihil horum fru&longs;trà e&longs;t, vt con&longs;tat;
&longs;i corporea,
vel e&longs;t corpus, vel forma; &longs;i corpus, vel elementum, vel mixtum;
vtrumque &longs;uum finem habet, & con&longs;tantem v&longs;um; &longs;i forma quamdiu
e&longs;t principium actionum compo&longs;iti fru&longs;trà non e&longs;t; quippe ad cum finem
e&longs;t in&longs;tituta; hinc optima ratio ducitur, cur forma materialis &longs;eparata
exi&longs;tere non po&longs;&longs;it citra miraculum, quia &longs;cilicet fru&longs;trà e&longs;&longs;et; cum enim
non po&longs;&longs;it agere ni&longs;i in &longs;ubiecto, &longs;i &longs;ubiectum non e&longs;t, fru&longs;trà e&longs;t; at verò
anima rationalis, quæ aliquas actiones in organicas habet, fru&longs;trà non
e&longs;t etiam &longs;eparata, igitur immortalis e&longs;t: vtramque rationem &longs;uo loco fu
sè demon&longs;trabimus; &longs;i verò accidens e&longs;t, haud dubiè alteri ine&longs;&longs;e debet
propter &longs;uum finem intrin&longs;ecum, quem alibi effectum formalem &longs;ecun
darium appellamus; quem &longs;cilicet præ&longs;tat in &longs;uo &longs;ubiecto, cui certè &longs;i ni
hil præ&longs;taret, in eo fru&longs;trà e&longs;&longs;et; &longs;ic caloris effectus &longs;ecundarius e&longs;t rare
factio, vel re&longs;olutio partium &longs;ui &longs;ubiecti, vel aliquid aliud; impetus,
motus &c. Igitur tunc effet fru&longs;trà accidens, cum &longs;uo illo effectu careret;
hinc rationem contrarietatis aliquando petemus, certi&longs;&longs;imam quidem,
licet nouam, & inde clari&longs;&longs;imè con&longs;tabit, cur, & quomodo vnum contra
rium ab alio de&longs;trui dicatur; &longs;ed non e&longs;t huius loci: cùm verò audis fi
nem: ne quæ&longs;o cogites aliquid morale, nec enim illum finem intelligo, ad
quem ab agente rationabili de&longs;tinatur: &longs;ed eum dumtaxat, ad quem na
tura ip&longs;a, vel e&longs;&longs;entia rei &longs;pectat, &longs;ed de his &longs;atis.
Huc reuoca Principium illud,
id e&longs;t quod &longs;uo fine careat intrin&longs;eco.
Dices fortè, multa videri e&longs;&longs;e fru&longs;trà, quæ tamen exi&longs;tunt; ad quid
enim vel tanta aquarum copia, vel tantus &longs;tellarum numerus, vel tot are
næ puncta? tot fluitantes atomi?
tot in&longs;ecta?
& vermiculi:
Re&longs;pondeo
quamlibet &longs;tellam, quodlibet in&longs;ectum, &longs;eu vermiculum &longs;uis pollere pro
prietatibus; igitur fru&longs;trà non e&longs;t, & quodlibet punctum, quamlibet ato
mum, & quamlibet guttulam aquæ e&longs;&longs;e partem huius vniuer&longs;itatis: quod
enim dices de vna, dicam de omnibus; equidem pauciores e&longs;&longs;e po&longs;&longs;ent;
attamen nulla e&longs;t fru&longs;trà, cum quælibet &longs;imul cum aliis totum hoc com
ponat.
qua proprietas &longs;en&longs;ibilis, quæ non pote&longs;t tribui ip&longs;i materiæ,
naturales con&longs;idero, nec &longs;uper naturales attingo, quæ &longs;uas regulas diui
næ fidei debent, non &longs;en&longs;ibus.
Hoc Axioma omninò certum e&longs;t, & per Ax. 3. confirmatur, vt enim
dicas aliquid di&longs;tinctum ab omni alio exi&longs;tere, vel debet id &longs;en&longs;u percipi,
vel aliqua ratione probari quod &longs;it; atqui formam &longs;ub&longs;tantialem &longs;en&longs;u
non percipis immediatè; igitur aliquem eius effectum &longs;en&longs;ibilem vel me
diatè, vel immediatè; qui certè &longs;i tribui po&longs;&longs;it materiæ, haud dubiè per il
lum formam non probabis, ni&longs;i formæ ip&longs;ius e&longs;&longs;e antè demon&longs;tres; &longs;i ve
to e&longs;t forma accidentalis, quam &longs;en&longs;u percipis; certè id tantùm accidit ex
lo illam percipis, quod clarum e&longs;t.
Huc reuoca vulgare illud principium,
test fieri per pauciora,quod ita in
telligi non debet, vt &longs;ine gutta aquæ Oceanus, &longs;ine &longs;tella cœlum, &longs;ine gra
nulo arenæ terra, &longs;ine altero oculorum homo &longs;tare non po&longs;&longs;int; quæ
omnia fal&longs;i&longs;&longs;ima e&longs;&longs;e con&longs;tat; &longs;ed tantùm quod illud dicatur exi&longs;tere &longs;iue
&longs;it &longs;ub&longs;tantia, &longs;iue accidens, quod vel experientia certa euincit, vel nece&longs;
&longs;itas, vel ratio, vel diuina fides (immò & humana in rebus humanis, non
tamen in &longs;cientiis.)
Igitur nunquam claudicat hic equus Okami, vt vulgò dicitur, &longs;i hoc
fræno regatur, & præ&longs;cripto ambulet pa&longs;&longs;u.
Ob&longs;eruabis &longs;eptem præmi&longs;&longs;a Axiomata, licet metaphy&longs;ica &longs;altem ali
qua ex parte e&longs;&longs;e videantur, ita pertinere ad Phy&longs;icam, vt plurimæ phy
&longs;icæ affectiones &longs;ine illis explicari, & demon&longs;trari non po&longs;&longs;int.
Primum certum e&longs;t etiam certitudine metaphy&longs;ica, &longs;eu geometrica.
Secundum, Quartum, & Quintum per Primum demon&longs;trari po&longs;&longs;unt.
Tertium e&longs;t veluti communis po&longs;itio, &longs;eu commune po&longs;tulatum, in quo
docti omnes conunciunt; quippe nihil &longs;ine ratione dici debet à philo&longs;o
pho; Sextum & Septimum probari po&longs;&longs;unt per Tertium; &longs;ed iam ad
alia, quæ propiùs ad phy&longs;icam accedunt, veniamus.
Id e&longs;t quid
quid incipit e&longs;&longs;e ab alio e&longs;t; quippe à &longs;e e&longs;&longs;e non pote&longs;t;
nihil enim à &longs;e
ip&longs;o dependere pote&longs;t &longs;eu produci; quia quod à &longs;e e&longs;t, nece&longs;&longs;ariò e&longs;t,
quod verò nece&longs;&longs;ariò e&longs;t, non e&longs;&longs;e non pote&longs;t, alioquin priùs e&longs;&longs;et, &
po&longs;terius, priùs vt cau&longs;a, po&longs;teriùs vt effectus: præterea quidquid produci
tur aliquando producitur, & alicubi, vt certi&longs;&longs;imum e&longs;t; &longs;ed quia hoc ali
qui negant, contendo tantùm in hoc rerum ordine, & naturaliter lo
quendo, quidquid producitur alicubi produci, & aliquando, quod nemo
negabit; Igitur &longs;i aliquid &longs;e producit; cur hîc potiùs quam illîc?
cur
nunc potius quam antè? cum enim antè nullibi e&longs;&longs;et, cur de&longs;init non
e&longs;&longs;e hîc & non illîc, nunc & non antè? hinc quod à &longs;e e&longs;t, vbique, &
&longs;emper e&longs;t, &longs;ed ne quis mihi litem intendat, licet hoc Axioma certitudi
nem geometricam habeat; &longs;ufficit modò habere phy&longs;icam, quod ex om
nibus hypothe&longs;ibus demon&longs;tratur; &longs;i enim aliquid de nouo produci
tur, quod certum e&longs;t, ab alio produci video: calor ab igne mediatè
vel immediatè, impetus à potentia motrice, vel ab alio impetu: cuncta
hæc &longs;i reuera producuntur de quo alibi, ab alio produci con&longs;tat; in Me
taphy&longs;ica hoc ip&longs;um geometricè demon&longs;trabimus; cum enim agere &longs;up
ponat e&longs;&longs;e; quippe omnis actio alicuius agentis e&longs;t;
& cum agere termi
netur ad effectum, nam fieri e&longs;t alicuius fieri; certè agens, & terminus,
cau&longs;a, & effectus di&longs;tinguuntur, igitur.
Hoc certum e&longs;t;
quia agere
&longs;upponit e&longs;&longs;e; quippe agere e&longs;t perfectio realis actu exi&longs;tens; igitur ali
cuius actu exi&longs;tentis; igitur certum e&longs;t etiam Geometricè, de quo in
Metaph. Iam vero &longs;ufficiat certum e&longs;&longs;e phi&longs;icè, vt con&longs;tat ex omnibus
hypoth. phy&longs;icis;
nihil enim videmus agere, ni&longs;i quod e&longs;t;
&longs;i enim age
ret quod non e&longs;t; cur potius hîc, & nunc quam alibi, & aliàs? cur in
hoc &longs;ubiecto potius quàm in alio?
Dices, finis qui non e&longs;t influit;
igitur agit;
Re&longs;pondeo finem non
agere, nec influere ni&longs;i obiectiuè; atqui quod non exi&longs;tit actu, id e&longs;t in
&longs;tatu entatiuo, & reali, pote&longs;t e&longs;&longs;e in &longs;tatu obiectiuo; id e&longs;t quod non
habet actum rei, pote&longs;t habere actum obiecti, id e&longs;t e&longs;&longs;e cognitum, &
volitum, de quo aliàs; porrò hîc tantùm intelligimus cau&longs;am efficien
tem, &c.
Dices, cau&longs;a principalis pulli exclu&longs;i pote&longs;t non e&longs;&longs;e;
hæc omnia di
&longs;cutiemus &longs;uo loco cum de generatione animalium; &longs;ufficiat dixi&longs;&longs;e non
e&longs;&longs;e cau&longs;am immediatam, de qua hîc tantum loquimur; idem re&longs;pon&longs;um
e&longs;to de rana vaga.
Cur enim potiùs hîc
quam illîc; in hoc &longs;ubiecto potiùs, quam in alio, in hac di&longs;tantia potiùs,
quam in alia? quidquid &longs;it, certum e&longs;t phy&longs;icè; nec enim ignis, qui e&longs;t
Romæ, calefacit Lugduni.
Dices dari fortè actionem in di&longs;tans;
Re&longs;pondeo negando, quod de
mon&longs;trabimus in Metaph. præterea, licet daretur in productione quali
tatum occultarum, & &longs;impathicorum quorundam effectuum, quos exa
minabimus &longs;uo loco; nemo tamen dubitat quin productio caloris, lu
minis, impetus; de quibus hic tantùm agimus, debeat e&longs;&longs;e ab applicata
cau&longs;a.
Dices impetum produci in extremitate perticæ, quæ non e&longs;t applica
ta, vel in globo tudiculario etiam non applicato; calorem & lucem
produci à Sole in terra non applicata. Re&longs;pondeo, e&longs;&longs;e applicationem
mediatam; nam &longs;i reuera hæ qualitates producuntur continuata propa
gatione, diffunduntur per medium, in quo non e&longs;t difficultas.
Dices etiam partes interiores cau&longs;æ v. g. Solis agunt, &longs;ed non agunt
per totum medium; alioquin agerent in alias partes Solis, à quibus
obteguntur. Re&longs;pondeo, diffu&longs;ionem vel propagationem actionis in
choari tantum ab ipsâ &longs;uperficie Solis; quippe omnes partes agunt
actione communi, de quo infrà; atqui actio communis à communi me
dio incipit.
Dices ignem produci in parte medij remota interrupta propagatio
ne, vt con&longs;tat, &longs;i vitro per refractionem, vel &longs;peculo per reflectionem
radios Solares colligas.
Re&longs;pondeo, ignem quidem accendi in data di&longs;tantia;
at non &longs;ine
quomo
do vero ignis accendatur, & quid &longs;it ignem accendi, explicabimus &longs;uo
loco; quidquid &longs;it, certum e&longs;t ad productionem impetus requiri ali
quam applicationem, vt patet etiam in magnete.
effectus, non e&longs;t ponenda alia &longs;cilicet æquiuoca. Non dico omnem cau&longs;am
e&longs;&longs;e vniuocam, &longs;ed tantùm vniuocam &longs;ufficientem, & applicatam e&longs;&longs;e
cau&longs;am, v. g. calor e&longs;t cau&longs;a &longs;ufficiens caloris, vt con&longs;tat in aqua calida;
igitur &longs;i calor e&longs;t applicatus &longs;ubiecto, in quo producitur calor non &longs;upe
rans vires caloris applicati; dicendum e&longs;t calorem illum ab hoc produ
ci; cum calor &longs;it cau&longs;a nece&longs;&longs;aria;
igitur &longs;i &longs;it applicatus &longs;ubjecto apto,
nece&longs;&longs;ariò agit; igitur quantum pote&longs;t; igitur effectus non e&longs;t tribuen
dus alteri cau&longs;æ, quam &longs;ufficientem e&longs;&longs;e ignoramus.
Ad hoc Axioma aliud reuoca.
per effectus aliquis, illud ip&longs;um cau&longs;a dici debet huius effectus; licet aliud &longs;it
coniunctum, ex quo &longs;eor&longs;im &longs;umpto applicato non &longs;equitur effectusv. g. ex
applicatione aquæ calidæ &longs;equitur productio caloris; ex applicatione &longs;o
lius aquæ non &longs;equitur; igitur dicendum e&longs;t calorem hunc produci ab
ip&longs;o calore, qui aquæ ine&longs;t, non verò ab ip&longs;a aquæ &longs;ub&longs;tantia; idem dico
de ferro frigido, &c.
Dices non e&longs;&longs;e certum calorem produci; Re&longs;pondeo, negando;
&longs;ed,
quidquid &longs;it, loquor tantùm hypotheticè; dixi enim &longs;i producatur, à
calore aquæ inhærente producitur.
Dices produci po&longs;&longs;e ab aliqua cau&longs;a ignota po&longs;ita dumtaxat tali, vel
tali conditione. Re&longs;pondeo, hoc reuera geometricè non probari, &longs;ed
tantùm phy&longs;icè; quidquid &longs;it, voco cau&longs;am id, ex cuius applicatione
&longs;equitur &longs;emper effectus, & nunquam aliàs; nam phy&longs;icè loquendo, &longs;iue
&longs;it alia cau&longs;a, &longs;iue non, eodem modo &longs;e habet, ac &longs;i e&longs;&longs;et cau&longs;a; quippe
certum e&longs;t phy&longs;icè ignem calefacere, Solem illuminare, quod &longs;atis e&longs;t.
quantum pote&longs;t. Hoc Axioma duas partes habet;
prima certa e&longs;t per hy
poth. 8. & per definitionem cau&longs;æ nece&longs;&longs;ariæ, quæ in hoc differt à libe
râ: Secunda pars probatur; quia &longs;i partem effectus omitteret, quam ta
men ponere po&longs;&longs;et; haud dubiè non e&longs;&longs;et cau&longs;a nece&longs;&longs;aria contra hypoth.
nam &longs;i vnam partem effectus omittat; cur vnam potiùs quam aliam?
cur non duas? cur non omnes?
denique video cau&longs;am eandem eidem
&longs;ubiecto eodem modo applicatam, eundem &longs;emper effectum producere
per Hyp. 8.
Quælibet pars maioris
ignis non habet calorem inten&longs;iorem, quàm quælibet pars minoris; idem nec enim libra plumbi coniuncta cum
alia habet diuer&longs;am grauitatem ab eâ, quam habet &longs;eparata.
Dixi ad intra;
quia ad extra multum iuuat exten&longs;io;
&longs;ic maior ignis
longiùs diffundit &longs;uum calorem; corpus grauiùs cadens majorem ictum
infligit; Ad hoc Axioma reuocatur i&longs;tud.
1.
eum modum quo illam explicabimus in Metaph. nec punctum Solis &longs;e
paratum ad eandem di&longs;tantiam &longs;uam lucem, caloremque &longs;uum diffunde
ret; ad quam diffundit coniunctum cum aliis;
idem dico de igne maiori,
& minori; de quibus omnibus &longs;uo loco. Huc etiam reuoca dicta illa
communia.
2.
3.
& vici&longs;&longs;im.
4.
perfectiorem, &longs;uppo&longs;itâ eâdem applicatione; &longs;i enim maior e&longs;t applicatio &longs;ine
ratione loci, &longs;iue ratione temporis; haud dubiè maior erit effectus, vt con&longs;tat.
Hoc Axioma geometricum e&longs;t;
Quod
enim e&longs;t à &longs;e, nece&longs;&longs;ariò e&longs;t; cùm à libertate &longs;eu voluntate alterius non
pendeat; cum enim primo in&longs;tanti quo res e&longs;t, non &longs;it à &longs;e per Axiom. 8.
de &longs;ecundo idem dici debet, quod de primo, vt patet: quippe id eo
primo in&longs;tanti non e&longs;t nece&longs;&longs;ariò, quia ita e&longs;t illo in&longs;tanti, vt po&longs;&longs;it non
e&longs;&longs;e; &longs;ed etiam &longs;ecundo in&longs;tanti ita e&longs;t vt po&longs;&longs;it non e&longs;&longs;e; igitur non e&longs;t
nece&longs;&longs;ariò, igitur pendet ab alio, quod pote&longs;t facere vt non &longs;it.
Dices po&longs;&longs;e de&longs;trui &longs;ecundo in&longs;tanti ab aliquo contrario, à quo tamen
non pendet per po&longs;itiuum influxum. Re&longs;pondeo, non videri quomo
do de&longs;trui po&longs;&longs;it, quod influxu po&longs;itiuo non indiget, vt &longs;it; quid enim
faceret contrarium, quod tantùm exigere pote&longs;t contrarij de&longs;tructio
nem, quid e&longs;t porro de&longs;trui, ni&longs;i de&longs;inere con&longs;eruari? quæ omnia fusè
in Metaphy&longs;ica demon&longs;trabimus; quidquid enim e&longs;t aliquo in&longs;tanti vel
e&longs;t à &longs;e, vel non à &longs;e; &longs;i primùm Deus e&longs;t; &longs;i &longs;ecundum ab alio e&longs;t:
quidquid &longs;it, hoc Axioma certum e&longs;t phy&longs;icè.
Huc reuoca Axiomata &longs;equentia, quæ ex hoc vno deducuntur.
1.
Hæc enim &longs;unt idem, vt con&longs;tat.
2.
natura, ne aliquid &longs;it fru&longs;trà. Hoc etiam ex hypothe&longs;ibus &longs;equitur;
cum
enim de&longs;trui &longs;it idem ac de&longs;inere con&longs;eruari; certè qui de&longs;init con&longs;er
uare in&longs;tanti A potiùs quam in&longs;tanti B, hoc facere non pote&longs;t ni&longs;i ali
quid hoc exigat; &longs;cilicet iuxta leges naturæ.
3.
Hoc &longs;equitur ex priori, id e&longs;t quandiu e&longs;t eadem ratio, cur &longs;it, & con
&longs;eruetur, quæ erat antè.
Nec enim alia regula e&longs;&longs;e pote&longs;t;
&longs;ic minor
calor minùs de&longs;truit frigoris; minor impetus minùs de&longs;truit impetus
contrarij (&longs;i contrarium habet) quæ omnia con&longs;tant ex hypothe&longs;ibus.
Ratio e&longs;t, quia plùs vel minùs contrarij de&longs;truere, multam habet ex
ten&longs;ionem. v.g. &longs;int duo contraria A & B, &longs;it A vt 20. &longs;it B vt 5. certè &longs;i
B de&longs;truat A &longs;upra ratam, vel &longs;upra id, quod &longs;ibi ex æquo re&longs;pondet, id
e&longs;t &longs;upra 5. cur potius 6. quam 7. 8. &c. Si infra, cur potius 4. quam 3.
2. &c. Igitur cum plures &longs;int termini tùm infra, tùm &longs;upra 5. cur potius
vnus quàm alius? atqui vnus tantùm ex æquo re&longs;pondet, &longs;cilicet 5. &longs;ed
quod vnum e&longs;t determinatum e&longs;t, per Axioma 5. igitur pugnant pro
rata. Nec dicas A totum de&longs;trui à B, quòd e&longs;t contra hypothe&longs;im, nam
modicum caloris non de&longs;truit totum frigus: in impetu res e&longs;t clari&longs;&longs;ima;
adde quod minor cau&longs;a minùs agit per Ax. 13. num. 3. igitur minùs exi
git; porrò cum dico vnum ab alio de&longs;trui, intelligo tantùm ex applica
tione vnius &longs;equi de&longs;tructionem alterius &longs;altem ex parte.
Ob&longs;eruabis hæc Axiomata &longs;altem maiori ex parte e&longs;&longs;e metaph.
quæ
nos fusè in Theorematis metaph. explicabimus, & demon&longs;trabimus;
&longs;ed
nobis hoc loco &longs;atis e&longs;t, &longs;i parem cum phy&longs;icis &longs;upponas habere cer
titudinem, quod nemo negabit; con&longs;tátque ex hypothe&longs;ibus, licèt ma
iorem etiam habeant, de qua &longs;uo loco.
Ob&longs;eruabis prætereà nos diutiùs hæ&longs;i&longs;&longs;e in præmittendis huic libro
Axiomatis, quod tamen in aliis libris non faciemus.
durationem &longs;en&longs;ibilem, &longs;patiumque &longs;en&longs;ibile, metiri, comparare, &c.
Demon&longs;tratur; Motus
e&longs;t in mobili, in quo antè non erat per hypoth. 3. & de&longs;init e&longs;&longs;e in mobili,
in quo antè erat per hypoth.4. igitur mobile e&longs;t, & non e&longs;t motus; igi
tur à motu &longs;eparatum; igitur realiter di&longs;tinctum per Ax. 2. præterea
moueri, & non moueri &longs;unt prædicata contradictoria, vt con&longs;tat; igi
tur eidem &longs;imul ine&longs;&longs;e non po&longs;&longs;unt per Ax. 1. igitur cum eo non &longs;unt
idem; alioquin &longs;imul e&longs;&longs;ent;
igitur alterum illorum e&longs;t di&longs;tinctum à
mobili; non quies, vt con&longs;tat, quæ e&longs;t tantùm negatio motus, &longs;eu per
&longs;euerantia in eodem loco; igitur nullam dicit mutationem; at verò
motus mutationem dicit, per Def. 1. hoc Theorema fusè demon&longs;trabo
in Metaph.
tus cau&longs;æ efficientis. Demon&longs;t.
Motus e&longs;t mutatio, &longs;eu tran&longs;itus ex loco
in locum per Def. 1. &longs;ed mutatio propriè non producitur; quippè pro
ductio tantùm terminatur ad ens; nihil enim ni&longs;i ens produci pote&longs;t;
præ&longs;ertim hæc, quæ tantùm dicit
terminum à quo, ide&longs;t locum relictum; & terminum ad quem, id e&longs;t lo
cum immediatum acqui&longs;itum; nam &longs;eparato quocunque alio ab ip&longs;o
mobili; modo &longs;imul, id e&longs;t eodem in&longs;tanti relinquat primum locum, &
nouum acquirat, omninò mouetur, &longs;ed concretum illud ex loco relicto,
& acqui&longs;ito produci non pote&longs;t; illud autem e&longs;t motus, qui certè non
dicit tantùm locum relictum &longs;ine acqui&longs;ito; alioqui &longs;i mobile de&longs;true
retur, diceretur moueri; nec etiam locum acqui&longs;itum &longs;ine priori relicto:
alioqui &longs;i mobile primò produceretur, diceretur moueri localiter; igitur
motus neutrum dicit &longs;eor&longs;im; &longs;i primum, diceretur de&longs;tructus; &longs;i &longs;ecun
dum, diceretur aliquo modo productus, vel potiùs acqui&longs;itus; at vtrum
que coniunctim, &longs;imulque e&longs;&longs;entialiter dicit motus; nec enim conci
pio aliud, dum concipio motum: porrò vtrumque &longs;imul &longs;umptum indi
ui&longs;ibiliter non pote&longs;t dici, vel de&longs;tructum propriè, vel productum; Di
xi propriè; nam impropriè dici pote&longs;t motus productus.
Dices Motus e&longs;t ens, non à &longs;e; igitur ab alio; igitur motus e&longs;t pro
ductus. Re&longs;pondeo Motum non e&longs;&longs;e ens ab&longs;olutum, &longs;ed e&longs;&longs;e mutatio
nem entis, quæ mutatio e&longs;t concretum quoddam ex ente & non ente;
quòd certè non pote&longs;t dici propriè productum, &longs;ed re&longs;ultans, vt relatio;
nam producatur, &longs;i fieri pote&longs;t; certè e&longs;t aliquid, quod tam facilè de
&longs;trui pote&longs;t, quam produci; igitur de&longs;truatur, & remaneat tantùm en
titas mobilis, quæ, quo in&longs;tanti priorem locum relinquit, nouum acqui
rat; certè dicitur adhuc moueri, & tamen non erit motus ex &longs;uppo&longs;itio
ne, quod ab&longs;urdum e&longs;t.
Dices potentia motrix e&longs;t actiua;
igitur agit; igitur producit, &longs;ed ni
hil ni&longs;i motum. Re&longs;p. potentiam motricem e&longs;&longs;e actiuam vt dicemus,
& ab eâ produci impetum, qui deinde exigit motum, vt dicemus
infrà.
Nec e&longs;t quod aliqui ita mirentur hæc à me dici;
cum certum &longs;it effe
ctus formales &longs;ecundarios principum ferè qualitatum tales e&longs;&longs;e, vt mini
mè producantur; &longs;ed qua&longs;i re&longs;ultent ab exigentia; v. g. effectus calo
ris in &longs;uo &longs;ubiecto e&longs;t eiu&longs;dem &longs;ubiecti rarefactio, quæ reuerâ non
producitur, vt con&longs;tat.
Demon&longs;tratur, quia
motus, qui non erat, incipit e&longs;&longs;e per hypothe&longs;im tertiam; &longs;ed quod
huiu&longs;modi e&longs;t, habet cau&longs;am di&longs;tinctam per Ax.8.
Ob&longs;eruabis motum localem e&longs;&longs;e duplicis generis;
primum genus mo
tus e&longs;t actio potentiæ motricis, quæ reuerà mouet, & cuius exercitium
dicitur motus, &longs;eu latio, &longs;eu motio, &longs;eu actio, qua reuerâ agit, produ
citque impetum, non motum; cum etiam &longs;ine motu defatigetur, vt cum
quis alium pellit, à quo pellitur æquali ni&longs;u; patet etiam in manu &longs;u
&longs;tinente aliquod pondus, quæ non mouetur; licet reuerâ etiam &longs;ummo
immò &longs;i potentia motrix produceret motum primum, non
impetum in corpore proiecto; nulla deinde e&longs;&longs;et cau&longs;a applicata ad pro
ducendum impetum: Itaque hic motus primi generis, &longs;i comparetur
cum potentia motrice, e&longs;t verè influxus, vel actio; &longs;i cum termino, e&longs;t
eius fieri, &longs;eu dependentia; &longs;i cum &longs;ubiecto, &longs;eu mobili e&longs;t pa&longs;&longs;io;
nec
propriè dicitur produci, ni&longs;i vt quo (vt vulgò loquuntur) nec enim
actio e&longs;t terminus, vel effectus, in quo &longs;i&longs;tat cau&longs;a; &longs;ed e&longs;t via, qua ten
dit ad terminum. Motus &longs;ecundi generis e&longs;t mutatio, &longs;eu tran&longs;itus ex
vno loco in alium; hoc e&longs;t finis, vel effectus formalis &longs;ecundarius,
quem exigit impetus; & fru&longs;trà ponitur alia entitas, quæ tantùm e&longs;&longs;et
in&longs;tituta ad exigendam i&longs;tam loci mutationem; Igitur &longs;i &longs;ufficienter
exigatur ab ip&longs;o impetu, de quo infrà, certè fru&longs;tra ponitur quodcun
que aliud per Ax.3. & 7.
gens, quæ reducitur ad formalem, quæ &longs;uum effectum formalem &longs;ecundarium,
id est &longs;uum finem intrin&longs;ecum exigit. Sic calor exigit rarefactionem, vel
re&longs;olutionem, impetus motum; cum enim non &longs;it cau&longs;a efficiens per Th.
2. &longs;it tamen cau&longs;a per Th.3. nec &longs;it materialis, nec finalis, vt con&longs;tat, de
bet e&longs;&longs;e formalis, vel exigens, &longs;eu exigitiua; vt patet ex ip&longs;a cau&longs;arum
enumeratione; non e&longs;t materialis, quia non recipit motum, ni&longs;i ab alio;
nec finalis, quæ &longs;upponit alias; cum ip&longs;a non &longs;it dum ponitur
effectus.
lapis proiectus per Po&longs;tul. haud dubiè &longs;ub&longs;tantia lapidis non e&longs;t cau&longs;a
huius motus; quia lapis tandem &longs;i&longs;tit per hypoth.4. igitur non e&longs;t cau&longs;a
motus, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria; igitur &longs;emper cau&longs;aret per Ax.12. præ
terea potentia motrix proiicientis verè agit, cum etiam defatigetur; igi
tur aliquid producit, non motum immediatè, qui produci non pote&longs;t pro
prièper Th. 2. Adde quod motus &longs;ecundi generis habet tantùm cau&longs;am
immediatam exigentem, &longs;ed potentia motrix non exigit; quia primò
non defatigaretur exigendo; &longs;ecundò quia lapis &longs;eparatus à manu etiam
mouetur, &longs;ed non ad exigentiam potentiæ motricis, vt patet; quia &longs;tatim
po&longs;t &longs;eparationem pote&longs;t illa potentia de&longs;trui, licèt lapis longo pò&longs;t
tempore moueatur; &longs;ed quod non e&longs;t, nihil exigit.
Aliquis fortè diceret potentiam motricem exigere primam partem
motus, quæ deinde &longs;ecundam exigit, & &longs;ecunda tertiam, tertia quar
tam, &c. Sed contra;
quæro quid &longs;it prima illa pars motus;
nec enim
aliud agno&longs;co ni&longs;i primam mutationem loci, quæ mutatio non pote&longs;t
exigere ni&longs;i quando e&longs;t; atqui quando e&longs;t, nihil reale e&longs;t actu ni&longs;i mo
bile, & nouus locus acqui&longs;itus, mobile ip&longs;um non exigit, vt demon&longs;tra
tum e&longs;t, & conce&longs;&longs;um, nec etiam locus de nouo acqui&longs;itus, in quo
&longs;cilicet mobile &longs;i&longs;tere pote&longs;t: quidquid pones aliud, impetum appellabo.
Dices cum graue aliquod mouetur deor&longs;um, vel leue &longs;ur&longs;um, vel
corpus animatum &longs;e ip&longs;um mouet, dici pote&longs;t &longs;ub&longs;tantia corporis cau&longs;a
immediata motus. Re&longs;p. negando, tùm quia omnis potentia motrix
agit; igitur producit aliquid aliud, quod e&longs;t cau&longs;a motus: præterea po
tentia motrix corporis animati, agit v&longs;que ad defatigationem, &longs;udorem,
licèt non &longs;it motus, igitur aliud producit, de corpore graui probabi
mus infrà.
Demon&longs;tro, Sub&longs;tantia mobilis non e&longs;t cau&longs;a imme
diata motus, per Th.5. ergo aliquid aliud; igitur impetus, nam quod di
&longs;tinctum e&longs;t à &longs;ub&longs;tantia mobilis, & exigit motum, e&longs;t impetus per
Def.3. &longs;ed quia hoc Theorema e&longs;t veluti princeps huius tractatus cardo,
in eo paulò diutius hærendum e&longs;t, igitur.
Demon&longs;tro primò dari impetum:
Quidquid e&longs;t, & antè non erat, non
e&longs;t à &longs;e, &longs;ed habet cau&longs;am per Ax.8. Motus de nouo e&longs;t per hypothe&longs;im
tertiam; igitur habet cau&longs;am, &longs;ed non aliam, quam impetum, quod pro
bo: Lapis cadens, vel impactus in alium lapidem mouet illum per hy
poth.7. &longs;ed &longs;ub&longs;tantia lapidis in alium impacti non e&longs;t cau&longs;a huius mo
tus, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt patet; igitur applicata eundem effe
ctum produceret per Ax.12. &longs;ed etiam applicata immediata non agit, vt
con&longs;tat experientia; igitur per idem Axioma non e&longs;t cau&longs;a.
Scio e&longs;&longs;e aliquas re&longs;pon&longs;iones, quas infrà refellemus; nunc &longs;ufficiat
dixi&longs;&longs;e lapidem impactum non producere motum, qui propriè non pro
ducitur per Th.2. nec exigere, vt con&longs;tat ex &longs;ecunda probatione Th. 5.
igitur &longs;i aliquid exigit, vel producit, voco impetum.
Secundò probatur; potentia motrix e&longs;t actiua, quia defatigatur, quis
hoc neget? igitur aliquid producit;
non motum, qui propriè non pro
ducitur per Th.2. igitur aliquid aliud; voco impetum; adde quod etiam
&longs;ine motu agit, & defatigatur vt iam dictum e&longs;t; igitur habet alium effe
ctum immediatum; denique mouere, pellere, trahere, proiicere, percu
tere, nihil ni&longs;i actionem &longs;onant.
Tertiò probatur; pila di&longs;iuncta à manu proiicientis diu adhuc mo
uetur per hypoth.6. igitur hic motus habet cau&longs;am per Ax. 8. quælibet
enim pars motus de nouo e&longs;t, neque duæ illius partes &longs;imul e&longs;&longs;e po&longs;&longs;unt.
atqui potentia motrix non e&longs;t cau&longs;a per Ax.10. immò pote&longs;t e&longs;&longs;e de&longs;tru
cta; igitur non e&longs;t cau&longs;a per Ax. 9. Non e&longs;t etiam cau&longs;a &longs;ub&longs;tantia pilæ
mobilis per Th.5.5. nec priores pattes motus per re&longs;p. ad primam in
&longs;tantiam Th 5. igitur aliquid aliud; voco impetum.
Quartò probatur;
pila proiecta &longs;en&longs;im &longs;ine &longs;en&longs;u tardiore motu
mouetur; donec tandem moueri omnino de&longs;inat per hypoth. 5. igitur
non e&longs;t &longs;emper æqualis, & eadem cau&longs;a huius motus per Ax. 12. & 13.
num.3. igitur cau&longs;a huius motus eodem modo debilitatur, &longs;eu remitti
tur, quo ip&longs;e motus; &longs;ed decre&longs;cit &longs;ub&longs;tantia mobilis, nec potentia mo-
ergo e&longs;t alia cau&longs;a præ&longs;ens, quæ mi
nuitur; voco impetum.
Quintò corpus graue deor&longs;um cadens accelerat &longs;uum motum, vt patet
experientia; quæ maximè clara e&longs;t in funependulis, de qua in &longs;equen
tibus libris; igitur debet e&longs;&longs;e cau&longs;a huius motus velocioris;
non e&longs;t au
tem &longs;ub&longs;tantia lapidis, nec grauitas per Ax. 12. nec aliud quidpiam ex
trin&longs;ecum, vt videbimus &longs;uo loco; igitur aliquid aliquid intrin&longs;ecum,
voco impetum. Igitur certum e&longs;t dari impetum; qui certè tribui non
pote&longs;t, vel vlli connotationi, vel alteri exigentiæ, vt con&longs;tat ex
dictis.
Diceret fortè alius hæc omnia e&longs;&longs;e dubia;
nam fieri pote&longs;t vt Deus
tantùm moueat; quod &longs;ine impetu fieri po&longs;&longs;e certum e&longs;t;
Re&longs;p. equi
dem per miraculum hoc fieri po&longs;&longs;e; &longs;ed quemadmadum certum e&longs;t phy
&longs;icè ignem applicatum calefacere, niuem frigefacere, & modò calamum
à me hæc &longs;cribente moueri, ita certum o&longs;t phy&longs;icè &longs;agittam à &longs;agittario
emitti, & pilam à proiiciente, &c. adde quod Deus, vt auctor naturæ
e&longs;t, agit tantùm; vel de&longs;init agere iuxta exigentiam cau&longs;arum &longs;ecunda
rum; denique cau&longs;am phy&longs;icè appello, ex cuius applicatione nunquam
non &longs;equitur effectus per Ax.11. num.1.
Dicerent alij hoc totum prouenire à corpu&longs;culis;
vel atomis, vel fila
mentis &longs;ine vlla actione; equidem non reiicio corpu&longs;cula, & perennia
corporum effluuia: Dico tamen primò globum quie&longs;centem humi ha
bere &longs;altem aliquas partes quie&longs;centes, vel immobiles; quis hoc neget?
immò maximam &longs;uarum partium partem; igitur cum deinde proiicitur
idem globus, illæ partes mouentur; dari igitur debet cau&longs;a huius motus
per Ax.8, igitur impetus: nec dicas moueri illas partes à corpu&longs;culis;
quia
antè erant eadem, immò plura corpu&longs;cula; & tamen non mouebant:
igi
tur non &longs;unt cau&longs;a huius motus per Ax.12. Dices excitari; &longs;ed quid hoc
e&longs;t excitari? vel enim mutantur, vel non mutantur;
&longs;ecundum dici
non pote&longs;t; quia vt excitentur, ex non excitatis mutari debent; igitur
per aliquid: deinde quid e&longs;t illa excitatio, ni&longs;i impul&longs;io; igitur &longs;i mouen
tur illa corpu&longs;cula, & excitantur à potentia motrice, etiam partes prius
immobiles mouebuntur, & excitabuntur per Ax.12. quia &longs;unt applicatæ
cau&longs;æ nece&longs;&longs;ariæ.
Dico &longs;ecundò minimum ex his corpu&longs;culis non &longs;emper moueri;
po
te&longs;t enim &longs;i&longs;tere; quis hoc neget? igitur &longs;i modò mouetur, modò quie&longs;
cit, motus ab eo di&longs;tinguitur per Th.1. igitur mouetur per impetum, de
quo infrà.
Igitur datur nece&longs;&longs;ariò impetus, &longs;ine quo non po&longs;&longs;unt explicari prædi
ctæ omnes hypothe&longs;es, contra quem &longs;unt quidem graui&longs;&longs;imæ difficultates,
quas &longs;en&longs;im in &longs;equentibus Theorematis, in quibus explicantur pro
prietates huius impetus, di&longs;cutiemus.
Diceret aliquis lapidem impul&longs;um ab aëre deinde propelli;
&longs;ed aër po
tius re&longs;i&longs;tit motui; vt con&longs;tat experientiâ; &longs;ed hoc &longs;oluemus infrà.
Demon&longs;tratur.
Quia &longs;ub&longs;tantia mobilis non e&longs;t cau&longs;a exigens motum per Th. 5. Impe
tus e&longs;t cau&longs;a exigens per Def. 3. & Th. 6. de eodem contradictoria dici
non po&longs;&longs;unt per Ax. 1. n. 3. Igitur impetus non e&longs;t idem cum &longs;ub&longs;tantià
mobilis; igitur di&longs;tinctus; deinde &longs;eparari pote&longs;t à &longs;ub&longs;tantia mobilis
per Hypoth. 4. igitur e&longs;t di&longs;tinctus per Ax. 2.
Quippe non e&longs;t corpus, nec forma &longs;ub&longs;tantia
lis; quia omne corpus, & omnis forma &longs;ub&longs;tantialis moueri pote&longs;t, &
non moueri, vt con&longs;tat ex po&longs;t. & ex Hypoth.
3. & 4. igitur di&longs;tingui
tur à motu; igitur & ab impetu per Ax. 2. igitur impetus non e&longs;t &longs;ub
&longs;tantia; igitur accidens.
Modus duplicis generis e&longs;&longs;e pote&longs;t:
Modus
primi generis e&longs;t entitas quædam diminuta, vt vulgò loquuntur, di&longs;tin
cta quidem modaliter, vt aiunt, à re, cui adhæret; ac proinde ab ca &longs;e
parari pote&longs;t, non tamen exi&longs;tere &longs;eparata. Modus &longs;ecundi generis non
e&longs;t entitas quidem di&longs;tincta; e&longs;t tamen &longs;tatus quidam corporis; &longs;ic &longs;e&longs;&longs;io
e&longs;t modus, conden&longs;atio, compre&longs;&longs;io, &c. His po&longs;itis Impetus non e&longs;t mo
dus primi generis; nihil enim probat impetum e&longs;&longs;e modum, quod etiam
non probet calorem, & lucem e&longs;&longs;e modos; dicere autem omnia acci
dentia e&longs;&longs;e modos non debemus, de quo &longs;uo loco; modus enim ita à na
turâ comparatus e&longs;t, vt &longs;ine &longs;ubiecto actuali &longs;eu fulcro non exi&longs;tere mo
dò, &longs;ed ne concipi quidem po&longs;&longs;it; v. g. actio non pote&longs;t concipi ni&longs;i &longs;it
alicuius actio; nec fieri &longs;ine facto;
nec via &longs;ine termino;
nec dependen
tia &longs;ine dependente; at verò po&longs;&longs;um concipere calorem, & impetum
&longs;ine alio, quod &longs;it actu; licèt enim calor exigat re&longs;olutionem partium
&longs;ui &longs;ubiecti, &longs;eu rarefactionem, & impetus motum; nihil tamen impe
dit, quin per miraculum calor, & impetus con&longs;eruari po&longs;&longs;int &longs;ine eo.
quod exigunt, hoc e&longs;t &longs;ine &longs;uo &longs;ine; igitur &longs;ine &longs;ubiecto;
non e&longs;t etiam
modus &longs;ecundi generis vt patet, &longs;ed de modis in Metaphy&longs;ica; vix enim
hoc Theorema ad rem Phy&longs;icam quicquam facit.
Sequitur ex dictis; cum nec &longs;it motus.
nec &longs;ub&longs;tantia, nec modus, nec quidquam negatiuum, alioquin exige
ret; igitur e&longs;t aliud accidens; vocetur qualitas.
Quia impetus e&longs;t di&longs;tinctus realiter à &longs;ue
&longs;ubiecto per Th. 7. E&longs;t enim &longs;eparabilis per Hypoth. 3. & 4. igitur di
&longs;tinctus per Ax. 2. &longs;ed qualitatem realiter di&longs;tinctam apello Phy&longs;icam;
præ&longs;ertim cum nec moralis &longs;it, nec Logica, &c.
Quia lapis proiectus etiam &longs;eparatus
mouetur aliquandiu per Hyp. 6. igitur durat eius cau&longs;a, &longs;cilicet impe
tus; igitur e&longs;t qualitas permanens.
Diceret fortè aliquis lapidem proiectum pelli ab aëre à tergo in&longs;tan
te, vt voluit Ari&longs;toteles pluribus in locis; &longs;ed præ&longs;ertim 8. Ph.c.vlt.& 7.
cap.2. 3.de Cœlo, cap. 3. Re&longs;pondeo hoc dici non po&longs;&longs;e; Primò quia non
modò non iuuat aër; &longs;ed etiam impedit motum proiecti, quod de omni
medio nece&longs;&longs;ariò dicendum e&longs;t, vt patet experientiâ; vnde quo cra&longs;&longs;ius,
&longs;eu den&longs;ius e&longs;t
per aquam; rationem à priori afferemus infrà, cum de re&longs;i&longs;tentia medij:
Secundò, quis dicat pilam rotatam in &longs;olo moueri aëris appul&longs;u? cum
alia corpora, quæ pila rotata præterlambendo qua&longs;i allambit, nullo mo
do moueantur; præ&longs;ertim granula pulueris. Tertiò, an fortè aër id præ
&longs;tare pote&longs;t &longs;ine vi impre&longs;&longs;a; igitur non minus ip&longs;i pilæ proiectæ, quam
aëri ambienti imprimi poterit: Quartò, nullus aër à tergo pellitur;
&longs;ed
potius ip&longs;a pila aduer&longs;us aëra pellit, dum emittitur manu; igitur &longs;i aër
&longs;uccedit à tergo, id totum accidit, vel metu vacui, vel ne aër compri
matur, vt videbimus infrà. Quintò denique, cum diu moueatur eadem
pars aëris, haud dubiè in ca manet vis impre&longs;&longs;a; igitur impetus erit ad
huc qualitas permanens.
Ad id quod obiicitur ex Ari&longs;totele;
aliqui putant inclina&longs;&longs;e in cam &longs;en
tentiam; cùm tam en no&longs;tram teneant illu&longs;tres Peripatetici, quorum no
minibus parco, ne tot citationes paginas impleant; vide apud Conim
bric. l. 7. Phy&longs;. cap. 2. Aliqui excu&longs;ant ip&longs;um Ari&longs;totelem, putantque
non e&longs;&longs;e locutum ex propriâ &longs;ententiâ: Alij dicunt Ari&longs;totelem quidem
tribui&longs;&longs;e aliquam vim extrin&longs;ecam aëri; non tamen nega&longs;&longs;e intrin&longs;ecam
impetus; quidquid &longs;it, ip&longs;a verba Ari&longs;totelis demon&longs;trant ip&longs;um agno
ui&longs;&longs;e vim motricem impre&longs;&longs;am aëri, hoc e&longs;t impetum (
quit) &longs;cilicet motrix,
id e&longs;t aëri &longs;ur&longs;um, & deor&longs;um; quid porrò e&longs;t illa vis motrix, ni&longs;i impetus.
Probatur, quia motus non dicitur pro
ductus per Th. 2. Adde &longs;i vis rationem metaphy&longs;icam; quia nihil cogit
dicere accidens aliquod, ex iis &longs;cilicet, quæ &longs;en&longs;u percipimus, agere ad
intra; quod videtur e&longs;&longs;e proprium &longs;ub&longs;tantiæ, &longs;altem naturaliter; vt
demon&longs;trabimus in Metaph.
quia cau&longs;a imme
diata motus e&longs;t tantum exigens, per Th. 4. &longs;ed impetus e&longs;t cau&longs;a motus
immediata per Th. 5. & 6. igitur e&longs;t cau&longs;a exigens, adde quod id tantùm
accidens &longs;en&longs;ibile præ&longs;tare pote&longs;t in &longs;uo &longs;ubiecto, vt aliquam illius mu
tationem præ&longs;tet, vel exigat; quæ vel e&longs;t localis, hoc e&longs;t fluxus quidam:
vel alteratiua, vt vulgò vocatur; quà &longs;cilicet vel re
&longs;oluuntur partes, vel rarefiunt, vel lique&longs;cunt, vel concre&longs;cunt &c. vel
demùm mutant &longs;en&longs;ibilem &longs;tatum; vel e&longs;t perfectiua aliquo modo, qua
tenus &longs;ubiectum nouam aliquam habitudinem acquirit ad &longs;en&longs;us; &longs;ic
lumen illuminando obiectum reddit illud vi&longs;ibile. &c.
de quibus aliàs.
Cum enim &longs;it cau&longs;a
exigens per Th. 121. Voco effectum formalem &longs;ecundarium, quem in
mobili exigit impetus; quippe, vt iam dictum e&longs;t, cau&longs;a exigens redu
citur ad formalem; nec enim cau&longs;at aliquid producendo, quod &longs;pectat ad
efficientem; nec mouendo, quod &longs;pectat ad finalem;
nec determinando,
quod &longs;pectat ad obiectiuam; nec recipiendo, quod &longs;pectat ad materia
lem; nec dirigendo, quod &longs;pectat ad idæalem, vel exemplarem; &longs;ed
exigendo; quatenus &longs;cilicet ad id à natura e&longs;t in&longs;tituta, vt ex eius in
&longs;ubiecto præ&longs;entia talis affectio, vel mutatio con&longs;equatur; vocatur au
tem effectus formalis &longs;ecundarius; non verò primarius, qui e&longs;t tantùm
concretum ex ip&longs;a formâ, & &longs;ubiecto.
Dum finem audis intrin&longs;ecum,
cogita quæ&longs;o aliquid phy&longs;icum; e&longs;t enim id, propter quod talis, vel ta
lis forma in&longs;tituta e&longs;t: quid enim aliud e&longs;&longs;e pote&longs;t;
finem enim rerum
naturalium ex ip&longs;o v&longs;u cogno&longs;cimus; immò idem e&longs;t finis cum ip&longs;o v&longs;u;
cum igitur impetus illum tantùm v&longs;um habeat, quem in ip&longs;o mobili
præ&longs;tare cernimus, &longs;cilicet motum; dicendum e&longs;t motum e&longs;&longs;e finem in
trin&longs;ecum impetus; adde quod cum fru&longs;trà &longs;it impetus ille, qui non præ
&longs;tat motum mediatè &longs;altem in &longs;uo &longs;ubiecto; quid enim aliud in &longs;uo &longs;ub
iecto præ&longs;taret, quem effectum, quam mutationem? certè &longs;i fru&longs;trà e&longs;t, non
e&longs;t, per Ax.6.igitur vt &longs;it, debet habere id, &longs;ine quo e&longs;&longs;e non pote&longs;t; igitur
maximum eius bonum e&longs;t, igitur finis, quem natiuâ vel innatâ velut
appetentiâ concupi&longs;cit, vel exigit. Dixi mediatè, vel immediatè;
num
reuera datur fortè aliquis impetus, vt dicemus infrà; &longs;cilicet primus na
turalis, qui &longs;cilicet duos fines habet di&longs;iunctiuè; quorum alter e&longs;t gra
uitatio, alter motus deor&longs;um.
Probatur quia motus e&longs;t finis intrin
&longs;ecus impetus per Th. 16. igitur &longs;i nullus motus e&longs;&longs;e po&longs;&longs;et, &longs;uo fine ca
reret impetus; igitur non e&longs;&longs;et, vt patet, igitur non e&longs;&longs;et;
quia quod
fru&longs;trà e&longs;t, non e&longs;t per Ax. 6. nec ob&longs;tat quod &longs;uprà indicatum e&longs;t de im
petu naturali primo vel innato (&longs;ic enim deinceps appellabimus vt recti
di&longs;tinguamus ab acqui&longs;ito quem vocabimus impetum accelerationis)
qui &longs;ine motu con&longs;eruatur in corpore grauitante; quia ni&longs;i po&longs;&longs;ibilis e&longs;
&longs;et motus deor&longs;um nulla e&longs;&longs;et grauitatio; quippe grauitare e&longs;t deor
&longs;um inclinari, motumque inclinationis impediri; hinc dicemus
impeditur à corpore &longs;u&longs;tinente? immò dicemus infrà primo in&longs;tanti,
quo e&longs;t impetus, nondum e&longs;&longs;e motum.
Ob&longs;eruabis autem certi&longs;&longs;imam regulam; &longs;cilicet ex impo&longs;&longs;ibilitate
effectus formalis, &longs;equi impo&longs;&longs;ibilitatem cau&longs;æ formalis, huiu&longs;que po&longs;&longs;i
bilitatem ex illius po&longs;&longs;ibilitate.
Quia ni&longs;i e&longs;&longs;et cau&longs;a, non
e&longs;&longs;et naturaliter effectus per Ax. 8. Impetus enim e&longs;t cau&longs;a motus per
Th.15. Deinde omnis motus e&longs;t ab aliqua potentia motrice, vt patet ex
omni hypothe&longs;i; &longs;iue &longs;it naturalis in grauibus, & leuibus, &longs;iue &longs;it vitalis
in viuentibus; &longs;iue &longs;it media in compre&longs;&longs;is, & dilatatis; &longs;iue alia quæli
bet: &longs;ed omnis potentia motrix e&longs;t actiua, quia mouet;
ergo agit, &longs;ed
motum non producit per Th. 2. Igitur impetum, qui deinde exigit mo
tum per Th. 14. Dixi naturaliter; quia non e&longs;t dubium, quin Deus &longs;ine
impetu aliquo modo mouere po&longs;&longs;it; ide&longs;t, facere &longs;ine impetu, vt corpus
mutet locum: nec dicas Deum non po&longs;&longs;e &longs;upplere vices cau&longs;æ formalis;
nam concedo id quidem pro effectu formali primario; nec enim Deus
pote&longs;t facere, vt aliquid &longs;it calidum &longs;ine calore; cum e&longs;&longs;e calidum &longs;it
idem, ac e&longs;&longs;e habens calorem; id tamen nego pro effectu &longs;ecundario,
quem &longs;cilicet cau&longs;a formalis exigit: Etenim &longs;icut pote&longs;t &longs;ummo iure non
&longs;atisfacere exigentiæ; ita pote&longs;t id
gentia conferre pote&longs;t; &longs;ic pote&longs;t corpus re&longs;oluere &longs;ine calore, mouere
fine impetu &c. quanquam vt verum fatear non e&longs;&longs;et propriè motus, &longs;ed
qua&longs;i continuæ reproductionis modus; nam motus dicit aliquam pa&longs;
&longs;ionem; &longs;cilicet actum entis in potentiâ, vt aiunt.
exigeret motum per &longs;uam entitatem; quia nullus impetus exigeret;
ergo
aliquid aliud, nihil di&longs;tinctum, alioquin e&longs;&longs;et impetus; ergo ip&longs;a corpo
ris entitas; quanquam non e&longs;&longs;et motus, vt iam dictum e&longs;t, quia non e&longs;
&longs;et pa&longs;&longs;io.
Quia e&longs;&longs;et &longs;emper ea
dem cau&longs;a nece&longs;&longs;aria motus, id e&longs;t, ip&longs;a entitas corporis; igitur idem
effectus per Axioma 12. igitur idem, vel æqualis motus: dixi per &longs;e pro
pter diuer&longs;um medium.
Probatur;
quia in tantum indiget mobile impetu vt impetus exigat motum; &longs;ed
corpus illud per &longs;uam e&longs;&longs;entiam exigeret motum; igitur non indigeret
impetu; po&longs;&longs;et tamen impediri eius motus, vt patet; immò e&longs;&longs;et capax
recipiendi impetus., &longs;iue quem in ip&longs;o produceret, &longs;iue quem ab alia
Pro
batur; quia, ni&longs;i e&longs;&longs;et motus, non e&longs;&longs;et impetus per Th. 17. igitur &longs;ubie
ctum incapax motus e&longs;t incapax impetus.
Quia
vbi non pote&longs;t e&longs;&longs;e cau&longs;a formalis, ibi non pote&longs;t e&longs;&longs;e effectus forma
lis, quod certum e&longs;t.
Probatur 1. pars; quia impetus in eo non e&longs;&longs;et fru&longs;trà; haberet enim
&longs;uum effectum formalem, & finem intrin&longs;ecum. Probatur 2.pars;
quia in
eo impetus non e&longs;&longs;et fru&longs;trà per Ax. 6. igitur haberet &longs;uum effectum;
igitur motum.
Probatur 1. pars;
quia non e&longs;t vbique, igitur pote&longs;t transferri è loco in locum; cur enim
non po&longs;&longs;et? Dices fortè quia affixum e&longs;&longs;et e&longs;&longs;entialiter tali, vel tali lo
co, &longs;ed contra; quia de&longs;truantur omnia, præter ip&longs;um corpus; certè
nulli affixum manet. Dices &longs;patio imaginario; apage i&longs;tas nugas:
de i&longs;to &longs;patio plura demon&longs;trabimus in Metaphy. Probatur 2. pars;
quia
&longs;i e&longs;t capax motus, e&longs;t capax impetus per Th. 24. Quod dixi de corpo
re; dicendum e&longs;t de omni re creata finita permanente.
Pro
batur, quia non e&longs;t moueri, ni&longs;i relinquat locum, & acquirat alium; &longs;ed
1. acquirere locum, e&longs;t 1. e&longs;&longs;e in illo loco; & relinquere locum e&longs;t,
1. non e&longs;&longs;e in eo loco; nec &longs;imul e&longs;t in vtroque, quia in duobus locis
idem &longs;imul e&longs;&longs;e non pote&longs;t; vt demon&longs;tramus in Metaphy&longs;ica; & phy
&longs;icè certum e&longs;t ex omni hypothe&longs;i; igitur moueri nunc, id e&longs;t, hoc in
&longs;tanti, id e&longs;t, 1. acquirere nouum locum, & 1. relinquere priorem,
&longs;upponit nece&longs;&longs;ariò antè fui&longs;&longs;e in loco nunc relicto; &longs;ed quod durat
tantùm in in&longs;tanti, non habet antè, neque po&longs;t; igitur quod durat tan
tùm vno in&longs;tanti, moueri non pote&longs;t; igitur e&longs;t incapax motus; igitur
& impetus.
Tum quia vbique, e&longs;t igitur
nouum locum acquirere non pote&longs;t; igitur nec moueri per Definitio
nem 1. tùm quia æternitas Dei tota &longs;imul e&longs;t; igitur nec fuit antè, ne
que po&longs;t in ca; igitur non pote&longs;t dici antè habui&longs;&longs;e locum, quo nunc
caret: & nunc non habere illum quo caret;
tùm quia immutabilitas
nam moueri, e&longs;t affici intrin&longs;ecè;
quia etiam de
&longs;tructis omnibus extrin&longs;ecis creatis moueri po&longs;&longs;em, & fru&longs;trà recurres
ad partes virtuales immen&longs;itatis Dei, quas ferè animus abhorret; apa
ge partes in Deo: quis hoc ferre po&longs;&longs;it? præterea &longs;i &longs;unt, &longs;unt e&longs;&longs;entia
liter immobiles; igitur valet &longs;emper ratio allata;
igitur Deus e&longs;t inca
pax motus; igitur & impetus.
Diceret aliquis Deum quantumuis Immen&longs;um in orbem conuolui
po&longs;&longs;e; igitur 1. ratio non probat de omni motu. Re&longs;pondeo adhuc va
lere, quia etiam in orbem conuolui non pote&longs;t, ni&longs;i mutetur intrin&longs;e
cè; atqui &longs;i e&longs;t immen&longs;us, non pote&longs;t mutari intrin&longs;ecè per motum;
quia nullum locum de nouo acquireret; &longs;ed de hoc motu aliàs, cum de
infinito; vel de puncto phy&longs;ico mobili; quidquid &longs;it.
valet &longs;altem
1. ratio pro motu recto, & aliæ duæ pro omni motu.
Quia cum tantùm dicat mutationem
loci; certè mutatio non e&longs;t in loco; dicit enim tantùm locum relictum
eo in&longs;tanti, quo nouus acquiritur. Præterea quod e&longs;t in loco dicit tan
tùm ens phy&longs;icum; &longs;ed mutatio dicit etiam non ens;
radoxum; illud non mouetur per quod cuncta mouentur, quæ mouentur.
Cum enim &longs;it &longs;ucce&longs;&longs;iua, fluit per partes,
igitur quælibet illius pars, &longs;eu quod durat vna in&longs;tanti tantùm e&longs;t inca
pax motus, per Th. 26.
cum enim actio per quam res con&longs;erua
tur, &longs;it eius duratio; vt con&longs;tabit ex iis, quæ demon&longs;trabimus in Me
taphy&longs;ica, & cum duratio moueri non po&longs;&longs;it, per Th. 29. certè neque
actio moueri pote&longs;t.
Hinc in tanta rerum creatarum multitudine &longs;unt tantùm duæ, quæ
&longs;unt e&longs;&longs;entialiter immobiles; &longs;cilicet motus, & actio; quorum ille cum
&longs;it mutatio non e&longs;t adæquatè aliquid po&longs;itiuum; &longs;ecus actio.
Hinc &longs;unt tantùm duo adæquatè po&longs;itiua, quæ moueri non po&longs;&longs;unt;
&longs;cilicet Deus, & actio; Deus, qui &longs;emper e&longs;t; actio, quæ tantùm vno
in&longs;tanti e&longs;t; Deus vbique e&longs;&longs;entialiter; actio hic tantum e&longs;&longs;entialiter;
Deus primum ens; actio infinitum ens; e&longs;t enim modus;
Deus primum
mouens; actio ip&longs;e motus; &longs;cilicet primi generis, de quo in &longs;ect. Th.3.
Hinc &longs;i res aliqua creata per actionem tantæ perfectionis, quæ mille
annis e&longs;&longs;entialiter re&longs;ponderet, con&longs;eruaretur; certè per totum illud
tempus moueri non po&longs;&longs;et; e&longs;&longs;et enim vnicum in&longs;tans, hoc e&longs;t duratio
&longs;ed eodem in&longs;tanti in pluribus locis e&longs;&longs;e non pote&longs;t; igitur
nec moueri; adde quod per cam actionem &longs;um in loco, per quam &longs;um
in tempore; igitur &longs;i hæc e&longs;t &longs;emper eadem, illam eandem e&longs;&longs;e nece&longs;&longs;e
e&longs;t; &longs;ed hæc &longs;unt metaphy&longs;ica, quæ obiter tantùm attingo, aliàs fusè
de mon&longs;trabo.
Ob&longs;eruabis primò ex dictis præclarum naturæ in&longs;titutum;
cum enim
corpus moueri &longs;emper non debeat, (quippe hoc e&longs;&longs;et maximè incom
modum) certè per &longs;uam entitatem moueri non exigit; alioquin &longs;emper
moueretur; igitur per aliud ab entitate di&longs;tinctum, id e&longs;t per impetum;
itaque licet per &longs;uam entitatem exigat fluxum in tempore, id e&longs;t con&longs;er
uari, & durare; id e&longs;t nouam &longs;emper actionem con&longs;eruatiuam;
quia
maximum eius bonum e&longs;t durare vel exi&longs;tere; Igitur per &longs;e ip&longs;um illud
exigit; quia &longs;emper exigit, non tamen per &longs;e ip&longs;um exigit fluxum in
loco, id e&longs;t motum; quia moueri non &longs;emper e&longs;t bonum.
Ob&longs;eruabis &longs;ecundò, cum idem corpus aliquando velociùs, tardiùs
aliquando moueri exigat; &longs;i per &longs;uam entitatem moueri exigeret, eo
dem &longs;emper ferretur motu; quia eadem &longs;emper e&longs;&longs;et exigentia;
igitur
debet e&longs;&longs;e aliquid aliud; illud autem e&longs;t impetus, qui aliquando maior
&longs;eu perfectior, aliquando verò minor e&longs;t; igitur maiorem &longs;eu
rem
cum enim motus &longs;it eius finis intrin&longs;ecus, vt re&longs;olutio e&longs;t finis caloris
vel rarefactio; quemadmodum maior calor maiorem exigit, &longs;eu præ
&longs;tat re&longs;olutionem; ita & maior, &longs;eu perfectior impetus maiorem, &longs;eu
velociorem motum exigit.
Ob&longs;eruabis tertiò aliud naturæ in&longs;titutum, quo &longs;cilicet in eo tan
tùm &longs;ubiecto recipi pote&longs;t cau&longs;a formalis, in quo recipi pote&longs;t eius effe
ctus formalis &longs;ecundarius: nec alia regula, præter eam excogitari pote&longs;t;
cum enim aliqua forma ad talem, vel talem finem à natura in&longs;tituta e&longs;t;
certè propter illum finem e&longs;t, igitur in eo non e&longs;t, in quo &longs;uum finem
con&longs;equi non pote&longs;t; alioquin fru&longs;trà e&longs;&longs;et;
& contra in eo e&longs;&longs;e pote&longs;t,
in quo fru&longs;trà non e&longs;t; cum &longs;cilicet in eo &longs;uum finem con&longs;equatur;
ad
de quod finis ille intrin&longs;ecus phy&longs;icus &longs;cilicet, non moralis, aliquis no
uus effectus e&longs;t; atqui nouus effectus &longs;ine &longs;ua cau&longs;a e&longs;&longs;e non pote&longs;t, neque
cau&longs;a nece&longs;&longs;aria &longs;ine effectu; igitur ibi, &longs;cilicet in hoc &longs;ubiecto, in quo
e&longs;t, vel e&longs;&longs;e pote&longs;t effectus formalis, cau&longs;a formalis e&longs;t, vel e&longs;&longs;e pote&longs;t,
e&longs;t inquam citra miraculum.
Ob&longs;eruabis quartò egregiam rationem;
propter quam res eadem in
pluribus locis naturaliter e&longs;&longs;e non pote&longs;t; quippe cum res fuerit primo
producta in aliquo loco, illa certè nouum locum acquirere non pote&longs;t
naturaliter; ni&longs;i per motum, atqui motus dicit nece&longs;&longs;ario priorem lo
tum relictum, & nouum acqui&longs;itum; igitur cum tot acquirantur loca
per motum, quot relinquuntur; &longs;i ante motum vnus tantùm erat eiu&longs;
dem rei locus, po&longs;t motum etiam vnus e&longs;t: quod autem producatur tan-
vel enim à cau&longs;a prima vel ab aliqua 2. pro
ducitur; &longs;i à 2. ergo ab aliqua aplicata;
igitur ex &longs;uppo&longs;itione quòd il
la cau&longs;a 2. in vno tantùm loco producta &longs;it, vni tantum applicari po
te&longs;t; quod autem cau&longs;a 1. in pluribus locis naturaliter eundem effectum
non producat, certum e&longs;t, & demon&longs;trabimus in Metaphy&longs;. quia &longs;in
gulis effectibus &longs;ingulæ &longs;ufficiunt actiones; &longs;ingulis terminis &longs;ingulæ
viæ; immò hoc requiri videtur, &longs;eu &longs;pectare ad huius vniuer&longs;itatis or
dinem; quippe &longs;i res eadem in pluribus locis e&longs;&longs;et; cur potius in duo
bus quam in tribus? deinde multiplex iure po&longs;&longs;et exi&longs;timari;
denique
quod vnum e&longs;t in entitate creata, &longs;eu dependente ab eadem cau&longs;a, vnum
e&longs;t etiam in dependentia; quæ e&longs;t actio, per quam dependet; &longs;ed de his
aliàs.
motrice applicata, licèt à fortiori moueri po&longs;&longs;it. Probatur, quia impetus
e&longs;t tantùm propter motum, qui eius effectus e&longs;t, & finis, per Th. 15.
& 16. Igitur vbi non e&longs;t motus, fru&longs;trà e&longs;t impetus; &longs;ed quod fru&longs;trà
e&longs;t, non e&longs;t; id e&longs;t non e&longs;t, quod fru&longs;trà e&longs;&longs;et, &longs;i e&longs;&longs;et per Ax. 6. Exci
pio tamen impetum naturalem innatum, qui nunquam e&longs;t fru&longs;trà, vt
dictum e&longs;t &longs;uprà in Theorem. 17. adde quod non pote&longs;t cogno&longs;ci
impetus, ni&longs;i vel ex motu, vel ex ictu, vel ex contrario ni&longs;u, vel
impul&longs;u; &longs;ed nihil horum cernitur in rupe quam ferio;
Igitur non
e&longs;t dicendum in ea produci impetum, cuius rationem afferemus infrà;
nunc &longs;atis e&longs;t Ax. 3. id manife&longs;tè probari; nam qui diceret in rupe im
mobili impetum imprimi; certè po&longs;itiuo argumento probare tenere
tur, quod tantùm duci pote&longs;t, vel ab experimento; atqui hîc nullum e&longs;t;
vel à nece&longs;&longs;itate, quæ nulla e&longs;t; vel ex alio quocumque capite, quod
nullum excogitari pote&longs;t; &longs;ed maiorem lucem huic Th. 3. ex proximè
&longs;equentibus accer&longs;emus; nec e&longs;t quòd aliqui dicant produci impetum
inefficacem; qui cum fru&longs;trà &longs;it, &longs;i e&longs;t, ex nullo capite probari pote&longs;t:
ad
de quòd de&longs;truitur impetus, ne &longs;it fru&longs;trà; Igitur non producitur, ne &longs;it
fru&longs;trà; nam con&longs;eruatio e&longs;t vera actio, vt dicemus &longs;uo loco;
Igitur &longs;i
hæc non ponitur, ne aliquid &longs;it fru&longs;trà; etiam 1. productio poni non
debet; vnde commentum illud impetus inefficacis pror&longs;us inefficax e&longs;t.
bilior e&longs;t. Probatur, & explicatur; quippe debilior potentia minorem ef
fectum producit per. Ax. 13.
æquales vni certæ per idem
mobilis, &longs;eu rupis, quàm impetus; cum vna pars impetus duobus parti
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;
non e&longs;t mirum &longs;i nullus impetus producatur; cum non po&longs;&longs;int tot partes
illius produci, quot e&longs;&longs;ent nece&longs;&longs;ariæ; vt &longs;altem &longs;ingulæ &longs;ingulis &longs;ubie
cti, &longs;eu rupis partibus di&longs;tribuerentur.
Ob&longs;eruabis autem nouum quoddam genús re&longs;i&longs;tentiæ;
nam &longs;ingulæ
partes rupis ab applicata potentiâ aptæ &longs;unt loco moueri per impre&longs;
&longs;um impetum, & maior potentia &longs;imul omnes loco moueret; at verò
omnes &longs;imul, & coniunctim con&longs;ideratæ; quatenus &longs;cilicet vna pars
non pote&longs;t moueri &longs;ine alia, & comparatæ cum illa potentia debili di
cuntur habere prædictam re&longs;i&longs;tentiam, quæ &longs;uperat potentiæ vires;
quòd &longs;cilicet à maiori moueri tantùm po&longs;&longs;int; quia plures partes im
petus po&longs;tulantur, quam &longs;int eæ, quæ à prædictâ potentiâ po&longs;&longs;unt pro
duci.
modò nulla fiat &longs;eparatio, neque compre&longs;&longs;io
bus partibus, quæ auolant ab ictu, produci impetum. Probatur igitur
1. quia &longs;i non producatur in omnibus partibus, & nulla &longs;eparetur ab
alijs; certè nulla mouetur, vt certum e&longs;t;
igitur nulla habet impetum;
quia ibi non e&longs;t cau&longs;a formalis, vbi non e&longs;t effectus formalis; alioquin
e&longs;&longs;et fru&longs;trà, contra Ax. 6.2. Tu dicis produci impetum in aliquot parti
bus; hoc dicis, hoc proba? an potes digno&longs;cere impetum ni&longs;i ex motu?
vel con&longs;eruaretur hîc impetus &longs;equentibus in&longs;tantibus, vel &longs;tatim &longs;ecun
do in&longs;tanti de&longs;trueretur. Primum dicere ab&longs;urdum e&longs;t;
quia &longs;i hoc e&longs;&longs;et
multisictibus repetitis tandem moueretur totum mobile; &longs;i verò de
&longs;trui dicatur. Secundo in&longs;tanti; eadem ratio probat non produci.
Pri
mo in&longs;tanti, quæ probat de&longs;trui. Secundo nam ideo de&longs;truitur.
Secun
do quia e&longs;t fru&longs;trà, &longs;ed non minus e&longs;t fru&longs;trà. Primo igitur non produ
citur. Primo 4. probatur;
quia cum non &longs;ufficiant partes impetus, quas
dixi produci, vt omnibus partibus &longs;ubiecti di&longs;tribuantur; certè non e&longs;t
vlla ratio, cur potiùs his quàm illis di&longs;tribui dicantur; cum vna &longs;it tan
tùm immediatè applicata. Igitur certum e&longs;t vel produci in omnibus, vel
in nulla, ni&longs;i forte aliquæ auolent, &longs;ed tunc &longs;eparantur.
Obiiciet aliquis 1. e&longs;&longs;e cau&longs;am nece&longs;&longs;ariam applicatam &longs;ubiecto ap
to: igitur agit per Ax. 12. Re&longs;pondeo e&longs;&longs;e impeditam;
nam re&longs;i&longs;tentia
&longs;ubiecti &longs;uperat vires potentiæ vt dictum e&longs;t; immò in ip&longs;o motu re
torqueo argumentum; licèt enim &longs;it applicata cau&longs;a nece&longs;&longs;aria mouens,
non tamen mouet.
Obiiciet 2. Ignis applicatus agit in nonnullas partes &longs;ubiecti, licèt
non agat in omnes; igitur & potentia motrix. Re&longs;pondeo non e&longs;&longs;e pa
ritatem; quia vna pars pote&longs;t calefieri, & re&longs;olui &longs;ine alia, vt con&longs;tat
non tamen vna moueri &longs;ine alia, cui coniuncta e&longs;t, ni&longs;i &longs;eparetur; igi
tur nec recipere impetum &longs;ine alia.
Obiiciet.
3. &longs;int duo trahentes idem mobile;
ita vt &longs;eor&longs;im neuter
trahere po&longs;&longs;it, coniunctim verò vterque po&longs;&longs;it; certè &longs;i alter non pro
ducit impetum &longs;eor&longs;im, nec etiam coniunctim producet; nec enim au
gentur eius vires ab altero: Re&longs;pondeo vtrunque agere actione com
muni; igitur non e&longs;t mirum &longs;i effectus maior e&longs;t, quem tamen neuter
Dices &longs;i vterque coniunctim producit effectum:
&longs;int v. g. 100. par
tes impetus; Igitur &longs;inguli producunt tantùm 50. Igitur cur potiùs in
in his partibus &longs;ubiecti, quàm in alijs, cum vtriu&longs;que potentia eidem
&longs;ubiecti parti po&longs;&longs;et e&longs;&longs;e applicata? Re&longs;pondeo &longs;ingulos producere 100.
actione &longs;cilicet communi indiui&longs;ibiliter; &longs;int enim duo trahentes A. &
B. A. producit 100. &longs;ed non &longs;olus; B. producit ea&longs;dem 100. &longs;ed non &longs;o
lus; &longs;ed explicabimus hunc modum actionis communis in Metaphys.
quod autem agant actione communi patet per Ax. 13.
Obiicies 4. producitur &longs;onus &longs;i ferias rupem;
igitur & impetus;
Re&longs;
pondeo ad &longs;onum &longs;olam aëris colli&longs;ionem &longs;ufficere, quam fieri certum
e&longs;t à prædicto ictu; deinde mallej motus impacti in rupem facit &longs;onum;
quidquid tandem &longs;it &longs;onus, de quo hîc non di&longs;puto: adde quod in ru
pe &longs;unt &longs;emper aliquæ partes tremulæ, quæ modico tantùm, eoque flexi
bili nexu cum alijs partibus copulantur; adde aliquam compre&longs;&longs;ionem,
ex qua modicæ vibrationes &longs;equuntur.
Obiicies 5. Quando aliquæ partes auolant ab ictu, haud dubiè auo
lant propter impetum impre&longs;&longs;um: Igitur prius e&longs;t imprimi impetum,
quàm auolare; igitur productus e&longs;t impetus in nonnullis partibus, &
non in aliis, cum quibus illæ &longs;unt coniunctæ. Re&longs;pondeo equidem im
petum produci in illis partibus antequam auolent; &longs;ed ideo produci vt
deinde auolent nam tota ratio cur non producatur, e&longs;t ne &longs;it fru&longs;trà; &longs;ed
&longs;i auolent aliquæ partes: certè in ijs non e&longs;t fru&longs;trà, in quibus habet
&longs;uum effectum, id e&longs;t, motum.
Dices;
igitur primo in&longs;tanti impetus ille e&longs;t fru&longs;trà;
in quo non
habet &longs;uum effectum; Re&longs;pondeo nunquam primo in&longs;tanti e&longs;&longs;e fru&longs;trà,
modò &longs;it motus &longs;ecundo cum etiam primo in&longs;tanti, quo e&longs;t impetus,
non po&longs;&longs;it e&longs;&longs;e motus, vt demon&longs;trabo infrà; immò ideo ponitur im
petus primo vt &longs;it motus &longs;ecundo exigendo pro in&longs;tant &longs;equenti, de
cum impetus ponat tantùm motum quo aliàs.
Dices;
&longs;ed potentia motrix ne&longs;cit an po&longs;&longs;it pars aliqua mobilis &longs;epa
rari; igitur non e&longs;t quòd aliquando producat impetum, aliquando
non producat. Re&longs;pondeo non &longs;tare per cau&longs;am nece&longs;&longs;ariam, quin &longs;em
per agat; &longs;ed per &longs;ubiectum, quod &longs;i aptum e&longs;t, & capax effectus;
haud
dubiè eo ip&longs;o cau&longs;a nece&longs;&longs;aria applicata in ip&longs;um aget; &longs;i verò ineptum.
haud dubiè non aget; nam ad hoc vt producatur effectus in &longs;ubiecto;
non &longs;atis e&longs;t cau&longs;am po&longs;&longs;e producere, ni&longs;i etiam &longs;ubiectum po&longs;&longs;it recipe
re; igitur cum &longs;it talis ordo à natura in&longs;titutus, ne aliquid &longs;it fru&longs;trà;
certè &longs;i impetus producibilis &longs;it futurus fru&longs;trà, hauddubiè non produ
cetur; &longs;ecus verò &longs;i fru&longs;trà non &longs;it futurus, in quo non e&longs;t difficultas.
Ob&longs;eruabis 1. vix fieri po&longs;&longs;e quin &longs;emper aliquæ partes &longs;eparentur,
comprimantur, vel dilatentur, vt patet experientiâ.
Ob&longs;eruabis 2. etiam maximam corporis molem à debili potentia mi-
quod etiam ob&longs;eruauit Galileus in &longs;uis dialo
gis de motu; quem certè motum ob&longs;eruabis etiam in&longs;en&longs;ibilem, tùm
operâ radij luminis repercu&longs;&longs;i, & ad aliquod interuallum proiecti; tùm
operâ &longs;eu pi&longs;orum in tympani membranâ tremulo qua&longs;i motu &longs;ub&longs;ul
tantium; quâ etiam arte deprehenditur in arce ob&longs;e&longs;&longs;a, &longs;ub quam muri
partem cuniculi agantur.
Hinc egregia ratio erui pote&longs;t, cur ingens corporis moles à debili po
tentia loco moueri non po&longs;&longs;it; cum enim tot &longs;altem requirantur partes
impetus, quot &longs;unt partes &longs;ubiecti: quia vel in omnibus, vel in nulla
producitur; certè cum &longs;int plures partes &longs;ubiecti, quàm vt in &longs;ingulis
ab ea dumtaxat potentiâ impetus produci po&longs;&longs;it; quid mirum e&longs;t, &longs;i mo
ueri non po&longs;&longs;it.
Hinc certa ratio alterius vulgaris effectus potentiæ motricis, quæ lapi
dem 40. librarum tardo tantùm motu impellit, etiam cum &longs;ummo ni&longs;u,
cum tamen &longs;axo vnius libræ velociorem motum imprimat; quia &longs;cilicet
partes impetus producti di&longs;tribuuntur pluribus partibus &longs;ubiecti in ma
iori lapide, & paucioribus in minori; igitur &longs;ingulæ partes minoris
habent plures partes impetus, vt manife&longs;tè con&longs;tat; ergo ille impetus
inten&longs;ior e&longs;t; igitur maiorem exigit &longs;eu perfectiorem motum per Ax.
13. num.2.
Hinc &longs;ublata ratione diuer&longs;æ re&longs;i&longs;tentiæ medij, dato pondere
mobilis vtriu&longs;que, datoque ni&longs;u communi potentiæ, pote&longs;t de
terminari certus velocitatis gradus vtriu&longs;que; nam ratio velocitatum
e&longs;t inuer&longs;a ponderum v. g. &longs;it pondùs 4. librarum; fit etiam 2. librarum
&longs;it impetus impre&longs;&longs;us vtrique &longs;uppo&longs;ito communi, & æquali ni&longs;u
potentiæ, & æquali tempore; haud dubiè velocitas mobilis 2. libra
rum erit dupla velocitatis mobilis 4. librarum; quia cum &longs;int duplo
plures partes &longs;ubiecti in hoc mobili quàm in illo (accipio enim vtrum
que eiu&longs;dem materiæ, vt omnes lites fugiam) igitur in minori e&longs;t duplo
inten&longs;ior impetus: Igitur duplo velocior motus; dixi, &longs;i fiat æquali
ni&longs;u, & æquali tempore; quia reuerâ non fit in tempore æquali, &longs;ed
inæquali, &longs;i &longs;upponatur idem arcus brachij v. g. iacientis; nam tempo
ra &longs;unt in ratione &longs;ubduplicata ponderum; vt demon&longs;trabimus lib. 10.
& velocitates &longs;unt vt tempora permutando.
Hinc facilè determinari pote&longs;t proportio impetus impre&longs;&longs;i cognitâ
grauitate mobilium; v. g. &longs;it mobile graue vt4. & aliud graue vt 2. haud
dubiè vt moueatur æquali gradu velocitatis, debet produci duplo
maior impetus in maiori mobili, hoc e&longs;t, iuxta rationem maioris ad mi
nus, quod clari&longs;&longs;imè &longs;equitur ex dictis; vt enim tot &longs;int gradus impetus
haud
dubiè impetus maioris habet eandem rationem ad impetum minoris;
quam habet maius ad minus.
Hinc quoque ducitur manife&longs;ta ratio &longs;eu re&longs;pon&longs;io ad illud præcla
rum certè quorundam philo&longs;ophorum
ma ponderis acce&longs;&longs;ione totam terræ molem inclinari, vt in nouo æqui
librio &longs;tatuatur; quod omninò fal&longs;um e&longs;t; nam ex &longs;uppotione quòd
terra non grauitet (vt vulgò dicitur, & aliàs à nobis
certè moueri non pote&longs;t ni&longs;i producantur tot partes impetus quot &longs;unt
partes &longs;ubiecti in tota terra; quæ certè maximas
cau&longs;a. Probatur;
quia non pote&longs;t e&longs;&longs;e motus, ni&longs;i &longs;it locus prior reli
ctus, & nouus acqui&longs;itus, igitur &longs;i eodem in&longs;tanti, quo e&longs;t impetus,
haberet motum, eodem in&longs;tanti e&longs;&longs;et in duobus locis, quod dici non
pote&longs;t; & iam diximus in Th. 26. igitur impetus primo in&longs;tanti quo
e&longs;t non habet &longs;uum motum.
Quia non pote&longs;t
moueri, ni&longs;i acquirat nouum locum, & priorem relinquat; igitur, vel &longs;i
mul in vtroque e&longs;t, quod dici non pote&longs;t; vel in relicto antè fuit; igitur
non e&longs;t primum in&longs;tans, contra &longs;uppo&longs;itionem.
quo est. Nam moueatur mobile quodlibet;
& dum mouetur, impella
tur, factâ &longs;cilicet acce&longs;&longs;ione noui impetus; haud dubiè hoc primo in
&longs;tanti, quo producitur impetus in dato mobili non mouetur per Th.
35. quo tamen in&longs;tanti mouetur prædictum mobile.
Hinc egregium paradoxon;
non moueantur illa omnia, que eidem &longs;ubiecto reuerâ in&longs;unt.
Hinc etiam aliud paradoxon;
&longs;uum finem, nec habere pote&longs;t
Hinc pote&longs;t aliquid dato in&longs;tanti carere &longs;uo fine;
licèt non &longs;it fru&longs;trà;
fru&longs;trâ enim tantùm dicitur ille impetus, qui pro in&longs;tanti &longs;equenti
non pote&longs;t habere motum.
Probatur 1. quia alioquin vna pars
impetus &longs;ufficeret ad mouendam ingentem rupem; quod ab&longs;urdum e&longs;t.
2. &longs;icut vna pars caloris non re&longs;oluit alias partes &longs;ubiecti; ita nec im
petus. 3. Ratio à priori e&longs;t; quia impetus non e&longs;t cau&longs;a efficiens motus
per Th. 13. &longs;ed tantùm cau&longs;a formalis per Th. 15. Igitur præ&longs;tat tantùm
&longs;uum effectum formalem in eo &longs;ubiecto, in quo e&longs;t.
Hinc partes impetus non cau&longs;ant motum in &longs;uo &longs;ubiecto actione, vel
exigentia communi; quia quælibet pars impetus exigit tantùm motum
&longs;ui &longs;ubiecti; id e&longs;t illius partis, quàm afficit; quod etiam probatur per
Ax. 13.
Hinc corpus grauius per&longs;e, &longs;altem eiu&longs;dem materiæ, non cadit velo
ciùs, quàm leuius, vti globus plumbeus 100. librarum, quàm globus
vnius libræ plumbeus; quia &longs;cilicet impetus vnius partis non iuuat mo
tum alterius: præterea tam facilè 2, partes impetus in 2. partibus &longs;ubie
cti receptæ ea&longs;dem mouent, quàm 100. alias 100. dixi per &longs;e; nam di
uer&longs;a e&longs;&longs;e pote&longs;t medij re&longs;i&longs;tentia; &longs;ed de his fu&longs;e in 2. lib.
mobile &longs;it ferrum calidum, recipitur in ip&longs;a &longs;ub&longs;tantia ferri; non verò
in ip&longs;o calore (ex &longs;uppo&longs;itione quod calor &longs;it accidens, vt aliàs demon
&longs;trabimus; nec in alijs accidentibus, &longs;i quæ &longs;unt, in eodem &longs;ubiecto;
pro
batur 1. quia &longs;i produceretur etiam impetus in accidentibus, quo plu
ra e&longs;&longs;ent accidentia in aliquo &longs;ubiecto; plures quoque partes impetus
producendæ e&longs;&longs;ent; igitur maiori potentiâ opus e&longs;&longs;et per Ax. 13. n. 4.
Igitur difficiliùs mouerentur, quod e&longs;t ab&longs;urdum. Diceret fortè ali
quis eundem impetum recipi &longs;imul in &longs;ub&longs;tantia & in ip&longs;is accidenti
bus; &longs;ed contra, nam reuera, &longs;i hoc e&longs;&longs;et, dum proijcitur ferrum cali
dum, & &longs;tatim frigefit, de&longs;trueretur totus impetus, de&longs;tructo &longs;cilicet
eius &longs;ubiecto: 2. qui hoc diceret, deberet probare;
nam eodem modo
mouetur corpus &longs;iue afficiatur pluribus accidentibus, &longs;iue paucioribus;
igitur non euincit experientia recipi in illis impetum, nec etiam ratio,
vt dicam paulò po&longs;t. Ratio à priori e&longs;&longs;e pote&longs;t;
quia accidens cum &longs;uo
&longs;ubiecto coniunctum exigit &longs;emper e&longs;&longs;e præ&longs;ens &longs;ubiecto, cum natura
liter extra &longs;ubiectum exi&longs;tere non po&longs;&longs;it; igitur cum exigat con&longs;erua
ri, & exi&longs;tere; eo tantùm modo, quo pote&longs;t naturaliter con&longs;eruari &
exi&longs;tere; certè exigit con&longs;eruari, & ine&longs;&longs;e &longs;ubiecto;
igitur exi&longs;tere in
eo loco, in quo exi&longs;tit &longs;ubiectum, vt patet; igitur, &longs;i &longs;ubiectum mutet
locum etiam accidens cum eo coniunctum mutare debet.
Dices, igitur &longs;imiliter dici pote&longs;t non recipi impetum in omni
bus partibus &longs;ubiecti mobilis, &longs;ed in vnâ dumtaxat; cui cum
aliæ &longs;int vnitæ, exigunt moueri &longs;ine impetu ad illius motum? cum
hoc ip&longs;um ad omnem vnionem &longs;pectare videatur; Re&longs;pondeo vnam
exi&longs;tere po&longs;&longs;it; igitur non e&longs;t par ratio;
præterea vna pars plumbi non
e&longs;t in loco alterius; nec enim inuicem penetrantur cum &longs;it compene
tratio accidentium cum &longs;ubiecto; deinde, quò plures &longs;unt partes vnitæ,
maior e&longs;t re&longs;i&longs;tentia, quæ ip&longs;o etiam &longs;en&longs;u percipitur; denique non vide
tur cur potius produceretur in vna parte, quam in alia; quæ omnia
iam &longs;uprà Th. 33. demon&longs;trauimus.
Adde quod &longs;i impetus produceretur in ip&longs;is accidentibus, etiam in
ip&longs;o impetu prius producto alius impetus produceretur; cum &longs;cilicet
noua fit impetus acce&longs;&longs;io; quod &longs;atis ridiculum e&longs;t; qua&longs;i verò impetus
indigeat impetu &c. hîc loquor tantùm de accidentibus in &longs;ubiecto;
non verò de Euchari&longs;ticis, quæ à &longs;ubiecto per miraculum &longs;eparata etiam
moueri po&longs;&longs;unt per impre&longs;&longs;um impetum.
Hinc manife&longs;tè patet, quid dicendum &longs;it de anima bruti, quæ moue
tur etiam &longs;ine impetu; quia exigit &longs;emper e&longs;&longs;e coniuncta corpori, à
quo di&longs;iuncta naturaliter exi&longs;tere non pote&longs;t, vt &longs;uo loco dicemus; igi
tur ad motum corporis, &longs;eu &longs;ubiecti moueri deber.
Idem quoque de Anima rationali dicendum e&longs;&longs;e videtur;
licèt
enim à corpore &longs;eparata naturaliter exi&longs;tere po&longs;&longs;it; tandiù tamen cum
corpore manet coniuncta, quandiu agere pote&longs;t in organis corporeis;
ac proinde exigit con&longs;eruari in corpore ip&longs;o, quandiu &longs;uas operatio
nes organicas in eo exercere pote&longs;t.
Hinc patet ratio manife&longs;ta ad quæ&longs;itum illud;
quomodo &longs;cilicet po
tentia motrix materialis v.g. Taurus &longs;uo cornu hominem ventilare po&longs;
&longs;it; nec vlla &longs;upere&longs;t difficultas, dum dicas impetum non produci in
anima.
Ob&longs;eruabis primò In hoc Theoremate dictum e&longs;&longs;e naturaliter; quia
per miraculum accidens &longs;eparatum ab omni &longs;ub&longs;tantia, dum &longs;it impe
netrabile, per impetum &longs;ibi impre&longs;&longs;um moueri pote&longs;t.
Ob&longs;eruabis &longs;ecundò de anima bruti per miraculum &longs;eparatâ, idem
pror&longs;us dicendum e&longs;&longs;e.
Ob&longs;eruabis tertiò etiam Animam rationalem &longs;eparatam, modò &longs;it
cum impenetrabilitate coniuncta, capacem e&longs;&longs;e impetus; quem etiam
à potentia motrice corporea recipere pote&longs;t; idem dictum e&longs;to de An
gelo; &longs;ed de vtroque aliàs.
haud du
biè impetus ille impre&longs;&longs;us ab aliqua cau&longs;a efficiente producitur
per Ax. 8.
Proba
tur; quia &longs;i produceretur, e&longs;&longs;et cau&longs;a nece&longs;&longs;aria vt
cata, & non impedita ageret per Ax. 32. quod e&longs;t contra experientiam.
Dicunt aliqui requiri nam motus
præuius non requiritur vt cau&longs;a, vt patet; quia cau&longs;a vt agat debet exi
&longs;tere per Ax. 9. Igitur requiritur, vt conditio, quod dici non pote&longs;t;
quia primo etiam conditio debet e&longs;&longs;e præ&longs;ens; &longs;ed motus præuius de
nihil pre&longs;enti e&longs;t &longs;ecundo quia non pote&longs;t excogitari aliud munus con
ditionis; ni&longs;i vel vt tollat impedimentum, vel vt applicet cau&longs;am &longs;ubie
cto apto; præterea motus præuius non e&longs;t;
igitur eodem modo &longs;e
habet, ac &longs;i nunquam extiti&longs;&longs;et; & &longs;i eo in&longs;tanti quo corpus impa
ctum primo tangit, amitteret totum impetum, ita vt expræterito motu
nihil reliquum e&longs;&longs;et, haud dubiè corpus aliud non pelleret.
Diceret alius impetum e&longs;&longs;e tantùm conditionem, quæ &longs;emper e&longs;t
de præ&longs;enti: ad hanc in&longs;tantiam non valet &longs;uperior re&longs;pon&longs;io;
& certè
&longs;i eo ip&longs;o in&longs;tanti contactus noua fieret impetus acce&longs;&longs;io; haud dubiè
maior e&longs;&longs;et ictus; licèt cum eodem motu præuio, & tamen idem e&longs;&longs;et
corpus
catione impetus &longs;emper &longs;equitur productio alterius impetus; dum &longs;cili
cet &longs;ubiectum, cui applicatur &longs;it capax motus; ex applicatione corporis
&longs;eu &longs;ubiecti ip&longs;ius non &longs;emper &longs;equitur; igitur dicendum e&longs;t impetum
ip&longs;um e&longs;&longs;e cau&longs;am alterius per Ax. 11. n. 1. voco enim illud cau&longs;am,
ex cuius applicatione &longs;emper &longs;equitur &longs;imilis effectus; alioquin &longs;i hoc
neges; proba mihi aliter ignem accendi ab alio igne;
dicam enim tibi
ignem applicatum e&longs;&longs;e tantùm conditionem, & produci à cœlo; proba
mihi aliter calorem produci à calore? quo enim medio, vel argu
mento id euinces? quo etiam non euincam impetum produci ab im
petu: Deinde affer rationem à priori, propter quam &longs;ub&longs;tantia
corporis producat impetum &longs;ur&longs;um? v. g. cum non exigat à &longs;e ip&longs;a mo
tum &longs;ursùm, qui violentus e&longs;t corpori graui; numquid certum e&longs;t, vt
dicemus infrà impetum produci ad extra, vt tollatur impedimentum
motus? igitur illius e&longs;t tollere impedimentum, cuius e&longs;t exigere motum,
corpus ip&longs;um graue non exigit motum &longs;ur&longs;um, &longs;ed impetus; igitur im
petus e&longs;t tollere impedimentum &longs;ui effectus; igitur producere impetum,
quo vno tolli tantùm pote&longs;t: En tibi rationem à priori, cutum nullam
habeas: Præterea, cur negas impetum e&longs;&longs;e cau&longs;am &longs;ufficientem alterius
impetus, cum ex eius applicatione ip&longs;o &longs;en&longs;u percipiamus produci alium
impetum? quæ ratio?
quid inde ab&longs;urdi, quid incommodi: Igitur tàm
certum e&longs;t, immò certius impetum produci ab alio impetu, quàm calo
rem à calore. Dices impetum iam habere alium effectum &longs;cilicet mo
tum; bella profecto ratio! &longs;ed numquid motus e&longs;t effectus formalis im
petus? prætereà e&longs;t-ne effectus ad extra?
deinde idem dico de calore;
factionem, quæ e&longs;t mutatio exten&longs;ionis; quemadmodum motus e&longs;t mu
tatio loci, vel vbicationis; igitur cum hoc | non ob&longs;tante, calor pro
ducat calorem ad extra; cur impetus non producit impetum? cuius pro
ductionem concedis virtuti corporum re&longs;i&longs;titiuæ, id e&longs;t vnioni, impe
netrabilitati, & cæteris huiu&longs;modi modorum &longs;uperfluorum qui&longs;quiliis;
de quibus plurimi tecum contendunt.
Ob&longs;eruabis nonnullas e&longs;&longs;e difficultates, quæ communes &longs;unt etiam
illi &longs;ententiæ, quam &longs;equuntur ij, qui exi&longs;timant impetum ad extra
produci à corpore impacto; quas tamen facilè &longs;oluemus infrà in conti
nuata no&longs;trorum Theorematum &longs;erie.
Probatur, quia aliquis
impetus producitur ad intra à potentia motrice, vt patet. 2. cum non
detur progre&longs;&longs;us in infinitum, nec impetus idem producatur à &longs;e ip&longs;o, ad
aliquem tandem vltimum &longs;eu primum deueniendum e&longs;t, qui ab alio im
petu non producatur.
Quia cum &longs;emper
ad illius productionem requiratur applicatio alterius impetus; certè
non e&longs;t ponenda alia cau&longs;a per Ax. 11.
&longs;cilicet cau&longs;æ exigentis ad intra
& efficientis ad extra
cum enim
motus &longs;it finis intrin&longs;ecus impetus; certè &longs;i nihil impediret motum,
haud dubiè gauderet impetus &longs;uo fine; igitur fru&longs;trà quidquam aliud
de&longs;ideraret; præterea licèt applicetur à tergo aliud mobile;
non tamen
propterea in eo producit, vt con&longs;tat experientiâ; denique cum tan
tùm impetum cogno&longs;camus per motum; cum nequidem e&longs;&longs;et impetus,
&longs;i non e&longs;&longs;et motus, per Th. 17. certè totus e&longs;t impetus propter motum
qui e&longs;t eius finis; igitur non agit ni&longs;i propter motum:
&longs;ed non pote&longs;t
excogitari, quid faciat propter motum, dum agit, ni&longs;i dicamus ideo
tantùm agere, vt tollatur impedimentum; cum certum &longs;it corpus im
mobile, in quod impingitur aliud mobile, impedire eius motum.
&longs;ui motus; cui &longs;i nullum corpus occurrit reuerà non agit,
quia li
cèt aliud corpus mobili admoueatur in alia linea; cum non impediat
eius motum, vt &longs;uppono; cum agat tantùm impetus ad extra, vt tollat,
&
cum impediatur tantùm in vna linea, in ca tantùm agit; igitur non
agit in orbem.
Ob&longs;eruabis primò, hanc primam e&longs;&longs;e difficultatem; cum in hoc im
petus maximè differat ab alijs qualitatibus &longs;i quæ &longs;unt, quæ agunt in or
bem, vt dicemus &longs;uo loco.
Ob&longs;eruabis &longs;ecundò, hanc etiam e&longs;&longs;e communem illorum &longs;ententiam,
qui dicunt impetum ad extrà produci ab ip&longs;o mobili, &longs;ed ita vt ab illis
vix &longs;olui po&longs;&longs;it; cum tamen à nobis facilè &longs;oluatur.
Ob&longs;eruabis tertiò, impetum in vtroque munere cau&longs;æ &longs;ube&longs;&longs;e tantùm
vni lineæ; &longs;cilicet exigit motum per vnam lineam;
cum per plures &longs;i
mul motus e&longs;&longs;e non po&longs;&longs;it; ne idem mobile &longs;imul e&longs;&longs;et in pluribus lo
cis; & producit impetum per vnam lineam; cum producat tantùm pro
pter motum.
Ob&longs;eruabis quartò, alias qualitates, &longs;i quæ &longs;unt, non agere ad extra,
vt tollant impedimentum &longs;ui effectus ad intra; qui &longs;cilicet ab impedi
mento extrin&longs;eco impediri non pote&longs;t; vt accidit in ip&longs;o impetu;
etenim
corpus non pote&longs;t moueri ni&longs;i nouum locum acquirat: neque nouum
locum acquirere ab alio corpore occupatum, ni&longs;i corpus hoc loco ce
dat, neque hoc loco cedere pote&longs;t &longs;ine motu, vel moueri &longs;ine impetu,
igitur cum impediat motum amoueri debet, accepto dumtaxat impetu
ab alio mobili.
Ob&longs;eruabis quintò nonnullos e&longs;&longs;e, qui volunt motum vnius corporis
transferri in aliud corpus; &longs;ed mera e&longs;t metaphora;
nihil enim pror&longs;us
e&longs;t quod ab vno in aliud tran&longs;eat, &longs;eu transferatur; nec aliud dici po
te&longs;t, ni&longs;i quod dictum e&longs;t, impetum &longs;cilicet nouum produci.
Hinc etiam reiicies commentum illorum, qui dicunt ideo vnum
corpus ab alio moueri, quia ab vno in aliud deriuantur corpu&longs;cula illa,
quæ faciunt lumen, & calorem; quia lumen, & calor &longs;unt veræ qualita
tes, non corpu&longs;cula, vt demon&longs;trabimus in 5. tractatu: Adde quod li
cet ferrum candens aliud frigidum impellat, etiam veloci&longs;&longs;imè; hoc ip
&longs;um æquè frigidum manet; denique in cra&longs;&longs;is tenebris nix &longs;eu glacies
frigidi&longs;&longs;ima pernici&longs;&longs;imè moueri pote&longs;t: &longs;ed apage i&longs;ta commenta.
Probatur
per Ax. 13. n. 1. ni&longs;i enim agerent actione communi &longs;ed quælibet &longs;uam
produceret; cur potius in hac parte &longs;ubiecti, quam in alia, deinde ap
plicatur tantùm vna immediatè; Igitur agunt omnes actione commu
ni; omnes inquam illæ, quæ impediuntur;
cum enim impetus agat
tantùm ad extrà vt tollat impedimentum &longs;ui motus; ille pro&longs;ectò age
re non debet, cuius motus vel effectus non impeditur.
cum in aliud corpus impinguntur maxima vi illud impellunt. Ratio e&longs;t;
quia cum &longs;int plures partes impetus in pluribus partibus &longs;ubiecti, &
omnes agant actione communi, non mirum e&longs;t &longs;i maiorem effectum
producant, per Ax. 13. n. 2.
Vides primò in hoc ca&longs;u compen&longs;ari inten&longs;ionem ab exten&longs;ione;
quippe quod præ&longs;tarent plures partes impetus in minore corporis mole
inten&longs;æ; hoc idem præ&longs;tare po&longs;&longs;unt exten&longs;æ in maiore mole.
Secundò &longs;icut maior moles aptior e&longs;t ad motum imprimendum, & mi
nùs apta ad recipiendum ita minor contrà aptior e&longs;t ad recipiendum, &
minùs apta ad imprimendum.
Tertiò, Hinc corpora illa, quorum partes vel nullo vel modico nexu
copulantur, minimo ferè impul&longs;u commouentur; &longs;ic aër & aqua mini
mo flante vento agitantur, nubes pelluntur; hinc tot procellæ tempe
&longs;tate&longs;que cientur; nec vlla e&longs;t alia ratio, cur minima ferè venti vis, cui
modicum &longs;axum re&longs;i&longs;tit, tantam aquæ, vel aëris molem commoueat, ni
&longs;i quia cum partes illorum corporum nullo ferè nexu coniunctæ &longs;int vna
&longs;ine alia moueri pote&longs;t, quod in aqua gelu concreta minimè accidit.
Quartò, Hinc &longs;i maxima rupes ita comminueretur vt tota in pulue
rem &longs;eu &longs;abulum abiret, minima vis impre&longs;&longs;a particulas illas moueret.
Quintò, Hinc diuino penè con&longs;ilio factum e&longs;t, vt partes terre&longs;tris
globi arctiore fibula copulentur; ne, &longs;i di&longs;iunctæ e&longs;&longs;ent, minimo flatu
di&longs;pergerentur: vt videre e&longs;t in puluere etiam graui&longs;&longs;imo, qui ab aura
flant e di&longs;pergitur.
Probatur per
Th. 44. hinc &longs;i aliud corpus affigas mobili à tergo, nullum impetum in
eo producet, cuius effectus, qui certè impetui &longs;ingularis e&longs;t, alia ratio
e&longs;&longs;e non pote&longs;t; tam enim corpus e&longs;t applicatum à tergo, quam in
ip&longs;a fronte; & nihil e&longs;t in vno, quod non &longs;it in alio, ni&longs;i quod in fronte
impedit motum, à tergo verò non impedit.
Hinc egregium paradoxon erui pote&longs;t;
quod &longs;cilicet cau&longs;a nece&longs;&longs;aria
etiam immediatè applicata, & non impedita in &longs;ubiecto apto non agit;
quod videtur e&longs;&longs;e contra Ax. 12. vnde vt agat cau&longs;a nece&longs;&longs;aria, debet
applicari debito modo; &longs;i agat in orbem, omnis applicatio &longs;ufficiens
e&longs;t: &longs;i verò agat tantùm per vnam lineam;
certè applicari debet in ca
linea; alioquin non aget defectu debitæ applicationis.
Hinc etiam aliud paradoxon non minus iucundum;
cau&longs;a nece&longs;&longs;aria
at verò agit impedita;
&longs;cilicet
impetus qui tantùm agit, vt tollat impedimentum; igitur, &longs;i non
impediatur non agit.
impeditur, plùs agit. Cum enim ideò agat ad extra, vt tollat impedi
mentum; certè &longs;i nullum e&longs;t, nihil agit, &longs;i minùs, minùs agit; igitur
agit pro rata, id e&longs;t, pro diuer&longs;a impedimenti ratione.
corporis, maximum est impedimentum,&longs;int enim duo globi,
A mobilis, & B. occurrens ip&longs;i A, &longs;itque linea directionis DE ducta
per centrum vtriu&longs;que AB, & punctum contactus &longs;it C; certè glo
bus B maximum ponit impedimentum, quod ab eo poni po&longs;&longs;it; Igitur
impetus globi A agit quantùm pote&longs;t in globum B; vt &longs;cilicet maxi
mum impedimentum remoueat.
diametrum globi immobilis: haud dubiè nihil impedit
&longs;it enim globus
mobilis A, Immobilis B, linea directionis &longs;it GA, ip&longs;i parallela FC;
certè globus B. non impedit motum globi A. cum nihil loci globi B
occupari debeat à globo A; Igitur impetus A non agit in globum B per
Th. 48.
est minus impedimentum.
immobilis BA; &longs;it linea motus GC cum impedimento, de qua in Th. 50.
&longs;it alia KB cum nullo impedimento, de qua in Th. 51. &longs;int aliæ HD,
IE; certè minus e&longs;t impedimentum in contactu D, quàm in C;
quia ca
dit obliquè in D, perinde atque &longs;i caderet in tangentem NO; Igitur
minus impeditur; in qua vero proportione, dicemus aliàs, cum de re
flexione, & de motu mixto.
in contactu
in
contactuin
quia in ea proportione producitur
plùs vel minùs impetus, quo plùs e&longs;t, vel minùs impedimenti per
Th. 49. &longs;ed minùs e&longs;t impedimentum in E, quàm in C; & in E, quàm
in D, per Th. 52; Igitur in D producitur minùs impetus, quàm in C,
& minùs in E, quàm in D.
id e&longs;t in eadem linea.
ui potest
cur &longs;cilicet cor
pus incidens in aliud perpendiculariter maximum ictum infligat; quia
&longs;cilicet maximum impetum producit, qui po&longs;&longs;it ab eo produci; cur
idem corpus obliquè incidens in aliud minorem ictum infligat; cuius
rei alia ratio e&longs;&longs;e non pote&longs;t. Huc etiam reuoca tormenta bellica, quæ
vel directo, vel obliquo ictu muros verberant; hinc perpendicularis
forti&longs;&longs;ima e&longs;t; licèt eadem ratio pro motu corporum non valeat, quæ
valet pro diffu&longs;ione, &longs;eu propagatione qualitatum.
Hinc pote&longs;t determinari quota pars impetus producatur, & quantus
&longs;it ictus; cognito &longs;cilicet & &longs;uppo&longs;ito eo impetus gradu, qui producitur,
cum totus producitur, vt fit in perpendiculari; quippe tota men&longs;ura
impetus continetur in arcu CB; quam proportionem nos infrà demon
&longs;trabimus.
& immobilis, impetus producit totum impetum quem pote&longs;t producere &longs;iue in
maiori globo, &longs;iue in minori, &longs;iue in æquali
e&longs;t; quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria;
Igitur idem impetus eodem mo
do applicatus æquali tempore, æqualem &longs;emper effectum producit, per
Ax. 12. igitur cum impetus agat tantùm, vt tollat impedimentum per
Th. 44. & cum in prædicta linea agat quantum pote&longs;t per Th. 50. cer
tè æqualem effectum producat nece&longs;&longs;e e&longs;t; &longs;iue in maiori &longs;iue in mino
ri, &longs;iue in æquali globo immobili.
& hæc e&longs;t altera difficul
tas; cum &longs;cilicet maior globus in minorem impingitur
cum enim omnes
partes impetus maioris globi agant actione communi per Th. 46. &
cum agant quantùm maximè po&longs;&longs;unt; in minore globo, tot partes pro
ducunt impetus, quot in maiore, vt patet; igitur in minore globo pau
cioribus partibus &longs;ubiecti di&longs;tribuuntur plures partes impetus; ergo in
qualibet parte &longs;ubiecti &longs;unt plures; &longs;ed hoc e&longs;t e&longs;&longs;e inten&longs;um, vt con&longs;tat,
igitur impetus remi&longs;&longs;us producit inten&longs;um; quod e&longs;t paradoxon egre
gium.
in maiorem inciditquia &longs;cilicet pauciores partes impetus di&longs;tribuun
tur pluribus partibus &longs;ubiecti; igitur quælibet &longs;ubiecti pauciores impe
tus habet; quæ omnia con&longs;tant ex dictis.
Ob&longs;eruabis primò, &longs;ingularem impetus proprietatem, quæ alijs qua
litatibus minimè competit; nam aliæ qualitates v. g. calor;
lumen in
eadem di&longs;tantia effectum &longs;emper æquè inten&longs;um producunt; &longs;ecus verò
impetus, qui pro maiori vel minori obice maiorem, vel minorem, hoc
e&longs;t inten&longs;iorem, vel remi&longs;&longs;iorem impetum in eadem di&longs;tantia producit;
cuius ratio ex eo capite petitur; quòd impetus agat tantùm ad extra
propter &longs;uum effectum ad intra, vt &longs;cilicet tollat impedimentum; igi
tur in totum, quod impedit, agit; igitur non habet certam, & deter
minatam &longs;phæram; cum tantùm agat in obicem, &longs;iue &longs;it maior, &longs;iue
minor: Quia verò e&longs;t cau&longs;a nece&longs;&longs;aria, æqualem effectum producit, id
e&longs;t tot partes impetus in maiore, quot in minore, ergo, cum in mino
re &longs;int pauciores partes &longs;ubiecti, & plures in maiore; haud dubiè quæli
bet pars minoris habebit plures partes effectus, & quælibet pars maio
ris pauciores; igitur effectus erit inten&longs;ior in minore, & remi&longs;&longs;ior in
maiore.
Prætereà, cum dixi omnes partes mobilis actione communi agere ad
extra; ita primò intelligi debet, vt omnes illæ partes moueantur:
&longs;ecun
dò, vt linea motus, &longs;eu directionis per centra grauitatis vtriu&longs;que glo
bi v, g. ducatur; alioquin, vel omnes actione communi non agunt, vel
minus agunt, de quo infrà; &longs;ufficit verò iuxta præ&longs;ens in&longs;titutum, vt
globus ita impellat alium vel æqualem, vel inæqualem, vt linea dire
ctionis ducatur per centrum grauitatis alterius; vide figuram. in qua
linea directionis e&longs;t DE.
directionis ducta per centra grauitatis vtriu&longs;que producit in eo æqualemPro
batur, quia impetus e&longs;t cau&longs;a nece&longs;&longs;aria, quæ tunc agit quantum pote&longs;t
per Th. 57. &longs;ed æqualis pote&longs;t producere æqualem: Probatur primò,
exemplo aliarum qualitatum; &longs;ecundò, quia ideo agit vt tollat impedi
mentum, hoc e&longs;t vt corpus illud amoueat loco; igitur æquali motu per
&longs;e; alioquin ni&longs;i æquali motu amoueret, non tolleret impedimentum,
vt pater; tertiò &longs;int 30. partes impetus, certè vel producent plures vel
pauciores, vel totidem, non plures; cur enim potius 31. quam 32.
nec etiam pauciores; cur enim potius 20. quam 18, &c. Igitur totidem;
quia cum &longs;int plures numeri plurium partium &longs;upra 30. & pauciorum
infra vt patet; &longs;itque tantùm vnicus numerus æqualium;
certè quod
vnum e&longs;t, determinatum e&longs;t, per Ax. 5. hæc ratio licèt videatur negati
ua e&longs;t tamen potenti&longs;&longs;ima: quartò, quia actus &longs;ecundus, re&longs;pondet actui
primo, id e&longs;t, effectus productus virtuti cau&longs;æ producentis; itaque cum
virtus agendi impetus &longs;it eius entitas, vt patet, certè impetus productus
e&longs;t per &longs;e æqualis impetui producenti per &longs;e; id e&longs;t remoto omni
impedimento, & facto eo contactu iuxta modum prædictum, ea quo-
moueantur æquali motu.
Hinc reijcis illos, qui volunt à globo æquali produci in æquali &longs;ub
duplum impetum; in &longs;ubduplo &longs;ubtriplum; in &longs;ubquadruplo &longs;ubquin
tuplum; ratio illorum e&longs;t; quia duo globi æquales in&longs;tanti contactus
perinde &longs;e habent, atque &longs;i conflarent vnum corpus; &longs;ed &longs;i conflarent
vnum corpus quilibet &longs;ubduplum impetum haberet; &longs;i verò globus cum
alio &longs;ubduplo faceret vnum mobile; haud dubiè minor, id e&longs;t, &longs;ubduplus
haberet tantùm &longs;ubtriplum impetum; atque ita deinceps; hoc totum
fal&longs;i&longs;&longs;imum e&longs;t; nam primò &longs;i globus æqualis acciperet tantùm &longs;ubdu
plum impetum ab alio, &longs;ubduplo tantùm motu ferretur; igitur &longs;ubdu
plum &longs;patium decurreret, quod e&longs;t contra experientiam, & Th. 47. Se
cundò, ratio propo&longs;ita nulla e&longs;t; quia quando globus impactus impellit
alium, e&longs;t veluti potentiâ, quæ cum tota &longs;ua vi, & cum impetu agit,
cuius nulla pars transfertur in alium globum; nec enim migrat de
de &longs;ubiecto in &longs;ubiectum, &longs;ed producit &longs;ibi æqualem: equidem &longs;i duo
globi æquales e&longs;&longs;ent vel coniuncti, vel contigui in linea directionis,
quilibet pro rata acciperet impetus producti partem à potentia applica
ta; &longs;i e&longs;&longs;ent æquales, qui&longs;que &longs;ubduplum: &longs;i alter &longs;ubduplus &longs;ubtri
plum, &c. &longs;ed hæc &longs;unt &longs;atis facilia.
Obijci fortè po&longs;&longs;et ab aliquo primò experientia;
videmus enim &longs;æpè
globum impul&longs;um in ludo Tudiculario moueri tardiùs globo impellen
te; re&longs;pondeo id &longs;æpè accidere;
tùm quia linea directionis non connec
tit centra vtriu&longs;que globi; igitur minor e&longs;t ictus per Th 52. tùm quia
globus impellens, vel impul&longs;us deficiunt à perfecta &longs;phæra; tùm quia
non e&longs;t perfecta æqualitas globorum; adde quod quò accuratiùs prædi
ctæ leges ob&longs;eruantur, ip&longs;i motus ad æqualitatem propiùs accedunt, vt
con&longs;tat experientia.
Obiici po&longs;&longs;et &longs;ecundò de&longs;trui aliquid impetus globi impellentis ab ip&longs;o
ictu, vt con&longs;tat experientia; igitur illa pars impetus, quæ de&longs;truitur, non
producit nouum impetum in globo impul&longs;o; Re&longs;pondeo de&longs;trui quidem
aliquid impetus in globo impacto, vt videbimus infrà; cum tamen de
&longs;truatur tantùm &longs;equenti po&longs;t ictum in&longs;tanti; certè cum exi&longs;tat adhuc
ip&longs;o in&longs;tanti contactus, nece&longs;&longs;ariò agit, quippe aliquid vltimo in&longs;tanti
pote&longs;t agere; adde quod illud ip&longs;um repugnat manife&longs;tæ experientiæ;
licèt enim aliquando de&longs;truatur totus impetus in globo impacto, quod
&longs;æpè accidit in ludo Tudiculario, nam illicò &longs;i&longs;tit pila eburnea; alius
tamen globus velociter mouetur, cuius effectus rationem infrà addu
cemus.
Obijci po&longs;&longs;et tertiò inde &longs;equi progre&longs;&longs;um in infinitum, nam globus
A impactus in globum B impellet cum æquali motu, & B in C etiam
æquali, C in D, atque ita deinceps; modò illi globi ita &longs;tatuantur, vt
linea directionis per omnium centra rectà ducatur; Re&longs;pondeo, vel il-
gant; vel aliquod &longs;patium inter &longs;ingulos intercipi;
&longs;i primum, produci
tur impetus à potentia motrice in omnibus, &longs;i &longs;ufficiens e&longs;t; non verò
vnus globus in alio, vt con&longs;tat; &longs;icut duo pondera &longs;imul attollo, quorum
vnum alteri incumbit: &longs;i verò non &longs;e tangant, dico antequam A im
pingatur in B, dum &longs;patium illud interiectum percurrit, amittere aliquid
impetus: idem dico de B, & C, vnde &longs;i nihil impetus in eo primo motu
periret & linea directionis omnium centra perfectè connecteret; ita vt
omnium ictus illi omnino &longs;ine vlla deflexione re&longs;ponderent; haud du
biè non po&longs;&longs;ent e&longs;&longs;e tot globi, quin po&longs;&longs;et alius addi, qui ab vltimo
pelleretur; &longs;ed vix illa omnia de quibus &longs;uprà po&longs;&longs;unt ob&longs;eruari;
Hinc
tamen facilè vna pars aëris aliam pellit, quod di&longs;tinctè videmus in
aqua; &longs;ed de his aliàs, &longs;ufficiat modò propo&longs;itam obiectionem inde
manere &longs;olutam.
velociorem motum per Th.
cipium illud mechanicum; id e&longs;t, nullum corpus po&longs;&longs;e maiorem veloci
tatis gradum alteri corpori imprimere; eo &longs;cilicet gradu, quem ip&longs;um
habet; nec enim inuenio Principium illud apud eos Mechanicos, qui
mechanica momenta &longs;uarum demon&longs;trationum momentis confirmant;
quî porro fieri pote&longs;t, vt principium illud admittatur, quod manife&longs;tæ
experientiæ repugnat? Quis enim non vidit vel maius &longs;axum in aliud
etiam tardo motu impactum maiorem motum, & impetum imprimere?
quis non vidit maiores illas onerarias naues etiam pigro, & tardo motu
labentes maximum impetum minori occurrenti cymbæ etiam impri
mere? Rationem habes in Th. 47. &longs;ed dices;
igitur aliquis velocitatis
gradus nullam habet cau&longs;am; igitur e&longs;t à nihilo, quod dici non pote&longs;t.
Re&longs;pondeo, plures partes impetus non produci in minore globo, quàm
&longs;int in maiore; igitur nulla pars e&longs;t impetus minoris globi, quæ &longs;ui
cau&longs;am &longs;ufficientem non habeat; &longs;ed cum partes impetus maioris globi
di&longs;tribuantur pluribus partibus &longs;ubiecti, faciunt remi&longs;&longs;um impetum, igi
tur & tardum; cum &longs;cilicet impetus vnius partis non iuuet motum alte
rius per Th. 37. at verò cum partes impetus producti in minore globo
di&longs;tribuantur paucioribus partibus &longs;ubiecti, faciunt inten&longs;iorem im
petum; igitur velociorem motum, quippe omnes producuntur ab
omnibus illis actione communi per Ax. 17. num. 1. quid clarius.
& æqualis, æquali æqualem
cedentia.
Ob&longs;eruabis primò, vtrumque globum e&longs;&longs;e eiu&longs;dem materiæ;
&longs;i enim
&longs;int diuer&longs;æ materiæ, &longs;ecùs accidit, quàm diximus; &longs;i v. g. æneus mi
nor pellatur ab eburneo maiore, maiorem motum hic illi non impri
met; licèt enim &longs;it maior exten&longs;io eburnei;
e&longs;t tamen minus pondus;
igitur pauciores partes.
Secundò, eos globos accipiendos e&longs;&longs;e, quorum partes, vel non auo
lent ab ictu, vel non comprimantur; comprimuntur in plumbeis,
æneis, & auolant in vitreis; cum enim &longs;it compre&longs;&longs;io, vel partium di
ui&longs;io, de&longs;truitur multùm impetus.
Tertiò reiice commentum illorum, qui dicunt corpus illud e&longs;&longs;e ma
joris velocitatis capax, quod plures habet partes materiæ &longs;ub eadem
quantitate; nam &longs;uppo&longs;ita eadem re&longs;i&longs;tentiæ ratione, omne corpus e&longs;t
capax illius velocitatis, cuius aliud e&longs;t capax; cum nullus &longs;it motus, quo
non po&longs;&longs;it dari velocior, & tardior, vt dicemus infrà; immò &longs;it glo
bus plumbeus 12. librarum, &longs;it eburneus eiu&longs;dem diametri 2. librarum,
v. g. haud dubiè eadem potentia producet inten&longs;iorem impetum in
eburneo, vt patet experientia, & ratio con&longs;tat ex dictis; qua&longs;i verò &longs;it
aliqua materiæ inertia, quæ motum re&longs;puat; licèt fortè maior &longs;it pro
portio re&longs;i&longs;tentiæ medij comparatæ cum globo eburneo, quàm compa
ratæ cum plumbeo; &longs;ed de re&longs;i&longs;tentia de percu&longs;&longs;ione, & de &longs;patio age
mus infra.
lociùs mouetur eo &c. in quem impingitur
po&longs;&longs;et, quod ex ip&longs;is terminis con&longs;tat.
quia impeditur eius motus, igitur vt tollat impedimentum, agit ad
extra per Th. 44.
immobiliquia &longs;i &longs;i&longs;teret, maius e&longs;&longs;et impedimentum, quia
totum motum impediret, cuius tantùm partem impedit, dum mouetur ,
licèt paulò tardius; igitur minus agit ad extra per Th. 49.
feratur nullum producit in eo impetum. Probatur, quia mobile quod præit,
non impedit motum &longs;ub&longs;equentis; igitur nullum impetum ab eo acci
pit per Th. 48.
globus percu&longs;&longs;us ab alio eadem
&longs;emper velocitate mouetur, &longs;iue moueretur in&longs;tanti percu&longs;&longs;ionis, &longs;iue &longs;i
&longs;teret.
dus velocitatis: id e&longs;t æqualis impetus impetui percutientis, iam verò
moueatur A, cum 20. grad. velocitatis, & B, qui mouetur cum 40.
impingatur, certè cum impediatur tantùm &longs;ubduplum motus, produce
tur tantùm &longs;ubduplum impetus, id e&longs;t 20. qui &longs;i addantur 20. grad. erunt
40. quæ omnia con&longs;tant per Th.49.48.&c.
Hinc æquale &longs;emper &longs;patium percu&longs;&longs;us globus conficit, &longs;iue ante per
cu&longs;&longs;ionem moueretur, &longs;iue quie&longs;ceret.
Hinc &longs;i &longs;ecundò percutiatur idem globus, &longs;patium totum, quod per
currit tùm à primò, tùm à &longs;ecundo ictu e&longs;t maius eo, quod à primo ictu
confeci&longs;&longs;et, &longs;i non fui&longs;&longs;et &longs;ecundò percu&longs;&longs;us; maius inquam &longs;egmento &longs;pa
tij interiecto inter primum & &longs;ecundum ictum.
Hinc reiicies aliquos, quorum &longs;ententiam habes apud Doctum Mer
&longs;ennium, &longs;i malleus pilam
currentem eodem, ac anteà modo percutiat, nonam &longs;ui motus partem; &longs;i verò
currentem tertia vice percutiat, vnam vige&longs;imam &longs;eptimam &longs;ui motus par
tem ei tribuet, atque ita deinceps. Supponit primò hæc &longs;ententia mal
leum e&longs;&longs;e duplum pilæ percu&longs;&longs;æ. Secundò, malleum imprimere pilæ &longs;ub
duplæ &longs;ubtriplum motum; quod fal&longs;um e&longs;t, vt con&longs;tat ex Th 6. & Co
roll. 1. Præterea, licètin primà percu&longs;&longs;ione imprimeret tantùm prædi
ctæ pilæ &longs;ubtriplum impetum, in &longs;ecunda percu&longs;&longs;ione maiorem impri
meret po&longs;t longiorem motum, vbi iam ad quietem propiùs accedit; mi
norem verò paulò po&longs;t initium motus, vt con&longs;tat ex dictis, & ex ip&longs;a ex
perientia; pote&longs;t quidem in aliquo puncto &longs;ui motus &longs;ecunda vice per
cuti, in quo &longs;ubtriplum tantùm motum imprimet; hoc e&longs;t eo in&longs;tanti
quo tantùm ami&longs;it tertiam fui impetus partem; tum deinde in tertia
percu&longs;&longs;ione pote&longs;t tantùm (1/27) motus partem illi tribuere; eo &longs;cilicet in
&longs;tanti, quo tantùm ami&longs;it (1/27) &longs;ui impetus partem; &longs;ed in alijs temporis
punctis longè alia erit impetus producti ratio; Igitur tota hæc progre&longs;
&longs;io gratis omninò fuit excogitata.
Hinc etiam po&longs;t &longs;ecundam percu&longs;&longs;ionem æquale &longs;patium conficiet al
teri, quod iam confecit po&longs;t primam æqualibus temporibus; igitur æqua
lis e&longs;t velocitas vtriu&longs;que motus; quia &longs;cilicet, &longs;i e&longs;t æqualis impetus, e&longs;t
qualis motus: Ex his maximam carum dubitationum partem &longs;oluere po
teris quæ in eadem Mer&longs;enni propo&longs;itione courinentur reliquas vero ex
dicendis infrà.
Ex dictis etiam colliges diuer&longs;as percu&longs;&longs;ionum rationes &longs;uppo&longs;ita di
uer&longs;a ratione ponderum globi percutientis, & percu&longs;&longs;i; cum enim impe
tus productus &longs;it æqualis per &longs;e impetui producenti, per Th.60. modò
debita fiat applicatio, de qua in Th.50. &longs;i percutiens &longs;it duplus percu&longs;&longs;i,
&longs;uppo&longs;ita eadem materia, motus percu&longs;&longs;i erit duplò velocior; quia im
petus erit duplò inten&longs;ior, vt con&longs;tat ex Th. 61. &longs;i verò &longs;it quadruplus,
quadruplo, &c. Igitur velocitates motuum &longs;unt in ratiòne ponderum
permutando.
dem corporis; producitur impetus in percu&longs;&longs;io æqualis impetui percutientis
&longs;ed
opus e&longs;t aliqua figura: Sit corpus AD, parallelipedum;
diuidatur æqua
liter in E ita vt E &longs;it centrum grauitatis; &longs;i percu&longs;&longs;io fiatin E per lineam
perpendicularem HE, producetur impetus in corpore AD æqualis im
petui corporis percutientis; quia &longs;cilicet à corpore AD non pote&longs;t maius
e&longs;&longs;e impedimentum; igitur agit quantùm pote&longs;t impetus corporis per
cutientis per Th.50. igitur producit æqualem per Th.69.
mentum, quàm per
recto, &longs;i per IF, etiam motu circulari circa aliquod centrum; &longs;ed hic
motus e&longs;t facilior quam ille; igitur minus e&longs;t impedimentum;
(&longs;uppono
autem cylindrum BC vtroque modo moueri po&longs;&longs;e ab applicata potentia)
igitur minùs impetus producitur, &longs;i percu&longs;&longs;io fiat per IF, quàm &longs;i fiat
per LK: In qua verò proportione &longs;it minus impedimentum, & minori
opus impetu, po&longs;ito eodem potentiæ ni&longs;u, determinabimus facilè aliàs;
vt etiam demon&longs;trabimus circa quod centrum hic circularis motus fieri
debeat.
Ex duobus capitibus minus e&longs;&longs;e pote&longs;t impedimentum;
primum e&longs;t,
quod petitur à puncto contactus, &longs;ecundum à linea incidentiæ; v. g. &longs;i
accipiatur punctum E, in quo e&longs;t centrum grauitatis corporis AD, & in
eo fiat percu&longs;&longs;io; maximum e&longs;t impedimentum ratione puncti conta
ctus, in quo fit percu&longs;&longs;io; &longs;i verò percu&longs;&longs;io fiat per lineam perpendicu
larem HE, maximum e&longs;t impedimentum, ratione lineæ; &longs;i autem ex
vtroque capite &longs;imul accidat impedimentum, maximum e&longs;t omnium;
iam verò &longs;i accipiatur punctum E, & linea percu&longs;sionis ME; minor e&longs;t
percu&longs;sio ratione lineæ non puncti; accipiatur punctum N, & linea
percu&longs;sionis MN, minor e&longs;t percu&longs;sio ratione puncti non lineæ, acci
piatur punctum N, & linea IN, minor e&longs;t percu&longs;sio ratione vtriu&longs;que;
&longs;i demum accipiatur punctum E, & linea ME, minor e&longs;t percu&longs;sio raaccipiatur punctum N linea percu&longs;&longs;ionis MN,
minor e&longs;t percu&longs;&longs;io ratione puncti non lineæ; &longs;i accipiatur punctum N,
& linea IN, minor e&longs;t percu&longs;&longs;io ratione vtriu&longs;que: &longs;i demum accipia
tur punctum E & linea HE, maior e&longs;t percu&longs;&longs;io ratione vtriu&longs;que; igi
tur &longs;unt quatuor coniugationes; &longs;eu quatuor cla&longs;&longs;es diuer&longs;arum percu&longs;
&longs;ionum.
Hinc compen&longs;ari pote&longs;t ratione vnius quod dee&longs;t ratione alterius,
v. g. &longs;i fiat percu&longs;&longs;io in puncto E per lineam ME, pote&longs;t &longs;ciri punctum
inter ED, in quo percu&longs;&longs;io per lineam perpendicularem &longs;it æqualis
percu&longs;&longs;ioni per lineam ME; &longs;ed de his infrà in lib. 10. cum de percu&longs;
&longs;ione, determinabimus enim vnde proportiones i&longs;tæ petendæ &longs;int, &
demon&longs;trabimus totam i&longs;tam rem, quæ multùm curio&longs;itatis habet, &
vtilitatis.
Determinabimus etiam dato puncto percu&longs;&longs;ionis F v.g. cum &longs;equatur
motus vectis, quodnam &longs;it centrum vectis &longs;eu huius motus.
Hinc demum &longs;equitur, ne hoc omittam, data minimâ percu&longs;&longs;ione per
lineam MN dari po&longs;&longs;e adhuc minorem per lineam IN, & alias incli
natas; & data percu&longs;&longs;ione per lineam quantumuis inclinatam, po&longs;&longs;e da
ri æqualem per lineam perpendicularem; & data per lineam perpendi
cularem extra centrum grauitatis E, po&longs;&longs;e dari æqualem; & in qualibet
data ratione per aliquam inclinatam, quæ cadat in E, &longs;ed de his fusè
&longs;uo loco.
motu recto per lineam directionis, quæ producta à puncto contactus ducitur per
centrum globi, dum fiat contactus in centro grauitatis parallelipedi, maximum
ictum infligit, &longs;eu agit quantùm pote&longs;t.
moueatur motu recto parallelo, lineis CD, HG, &c. &longs;itque globus in
D; haud dubiè agit quantùm pote&longs;t, quia &longs;cilicet e&longs;t maximum impedi
mentum per Th.68. Tam enim globus in D impedit motum paralleli
pedi, quàm parallelipedum motum globi impacti per lineam ID; impedit
inquam ratione oppo&longs;itionis; quia centra grauitatis vtriu&longs;que con
currunt in eadem linea; igitur &longs;i maximum e&longs;t impedimentum, agit
quantùm pote&longs;t Th. 50. hinc producitur impetus æqualis per Th.60.
& invt con&longs;tat ex dictis in &longs;uperioribus Theorematis;
in qua vero proportione determinabimus aliàs.
non,eiu&longs;que motus
&longs;it parallelus lineis ED, BC: &longs;it autem DA dupla DE;
&longs;itque diui&longs;a to
ta DA æqualiter in C, in C non erit maximus ictus; quia in C non
vt autem habeatur centrum impre&longs;&longs;io
nis; a&longs;&longs;umatur AN media proportionalis inter totam AD, & &longs;ubdu
plum AC; certè cum triangulum ANO &longs;it &longs;ubduplum totius ADE,
vt con&longs;tat ex Geometria, & æquale trapezo ND EO; erit impetus in
vtroque æqualis; igitur in N erit centrum impre&longs;&longs;ionis, vel impetus;
vt
autem habeatur centrum percu&longs;&longs;ionis; in quo &longs;cilicet maximus ictus in
fligitur, inueniatur centrum grauitatis H, ducaturque KHI parallela
DE, centrum percu&longs;&longs;ionis erit in I; quippe in I totus impeditur impetus
grauitatis vtrimque, cum &longs;it in æquilibrio; quomodo verò inueniatur
punctum H facilè habetur ex Archimede, ductis &longs;cilicet AF, DB, quæ
diuidant bifariam æqualiter DE, EA; vel a&longs;&longs;umpta AI dupla ID, quod
demon&longs;trabimus in Mechan.
ratio percu&longs;&longs;ionum ab ea, quàm &longs;uprà propo&longs;uimusmoueatur enim circa
centrum C, fitque CA diui&longs;a bifariam in B, haud dubiè punctum A
faciet arcum AE eo tempore, quò punctum B faciet BD &longs;ubduplum
AE; igitur punctum A duplò velociùs mouetur quàm B, vt con&longs;tat;
igi
tur habet duplò maiorem impetum; cum effectum habeat duplò maio
rem per Ax. 13. n. 4. igitur cum totus motus &longs;egmenti AB &longs;it ad to
tum motum &longs;egmenti BC, vt &longs;patia acqui&longs;ita; certè &longs;patia acqui&longs;ita
&longs;unt vt arcus; igitur & trapezus BAED, continet 3/4 totius CAE, vt
con&longs;tat; &longs;unt enim &longs;ectores &longs;imilis in ratione duplicata radiorum;
igi
tur totus motus &longs;egmenti BC &longs;ubquadruplus motus totius CA; igitur
& impetus; vt autem habeatur centrum impre&longs;&longs;ionis, vel impetus;
&longs;it &longs;e
ctor CHI, &longs;ubduplus totius CAE quod quomodo fiat, patet ex Geo
metria; accipiatur tantùm &longs;ubdupla diagonalis quadrati lateris CA, igi
tur in puncto H e&longs;t centrum impre&longs;&longs;ionis, &longs;eu media proportionalis in
ter totam CA, & &longs;ubduplam CB: vt autem habeatur percu&longs;&longs;ionis, a&longs;
&longs;umatur CY dupla YA; Dico punctum Y e&longs;&longs;e centrum percu&longs;&longs;ionis;
quia perinde &longs;e habet, atque &longs;i e&longs;&longs;et trianguli cadentis ictus, vt demon
&longs;trabimus aliàs nunc tantùm indica&longs;&longs;e &longs;ufficiat.
Hinc etiam &longs;oluetur, quod proponunt aliqui; &longs;eu potiùs quærunt;
in quà &longs;cilicet parte maiorem ictum infligat en&longs;is; &longs;i enim &longs;it eiu&longs;dem
cra&longs;&longs;itiei in omnibus &longs;uis partibus, idem dicendum e&longs;t quod de cylin
dro CA; &longs;i verò in mucronem de&longs;inat, inueniemus etiam centrum
percu&longs;&longs;ionis.
Huc etiam reuoca clauarum ictus, vel aliorum corporum, quæ ad in
&longs;tar &longs;eu conorum, &longs;eu pyramidum ver&longs;us mucronem maiora &longs;unt, vel
den&longs;iora; quippe ex iacto &longs;uprà principio i&longs;torum omnium effectuum
rationes demon&longs;trabimus.
Colligemus etiam quid dicendum &longs;it de malleorum ictu;
&longs;it enim
malleus F æqualis malleo G (in his vna fere manubrij longitudinis ha
betur ratio) ducatur arcus NM, itemque OG; ictus mallei G e&longs;t ferè
&longs;ubduplus alterius, dum vterque malleus &longs;it æqualis; dixi ferè, quia
motus totius mallei G non e&longs;t omninò &longs;ubduplus motus mallei F, quia
&longs;cilicet trapezus OD e&longs;t minor &longs;ubduplo alterius NE; quotâ vero parte
&longs;it minor facilè pote&longs;t &longs;ciri opera Geometriæ: &longs;ed hæc omnia determi
nabimus.
in infinitumpater, quia quocumque dato motu pote&longs;t dari velocior in
infinitum; igitur pote&longs;t dari impetus inten&longs;ior, & inten&longs;ior in infinitum.
Hîc ob&longs;erua nouum di&longs;crimen, quod intercedit inter impetum, &
alias qualitates; quæ fortè non po&longs;&longs;unt intendi in infinitum, ratio di&longs;
criminis e&longs;t, quia totus calor exten&longs;us in maiore &longs;ubiecto non pote&longs;t
produci in minore, in quo eadem cau&longs;a eumdem &longs;emper effectum pro
ducit; quia &longs;cilicet agit vniformiter difformiter; at verò impetus exten
&longs;us in magno
ferè pilâ.
ten&longs;ione, non pote&longs;t intendere alium &longs;imilemProbatur, quia agit tantùm ad
extra, vt tollat impedimentum per Th. 44. &longs;ed eorum mobilium, quæ
ver&longs;us
terius motum, vt con&longs;tat; igitur impetus &longs;imilis, &c.
Ob&longs;erua de impetu &longs;imili id tantùm dici;
&longs;imili inquam id e&longs;t non
modò eiu&longs;dem inten&longs;ionis; &longs;ed etiam eiu&longs;dem lineæ:
&longs;i enim alterum
de&longs;it, haud dubiè &longs;imilis impetus non e&longs;t; &longs;ic impetus quatuor grad. in
tendere pote&longs;t impetum duorum graduum; licèt vterque ad
neam &longs;it determinatus; &longs;i verò ad diuer&longs;as lineas determinentur; etiam
impetus vt duo pote&longs;t intendere impetum vt quatuor.
Ob&longs;eruabis præterea hoc Theorema ita e&longs;&longs;e intelligendum, vt impe
tus mobilis præeuntis nullo modo impediatur; alioquin mobile &longs;ucce
dens omninò aliud vrgeret, vt con&longs;tat.
Hinc &longs;imile pote&longs;t in aliquo ca&longs;u agere in &longs;imile;
vnde rectè colligo
id tantùm dictum e&longs;&longs;e ab Ari&longs;totele de qualitatibus alteratiuis; quid
verò accidat, cum mobile graue mobili alteri &longs;uperponitur; dicemus
infrà.
cum
enim extra &longs;ubjectum e&longs;&longs;e non po&longs;&longs;it, cum &longs;it qualitas; certè ibi e&longs;t, vbi
&longs;ubjectum e&longs;t; nam penetratur accidens cum ip&longs;o
Ob&longs;eruabis qualitatem omnem ita &longs;uo &longs;ubjecto coëxtendi, vt æqua
lem omnino quodlibet eius punctum, &longs;eu pars extentionem habeat ex
tentioni puncti, &longs;eu partis &longs;ui &longs;ubjecti; nec enim aliud e&longs;t, vnde po&longs;&longs;it
determinari extentio qualitatum, præter ip&longs;am exten&longs;ionem &longs;ubjecti;
quod maximè in impetu videre e&longs;t, cuius partes in mobili den&longs;o minori
extentioni &longs;ubjacent, quàm in mobili raro; cum ex maiore ictu &longs;eu per
cu&longs;&longs;ione in mobili den&longs;o plures impetus agentis partes e&longs;&longs;e con&longs;tet; quia
&longs;cilicet &longs;unt plures partes &longs;ubiecti.
dum entitatem; quia iam dictum e&longs;t &longs;uprà dari perfectiorem &longs;ecundum
inten&longs;ionem; huius Theorematis veritas mihi maximè demon&longs;tranda
e&longs;t, ex quo tàm multa infrà deducemus; &longs;ic autem probamus;
Quotie&longs;
cunque mouetur corpus, producuntur &longs;altem tot partes impetus quot
&longs;unt partes mobilis per Th. 33. Quotie&longs;cunque producuntur in mobili
tot partes impetus quot &longs;unt in mobili partes &longs;ubjecti, mouetur mobile,
modó non impediatur; quia po&longs;ita cau&longs;a nece&longs;&longs;aria, & non impedita per
Ax. 11. ponitur effectus, quod de omni cau&longs;a, &longs;ed de formali poti&longs;&longs;imum
dici debet; præterea datur aliquod pondus, quod data potentia &longs;ine me
chanico organo mouere non pote&longs;t, licèt cum organo facilè moueat; hæc
hypothe&longs;is certa e&longs;t; igitur cum mouet, producit tot partes impetus quot
&longs;unt nece&longs;&longs;ariæ, vt omnibus partibus mobilis di&longs;tribuantur per idem Th.
33. cum verò non mouet, non producit tot partes impetus vt con&longs;tat ex
dictis; igitur producit plures cum organo in mobili, quàm &longs;ine organo;
igitur imperfectiores, quod demon&longs;tro: &longs;it enim vectis BF, cuius cen
trum &longs;eu fulcrum &longs;it in A, potentia in B, pondus G, quod attollitur in F;
plures partes impetus produci po&longs;&longs;unt in F, vel in E, quàm in B, &longs;cilicet
in ip&longs;o pondere; quia pondus quod non pote&longs;t attolli in B, attollitur in
E, vel in F, vt patet ex dictis; præterea punctum F mouetur tardius, quàm
B; quia motus &longs;unt vt arcus, arcus vt &longs;emidiametri, hæ demum vt AF,
ad AB; igitur motus puncti F, e&longs;t tardior, vel imperfectior;
igitur im
petus puncti F, e&longs;t imperfectior impetu puncti B, per Ax. 13 num.4. atqui
non e&longs;t imperfectior ratione numeri partium, igitur ratione entitatis,
quæ imperfectior e&longs;t; igitur datur impetus altero impetu imperfectior.
Ob&longs;eruabis primò multa hîc &longs;upponi &longs;eu de&longs;iderari, quæ pertinent
ad propagationem impetus, de quibus infrà; Secundò hoc Theorema
perfectior, vel imperfectior.
Tertiò, &longs;i dato quocunque motu pote&longs;t dari tardior: igitur dato quo
cunque impetu pote&longs;t dari imperfectior.
Quartò, &longs;i daretur punctum impetus in inten&longs;ione: non po&longs;&longs;et dari
motus tardior in infinitum &longs;ine diuer&longs;is gradibus perfectionis.
Quintò, &longs;ine hac diuer&longs;a impetus perfectione non po&longs;&longs;et explicari
productio continua impetus, quæ &longs;it temporibus inæqualibus, neque de
&longs;tructio eiu&longs;dem impetus; nec motus in diuer&longs;is planis inclinatis, vel in
diuer&longs;is lineis citra perpendicularem, &longs;ed de his omnibus &longs;uo loco.
Sextò, Denique ratio propo&longs;ita rem i&longs;tam euincit;
cum enim in motu
vectis plures partes producantur ver&longs;us centrum, &longs;cilicet, in maiori pon
dere, quod attollitur; & cum hæ habeant motum tardiorem, &longs;equitur ne
ce&longs;&longs;ariò e&longs;&longs;e imperfectiores.
to quocumque motu dari pote&longs;t tardior, ergo dato quocumque impetu
imperfectior.
pauciores &longs;cilicet eiu&longs;dem impetus partes. Primò, quia cum retardari po&longs;&longs;it
hic motus, & de&longs;trui &longs;ucce&longs;&longs;inè hic impetus; cumque in&longs;tantia motus
velocioris &longs;int breuiora; certè initio motus, breuiori &longs;cilicet tempore
imperfectior impetus de&longs;trui tantùm pote&longs;t; cum enim æqualis æquali
bus temporibus; certè inæqualis inæqualibus. Secundò quia vix explica
ri pore&longs;t quomodo duæ formæ homogeneæ in eodem &longs;ubiecti puncto
exi&longs;tere po&longs;&longs;int, quod etiam in commune e&longs;t calori, lumini, &c.
&longs;us circumferentiam in omnibus partibus, &longs;ed maior ver&longs;us eandem circumfe
rentiam,
Hinc difficiliùs attollitur pertica CA ex puncto C motu circulari,
quàm ex puncto B motu recto; quia &longs;cilicet, cum motu recto ex puncto B
attollitur, omnes partes mouentur motu æquali; igitur impetus æqualiter
omnibus di&longs;tribuitur; igitur modò producantur tot partes impetus, quot
&longs;unt partes in mobili; haud dubiè attolletur:
at verò, cum motu circulari
ex puncto C attollitur, omnes partes inæquali motu attolluntur; igitur
plures &longs;unt nece&longs;&longs;ariæ, vt attollatur motu circulari; igitur difficiliùs iuxta
experimentum; adde quod cum applicatur potentia in C, punctum A,
maius momentum habet, de quo aùàs.
Hinc ratio euidens illius experimenti, quo manife&longs;tè con&longs;tat perti-
licet cuiu&longs;dam qua&longs;i reflexionis opera eodem tempore vtraque extremi
tas æquali motu attollitur.
A,
eiu&longs;dem perfectionis proportionaliter, cuius e&longs;t ab
ver&longs;us B, producitur quidem vnum punctum, vel vnus gradus impetus
eiu&longs;dem perfectionis cum eo, qui productus e&longs;t in F, & in E (&longs;upponi
tur enim ex. gr. vnus tantùm gradus in F, & in E, productus) at verò
producuntur alij imperfectiones. v.g. in D, præter æquè perfectum pro
ducuntur 3. alij adæquantes perfectionem prioris; in C verò, præter 4.
&longs;imiles ijs, qui &longs;unt in D, producuntur 5. alij adæquantes prioris perfe
ctionem in B7; atque ita deinceps per numeros impares, & quadrata,
nullus tamen producitur perfectioris entitatis.
quatenus fit tali modoquæ non pote&longs;t explicari per impetum remi&longs;&longs;io
rem, vel inten&longs;iorem; nam cum &longs;it tantùm impetus in&longs;titutus propter
motum; certè ille tantùm impetus produci pote&longs;t, ex quo pote&longs;t &longs;equi
motus; igitur &longs;i tali tantùm motu data pars mobilis moueri pote&longs;t; haud
dubiè talis tantùm impetus, ex quo &longs;equitur talis motus, in ea produ
cetur, & tali modo.
&longs;eor&longs;im entitas eiu&longs;dem impetus; &longs;ed debet comparari tota collectio omniu&mtail;
partium impetus, quæ in&longs;unt datæ parti &longs;ubiecti, cum tota collectione partium
quæ alteri parti mobilis in&longs;untquippe plures partes impetus po&longs;&longs;unt ha
bere eum motum, vel potius eam motus perfectionem, quam pauciores
haberent; igitur perfectio illarum e&longs;t ab ip&longs;o motu, quatenus cum ip&longs;o
partium numero comparatur.
nam po
tentia, &longs;en pondus extremitati appen&longs;um producit in &longs;e impetum, à quo
deinde impetus in toto vecte producitur per Th.42. &longs;ed impetus pon
deris appen&longs;i e&longs;t eiu&longs;dem perfectionis cum impetu producto in ip&longs;a ve
ctis extremitate, ex qua pendet; cum &longs;it vtriu&longs;que æqualis motus; &longs;ed
ver&longs;us centrum eiu&longs;dem vectis producitur impetus imperfectior per
Th.82. igitur imperfectus à perfecto producitur.
adde quod nunquam effectus perfectio &longs;uperat perfectionem cau&longs;æ; dixi
perfectum ab imperfecto; &longs;cilicet &longs;i con&longs;ideretur perfectio ratione en-
cum reuerâ, vt dictum e&longs;t &longs;uprà, remi&longs;&longs;us producat inten&longs;um,
quod in vecte clari&longs;&longs;imum e&longs;t; quippe momentum applicatum in F, quod
tardiùs mouetur deor&longs;um, quàm B, &longs;ur&longs;um, vt patet, habet impetum re
mi&longs;&longs;iorem, qui tamen producit in B, inten&longs;iorem: Pro quo, ob&longs;eruabis
impetum imperfectum cum alio perfecto actione communi agentem
po&longs;&longs;e concurrere ad producendum perfectum, vt patet; non tamen in
ratione cau&longs;æ totalis: &longs;imiliter plures imperfecti &longs;imul concurrentes
po&longs;&longs;unt producere perfectum; quia plures imperfecti conjunctim adæ
quant perfectionem alterius perfectioris &longs;inguli &longs;eor&longs;im.
Ob&longs;eruabis &longs;ecundò præclarum naturæ in&longs;titutum, quo factum e&longs;t;
vt cum vires hominum maiora pondera leuare non po&longs;&longs;int, &longs;i &longs;eor&longs;un
con&longs;iderentur; cum organis tamen mechanicis conjunctæ nullum pon
dus quantumuis immane leuare non po&longs;&longs;int; quod certè nullo modo ac
cideret, ni&longs;i plures partes impetus producerent neque plures producere
po&longs;&longs;ent, ni&longs;i minoris perfectionis e&longs;&longs;ent; quia faciliùs producitur effe
ctus imperfectus, quam perfectus per Ax. 13.num.4.
Tertiò hinc optimè à natura proui&longs;um e&longs;t, vt motus tardior in infi
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
alio imperfectior.
Quartò, hinc quoque benè explicatur diuer&longs;itas impetus, quæ oritur
tum à diuer&longs;o medio, tùm à plano inclinato, tùm ab aliis impedimentis,
tùm à diuer&longs;o ni&longs;u eiu&longs;dem potentiæ, tùm maximè à diuer&longs;o applicatio
nis modo; de quibus aliàs.
Quintò, &longs;i potentia applicata mobili immediatè illud moueat motu
recto, vel in &longs;ingulis punctis mobilis producitur vnum punctum impe
tus, vel plura; &longs;i primum, erit primus tantùm gradus maximæ perfectio
nis; ita vt perfectiorem producere non po&longs;&longs;it, ad quem e&longs;t determinata
potentia; imperfectiorem tamen impetu innato, de quo infrà; &longs;i verò
&longs;ecundum, producet in &longs;ingulis partibus
mum cum aliis pluribus, vel paucioribus heterogeneis, & imperfectio
ribus.
punctum impetus in quolibet puncto &longs;ubiecti; &longs;i tamen impetum producit, quod
definiam lib.
pertinetProbatur, quia fru&longs;trà e&longs;&longs;ent plura puncta impetus; nec enim
&longs;unt multiplicandæ formæ &longs;ine nece&longs;&longs;itate, ratione &c. per Ax. 7. & 3.
n. 1. Præterea non e&longs;t, cur potius produceret 2. quàm 3. 4. &c.
atqui
quod vnum e&longs;t, determinatum e&longs;t per Ax. 5.
impetus in eadem parte &longs;ubiecti producere potestProbatur in proiectis,
quorum impetus aliquando plùs, aliquando minùs durat licèt &longs;en&longs;im
&longs;ingulis in&longs;tantibus aliquid illius de&longs;truatur; determinatur autem
motrix; quia modò maior e&longs;t ni&longs;us, modò minor.
ne producitaccipiatur enim totum illud tempus, quo vnicum tantùm
punctum impetus producit (vocetur in&longs;tans) de quo in Th. 86; certè
&longs;i in minori tempore agat, minùs aget, per Ax. 13. num. 4. &longs;ed non
pote&longs;t minùs agere ratione numeri, vt patet; igitur ratione perfectio
nis.
Ob&longs;eruabis &longs;ine hoc Theoremate explicari non po&longs;&longs;e accelerationem
motus naturalis, vel augmentum impetus, vt videbimus.
&longs;tantiis medij, & re&longs;i&longs;tentiæ, minori tempore minùs de&longs;truitur; plus verò ma
jori:
exigentiam alicuius de&longs;truitur, per Ax. 14. num. 2. igitur minori
tempore minùs de&longs;truitur per Ax. 13. 4. alioquin totus &longs;imul debe
ret de&longs;trui.
Ob&longs;eruabis etiam &longs;ine hoc Theoremate non po&longs;&longs;e explicari de&longs;tru
ctionem impetus violenti, vt videbimus infrà.
Hinc, quò potentia diutiùs manet applicata (putà malleo) percu&longs;&longs;io ma
ior e&longs;t.
Hinc, quò impedimentum diutiùs manet applicatum, illa de&longs;tructio
e&longs;t maior.
Hinc præclara eruitur ratio, cur maior lapis, quàm minor impactus
maiorem ictum infligat; licèt tot partes impetus eodem in&longs;tanti produ
cantur in vno, quot in alio: quia &longs;cilicet diutiùs manet applicatus po
tentiæ; &longs;ed hanc rationem explicabimus fusè lib. 10. cum de percu&longs;
&longs;ione.
Probatur;
quia cum omnes eius partes moueantur, nec vlla &longs;ine im
petu moueri po&longs;&longs;it per Th. 18. & 33. cum etiam potentia motrix non
&longs;it omnibus immediatè applicata, vt con&longs;tat; certè &longs;ine propagatione,
vel diffu&longs;ione non pote&longs;t explicari productio huius motus.
Ob&longs;eruabis propagationem impetus, vel alterius qualitatis e&longs;&longs;e tan
tùm continuatam eiu&longs;dem productionem, quæ incipit ab ea parte, cui
potentia e&longs;t immediatè applicata, & propagatur, &longs;eu diffunditur per
omnes alias donec ad vltimam perueniat eo modo, quo iam definio.
prima parte &longs;ubiecti tran&longs;eat ad &longs;ecundam,
cidens per Th. 8. de &longs;ubiecto in &longs;ubiectum tran&longs;ire non pote&longs;t per deff.
accidentis; de qua in Metaphy&longs;icâ; nec e&longs;t quod aliqui dicant &longs;e
concipere, quomodo id fiat &longs;ine motu locali; cum ip&longs;is etiam oculis
qua&longs;i cernatur; cum enim percutis corpus oblongum AE, & cadit ictus
in extremitatem A, corpus ip&longs;um totum &longs;imul moues; igitur pars impe
tus, quæ recipitur in A, non migrat in E, &longs;ed hæc producitur in A, &
alia in B, alia in C, atque ita deinceps.
Ob&longs;eruabis ex hac propagatione impetus per analogiam rectè om
ninò explicari propagationem luminis, & aliarum qualitatum, de qui
bus &longs;uo loco.
hæcPrimò, quia &longs;i hoc e&longs;&longs;et,
omne corpus po&longs;&longs;et moueri à qualibet potentia; nam modò po&longs;&longs;et pro
duci vnum punctum impetus, hoc etiam aliud produceret, & hoc aliud,
atque ita deinceps. Secundò, Minimum granum &longs;uperpo&longs;itum rupi, to
tam ip&longs;am rupem mouere po&longs;&longs;et. Tertio, Quia vel in omnibus, vel in
nulla parte impetus producitur per Th.33. Quartò, quia impetus mobi
lis projecti intenderetur; nam impetus vnius partis impetum alterius
intenderet. Quintò, quia impetus partis B, tàm ageret in A, trahendo,
quàm in C pellendo; cum impetus vtroque modo propagetur. Sextò, &longs;i
applicaretur potentia in C, non video, cur impetus partis C, ageret po
tius versùs E, quàm versùs A? alioquin eadem pars impetus plures pro
ducere po&longs;&longs;et; igitur impetus potentiæ motricis &longs;ufficiens erit cau&longs;a ad
producendum totum alium. Septimò, tractionis impetus explicari non
pote&longs;t, &longs;i impetus vnius partis producat in alia impetum; alioquin dare
tur mutua actio infinities repetita, vt con&longs;ideranti patebit. Octauò, &longs;i
impetus vnius partis producit in alia; &longs;int duo globi contigui; igitur il
le, qui impellit alium, reflecti po&longs;&longs;et, quod nunquam accidit quando
&longs;unt contigui.
Ob&longs;eruabis illud quidem verum e&longs;&longs;e in motu recto, &longs;ecus in circulari;
nam cum cylindrus circa alteram extremitatem vibratus deor&longs;um cadit;
partes, quæ propiùs ad extremitatem immobilem accedunt iuuant mo
tum aliarum, quæ longiùs ab eadem recedunt.
Proba
tur; &longs;it enim applicata potentia in A, dico &longs;imul produci impetum in
BCDE; quia &longs;i primo in&longs;tanti produceretur in A, & &longs;ecundo in B, vel
A moueretur ante B, vel impetus in A e&longs;&longs;et fru&longs;trà; vtrumque e&longs;t ab&longs;ur
dum; nam totum AE, &longs;imul mouetur.
Primò
&longs;i æqualiter omnibus partibus &longs;ubjecti di&longs;tribuatur; id e&longs;t vniformiter.
Secundò, &longs;i plùs partibus propioribus, & minùs remotioribus. Tertiò, è
contra, &longs;i plùs remotioribus, & minùs propioribus; tribus etiam ratione
perfectionis eo modo, quo diximus de inten&longs;ione; at verò nouem mo
dis propagari pote&longs;t ratione vtriu&longs;que; patet ex regula combinationum;
&longs;i enim 3. ducantur in 3. habebis 9. Iam &longs;upere&longs;t, vt videamus, an reue
rà omnibus i&longs;tis modis impetus re ip&longs;a propagetur; quod licèt difficile
&longs;it, & vix hactenus explicatum: Audeo tamen polliceri meum &longs;uper hac
re conatum non pror&longs;us inutilem fore.
æquali motuprobatur, quia impetus non cogno&longs;citur ni&longs;i per motum;
igitur vbi e&longs;t æqualis motus, debet e&longs;&longs;e æqualis impetus in omnibus par
tibus, id e&longs;t æqualis graduum heterogeneorum collectio, in quo non
e&longs;t difficultas.
Ob&longs;eruabis illud mobile moueri motu æquali &longs;ecundum omnes &longs;ui
partes, quod mouetur motu recto; quippe fieri non pote&longs;t, quin omnes
partes, quæ mouentur motu recto &longs;implici, motu etiam æquali mouean
tur.
v. g. globi A & B, æquales &longs;ibi inuicem contigui in C, &longs;it applicata po
tentia in D, non modò producet impetum in globo A, &longs;ed etiam in B:
probatur primò, quia &longs;e habent per modum vnius, vt patet ex re&longs;i&longs;ten
tia, nec enim A moueri pote&longs;t &longs;ine B per lineam DE, quod certè cla
ri&longs;&longs;imum e&longs;t; probatur &longs;ecundò quia &longs;i A produceret impetum in B, duo
globi, vel 3. vel 5. vel infiniti tantùm re&longs;i&longs;terent, quantùm vnicus glo
bus, quod fal&longs;um & ab&longs;urdum e&longs;t. Tertiò, Ratio à priori e&longs;t;
quia ideo
producitur, & propagatur impetus in toto A; quia vna pars non pote&longs;t
moueri &longs;ine alia per Th. 33. &longs;ed non pote&longs;t A moueri ni&longs;i moueatur B;
igitur in vtroque &longs;imul, & æqualiter propagatur impetus.
Hinc ratio manife&longs;ta cur maior &longs;it re&longs;i&longs;tentia duorum quàm vnius.
Hinc eadem vis requiritur ad &longs;u&longs;tinenda duo pondera; &longs;iue vtrum
que &longs;eor&longs;im humeris incubet, &longs;iue alterum alteri &longs;uperponatur.
Hinc percu&longs;&longs;io vel ictus globi B, cui alter A à tergo immediatè in
&longs;i&longs;tit maior e&longs;t.
Hinc pondus alteri &longs;uperpo&longs;itum actione communi cum alio graui
tat in &longs;uppo&longs;itam manum. v. g.
Hinc potentia applicata in D, minùs impetus &longs;ingulis imprimit.
Hinc demum licèt impetus ratione inten&longs;ionis &longs;it æqualis in vtroque
globo; attamen, &longs;i accipiatur numerus partium vtriu&longs;que impetus, im
petus &longs;unt vt globi v. g. &longs;i B e&longs;t æqualis A impetus productus in B e&longs;t
æqualis producto in A, &longs;i B &longs;it &longs;ubduplus, vel &longs;ubtriplus, impetus e&longs;t
&longs;ubtriplus, vel &longs;ubduplus; quorum omnium rationes patent ex Th.96.
Hinc etiam colligi pote&longs;t manife&longs;tum di&longs;crimen, quod intercedit inter
propagationem impetus, & aliarum qualitatum, quæ (vt vulgò dicitur)
vniformiter difformiter propagantur, id e&longs;t, æqualiter in æquali
di&longs;tantia, & inæqualiter inæquali.
Hinc demum colligi pote&longs;t non modò impetum produci in globo B
v. g. verùm etiam in aëre ambiente, cui &longs;cilicet globus contiguus e&longs;t;
qui reuera aër facilè amouetur; tùm quia propter raritatem pauci&longs;&longs;imæ
partes mouendæ &longs;unt; tùm quia facilè diuiduntur, de quibus alias;
tùm
quia, ne detur vaçuum, &longs;patium à tergo relictum occupare debet, quod
reuerà præ&longs;tat breui peracto circuitu, vt videre e&longs;t in aqua; nec enim
totus aër agitari debet; quis enim id con&longs;equi po&longs;&longs;et; tum denique, quia
aër non grauitat in aëre, igitur cum non re&longs;i&longs;tat vlla grauitatio, facilè
moueri pote&longs;t.
plures partes impetus continuò producantur ver&longs;us
cylindrus CA, fig. Th. 73. &longs;it centrum motus C;
haud dubiè plures
partes impetus producuntur in B, quàm in C, & plures in A, quam in B;
quia, cum pars B moueatur velociùs, quàm C, & A quàm B; certè, vbi e&longs;t
maior motus, vel effectus, ibi debet e&longs;&longs;e maior impetus, vel cau&longs;a per
Ax. 13. n. 4. quod autem &longs;it maior motus, con&longs;tat ex maioribus &longs;patiis,
vel arcubus æquali tempore confectis; quod verò &longs;it impetus inten&longs;ior
tro motus&longs;int enim punctum B, & punctum A:
ita &longs;e habet inten&longs;io
impetus puncti A ad inten&longs;ionem impetus puncti B, vt di&longs;tantia AC
ad BC. Probatur, quia cum impetus &longs;int vt motus, motus vt &longs;patia, &longs;patia
verò &longs;int arcus AE. BD; arcus &longs;unt, vt &longs;emidiametri AC, BC; igitur vt
di&longs;tantiæ quòd erat demon&longs;trandum.
Hinc &longs;i di&longs;tantia CA e&longs;t dupla di&longs;tantiæ CB, impetus in A e&longs;t du
plus impetus in B: at verò impetus &longs;egmenti e&longs;t ad impetum alterius,
vt diximus in Th. 73.
Hinc hæc propagatio fit iuxta progre&longs;&longs;ionem arithmeticam id e&longs;t, &longs;i
in primâ parte ver&longs;us centrum producitur impetus vt 1. in &longs;ecunda pro
ducitur vt duo, in tertiâ vt tria, atque ita deinceps; quia proportio
arithmetica e&longs;t laterum, &longs;eu linearum.
Hinc hæc propagatio e&longs;t omninò inuer&longs;a illius, quæ aliis qualitatibus
competit, vt patet.
Hinc etiam manife&longs;ta ratio &longs;equitur illius experimenti, quod propo
&longs;uimus corol. 2. Th. 80.
Hinc &longs;i tantùm habeatur ratio impetus, facilè pote&longs;t determinari in
qua proportione cylindrus faciliùs moueatur motu recto, quàm motu
circulari; po&longs;ito &longs;cilicet centro motus in altera extremitate, cui applica
tur potentia; quippe impetus propagatus in motu circulari e&longs;t &longs;umma
terminorum; propagatus verò in motu recto e&longs;t vltimus terminorum,
v.g. &longs;int &longs;ex puncta &longs;ubiecti; in quolibet producatur impetus vt vnum;
haud dubiè erit motus rectus; vt verò &longs;it motus circularis in primo
puncto; producatur vt 1. in &longs;ecundo vt 2. in tertio, vt 3. atque ita dein
ceps; &longs;umma erit 21. cum tamen in motu recto e&longs;&longs;ent tantùm 6. igitur
vt &longs;e habent 21. ad 6. ita &longs;e habet facilitas motus recti ad facilitatem
motus circularis.
Dixi, &longs;i tantùm habeatur ratio impetus;
quia &longs;i addatur ratio graui
tationis, &longs;eu momenti; haud dubiè maior erit adhuc difficultas, de
quo infrà in Schol.
Hinc quò longior e&longs;t cylindrus, v. g. cre&longs;cit proportio maioris illius
facilitatis, vt patet inductione; nam &longs;i &longs;int tantùm 2. puncta, proportio
erit 3. ad 2.; &longs;it tria 6. ad 3.;
&longs;i 4. 10. ad 4. &longs;i 5. 15. ad 5.;
&longs;i 6. 21. ad 6.
&longs;i 8. 36. ad 8;
&longs;i 9. 45. ad 9; atque ita deinceps; ex quibus primò
vides cre&longs;cere &longs;emper proportionem. Secundò inter duplam, & triplam
rationem, &longs;cilicet 6. ad 3. & 15. ad 5. intercedere 2 1/2; inter triplam &
quadruplam intercedere 3. 1/2; inter quadruplam & quintuplam inter
cedere 4 1/2; atque ita deinceps.
Colligo denique po&longs;&longs;e in motu recto cum maiore ni&longs;u produci inten
&longs;iorem impetum in data ratione; &longs;it enim cylindrus AB, qui moueatur
circa centrum A, percurrátque B, arcum BD; qui accipiatur vt recta,
quæ à minimis arcubus &longs;en&longs;u di&longs;tingui non pote&longs;t; haud dubiè &longs;i eo
tempore, vel æquali, quo AB tran&longs;it in AD; eadem AB, vel æqualis
motu recto tran&longs;eat in FD, Dico impetum huius motus e&longs;&longs;e duplò in
ten&longs;iorem impetu illius; quia impetus &longs;unt vt motus;
motus verò vt
&longs;patia, quæ percurruntur æqualibus temporibus; &longs;ed &longs;patium rectanguli
AD, e&longs;t duplum trianguli ADB; igitur & motus;
igitur & impetus;
&longs;i
verò AB tran&longs;eat in EL, ita vt AF, &longs;it dupla AE; impetus erunt
æquales; quia rectangulum AC, e&longs;t æquale triangulo ABD.
Dixi arcum BD, accipi vt lineam rectam;
Si enim accipiatur vt ar
cus; haud dubiè motus cylindri AB, dum transfertur in FD, e&longs;t ad mo
tum eiu&longs;dem AB, dum transfertur in AD, vt rectangulum AD, ad &longs;e
ctorem, cuius arcus &longs;it æqualis rectæ BD, & radius ip&longs;i AB.
Ob&longs;eruabis primò, id quod &longs;uprà dictum e&longs;t ita e&longs;&longs;e intelligendum,
vt momentum grauitationis nullo modo con&longs;ideretur, & prædictus
cylindrus cen&longs;eatur potiùs moueri in plano horizontali, à quo &longs;u&longs;tinea
tur, quàm in circulo verticali, in quo libera &longs;it eius libratio, &longs;eu gra
uitatio.
Secundò, non po&longs;&longs;e &longs;u&longs;tineri cylindrum horizonti parallelum, ni&longs;i
aliqua eius portio &longs;eu manu, &longs;eu forcipe, vel alio quouis modo accipia
tur, v.g. &longs;it cylindrus AG horizonti parallelus; vt in hoc &longs;itu reti
neatur, debet aliqua eius portio putà AB, manu teneri, alioqui ne à po
tentiâ quidem infinita &longs;u&longs;tineri po&longs;&longs;et.
Tertiò, &longs;i &longs;upponatur fulcitus in B;
vt retineatur in æquilibrio, debet
addi momentum in A; &longs;eu debet retineri ab ip&longs;a potentiâ applicata
in A.
Quartò, pondus in G &longs;e habet ad idem pondus in A, &longs;tatuto centro in
B, vt &longs;egmentum GB, ad BA, id e&longs;t, vt 5. ad 1.
Quintò, &longs;i proprio pondere frangeretur BG, haud dubiè in B frange
retur; e&longs;t autem momentum ponderis BG, vt &longs;ubduplum eiu&longs;dem BG
po&longs;itum in G, vt demon&longs;trat Galileus prop.1.de re&longs;i&longs;tentia corp.&longs;it enim
BG, duarum librarum, &longs;itque BG, diui&longs;a bifariam in H; haud dubiè
pondus in H, facit momentum &longs;ubduplum eiu&longs;dem in G, vt patet; &longs;unt
enim vt di&longs;tantiæ; igitur cum &longs;egmentum HG tantùm addat momenti
&longs;upra H, quantùm detrahit HB; certè momentum totius ponderis BG,
itaque &longs;it BG, 10. librarum,
æquiualet 5. libris &longs;tatutis in G, & AB, vni libræ po&longs;itæ in A; &longs;ed hæc
libra in A, habet tantùm &longs;ubquintuplum momentum eiu&longs;dem in G, igi
tur 5. libræ in A, æquiualent vni in G; igitur vt &longs;tatuatur æquilibrium,
debent e&longs;&longs;e 24. libræ in A, &longs;eu vires æquiualentes; quibus adde pondus
ab&longs;olutum 12. librarum; erunt 36. igitur re&longs;i&longs;tentia ad motum circula
rem verticalem ex triplici capite oritur. Primò ex ip&longs;o pondere ab&longs;olutè
&longs;umpto, quæ communis e&longs;t motui propagationis. Secundò, ex momento
eiu&longs;dem ponderis; Tertiò, ex tali genere propagationis, de quo &longs;uprà;
quæ omnia &longs;unt apprimè tenenda, ne quis error &longs;ubrepat.
ita propagatur
impetus, vt plures partes ver&longs;us centrum motus producantur in pondere, quod
attollitur&longs;it enim idem cylindrus CA;
&longs;itque applicata potentia in
A, dico ver&longs;us C, plures partes produci in pondere, Probatur, quia attol
litur pondus in C, quod moueri non pote&longs;tin A, operâ vectis AC, vt con
&longs;tat ex certa hypothe&longs;i; igitur plures partes impetus producuntur per
rationem 6. & 7. Th.77,
Scio quidem hoc ip&longs;um à nemine hactenus, quod &longs;ciam, explicatum
e&longs;&longs;e; atque fore vt à multis tanquam nouum, & in&longs;olens minùs fortè
probetur: quamquam illa hypothe&longs;is hoc ip&longs;um euincit, vulgaris certè,
& nemini qua&longs;i non nota; qua nempè dicimus in omnibus partibus mo
bilis, quod actu mouetur, impetum produci; & &longs;i quando accidat corpo
ris ingentem molem ab applicata potentia non po&longs;&longs;e moueri, illud e&longs;&longs;e
tantùm, quòd non po&longs;&longs;int produci tot partes impetus, quot &longs;unt nece&longs;&longs;a
riæ, vt omnibus partibus &longs;ubjecti di&longs;tribuantur; igitur ex hac hypothe
&longs;i, quæ ex manife&longs;tis ducitur experimentis, nece&longs;&longs;ariò dicendum e&longs;t plu
res partes impetus versùs centrum vectis produci in pondere, quod at
tollitur, cuius propagationis proportionem infrà demon&longs;trabimus.
mero, decre&longs;cit tamen in perfectione. Probatur per Th.81. ex motu imper
fectiore, cui re&longs;pondet impetus imperfectior per Ax. 17.num.4. non ratio
ne numeri, qui maior e&longs;t per Th.99. igitur ratione entitatis, &longs;eu perfe
ctionis entitatiuæ.
ad aliam collectionem alterius puncti in perfectione, vt distantia illius puncti
à centro, ad di&longs;tantiam huiusprobatur, quia perfectio vnius collectionis
e&longs;t ad perfectionem alterius, vt motus ad motum; motus verò &longs;unt vt
&longs;patia, &longs;patia vt arcus, arcus vt &longs;emediametri, hæ demum, vt di&longs;tantiæ.
ver&longs;us centrum vectisprobatur, quia pondus ver&longs;us centrum mouetur
minore motu, vt con&longs;tat; igitur ab imperfectiore impetu; &longs;ed non e&longs;t
imperfectior tantùm ratione numeri, id e&longs;t, pauciorum partium impe
tus; quia &longs;i hoc e&longs;&longs;et, &longs;it vectis AC, motus B, e&longs;t &longs;ubduplus motus
A; igitur &longs;i e&longs;t impetus eiu&longs;dem perfectionis entitatiuæ, vt &longs;ic loquar;
ita &longs;e habet numerus partium impetus in B, ad numerum partium in A,
vt motus B, ad motum A; & hic vt arcus BD, ad arcum AE;
& hic vt
BC, ad AC; igitur e&longs;t &longs;ubduplus;
igitur æqualis omninò producitur
impetus ab eadem potentia in vecte AC, &longs;iue applicetur centro C, &longs;iue
circumferentiæ A; igitur æquè facilè; quod e&longs;t contra experientiam;
probatur &longs;ecundò, quia &longs;i hoc e&longs;&longs;et, pondus idem tàm facilè attolleretur
in A, quàm in B; quia idem impetus produceretur, quod e&longs;t contra ex
perientiam.
tia ad centrum, quàm à centro ad circumferentiam, & cur longior vectis ab
eadem potentia moueri po&longs;&longs;it primo modo, non &longs;ecundo, quod clarum est.
probatur
quia decre&longs;cit iuxta rationem motuum; & hæc iuxta rationem di&longs;tan
tiarum.
probatur, quia cum à circumferentia ad centrum ita propagetur impe
tus, vt vnicum tantùm punctum producatur in ip&longs;a extremitate mobilis;
certè non pote&longs;t minùs impetus produci ver&longs;us centrum ratione nume
ri; igitur non decre&longs;cit numerus; hinc producitur nece&longs;&longs;ariò imperfe
ctior ver&longs;us centrum.
&longs;unt plures in puncto vectis propiùs ad centrum accedente, quàm in co; quod
longiùs distat:Secundò, cur
potiùs in vna proportione, quàm in alia?
a circumferentia ad centrum, cum &longs;cilicet applicatur potentia circumferentiæ
probatur, quia non producitur numerus minor per Th.105. neque maior
per Th. 106. igitur æqualis; adde quod res explicari non pote&longs;t per ma
iorem, neque per minorem; ita vt &longs;cilicet pondera, quæ à data potentia
leuantur, &longs;int vt di&longs;tantiæ, de quo &longs;uprà.
Ob&longs;eruabis, quod aliquando in mentem venerat;
&longs;cilicet, ver&longs;us cen
trum produci maiorem numerum in ratione di&longs;tantiarum permutando;
& imperfectiorem in ratione duplicata earumdem di&longs;tantiarum, etiam
permutando, v. g. &longs;it idem vectis AC &longs;ectus bifariam in B; in puncto
B producitur numerus duplus producti in A; at verò perfectio impetus
in B e&longs;t ad perfectionem impetus in A, vt quadratum BC ad quadra
tum AC; vel in ratione &longs;ubquadrupla, licèt tota collectio impetus B
&longs;it tantùm &longs;ubdupla perfectione collectionis impetus A; &longs;ed hoc profe
ctò dici non pote&longs;t; nam &longs;int in A 4. partes impetus; igitur in B erunt
8. applicetur autem pondus in B. Primò producentur in eo partes 8.
impetus perfectionis &longs;ubquadruplæ; &longs;i comparentur cum partibus A,
tum producentur 16. quæ æquiualent 4 A; igitur 24. at verò in A pro
ducentur primò 4. tum deinde 2. quæ æquiualent 8. productis in B; igitur
6. igitur pondus, quod leuari pote&longs;t in B, e&longs;t ad pondus, quod leuari pote&longs;t
in A, vt 24. ad 6.id e&longs;t, in ratione quadrupla quod omninò fal&longs;um e&longs;t.
lantur ex diuer&longs;is punctis vectis
tur.v.g. in &longs;ingulis punctis vectis &longs;ingula puncta impetus, &longs;ed diuer&longs;æ
perfectionis; haud dubiè plures partes impetus imperfecti po&longs;&longs;unt face
re impetum æqualem in perfectione alteri, qui con&longs;tat paucioribus, &longs;ed
perfectioribus; igitur cum impetus B &longs;it imperfectior duplò quàm im
petus in A, duplò plures partes impetus producentur in B, quàm in A, er
go duplò maius pondus mouebitur; atque ita deinceps; eum enim ap
ponitur pondus in B, producuntur in eo partes impetus omnes eiu&longs;dem
perfectionis; quæ &longs;cilicet re&longs;pondet B, id e&longs;t, quæ e&longs;t &longs;ubdupla perfectio
nis impetus A; igitur plures partes producuntur, quàm &longs;i e&longs;&longs;ent perfe
ctionis A; &longs;ed pauciores quàm &longs;i e&longs;&longs;ent perfectionis O, quæ minor e&longs;t;
quippe eadem potentia, &longs;eu cau&longs;a, quæ agit quantum pote&longs;t (quod &longs;up
pono modò) producit æqualem effectum in perfectione, per Ax. 13. n.
4. &longs;ed æqualis perfectio pote&longs;t con&longs;tare pluribus, vel paucioribus parti
bus perfectionis, nam 4. pattes perfectionis vt 4. faciunt æqualem effe
ctum alteri qui con&longs;tat 8. partibus perfectionis vt 2. quod certum e&longs;t; &longs;ed
de his plura aliàs.
vectis, &longs;eu distantiarum.
centrum C; vt impetus puncti B &longs;it &longs;ubduplus in perfectione, puncti R
&longs;ubtriplus: iam verò &longs;it vectis &longs;ubduplus prioris BC, &longs;ectus bifariam in
Z; &longs;i impetus productus in B, qu&etail; e&longs;t extremitas minoris vectis B &longs;it æqua
lis perfectionis cum impetu producto in A (& reuera &longs;unt æquales) &longs;i
æquali tempore percurrant arcus æquales, &longs;cilicet AV, & BD) certè im-
iorem vectem; quia vt AC totus maior vectis e&longs;t ad BC ita BC ad
ZC: igitur decre&longs;cit perfectio versùs centrum iuxta rationem longi
tudinum.
centro, vnam tantùm partem, vel vnum punctum impetus producit
enim minùs produci pote&longs;t, po&longs;ito quod potentia applicata ad talem gra
dum perfectionis &longs;it determinata, id e&longs;t ad producendum impetum talis
perfectionis in ea parte &longs;ubjecti, cui applicatur immediatè, vt &longs;uprà di
ctum e&longs;t.
uendum &longs;ufficiens motu circulari est ad aliam &longs;ufficientem ad illum mouen
dum motu recto, vt
&longs;i 5. vt 3. ad 5. &longs;i 6. vt 3. 1/2 ad 6. atque ita deinceps iuxta hanc propor
tionem in quo non e&longs;t difficultas, cum hoc totum &longs;equatur ex Th. 109.
Ob&longs;erua tamen quacumque data potentia po&longs;&longs;e dari minorem;
quia
quocumque dato motu, etiam recto, pote&longs;t dari tardior; igitur quocum
que impetu imperfectior; igitur quando appellaui potentiam minimam;
intellige illam quæ comparatur cum vnico puncto impetus talis perfe
ctionis; hæc enim reuera minima e&longs;t illarum omnium, quæ po&longs;&longs;unt pro
ducere impetum talis perfectionis, &longs;i verò comparetur cum impetu im
perfectiore, haud dubiè minima non e&longs;t.
Ob&longs;erua præterea &longs;uppo&longs;itum e&longs;&longs;e hactenus in extremitate vectis &longs;iue
maioris, &longs;iue minoris, produci impetum eiu&longs;dem perfectionis, eiu&longs;que
vnicum punctum, &longs;eu partem, vnde potentia quæ applicatur maiori vecti
conuenit quidem cum ea, quæ applicatur minori in eo, quòd vtraque in
extremitate &longs;ui vectis producat vnum punctum impetus eiu&longs;dem perfe
ctionis; differt tamen in eo, quòd illa, quæ applicatur maiori vecti, &longs;it
maior iuxta rationes prædictas in Theoremate. v. g. illa, quæ applicatur
vecti. 2. punctorum e&longs;t ad eam, quæ applicatur vecti trium punctorum,
&longs;cu partium, vt 1. 1/2 ad 2. & &longs;i vectis &longs;it 4. punctorum ad 2. 1/2; &longs;i 5. ad 3.
&longs;i 6. ad 3. 1/2; &longs;i 7. ad 4. &longs;i 8. ad 4. 1/2. Vides egregiam progre&longs;&longs;ionem;
&longs;it
enim vectis 2. punctorum AB, in puncto A, quod e&longs;t extremitas, produ
catur punctum impetus datæ perfectionis, in B producetur aliud, cuius
perfectio e&longs;t &longs;ubdupla prioris per Th. 109. igitur caracter, &longs;eu momen
tum totius impetus e&longs;t 1. 1/2. &longs;it porrò vectis 4. punctorum CDEF, in
C, quod e&longs;t extremitas; producatur vnum punctum impetus eiu&longs;dem
perfectionis cum eo, quod productum e&longs;t in A; certè in D producetur
aliud cuius perfectio erit ad priorem vt 3.ad 4. per idem Th. &longs;ic autem
notetur 1/4, in E 2/4, in F 3/4, in C vero 4/4; perfectiones enim &longs;unt vt lon-
quæ &longs;i colligantur, habebis characterem totius impetus, 2 1/2:
igitur totus impetus productus in minore vecte, qui con&longs;tat 2. punctis,
e&longs;t ad impetum, qui producitur in maiore con&longs;tante 4.punctis, vt 1. 1/2 ad
2. 1/2; igitur vectis maior maiorem potentiam ad mouendum ip&longs;um ve
ctem requirit; non certè in de&longs;cen&longs;u; quippe &longs;uo pondere de&longs;cendit, &longs;ed
in plano horizontali; ni&longs;i enim potentia po&longs;&longs;it mouere vectem; haud
dubiè nullum pondus vecte mouebit.
At verò &longs;i potentia &longs;it tantùm dupla minimæ, quæ datum vectem mo
uere po&longs;&longs;it; haud dubiè dato illo vecte datum ferè quodcumque pondus
mouere poterit; cum ip&longs;e vectis con&longs;tet ferè infinitis punctis in longi
tudine, vt patet ex dictis, & con&longs;ideranti patebit.
Ob&longs;eruabis demum in mechanicis nullam ferè haberi rationem pon
deris ip&longs;ius vectis; parum enim pro nihilo computatur:
Ex his tamen
erui po&longs;&longs;unt veri&longs;&longs;imæ rationes Phy&longs;icæ proportionum vectis AH; &longs;ia
que A extremitas, H centrum; &longs;itque BH 1/2. CH 1/4, DH 1/2, EH (1/16),
FH (1/32), GH (1/64) pondus I applicetur in A, & moueatur; certè in B moue
bitur pondus K duplum I; quia, cum impetus productus in B, &longs;it &longs;ubdu
plus in perfectione illius, qui producitur in A; vt æqualis producatur in
B, & in A, debent produci in B duplò plures partes impetus; igitur du
plò maius pondus mouebit; at verò in C mouebitur pondus L quadru
plum I, in D octuplum, atque ita deinceps; donec tandem in G mouea
tur pondus, quod &longs;it ad I vt 64. ad 1. & cum adhuc po&longs;&longs;int accipi inter
GH, partes aliquotæ minores, & minores ferè in infinitum, non mirum
e&longs;t &longs;i pondus maius po&longs;&longs;it adhuc moueri.
Ob&longs;eruabis etiam in omni vecte ab&longs;trahendo ab eius pondere, & ap
plicata eadem potentia, hoc e&longs;&longs;e commune; vt po&longs;&longs;it quodcumque pon
dus attolli, licèt difficiliùs in minore; quia hic non pote&longs;t in tam mul
tas partes aliquotas &longs;en&longs;ibiliter diuidi, in medio tamen vecte duplum
&longs;emper pondus mouetur; &longs;iue ip&longs;e vectis &longs;it maior, &longs;iue minor.
Ob&longs;eruabis deinde, &longs;i centrum vectis non &longs;it in altera extremitate,
&longs;ed. v.g. in C;
haud dubiè producitur in H, & in B impetus æqualis;
quia
æqualiter di&longs;tat vtrumque punctum à centro C; igitur æquale pondus
mouebitur in B, & in H; propagatur tamen nouo modo à C ver&longs;us H, de
quo iam &longs;uprà dictum e&longs;t.
Ob&longs;eruabis denique triplicem propagationem impetus e&longs;&longs;e legiti
mam. Prima e&longs;t in motu recto, cum propagatur per partes æquales, tùm
in perfectione, tùm in numero in &longs;ingulis partibus &longs;ubjecti per gradus,
&longs;cilicet heterogeneos. Secunda e&longs;t in motu circulari, applicata &longs;cilicet
potentia centro; cum propagatur per partes æquales in perfectione, &
inæquales in numero. Tertia e&longs;t in vecte, cum propagatur per partes
æquales in numero, & inæquales in perfectione.
probatur, quia
non pote&longs;t e&longs;&longs;e impetus, ni&longs;i exigat motum per Th.14. nec exigere mo-&longs;ed hoc e&longs;t impetum e&longs;&longs;e de
terminatum ad aliquam lineam motus; præterea &longs;i non e&longs;t determina
tus ad aliquam lineam; igitur indeterminatus, & indifferens per Ax.1.
&longs;ed indifferens manere non pote&longs;t; cur enim potius haberet motum
per vnam lineam, quàm per aliam? igitur debet determinari.
petus pilæ in aliam impactæ producit in ea impetum, qui pro diuer&longs;o
contactu ad diuer&longs;am lineam determinari pote&longs;t; præterea corpus graue
in diuer&longs;is planis inclinatis de&longs;cendit; igitur per diuer&longs;as lineas;
deinde
pila reflectitur propter impetum priorem, qui tantùm mutat lineam, vt
dicemus infrà; adde quod funependuli vibrati impetus &longs;ine reflexione
mutat lineam motus; igitur idem impetus ad plures lineas &longs;eor&longs;im e&longs;t
indifferens.
quia ad
eas pote&longs;t determinari, ad quas e&longs;t indifferens, vt patet; &longs;ed ad multas
e&longs;t indifferens per Theorema 113. igitur ad multas pote&longs;t determi
nari.
Ob&longs;eruabis primò determinationem hanc nihil e&longs;&longs;e aliud, ni&longs;i ip&longs;um
impetum cum tali linea comparatum, &longs;eu coniunctum; vnam verò li
neam differre ab alia ratione terminorum v. g. illa quæ tendit ver&longs;us
ortum differt ab ea, quæ tendit ver&longs;us au&longs;trum, vel occa&longs;um, &longs;cilicet
ratione terminorum, &longs;unt enim duo termini, nempè à quo, & ad quem;
4. autem modis differunt termini lineæ, vel enim neuter communis e&longs;t
vt AB. DC, vel terminus à quo vtrique lineæ communis e&longs;t, vt BA.
BE, vel terminus ad quem vt AB, EB; vel denique vici&longs;&longs;im commu
tantur termini, vt BE, EB, & hæc terminorum coniugatio facit oppo
&longs;itionem maximam, id e&longs;t diametralem.
Secundò ob&longs;eruabis aliquando videri e&longs;&longs;e vtrumque terminum com
munem licèt differant lineæ; &longs;it linea recta BE, habet communes ter
minos cum curua BFE, licèt omninò differat ab illa; at profectò licèt
BE videatur e&longs;&longs;e vnica &longs;implex linea duobus terminis clau&longs;a; con&longs;tat
ramen ex pluribus aliis continuata, rectáque &longs;erie iunctis; vnde, vt
linea dicatur eadem e&longs;&longs;e cum alia, debet vna cum aliâ conuenire; ita vt
alteri &longs;uperpo&longs;ita nec excedat, nec deficiat.
Tertiò linea motus non differt ab ip&longs;o motu continuo tractu, &longs;eu
fluxu qua&longs;i labenti: Porrò vnus motus differt ab alio, vel ratione velo
citatis, vel ratione terminorum; &longs;ed hæc parum difficultatis habent.
v. g.
probatur quia
motus deor&longs;um e&longs;t finis huius impetus; quia ideo corpus graue produ
cit in &longs;e impetum (&longs;i tamen producit) vt tendat deor&longs;um, vt certum e&longs;t;
tàm enim omne graue non impeditum tendit deor&longs;um, quàm omnis
ignis e&longs;t calidus; igitur &longs;i e&longs;t proprietas omnis ignis e&longs;&longs;e calidum, quia
omni competit; ita omni graui competit tendere infrà leuius, modò
non impediatur; igitur e&longs;t eius proprietas;
igitur ille impetus e&longs;t de
terminatus ad lineam quæ tendit deor&longs;um; &longs;ed de hoc impetu naturali
innato fusè agemus infrà in &longs;ecundò libro; nunc &longs;ufficiat dixi&longs;&longs;e po&longs;&longs;e
dari aliquem impetum ita determinatum ad certam lineam, vt ad aliam
determinari non po&longs;&longs;it naturaliter, nulla e&longs;t enim repugnantia.
pro
batur, quia primus impetus ab ip&longs;a potentia productus &longs;ine impedimen
to ab alio determinari non pote&longs;t; potentia porrò motrix vel e&longs;t gra
uium, vel leuium, vel animantium, vel proiectorum, vel compre&longs;&longs;o
rum, &c.
ip&longs;a &longs;ua natura&longs;ic grauitas &longs;eu potentia motrix grauium determinata
e&longs;t ad motum deor&longs;um perpendicularem, dum in medio libero corpus
graue mouetur; vel à plano inclinato; pro cuius diuer&longs;a inclinatione
diuer&longs;a e&longs;t linea motus deor&longs;um; vel ab ip&longs;a via, &longs;eu exitu patefacto;
&longs;ic potentia motrix compre&longs;&longs;orum &longs;uas vires exerit, & mobile ip&longs;um
agit, quâ patet viâ, &longs;ur&longs;um, deor&longs;um &c. vel ab appetitu &longs;eu libero, &longs;eu
&longs;en&longs;itiuo; &longs;ic potentia progre&longs;&longs;iua animantium cò corpus agit, quò iu
bet appetitus, vel ab aliqua affectione intrin&longs;eca intrin&longs;ecùs vel extrin
&longs;ecùs adueniente; &longs;ic dilatatur pupilla, vel contrahitur pro diuer&longs;a lu
minis appul&longs;i vi, vel obiecti di&longs;tantia: Huc reuoca motus illos natura
les, qui animalibus competunt v. g. tu&longs;&longs;is, &longs;ingultus, &longs;ternutationis, &c.
de quibus fusè &longs;uo loco.
&longs;ic impetus corporis proiecti determinatur ab impetu vel organi vel
manus proiicientis; quia nihil e&longs;t aliud à quo determinari po&longs;&longs;it, vt
patet; adde figuram organi, di&longs;po&longs;itionem &longs;eu &longs;itum mobilis, quod ma
nu tenetur; impedimenti etiam habetur ratio v. g. corpus oblongum
proiici pote&longs;t, vel motu recto ad in&longs;tar teli, vel motu mixto ex recto
& circulari; cum &longs;cilicet diuer&longs;imodè vibratur: &longs;i enim altera extremi
tas adhuc hæreat in manu, dum altera mouetur, vt cum quis baculo
ferit; tunc certè e&longs;t aliquòd impedimenti genus, ex quo oritur talis li
nea motus; illud autem impedimentum emergit ex diuer&longs;a applicatione
diuer&longs;aque brachij vibratione, quæ omnia &longs;unt &longs;atis clara.
minatuvt patet in corpore reflexo; nec enim dici pote&longs;t totum prio
rem impetum in ip&longs;o reflexionis puncto de&longs;trui, vt demon&longs;trabimus
aliàs. Probatur etiam ex impetu proiectorum, quæ mutant lineam mo
tus manente adhuc priore impetu &longs;altem ex parte.
ratione puncti contactusSit enim, ne multiplicemus figuras, globus,
cuius linea directionis &longs;it DC, punctum contactus C, ita globus A im
pellet globum B, vt linea motus, ad quam determinatur, &longs;it CB, id e&longs;t
ducta à puncto contactus ad centrum globi impul&longs;i; &longs;it etiam globus
P impactus in globum A punctum contactus &longs;it D, linea motus, ad
quam determinatur, e&longs;t DA, quæ &longs;cilicet à puncto contactus ducitur
per centrum grauitatis corporis impul&longs;i: experientia huius rei certa
e&longs;t, nec ignorant qui in ludo minoris tudiculæ ver&longs;ati &longs;unt; ratio au
tem inde tantùm duci pote&longs;t, quod &longs;cilicet ab ip&longs;o puncto contactus ita
diffunditur impetus, vt hinc inde æqualiter in vtroque hemi&longs;phærio
diffundatur; coniungitur autem vtrumque hemi&longs;phærium circulo A,
vel B, in priore figura, e&longs;tque vtriu&longs;que communis &longs;ectio; cum autem
vtrimque &longs;it æqualis impetus, nulla e&longs;t ratio, cur linea directionis in
clinet potiùs in vnum hemi&longs;phærium, quàm in aliud: præterea cum
motus orbis globi determinetur à motu centri; cum &longs;cilicet globus in
globum impingitur; haud dubiè non pote&longs;t e&longs;&longs;e alius motus centri, ni&longs;i
qui determinatur à puncto contactus, à quo vnica tantùm linea ad cen
trum duci pote&longs;t, vt con&longs;tat; & hæc ratio veri&longs;&longs;ima e&longs;t, & totam rem
ip&longs;am euincit.
ctum contactus ad
cet globus P. eiu&longs;dem figuræ tangat globum A in D per lineam PD &longs;iue
per lineam HD &longs;iue per quamlibet aliam, globus A mouebitur &longs;emper
per lineam directionis DA propter rationem propo&longs;itam, quod etiam
mille experimentis conuincitur.
uam lineampatet experientiâ in pilâ reflexâ; reflexionis autem ratio
nem afferemus in lib. de motu reflexo.
Probatur, quia cum
eodem puncto contactus pote&longs;t e&longs;&longs;e determinatio ad diuer&longs;am lineam,
vt manife&longs;tum e&longs;t; &longs;it enim reflexio per angulum æqualem incidentiæ,
&longs;ed diuer&longs;i anguli po&longs;&longs;unt in idem punctum coire, vt patet.
incidentiæ
uer&longs;is lineis motus reflexi, vt patet.
tis
reflexi e&longs;&longs;e po&longs;&longs;unt, vt con&longs;tat.
paratæ cum plano reflectente,
lo incidentiæ, cuius effectus rationem aliàs afferemus, cum de motu
reflexo; & verò multa hîc cur&longs;im tantùm per&longs;tringimus, quæ in libro
de motu reflexo accurati&longs;&longs;imè demon&longs;trabimus; Hìc tantùm dixi&longs;&longs;e &longs;uf
ficiat determinari mobile in reflexionis puncto ad nouam lineam motus,
quod nemo in dubium reuocare pote&longs;t, & propter quid fiat loco citato
demon&longs;trabimus.
centra vtriu&longs;que ducatur, determinatio noua e&longs;t æqualis prioriPatet ex
perientia in pilis illis eburneis, quas de&longs;iderat ludus minoris tudiculæ;
nec e&longs;t vlla ratio, cur determinatio &longs;it maior potiùs, quàm minor, cum
vtraque pila &longs;it æqualis; &longs;i enim maior e&longs;&longs;et, vel minor; cur potiùs vno
gradu, quàm duobus? quàm tribus?
Præterea, cum re&longs;i&longs;tens, vel im
pediens e&longs;t æquale agenti; certe &longs;icut agens refundit in pa&longs;&longs;um totum
id, quod habet, id e&longs;t æqualem impetum in inten&longs;ione, & æquè velo
cem motum per Th. 60. Ita re&longs;i&longs;tens, vel impediens refundit æquale
impedimentum, quod tantùm &longs;umi pote&longs;t ex æqualitate mobilium; &longs;ed
ex æquali impedimento duci tantùm pote&longs;t æqualis determinatio priori;
denique pote&longs;t dari determinatio noua æqualis priori, vt con&longs;tat, &longs;ed
aliunde duci non pote&longs;t quàm ex ip&longs;a mobilium æqualitate, modò fiat
contactus per lineam connectentem centra.
quippe hæc quie&longs;cet illicò ab ictu; quia &longs;cilicet, cum noua determina
tio &longs;it æqualis priori, non e&longs;t vlla ratio, cur alterutra præualeat; nec
etiam pote&longs;t e&longs;&longs;e determinatio communis, &longs;eu mixta; cur enim potius
dextror&longs;um quam &longs;ini&longs;tror&longs;um? de quo infrà.
globi, determinatur noua linea motus tùm à priore linea incidentiæ, tùm à
connectente centra, quæ &longs;cilicet per punctum contactus à centro impacti globi
po&longs;&longs;it, vt patet; non determinatur etiam ab alterutra &longs;eor&longs;im, vt con
&longs;tat, igitur ab vtraque conjunctim; in qua verò proportione dicemus,
& demon&longs;trabimus in libro de motu reflexo; &longs;unt enim mirificæ quæ
dam reflexionum proportiones, quas ibidem explicabimus.
ratio e&longs;t, quia vtraque linea
determinationis cum angulum faciat, in communem lineam abit; nam
ex duabus lineis motus minimè oppo&longs;itis ex diametro, fit alia tertia me
dia pro rata; hîc etiam latent my&longs;teria, de quibus loco citato.
nis, determinatur ad nouam lineam motus reflexiexperientia clara e&longs;t; ra
tio e&longs;t, quia maior globus maius e&longs;t impedimentum, hinc nunquam
quie&longs;cit minor globus impactus.
ctat centra, &longs;eruat patet etiam experientiâ, cuius ratio e&longs;t
minor re&longs;i&longs;tentia minoris globi; &longs;i verò &longs;it alia linea directionis, omni
nò reflectitur &longs;uo modo; id e&longs;t mutat lineam;
&longs;ed de his omnibus fusè
aliàs; hîc tantùm &longs;ufficiat indica&longs;&longs;e;
(&longs;uppo&longs;ita linea directionis cen
trali &longs;eu connectente centra, &longs;ic enim deinceps eam appellabimus, in
quo ca&longs;u duplex determinatio tertiam mediam conflare non pote&longs;t) in
dica&longs;&longs;e inquam &longs;ufficiat nouam determinationem, vel e&longs;&longs;e æqualem prio
ri, vel maiorem, vel minorem; &longs;i æqualis e&longs;t, globus impactus &longs;i&longs;tit; &longs;i
maior, reflectitur; &longs;i minor,
&longs;equitur.
bili, &longs;ique illæ &longs;int ex diametro oppo&longs;itæ &longs;i&longs;tere debet mobile&longs;it enim
globus vtrimque gemino malleo percu&longs;&longs;us æquali ictu; haud dubiè &longs;i&longs;tit;
cur enim potiùs in vnam partem quam in aliam? cum &longs;imul in vtramque
moueri non po&longs;&longs;it.
terminatio præualebit pro rataratio e&longs;t, quia im
petus fortior debiliorem vincit; pugnant enim pro rata per Ax. 15.
hinc &longs;i &longs;it duplò inten&longs;ior, &longs;ubduplum &longs;uæ velocitatis amittet, &longs;i triplè
&longs;ubtriplum, &c. de quo aliàs.
centra, reflectitur vterque æquali motu, quo antè. Probatur;
&longs;unt enim globi
tactus &longs;it C, haud dubiè globus A impactus in B amittit totum &longs;uum im
petum per Th.127. & 128. B, item impactus in A amittit totum &longs;uum per
eandem rationem; globus A producit impetum in B æqualem &longs;uo per
Th.60. item B producit in A æqualem per idem Th. igitur tantùm perit
impetus quantùm accedit; igitur in vtroque globo remanet æqualis im
petus priori; igitur æquali motu vterque mouetur, quod erat dem. & hæc
e&longs;t ratio veri&longs;&longs;ima toties probatæ experientiæ.
&longs;et motu directo, &longs;i propagatus fui&longs;&longs;et &longs;ine obicenam æquali motu æquali
tempore in eodem plano &longs;eu medio idem &longs;patium decurritur; quid verò
accidat in aliis punctis contactus dicemus infrà, cum de reflexione.
ad duas lineas &longs;it determinatus quæ conjunctæ faciant angulum, determinatur
vterque ad tertiam lineam mediam&longs;it enim mobile in A. v. g. globus,
cui &longs;imul imprimatur impetus determinatus ad lineam AD, in plano
horizontali AF; &longs;i vterque &longs;it æqualis, ad nouam lineam determinabi
tur AE; quippe tantùm debet acquirere in horizontali AB, vel in eius
parallela DE, quantum acquirit in alia horizontali AD, vel in eius pa
rallela BE; igitur debet ferri in E;
igitur per diagonalem AE;
clara e&longs;t
omninò experientia; cuius ratio à priori hæc e&longs;t, quòd &longs;cilicet impetus
po&longs;&longs;it determinari ad quamlibet lineam ab alio impetu per Th.118.119.
igitur in eodem mobili pro rata quilibet alium determinat; igitur &longs;i
vterque æqualis e&longs;t, vterque æqualiter; igitur debet tantum &longs;patij acqui
ri in linea vnius, quantum in linea alterius.
Si verò impetus per AC &longs;it duplus impetus per AD;
accipiatur AC
dupla AD, ducatur DF æqualis & parallela AC; linea motus noua
erit diagonalis AF, quia vtraque determinatio concurrit ad nouam pro
rata; igitur debet &longs;patium acqui&longs;itum in AC e&longs;&longs;e duplum acqui&longs;iti
in AD.
mutabitur linea; &longs;ed cre&longs;cet motus & &longs;patium
per AB, quo dato tempore percurratur &longs;patium AB; deinde produca
tur &longs;imul alius impetus æqualis priori in eodem mobili per lineam AB;
Dico quod eodem tempore percurretur tota AE, dupla &longs;cilicet AB;
quia &longs;cilicet dupla cau&longs;a non impedita duplum effectum habet per Ax.
13. num.1. duplus impetus duplum motum; igitur duplum &longs;patium; &longs;i
verò &longs;it triplus impetus, triplum erit &longs;patium, &c.
&longs;int duæ lineæ IK IL, mobili &longs;cilicet &longs;tatuto in I;
haud dubiè noua linea erit IM; & quo angulus KIL, erit acutior (&longs;up
po&longs;itis æqualibus &longs;emper lateribus IK IL) Diagonalis IM, erit ma
ior; donec tandem IL & IK coeant in eandem lineam;
tunc enim li
nea erit dupla IK per Th. &longs;uperius: quandiu verò e&longs;t aliquis angulus in
I quantumuis acutus, linea motus erit minor dupla IK, ad quam tamen
propiùs &longs;emper accedit; quæ omnia con&longs;tant ex elementis.
breuius, & eò breuius quò angulus e&longs;t obtu&longs;ior&longs;int enim
AB mobili &longs;tatuto in A, noua linea erit AC per Th. 137. & &longs;i accipia
tur angulus obtu&longs;ior HEF; noua linea erit EG, eo rectè breuior,
quò angulus e&longs;t obtu&longs;ior, non tamen iuxta rationem angulorum; donec
tandem de&longs;inat angulus, & ED EF coëant in vnam lineam; tunc enim
nullum erit &longs;patium, quia &longs;i&longs;ter omninò mobile per Th.133.quæ omnia
ip&longs;a luce clariora e&longs;&longs;e con&longs;tat; quippe quæ cum certis experimentis, &
clari&longs;&longs;imis principiis con&longs;entiant; &longs;ed de his plura infrà.
terminatus non habeat motum duplum, & con&longs;equenter &longs;patium duplumnec
enim AE e&longs;t dupla AB, vt con&longs;tat; nam &longs;i lineæ &longs;int oppo&longs;itæ ex
diametro vt BA BE totus de&longs;truitur impetus, per Th.133. &longs;i verò vna
in
tur per Th.138. igitur quà proportione propiùs accedet ad oppo&longs;itas;
plùs de&longs;truetur, & minus erit &longs;patium; & quâ proportione accedent
propiùs ad coëuntes, minùs de&longs;truetur, & maius erit &longs;patium, vt con&longs;tat
ex dictis.
nùs pugnent, quorum lineæ propiùs accedunt ad coëuntes; plùs verò, quorum
lineæ propiùs accedunt ad oppo&longs;itas, idque iuxta proportiones Diagonalium,
quod totum &longs;equitur ex dictis.
Ob&longs;eruabis vt faciliùs concipias duos impetus ad duas lineas deter
minatos; finge tibi nauim à diuer&longs;is ventis impul&longs;am, &longs;eu lapidem pro
jectum è naui mobili; &longs;ed de his plura in lib.4. cum de motu mixto.
Probatur,
quia non pote&longs;t e&longs;&longs;e effectus, ni&longs;i &longs;it eius cau&longs;a per Ax. 8. igitur &longs;i e&longs;t mo
tus, e&longs;t impetus.
Probatur; quia proii-
trice per hypoth. 6. igitur non con&longs;eruatur à potentia motrice per Ax.
10. igitur nec à causâ primò productiua.
tus non e&longs;t à &longs;e, quia de&longs;truitur aliquando per Ax. 14. igitur con&longs;eruatur
ab alio per Ax.14. num. 1. non à cau&longs;a primò productiua per Th.144.igi
tur ab alia, eaque applicata per Ax. 10. quæcumque tandem illa &longs;it, ali
quando cau&longs;am primam e&longs;&longs;e demon&longs;trabimus; nunc verò &longs;ufficiat dixi&longs;
&longs;e dari aliquam cau&longs;am reuerâ applicatam, quæ ip&longs;um con&longs;eruat impe
tum; immò ex hac ip&longs;a rerum con&longs;eruatione argumentum aliquando
ducemus, quo Deum ip&longs;um exi&longs;tere demon&longs;trabimus.
tur, quamdiu e&longs;&longs;et applicata. Demon&longs;tratur, quia e&longs;&longs;et cau&longs;a nece&longs;&longs;aria
(nam de hac ip&longs;a loquor) igitur &longs;emper ageret, igitur &longs;emper con
&longs;eruaret, quod e&longs;t contra experientiam; nam reuerâ impetus pro
ductus deor&longs;um à corpore graui motu naturaliter accelerato de&longs;truitur,
vt patet; præterea &longs;i corpus graue con&longs;eruaret impetum primò produ
ctum, non produceret nouum contra experientiam; quippe cau&longs;a ne
ce&longs;&longs;aria non plùs agit vno in&longs;tanti quàm alio, per Ax.12. adde quod im
petus de&longs;truitur ad exigentiam alterius, quidquid tandem illud &longs;it per
Ax.14. num.2. & 3. &longs;ed cau&longs;a primò productiua impetus non nouit rerum
exigentiam; igitur illi facere &longs;atis non pote&longs;t; ex hoc etiam capite cau
&longs;æ primæ exi&longs;tentiam &longs;uo loco demon&longs;trabimus.
Ob&longs;eruabis primò rem quamlibet ideo de&longs;trui, quia ce&longs;&longs;at cau&longs;a con
&longs;eruans illam con&longs;eruare; quippe quod de&longs;truitur eo in&longs;tanti dicitur de
&longs;trui, quo primò non e&longs;t, &longs;eu quo incipit primò non e&longs;&longs;e; atqui incipit
primò non e&longs;&longs;e &longs;eu de&longs;init e&longs;&longs;e, cum de&longs;init con&longs;eruari.
Secundò ob&longs;eruabis præclarum naturæ in&longs;titutum, quod etiam ex ip&longs;is
hypothe&longs;ibus con&longs;tat, quo fit vt qualitates quæ carent contrario à cau&longs;a
primò productiua con&longs;eruentur, vt lumen; ne &longs;i ab alia con&longs;eruarentur,
de&longs;truerentur vmquam; cum earum de&longs;tructionem nihil exigeret per
Ax.14.n.2. & 3. at verò qualitates, quæ contrarias habent: &longs;i quæ &longs;unt,
à cau&longs;a primò productiua minimè con&longs;eruantur; cum enim ideo con
trarium dicatur de&longs;truere contrarium, quia exigit eius de&longs;tructionem, id
e&longs;t, ne con&longs;eruetur amplius; certè vt cau&longs;a con&longs;eruans ce&longs;&longs;et con&longs;eruare,
debet no&longs;&longs;e illam exigentiam; atqui nulla cognitione pollent cau&longs;æ illæ
motrices naturales, de quibus e&longs;t quæ&longs;tio.
quia
de&longs;truitur tantùm ad exigentiam alicuius, quidquid tandem illud &longs;it, de
e&longs;t, quod exigat eius de&longs;tructionem; igitur tamdiu con&longs;eruatur per Ax.
14.num.3.
Inde certa ducitur ratio, cur mobile etiam &longs;eparatum à manu mouea
tur; quia &longs;cilicet ip&longs;i adhuc ine&longs;t impetus, qui e&longs;t cau&longs;a motus;
quippe
&longs;uppo&longs;ui iam antè de hac hypothe&longs;i quod &longs;it, non tamen propter quid &longs;it;
igitur hæc e&longs;t germana illius ratio & cau&longs;a.
Hinc etiam rationem ducemus æquè præclaram in lib.2. motus natu
raliter accelerati.
antè mouebatur, de&longs;init tandem moueri per hyp. 4. igitur de&longs;truitur
impetus; alioqui &longs;i remaneret, e&longs;&longs;et cau&longs;a nece&longs;&longs;aria &longs;ine effectu contra
Ax.12. ideo porrò de&longs;truitur, quia aliquid exigit eius de&longs;tructionem,
quippe hæc e&longs;t vnica de&longs;tructionis ratio per Ax.14. num.2.
&longs;it enim globus
proiectus ver&longs;us au&longs;trum; cui deinde imprimatur nouus impetus ver
&longs;us Boream; de&longs;truitur prior vt con&longs;tat, igitur ad exigentiam alicuius,
&longs;ed nihil e&longs;t quod po&longs;&longs;it exigere, ni&longs;i nouus impetus, &longs;cilicet mediatè;
nihil enim aliud e&longs;t applicatum, igitur nihil aliud exigit per Ax. 10.
hæc porrò exigentia non e&longs;t immediata, &longs;ed mediata, vt dixi.
ad diuer&longs;am lineam corpori graui impre&longs;&longs;us e&longs;t &longs;cilicet mediatè,
certa e&longs;t in proiectis, quæ tandem quie&longs;cunt; igitur ad exigentiam ali
cuius, &longs;ed illud tantùm e&longs;t impetus innatus; nec enim e&longs;t &longs;ub&longs;tantia
corporis; tùm quia qualitas &longs;ub&longs;tantiæ non opponitur;
tùm quia nulla
e&longs;&longs;et ratio, cur &longs;ub&longs;tantia de&longs;trueret potiùs vno in&longs;tanti vnum gradum,
quàm duos, quàm tres; adde quod ex duobus violentis oppo&longs;itis alte
rum de&longs;truit; igitur impetus e&longs;t cau&longs;a &longs;ufficiens de&longs;tructiua impetus,
igitur non e&longs;t ponenda alia, eo &longs;cilicet modo, quo diximus.
rientia, &longs;iue propter nouam determinationem, &longs;iue propter attritum,
vel pre&longs;&longs;ionem partium, de quo infrà.
terminationem nece&longs;&longs;ariam, & quam nunquam mutat, pugnat cum omni
tat lineam perpendicularem deor&longs;um, de quo infrà; &longs;i hunc igitur excipias,
omnes aly pugnant tantùm ratione diuer&longs;æ lineæ, &longs;eu determinationis, in eodem
mobili:
dem lineam determinetur.
Ob&longs;eruabis primò, præclarum naturæ in&longs;titutum, quo fit, vt impe
tus perennis non &longs;it; vnde certè infinita propemodum emergerent ab
&longs;urda, & incommoda.
Secundò, faciliorem modum de&longs;tructionis impetus in&longs;titui non po
tui&longs;&longs;e, immò nec excogitari po&longs;&longs;e; quàm enim facilè, vel impetus op
po&longs;itus in mobili producitur, vel corpus durum opponitur &c.
Tertiò, præcipuam rationem huius de&longs;tructionis ducendam e&longs;&longs;e ex
Ax.6. in quo dicimus nihil e&longs;&longs;e fru&longs;trà, cumque ordinem à natura e&longs;&longs;e
in&longs;titutum, vt potiùs aliquid de&longs;truatur, & de&longs;inat e&longs;&longs;e, quàm fru&longs;trà &longs;it,
& dicimus de&longs;trui ad exigentiam totius naturæ.
Quartò, cum impetus &longs;uo fine caret, fru&longs;trà e&longs;t;
finis impetus e&longs;t mo
tus, vt &longs;æpè diximus, &longs;ic cum globus impactus in alium æqualem &longs;tatim
ab ictu &longs;i&longs;tit immobilis; certe ne fru&longs;trà &longs;it impetus, de&longs;truitur per Ax.6.
& per Ax. 14. num.2. cum verò determinatio altera maior e&longs;t, certè præ
ualet tantùm pro rata; igitur minor e&longs;t motus;
igitur, ne aliqui gradus
impetus &longs;int fru&longs;trà, de&longs;truuntur, cum verò &longs;unt duo impetus in eodem
mobili, vt in naui mobili ad lineas oppo&longs;itas determinati; haud dubiè
maior impetus præualet pro rata per Ax. 15. Igitur non modò totus
impetus minor perit, ne &longs;it fru&longs;trà; &longs;ed etiam aliquot gradus maioris, ne
&longs;int etiam fru&longs;trà; nec enim in communem lineam coïre po&longs;&longs;unt.
Denique quando &longs;unt duo impetus ad lineas diuer&longs;as determinati,
&longs;ed non oppo&longs;itas ex diametro, pugnant pro diuer&longs;o oppo&longs;itionis gradu,
vt &longs;uprà fusè dictum e&longs;t. Igitur cum totus impetus non habeat totum
motum, quod duplex illa determinatio impedit, ne aliqui gradus
&longs;int fru&longs;trà, de&longs;truuntur; igitur vides impetum impre&longs;&longs;um ab ex
trin&longs;eco de&longs;trui tantùm ne &longs;it fru&longs;trà; faceret enim vt e&longs;&longs;et fru&longs;trà vel
nouus impetus, vel determinato noua, & in hoc &longs;en&longs;u dicitur impetus
de&longs;trui ab impetu.
Quintò, &longs;i de&longs;trueretur mobile, etiam de&longs;trueretur impetus per idem
Ax. 6. quia e&longs;&longs;et fru&longs;trà &longs;eparatum; immò ex hoc vno principio demon
&longs;tramus accidentia & formas &longs;ub&longs;tantiales materiales non po&longs;&longs;e natura
liter con&longs;eruari extra &longs;uum &longs;ubiectum, quia &longs;cilicet e&longs;&longs;ent fru&longs;trà; quip
pe finem &longs;uum habent in &longs;ubiecto.
Sextò, Impetus naturalis innatus nunquam de&longs;truitur;
quia nunquam
e&longs;t fru&longs;trà; quippe &longs;emper habet alterum &longs;uorum effectuum formalium,
id e&longs;t vel motum deor&longs;um, vel grauitationem, adde quod fru&longs;trà de
&longs;trueretur, cum &longs;it &longs;emper applicata potentia, id e&longs;t ip&longs;a grauitas, &longs;ed de
his infrâ fusè.
Septimò, Impetus &longs;ur&longs;um de&longs;truitur etiam, quia e&longs;t fru&longs;trà;
quippe
naturalis detrahit aliquid &longs;patij pro rata; igitur ne aliquid impetus &longs;it
fru&longs;trà, de&longs;truitur; idem dico de impetu per inclinatam &longs;ur&longs;um, licèt
minùs de&longs;truatur quàm in perpendiculari &longs;ur&longs;um; idem de impetu per
inclinatam deor&longs;um, &longs;ed minùs adhuc, &longs;ed hæc acuratiori meditationi
&longs;unt relinquenda; quod reuerâ præ&longs;tabimus in lib.4. de motu mixto;
quidquid &longs;it, con&longs;tat ex dictis per idem Principium probari po&longs;&longs;e de
&longs;tructionem impetus, &longs;cilicet ne &longs;it fru&longs;trà; &longs;ed de his aliàs fusè.
terminationis, &longs;eu diuer&longs;æ lineæProbatur primò, quia vterque ad omnem
lineam e&longs;t indifferens per Th.113. igitur vnus non e&longs;t alteri contrarius
ratione entitatis; cùm vterque &longs;imilem motum, immò
po&longs;&longs;it, vt patet ex dictis: Igitur ratione tantùm lineæ vnus alteri e&longs;t
contrarius; hinc minùs e&longs;t contrarietatis, quo minùs e&longs;t oppo&longs;itionis
inter lineas & contrà.
ne lineæ. Probatur eodem modo; quia determinari pote&longs;t ad omnem li
neam, vt patet ex reflexione grauis cadentis.
&longs;cilicet non pote&longs;t determinari ad omnem lineam, patet, alioquin cor
pus graue, quod &longs;ur&longs;um po&longs;t ca&longs;um reflectitur non de&longs;cenderet amplius,
de quo aliàs, hæc enim cur&longs;im tantùm per&longs;tringo, ne quid aliis libris
detrahatur.
vt patet in motu re
flexo grauium; ratio e&longs;t. quia mutatur linea.
vt patet in eodem ca&longs;u;
nam impetus naturalis innatus, qui in de&longs;cen&longs;u non erat contrarius
acqui&longs;ito, in motu &longs;ur&longs;um reflexo fit contrarius.
lineæ,
trarius, id tantùm e&longs;t ratione propagationis impetus acqui&longs;iti, vel ac
celerationis motus; quod reuerà multa, & benè longâ explicatione indi
get, quam con&longs;ule in lib.4.
Ob&longs;eruabis cogno&longs;ci tantùm contrarietatem qualitatum ex mutua de
&longs;tructione; cur verò vna qualitas dicatur de&longs;truere aliam, & cur illam
maximum my&longs;terium e&longs;t, quod alibi enucleabi
mus; quàm multa enim &longs;uper hac re tacuere Philo&longs;ophi!
in qua de&longs;truitur impetus ex parte propter diuer&longs;as determinationes;
cum &longs;cilicet corpus reflectens mouetur; igitur impetus prout determina
tus ad lineam incidentiæ e&longs;t aliquo modo &longs;ibi ip&longs;i contrarius, prout e&longs;t
determinatus ad lineam reflexionis.
Iam ferè tumultuatim, &longs;i quæ &longs;unt reliqua, Theoremata congeremus.
deor&longs;um.
Idem impetus pote&longs;t
intendere. v. g. 4. gradus impetus additi aliis 4. per
iidem ei&longs;dem, minùs intendunt, vt iam &longs;uprà &longs;atis fusè dictum e&longs;t.
num.2. vt con&longs;tat ex multis Theorematis &longs;uperioribus.
hoc in&longs;tanti primo e&longs;t, immediatè antecedenti vltimo non fuit, & quod
primo non e&longs;t hoc in&longs;tanti, immediatè antè vltimo fuit, nec pote&longs;t e&longs;&longs;e
immediatè pò&longs;t, ni&longs;i &longs;it immediatè antè, & vici&longs;&longs;im.
quia
potentia motrix e&longs;t determinata ad tale indiuiduum &longs;iue à &longs;e, &longs;iue ab
alio; idem enim de illa dicendum e&longs;t, quod de aliis cau&longs;is naturalibus;
porrò idem dici debet de de&longs;tructione, quod de productione.
Ob&longs;eruabis breuiter aliqua, quæ fortè in no&longs;tris Theorematis fuere
omi&longs;&longs;a.
Primò qualitates, quæ à cau&longs;a primò productiua con&longs;eruantur, ab ea
intendi non po&longs;&longs;e; quia &longs;ingulis in&longs;tantibus nouum effectum non pro
ducit; exemplum habes in luce; &longs;ecus vero de iis dicendum e&longs;t, quæ à
cau&longs;a primò productiua non con&longs;eruantur.
Secundò qualitates, quæ contrarias habent, etiam de&longs;trui po&longs;&longs;e ab
alio, quam ab iis, &longs;cilicet ad exigentiam totius naturæ; ne &longs;cilicet &longs;int
fru&longs;trà.
Tertiò aliqua carere contrario, non tamen con&longs;eruari à cau&longs;a primò
productiua. v.g. anima bruti, quæ de&longs;truitur ad exigentiam totius natu
ræ, nç &longs;it fru&longs;trà.
Quartò, impetum inten&longs;iorem in projectis diutiùs durare;
quia cum
&longs;en&longs;im de&longs;truatur; certè plures partes maiori tempore de&longs;truuntur, quàm
pauciores.
Quintò, &longs;i totus impetus de&longs;trueretur vno in&longs;tanti, minima re&longs;i&longs;tentia
&longs;ufficeret ad motum impediendum: adde quod contraria pugnant pro
rata per Ax.15.
Sextò, ob&longs;eruabis plurima in hoc libro qua&longs;i obiter e&longs;&longs;e indicata, quæ
in aliis fusè explicata maiorem lucem accipient.
Septimò, denique totam rem i&longs;tam, quæ pertinet ad impetum paulò
fu&longs;ius pertractatam in hoc primo libro; quòd &longs;cilicet ab ea reliqua ferè
omnia pendeant, quæ in hoc tractatu habentur; &longs;ed de his &longs;atis.
MOtus localis naturalis latè &longs;umptus e&longs;t,
qui ab aliqua causâ naturali ponitur;
&longs;trictè verò &longs;umitur pro motu grauium
deor&longs;um, à principio intrin&longs;eco &longs;altem
&longs;en&longs;ibiliter; In hoc vltimo &longs;en&longs;u mo
tum naturalem v&longs;urpabo; &longs;it ergo.
tio vix aliqua explicatione indiget; dicitur e&longs;&longs;e à grauitate,
quidquid &longs;it grauitas, &longs;iue qualitas di&longs;tincta, &longs;iue non.
curruntur &longs;patia ab eodem mobili.
ius &longs;patium acquiritur, & tertio, quàm &longs;ecundo, & quarto quàm tertio, atque
ita deinceps; nulla &longs;cilicet addita vi ab extrin&longs;eco &longs;altem &longs;en&longs;ibiliter.
Definit aliter hunc motum Galileus;
dicit enim eum e&longs;&longs;e, qui æquali
bus temporibus æqualia acquirit velocitatis momenta; &longs;ed profectò non
conuenit hæc definitio omni motui naturaliter accelerato, v. g. motui
de&longs;cen&longs;us funependuli, vel in orbe cauo, vel etiam in plano decliui ma
ximæ longitudinis; definitio no&longs;tra clarior e&longs;t.
infligit quam &longs;i caderet ex minore
nife&longs;ta experientia.
percurruntur
probare conetur.
Globus per planum inclinatum læuigatum de&longs;cendens &longs;ecundum &longs;pa
tium citiùs percurrit, quàm primum; quod etiam &longs;en&longs;u percipi pote&longs;t,
& tam &longs;æpè probatum e&longs;t, vt nemo iam negare audeat motus naturalis
accelerationem.
idem dico de &longs;patio,
negare au&longs;it; alioquin &longs;i quis negaret, dicat mihi quæ&longs;o quot &longs;int in mi
nuto horæ in&longs;tantia? quot in apice acus puncta?
& inten&longs;iorem impetum
facit, & vici&longs;&longs;im.
modo eidemque &longs;ubjecto &longs;it applicata,
portione illa decre&longs;cit, hic decre&longs;cit, & vici&longs;&longs;im.
temporibus æqualem effectum producit, & contrà. Probatur per Ax.12.l.
1. &
vici&longs;&longs;im æqualis effectus &longs;upponit æqualem cau&longs;am.
nulla cau&longs;a extrin&longs;eca e&longs;t applicata, producitur ab intrin&longs;eco
habere debet aliquam cau&longs;am per Ax.8.
verò, quæ maiorem, quæ demum æqualem, æquali proportione agit.
cuius virtus, vel actiuitas e&longs;t vt 20. & re&longs;i&longs;tentia vt 10. agit in maiori
proportione, quàm illa cuius actiuitas e&longs;t 30. & re&longs;i&longs;tentia 20. in minori
verò quàm ea, cuius actiuitas e&longs;t vt 3. & re&longs;i&longs;tentia vt 1. in æquali de
nique cum illa, cuius actiuitas e&longs;t vt 4. & re&longs;i&longs;tentia vt 2.
Hoc Axioma certi&longs;&longs;imum e&longs;t;
quippe 20. faciliùs &longs;uperabunt 10. quàm
30. 20. & difficiliùs quam 3. 1. & æquè facilè, ac 4. 2. In motu locali
res e&longs;t clari&longs;&longs;ima; quippe vires vt 12. tam facilè mouebunt 12. libras,
quàm vires vt 4. 4.libras; &longs;ed faciliùs, quàm vires vt 20. 30.libras, & dif
ficiliùs quàm vires vt 4. 3. libras; quid clarius? Igitur illa cau&longs;a faciliùs
virium cum re&longs;i&longs;tentia, quàm quæ minorem.
Si quando appellandum erit aliquod Axioma vel Theorema lib. 1.ci
tabitur Liber.
Probatur; corpus gra
ue mouetur localiter deor&longs;um per hypoth. hic motus e&longs;t ab intrin&longs;eco,
quod probatur; non e&longs;t ab vllâ causâ extrin&longs;ecâ; igitur e&longs;t ab intrin&longs;eca
per Ax.4. antecedens probatur inductione factâ omnium extrin&longs;ecorum.
Primò non e&longs;t à cau&longs;a prima, vt aliquis fortè minùs prudenter, & magis
piè, quàm par &longs;it, diceret; quia ille effectus tribui tantùm debet cau&longs;æ
primæ, qui nullam habere pote&longs;t cau&longs;am &longs;ecundam applicatam, vt patet;
&longs;ed hic effectus pote&longs;t habere cau&longs;am &longs;ecundam applicatam, quam a&longs;&longs;i
gnabimus infrà; deinde cau&longs;a prima agit tantùm naturaliter iuxta exi
gentiam cau&longs;arum &longs;ecundarum; igitur ideo moueret corpus graue deor
&longs;um; quia tunc motum corpus graue exigeret;
&longs;ed hoc mihi &longs;ufficit, vt
dicatur hic motus e&longs;&longs;e ab intrin&longs;eco; præterea, &longs;i dicatur Deus mouere
corpus graue deor&longs;um iuxta illius exigentiam, dicetur etiam tùm cale
facere, tùm illuminare, ad exigentiam ignis; quippe tàm mihi &longs;en&longs;ibile
e&longs;t corpus graue de&longs;cendere &longs;ine vi impre&longs;&longs;a ab extrin&longs;eco, quàm ignem
calefacere, & &longs;olem lucere &longs;ine vi extrin&longs;eca; adde quod illud &longs;olenne
e&longs;t naturæ in&longs;titutum, vt id, quod exigit res aliqua ad finem &longs;uum con&longs;e
quendum, per virtutem intrin&longs;ecam po&longs;&longs;it ponere, &longs;i dumtaxat excipias
concur&longs;um diuinum, & ip&longs;am con&longs;eruationem; &longs;ic animal exigit vide
re, audire, &longs;entire, moueri; igitur habet virtutem intrin&longs;ecam, per quam
videat, audiat, & moueatur; &longs;ic ignis exigit calefacere, lucere;
aër, vel aqua
frigefacere, quidquid tandem &longs;int i&longs;tæ qualitates, de quibus alibi; &longs;ic
demum corpus graue exigit moueri deor&longs;um; quis enim neget corpori
graui tàm natiuum e&longs;&longs;e tendere deor&longs;um, cum &longs;cilicet corpus leuius &longs;ub
e&longs;t, quàm &longs;it animali progredi, vrere igni, lucere, &c.
Denique &longs;atis e&longs;t mihi, vt dicatur aliquid cau&longs;a, Phy&longs;icè loquendo, &longs;i
ex illius applicatione &longs;emper &longs;equatur effectus; nam non nego po&longs;&longs;e fie
ri effectus omnes, qui no&longs;tris &longs;en&longs;ibus &longs;ubiiciuntur, &longs;altem extrin&longs;ecos,
e&longs;&longs;e à cau&longs;a prima, quippe &longs;i &longs;emper ex ignis applicatione Deus diffun
deret lucem, & calorem, quem &longs;olus ip&longs;e produceret, igne ip&longs;o inerte re
licto, nullam pror&longs;us mutationem perciperemus; & nemo e&longs;&longs;et, qui non
exi&longs;timaret lucem hanc & calorem hunc e&longs;&longs;e ab igne; igitur Phy&longs;icè lo
quendo cau&longs;am appellamus id, ex cuius applicatione &longs;emper &longs;equitur
effectus, vt iam diximus in Ax. 11.l.1. n.1. Igitur cum ex corpore graui
po&longs;ito in aëre libero &longs;equatur motus deor&longs;um; dicendum e&longs;t, Phy&longs;icè lo
quendò, e&longs;&longs;e huius motus cau&longs;am, id e&longs;t in ordine ad Phy&longs;icam, perinde
omninò &longs;e habere, atque &longs;i e&longs;&longs;et cau&longs;a, licèt cau&longs;a non e&longs;&longs;et.
Secundò hic motus non e&longs;t ab aëre ambiente;
probatur, ruderet aër
deor&longs;um corpus graue, quia leuior e&longs;t, id e&longs;t ne &longs;uprà &longs;e corpus grauius
haberet; &longs;ed eâdem ratione corpus graue debet remouere &longs;ur&longs;um aëra,
id e&longs;t corpus leue, ne infrà &longs;e habeat corpus leuius; e&longs;t enim par omni
nò ratio: Præterea &longs;i aër trudit deor&longs;inn corpus graue, quia ip&longs;i loco
cedit; certè ip&longs;e aër mouetur, igitur ab intrin&longs;eco;
&longs;i enim vna pars aë
ris pellit aliam, & hæc aliam, tandem ad aliquam peruenitur, quæ &longs;e ip
&longs;am mouet; igitur motus illius e&longs;t ab intrin&longs;eco;
igitur motus natura
lis; deinde non modò lapis de&longs;cendit per aëra, &longs;ed per mediam aquam;
igitur &longs;i ab aëre truditur deor&longs;um, idem dicendum e&longs;t de aquâ, a qui
haud dubiè maiore vi truderetur; nam corpus den&longs;um maiore vi pellit,
quàm rarum, vt con&longs;tat exprientiâ; cum tamen corpus graue per me
dium den&longs;ius difficiliùs decendat; igitur medium ip&longs;um re&longs;i&longs;tit motui,
quis hoc neget? igitur non e&longs;t cau&longs;a motus, quem impedit.
Denique &longs;i corpus graue non tendit, fertur que deor&longs;um &longs;uá &longs;ponte,
&longs;ed ab aëre extru&longs;um; igitur dum vix &longs;u&longs;tineo manu; o.
libras ferri, &longs;eu
plumbi; hæc vis illata manui, quam probè &longs;entio, e&longs;t ab aëre impel
lente plumbum, quod e&longs;t ridiculum, cum eadem quantitas aëris incu
ber, & &longs;ub&longs;it manui, &longs;iue &longs;u&longs;tineat plumbum, &longs;iue &longs;it vacua; ex hoc, ni
fallor, euincitur pondus ip&longs;um &longs;ui &longs;ponte deor&longs;um tendere.
Tertiò non de&longs;unt, qui dicant corpus graue trahi ab ip&longs;a vi quadam
magneticâ, quod triplici modo fieri pote&longs;t; Primò per qualitatem
quamdam diffu&longs;am, quod dici non pote&longs;t; quia capillus traheretur faci
liùs, quàm ingens &longs;axum, quàm ma&longs;&longs;a, &longs;eu lamina; & faciliùs eadem po
tentia motrix minus pondus moueret quàm maius, cæteris paribus; præ
terea manum meam æqualiter traheret, &longs;iue &longs;it cum aliquo pondere con
iuncta, &longs;iue &longs;it nuda &longs;ine pondere; deinde illa virtus tractrix ita diffun
ditur, vt in maiori di&longs;tantia &longs;it infirmior, fortior in minori; alioqui
diffunderetur in infinitum, quod dici non pote&longs;t; igitur &longs;i idem lapis
demittatur ex maiore altitudine, tum ex minore; haud dubiè morus ille
primus initio e&longs;&longs;et tardior i&longs;to contra experientiam; deinde in &longs;pecu al
ti&longs;&longs;ima &longs;ubterranea trahi po&longs;&longs;et corpus vndequaque, &longs;icut in magnete;
quæ omnia intelligi non po&longs;&longs;unt; denique virtutes illas &longs;eu qualitates
tractrices refellemus &longs;uo loco.
Secundò, aliqui dicunt hoc totum fieri per vim quamdam &longs;ympathi
cam, quod etiam fal&longs;i&longs;&longs;imum e&longs;t; tùm quia hæc &longs;ympathia explicari
non pote&longs;t; tùm quia vel terra ip&longs;a producit aliquid in corpore graui,
quod in aëre libratur; vel corpus in &longs;e ip&longs;o; &longs;i primum;
refellitur ii&longs;
dem omninò rationibus, quibus ip&longs;am vim terræ tractricem &longs;uprà expu
gnauimus; &longs;i verò &longs;ecundum, hoc ip&longs;um e&longs;t, quod &longs;uprà diximus.
Tertiò, Dixere aliqui &longs;ubtiliùs profectò quàm veriùs, corpus graue
trahi deor&longs;um, non vi quadam occultâ, vt &longs;uprà dictum e&longs;t; &longs;ed filamen
tis quibu&longs;dam, &longs;eu ductili terræ profluuio, quod illius capillitium vo
cant; idque tantùm fieri probant ducta ab electro analogiâ, quod pa
leam & minutiora corpu&longs;cula hac eâdem arte trahit; &longs;ed profectò gra-
nam primò cor
pus leuius ab his filamentis abripi faciliùs po&longs;&longs;et, vt con&longs;tat in electro;
igitur citiùs de&longs;cenderet.
Secundò, corpus vicinius etiam faciliùs abriperetur.
Tertiò, numquid flante vento, vel imbre cadente di&longs;&longs;ipantur hæc fi
lamenta? quod etiam videmus in electro.
Quartò, manum meam æquè facilè traheret terra his funiculis &longs;eu
pondere grauatam, &longs;eu vacuam.
Quintò, quemadmodum electrum ex omni parte trahit, ita terra ip&longs;a
per omnem lineam traheret; immò etiam &longs;ur&longs;um in &longs;ubterranea &longs;pecu,
quod e&longs;t ab&longs;urdum.
Sextò, hæc filamenta, quæ deinde reducuntur, debent habere cau
&longs;am huius reductionis non extrin&longs;ecam; igitur intrin&longs;ecam; igitur datur
motus naturalis.
Septimò, hæc filamenta per mediam flammam non traherent, quod
etiam fieri videmus in electro.
Quartò, motus naturalis non e&longs;t à virtute quadam pellente, quam
cælo quidam affingunt; nam vel ab omni parte cæli deor&longs;um trudere
tur, vel ab vnâ; &longs;i ab vna; igitur in omni cæli plaga corpus non fertur
deor&longs;um; &longs;i ab omni, ergo cum pellatur corpus per plures lineas etiam
oppo&longs;itas moueri non pote&longs;t: Præterea debilior e&longs;&longs;et hæc vis in maiori
di&longs;tantiâ; denique vapores, & alia minutiora corpu&longs;cula in aëre fluitan
tia faciliùs deor&longs;um truderentur, contra experientiam.
Sed non e&longs;t omittendum, quod aliqui putant ex illis filamentis con
texi po&longs;&longs;e legitimam rationem, cur atomi etiam plumbeæ materiæ non
ita facilè de&longs;cendant; quòd &longs;cilicet propter &longs;uam tenuitatem ab illis fi
lamentis non ita intercipi vel implicari po&longs;&longs;int; &longs;ed qua&longs;i pi&longs;ces per fo
ramina retium euadant; &longs;ed profectò longè alia ratio e&longs;t, quàm &longs;uo loco
afferemus, nam etiam plumæ, fe&longs;tucæ, paleæ, & alia corpu&longs;cula longio
ra, &longs;ed leui&longs;&longs;ima iis filamentis implicarentur, vt videre e&longs;t in electro.
Quintò, aliqui recentiores exi&longs;timant corpora deor&longs;um trudi ab
ip&longs;a luce, quæ nihil e&longs;t aliud, quam motio æthereæ cuiu&longs;dam &longs;ub&longs;tan
tiæ per poros aëris traductæ, vt ip&longs;i volunt; &longs;ed neque hoc probari po
te&longs;t. Primò quia de nocte corpora æquali motu deor&longs;um feruntur; pe
rinde atque de die, nec minùs in ob&longs;curi&longs;&longs;imo conclaui, quàm &longs;ub dio,
vel aperto cælo. Secundò, in &longs;ubterraneis locis etiam grauia æquè veloci
ter de&longs;cendunt; licèt eò lumen non penetret;
quod &longs;i aliquis ob&longs;tinatè,
id a&longs;&longs;ereret; haud dubiè per medium aëra maior huius materiæ copia
diffunditur, quàm per medias rupes, quis hoc neget; igitur pauci&longs;&longs;imi
radij v&longs;que ad interius & inferius antrum perueniunt. Tertiò, manum
meam &longs;iue ponderi coniunctam &longs;iue ab eo
materiæ deor&longs;um pelleret, vt patet; igitur æquali motus vi. Quartò, cor
pus diaphanum, per cuius poros facilè traiicitur hæc materia, e&longs;&longs;et leuius
alio quod tamen fal&longs;um e&longs;t, vt videre e&longs;t in vitro, cry&longs;tallo, adamante,
glacie. Quintò maxima huius materiæ copia collecta &longs;eu &longs;peculi opera
quia maior cau&longs;a maio
rem effectum producit per Ax.2. Sextò po&longs;t refractionem lineam mutat
radius luminis; igitur deor&longs;um rectà non pelleret. Septimò radij traie
cti per vitrum maiore vi deor&longs;um pellerent quàm per lignum, vel &longs;pon
giam; quippè per hæc corpora traiecti &longs;ecundum authores huius &longs;enten
tiæ di&longs;trahuntur propter obliquitatem pororum. Octauò denique radij
profecti à Sole iuxta ortum, vel occa&longs;um &longs;unt valdè obliqui; igitur non
truderent deor&longs;um rectà.
Nec e&longs;t quod prædicti àuthores confugiant ad experientiam, qua
&longs;cilicet videmus tripudiantes atomos in radio &longs;olari immer&longs;as; igitur
agitantur ab ip&longs;o radio, quod maximè accidit in linea v&longs;toria, cuius
effectus veri&longs;&longs;imam rationem &longs;uo loco afferemus, cum de lumine.
Sextò, &longs;unt denique multi,
exi&longs;timant grauia moueri deor&longs;um à generante, quod expre&longs;&longs;is verbis
traditum e&longs;t ab
&longs;ali&longs;&longs;imum; ab alio mouetur
&longs;ed profectò ij ip&longs;i, qui
motum grauium generanti tribuunt, tanquam principi cau&longs;æ, non ne
gant ine&longs;&longs;e grauibus grauitatem, quæ &longs;it principium actiuum minus
principale motus; ad quem etiam, vt ip&longs;i exi&longs;timant, forma &longs;ub&longs;tantialis
concurrit; In hoc quippe conueniunt omnes tùm &longs;ectarum Principes,
tùm recentiores: quidquid &longs;it etiam ex iis ip&longs;is datur motus naturalis,
qui e&longs;t à virtute proxima intrin&longs;eca; hoc ip&longs;um etiam &longs;en&longs;it Ari&longs;toteles
lib.4. de cælo cap. 3. t. 25. vbi ait grauibus & leuibus ine&longs;&longs;e principium
actiuum &longs;uorum motuum; immò &longs;i totum cap.4. l.8. phy&longs;. attentè lega
tur, vbi dicit moueri à generante, haud dubiè intelligetur nihil aliud in
tendi&longs;&longs;e Ari&longs;totelem quàm grauia à generante, in&longs;tanti, quo generan
tur, accipere actum primum huius motus; id e&longs;t virtutem, à qua po&longs;
&longs;int reduci ad actum &longs;ecundum, id e&longs;t ad ip&longs;um motum, de cuius rei ve
ritate iam mihi non e&longs;t laborandum.
Igitur non mouetur corpus graue à cau&longs;a primâ, licèt hæc concurrat
cum aliâ ad eius motum, nec ab aëre, nec à virtute magnetica, quæ in
&longs;it terræ, nec adductis, reducti&longs;que filamentis, nec à cælo pellente, nec
à vi &longs;ympathicâ, nec à generante proximè & immediatè; quia fortè iam
interiit, nec ab vllo alio extrin&longs;eco, vt con&longs;tat inductione; igitur ab ali
quâ vi intrin&longs;ecâ, quidquid &longs;it, de qua alibi: hæc omnia paulò fu&longs;iùs
tractauimus, quia in hoc vno Theoremate totam motus naturalis rem
verti iudicamus.
mobile ip&longs;um aliquando quie&longs;cit per hypoth.4.lib.1. igitur e&longs;t &longs;ine mo
tu; igitur &longs;eparatum à motu;
igitur realiter di&longs;tinctum per Ax.2. lib.1.
hoc etiam probatus per Th. 1.lib. 1. Et certè mirari &longs;atis non po&longs;&longs;um
aliquos recentiores non po&longs;&longs;e concipere, vt ip&longs;i aiunt, motum e&longs;&longs;e ali
quid ab ip&longs;o mobili di&longs;tinctum; nam quotie&longs;cunque duo prædicata, vel
jecto diuer&longs;is temporibus ine&longs;&longs;e dicuntur, haud dubiè alterum &longs;altem ab
eo di&longs;tingui realiter nece&longs;&longs;e e&longs;t; alioqui &longs;i vtrumque idem e&longs;&longs;e cum vno
tertio vere dicitur;
ctoria; igitur vel moueri, vel non moueri dicit di&longs;tinctum realiter à mo
bili; Secundum e&longs;t mera negatio; nam eo ip&longs;o, quod mobile e&longs;t &longs;ine vllo
addito, non mouetur; igitur &longs;uprà ip&longs;um mobile dicit puram putam ne
gationem motus; igitur moueri, dicit aliquid di&longs;tinctum.
Præterea quotie&longs;cunque prædicatum aliquod tribuitur in propo&longs;i
tione affirmatiua falsâ; certè prædicatum illud non ine&longs;t &longs;ubiecto;
alio
quin e&longs;&longs;et vera, vt patet; igitur di&longs;tinguitur à &longs;ubiecto realiter;
&longs;ed hæc
propo&longs;itio,
ine&longs;t mobili, igitur ab eo di&longs;tinguitur realiter, &longs;eu modaliter, quæ e&longs;t
di&longs;tinctio realis minor.
vnde &longs;it hic motus:
ictum infligit perhypoth. 1. maior e&longs;t effectus, igitur maior cau&longs;a, id e&longs;t
motus; igitur cau&longs;a motus per Ax.2. &longs;ed e&longs;t eadem entitas mobilis, vt
patet; igitur non e&longs;t cau&longs;a immediata motus; Præterea globus per pla
num inclinatum deuolutus &longs;uum motum accelerat per hypotl. 3. & fune
pendulum &longs;uam vibrationem per hypoth. 2. igitur debet e&longs;&longs;e cau&longs;a huius
maioris, &longs;eu velocioris motus per Ax.8. lib. 1. hæc porrò non e&longs;t &longs;ub
&longs;tantia ip&longs;ius corporis, quæ &longs;emper eadem e&longs;t, tùm initio, tùm in fine
motus per Ax.2.
Probatur, &longs;int
enim eædem hypoth.1.2.3. igitur maior ictus in fine motus, & velocior
motus debent habere cau&longs;am; &longs;ed hæc grauitas non e&longs;t, quæ &longs;emper ea
dem e&longs;t, vt patet, vtrum verò di&longs;tinguatur grauitas ab ip&longs;a corporis
&longs;ub&longs;tantia di&longs;cutiemus in tractatu &longs;equenti. Fuit aliquis non infimæ no
tæ Philo&longs;ophus, qui diceret maiorem illum ictum e&longs;&longs;e ab ipsâ corporis
&longs;ub&longs;tantiâ; &longs;ed hoc iam refellimus Theoremate 4. lib.1. Adde quod im
petu, ad extra producitur ab alio impetu per Th.42.lib.1. Dicebat etiam
velociorem motum e&longs;&longs;e ab ipsâ grauitate connotante præuium motum,
quod etiam refellemus infrà.
Probatur; e&longs;t ab aliqua cau&longs;a per
Ax.8. lib.1. ab aliqua intrin&longs;eca per Th. 1. non à &longs;ub&longs;tantia corporis
grauis per Th. 3. non à grauitate per Th. 4. igitur ab impetu, quia
nihil aliud e&longs;&longs;e pote&longs;t intrin&longs;ecum, à quo &longs;it motus per definitionem
3. lib. 1.
Probatur, quia quidquid de no
uo e&longs;t, habet cau&longs;am per Ax.8. lib. 1.
Producitur ab aliqua cau&longs;a intrin&longs;eca, quia non producitur ab aliqua
extrin&longs;eca; alioquin motus naturalis e&longs;&longs;et ab extrin&longs;eco contra definitio
nem primam, & Th.1.
nam graui
tas e&longs;t ip&longs;e impetus innatus, de qua infrà:quia nihil e&longs;t aliud in
trin&longs;ecum, à quo produci po&longs;&longs;it; quòd autem non producatur ab alio im
petu ad intra, patet per Th.41. lib. 1.
Probatur pri
mò; quia &longs;emper habet &longs;uum effectum formalem; vel grauitationis, &longs;i
impeditur; vel motus in medio libero;
igitur non e&longs;t fru&longs;trà;
igitur
non de&longs;truitur per Th.162.lib.1. nihil enim exigit de&longs;tructionem; non
tota natura, quia non e&longs;t fru&longs;trà per Ax. 6. non à contrario impetu, qui
&longs;æpè abe&longs;t, vt cum liberè mouetur corpus graue in aëre, vel &longs;u&longs;tinetur,
v.g. glans plumbea ab ingenti rupe: adde quod, licèt producatur in cor
pore graui impetus violentus &longs;ur&longs;um, non de&longs;truitur, tamen innatus; alio
quin nihil e&longs;&longs;et, quod de&longs;trueret violentum per Th.150. & Schol. Th.
152.num.6.lib.1.
&longs;a
Probatur per Th.144. lib.1.
alioquin non po&longs;&longs;et intendi ab eadem cau&longs;a per Th. 146. lib 1. quippè
con&longs;eruatio nihil e&longs;t aliud, quàm repetita productio, vt con&longs;tat; nam
cau&longs;a con&longs;eruans verè influit; igitur &longs;i e&longs;t cau&longs;a nece&longs;&longs;aria primo, & &longs;e
cundo in&longs;tanti æquali ni&longs;u influit; influit enim quantum pote&longs;t per Ax.
12.lib.1.quòd autem impetus intendatur, demon&longs;trabimus infrà; con&longs;ule
Schol.Th.146.lib.1.in quo habes rationem præclari natura in&longs;tituti; quo
&longs;cilicet factum e&longs;t, vt qualitates, quæ contrario carent à causâ primò pro
ductiua; aliæ verò, quæ contrarium habent, ab alia causà con&longs;er
uentur.
Hinc ab aliâ causâ con&longs;eruari nece&longs;&longs;e e&longs;t, vt patet, eáque aplicatâ per
Ax.10.lib.1. quæcumque tandem illa &longs;it; nos aliquando cau&longs;am primam
e&longs;&longs;e dicemus.
do instanti producitur nouus impetus, itemque tertio, quarto, quinto. &c.
Pro
batur primò; quia &longs;ecundo in&longs;tanti e&longs;t eadem cau&longs;a quæ primo non ma
gis impedita, eáque nece&longs;&longs;aria; igitur nece&longs;&longs;ariò agit per Ax. 12. lib.1.
igitur aliquem effectum producit; &longs;ed hic effectus non e&longs;t impetus pro
ductus primo in&longs;tanti, quia non con&longs;eruatur à cau&longs;a primò productiua
per Th.11. igitur e&longs;t nouus. Probatur &longs;ecundò; cre&longs;cit motus grauium in
libero medio per hypoth. 1.2.3. igitur cre&longs;cit impetus;
quia cum motus
naturalis &longs;it ab impetu per Th.5. quâ proportione cre&longs;cit effectus, &longs;cilicet
formalis, & exigentiæ; &longs;ic enim motus e&longs;t effectus impetus per Th. 15.
lib.1.eàdem cre&longs;cit cau&longs;a per Ax.2. Probatur tertiò, quia corpus graue ex
maiore altitudine cadens maiorem quoque ictum infligit per hypoth.1.
igitur maior impetus imprimitur in corpore percu&longs;&longs;o; &longs;ed impetus ad ex
tra producitur ab alio impetu per Th.42.lib.1. igitur &longs;i cre&longs;cit productus
inpetus, cre&longs;cit impetus producens.
Hinc reiicies quorumdam placitum, qui volunt cau&longs;am velocioris
motus e&longs;&longs;e grauitatem ip&longs;am, quatenus connotat motum præuium; quia
&longs;cilicet grauitas non producit illum maiorem impetum ad extra, vt con
&longs;tat; nec &longs;ub&longs;tantia ip&longs;ius corporis grauis per Th.40.lib.1.igitur ip&longs;e im
petus: præterea &longs;i hoc e&longs;&longs;et, fru&longs;trà requireretur impetus contra Th. 5.
Denique motus præuius nihil e&longs;t amplius, cum alius &longs;uccedit: Vide Th.
40.lib.1. vbi hæc fusè di&longs;cu&longs;&longs;imus.
productus tertio con&longs;eruatur, quarto, atque ita deincepsquia &longs;cilicet nec con
&longs;eruantur à cau&longs;a primo productiua per Th.144.libri: nec aliquid exigit
de&longs;tructionem; non contrarius impetus, quia nullus e&longs;t applicatus, vt
con&longs;tat; non re&longs;i&longs;tentia medij, quæ quidem alicuius momenti e&longs;t;
&longs;ed
non tanti, vt impedire po&longs;&longs;it motum omninò, vt con&longs;tat; nam &longs;uppono
liberum medium, igitur nec de&longs;truere impetum; cum tamdiu duret cau
&longs;a quamdiu durat effectus, vt patet; igitur nihil e&longs;t quod exigat impe
tus huius de&longs;tructionem; igitur non de&longs;truitur per Ax. 14. lib.1.
infrà.
cundo instanti non producitur nouus impetus. Probatur primò, non cre&longs;cit
corporis grauis &longs;eu grauitas, &longs;eu grauitatio, vt con&longs;tat experientiâ; igitur
non cre&longs;cit impetus; alioquin &longs;i cre&longs;ceret cau&longs;a, cre&longs;ceret effectus per
Ax.2. igitur de re, quòd &longs;it, certum e&longs;t, atque euidens; iam demon&longs;tratur
propter quid &longs;it; impetus &longs;ecundo in&longs;tanti productus e&longs;&longs;et fru&longs;trà;
careret
quod fru&longs;trà e&longs;t, non e&longs;t per Ax.6.l.1.
Ob&longs;erua quæ&longs;o, quod iam &longs;uprà indicatum e&longs;t, e&longs;&longs;e tres veluti &longs;pecies
impetus. Prima e&longs;t impetus naturalis innati.
Secunda naturalis acqui&longs;iti.
Tertia violenti; innatus e&longs;t qui vel à generante &longs;imul cum corpore
graui productus e&longs;t; qui&longs;quis tandem &longs;it generans, de quo aliàs;
vel ab
ip&longs;o graui qua&longs;i profunditur, id e&longs;t, producitur in &longs;e ip&longs;o &longs;tatim initio,
quo e&longs;t; porrò cum in corpore graui duplex qua&longs;i proprietas &longs;en&longs;ibilis
e&longs;&longs;e videatur, &longs;cilicet grauitas, &longs;eu pondus & motus deor&longs;um; certè de
bet e&longs;&longs;e in eo aliquid per quod tùm cogno&longs;ci po&longs;&longs;it eius pondus, tùm in
cipiat moueri deor&longs;um; quippe maximè corpora ex pondere cogno&longs;ci
mus, vnumque ab alio di&longs;tinguimus; igitur debet e&longs;&longs;e aliquid, quod &longs;en
&longs;um afficiat, vt cogno&longs;ci po&longs;&longs;it; atqui illud ip&longs;um non e&longs;t &longs;ub&longs;tantia cor
poris; nam corpus graue meæ manui &longs;u&longs;tinenti impetum imprimit;
immò vim alterius impetus infringit; igitur operâ alterius per Th. 40.
& 42.lib.1. Præterea illud ip&longs;um, quod agit, &longs;eu deor&longs;um pellit &longs;u&longs;tinen
tem manum, e&longs;t illud ip&longs;um quod inclinat corpus graue deor&longs;um imme
diatè, &longs;eu quod exigit motum naturalem deor&longs;um; illud autem quod
immediatè præ&longs;tat hunc motum, nec e&longs;t &longs;ub&longs;tantia corporis grauis per
Th.3. igitur ip&longs;e impetus per Th.5. adde quod primo in&longs;tanti, quo e&longs;t im
petus, non e&longs;t motus ille, quem exigit per Th.34. lib.1. igitur præexi&longs;tit
&longs;emper impetus, qui ne &longs;it fru&longs;trà, habet primum effectum &longs;uum forma
lem, id e&longs;t grauitationem: Ex his dicendum e&longs;t hunc impetum natiuum
nunquam de&longs;trui, quia nunquam e&longs;t fru&longs;trà, habet enim &longs;emper aliquem
effectum, primum quidem &longs;i caret &longs;ecundo; &longs;ecundum verò &longs;i caret pri
mo; quippe vtrumque &longs;imul habere non pote&longs;t; nam corporis pondus
cogno&longs;ci non pote&longs;t, dum fertur deor&longs;um accelerato motu, quot verò,
& quanta commoda ex cognitione ponderis cuiu&longs;libet materiæ proce
dant, vix explicari pote&longs;t.
Ex his verò concludendum &longs;upere&longs;t impetum innatum e&longs;&longs;e proprie
tatem quarto modo, vt vulgò aiunt, corporis grauis; ac proinde ab illo
in&longs;eparabilem; quid verò fiat de illo, cum corpus graue fit leue; &longs;i tamen
hoc aliquando accidit, dicemus cum de grauitate, & grauitatione, iam
verò &longs;atis e&longs;t ad præ&longs;ens in&longs;titutum impetum innatum ab ip&longs;o corpori
graui nunquam &longs;eparari, quandiu remanet graue.
Impetus naturalis acqui&longs;itus producitur ab eodem principio intrin
&longs;eco; hinc dicitur naturalis:
dicitur verò acqui&longs;itus, quia non e&longs;t inna
tus; &longs;ed &longs;eparatur à corpore graui;
quod &longs;emper eo caret, quandiu
quie&longs;cit: &longs;ed innato tantùm accedit ad motus accelerationem, & ad alia
quamplurima, quæ ex ea &longs;equuntur; putà maiorem percu&longs;&longs;ionem, re&longs;i
&longs;tentiam, vim, & ad tollendum totius naturæ languidiorem; quo certè af
ficeretur, &longs;i corpus graue tardi&longs;&longs;imo motu deor&longs;um ferretur, de quo in
frà; Porrò impetus acqui&longs;itus in multis differt ab innato; primò quia
Secundò, quia determinari pote&longs;t ad omnem lineam.
Impetus violentus e&longs;t, qui e&longs;t ab extrin&longs;eco, de quo agemus infrà, &
iam &longs;uprà in lib.1. multa &longs;unt de eo demon&longs;trata.
medio libero
tio, quarto, &c. in&longs;tantibus per Th.12. &longs;ed productus in primo con&longs;er
uatur &longs;ecundo, per Th.9. productus &longs;ecundo con&longs;eruatur tertio, produ
ctus tertio con&longs;eruatur quarto per Th.13. igitur &longs;ecundus additur tertio,
tertius primo, &longs;ecundo, quartus primo, &longs;ecundo, & tertio, &c.&longs;ed impetus
additus alteri facit inten&longs;iorem impetum per Ax.1. igitur impetus natu
ralis intenditur, quod crat demon&longs;trandum.
hoc ip&longs;um &longs;uppo&longs;ui &longs;uprà
Quod e&longs;&longs;et in hyp.1.2.3. iam verò demon&longs;tro propter quid e&longs;t; &longs;ie enim
hypothe&longs;is in Theorema conuerti pote&longs;t, vt &longs;æpè monuimus in metho
do; igitur probatur hoc Theorema facilè;
cre&longs;cit impetus in corpore gra
ui, quod tendit deor&longs;um in libero medio per T. 15. igitur cre&longs;cit cau&longs;a
motus; nam impetus e&longs;t cau&longs;a immediata motus naturalis per Th. 51.
&longs;ed quâ proportione cre&longs;cit cau&longs;a, debet cre&longs;cere effectus per Ax.2. igi
tur motus naturalis deor&longs;um cre&longs;cit, id e&longs;t acceleratur, id e&longs;t fit velo
cior, quod erat dem: nec e&longs;t quod aliquis exi&longs;timet hic à me committi
vitio&longs;um argumentationis circulum; quippe probaui &longs;uprà cre&longs;cere im
petum, quia cre&longs;cit motus; iam verò probo cre&longs;cere motum, quia cre&longs;
cit impetus; nam primò probaui produci nouum impetum in Th.12. eo
quod &longs;ecundo in&longs;tanti. v.g. &longs;it eadem cau&longs;a nece&longs;&longs;aria applicata non im
pedita, igitur tàm debet agere &longs;ecundo quàm primo in&longs;tanti, hæc fuit
mea probatio à priori; &longs;ecundò verò probaui ex hypothe&longs;i certa;
quia
&longs;cilicet cre&longs;cit motus, cuius veritatem cogno&longs;co &longs;en&longs;ibiliter in &longs;e, vnde
&longs;uppono tantùm de illa quod &longs;it; igitur nullus committitur circulus; nam
diuer&longs;a e&longs;t omninò cognitio. Prima &longs;cilicet qua cogno&longs;co de motu na
turaliter accelerato quod &longs;it, quæ mihi, & ru&longs;tico communis e&longs;t. Secun
da verò qua non modò cogno&longs;co de motu illo quod &longs;it acceleratus, ve
rùm propter quid &longs;it acceleratus, id e&longs;t cau&longs;am huius accelerationis, id
e&longs;t propter quam attributum hoc ine&longs;t &longs;ubiecto, & hæc e&longs;t vera demon
&longs;tratio à priori; porrò in Phy&longs;ica de effectu &longs;en&longs;ibili &longs;upponi debet quod
&longs;it, hoc enim percipitur &longs;en&longs;u. v. g. &longs;upponam in Phy&longs;ica quod &longs;it motus
acceleratus, quod ignis &longs;it calidus, Sol lucidus, nix candida, vinum ru
brum, &c. at verò demon&longs;trabo propter quid hæc &longs;int, &longs;ed de his
&longs;atis.
Ob&longs;eruabis etiam aliud naturæ in&longs;titutum, quo &longs;cilicet factum e&longs;t, vt
&longs;i enim motu
ferrentur æquabili, vel e&longs;&longs;et æqualis illi quem initio &longs;ui de&longs;cen&longs;us ha
bent, qui e&longs;t tardi&longs;&longs;imus, vt con&longs;tat ex ip&longs;a ictuum differentia; atque
ita infinitum ferè tempus ponerent grauia in minimo etiam de&longs;cen&longs;u,
quod e&longs;&longs;et maximè incommodum; &longs;i verò motus ille e&longs;&longs;et æqualis mo
tui v.g. quem acqui&longs;iuit in &longs;patio 3. vel 4. perticarum, pondera corpo
rum cre&longs;cerent in immen&longs;um, ide&longs;t in ea proportione, qua ictus, qui in
fligitur à corpore graui confecto 4. perticarum &longs;patio maior e&longs;t ictu, qui
infligitur po&longs;t decur&longs;um minimum omnium &longs;patiorum, quod valdè in
commodum e&longs;&longs;et.
tio, idemque impedimentumprobatur, quia cau&longs;a huius impetus e&longs;t ne
ce&longs;&longs;aria; &longs;ed eadem cau&longs;a nece&longs;&longs;aria æqualibus temporibus æqualem
impetum producit per Ax.3.
ne cre&longs;cit cau&longs;a, etiam cre&longs;cit effectus per Ax.2.
locitatis, vel accelerationis momentahoc ip&longs;um e&longs;t quod definitionis lo
co Galileus in dialogo tertio de motu naturali a&longs;&longs;umit; quod tamen
meo iudicio fuit antè demon&longs;trandum quàm &longs;upponendum; quare &longs;ic
demon&longs;tramus, quâ proportione cre&longs;cit impetus, cre&longs;cit motus per Th.
18. &longs;ed temporibus æqualibus acquiruntur æquales impetus gradus per
Th.17. igitur æqualia velocitatis momenta, vel incrementa.
Probatur per Def.2.
patet; cùm enim impetus &longs;int vt motus per Ax. 2. motus &longs;unt vt &longs;patia;
quippe vt ex impetu &longs;equitur motus, ita ex motu confectum &longs;pa
tium.
permutande
minori tempore per Def.2. l. 1. Igitur eò velocior, quò minori tem
pore.
quod percurritur minori æquè veloci motu in ea ratione, qua vnum tempuspatet, quia æqualia &longs;unt æqualibus temporibus per Th.
20. igitur inæqualibus inæqualia iuxta rationem temporum; item &longs;pa
tium, quod idem percurritur minori tempore minus e&longs;t.
quò minus conficitur iuxta rationem &longs;patiorum:
pora, igitur tempora &longs;unt vt &longs;patia; item tempus, quo minus &longs;patium
percurritur e&longs;t minus co, quo maius.
ritur æquali certè tempore, &longs;ed tardiore motu,2. l. 1.
imò e&longs;t maius iuxta rationem velocitatis maioris, item e&longs;t minus iuxta
rationem tarditatis maioris.
quo conficitur tardioreProbatur per def.2. & per Th.22. idque in ratio
ne velocitatum permutando; item tempus quo conficitur &longs;patium æqua
le tardiore motu e&longs;t maius eo, quo conficitur velociore, patet.
tuum erunt vt &longs;patia, & vici&longs;&longs;im &longs;patia vt tempora. Probatur per Th.
24. & 23.
&longs;ed inæquali velocitate; &longs;patia erunt vt velocitates, & hæ vt illa;
imò &longs;i
&longs;patia &longs;unt vt velocitates, tempora erunt æqualia
Th.25.
locitates erunt in ratione permutata temporum, ide&longs;t maior velocitas re&longs;pon
debit minori tempori, & minor maiori
libus temporibus, &longs;patia &longs;unt in ratione compo&longs;ita ex ratione temporum, & ex
ratione velocitatum,
citates per Th.25. &longs;i æquales &longs;int velocitates, &longs;patia erunt vt tempora, per
Th.29. igitur &longs;i nec æquales velocitates, nec æqualia tempora, erit ratio
&longs;patiorum compo&longs;ita ex ratione temporum, & ex ratione velocitatum;
&longs;it ratio temporum 3/2 ratio velocitatum 2/3 compo&longs;ita ex vtraque erit 6/2
&longs;eu 3. vt con&longs;tat ex ip&longs;is elementis.
citate, tempora erunt in ratione compo&longs;ita ex ratione &longs;patiorum & ratione
velocitatum permutataprobatur eodem modo quo &longs;uperius Th. 30. &longs;it
ratio &longs;patiorum 4/1, velocitatum 4/2; permutetur hæc 1/4; componetur ex
vtraque 4/1, ide&longs;t 1/2, quæ e&longs;t ratio temporum.
temporibus; ratio velocitatum erit compo&longs;ita ex ratione &longs;patiorum, & ex ra
tione temporum permutata
4/2 temporum 1/2, permutetur 2/1, compo&longs;ita ex vtraque erit 2/2, ide&longs;t 4.
quæ e&longs;t ratio velocitatum.
Ob&longs;eruabis hæc omnia à vige&longs;imo Theoremate maiori ex parte tradi
à Galileo &longs;uo modo, optimo quidem, &longs;ed fortè longiore quàm par &longs;it,
nulla habita ratione cau&longs;arum phy&longs;icarum.
bus
tur &longs;ingulis in&longs;tantibus aliquem effectum producit, per Ax. 12. l.1. &longs;ed
priorem non con&longs;eruat, vt dictum e&longs;t &longs;uprà, igitur nouum producit.
pe e&longs;t æqualis, imò eadem cau&longs;a, igitur æqualem effectum producit per
Ax.12. l.1.
in&longs;tantibus producatur nouus, & prior con&longs;eruetur, cui cum addatur,
intenditur per Ax. 1.
34. igitur æqualiter etiam &longs;ingulis in&longs;tantibus cre&longs;cit velocitas motus
per Ax.2.
Ob&longs;eruabis
aliter explicari non pote&longs;t, quàm per in&longs;tantia finita, vt demon&longs;trabimus
in Metaphy&longs;ica; quid quid &longs;it, voco in&longs;tans totum illud tempus, quo res
aliqua &longs;imul producitur, &longs;iue &longs;it maius, &longs;iue minus, &longs;iue &longs;it pars maior,
vel minor, quod ad rem no&longs;tram nihil facit penitus; nam dato quocun
que tempore finito pote&longs;t dari maius & minus, quod certum e&longs;t; igitur
totum illud tempus, quo producitur primus impetus acqui&longs;itus, vo-
pora.
cum &longs;ingula in
&longs;tantia æqualem impetum addant&longs;i primo in&longs;tanti &longs;it vnus gradus, erunt
duo; productus &longs;cilicet alteri additus qui con&longs;eruatur, tertio erunt;.
quarto 4. quinto 5. &c. igitur cre&longs;cit &longs;ecundum progre&longs;&longs;ionem arith
meticam.
tur velocitatis momenta
in&longs;tanti motus e&longs;t velocior per Th.36. igitur maius conficitur &longs;patium,
tempore &longs;cilicet æquali per Def. 2. l. 1. idem dico de tertio, quar
to, &c.
primo vt velocitas, quæ e&longs;t &longs;ecundo ad velocitatem, quæ e&longs;t primo. Patet per
Th.28. quia cum tempora illa &longs;int æqualia, &longs;patia &longs;unt nece&longs;&longs;ariò vt ve
locitates; quippe æquali velocitati æquale &longs;patium re&longs;pondet tempore
æquali, igitur inæquale inæquali, igitur maius maiori, idem dico de
aliis in&longs;tantibus.
quiritur primo. Probatur, quia velocitas e&longs;t dupla per Th 38. igitur &longs;pa
tium duplum, & triplum tertio, quadruplum quarto, &c.
quia &longs;patia cre&longs;cunt vt motus, &longs;eu vt velocitates; hæ cre&longs;cunt æqualiter
&longs;ingulis in&longs;tantibus æqualibus per Th.36. igitur æqualiter cre&longs;cunt &longs;in
gula &longs;patia per Th.40.
nem arithmeticam
per Th.38. hic demum iuxta progre&longs;&longs;ionem arithmeticam per Th. 37.
igitur &longs;i &longs;patium acqui&longs;itum primo in&longs;tanti &longs;it 1. acqui&longs;itum &longs;ecundo erit
2. tertio 3. quarto 4. &c. hinc &longs;patia acqui&longs;ita &longs;ingulis in&longs;tantibus &longs;unt
vt &longs;eries numerorum, qui componunt progre&longs;&longs;ionem &longs;implicem, &longs;cilicet
1.2.3.4.5.6. &c. dixi &longs;ingulis in&longs;tantibus æqualibus, quod e&longs;t apprimè
tenendum; &longs;i enim a&longs;&longs;umantur partes temporis maiores, perturbatur
hæc progre&longs;&longs;io, de quo infrà.
quod illo in&longs;tanti decurriturquod certè verum e&longs;t, dum intelligatur legi
timus horum verborum &longs;en&longs;us; quidquid reclamet Saluiatus apud
Galil. dialogo 3. modò a&longs;&longs;umatur progre&longs;&longs;io incrementi in &longs;ingulis in
&longs;tantibus, in quibus reuerà fit; cur enim potiùs in vno quàm in alio?
quippe &longs;i comparetur velocitas vnius in&longs;tantis cum velocitate alterius;
haud dubiè erit eadem vtriu&longs;que ratio, quæ &longs;patiorum; &longs;i enim vno in
&longs;tanti percurritur vnum &longs;patium cum vno velocitatis gradu; certè in
&longs;tanti æquali acquiritur duplum &longs;patium cum duobus velocitatis gradi
bus, nec obe&longs;t, quod obiicit Galileus tunc motus e&longs;&longs;e æquabiles; quia
motus qui fit in in&longs;tanti debet con&longs;iderari vt æquabilis; appello enim
in&longs;tans totum illud tempus, quo &longs;imul acquiritur aliquid impetus, ali
quid enim &longs;imul acquiri nece&longs;&longs;e e&longs;t; nec demum ob&longs;tat quod dicit, dari
non po&longs;&longs;e motum in&longs;tantaneum, quod multi haud dubiè negabunt; ego
in Metaphy&longs;ica explicabo quonam pacto dari po&longs;&longs;it motus in&longs;tanta
neus, qui reuerà datur actu, non potentiâ; quia quacunque duratione
data pote&longs;t dari minor; igitur quocunque dato motu pote&longs;t dari minor.
Ob&longs;eruabis primò hanc &longs;patiorum rationem, quæ e&longs;t eadem cum ra
tione velocitatum a&longs;&longs;umendam tantùm e&longs;&longs;e in iis &longs;patiis, quæ acquirun
tur &longs;ingulis in&longs;tantibus; &longs;i enim accipiantur partes temporis maiores, quæ
conflentur ex multis in&longs;tantibus; haud dubiè maior erit ratio &longs;patio
rum, quàm velocitatum.v.g.&longs;i primo in&longs;tanti acquiratur primo &longs;patium,
&longs;ecundo, 2.tertio, 3.quarto 4.igitur &longs;i
cum velocitate quarti æqualis erit, vt ratio &longs;patiorum, id e&longs;t, vt 1. ad 4.
At verò &longs;i accipiatur pars temporis con&longs;tans duobus in&longs;tantibus, hæc 4.
in&longs;tantia conflabunt tantùm 2. partes temporis æquales; in prima ac
quirentur 3.&longs;patia, in &longs;ecunda 7.vt patet: &longs;ed quia velocitas primæ par
tis temporis non e&longs;t æquabilis, nec etiam velocitas &longs;ecundæ; addantur
velocitates primi & &longs;ecundi in&longs;tantis, itemque &longs;eor&longs;im velocitates tertij,
& quarti; certè ratio collectorum erit vt ratio &longs;patiorum; &longs;i enim velo
citas &longs;ecundi in&longs;tantis comparetur cum velocitate quarti e&longs;t tantùm
1/2 cum tamen primum &longs;patium &longs;it ad &longs;ecundum in ratione 3/7.
Secundò, &longs;i comparentur &longs;patia cum temporibus e&longs;t alia ratio v.g.&longs;pa
tium acqui&longs;itum vno in&longs;tanti &longs;e habet ad &longs;patium acqui&longs;itum in duobus
in&longs;tantibus, vt 1, ad 3.in tribus vt 1.ad 6.in 4. vt 1. ad 10.
Tertiò ob&longs;eruabis, non po&longs;&longs;e &longs;en&longs;u percipi in&longs;tans, imò neque tempo
ris partem ex mille in&longs;tantibus conflatam; nec etiam &longs;patium quod ac
quiritur primo in&longs;tanti; adhibenda &longs;unt tamen in&longs;tantia nece&longs;&longs;ariò ad
explicandam proportionem huius accelerationis, quæ fit in &longs;ingulis in
&longs;tantibus; vt verò rem i&longs;tam reuocemus ad &longs;en&longs;ibilem praxim, a&longs;&longs;ume
mus proportionem aliam &longs;en&longs;ibilem, quæ proximè ad veram accedit, nec
ferè &longs;en&longs;ibiliter fallere pote&longs;t, de qua infrà.
Cùm enim ratio &longs;patiorum &longs;it vt ratio velocitatum;
dum &longs;cilicet hæc
progre&longs;&longs;io accipitur in in&longs;tantibus, & ratio velocitatum vt ratio incre
menti impetuum; vt con&longs;tat ex dictis, & hæc &longs;equatur &longs;implicem
progre&longs;&longs;ionem 1. 2. 3. 4. &c. certè collectio &longs;patiorum e&longs;t &longs;umma ter
minorum.
in&longs;tanti & &longs;patio quod percurritur vltimo instanti, cogno&longs;citur &longs;umma, id e&longs;t
collectio &longs;patiorum, id e&longs;t, totum &longs;patium confectum.
&longs;ecundus S.igitur &longs;umma e&longs;t 36. quippe vltimus terminus indicat nume
rum terminorum, quia primus e&longs;t &longs;emper vnitas, & progre&longs;&longs;iuus etiam
vnitas.
tium æqualium, qui &longs;emper est idem cum numero terminorum, cogno&longs;citur
etiam primus terminus, id e&longs;t &longs;patium quod primo instanti percurritur, cogno
&longs;cuntur etiam gradus velocitatisquippe hæc omnia &longs;unt in eadem ratio
ne; quæ omnia con&longs;tant ex regulis arithmeticis præter alia multa data,
quæ lubens omitto; tùm quia Phy&longs;icam non &longs;apiunt, tùm quia hypothe
&longs;is illa e&longs;t impo&longs;&longs;ibilis phy&longs;icè; quis enim &longs;en&longs;u percipere po&longs;&longs;it & di
&longs;tinguere vnum temporis in&longs;tans, vel &longs;patij punctum? licèt recen&longs;enda
fuerit hæc accelerati motus proportio in in&longs;tantibus, vt ad &longs;ua phy&longs;ica
principia reduceretur.
&longs;i inueniatur numerus, per quem diuidatur, qui &longs;uperet tantùm vnitate du
plum quotientisquippe habebis in duplo quotientis numerum termino
rum v.g. &longs;it &longs;umma 10. diui&longs;or &longs;it 5. quotiens 2. duplus 4. hic e&longs;t nume
rus terminorum datæ &longs;ummæ; &longs;it alia &longs;umma 21. diui&longs;or &longs;it 7.quotiens 3.
numerus terminorum 6. &longs;it alia &longs;umma 36. dini&longs;or &longs;it 9. quotiens 4. nu
merus terminorum 8. &longs;it alia &longs;umma 45. partitor &longs;it 10. quotiens 4 1/2,
numerus terminorum 9. quomodo verò hic partitor inueniri po&longs;&longs;it, vi
derint Arithmetici; nec enim e&longs;t huius loci, quamquam datâ &longs;ummâ
huius progre&longs;&longs;ionis &longs;implicis facilè cogno&longs;ci pote&longs;t numerus termino
rum; duplicetur enim, & radix 9. neglecto re&longs;iduo dabit numerum ter
minorum v.g. &longs;it &longs;umma 21. duplicetur, erit 42. rad. 9. 6. dat numerum
terminorum; &longs;it &longs;umma 36. duplicetur, erit 72.rad.9.8. dabit numerum
terminorum.
velocitatis &longs;ecundi in&longs;tantis ad primum quàm tertij ad &longs;ecundum, & maior&longs;it enim
primo in&longs;tanti velocitas vt 1.&longs;ecundo erit, vt 2.tertio, vt 3.quarto, vt 4.
&longs;ed maior e&longs;t proportio 2.ad 1.quàm 3.ad 2. & hæc maior quàm 4. ad 3.
atque ita deinceps; &longs;imiliter maior e&longs;t proportio &longs;patij quod percurritur
&longs;ecundo in&longs;tanti ad &longs;patium, quod percurritur primo, quàm &longs;patij, quod
percurritur &longs;ecundo in&longs;tanti ad &longs;patium, quod percurritur primo quàm
&longs;patij quod percurritur tertio ad &longs;patium, quod percurritur &longs;ecundo, at
que ita deinceps; e&longs;t enim eadem ratio &longs;patiorum quæ &longs;ingulis in&longs;tanti
bus re&longs;pondent, quæ velocitatum, vt demon&longs;tratum e&longs;t &longs;uprà.
tum &longs;patium, quod acquiritur vno, quàm &longs;it illius, quod acquiritur quatuor in
&longs;tantibus ad aliud, quod acquiritur duobus&longs;i enim primo
in&longs;tanti acquiritur vnum &longs;patium, &longs;ecundo acquiruntur 2.igitur duobus
&longs;imul acquirantur 3. igitur proportio e&longs;t vt 3.ad 1.Sed &longs;i duobus acqui
runtur 3. &longs;patia; certè 4.in&longs;tantibus acquiruntur 10. igitur proportio e&longs;t
vt 10.ad 3. &longs;ed proportio 10/3 e&longs;t maior 3/1, erit adhuc maior proportio &longs;pa
tij quod acquiretur 6. in&longs;tantibus ad illud quod acquiritur tribus; e&longs;t
enim (21/6) vt patet.
ma, æquali tempore, idem &longs;patium percurretur hoc motu naturaliter accelera
to&longs;it enim maxima velocitas vt 6. minima vt 1. motu naturaliter acce
lerato percurrentur &longs;patia 21. cuius &longs;ummæ termini &longs;unt 6.igitur 6. in
&longs;tantibus con&longs;tat hic motus; accipiatur &longs;ubduplum maximæ, & minimæ
velocitatis, &longs;cilicet 3 1/2. sítque velocitas motus æquabilis in&longs;tantium 6.
haud dubiè &longs;i ducantur 3 1/2 in 6 erunt 21.ratio ex eo petitur quod &longs;cili
cet, vt habeatur &longs;umma progre&longs;&longs;ionis arithmeticæ, debet addi primus
terminus maximo, & a&longs;&longs;umi &longs;ubduplum totius; illudque ducere in nu
merum terminorum per regulam arithmeticam; atqui eadem e&longs;t ratio
velocitatum, quæ &longs;patiorum; vt dictum e&longs;t &longs;uprà; &longs;cilice, in &longs;ingulis
in&longs;tantibus.
constent, &longs;erueturque eadem accelerationis progre&longs;&longs;io arithmetica, &longs;patium
quod ex &longs;umma huius progre&longs;&longs;ionis re&longs;ultabit, erit minus vero,
&longs;tantia, & cuilibet iuxta progre&longs;&longs;ionem prædictam &longs;uum &longs;patium re&longs;pon
deat, haud dubiè &longs;patium &longs;ecundi erit duplum &longs;patij primi, & tertium
triplum, &c. vt con&longs;tat ex dictis;
igitur erunt &longs;patia 21. iam verò a&longs;&longs;u
mantur 3. partes temporis, quarum quælibet ex 2. con&longs;tet in&longs;tantibus;
primæ parti tria ex prædictis &longs;patiis re&longs;pondeant; certè &longs;i &longs;eruetur pro
gre&longs;&longs;io arithmetica, &longs;ecundæ re&longs;pondebunt 6. & tertiæ 9. igitur totum
&longs;patium erit 18. minus vero quod erat 21. &longs;i verò a&longs;&longs;umantur tantùm 2.
partes, quarum quælibet tribus in&longs;tantibus con&longs;tet; primæ parti re&longs;pon-
21.hinc vides &longs;uppo&longs;ito eodem in&longs;tantium numero &longs;patium e&longs;&longs;e &longs;emper
æquale, &longs;iue a&longs;&longs;umantur partes maiores temporis, &longs;iue minores, v. g. &longs;up
po&longs;itis 6.in&longs;tantibus, ex quibus totum &longs;patium 21.con&longs;equitur, &longs;iue a&longs;&longs;u
mantur tres partes, quarum quælibet con&longs;tet 2. in&longs;tantibus, &longs;iue duæ,
quarum quælibet con&longs;tet tribus, &longs;patium quod ex illis re&longs;ultat, e&longs;t &longs;em
per idem &longs;cilicet 18. a&longs;&longs;umptis verò 8. in&longs;tantibus, & totali &longs;patio, quod
illis re&longs;pondet 36. &longs;patium quod ex partibus re&longs;ultabit erit 30. &longs;iue &longs;int
duæ partes, quarum quælibet con&longs;tet 4. in&longs;tantibus, &longs;iue &longs;int 4. quarum
quælibet con&longs;tet duobus: hinc rur&longs;us vides a&longs;&longs;umpto maiori in&longs;tantium
numero &longs;patium verum habere maiorem rationem ad non verum, quàm
a&longs;&longs;umpto minori in&longs;tantium numero, v.g.a&longs;&longs;umantur 4.in&longs;tantia, &longs;umma
&longs;patiorum erit 10. &longs;i verò a&longs;&longs;umantur 2.partes temporis, quarum quæli
bet duobus in&longs;tantibus re&longs;pondeat; &longs;umma &longs;patij erit 9.igitur ratio ve
ri &longs;patij ad non verum e&longs;t (10/9). a&longs;&longs;umantur 6. in&longs;tantia &longs;patij veri, &longs;umma
erit 21.non veri 18. igitur ratio (21/18) &longs;eu 7/6 quæ maior e&longs;t priori: denique
a&longs;&longs;umantur 8. in&longs;tantia &longs;patij veri, &longs;umma erit 36. non veri 30 igitur ra
tio (36/30) &longs;eu 6/3 quæ maior e&longs;t prioribus, atque ita deinceps.
matius &longs;patium re&longs;pondebit &longs;ecundæ quo vtraque in plures partes minores diui
detur, &longs;uppo&longs;ita &longs;emper eadem progre&longs;&longs;ione arithmetica in ip&longs;o incremento
&longs;int enim duæ partes temporis &longs;en&longs;ibiles æquales AG. GH. & &longs;pa
tium quod percurritur prima parte temporis AG &longs;it HI; in &longs;ecunda
percurretur IO, id e&longs;t, duplum HI; at verò diuidatur pars temporis
AG in duas æquales AF, FG, & con&longs;equenter totum tempus AH in 4.
æquales; haud dubiè in prima AF percurretur NP &longs;ubtripla HI, & in
&longs;ecunda FG percurretur PK dupla NP; igitur in 4. partibus temporis
AH percurretur &longs;patium decuplum PN, &longs;ed HO e&longs;t tantùm nonecupla
NP; igitur re&longs;ultabit maius &longs;patium in 4.partibus temporis, quam in dua
bus; licèt duæ æquiualeant 4. iuxta progre&longs;&longs;ionem arithmeticam.
Similiter AF diuidatur bifariam in E. & tota AH in 8. æquales AE;
certè primis 4.percurretur idem &longs;patium ML æquale NK & HI; igitur
in prima AE percurretur MR. cuius ML &longs;it decupla; nam 4. terminis
re&longs;pondet &longs;umma 10. &longs;ed 8. terminis id e&longs;t 8.partibus temporis re&longs;pon
det &longs;umma; 6. æqualium RM;
&longs;ed HO tripla ML e&longs;t tantum 30.
æqualium MR; igitur in 8.partibus re&longs;ultabit maius &longs;patium, quàm in
4.quæ æquiualent 8.
Ex quibus etiam con&longs;tat quo plures accipientur partes temporis ma
ius &longs;patium re&longs;ultare, donec tandem perueniatur ad vltima in&longs;tantia, ex
quibus re&longs;ultat maximum; & &longs;i accipias AG partes temporis AG. GH.
habebitur HO; &longs;i verò 4.æquales AF, cre&longs;cet &longs;patium &longs;eu &longs;umma 1/9 HO;
&longs;i autem 8. æquales AE cre&longs;cet 1/5 HO; &longs;i porrò 16. æquales AD cre&longs;
cet (22/108) &longs;i 32. æquales AC cre&longs;cet (120/408); &longs;i 64. æquales AB cre&longs;cet (496/1584).
qualiter nunquam &longs;umma po&longs;terioris &longs;egmenti e&longs;t tripla prioris
piantur duo termini e&longs;t tantùm 2/1, &longs;i 4. e&longs;t 7/3 &longs;i 6. e&longs;t (15/6), &longs;i 8. e&longs;t (26/10), &longs;i 10
(40/15), &longs;i 12. (57/21), &longs;i 14. (77/28), atque ita deinceps.
Ex quo ob&longs;erua mirabilem con&longs;equutionem;
quippe &longs;i a&longs;&longs;umantur
tantùm duo termini, & diuidantur bifariam, &longs;umma po&longs;terioris medie
tatis e&longs;t tripla primæ minùs vnitate; &longs;i accipiantur 4. e&longs;t tripla minùs
2. &longs;i 6. minùs 3. &longs;i 8. minùs 4. &longs;i 10. minùs 5. &longs;i 12. minùs 6. &longs;i 14. mi
nùs 7. atque ita deinceps; vnde &longs;umma po&longs;terioris medietatis e&longs;t &longs;emper
tripla minùs numero &longs;uorum terminorum, vel quod clarum e&longs;t minùs
&longs;ubduplo vltimi, &longs;eu maximi termini, vel numeri terminorum totius
progre&longs;&longs;ionis, quod probè omninò tenendum e&longs;t, vt omnes experientiæ
explica ri po&longs;&longs;int, quod infrà faciemus.
pacti quod cadit ex maiori altitudine1. &longs;ed ideò e&longs;t maior ictus,
quia maior imprimitur impetus, vt patet, at ideò maior impetus impri
mitur, quia maior e&longs;t imprimens per Ax. 2. cre&longs;cit enim impetus, vt
con&longs;tat ex dictis.
tis inflictæ
aquis tanquam iacto fundamini &longs;uperædificatur ingens &longs;æpè ædificij
moles.
fligit
acce&longs;&longs;ione maximus ferè euadit.
ab eodem mobiliquia &longs;cilicet quantumuis tempore breui&longs;&longs;imo mouea
tur, plurimis tamen eius motus durat in&longs;tantibus; atqui quolibet in&longs;tan
ti motus acquiritur impetus æqualis primo impetui grauitationis, vt
con&longs;tat ex dictis. v. g. &longs;it mobile quod moueatur per mille in&longs;tantia
(modicum certè tempus & minimè &longs;en&longs;ibile) po&longs;t hunc motum impetus
erit millecuplus; igitur effectus etiam millecuplus; quæ omnia con&longs;tant
ex dictis.
quia
cùm &longs;ingulis in&longs;tantibus cre&longs;cat impetus per partes æquales, & cùm per
cu&longs;&longs;io &longs;it vt impetus; etiam erit vt tempus; igitur percu&longs;&longs;io, quæ fit po&longs;t
duo in&longs;tantia motus eiu&longs;dem corporis grauis deor&longs;um cadentis e&longs;t du-
4. quadrupla, atque ita deinceps; cùm enim æqualibus temporibus æqua
lia acquirantur velocitatis momenta, id e&longs;t æquales impetus, impetus
erunt vt tempora, percu&longs;&longs;iones vt impetus, igitur percu&longs;&longs;iones vt tem
pora.
Dixi in primo in&longs;tanti contactus; nam reuerâ &longs;ecundò in&longs;tanti con
tactus, ni&longs;i fiat reflexio, augetur vis ictus, quia cau&longs;a nece&longs;&longs;aria e&longs;t ap
plicata.
dum cadunt deor&longs;um&longs;i enim cadunt æqualibus temporibus, percu&longs;&longs;io
nes erunt vt corpora &longs;eu grauitates, vt patet v.g. corpus 2. librarum po&longs;t
2. in&longs;tantia motus infligit duplam percu&longs;&longs;ionem illius, quam infligit cor
pus vnius libræ po&longs;t 2. in&longs;tantia motus; &longs;i verò tempora motus &longs;unt inæ
qualia, & grauitates æquales, percu&longs;&longs;iones erunt vt tempora; &longs;i demum
grauitates inæquales, & tempora motus inæqualia, percu&longs;&longs;iones erunt
in ratione compo&longs;ita ex ratione grauitatum & temporum, quæ omnia
patent ex dictis in Th. &longs;uperioribus, v. g. &longs;it corpus duarum librarum,
& alterum trium librarum; primum moueatur per 5. in&longs;tantia, & &longs;ecun
dum 2.per 5. ratio grauitatum e&longs;t 3/2; ratio temporum e&longs;t 7/5;
compo&longs;ita
ex vtraque erit (21/10); & hæc e&longs;t ratio percu&longs;&longs;ionum.
& grauitationis eiu&longs;dem mobilis in pri
mo in&longs;tanti vtriu&longs;que, &longs;i cogno&longs;catur numerus in&longs;tantium motuscum enim
&longs;ingulis in&longs;tantibus æqualis impetus accedat, vt &longs;æpè dictum e&longs;t; certè
erit percu&longs;&longs;io ad grauitationem, vt numerus in&longs;tantium motus ad vnita
tem, v.g. grauitatio &longs;it vt 4.&longs;it&qacute;ue motus eiu&longs;dem corporis per 8. in&longs;tan
tia; percu&longs;&longs;io erit ad grauitationem, vt 32. ad 4.vel vt 8.ad 1.quæ om
nia con&longs;tant ex dictis.
dari po&longs;&longs;et grauitatio ip&longs;i æqualisv.g. &longs;it percu&longs;&longs;io dati corporis cadentis
per 8.in&longs;tantia, eius percu&longs;&longs;io e&longs;t octupla grauitationis eiu&longs;dem per Th.
56. igitur &longs;i detur grauitatio octupla huius, erit æqualis datæ percu&longs;
&longs;ioni; dabitur autem grauitatio octupla, &longs;i detur corpus eiu&longs;dem mate
riæ octuplò grauius, vt con&longs;tat.
ma vis e&longs;t, quæ con&longs;equentibus in&longs;tantibus augetur, hinc licèt corpus
breui tempore quis &longs;u&longs;tineat, paulò po&longs;t tamen ponderi cedit, ratio e&longs;t
clara ex dictis.
Ob&longs;eruabis primò numerum in&longs;tantium non po&longs;&longs;e à quoquam &longs;en&longs;u
percipi, nec in calculos vocari, vt patet; vnde Theoremata non po&longs;&longs;unt
ad praxim reduci defectu huius cognitionis; quam &longs;upra adhibui hypo
the&longs;eos loco.
Secundò non pote&longs;t ad amu&longs;&longs;im tempus cum tempore componi ad
æqualitatem, vel aliam datam rationem; licèt enim vnum tempus &longs;en&longs;i
bile haberet mille in&longs;tantia &longs;upra aliud; illa tamen inæqualitas &longs;en&longs;u
minimè perciperetur; idem dico de aliis rationibus, in quo, ni fallor,
maximè peccant, qui temporum æqualitatem perfectam ob&longs;eruari po&longs;&longs;e
contendunt.
Tertiò, idem dico de percu&longs;&longs;ionum ratione;
quippe non pote&longs;t &longs;en&longs;u
percipi inæqualitas duarum percu&longs;&longs;ionum, licèt vires vnius præualeant
mille punctis &longs;eu gradibus in&longs;en&longs;ibilibus; quippe non pote&longs;t di&longs;tingui
ab alia ni&longs;i vel ex &longs;patio; atqui di&longs;cerni non pote&longs;t, an vnum &longs;patium
&longs;uperet aliud mille punctis; vel ex &longs;ono; atqui &longs;onus pote&longs;t diuidi in in
finitos ferè gradus &longs;en&longs;u minimè perceptibiles; igitur nulla hypothe&longs;is
in his experimentis &longs;tatui pote&longs;t, quibus æqualitas vel temporum, vel
&longs;patiorum cogno&longs;ci dicatur; nec dicas aliquot in&longs;tantia parùm di&longs;eri
minis importare, nam cùm &longs;ingulis in&longs;tantibus fiat æqualis impetus ac
ce&longs;&longs;io, mille in&longs;tantia reddunt percu&longs;&longs;ionem millecuplam grauitationis;
hinc certum e&longs;t ex numero in&longs;tantium cognito cogno&longs;ci tantùm po&longs;&longs;e
numerum punctorum, & vici&longs;&longs;im; at certè neuter &longs;en&longs;u percipi pote&longs;t; ne
que tanti e&longs;t hoc &longs;cire.
in&longs;tantis; certè vix modicum &longs;patium post multos annos decurreret
&longs;uppo
namus enim quod plures habent, licèt accuratè experimento &longs;ubii
ci non po&longs;&longs;it, &longs;cilicet vno &longs;ecundo minuto temporis decurri à corpore
graui deor&longs;um 12. pedes &longs;patij; in &longs;ecundo minuto &longs;upponamus e&longs;&longs;e
mille in&longs;tantia, quamuis infinita penè contineat; &longs;itque in primo in
&longs;tanti motus vnus gradus impetus; &longs;ic enim vocetur illa pars impetus, que
producitur primo in&longs;tanti; certè po&longs;t mille in&longs;tantia motus, erunt mille
gradus impetus; iam vcrò &longs;i accipiatur &longs;ubduplum maximæ & minimæ
velocitatis; id e&longs;t vnius gradus, & mille graduum, &longs;cilicet 500. 1/2 tri
buaturque motui æquabili; haud dubiè vno fecundo minuto percur
rentur 12. pedes &longs;patij per Th. 46. Igitur &longs;i cum velocitate vt 500, 1/2
percurrentur 12. pedes 1.minuto, cum velocitate vt 1. percurrentur
12. pedes 500.&longs;ecundis minutis, &; 30. tertiis; &longs;i verò accipiantur plura
in&longs;tantia, v.g. 1000000.in&longs;tantia, percurrentur 12. pedes 500000. &longs;e
cundis minutis; &longs;i verò 1000000000000. percurremur 500000000000.
&longs;ecundis, id e&longs;t 8333333333. minutis, id e&longs;t 138888888. horis
gè adhuc plura in vno minuto continentur in&longs;tantia.
stantis, duplum ferè &longs;patium æquali tempore conficeret illius quod conficit
motu accelerato, duplum inquam ferè &longs;cilicet paulò minùsquia conficit
idem motu æquabili; cuius velocitas e&longs;t &longs;ubdupla maximæ & minimæ;
&longs;ed minima velocitas primi in&longs;tantis pro nihilo reputatur; igitur acci
piatur tantùm &longs;ubduplum maximæ, igitur cum velocitate æquali maxi
mæ, eodem tempore duplum &longs;patium percurretur; igitur in vno minuto
&longs;ecundo, v.g. 24. pedes; igitur in vno minuto primo eodem motu æqua
bili 1440. pedes percurrentur; igitur in vna hora 86400. pedes; hinc
non e&longs;t quod aliqui adeo mirentur, &longs;eu potiùs reiiciant hanc motus
accelerationem quod ex ea tùm tardi&longs;&longs;imus motus, tùm veloci&longs;&longs;imus
con&longs;equatur.
dusquia tot &longs;unt huius propagationis gradus, quot &longs;unt in&longs;tantia,
quibus durat hic motus, cum &longs;ingulis in&longs;tantibus noua fiat impetus ac
ce&longs;&longs;io, &longs;ed non &longs;unt infinita in&longs;tantia, vt demon&longs;trabimus in Metaphy
&longs;ica; prætereà licèt e&longs;&longs;ent infinita in&longs;tantia, non fieret adhuc per omnes
tarditatis gradus hæc propagatio; quia daretur aliquis gradus tarditatis,
quem non comprehenderet hæc graduum &longs;eries; nam incipit moueri
tardiùs in plano inclinato quàm in libero medio rectà deor&longs;um, vt con
&longs;tat, & in medio den&longs;o quàm in raro v.g. in aqua quàm in aëre; igitur
hic tarditatis gradus, quo incipit moueri in plano tantillùm inclinato,
non continetur inter illos, quibus mouetur rectà deor&longs;um.
Hinc duplici nomine reiice Galilæum qui hoc a&longs;&longs;erit.
Primò, quia
fru&longs;trà ponit infinita in&longs;tantia &longs;ine nece&longs;&longs;itate; &longs;ecundò, quia ratio, quam
habet, non conuincit; vocat enim quietem tarditatem infinitam;
à qua
dum recedit mobile, haud dubiè per omnes tarditatis gradus propagari
pote&longs;t eius motus; &longs;ed contrà primò, nam reuerà quies non e&longs;t tarditas,
quæ motui tantùm ine&longs;&longs;e pote&longs;t. Secundò, quia tàm ex quiete &longs;equi po
te&longs;t immediatè velox motus, quàm tardus, vt patet in proiectis. Tertiò,
quia motus incipit; igitur per aliquid &longs;ui, igitur ille primus motus à
quiete infinitè non di&longs;tat; denique rationes &longs;uprà propo&longs;itæ rem i&longs;tam
euincunt.
Ob&longs;eruabis con&longs;ideratum e&longs;&longs;e hactenus hunc motum nulla habita
ratione re&longs;i&longs;tentiæ medij, quæ haud dubiè hanc propo&longs;itionem motus
accelerati tantillùm impedit, &longs;ed de re&longs;i&longs;tentià medij agemus infrà.
Ex dictis facilè reiicies primò &longs;ententiam illorum, qui negant mo-
&longs;en&longs;ibile experimentum omninò conuincere pote&longs;t.
Secundò reiicies illos, qui volunt accelerationem motus e&longs;&longs;e, vel à vi
magnetica, quâ terra trahit ad &longs;e omnia grauia; vel ab alia vi occulta,
quâ cœlum pellit deor&longs;um; vel à cœle&longs;ti illa, imò potiùs fabulosâ mate
riâ; vel demum ab ip&longs;a vi &longs;ympathicâ, quâ corpus &longs;uo centro propiùs
factum totas &longs;uas vires exerit, vt ei &longs;e conjungat; quæ omnia gratis di
cuntur, & ex dictis plu&longs;quam efficaciter refelli po&longs;&longs;unt, ne fru&longs;trà tempus
in iis iterum refellendis teramus.
Tertiò reiicies, qui volunt motum accelerari ex aëris à tergo impel
lentis appul&longs;u, quod ridiculum e&longs;t: licèt enim Ari&longs;toteles videatur illud
&longs;en&longs;i&longs;&longs;e de projectis, quod examinabimus &longs;uo loco; nunquam tamen hoc
dixit de motu naturali; quin potiùs antiquorum fuit omnium hic &longs;en
&longs;us, fieri hinc
dictum illud vulgare, nihil porrò intelligi pote&longs;t
nomine virium, ni&longs;i id, ex quo maior ictus, &longs;eu percu&longs;&longs;io &longs;equitur; illud
autem e&longs;&longs;e impetum con&longs;tat.
Quartò ex his &longs;ententia Ari&longs;totelica de motu accelerato optimè vin
dicatur; quòd &longs;cilicet grauia &longs;ub finem &longs;ui motus velociùs &longs;erantur ver
sùs centrum; quod ex dictis, & &longs;implici&longs;&longs;imis, certi&longs;&longs;imi&longs;que principiis
demon&longs;tratum fuit.
Quintò reiicies etiam illorum &longs;ententiam, qui hanc accelerationem
tribuunt vel medio minùs re&longs;i&longs;tenti, vel grauitatis augmento, vel impe
tui violento priùs impre&longs;&longs;o dum corpus graue attollitur, quod meo iudi
cio ridiculum e&longs;t; qua&longs;i verò fru&longs;tum rupis deci&longs;um, deor&longs;umque ruens
impetum violentum aliquando habuerit.
Sextò reiicies illorum &longs;ententiam, qui volunt accelerationem motus
naturalis ita fieri, vt &longs;patia temporibus æqualibus acqui&longs;ita &longs;equantur &longs;e
riem numerorum imparium 1.3.5.7.9.11.13. &c. & &longs;patia &longs;int vt
quadrata temporum v. g. &longs;i primo in&longs;tanti acquiritur 1.&longs;patium: &longs;ecundo
acquiruntur 3. tertio 5. quarto 7. &c. fique vno in&longs;tanti acquiritur 1.
&longs;patium, duobus acquiruntur 4. tribus 9. quatuor 16. atque ita deinceps
per quadrata, quæ omnia ex dictis fal&longs;a e&longs;&longs;e con&longs;tat; quippe &longs;i æqualibus
temporibus acquiruntur æqualia velocitatis momenta; igitur &longs;i primo
in&longs;tanti e&longs;t 1.gradus, &longs;ecundo erunt 2. igitur &longs;ecundo tempore cum duo
bus gradibus velocitatis vel impetus percurrentur duo tantùm &longs;patia, &longs;i
primò in&longs;tanti æquali cum vno gradu percurritur vnus, &longs;ed de his fusè
infrà.
Septimò reiicies etiam aliquos recentiores, qui volunt fieri hanc pro
gre&longs;&longs;ionem &longs;patiorum æqualibus temporibus re&longs;pondentium &longs;ecundùm
progre&longs;&longs;ionem Geometricam, duplam, &longs;cilicet iuxta hos numeros 1. 2. 4.
8. 16. 32. &c. quod etiam ex eadem ratione facilè confutatur:
reiicies
etiam alium recentiorem, qui vult hanc progre&longs;&longs;ionem &longs;umi ex linea
proportionaliter &longs;ectâ, id e&longs;t in mediam & extremam rationem; &longs;ed de
his omnibus in di&longs;&longs;ertatione &longs;equenti fusè di&longs;putamus; quippe rem hanc
tanti e&longs;&longs;e putamus, vt nihil omittendum &longs;it, quod ad eius pleni&longs;&longs;imam
confirmationem pertineat.
DVæ &longs;unt poti&longs;&longs;imùm in hac materia celebres &longs;ententiæ;
Prima e&longs;t
Galilei, & ferè omnium recentiorum, qui po&longs;t Galileum de motu
&longs;crip&longs;erunt; inter quos, ne omittam Genuen&longs;em Patricium, Balianum;
Doctus Mer&longs;ennus, & eruditus Ga&longs;&longs;endus primum locum obtinent;
quorum ille hanc &longs;ententiam multis in locis, &longs;cilicet in &longs;uis quæ&longs;tioni
bus Phy&longs;icis, in &longs;ua Galilei ver&longs;ione, in harmonia vniuer&longs;ali, & demum
in &longs;ua Bali&longs;tica pa&longs;&longs;im, tùm fusè proponit, & explicat, tùm etiam &longs;uis ra
tionibus confirmat; Galileus verò illam habet tùm in gemino &longs;y&longs;tema
te, tùm in dialogo tertio de motu locali.
Secunda &longs;ententia no&longs;tra e&longs;t, de qua non &longs;emel di&longs;putandum fuit à
Magi&longs;tro, tùm verbis tùm etiam litteris &longs;criptis; & ne quid fortè di&longs;&longs;imu
lem, illa e&longs;t &longs;ententia quam anonimo Philo&longs;ophe (quem non &longs;ine laude
appellat idem Mer&longs;ennus) tribuit. prop.18.&longs;uæ Bali&longs;ticæ &longs;ub finem; illa
e&longs;t inquam &longs;ententia, quam hactenus meo iudicio &longs;atis luculenter de
mon&longs;trauimus.
Sunt tres aliæ &longs;ententiæ, quæ ab eodem Mer&longs;enno referuntur; prima
e&longs;t quæ progre&longs;&longs;ionem &longs;patiorum
nuum ver&longs;orum, centro quadrantis po&longs;ito in centro terræ, & altero ex
tremo &longs;inus totius in eo punctò, in quo incipit motus. Secunda e&longs;t quo
rumdam, qui volunt progre&longs;&longs;ionem &longs;patiorum, quæ &longs;ingulis temporibus
re&longs;pondent, e&longs;&longs;e in progre&longs;&longs;ione geometrica dupla iuxta hos numeros,
1.2.4.8.32. Tertia e&longs;t alicuius, qui voluit e&longs;&longs;e iuxta proportionem lineæ
&longs;ectæ in mediam, & extremam rationem.
Tres vltimæ &longs;ententiæ nullo pror&longs;us nituntur fundamento; igitur vel
inde maximè confutantur, quòd gratis &longs;ine vllo pror&longs;us vel rationis vel
experimenti momento excogitatæ &longs;int. Igitur in hac di&longs;&longs;ertatione duæ
tantùm primæ di&longs;cutiendæ &longs;unt Sententiæ Galilei &longs;chema hic habes
in linea AF, in qua a&longs;&longs;umitur AB, &longs;patium &longs;cilicet, quod dato tempore & &longs;ecundo tempore æquali BC, quæ
tripla e&longs;t AB, tertio CD quintupla quarto DE &longs;eptupla, quinto EF
nonecupla; vides primò &longs;eriem numerorum imparium 1. 3. 5. 7. 9.atque
ita deinceps. Secundò vides &longs;patia e&longs;&longs;e in ratione duplicata temporum,
hoc e&longs;t vt temporum quadrata. v.g. &longs;i accipiatur &longs;patium AB primo tem
pore peractum, & &longs;patium AC duobus temporibus confectum: ratio hu
ius ad illud e&longs;t vt 4.ad 1.id e&longs;t vt quadratum 2.ad quadratum 1. &longs;imiliter,
&longs;i accipiatur &longs;patium AD confectum tribus temporibus, erit 9.id e&longs;t qua
dratum 3, &longs;patium AE confectum 4.temporibus erit 16.id e&longs;t quadratum
4. & AF 25. quadratum 5.
Hæc &longs;ententia ingeniosè à Galileo excogitata ex duplici capite à &longs;uis
auctoribus confirmatur; primò experientiâ, &longs;ecundò ratione. Experien
tia tribus poti&longs;&longs;imum experimentis fulcitur; primum e&longs;t in motu deor
&longs;um per lineam perpendicularem. v. g. in linea AF;
nam reuerà multi
&longs;unt, iique graui&longs;&longs;imi auctores in rebus tùm philo&longs;ophicis, tùm mathe
maticis ver&longs;ati&longs;&longs;imi, qui &longs;æpiùs &longs;en&longs;u ip&longs;o probarunt, repetitis v&longs;que ad
nau&longs;eam experimentis, tempore vnius &longs;ecundi minuti corpus graue in
libero aëre 12. pedes &longs;patij motu naturali deor&longs;um percurrere; in 2.ve
rò &longs;ecundis 48. in 3.&longs;ecundis 108.&longs;ed &longs;patia i&longs;ta &longs;unt vt temporum qua
drata, vt con&longs;tat.
Secundum experimentum e&longs;t in plano inclinato, in quo corpus graue
de&longs;cendit iuxta prædictam progre&longs;&longs;ionem, quod expre&longs;&longs;is verbis te&longs;tatur
Galileus à &longs;e fui&longs;&longs;e probatum &longs;æpiùs, nec vnquam à vero ne tantillùm
quidem aberra&longs;&longs;e. &longs;ed in perpendiculari deor&longs;um eadem proportione
cre&longs;cit motus, quâ in plano inclinato; licèt in plano inclinato tardior &longs;it
motus, vt demon&longs;trabimus aliàs.
Tertium experimentum petitur ex funependulis;
in quibus &longs;æpiùs
ob&longs;eruatum e&longs;t longitudinem funis, & con&longs;equenter arcum quadrantis
longioris funependuli e&longs;&longs;e ad longitudinem, &longs;eu quadrantem alterius
breuioris, vt quadratum temporis, quo perficitur vibratio maioris ad
quadratum temporis, quo perficitur vibratio minoris.v.g.&longs;it longitudo
funependuli maioris, CG minoris verò &longs;ubquadrupla CF; eleuetur vter
que funis, cui pondus æquale &longs;it appen&longs;um v&longs;que ad horizontalem
CDE & alterum ex D; alterum verò ex E demi&longs;&longs;um cadat deor&longs;um; haud
dubiè funependulum CE duplum temporis collocabit in decurrendo
quadrante EG, & funependulum ED &longs;ubduplum. v. g. &longs;i CD conficit
&longs;uam vibrationem DF vno &longs;ecundo, EG conficiet &longs;uam EG duobus, vt
centies ob&longs;eruatum e&longs;t; &longs;ed EG e&longs;t quadruplus DF, vt patet; igitur EG
& DF &longs;unt vt quadrata temporum, quibus percurritur EG & DF &longs;ed vt
de&longs;cendit graue per DF & EG, ita de&longs;cendit per CF & CG, quippe
DF & EG habent rationem plani inclinati deor&longs;um.
Adde quod, vt &longs;e habet tempus, quo de&longs;cendit per totum quadrantem
DF, ad tempus, quo de&longs;cendit per totum quadrantem EG. &longs;ic &longs;e habet
tempus, quo de&longs;cendit per arcum DL &longs;ubduplum DF ad tempus, quo
de&longs;cendit per arcum EI &longs;ubduplum EG; item tempus, quo de&longs;cendit
EK &longs;ubquadruplum EG; denique vt tempus, quo per minimum ar
cum quadrantis DF, ad tempus, quo de&longs;cendit per alium proportiona
lem, &longs;cilicet quadruplum in quadrante EG; atqui tam parui arcus po&longs;
&longs;unt a&longs;&longs;umi, vt &longs;int ad in&longs;tar lineæ rectæ deor&longs;um tangentis &longs;cilicet in D
& in E; igitur in his rectis de&longs;cendunt grauia iuxta progre&longs;&longs;ionem præ
dictam; id e&longs;t, cum arcus minimus a&longs;&longs;umptus ab E, qui æquiualet rectæ,
&longs;it quadruplus arcus minimi a&longs;&longs;umpti à puncto D, tempus, quo percurri
tur ille primus, e&longs;t ad tempus, quo percurritur hic &longs;ubquadruplus, vt tem
pus, quo percurritur EG ad tempus, quo percurritur DF vt dictum e&longs;t;
&longs;ed tempus, quo percurritur EG e&longs;t duplum illius, quo percurritur DF;
igitur tempus, quo percurritur minimus arcus a&longs;&longs;umptus ab E, & qui e&longs;t
ad in&longs;tar rectæ, e&longs;t duplum temporis quo percurritur minimus arcus a&longs;
&longs;umptus à puncto D &longs;ubquadruplus prioris, & qui e&longs;t etiam ad in&longs;tar re
ctæ; igitur &longs;patia &longs;unt vt temporum quadrata.
Quod autem tempus, quo percurritur EG &longs;it duplum illius, quo per
curritur DF, patet experientiâ; nam &longs;i numerentur ducentæ vibrationes
funependuli CD; eodem tempore numerabuntur centum vibrationes
maioris CE; igitur vibrationum minoris numerus e&longs;t duplus numeri vi
brationum maioris, dum &longs;imul vibrantur; igitur eo tempore, quo fiunt
100.maioris, fient 200. minoris; nam omnes vibrationes eiu&longs;dem fune
penduli &longs;unt æquò diuturnæ, licèt fiant per arcus inæquales eiu&longs;dem.
quadrantis, vt &longs;æpè ob&longs;eruatum e&longs;t. In his tribus poti&longs;&longs;imum experimen
tis fundatur hæc hypothe&longs;is Galilei, quæ nec clariùs meo. iudicio, nec
&longs;inceriùs exponi po&longs;&longs;unt.
Antequam rationes, quæ pro hac &longs;ententia facere videntur, propona
mus, refellamu&longs;que; o&longs;tendo primò quomodo cum his experimentis
&longs;tare po&longs;&longs;it no&longs;tra hypothe&longs;is; igitur ex iis hypothe&longs;is Galilei rectè de
duci non pote&longs;t: quippe hæc e&longs;t certi&longs;&longs;ima regula, quam nemo Philo&longs;o
phus negare au&longs;it: Quotie&longs;cumque aliquod experimentum tale e&longs;t, vt
cum eo &longs;tare po&longs;&longs;int contrariæ hypothe&longs;es; ex eo certè neutra deduci po
te&longs;t; igitur ex propo&longs;itis experimentis &longs;uam hypothe&longs;im Galileus non
legitimè deducit, quod vt clari&longs;&longs;imè o&longs;tendam.
Suppono, quando dicitur &longs;ecundum &longs;patium e&longs;&longs;e triplum primi &longs;up
po&longs;itis æqualibus temporibus, non ita Geometricè, certaque, & acuratâ
a&longs;&longs;ertione hoc dici; quin vel aliqua puncta in &longs;patiis, vel in&longs;tantia in
temporibus de&longs;int, vel &longs;uper&longs;int; &longs;i enim quis diceret &longs;patium e&longs;&longs;e tri
plum primi minus 100000. punctis, vel &longs;ecundum tempus e&longs;&longs;e maius
primo 100000. in&longs;tantibus; quis hanc, vel &longs;patij, vel temporis differen
tiam &longs;en&longs;u percipiat? cum tamen experimentum omne phy&longs;icum &longs;en&longs;ui
&longs;ube&longs;&longs;e po&longs;&longs;it; nec e&longs;t quod aliquis dicat hoc idem toties ob&longs;eruatum
e&longs;&longs;e, tam multis locis temporibus, totque ac tantis etiam te&longs;tibus, vt mi
nimè fraus aliqua, vel error &longs;ubrepere potuerit; nam cum parua &longs;it, &
in&longs;en&longs;ibilis tùm &longs;patiorum, tùm temporum differentia, maius vel minus
æquali tempus, pro æquali, maius.vel minus triplò &longs;patium pro triplo
Adde quod non de&longs;unt viri graui&longs;&longs;imi qui dicant &longs;e vix ob&longs;eruare po
tui&longs;&longs;e hanc &longs;patiorum progre&longs;&longs;ionem; plures appellare po&longs;&longs;em;
vnus
Ga&longs;&longs;endus e&longs;t in&longs;tar omnium; qui &longs;anè in ob&longs;eruando fuit acurati&longs;&longs;imus,
qui literis &longs;criptis, quas ego vidi, expre&longs;&longs;is verbis a&longs;&longs;erit progre&longs;&longs;ionem
hanc non e&longs;&longs;e omninò iuxta hos numeros 1.3.5.7. &longs;ed &longs;ingulis addendas
e&longs;&longs;e &longs;uas minutias, quas ip&longs;e habet; &longs;ed ego omitto, quia etiam &longs;ua incer
titudine laborant; igitur nullo experimento ad amu&longs;&longs;im concludes,
vel
portionem: Equidem &longs;en&longs;u percipio practicam hanc e&longs;&longs;e maiorem pede;
at tot lineis vel
neret: Sed efficaciter, meo iudicio, hanc Galilei hypothe&longs;im refello;
&longs;int
2.partes temporis æquales AE, EF, eæque &longs;en&longs;ibiles; nec enim aliæ a&longs;
&longs;umi po&longs;&longs;unt; &longs;intque minimæ omnium &longs;en&longs;ibilium;
haud dubiè con&longs;tant
&longs;ingulæ infinitis ferè aliis in&longs;en&longs;ibilibus, vt patet; igitur &longs;ic ratiocinatur
Galileus; in prima parte temporis AE corpus graue percurrit &longs;patium
GH, & in &longs;ecunda æquali EF percurrit &longs;patium HL triplum prioris;
igitur &longs;patia &longs;unt vt quadrata temporum, rectè; &longs;ed antequam vlterius
progrediar; Quæro vel à Galileo, vel à quolibet alto, vtrum &longs;patium
HL &longs;it omnino triplum? & &longs;i aliquis contenderet dee&longs;&longs;e (1/1000000) GH
vtrum experimento præ&longs;enti conuinci po&longs;&longs;it? nemo, vt puto, id a&longs;&longs;erere
au&longs;it; hoc po&longs;ito, a&longs;&longs;umptaque progre&longs;&longs;ione arithmetica
tentia in &longs;patiis ad&longs;truit; &longs;i prima parte temporis AE percurratur &longs;pa
tium GH, &longs;ecunda EF. percurretur tantùm HK duplum GH; igitur
minus e&longs;t hoc &longs;patium vero &longs;patio 1/4. &longs;cilicet tota KL; res pror&longs;us de
mon&longs;trata e&longs;&longs;et, &longs;i termini proportionis vnius e&longs;&longs;ent tantùm 2. id e&longs;t, &longs;i
progre&longs;&longs;io fieret in partibus temporis &longs;en&longs;ibilibus; at po&longs;ito quod &longs;int
plures termini, vt reuerâ &longs;unt; nam in totidem terminis fit progre&longs;&longs;io, in
quibus fit augmentum impetus, vel accelerationis acce&longs;&longs;io; atqui hæc
fit in &longs;ingulis in&longs;tantibus, licèt finitis, igitur & progre&longs;&longs;io; Quare duæ
partes temporis AE, EF diuidantur in 4. æquales AD; certè in duabus
primis percurretur &longs;patium. VQ æquale GH; igitur duabus vltimis per
curretur QK, quæ &longs;it ad QV vt 7. ad 3. nam prima parte percurritur 1.
&longs;patium. &longs;ecunda 2. igitur QV continet tria &longs;patia;
tertia verò 3. quarta
4.ergo hæ duæ vltimæ 7. &longs;ed QM e&longs;t dupla QV; igitur continet 6. igi
tur MK e&longs;t 1/3 VQ, vel KL; igitur KM e&longs;t (1/12) GL;
igitur 12. L (1/10), vel
1/6, igitur VK e&longs;t ad GL vt 10.ad 12. igitur totum &longs;patium VK e&longs;t mi
nus vero 1/6. Præterea 2. partes temporis AE EF diuidantur in 8. partes
æquales AE; haud dubiè 4. primis percurretur &longs;patium XT æquale
GH, quod debet diuidi in 10. &longs;patia; nam 4. terminis, &longs;eu temporibus
re&longs;pondent &longs;patia 10. quibus æqualia &longs;unt 40. in teta GL, cuius XT e&longs;t
(1/14), &longs;ed &longs;i in 4.primis acquiruntur 10. 4. vltimis EF acquiruntur 26.&longs;cili
cet T 5; igitur tota X 5. e&longs;t 6. igitur e&longs;t ad GL vt 36. ad 40. &longs;eu 9. ad
10. igitur X 5. e&longs;t &longs;patium minus vero (1/10).
Præterea diuidatur tempus AF in 16. partes æquales AB;
haud dubiè
tiola 36, quæ re&longs;pondent 8. temporibus, &longs;eu terminis huius progre&longs;&longs;io
nis, quibus æqualia &longs;unt 144. in GL, cuius YS e&longs;t 1/4, &longs;ed &longs;i in 8. primis
acquiruntur 36. in 8. vltimis acquirentur 100. igitur S 6. e&longs;t 100. igitur
Y6. e&longs;t 136. igitur e&longs;t ad GL vt 136. ad 144.&longs;eu 17.ad 18.igitur Y6.e&longs;t
&longs;patium totale minus vero (1/18).
Deinde diuidatur adhuc tempus AF in partes 32. æquales, 16. pri
mis acquiritur ZR æquale GH, quod debet diuidi in &longs;patiola 136.quæ
re&longs;pondent 16. temporibus quibus æqualia &longs;unt 544. in tota GL, cuius
ZR e&longs;t 1/4 &longs;ed &longs;i in 16. primis temporibus acquiruntur 136. in vltimis
16. acquiruntur 392. igitur R 7. e&longs;t 392. & ZR 136. igitur Z 7.528.
igitur Z 7. e&longs;t ad GL, vt 528. ad 544. &longs;eu vt 33. ad 34. igitur Z 7 e&longs;t
&longs;patium minus verò (1/34)
Denique &longs;i diuidatur tempus AF in partes 64.&longs;patium acqui&longs;itum erit
minus vero, a&longs;&longs;umpto &longs;cilicet tota HL (1/66), &longs;i diuidatur in 128. partes, erit
minus (1/130) &longs;i diuidatur in 256. partes, erit minus (1/258) &longs;ed temporis par
tes 2.AE. EF minimè &longs;en&longs;ibilium diuidi po&longs;&longs;unt in infinita ferè in&longs;tan
tia; &longs;int tantùm ex.g. 1000000. igitur &longs;patium tunc acqui&longs;itum erit mi
nus &longs;uppo&longs;ito vero HL (1/1000002), quæ &longs;i de&longs;it tantùm &longs;patio KL vt &longs;it 1/4
totius GL, quis hoc di&longs;cernat? igitur etiam &longs;uppo&longs;ita progre&longs;&longs;ione arith
metica, quæ fiat in finitis in&longs;tantibus; &longs;i ob&longs;eruetur acurati&longs;&longs;imè &longs;patium,
quod percurritur in vna parte temporis &longs;en&longs;ibili v. g. &longs;patium GH in
parte temporis AE; &longs;patium, quod acquiretur in tempore &longs;ecundo æqua
li tàm propè accedet ad &longs;patium HL, id e&longs;t ad triplum prioris GH, vt
nullus mortalium di&longs;cernere po&longs;&longs;it; igitur cum hoc experimento tàm
pote&longs;t &longs;tare no&longs;tra hypothe&longs;is, quàm alia Galilei, igitur neutra ex eo tan
tùm euinci pote&longs;t.
Hinc obiter ob&longs;erua progre&longs;&longs;ionem differentiarum;
quippe &longs;i &longs;int
tantùm 2. partes temporis, differentia e&longs;t 1/4; &longs;i 4.1/6 &longs;i 8. (1/10); &longs;i 16.(1/18); &longs;i 32.
(1/34); &longs;i 64.(1/66) nam primò denominator fractionis &longs;uperat tantùm binario
numerum partium temporis; &longs;ecundò differentiæ denominatorum &longs;unt
in progre&longs;&longs;ione geometrica dupla numerorum 2. 4. 8. 16. 32. 64.
128. &c.
Eodem modo &longs;oluendum e&longs;t &longs;ecundum experimentum rotati globi in
plano decliui; præ&longs;ertim cum globus ab incur&longs;u a&longs;periorum partium
tùm globi, tùm plani &longs;altuatim de&longs;cendat; quod dubium e&longs;&longs;e non pote&longs;t,
& quò decliuius erit, faciliùs re&longs;iliet a plano, vt patet; &longs;ed de motu in
planis inclinatis fusè agemus infrà libro integro.
Quod &longs;pectat ad tertium experimentum;
multa in eo &longs;upponuntur
vel fal&longs;a, vel &longs;altem dubia: vel ea quæ cum no&longs;tra hypothe&longs;i optimè con
ueniant. Primum e&longs;t, quando dicuntur omnes vibrationes eiu&longs;dem fune
penduli, &longs;iue maiores, &longs;iue minores e&longs;&longs;e æquediuturnæ, quod manife&longs;tis
experimentis repugnat; quippe vibratio maior plùs temporis;
minor ve
rò minùs in &longs;uo de&longs;cen&longs;u ponit; dimittantur enim duo funependula æ
qualia; alterum quidem ex altitudine 90.graduum, alterum ex altitudine
ita vt &longs;imul vibrationes &longs;uas incipiant;
numerentur
vibrationes vtriu&longs;que, vbi 100. è minoribus numerat&ecedil; fuerint, numera
buntur circiter 97. è maioribus, quod &longs;æpiùs ob&longs;eruaui te&longs;tibus etiam
adhibitis; hoc ip&longs;um etiam ob&longs;eruarunt alij;
atque adeo ip&longs;e P.Mer&longs;en
nus, qui L. 2. &longs;uæ ver&longs;ionis, Ar.17. Galileum arguit parùm acurati &longs;tu
dij in his ob&longs;eruationibus adhibiti: rationem huius effectus in libro de
funependulis explicabimus; immò &longs;i omnes vibratìones maiores primæ
vibrationi 90. grad. e&longs;&longs;ent æquales, & aliæ minores alterius funependu
li &longs;en&longs;un, vt &longs;it, minuerentur; vix 90. maiores numerare po&longs;&longs;es, iam enu
meratis 100. ex minoribus; &longs;ed de his omnibus &longs;uo loco;
in vna tamen
vel altera vibratione vix aliquod di&longs;crimen ob&longs;eruatur; quod tamen ob
&longs;eruari facilè po&longs;&longs;et in maioribus funependulis.
Secundum, quod &longs;upponitur, e&longs;t quod longitudines funependulorum
&longs;int pror&longs;us, vt quadrata temporum, quibus vibrationes &longs;ingulorum
fiunt, v.g. funependulum longitudinis 4. pedum facere vnam vibratio
nem eo tempore, quo funependulum longitudinis vnius pedis facit duas;
quod primò in multis vibrationibus non tàm accuratè ob&longs;eruatur;
dò
&longs;ingulis vibrationibus e&longs;&longs;e tantùm di&longs;crimen; quod etiam in multis &longs;en&longs;i
bile non e&longs;t; &longs;i enim di&longs;crimen primarum vibrationem v.g.&longs;it (1/100000000)
certè vltimarum adhuc in&longs;en&longs;ibile erit.
Tertium &longs;uppo&longs;itum fuit, minimum arcum minoris quadrantis a&longs;&longs;um
ptum, & alium minoris quadrantis e&longs;&longs;e ad in&longs;tar perpendicularium; cùm
tamen diuer&longs;a &longs;it inclinatio minoris, & maioris quadrantis: quippe
principium maioris accedit propiùs ad perpendicularem; facit enim
angulum contingentiæ minorem; alia verò extremitas accedit propiùs
ad horizontalem propter rationem prædictam; hinc illa extremitas ma
ioris, vnde e&longs;t initium motus, planum decliuius facit; altera verò minùs
decliue; &longs;ed hæc fusè pro&longs;equar &longs;uo loco.
Quartum, quod &longs;upponitur e&longs;t, accelerationem motus fieri in qua
drante in ea ratione, in qua fit per plana chordarum inclinata, quod
etiam fal&longs;um e&longs;t; quia in eodem plano inclinato &longs;upponitur eadem
inclinatio; &longs;ecus in quadrante, cuius &longs;ingula puncta nouam faciunt in
clinationem: adde quod quarta pars quadrantis maioris EK non facit
eandem inclinationem, quam totus quadrans minor DF ip&longs;i EK æqua
lis; quamquam hoc ip&longs;i vltrò concedent aduer&longs;arij.
Præterea, &longs;it ita vt &longs;upponitur;
ita vt &longs;en&longs;ibiliter differentia huius
progre&longs;&longs;ionis percipi non po&longs;&longs;it, &longs;intque numeratæ omnes vibrationes
&longs;en&longs;ibiles dati funependuli ex altitudine 90, grad. demi&longs;&longs;i; quæ vix e&longs;&longs;e
po&longs;&longs;unt 1800; &longs;int autem plures &longs;cilicet 2000. dicis confectas e&longs;&longs;e 2000
minoris funependuli eo tempore, quo 1000. tantùm in quadruplo fune
pendulo numerantur; annuo quidem, &longs;i res tantùm &longs;en&longs;ibiliter con&longs;ide
retur; &longs;in verò &longs;ecùs, id pernego; &longs;ed dico dee&longs;&longs;e v. g. 1000000. puncta
&longs;patij, quæ di&longs;cerni non po&longs;&longs;unt, ita vt primæ vibrationi 1000. pun
cta &longs;ecundæ, 2000. tertiæ 3000. &c. vltimæ verò, &longs;eu mille&longs;imæ
cultatem, quàm in motu perpendiculari, de quo &longs;uprà; etiam conce&longs;&longs;is
vltrò omnibus experimétis propo&longs;itis. Igitur &longs;uppo&longs;itâ progre&longs;&longs;ione &longs;pa
tiorum arithmetica in in&longs;tantibus, tàm propè accedit ad aliam, quàm
Galileus ponit, &longs;iue in perpendiculari deor&longs;um, &longs;iue in quadrante fune
penduli; a&longs;&longs;umptis &longs;cilicet partibus temporis &longs;en&longs;ibilibus, vt differentia
di&longs;cernit non po&longs;&longs;it; immò nec duplum differentiæ, nec centuplum, nec
millecuplum; &longs;ed de his &longs;atis quæ ex dictis &longs;uprà facilè intelligi po&longs;&longs;unt:
quare veniemus iam ad rationes.
Prima ratio, quam affert Galileus e&longs;t;
quia cum natura in &longs;uis opera
tionibus adhibeat &longs;implici&longs;&longs;ima media; & cum acceleratio motus natu
ralis non po&longs;&longs;it fieri iuxta faciliorem, vel &longs;impliciorem progre&longs;&longs;ionem,
quàm &longs;it-ea quæ fit per quadrata; non e&longs;t dubium, quin iuxta illam pro
gre&longs;&longs;io motus naturaliter accelerati fieri debeat; præ&longs;ertim cùm omni
bus experimentis con&longs;entiat, & in ea omnia phænomena explicari
po&longs;&longs;int.
Re&longs;p. Primò progre&longs;&longs;ionem arithmeticam &longs;implicem iuxta hos nu
meros 1.2.3.4. longè &longs;impliciorem e&longs;&longs;e alia quæ fit iuxta illos 1.3.5.7.vt
nemo non iudicabit. Secundò
omnibus experimentis &longs;eu phænomonis, debet e&longs;&longs;e aliqua ratio, cur ad
hibeatur vna potiùs quàm alia; &longs;ed nulla e&longs;t ratio, cur Galileus adhibeat
&longs;uam, vti videbimus; nos verò ratione demon&longs;tratiuâ probamus no&longs;tram;
igitur no&longs;tra e&longs;t præferenda pro theorica rei veritate; quia verò alia in
temporibus &longs;en&longs;ibilibus proximè ad verum accedit eam adhibendam e&longs;&longs;e
decernemus infrà ad praxim, & communem i&longs;torum motuum men
&longs;uram.
Secunda ratio e&longs;t;
quia, &longs;i accipiatur &longs;ubduplum maximæ, & minimæ
velocitatis; &longs;itque ex his qua&longs;i conflata velocitas motus æquabilis, hoc
motu æquabili æquali tempore pèrcurretur &longs;patium idem, quod antè
motu naturaliter accelerato v.g. &longs;int numeri datæ progre&longs;&longs;ionis 1.3.5.7.
9.11. certè &longs;umma terminorum &longs;eu totum &longs;patium erit 36. accipiatur
&longs;ubduplum primi 1/2 & &longs;exti 5. 1/2 habebitur velocitas vt 6. igitur cum
velocitate vt 6. æquali tempore percurretur &longs;patium 36. quod rectè de
mon&longs;trauit Galileus.
Re&longs;pondeo non minùs no&longs;tram hypothe&longs;im cum hoc ip&longs;o &longs;tare, quàm
&longs;tet hypothe&longs;is Galilei: &longs;int enim 6. in&longs;tantia, & &longs;ingulis &longs;ua tribuantur
&longs;patiola more dicto 1 2 3 4 5 6. &longs;umma &longs;patiorum e&longs;t 21. a&longs;&longs;umatur &longs;ub
duplum velocitatis primi in&longs;tantis 1/2, & &longs;ubduplum &longs;exti in&longs;tantis, &longs;cili
cet 3. conflatum ex vtroque 3 1/3; ducatur in 6.id e&longs;t in numerum termi
norum, vel in&longs;tantium; &longs;umma erit 21. igitur quod tribuit Galileus &longs;uæ
progre&longs;&longs;ioni, etiam no&longs;træ competit.
Tertia ratio petitur ex mathe&longs;i &longs;it enim linea AE diui&longs;a in quatuor
partes æquales, quæ nobis repre&longs;entent 4. partes temporis æquales;
haud dubiè, cùm acquirantur temporibus æqualibus æqualia velocitatis
momenta; haud dubiè, inquam, his 4. temporibus AB, BC, CD, DE, ac-
&longs;it autem BI, men&longs;ura velocitatis,
quam acquirit mobile cadens ex &longs;ua quiete in fine primæ partis tempo
ris AB; certè in fine &longs;ecundæ partis temporis BC acquiret velocitatem,
quæ coniuncta cum priore BI faciet duplam CH, & in fine tertiæ par
tiæ CD triplam DG; denique in fine quartæ DE quadruplam EF;
quip
pe cum in parte BC remaneat tota velocitas B, & acquiratur æqualis;
certè in fine BC e&longs;t velocitas CH dupla illius quæ commen&longs;uratur BI.
&longs;imiliter in parte CD remanebit vtraque, & accedet altera; igitur e&longs;t ve
locitas DG tripla BI, & EF e&longs;t quadrupla: Similiter ita &longs;e ratio habet
cuiu&longs;libet alterius partis inter AB ad aliam alterius partis inter BC, vt
lineæ ductæ parallelæ BICH, &c. igitur cum &longs;patium acqui&longs;itum re&longs;
pondeat exercitio huius velocitatis; &longs;itque in&longs;tanti B vt BI, & in&longs;tanti
C vt CH; certè tempore AB e&longs;t vt triangulum AIB;
nam &longs;patium AIB
e&longs;t collectio omnium linearum, quæ duci po&longs;&longs;unt parallelæ in tempore
AB; idem dico de trapezo CBIH, qui e&longs;t triplus trianguli IBA;
& de
trapezo GDCH, qui e&longs;t quintuplus; igitur triangulum HCA e&longs;t qua
druplum IBA; quia hæc triangula &longs;unt vt quadrata laterum;
igitur &longs;pa
tium acqui&longs;itum temporibus AB, BC, e&longs;t ad &longs;patium acqui&longs;itum tempo
re AB, vt triangulum HCB ad triangulum IBA; igitur vt quadratum
AB ad quadratum AC; igitur vt quadratum temporis AB ad quadra
tum temporis AC; igitur &longs;patia diuer&longs;is temporibus decur&longs;a &longs;unt vt qua
drata temporum, quibus &longs;ingula decurruntur.
Hæc ratio ad &longs;peciem videtur e&longs;&longs;e demon&longs;tratiua, deficit tamen à ve
ra demon&longs;tratione; primo, quia &longs;upponit in&longs;tantia infinita, quæ multi
pa&longs;&longs;im negabunt in tempore; immò aliquis vltrò demon&longs;trare tentaret
non e&longs;&longs;e infinita; itaque ex &longs;uppo&longs;itione quod &longs;int tantùm finita in&longs;tan
tia a&longs;&longs;umantur 4. æqualia AC, CD, DE, EF, certè cum in&longs;tans &longs;it to
rum &longs;imul, velocitatem habet æquabilem &longs;ibi toti re&longs;pondentem; igitur
in&longs;tanti AC re&longs;pondeat velocitas, cuius men&longs;ura &longs;it ABCG; haud du
biè in&longs;tanti CD re&longs;pondebit velocitas CH, &longs;cilicet dupla AB; nam re
manet primus velocitatis gradus acqui&longs;itus primo in&longs;tanti: &longs;ed alter æ
qualis acquiritur; igitur e&longs;t duplus prioris;
igitur re&longs;pondet lineæ DK.
quæ tripla e&longs;t AB, & quarto lineæ FN, quæ e&longs;t quadrupla AB; igitur
cre&longs;cit &longs;patium, vt rectangula CB, DH, EK, FM; &longs;ed hæc cre&longs;cunt iuxta
progre&longs;&longs;ionem numerorum 1.2.3.4. nec aliter res e&longs;&longs;e pote&longs;t ex &longs;uppo&longs;i
tione quod &longs;int in&longs;tantia finita; quod alibi ex profe&longs;&longs;o tractamus:
quippe
illa quæ&longs;tio pertinet ad Metaphy&longs;icam, non verò ad phy&longs;icun; nam vel
&longs;ingula aliquid addunt, vel nihil: aliquid addunt haud dubiè; igitur con
&longs;iderantur tantùm 4. in&longs;tantia prima AC, CD, DE, EF, in &longs;ua &longs;crie; certè
non po&longs;&longs;unt aliam progre&longs;&longs;ionem facere quàm eam, quæ e&longs;t iuxta hos
numeros 1.2.3.4.vnde non fit per triangula &longs;ed per rectangula minima;
igitur linea AF præcedentis figuræ non e&longs;t recta, &longs;ed denticulata, qualis
e&longs;&longs;et ABGHIKLMN, &longs;ed longè minoribus gradibus, &longs;eu denticulis.
Hinc quò rectangula CB, DH, &c. fient maiora in partibus &longs;cilicet tem
poris &longs;en&longs;ibilibus, &longs;eruata &longs;cilicet in illis progre&longs;&longs;ione numerorum 1.2.3. vt &longs;uprà iam totius repetitum fuit:
quippe hæc progre&longs;&longs;io in puris in&longs;tantibus fieri tantùm pote&longs;t, cum &longs;in
gulis in&longs;tantibus noua fiat acce&longs;&longs;io velocitatis, in hoc enim e&longs;t error,
quòd in tota parte temporis AC ponatur æquabilis velocitas, eiu&longs;que
principium A, &longs;it æquale fini C; nam AB, & GH &longs;unt æquales;
cùm ta
men &longs;it minor velocitas in A, quàm in C, ni&longs;i AC &longs;it tantùm vnde
tota velocitas in hypothe&longs;i Galilei acqui&longs;ita in 4.partibus temporis a&longs;
&longs;umptis e&longs;t, vt triangulum AFN; acqui&longs;ita verò in no&longs;tra hypothe&longs;i e&longs;t vt
&longs;umma rectangulorum CB, CI, EK, EN, quæ &longs;umma e&longs;t ad triangulum
AFN, vt 10, ad 8. vel vt 5.ad 4. igitur maior 1/4; nam prima pars tempo
ris addit triangulum ABG, &longs;ecunda GHI. &c.
Si tamen diuidantur i&longs;tæ partes temporis in minores v. g. in 8. tunc
&longs;umma rectangulorum erit tantùm maior 1/8; &longs;i in 16. (1/16) &longs;i in 32. (1/32);
&longs;i in
64.(11/64), cuius &longs;ehema hîc habes; &longs;int enim 3.partes temporis &longs;en&longs;ibiles A
CDFE, & &longs;patium vt triangulum AFN, &longs;patia verò acqui&longs;ita in &longs;ingulis
partibus, vt portiones trianguli prædicti, quæ ip&longs;is re&longs;pondent v. g. ac
qui&longs;itum in prima parte ad acqui&longs;itum in &longs;ecunda tantùm, vt triangu
lum ACG ad trapezum GCDI &c. denique acqui&longs;itum in temporibus
inæqualibus, vt quadrata temporum v. g. acqui&longs;itum in prima parte ad
acqui&longs;itum in duabus, vt triangulum ACG ad triangulum ADI; id e&longs;t
quadratum CA ad quadratum DA; in no&longs;tra verò hypothe&longs;i, &longs;i velocitas
in tota prima parte AC ponatur vt CG æquabiliter; haud dubiè &longs;patium
acqui&longs;itum in prædictis 4. temporibus erit, vt &longs;umma rectangulorum C
B, CI, EK, EN, quæ maior e&longs;t toto triangulo, AFN, 4. triangulis ABG,
GHI, IKL, LMN, ie e&longs;t 1/4 totius trianguli AFN; atque ita &longs;umma re
ctangulorum continet 10. quadrata æqualia quadrato CB, & triangu
lum AFN, continet. tantùm 8.
Iam verò diuidantur 4. partes temporis AF, in 8. æquales;
in &longs;enten
tia Galilei totum &longs;patium erit &longs;emper triangulum AFN, id e&longs;t vt &longs;ubdu
plum quadrati &longs;ub AF; quæ cùm &longs;it 8. quadratum erit 64.& &longs;ubduplum
quadrati 32. at verò &longs;umma rectangulorum e&longs;t 36. id e&longs;t continet 36.
quadrata æqualia quadrato XA; cùm tamen triangulum AFN, conti
neat tantùm 32. igitur &longs;umma prædicta e&longs;t ad triangulum AFN, vt 36.
ad 32. id e&longs;t vt 9.ad 8. igitur &longs;umma e&longs;t maior triangulo 1/8, quæ omnia
con&longs;tant.
Præterea diuidatur vlteriùs tempus AF in 16. æquales partes;
qua
dratum 16. cum &longs;it 256. accipiatur &longs;ubduplum id e&longs;t 128. & erit trian
gulum AFN, cui &longs;emper re&longs;pondet totum &longs;patium acqui&longs;itum in &longs;enten
tia Galilei; at verò &longs;umma rectangulorum erit 136. igitur &longs;umma e&longs;t ad
&longs;ummam vt 136.ad 128.id e&longs;t vt 17.ad 16. igitur e&longs;t maior &longs;umma trian
gulo (1/16) atque ita deinceps; &longs;i vlteriùs diuidas prædictum tempus in par
tes minores: quot porrò erunt, antequam fiat tota re&longs;olutio in in&longs;tan
tia, &longs;int enim v. g. in tempore AF in&longs;tantia 1000000. &longs;umma quæ re&longs;
pondet no&longs;træ progre&longs;&longs;ioni, erit maior altera, quæ re&longs;pondet progre&longs;&longs;io
ni Galilei (1/1000000) quis hoc percipiat?
Si verò in no&longs;tra hypothe&longs;i &longs;patium, quod re&longs;pondet primæ parti tem
poris AC &longs;it idem cum illo, quod re&longs;pondet eidem parti in &longs;ententia
Galilei, id e&longs;t æquale triangulo CAG, &longs;umma &longs;patiorum erit minor in
no&longs;tra hypothe&longs;i triangulo AFN &longs;ex triangulis æqualibus triangulo
ACG; igitur erit vt 10.ad 16. igitur minor 1/8. &longs;i verò diuidantur in 8.
temporis partes, triangulum AFN continebit 64. triangula æqualia
AXQ: at verò &longs;umma quæ re&longs;pondet no&longs;træ hypothe&longs;i 36.igitur minor
(7/16). denique &longs;i diuidantur in 16. partes, triangulum AFN continebit
256. triangula æqualia AYZ; at verò &longs;umma no&longs;tra 136. igitur minor
(15/52) &longs;ed nunquam erit minor 1/2.
Ob&longs;eruabis obiter dictum e&longs;&longs;e &longs;uprà &longs;ummam rectangulorum CB CI
EK EN e&longs;&longs;e maiorem triangulo AFN, 2.quadratis æqualibus CB; &longs;i
verò diuidatur tempus in 8. partes, &longs;umma rectangulorum e&longs;t minor præ
cedenti &longs;ummâ, toto quadrato æquali CB, id e&longs;t 4.quadratis æqualibus
XB, id e&longs;t 1/2 primæ differentiæ, quæ e&longs;t &longs;umma duorum quadratorum
æqualium CB; at &longs;i diuidatur in 16. partes, tempus AF, &longs;umma rectan
gulorum e&longs;t minor præcedente 8. quadratis æqualibus QZ, vel &longs;ubdu
plo quadrati CB, id e&longs;t 1/4 primæ differentiæ quæ e&longs;t &longs;umma duorum
quadratorum æqualium CB; &longs;i 4. partes temporis diuidantur in 8. de
trahitur 1/2 differentiæ, quæ e&longs;t inter &longs;ummam primam rectangulorum,
& triangulum AFN; &longs;i diuidantur in 16. detrahitur 1/4 eiu&longs;dem diffe
rentiæ; &longs;i diuidantur in 32. detrahitur 1/8, &longs;i in 64. (1/16);
atque ita deinceps,
& nunquam hæ minutiæ &longs;ubtractæ in infinitum totam differentiam ex
haurient; hinc minutiæ i&longs;tæ 1/2 1/4 1/8 (1/16) (1/32) (1/64) &c. in infinitum non fa
ciunt vnum integrum; &longs;ed hæc &longs;unt facilia.
Quarta ratio, quam afferunt aliqui, e&longs;t;
quia &longs;i cum eadem velocita
te acqui&longs;ita in fine temporis dati &longs;ine augmento nouo moueatur mobi
le; haud dubiè acquiret duplum &longs;patium tempore æquali tempori dato;
v. g. &longs;it triangulum AFE; &longs;itque velocitas acqui&longs;ita EF in 4. parti
bus temporis AE, vt iam &longs;uprà dictum e&longs;t, ne cogar repetere: certè &longs;i du
catur velocitas EF in tempus AE, vel EL æquale; habebitur rectan
gulum EK duplum trianguli AFE: &longs;ed triangulum AFE e&longs;t &longs;umma
&longs;patiorum motus accelerati tempore AE, & rectangulum EK e&longs;t &longs;um
ma &longs;patiorum motus æquabilis cum velocitate EF; igitur duplum e&longs;t
&longs;patium motus æquabilis, quod erat demon&longs;trandum. Præterea &longs;i diui
datur velocitas EF, & eius &longs;ubdupla ducatur in tempus AE; habebitur
rectangulum æquale triangulo AFE, vt con&longs;tat. Re&longs;pondeo facilè ex di
ctis, hoc ip&longs;um etiam ex no&longs;tra hypothe&longs;i proxime &longs;equi; &longs;int enim duo
in&longs;tantia; haud dubie &longs;i non cre&longs;cit velocitas, &longs;ecundo in&longs;tanti æquale
&longs;patium percurretur; &longs;i vero &longs;ecundo in&longs;tanti cre&longs;cat, percurrentur illo
motu 3.&longs;patia; & cùm velocitas
in&longs;tantis, primo in&longs;tanti &longs;it 1.gradus v.g. &longs;ecundo erunt 2. gradus; igi
tur moueatur per duo in&longs;tantia motu æquabili veloci vt 2. percurrentur
4. &longs;patia; igitur totum &longs;patium, quod percurritur motu veloci vt 2. per
2.in&longs;tantia e&longs;t ad totum &longs;patium, quod percurritur æquali tempore mo-&longs;i verò
&longs;int 3. in&longs;tantis continet illud, 1/2; &longs;i 4. continet 1. 3/5, &longs;i 5. continet 1.2/3
&longs;i 5. continet 1 2/3. &longs;i 6. continet 1 5/7. &longs;i 7. continet 1 3/4. &longs;i 8. continet
1 7/9. &longs;i 9. continet 1 (4/11). &longs;i 10. continet 1 9/5 &longs;ic quo plura erunt in&longs;tantia
accedet propiùs ad rationem duplam, nunquam tamen ad illam perue
niet. Ex dictis multa tumultuatim Corollaria congeri po&longs;&longs;unt;
Etiam&longs;i non &longs;int partes infinitæ temporis;
in ordine tamen ad praxim
eodem modo &longs;e habent, ac &longs;i e&longs;&longs;ent infinitæ; quia licèt finitæ &longs;int, nume
rari tamen non po&longs;&longs;unt.
Etiam &longs;i non &longs;int infiniti tarditatis gradus, vt con&longs;tat ex dictis, &longs;ed fi
niti; in ordine tamen ad praxim eodem modo &longs;e habent, ac &longs;i e&longs;&longs;ent in
finiti; quia non pote&longs;t di&longs;tingui primus, & minimus ab omnibus
aliis.
Licèt hypothe&longs;is Galilei &longs;it fal&longs;a in hypothe&longs;i in&longs;tantium finitorum;
nam &longs;ingulis in&longs;tantibus noua fit velocitatis acce&longs;&longs;io; phy&longs;icè tamen lo
quendo eodem modo &longs;e habet, ac &longs;i e&longs;&longs;et vera; quia cum non po&longs;&longs;it pro
bari, ni&longs;i in partibus temporis &longs;en&longs;ibilibus; certà, cùm quælibet pars
&longs;en&longs;ibilis innumera ferè in&longs;tantia contineat, in quibus fit progre&longs;&longs;io;
differentia vtriu&longs;que &longs;en&longs;ibilis e&longs;&longs;e non pote&longs;t; igitur linea denticulata
eodem modo &longs;e habet phy&longs;icè, hoc e&longs;t &longs;en&longs;ibiliter, ac &longs;i e&longs;&longs;et recta; &longs;ic
que progre&longs;&longs;io arithmetica in multis terminis reducitur &longs;en&longs;ibiliter ad
Geometriam in paucioribus terminis; immò in communi illa &longs;ententia.
in qua dicitur tempus con&longs;tare ex partibus actu infinitis, progre&longs;&longs;io Ga
lilei tantùm locum habere pete&longs;t; igitur hæc e&longs;to clauis huius difficul
tatis; progre&longs;&longs;io &longs;implex principium phy&longs;icum habet, non experimen
tum; progre&longs;&longs;io numerorum imparium experimentum non principium;
vtramque cum principio & experimento componimus; prima enim &longs;i.
a&longs;&longs;umantur partes temporis &longs;en&longs;ibiles tran&longs;it in &longs;ecundam, &longs;ecunda in
primam, &longs;i vltima a&longs;&longs;umantur in&longs;tantia.
Cognito &longs;patio quod percurritur in data parte temporis &longs;en&longs;ibili, co
gno&longs;ci pote&longs;t &longs;patium quod in duabus æqualibus vel 3.vel 4.&c.percurri
pote&longs;t.v.g. multi probarunt &longs;æpiùs primo &longs;ecundo minuto corpus graue
percurrere 12. pedes; igitur duobus percurreret 48. accipe enim 9. 2.
id e&longs;t 4. & in 4. duces 12. vt habeas 48. 4. verò minutis percurret 192.
nam accipe 9. 4. id e&longs;t 16. & in 16. duces 12.vt habeat 192. res omninò
facilis.
Similiter cognito &longs;patio quod percurrit 4. &longs;ecundis minutis, cogno
&longs;ces &longs;patium, quod percurret 2. vel 1. v.g. percurrit 4. &longs;ecundis 192. pe-accipe 9.4. id e&longs;t 16. & per 16. diuide 192. quotíens dabit 12. pro
primo &longs;ecundo: accipe 9.2. id e&longs;t, 4. & diuide 192. per 4.quotiens dabit
48. pro duobus minutis, atque ita deinceps.
Similiter cognito tempore cogno&longs;ci pote&longs;t &longs;patium decur&longs;um;
quia
&longs;patia &longs;unt vt quadrata temporum; vel cognito &longs;patio cogno&longs;ci pote&longs;t
tempus; quia tempora &longs;unt, vt radices &longs;patiorum, hæc elementa &longs;altem
Arithmetices de&longs;iderant.
Sed iam re&longs;tat, vt &longs;oluamus objectiones aliquas, quæ contra motus ac
celerationem pugnare videntur.
Prima objectio e&longs;t;
&longs;i motus acceleratio fieret in in&longs;tantibus, &longs;ecundo
in&longs;tanti idem corpus e&longs;&longs;et in duobus locis adæquatis quod &longs;ic o&longs;tendo:
&longs;it &longs;patium AB quod percurrit corpus graue primo in&longs;tanti; haud du
biè AB, e&longs;t eius locus adæquatus; &longs;ecundo in&longs;tanti percurrit BC duplum
AB; igitur eodem in&longs;tanti re&longs;pondet loco BD, & DC, quorum vterque
e&longs;t æqualis AB; igitur &longs;ecundo in&longs;tanti e&longs;t in duobus locis, &longs;cilicet BD
& DC, quod dici non pote&longs;t.
Hæc objectio impugnat omnem velocitatem;
hoc e&longs;t, non modò eam,
quæ motui naturaliter accelerato competit; verùm etiam illam, quæ
ine&longs;t motui violento; igitur vt re&longs;pondeam faciliùs; &longs;uppono punctum
phy&longs;icum, mobile &longs;cilicet A; aut &longs;i mauis Angelum coëxten&longs;um quadra
to A; qui &longs;cilicet moueatur motu accelerato, & primo in&longs;tanti acquirat
locum immediatum æqualem priori, &longs;cilicet AB; licèt enim po&longs;&longs;et ac
quirere vibrationem participantem de priori; quia tamen acquireret
tandem non participantem, id e&longs;t, quæ tota &longs;it extra illam, cui e&longs;t imme
diata, qualis e&longs;t AB. &longs;uppono hîc acquiri vibrationem non participan
tem de priori, id e&longs;t &longs;patium AB, æquale priori, in quo erat A, & pror
&longs;us extra illud po&longs;itum licèt immediatum; hoc po&longs;ito, primo in&longs;tanti pun
ctum A acquirit AB tanquam locum adæquatum, vt certum e&longs;t: certum
e&longs;t etiam loca BC, CD, e&longs;&longs;e adæquata: igitur &longs;imul, id e&longs;t eodem in
&longs;tanti in vtroque e&longs;&longs;e non pote&longs;t; nam in&longs;tans &longs;imul totum e&longs;t;
igitur
&longs;ecundo in&longs;tanti non percurrit BC, &longs;ed &longs;ecundo tempore æquali primo;
hoc enim &longs;ecundum tempus con&longs;tat duobus in&longs;tantibus, quod &longs;imul
vtrumque re&longs;pondet primo: quippe dari po&longs;&longs;unt in&longs;tantia phy&longs;ica;
igitur
primum in&longs;tans quo percurritur AB e&longs;t æquale duobus aliis, quibus
percurruntur BD, & CD; vnde quando dixi primo in&longs;tanti acquiri &longs;pa
tium duplum primi, idem e&longs;t, ac &longs;i dixi&longs;&longs;em &longs;ecundo tempore æquali pri
mo, quod reuerà tempus con&longs;tat 2. in&longs;tantibus, quorum alterum re&longs;pon
det &longs;patio BC, & alterum &longs;patio DC.
Secunda objectio;
Sed inquiet aliquis, igitur non e&longs;t continua acce
leratio motus; nam in&longs;tans quo percurritur &longs;ecundum &longs;patium BD, cùm
&longs;it æquale in&longs;tanti quo percurritur tertium &longs;patium DC, in vtroque &longs;pa
tio e&longs;t æquabilis motus. Re&longs;pondeo in&longs;tans quo percurritur &longs;ecundum
&longs;patium BD, e&longs;&longs;e maius in&longs;tanti, quo percurritur tertium &longs;patium DC;
tà tamen lege, vt vtrumque &longs;imul &longs;umptum &longs;it omninò equale in&longs;tanti, &longs;imiliter totum &longs;patium CG ita
percurritur tertio tempore, vt &longs;ingula &longs;patia CE. EI. FG. &longs;ingulis in
&longs;tantibus percurrantur; &longs;ed hæc tria in&longs;tantia &longs;imul &longs;umpta &longs;unt æqualia
primo in&longs;tanti, quo percurritur &longs;patium; licèt primum quo percurritur
CE &longs;it maius &longs;ecundo, quo percurritur EF, & hoc maius tertio, quo per
curritur FG, atque ita deinceps.
Ob&longs;eruabis po&longs;&longs;e velocitatem motus explicari duobus modis.
Primò,
&longs;i a&longs;&longs;umantur tempora æqualia, & &longs;patia inæqualia in ea progre&longs;&longs;ione,
quam hactenus explicuimus. Secundò &longs;i accipiantur &longs;patia æqualia &
tempora inæqualia, quod duobus modis fieri tantùm pote&longs;t. Primò &longs;i ac
cipiantur &longs;patia æqualia primo &longs;patio, quod percurritur primo in&longs;tanti.
Secundò &longs;i accipiantur &longs;patia æqualia alteri &longs;patio, quod in parte tempo
ris &longs;en&longs;ibili percurritur; in qua verò proportione tempora fiant &longs;emper
minora, 'dicemus infrà; nec dicas durum e&longs;&longs;e dicere in&longs;tans e&longs;&longs;e po&longs;&longs;e
minus in&longs;tanti; nam equidem fateor in&longs;tanti mathematico nihil e&longs;&longs;e
po&longs;&longs;e minus; &longs;ecus verò in&longs;tanti phy&longs;ico, quod e&longs;t diui&longs;ibile potentiâ, vt
dicemus aliàs; nomine in&longs;tantis phy&longs;ici intelligo durationem indiui&longs;i
bilem, hoc e&longs;t, cuius entitas tota &longs;imul e&longs;t.
Tertia objectio.
Sed inquies, igitur &longs;ecundo tempore æquali primo
acquiruntur 2.gradus velocitatis, vel impetus; igitur tria &longs;patia &longs;ecun
do tempore percurruntur, quod e&longs;t contra hypothe&longs;im; quippe duo gra
dus impetus accedunt primo, &longs;imiliter tertio tempore producentur tres
gradus impetus; qui &longs;i iungantur tribus præcedentibus, erunt 6. Igitur
percurrentur tertio tempore 6. &longs;patia, & quarto 10.quinto 15. quia &longs;in
gulis in&longs;tantibus debet produci impetus; e&longs;t enim cau&longs;a nece&longs;&longs;aria ap
plicata.
Re&longs;pond&etail;o, equidem eo in&longs;tanti, quo percurritur &longs;patium BD, pro
duci aliquid impetus, & aliquid eo in&longs;tanti, quo percurritur &longs;patium
DC; ita vt tamen totus ille impetus, qui producitur his duobus in&longs;tan
tibus, &longs;it æqualis illi, qui producitur primo in&longs;tanti, quo &longs;cilicet percurri
tur &longs;patium AB; quia duo illa in&longs;tantia &longs;imul &longs;umpta faciunt tempus
æquale primo in&longs;tanti; atqui temporibus æqualibus eadem cau&longs;a nece&longs;
&longs;aria non impedita æqualem effectum producit per Ax.3.hinc vides &longs;in
gulis in&longs;tantibus eadem proportione decre&longs;cere impetum in perfectio
ne, qua tempus e&longs;t breuius, &longs;eu velocior motus; &longs;ed de hoc infrà.
Quarta objectio;
&longs;i impetus &longs;ingulis in&longs;tantibus cre&longs;ceret, vel intende
retur, augeretur grauitatio: quippe &longs;i grauitas primo in&longs;tanti producat
vnum gradum impetus; &longs;ecundo æqualem producet, & tertio, atque ita
deinceps, quod e&longs;&longs;et ab&longs;urdum; alioqui minima atomus quodlibet cor
pus graue adæquaret, quod e&longs;t ab&longs;urdum.
Re&longs;pondeo nunquam impetum intendi, ni&longs;i &longs;it motus, qui e&longs;t illius fi
nis; alioquin fru&longs;tra e&longs;&longs;et per plura in&longs;tantia;
igitur de&longs;trui deberet;
nec
dicas impetum naturalem etiam fru&longs;trà e&longs;&longs;e &longs;ine motu; quia cum mo
tus non &longs;it eius finis adæquatus; non mirum e&longs;t &longs;i po&longs;&longs;it e&longs;&longs;e &longs;ine motu;
atqui iam diximus &longs;uprà habere duos fines, quorum alterum &longs;emper ha-primus e&longs;t grauitatio, &longs;eu ni&longs;us ver&longs;us centrum; &longs;ecundus motus
deor&longs;um; cùm tamen impetus additivius motum tantùm pro fine habeat;
igitur &longs;i impeditur totus motus, non producitur hic impetus.
Quinta objectio;
&longs;i impetum &longs;uum intendit corpus graue;
&longs;imiliter
Ignis diceretur intendere calorem; Sol lucem, &c. Re&longs;pondeo primò de
luce &longs;ingularem e&longs;&longs;e rationem; quia &longs;cilicet con&longs;eruatur à cau&longs;a &longs;ua pri
mo productiua; quidquid &longs;it; &longs;i viderem effectum caloris, vel frigoris
perpetuò cre&longs;cere; haud dubiè dicerem etiam cau&longs;as ip&longs;as intendi;
atqui
hoc ip&longs;um video in motu naturali, qui effectus impetus e&longs;t; adde quod
argumentum à pari debile e&longs;t; cum enim &longs;int diuer&longs;i naturæ fines, diuer
&longs;æ &longs;unt viæ quibus &longs;uos fines con&longs;equítur; denique contrarietas caloris,
& frigoris impedit fortè, ne vlterius vtraque qualitas intendatur, de qua
fusè &longs;uo loco; porrò dicemus Tomo &longs;exto calorem con&longs;eruari à cau&longs;a &longs;ua
primo productiua; quo po&longs;ito ce&longs;&longs;at difficultas; quod licèt alicui durum
videri po&longs;&longs;it, demon&longs;trabo tamen.
Sexta objectio; igitur &longs;i ex infinita di&longs;tantia lapis de&longs;cenderet, inten
deret etiam &longs;uum motum. Re&longs;pondeo primò, non po&longs;&longs;e dari infinitam il
lam di&longs;tantiam. Secundò etiam&longs;i daretur lapis, ex ea non caderet;
fru&longs;trà
enim e&longs;&longs;et ille motus: Tertiò, &longs;i daretur motus infinitus, haud dubiè e&longs;&longs;et
æquabilis; qualis e&longs;t motus circularis corporum cœle&longs;tium;
at verò
motus naturalis deor&longs;um corporum grauium debet e&longs;&longs;e acceleratus ne
vel de&longs;cenderent tardiùs, &longs;i cum primo tantùm velocitatis gradu de&longs;cen
derent; vel &longs;u&longs;tineri vix po&longs;&longs;ent, &longs;i impetum innatum intentiorem habe
rent; vtrum verò &longs;emper intendatur, & ex quacumque altitudine cadat
corpus graue, videbimus infrà.
Ex dictis hactenus facilè refelluntur aliæ &longs;ententiæ de proportione
motus naturaliter accelerati.
Et primò quidem illa, quæ vult fieri &longs;ecundum rationem &longs;inuum
ver&longs;orum, licèt initio tàm propè accedat ad proportionem Galilei, vt
di&longs;cerni &longs;en&longs;ibiliter ab ea non po&longs;&longs;it; quare tutò &longs;atis a&longs;&longs;umi po
terit, &longs;i quando &longs;it opus illius loco, quod nos in explicandis motibus cœ
le&longs;tibus præ&longs;tabimus; interim quia faciliùs explicatur in motu recto per
rationem quadratorum quàm &longs;inuum, illam retinebimus; præ&longs;ertim cùm
vtraque ad no&longs;tram reducatur; modò progre&longs;&longs;io fiat in in&longs;tantibus.
Secundò reiicitur &longs;ententia illorum qui volunt hanc progre&longs;&longs;ionem fie
ri iuxta proportionem geometricam, quam vides in his numeris 1.2.4.8.
16. quæ licèt initio minùs recedat à vera, in progre&longs;&longs;u tamen multùm
aberrat, nec e&longs;t vlla ratio quæ pro illa faciat: Et verò nulla in mentem
venire pote&longs;t; ni&longs;i fortè dicatur, cùm &longs;ecundo in&longs;tanti &longs;it dupla velocitas,
tertio octuplam;
quia vt velocitas pri
mi in&longs;tantis e&longs;t ad velocitatem &longs;ecundi, ita velocitas huius ad velocita
tem tertij, & velocitas huius ad velocitatem quarti; igitur &longs;equitur pro
gre&longs;&longs;ionem rationis geometricæ duplæ; cur enim e&longs;&longs;et maior ratio pri
mi in&longs;tantis ad &longs;ecundum quàm &longs;ecundi ad tertium tertij ad quartum? &c.
&longs;ed profectò vix vlla apparet rationis &longs;pecies, cùm nulla &longs;it cau&longs;a,
igitur e&longs;t peculiaris
cau&longs;a huius inæqualitatis rationum; quòd &longs;cilicet æqualibus temporibus
æqualia acquirantur velocitatis momenta; vt &longs;uprà demon&longs;trauimus;
quippe id præ&longs;tari debet in explicandis inæqualitatibus motuum recto
rum naturalium, quod præ&longs;tant A&longs;tronomi in explicanda inæqualitate
motuum cæle&longs;tium; qui &longs;emper æqualitatem aliquam &longs;upponunt, nec e&longs;t
quòd hanc &longs;ententiam nonnullis experimentis ictuum qui&longs;quam con
firmet, in quibus multa fraus &longs;ube&longs;&longs;e pote&longs;t.
Tertiò reiicitur illa quoque &longs;ententia, quæ proportionem lineæ &longs;ectæ
in mediam, & extremam rationem huic lineæ tribuit, quam ferè in his
numeris vides 1.2.3.5.8, 13. 21. 34. 55. quæ &longs;ub finem etiam longi&longs;&longs;imè
aberrat, vt videre e&longs;t, quare ii&longs;dem rationibus impugnatur, quibus iam
aliam impugnauimus.
Scio e&longs;&longs;e alias multas rationes, quibus aliqui recentiores motus natu
ralis accelerationem explicare nituntur, &longs;ed iam &longs;uprà &longs;atis &longs;uperque re
iectæ fuerunt, vel profectò eæ &longs;unt, quæ ne quidem inter fabulo&longs;a poë
tarum commenta locum aliquem habere po&longs;&longs;int: Et verò ni&longs;i me ani
mus fallit in re clari&longs;&longs;ima, rationem huius effectus ex communibus
principiis deductam cum ip&longs;is etiam experimentis con&longs;entire hactenus
ita demon&longs;trauimus, vt iam vix vllus dubitationi locus relinquatur; &longs;ed
interruptam Theorematum &longs;eriem tandem repetimus.
in&longs;tantia &longs;unt inæqualia in motu natur aliter acceleratoprobatur, quia &longs;e
cundum &longs;patium æquale primo percurritur motu velociore, quàm pri
mo, & tertium quam &longs;ecundo: ergo minori tempore per Def.2.l.1. &longs;ed
primum &longs;patium conficitur vno in&longs;tanti; igitur &longs;ecundum vno in&longs;tanti,
&longs;ed minore; idem dico de tertio.
&longs;ecundum tertio, tertium quarto, quartum quinto, quintum &longs;exto,
atque ita deinceps; ita vt &longs;ecundum & tertium &longs;imul &longs;umpta, item quar
tum, quintum, &longs;extum, &longs;eptimum, item octauum, nonum, decimum, &longs;imul
&longs;umpta adæquent primum, hoc e&longs;t vt vnum, duo, tria, quatuor, quinque,
&longs;ex, &c. faciant &longs;emper tempora æqualia, quia temporibus æqualibus æ
qualia acquiruntur velocitatis momenta? igitur &longs;i primo in&longs;tanti per
curritur vnum &longs;patium; &longs;ecundo tempore æquali percurruntur duo &longs;pa
tia æqualia primo, & tertio, tria; atque deinceps; &longs;ed vt &longs;uprà dictum e&longs;t
in re&longs;pon&longs;. ad obiect. primam, vno, &
pus percurrere duo &longs;patia, ne &longs;imul e&longs;&longs;et in duobus locis; igitur &longs;ingula
&longs;patia re&longs;pondent &longs;ingulis in&longs;tantibus licèt minoribus; &longs;ed &longs;ecundo tem
pore æquali primo in&longs;tanti percurruntur duo &longs;patia æqualia primo &longs;pa
tio; igitur &longs;ecundum, & tertium in&longs;tans debent &longs;imul &longs;umpta adæquare
alioquin duobus illis in&longs;tanti
bus motus e&longs;&longs;et æquabilis; igitur &longs;ecundum e&longs;t maius tertio, ita vt tamen
ex vtroque tempus fiat æquale primo in&longs;tanti.
mediam rationem propagatam; ita vt primum &longs;it ad &longs;ecundum, vt &longs;ecundum
ad tertium, tertium ad quartum, quartum ad quintum at que ita deinceps
&longs;it enim aliqua &longs;eries numerorum, qui aliquo modo accedant ad prædi
ctam proportionem 1.2.3.5.8.13.21.34.55. &longs;itque primum in&longs;tans vlti
mus numerus 55. &longs;ecundum 34.tertium 21. atque ita deinceps: Equidem
&longs;ecundum, & tertium adæquant primum; at verò quartum, quintum,
&longs;extum nullo modo adæquant; immò ne quidem eius &longs;ubduplum, &
multò minus 3. alij addito primo: immò &longs;i linea data duodecies propor
tionaliter diuidatur, vltimum &longs;egmentum vix e&longs;&longs;et &longs;ubcentuplum primi,
vt con&longs;tat; igitur reiici debet hæc propo&longs;itio.
nec ad
tertium, quartum, quintum, &longs;extum, &c.
excogitari pote&longs;t quo de&longs;ignari po&longs;&longs;it quantitas, &longs;eu perfectio, &longs;eu va
lor i&longs;torum in&longs;tantium; &longs;it enim primum in&longs;tans &longs;ecundum &longs;it 3/5. tertium
2/5 quartum 4/9 quintum 2/9 &longs;extum 2/9. Equidem &longs;ecundum, & tertium ad&etail;
quant primum; adde quod non pote&longs;t amplius &longs;eries propagari per nu
meros rationales; &longs;it autem &longs;ecundum (6/11) 3. (5/11) cum tribus aliis 4/9 1/9 7/9;
equidem &longs;i reducantur hæ 5. minutiæ, re&longs;pondebunt his (54/99) (45/99) (44/99) (12/99) (26/99):
igitur &longs;ecunda erit maior quarta; at prima &longs;uperat &longs;ecundam (9/999) &longs;ecunda
tertiam (1/99) tertia quartam (11/99) quarta quintam (12/99). Cur porrò hæc inæqua
litas, igitur numeri po&longs;&longs;unt a&longs;&longs;ignari; non po&longs;&longs;unt etiam poni in &longs;erie
geometrica &longs;ubdupla 1. 1/2 1/4 1/8 &c. quia &longs;ecunda.
& tertia non adæquant
primam idem dicendum e&longs;t potiori iure de tribus aliis; nec etiam in &longs;e
rie arithmetica &longs;implici 1. 1/2 1/3 1/4 2/5 1/6; quia &longs;ecunda, & tertia &longs;unt mi
nores prima 1/6, vt quarta, quinta, &longs;exta &longs;unt minores prima (26/74).
minorum
tertius quartum, &c. vt &longs;ecundus, & tertius adæquent primum, item
quartus, quintus, &longs;extus. item 4. alij, qui &longs;equuntur, item 5. item 6. &c.
v. g. pote&longs;t dari linea AG con&longs;tans tribus partibus æqualibus, &longs;cilicet
AB, BC, CG, & &longs;ecunda BC duabus BD maiore, & DC minore, & ter
tia tribus prima CE minore ED, &longs;ed maiore EF, &longs;ecunda EF maiore F
G, atque ita deinceps; addi pote&longs;t quartum &longs;egmentum æquale AB;
quod
&longs;ubdiuidetur in 4. partes, quarum prima &longs;it maior &longs;ecunda, &
& hæc quarta, & omnes minores FG; ita autem &longs;uperant primæ &longs;equen
tes, vt differentia primæ, & &longs;ecundæ &longs;it maior differentia &longs;ecundæ, &
aliter res e&longs;&longs;e pote&longs;t.
& DC accedunt propiùs ad æqualitatem quàm AB, BD, & DC, CE, pro
piùs quàm CD, DB, & CE, EF, quàm EC, CD, atque ita deinceps, vt patet;
hinc po&longs;t aliquot in&longs;tantia motus, æqualia ferè redduntur in&longs;tantia, vt
con&longs;tat.
impetusquia temporibus æqualibus eadem cau&longs;a nece&longs;&longs;aria æqualem ef
fectum producit per Ax. tertium igitur inæqualem inæqualibus, per Ax.
13. num.4. igitur minorem minore tempore; igitur minorem in eadem
proportione, in qua tempus e&longs;t; igitur qua proportione, &c.
cauimus lib.
quat impetum productum primo, quem etiam adæquat productus quar
to, quinto, &longs;exto, item productus &longs;eptimo, octauo, nono; decimo, atque ita
deinceps; hinc e&longs;t eadem differentia impetuum, quæ
hinc &longs;in
gulis &longs;patiis æqualibus primo &longs;patio, quod percurritur primo in&longs;tanti;
re&longs;pondent &longs;ingula in&longs;tantia, & &longs;ingulis in&longs;tantibus &longs;inguli, & &longs;ingulares
impetus; hinc non e&longs;t quod primo in&longs;tanti dicantur produci plura pun
cta impetus in eodem puncto corporis grauis; &longs;ed vnicum tantùm pun
ctum talis perfectionis &longs;cilicet phy&longs;icum; cur enim potius duo puncta,
quam tria? &longs;ed quod vnum e&longs;t determinatum e&longs;t per Ax. 5. lib. 1. hinc
optima ratio cur potius tali in&longs;tanti producatur impetus talis perfectio
nis quàm alterius? quippe perfectio impetus &longs;equitur perfectionem in
&longs;tantis quo producitur; hinc dicendum videtur omnia puncta impetus
e&longs;&longs;e diuer&longs;æ perfectionis, vel heterogenea; vt vulgò aiunt Philo&longs;ophi;
cuius rationem demon&longs;tratiuam afferemus lib. &longs;equenti cum de motu
violento; hinc vides duplicem progre&longs;&longs;ionem;
primam &longs;cilicet, qua ex
&longs;uppo&longs;itis temporibus æqualibus acquiruntur &longs;patia inæqualia, de qua
fusè &longs;uprà; in hac enim velocitas eadem proportione cum impetu cre&longs;
cit, & cum ip&longs;o tempore; hoc e&longs;t tempore triplo e&longs;t tripla, quadruplo
quadrupla; item impetus in duplo tempore e&longs;t duplus, in triplo triplus;
modò progre&longs;&longs;io fiat in temporibus primo in&longs;tanti æqualibus; &longs;ecunda
progre&longs;&longs;io e&longs;t qua ex &longs;uppo&longs;itis &longs;patiis æqualibus tempora fluunt inæ
qualia, hoc e&longs;t minora & minora; quibus etiam re&longs;pondet impetus im
perfectior in eadem proportione temporum; prima fit per differentias
æquales, & proportiones inæquales, &longs;ecunda verò per differentias inæ
quales, & proportiones inæquales.
plicata &longs;patiorumcrun enim &longs;patia &longs;int vt quadrata
certè tempora &longs;unt, vt radices i&longs;torum quadratorum, &longs;cilicet &longs;patiorum;
&longs;int enim quæcunque &longs;patia æqualia in linea AF; &longs;intque &longs;patia AC 4.
AE 16. radix quadr.4. e&longs;t 2.16. verò 4. igitur tempora &longs;unt vt 4.2.&longs;i ve
rò accipiatur primum &longs;patium, quod vno tempore percurritur; tempus
quo percurruntur duo &longs;patia æqualia primum e&longs;t v.2.quo percurruntur
tria v.3.quo percurruntur 4.&longs;patia, 2. atque ita deinceps; igitur in praxi
quæ tantùm fit in &longs;patiis &longs;en&longs;ibilibus hæc progre&longs;&longs;io adhibenda e&longs;t, il
lamque deinceps, &longs;i quando opus e&longs;t, adhibebimus.
&longs;eruarenturquia &longs;cilicet nullum e&longs;&longs;e impedimentum;
at verò &longs;i aliquod
intercedit impedimentum; haud dubiè non &longs;eruantur accuratè; e&longs;t autem
aliquod impedimentum in medio, quantumuis liberum e&longs;&longs;e videatur,
quæ omnia con&longs;tant.
quippe pila
deor&longs;um cadens tandem quie&longs;cit, licèt à terra reflectatur ratione impe
dimenti, ex quo re&longs;ultat duplex determinatio, ratione cuius idem im
petus &longs;ibi aliquo modo redditur &longs;ed de his fusè in primo libro
à Th.148. ad finem v&longs;que libri: nam reuerâ duæ determinationes op
po&longs;itæ pugnant pro rata per Ax. 15.l.1. & quotie&longs;cunque idem impetus
e&longs;t ad lineas oppo&longs;itas determinatus eodem modo &longs;e habet, ac &longs;i duplex
e&longs;&longs;et, & quilibet &longs;uæ &longs;ube&longs;&longs;et determinationi; atqui &longs;i duplex e&longs;&longs;et oppo
&longs;itus, pugnarent pro rata; igitur tàm pugnant duæ determinationes op
po&longs;itæ in eodem impetu, quàm duo impetus ad oppo&longs;itas lineas deter
minati; igitur impetus naturalis aduentitius de&longs;truitur, &c.
Probatur, quia nihil e&longs;te
quod exigat eius de&longs;tructionem, quia &longs;cilicet nunquam e&longs;t fru&longs;trà; nam
vel habet motum deor&longs;um, vel grauitationis effectum, vel de&longs;truit impe
tum extrin&longs;ecum in motu violento; igitur nunquam e&longs;t fru&longs;trà, cum &longs;em
per habeat aliquem effectum.
Dices lignum vi extrin&longs;eca in aqua immer&longs;um &longs;ua &longs;ponte a&longs;cendit;
igitur ille gradus impetus grauitationis de&longs;truitur, & alius producitur;
hæc quæ&longs;tio ad præ&longs;ens in&longs;titutum non pertinet, &longs;ed ad librum de gra
uitate, & leuitate. Igitur breuiter re&longs;pondeo illum impetum nunquam
de&longs;trui, quandiu mobile grauitat, vel grauitatione &longs;ingulari, (&longs;ic corpus
grauitat in manum &longs;u&longs;tinentis,) vel grauitatione communi, (&longs;ic lignum
humori innatans grauitat, non quidem in aquam, at &longs;imul cum aqua;)
&longs;ed de grauitate, & grauitatione in Tomo de &longs;tatibus corporum &longs;en&longs;ibi-
extenditur, & ideo corpora &longs;ur&longs;um ire, quia alia deor&longs;um eunt.
motusProbatur primò experientiâ, quæ certa e&longs;t;
tàm enim aër impe
dit motum deor&longs;um, quàm &longs;ur&longs;um, vt videre e&longs;t in mobili leuiore &longs;eu ra
riore, quod etiam flante vento ob&longs;eruare omnes po&longs;&longs;unt; quomodo ve
rò impediat, dicemus aliàs; &longs;ecundò corpus immobile, in quod mobile
impingitur, motum illius impedit; &longs;ed in diuer&longs;as partes aëris corpus
graue impingitur in de&longs;cen&longs;u; igitur aliquantulum impeditur eius
motus.
præ&longs;tare pote&longs;t huiu&longs;modi impedimentum, ni&longs;i aliquam retardationem;
igitur motus inde redditur tardior.
fectus requiritur imperfectio cau&longs;æ per Ax. 13.l. 1. & quâ proportione
e&longs;t tardior motus eâdem impetus e&longs;t imperfectior, per Ax. 5. excipe ta
men impetum innatum, qui &longs;emper habet eundem effectum grauitatio
nis, vel &longs;ingularis, quâ grauitas cum ip&longs;o medio, &longs;i reuerâ medium gra
uitat, de quo aliàs.
Probatur, quia &longs;i
motum impedit; certè non totum; quis enim hoc dicat; &longs;ed eæ dumta
xat partes, quibus incubat corpus graue; igitur quò &longs;unt plures huiu&longs;
modi partes, maius e&longs;t impedimentum; &longs;ed in medio den&longs;iori plures &longs;unt
cum minore exten&longs;ione; hoc enim e&longs;t, quod voco den&longs;ius; igitur me
dium den&longs;ius plùs impedit.
aëra,
Ob&longs;erua e&longs;&longs;e aliqua corpora minus den&longs;a, quæ motum omninò im
pediunt; quippe certum e&longs;t aquam e&longs;&longs;e den&longs;iorem ligno;
atqui li
gnum de&longs;cen&longs;um lapidis impedit, non verò aqua; quia &longs;cilicet lignum
non e&longs;t medium, vt aqua; vt enim aliquod corpus &longs;it medium, debet e&longs;&longs;e
liquidum, vt, aqua & alij liquores; vel &longs;pirabile vt aër, vapor, &c. ratio
e&longs;t, quia partes ligni, vel alterius corporis durioris, ita &longs;unt inter &longs;e con
junctæ, vel implicatæ, vt omnem tran&longs;itum intercludant, ni&longs;i corpus ip
&longs;um graue valido ictu vel impetu &longs;ibi viam aperiat; igitur vt corpus ali
quod vice medij defungatur, debet in eo &longs;tatu e&longs;&longs;e, in quo eius partes &longs;ed de his &longs;tatibus cor
porum fusè agemus Tomo 5. adde quod ad medium &longs;ufficit vacuum &longs;i
motus in vacuo e&longs;&longs;e pote&longs;t, de quo alibi; quod certè e&longs;t omnium me
diorum optimum, cum nullo modo re&longs;i&longs;tar mobili.
dior e&longs;t motus, ex cuius tarditate arguitur imperfectio impetus per Ax.
13.num.4.
Ob&longs;erua den&longs;itatem medij cogno&longs;ci ex eius grauitate;
illud enim
den&longs;ius e&longs;t, quod e&longs;t grauius & vici&longs;&longs;im; quod fusè explicabimus &longs;uo lo
co; e&longs;t enim grauitas quædam
9.
periùs;
graue & leue; molle & durum den&longs;itates quædam e&longs;&longs;e, & raritates videntur,
quæ adnotare volui, vt vel inde con&longs;tet doctrinam hanc cum Peripate
tica optimè con&longs;entire.
Ob&longs;eruabis etiam hîc à me non di&longs;cuti, in quo con&longs;i&longs;tat den&longs;itas, vel
raritas, grauitas, vel leuitas; &longs;uppono tantùm graue illud e&longs;&longs;e, quod ten
dit deor&longs;um; leue illud, quod tendit &longs;ur&longs;um &longs;iue pellatur à grauiori, &longs;iue
non, den&longs;um verò e&longs;&longs;e id quod multùm materia habet &longs;ub parua exten
&longs;ione, rarum è contrario; quorum omnium cau&longs;as, & rationes &longs;uo loco
explicabimus.
certa e&longs;t hypothe&longs;is, ni&longs;i for
tè aliquando per accidens &longs;ecus accidat; ratio porrò petitur ex ip&longs;a
grauitatis natura, quâ corpus graue tendit deor&longs;um; nihil enim aliud
grauitas e&longs;t, quidquid tandem illa &longs;it; quippe corpus graue de&longs;cendit,
quando medium liberum habet, idemque leuius, per quod de&longs;cendat;
quod certè &longs;i grauius e&longs;&longs;et, haud dubiè non de&longs;cenderet; &longs;ic ferrum, &
&longs;axum plumbo liquato innatant; cum tamen per mediam aquam de
&longs;cendant; fic lignum aquæ &longs;upernatat, quod per liberum aëra de&longs;cendit;
ratio e&longs;t, quia grauius de&longs;cendit &longs;ub medium leuius; cur autem id fiat
fusè alibi explicabo; id tantùm obiter indico. Omnis motus, qui fit à
principio intrin&longs;eco per lineam rectam propter locum e&longs;t, vt patet; quis
enim neget corpus graue ideo de&longs;cendere &longs;ub leuius, vt occupet aliquem
locum quo prius carebat, qui tamen illi connaturalis e&longs;t in hoc rerum
ordine? cum à natura acceperit vim illam intrin&longs;ecam, quâ in eum lo
cum &longs;e&longs;e recipere pote&longs;t; quam certè vim intrin&longs;ecam nunquam à na
tura rebus creatis in&longs;itam e&longs;&longs;e con&longs;tat, ni&longs;i ad eum finem con&longs;equendum,
cui à natura de&longs;tinantur; cur verò locus connaturalis corporis grauio
ris &longs;it ille, in quo leuiori &longs;ube&longs;t, non diu hærebit animus, quin &longs;tatim ra
tio affulgeat; cum enim corpus, quod e&longs;t &longs;uprà, &longs;u&longs;tineatur ab eo quod e&longs;t
infrà; illud certè infra e&longs;&longs;e connaturalius e&longs;t, quod aptius e&longs;t ad &longs;u&longs;tinen-
atqui den&longs;um aptius e&longs;t ad id munus, quia plures partes &longs;u&longs;tinentis
pauciores &longs;u&longs;tinent alterius leuioris, &longs;eu rarioris, vt con&longs;tat; v.g. certum
e&longs;t &longs;ed de hoc
alias fusè; hæc interim &longs;ufficiat indica&longs;&longs;e, vt vel aliqua ratio affulgeat;
cur &longs;cilicet corpus graue &longs;ub medium leuius &longs;ua &longs;ponte de&longs;cendat; adde
quod cum omne corpus graue tendat deor&longs;um, tunc vnum infra aliud de
&longs;cendit, cum &longs;unt plures partes pellentis, quàm pul&longs;i; denique per va
cuum modicum &longs;ine vlla re&longs;i&longs;tentia de&longs;cenderet.
tatv.g. lignum aquæ, ferrum plumbo liquato;
certa e&longs;t hypothe&longs;is:
ratio
e&longs;t, quia ideo de&longs;cendit graue &longs;ub medium, quia grauius &longs;eu den&longs;ius e&longs;t
medio; igitur, &longs;i den&longs;ius e&longs;t ip&longs;um medium, non de&longs;cendet; clarum e&longs;t;
cur verò a&longs;cendat &longs;upra medium. v.g. cur lignum aquæ immer&longs;um tan
dem emergat hîc non di&longs;cutio, &longs;ed tantùm indico ab ip&longs;a aqua &longs;ur&longs;um
extendi; quanta verò parte lignum emergat, dicemus aliàs, cum de in
natantibus humido.
ra
tio e&longs;t, quia ideo de&longs;cendit &longs;ub medium, quia medium leuius e&longs;t, ideo
a&longs;cendit &longs;upra, quia medium grauius e&longs;t; igitur &longs;i nec &longs;it grauius nec
leuius, non e&longs;t quod a&longs;cendat vel de&longs;cendat; nihil tamen illius &longs;upra
primam medij &longs;uperficiem extare poterit; alioqui e&longs;&longs;et leuius medio,
contra &longs;uppo&longs;itionem.
quod iam à nullo in dubium reuocari po
te&longs;t; nam &longs;i comprimatur intra vas æneum v.g. etiam minimæ cra&longs;&longs;itu
dinis; &longs;i deinde ponderetur, maius e&longs;t haud dubiè pondus, quo maior
e&longs;t aëris copia intru&longs;a; atqui non modo triplum totius aëris, qui ante
compre&longs;&longs;ionem totam va&longs;is capacitatem occupabat intrudi pote&longs;t, vel
decuplum; verùm etiam vigecuplum;
immò centuplum, & millecuplum
adhibita cochleâ, vel alio mechanico organo, & aucta va&longs;is cra&longs;&longs;itudine,
de quo aliàs: quanta verò &longs;it grauitas aëris comparata cum grauitate
aquæ, cen&longs;et Galileus e&longs;&longs;e ferè vt 1. ad 400. Mer&longs;ennus verò vt 1. ad
1356. vel &longs;altem vt 1.ad 1300. Nos maiorem illà; hâc vero minorem
e&longs;&longs;e ob&longs;eruauimus, de quo aliàs; nec enim e&longs;t præ&longs;entis in&longs;tituti, pro
quo &longs;ufficiat modò, aëri aliquam ine&longs;&longs;e grauitatem; nec dicas aëra le
uem e&longs;&longs;e; nam reuerâ leuis e&longs;t, &longs;i comparetur cum aqua;
grauis autem &longs;i
comparetur cum a&longs;cendente halitu, vel fortè cum vacuo; nec e&longs;t quod
aliquis fortè metuat, ne &longs;i aër &longs;it grauis, ab eo tandem opprimatur, nam
etiam&longs;i aqua &longs;it grauis non tamen opprimit vrinatores, cuius rei veri&longs;&longs;i
mam rationem &longs;uo loco afferemus; denique non e&longs;t quod aliqui &longs;atis
incautè re&longs;pondeant, ip&longs;um aëra non e&longs;&longs;e grauem, &longs;ed tantùm &longs;entiri ali
quod pondus cra&longs;&longs;ioris vaporis immixti; nam de alio aëre non affirmo
nos ambit: adde quod Ari&longs;toteles l.4.
uitatem his verbis;
grauitatem.
grauitationem &longs;ingularem; hoc e&longs;t corpus graue in medium æquè graue non
grauitatquia &longs;i grauitaret de&longs;cenderet;
&longs;ic pars aquæ in aliam partem
aquæ non grauitat, & &longs;i aqua ponderetur in aqua, nullius ponderis e&longs;t;
cum enim nulla &longs;it ratio cur vna &longs;it infrà potiùs, quàm alia, vna certè al
terius locum non ambit; igitur caret grauitatione &longs;ingulari.
riscerta e&longs;t hypothe&longs;is;
nec enim plumbum e&longs;t eius ponderis &longs;ingula
ris in aqua, cuius e&longs;t in aëre; dixi &longs;ingularis; nam &longs;i plumbum & ip&longs;a
aqua &longs;imul appendantur, haud dubiè totum habebis pondus plumbi, &
totum pondus aquæ; ratio verò huius effectus non e&longs;t huius loci;
quid
quid &longs;it, &longs;i æqualis grauitas medij tollit totam æqualem alterius corpo
ris; certè maiorem alterius corporis totam non tollit per Th. 80. &longs;ed
tantùm aliquid illius, quod quomodo fiat, dicemus Tomo quinto cum de
graui, & leui.
e&longs;t æqualis &longs;uæ grauitationi.
&longs;ubduplum grauitationis; &longs;i &longs;ubdecupla, &longs;ubdecuplum, atque ita dein
ceps; hoc iam olim &longs;uppo&longs;uit magnus Archim. &longs;upponunt etiam reliqui
omnes, præ&longs;ertim recentior Galileus; &longs;i enim æqualis &longs;uperat æqualem,
ergo inæqualis pro rata; &longs;cilicet &longs;ubdupla &longs;ubduplum &longs;ubtripla, &c. Præ
terea, cum detrahat aliquam partem grauitationis maioris per Th.85.nec
detrahat inæqualem maiorem, per Th.80.nec inæqualem minorem; cur
enim potius vnam minorem quam aliam? certè æqualem tantùm
detrahere pote&longs;t, quod &longs;uo loco per Principium po&longs;itiuum demon&longs;tra
bimus.
aëre, quàm in vacuohinc etiam maioris &longs;unt ponderis in aëre quam in
aqua; hinc &longs;i grauitas alicuius corporis &longs;it ad grauitatem aëris vt 100.
ad 1. haud dubiè decre&longs;cet eius pondus in aëre (1/100); id e&longs;t, &longs;i penderet 100.
libras in vacuo, in aëre penderet 99. & eo tempore quo in vacuo decur
reret 100. pa&longs;&longs;us, in aëre decurreret 99. &longs;i nulla &longs;it aliunde re&longs;i&longs;tentia,
qualis reuerâ e&longs;t, vt dicam infrà; &longs;imiliter &longs;i grauitas alicuius corporis
&longs;it ad grauitatem aquæ, vt 10. ad 1. decre&longs;cet eius pondus in aqua (1/10), &
eo tempore quo decurreret in vacuo 10. palmos &longs;patij, in aqua decurre quanta verò &longs;it
grauitas omnium corporum tùm duriorum, qualia &longs;unt metalla, tùm li
quidorum, tùm &longs;pirabilium, dicemus &longs;uo loco; illorum tabulas habes
apud Gethaldum, & Mer&longs;ennum.
dere vtriu&longs;que, medij & corporis grauis, &longs;patio, quod in vno illorum conficit,
cogno&longs;ci po&longs;&longs;et &longs;patium, quod in alio conficeret æquali tempore
mus grauitatem aquæ e&longs;&longs;e ad grauitatem aëris vt 400. ad 1. &longs;itque corpus,
cuius grauitas &longs;it dupla grauitatis aquæ; haud dubiè eo tempore, quo
conficit in aëre 799. &longs;patia, in aqua conf;iciet tantùm 400. quia in vacuo
conficeret 800. aër autem detrahit (1/800), & aqua 1/2, vt con&longs;tat ex dictis; &longs;i
militer cognitis &longs;patiis in vtroque medio confectis, & grauitate vtriu&longs;que
medij cogno&longs;ceretur grauitas corporis de&longs;cendentis; quia tamen e&longs;t alia
re&longs;i&longs;tentiæ ratio, hîc non hæreo.
Ob&longs;eruabis dictum e&longs;&longs;e hactenus;
&longs;i nihil aliud de&longs;cen&longs;um corporis
grauis impedit; nam certè aliud e&longs;t, de quo infrà, ex cuius ignoratione
plures haud dubiè in inue&longs;tigandis grauitatum medij rationibus hallu
cinarentur; cum enim ob&longs;eruatum &longs;it globum plumbeum, cuius graui
tas e&longs;t ferè dodecupla grauitatis aquæ, conficere in libero aëre 48. pedes
&longs;patij tempore duorum &longs;ecundorum, in aqua verò 12. pedes eodem tem
pore; certè in vacuo ip&longs;o moueretur tardiùs quàm in aëre;
quia eo tem
pore, quo conficit in aqua 12.pedes in vacuo conficeret (13 1/21), &longs;i tantùm
detrahitur (1/12) grauitationis, & de&longs;cen&longs;us; atqui in aëre eodem tempore
conficit 48. pedes; igitur velociùs moueretur in aëre quàm in vacuo;
igitur e&longs;t aliquid aliud quod impedit motum; vt enim optimè monet
Mer&longs;ennus, &longs;i grauitas aquæ &longs;it ad grauitatem aëris vt 400 ad 1.& graui
tas plumbi ad grauitatem aquæ vt 12. ad 1.eadem grauitas plumbi e&longs;t ad
grauitatem aëris vt 4800. igitur &longs;i &longs;patium, quod decurrit plumbum in
vacuo diuidatur in 4800. partes, decurret in aëre 4799. partes; in aqua
verò 4400. quod e&longs;t contra experientiam; nam &longs;patium, quod decurrit
in aëre e&longs;t maius &longs;patio, quod decurrit in aqua 3/4; quippe conficit 12.
pedes in aqua eodem tempore, quo in aëre conficit 48; igitur in aqua
amittit 3/4 &longs;uæ grauitationis, & &longs;ui motus; igitur 3600. partes; igitur
plumbi grauitas e&longs;&longs;et ad grauitatem aquæ vt 4.ad 3.& ad grauitatem aë
ris vt 3600. ad 1. atqui vtrumque fal&longs;um e&longs;&longs;e con&longs;tat; igitur e&longs;t aliquid
aliud, quod etiam impedit motum; nec ex motu diuer&longs;o per diuer&longs;a me
dia cogno&longs;ci pote&longs;t eorum grauitas.
corporis de&longs;cendentis per diuer&longs;a media pro diuer&longs;a ratione grauitatum vtriu&longs;
que medynam aqua &longs;it ad grauitatem aëris vt
400. ad 1. deberet omne corpus de&longs;cendere velociùs in aëre quadrin-
do de&longs;cendat in aqua, quod de&longs;cendit in aëre, vt lignum.
medium loco cedat, vel aliquæ partes eiu&longs;dem medij,
pus non pote&longs;t penetrari cum alio.
patet etiam, tùm
quia ad mouendum totum medium exigua vis corporis grauis non &longs;uffi
cit; tùm quia tàm facilè per medium durum eiu&longs;dem grauitatis de&longs;cen
deret; denique patet manife&longs;tâ experientiâ.
medium, vel aliquæ eius partes, per Th.90.non primum per Th.91. igitur
&longs;ecundum, in his certè non e&longs;t vlla difficultas.
nec moueri &longs;ine impetu, nec
habere impetum, ni&longs;i producatur in illis à cau&longs;a aliqua applicata; quæ certè
alia none&longs;t quàm impetus corporis de&longs;cendentis,
mus primo lib.
&longs;eparantur, & &longs;uo appul&longs;u, vel impul&longs;u alias multas impellunt, ac &longs;eparant,
atqui &longs;eparari non po&longs;&longs;unt ab aliis, ni&longs;i &longs;oluatur vnio, &longs;eu nexus,
quo cum aliis deuinciuntur; quidquid tandem &longs;it illa vnio, de qua
aliàs.
impetu opus e&longs;t, vt &longs;olui po&longs;&longs;it, vt con&longs;tat.
arctiorem nexum partium ligni, qui ab impetu plumbi quantumuis gra
ui&longs;&longs;imi &longs;uperari non pote&longs;t; hinc corpus illud, medium tantùm appello
in quo po&longs;&longs;int corpora moueri, cuius nexus &longs;uperari pote&longs;t à corpore
grauiori in aliqua &longs;altem figura, vel &longs;itu; hinc corpora dura non po&longs;&longs;unt
e&longs;&longs;e medium; immò neque mollia, vt cera, argilla; &longs;ed vel liquida, vel
&longs;pirabilia.
beat arctiorum partium implicationem & nexumquia non modo partes
verùm etiam nexus ille partium &longs;ol
uendus; igitur ex vtroque capite impeditur motus.
quibu&longs;dam filamentisigitur cum aëris partes &longs;int magis lubricæ,
quàm partes aquæ, & faciliùs per obuia quæque foramina irrepere po&longs;
&longs;int, non po&longs;&longs;unt ita contineri; &longs;ic videmus multùm aquæ hauriri, dum
arctioribus retibus attollitur; immò dum aquam manu &longs;tringimus, ali
quam re&longs;i&longs;tentiam &longs;en&longs;u percipimus; quæ certè nulla e&longs;t, dum aëra &longs;trin
gimus.
Ob&longs;eruabis vnionem continuatiuam corporum aliquando po&longs;itam
e&longs;&longs;e in plexu, vel implicatione partium, vt videmus in fune, ligno, carne,
o&longs;&longs;ibus, &c. aliquando in vacui metu;
&longs;ic aqua, vt &longs;uo va&longs;i adhæreat,
a&longs;cendit, vel &longs;ur&longs;um attollitur, ne detur vacuum; aliquando in coitione
quadam magnetica; porrò hic plexus con&longs;tat ex infinitis ferè tenui&longs;&longs;i
morum filamentorum voluminibus, vel aduncis &longs;iue hamatis partibus,
&longs;eu corpu&longs;culis: Vtrum verò præter hæc requiratur alius vnionis mo
dus, di&longs;cutiemus fusè Tomo 5. quidquid &longs;it; certum e&longs;t medium illud,
cuius partes arctiori maiorique nexu copulantur, longè difficiliùs per
curri po&longs;&longs;e, &longs;eu perrumpi.
uitas &longs;it dedecupla grauitatis aquæ, verùm etiam propter re&longs;istentiam petitam
ex alio capite aliquid adhuc detrahere pote&longs;t&longs;cilicet quia partes aquæ non
po&longs;&longs;unt amoueri, ni&longs;i ab aliis &longs;eparentur; atqui maiore vi opus e&longs;t ad
&longs;oluendum &longs;trictiorem nexum; immò licèt partes aquæ nullo penitus
nexu vniantur, &longs;ed tantùm vel vacui metu, vel alio modo, quod alibi ex
plicabimus; omninò detraherent adhuc plumbo (1/12) motus;
igitur, &longs;i
præter illud impedimentum, quod petitur à comparatione grauitatis
corporis mobilis cum grauitate medij, addatur aliud longè robu&longs;tius;
non mirum e&longs;t, &longs;i maior inde &longs;equatur effectus, id e&longs;t maior imminutio
motus, qui qua&longs;i frangitur ab impedimento.
currere plumbeam pilam in aërehinc
tenui nexu partes aëris copulantur; partes verò aquæ firmiori;
hinc aër
minùs re&longs;i&longs;tit etiam motibus violentis; hinc vix pote&longs;t qui&longs;piam in aqua
currere propter maiorem aquæ re&longs;i&longs;tentiam; hinc pote&longs;t dici quota parte
firmior &longs;it nexus vnius corporis quàm alterius; hinc non tantùm copu
lantur partes metu vacui; alioquin æquè re&longs;i&longs;terent partes aëris, ac par
tes aquæ ratione nexus; hinc videntur guttulæ illæ &longs;phericæ inuolui te
nui qua&longs;i membranula, &longs;eu &longs;uperficie, cuius analogiam videmus in aqua in bullis, quæ ex guttis pluuiæ re&longs;ilientibus na&longs;ci videntur;
in
bullis etiam illis &longs;aponariis, quas leui calamo pueri inter ludendum in
flant; hinc ex minimo ferè contactu guttula &longs;pargitur, ni&longs;i fortè cum
multo a&longs;per&longs;a puluere cru&longs;tam quamdam induit &longs;olidiorem; &longs;ic bullæ il
læ ad minimum etiam contactum di&longs;&longs;ipantur; hinc ip&longs;a &longs;uperficies
aquæ plus videtur re&longs;i&longs;tere quod multis experimentis comprobatur; &longs;ed
illo maximè, quo videmus findi à remo cum quodam qua&longs;i &longs;tridulo cre
pitu re&longs;i&longs;tentiæ maioris te&longs;te; immò cum ab ip&longs;a naui qua&longs;i &longs;ulcatur,
idem &longs;tridor auditur, maximè in iis tractibus; in quibus nullis fluctibus
agitata læuigati&longs;&longs;imam faciem præfert; habes analogiam in illa cru&longs;ta,
quæ concre&longs;cit in &longs;uperficie liquorum, &longs;ed præ&longs;ertim o&longs;&longs;arum: adde quod
aër paulò compre&longs;&longs;ior vndique guttulam premens æquali ni&longs;u eam miri
ficè tornat: hæc tantùm tumultuatim conge&longs;ta alibi fusè pertractabi
mus, & ex &longs;implici&longs;&longs;imis principiis demon&longs;trabimus; plura hîc de graui
tate crant dicenda, & de grauitatione, quæ tantùm indica&longs;&longs;e &longs;ufficiat, vt
deinde Tomo quinto fusè explicentur.
modo comprimi nece&longs;&longs;e e&longs;t, vel in&longs;en&longs;ibilitercum enim tantus relinquatur
locus retrò, quantus acquiritur antè, nulla opus e&longs;t compre&longs;&longs;ione; &longs;ed
partes à fronte pul&longs;æ factâ circuitione retror&longs;um eunt, non certè tramite
recto; &longs;i enim frons ip&longs;ius lata &longs;it, haud dubiè partes pul&longs;æ alias pellunt,
& hæ vici&longs;&longs;im alias longo circuitu, vt patet experientia; nulla tamen, vel
modica fieri videtur compre&longs;&longs;io.
&longs;tentiaigitur maiore vi opus e&longs;t, igitur maiore grauitate;
&longs;ed in medio
den&longs;iore ab eodem mobili plures &longs;eparantur quàm in rariore; quia &longs;ci
licet corpus den&longs;um plures habet &longs;ub minori exten&longs;ione, & rarum è con
trario, vt videbimus &longs;uo loco; igitur in medio den&longs;iore idem mobile ma
jorem re&longs;i&longs;tentiam inuenit, quàm in rariore; licèt vtriu&longs;que partes
æquali nexu &longs;eu fibula copulentur; quia &longs;cilicet plures &longs;unt diuidendæ
in den&longs;iore; quia plures &longs;cilicet in æquali &longs;patio occurrunt, quàm in ra
riore; igitur maiore vi grauitatis opus e&longs;t.
Primum e&longs;t in
grauitate, vel den&longs;itate, nam reuerâ ex maiori den&longs;itate maiorem gra
uitatem reducimus; Secundum e&longs;t in maiori, vel minori partium nexu,
ex quo 4. &longs;equuntur combinationes 2.mediorum; nam vel &longs;unt eiu&longs;dem
grauitatis, & mollitiei; vel eiu&longs;dem grauitatis & diuer&longs;æ mollitiei;
vel
eiu&longs;dem mollitiei, & diuer&longs;æ grauitatis; vel diuer&longs;æ grauitatis, & eiu&longs;
dem mollitiei; mollius autem illud appello, cuius partes laxiori nexu
copulantur; porrò 4. i&longs;tæ combinationes &longs;upponunt
medio; &longs;i &longs;it prima combinatio, motus e&longs;t æqualis in vtroque;
&longs;i &longs;ecunda
&longs;i tertia maior in grauiori;
&longs;i verò quarta &longs;ubdi
uidi pote&longs;t in duas; nam vel grauius e&longs;t conjunctum cum maiori molli
tie, vel leuius; &longs;i leuius, haud dubiè maior e&longs;t motus in leuiore;
&longs;i gra
uius & mollities compen&longs;et grauitatem, id e&longs;t, &longs;i vt &longs;e habet grauitas gra
uioris ad leuitatem leuioris; ita &longs;e habet mollities illius ad mollitiem
huius, æqualis e&longs;t in vtroque; &longs;i &longs;ecus, pro rata; hinc pote&longs;t e&longs;&longs;e æqualis
motus in grauiore & leuiore medio, & in æquè graui pote&longs;t e&longs;&longs;e maior
in grauiore; & minor; maior quidem, &longs;i maior &longs;it ratio mollitiei gra
uioris ad mollitiem leuioris, quàm grauitatis ad grauitatem; minor ve
rò, &longs;i maior &longs;it ratio grauitatis ad grauitatem, quàm mollitiei ad molli
tiem; æqualis denique &longs;i æqualis ratio;
& his regulis cuncta facilè ex
plicari po&longs;&longs;unt; hîc porrò &longs;uppono idem mobile, quod per vtrumque me
dium de&longs;cendere po&longs;&longs;it, id e&longs;t, quod &longs;it vtroque grauius, medium autem
appello illud, per quod mobile grauius per &longs;e de&longs;cendit; dixi per &longs;e quia
nonnunquam accidit, vt vel ratione figuræ, vel alterius impedimenti non
de&longs;cendat.
prima, &longs;i &longs;it idem mobile
cum diuer&longs;is mediis; &longs;ecunda, &longs;i idem medium cum diuer&longs;is mobilibus;
tertia &longs;i diuer&longs;a mobïlia cum diuer&longs;is mediis; de primâ actum e&longs;t iam
&longs;uprà; &longs;ecunda &longs;ube&longs;t 4. combinationibus. Prima &longs;i mobilia &longs;int eiu&longs;
dem materiæ, &longs;ed diuer&longs;æ figuræ; Secunda eiu&longs;dem figuræ & diuer&longs;æ
materiæ. Quarta diuer&longs;æ materiæ & figuræ;
&longs;i prima & &longs;ecunda, vel &longs;unt
figuræ æquales, vel inæquales; &longs;i primum &longs;unt eiu&longs;dem grauitatis; &longs;i &longs;e
cundum diuer&longs;æ; quippe figuræ &longs;imiles po&longs;&longs;unt e&longs;&longs;e æquales, vel inæ
quales; & figuræ æquales po&longs;&longs;unt e&longs;&longs;e &longs;imiles, vel di&longs;&longs;imiles;
&longs;i &longs;it tertia
combinatio, in qua &longs;int eiu&longs;dem figuræ, & diuer&longs;æ materiæ, diuer&longs;æ in
quam in grauitate; &longs;i figuræ &longs;unt æquales, &longs;emper e&longs;t diuer&longs;a grauitas;
&longs;i
inæquales pote&longs;t e&longs;&longs;e vel eadem, vel tertia; in quarta combinatione di
uer&longs;a compen&longs;atio fieri pote&longs;t; idem dicendum e&longs;t de tertia combinatio
ne diuer&longs;orum mobilium, & mediorum, de quibus omnibus &longs;eor&longs;im iam
dicemus.
&longs;cendant, æquali motu ferunturvbi e&longs;t eadem proportio cau&longs;æ & re&longs;i
&longs;tentiæ ibi e&longs;t idem effectus, per Ax. 5. &longs;ed in hoc ca&longs;u eadem e&longs;t illa pro
portio; nam e&longs;t æqualis cau&longs;a, &longs;cilicet grauitas;
idem medium æqualiter
vtrique re&longs;i&longs;tens, cum non plures medij partes re&longs;i&longs;tant vni, quam alteri;
igitur æqualis proportio.
eius partes re&longs;istunt quàm cum pauciores
&longs;pondet pluribus partibus cau&longs;æ per Ax.13.l.1. num.2.
quip
pe in tantum re&longs;i&longs;tunt, in quantum ab aliis &longs;eparantur; atqui in tantum
&longs;eparantur, in quantum amouentur è &longs;uo loco; &longs;ed ideo amouentur è
&longs;uo loco, in quantum pelluntur; igitur cum plures pelluntur tunc plures
re&longs;i&longs;tunt; igitur tunc maior e&longs;t re&longs;i&longs;tentia.
parallela horizonti.
tot pelluntur quot re&longs;pondent primæ faciei, &longs;eu primo plano, quod e&longs;t in
fronte.
terum, & ratio &longs;olidorum triplicata,
GK, nam in gratiam eorum qui Geometriam ignorant hoc ip&longs;um ocu
lis &longs;ubiiciendum e&longs;&longs;e videtur; diuidantur 6. eius facies in 4. quadrata
æqualia v. g. facies AI in quad. AE. EC. EG. EI. idem fiat in aliis
5. faciebus, quarum duæ hîc tantum apparent; &longs;cilicet AK. KL; &longs;ed
tribus aliis parallelis; his tribus cædem diui&longs;iones re&longs;pondent;
haud
dubiè erunt cubi minores, quorum latus &longs;it æquale AB, & quælibet fa
cies æqualis quadrato AE, &longs;ed facies maior AI, e&longs;t quadrupla minoris
AE, ergo AI e&longs;t ad AE vt quadratum lateris AG ad quadratum lateris
AD; &longs;ed hæc e&longs;t ratio duplicata laterum 1. 2. 4. &longs;imiliter cubus maior
GK e&longs;t octuplum minoris DN, igitur vt cubus lateris AG ad cubum
lateris AD. &longs;ed hæc e&longs;t ratio triplicata. 1.2.4.8.
quàm latuspatet ex dictis, nam latus minoris cubi e&longs;t tantùm &longs;ubdu
plum lateris maioris, & facies &longs;ubquadrupla; &longs;olidum verò &longs;ub
octuplum.
quia grauitas
re&longs;pondet &longs;olido, & re&longs;i&longs;tentia prim&etail; faciei; re&longs;i&longs;tentia
tium medij; &longs;ed &longs;olidum plus minuitur quàm facies, vt dictum e&longs;t;
igitur
plus minuitur grauitas, quæ e&longs;t cau&longs;a virium quàm hæc re&longs;i&longs;tentia; ergo
decre&longs;cunt vires in maiore proportione quàm hæc re&longs;i&longs;tentia, quod be
nè ob&longs;eruauit Galileus in dìalogis.
Hinc concludit Galileus duos cubos eiu&longs;dem materiæ, &longs;ed inæquales
de&longs;cendere inæquali motu; maiorem &longs;cilicet velociùs minori;
demon
&longs;trare videtur, quia maior habet maiorem proportionem virium ad re
&longs;i&longs;tentiam, quàm minor; igitur maiorem habet effectum per Ax. 5. igi
tur maiorem, & velociorem motum.
Scio non dee&longs;&longs;e multos viros doctos qui acriter in hanc &longs;ententiam
Obiicient fortè primò, experientiam e&longs;&longs;e contrariam;
&longs;i enim
accipiantur duo cubi maior, & minor eiu&longs;dem materiæ, & dimittantur
ex eadem altitudine eodem pror&longs;us momento terram ferient; Re&longs;ponde
ri pote&longs;t momentum illud &longs;en&longs;u percipi non po&longs;&longs;e; &longs;i enim dicam ma
iorem tangere terram 1000. in&longs;tantibus ante minorem, an fortè &longs;en&longs;u
hoc percipies, vi&longs;u &longs;cilicet vel auditu? igitur in maxima altitudine hæc
&longs;patiorum inæqualitas, & temporum &longs;en&longs;u percipi po&longs;&longs;et, quæ in minori
&longs;ub &longs;en&longs;um non cadit: præterea accipe pulueris granulum eiu&longs;dem ma
teriæ, tuncque etiam &longs;en&longs;ibilem motuum differentiam videbîs, atqui
e&longs;t eadem ratio de omni minore.
Secundò obiicient, &longs;i &longs;uperponatur cubus minor maiori in &longs;uo motu
nunquam &longs;eparantur; igitur æquali motu de&longs;cendunt. Re&longs;p. videri po
te&longs;t equidem æquali motu de&longs;cendere quia &longs;unt veluti partes eiu&longs;dem
corporis, & grauitant grauitatione communi, neque minor habet &longs;ingu
larem re&longs;i&longs;tentiam &longs;uperandam; immò &longs;i &longs;uperponatur minor maiori,
vel maior minori, motus e&longs;t velocior quàm e&longs;&longs;et &longs;olius maioris; quia
cum non &longs;it maior re&longs;i&longs;tentia, maiores illi vires opponuntur; igitur fa
ciliùs &longs;uperatur.
Tertiò obiicient;
e&longs;t eadem &longs;pecie grauitas;
igitur eadem grauitatio,
idemque motus deor&longs;um; Re&longs;ponderi po&longs;&longs;et concedendo antecedens,
vnde in vacuo omnia grauia æquè velociter de&longs;cenderent, &longs;i in eo mo
tus e&longs;&longs;et; at verò altera duarum cau&longs;arum eiu&longs;dem &longs;peciei, quæ habet mi
norem proportionem actiuitatis ad re&longs;i&longs;tentiam, profectò minùs agit,
quod certum e&longs;t.
Quartò obij:igitur motus po&longs;&longs;et e&longs;&longs;e velocior, & velocior in infini
tum; &longs;i enim maior cubus de&longs;cenderet velociùs;
igitur &longs;i detur maior ad
huc velociùs, atque ita deinceps: Re&longs;p. inanem pror&longs;us e&longs;&longs;e difficulta
tem; quia cubus ille quantumuis maximus in vacuo de&longs;cendit velociùs
quàm in aliquo medio v.g.in aëre, igitur nunquam augmentum veloci
tatis infinitum e&longs;t; quippe inter duos gradus velocitatis infiniti &longs;unt
po&longs;&longs;ibiles. v. g. &longs;it velocitas, quam habet in vacuo vt 2. illa verò quàm
habet in aëre vt 1. &longs;i cre&longs;cat velocitas iuxta has minutias &longs;ingulis in&longs;tan
tibus 1/2 1/4 1/8 (1/16) (1/32), atque ita deinceps; quàm porrò multæ &longs;unt huiu&longs;modi
progre&longs;&longs;iones 1/3 1/6 (1/12) (1/24) &c. igitur obiectiones illæ non euertunt Gali
lei &longs;ententiam.
Inde idem Galileus o&longs;tendere videtur cur atomi materiæ etiam gra
ui&longs;&longs;imæ, &longs;eu granula pulueris motu tardi&longs;&longs;imo de&longs;cendant in aëre vel in
aqua; quia &longs;cilicet per illam diui&longs;ionem ita imminutæ &longs;unt vires graui
tatis, vt iam re&longs;i&longs;tentiam medij &longs;uperare non po&longs;&longs;int.
Sed videtur e&longs;&longs;e graui&longs;&longs;ima difficultas, &longs;int enim duo cubi, maior B
F, minor GM, & vterque innatet medio liquido duplo grauiori; certè ex
tabit maior toto rectangulo CA æquali CF, & minor toto rectangulo
KH æquali KM; igitur e&longs;t eadem proportio grauitatis maioris ad re&longs;i
&longs;tentiam medij in grauitatione, quæ e&longs;t minoris; igitur & in motu.
Re&longs;ponderi pote&longs;t e&longs;&longs;e maximam di&longs;paritatem inter grauitationem, &
&longs;it enim cubus BD qui de&longs;cendat per totam AH;
haud dubiè
cum &longs;patium DI, contineat 3. cubos medij æquales DB, eos debet remo
uere in &longs;uo de&longs;cen&longs;u; &longs;it autem cubus BG;
haud dubiè, cum &longs;it eadem pro
portio cubi AE ad cubum medij DM, quæ e&longs;t cubi BG ad cubum me
dij FL, eodem tempore vterque cubum medij &longs;uppo&longs;iti è &longs;uo loco extru
det; igitur eo tempore, quo AE expellet 3. DI, FL extrudet 3. EO, ergo
æquabili tempore inæquale &longs;patium percurrunt.
Dices ergo &longs;patia &longs;unt vt latera:
Re&longs;ponderi pote&longs;t hoc reuerâ per &longs;e
e&longs;&longs;e debere; &longs;ed quia cubus DM vt extrudatur, maiorem debet facere cir
cuitionem, vt à fronte retrò eat, velociori motu extrudi debet; igitur vi
res &longs;uas in eo con&longs;umit maiori ex parte cubus AE; hinc compen&longs;atio e&longs;&longs;e
videtur.
Vt &longs;olui po&longs;&longs;it præ&longs;ens difficultas, quæ cettè maxima e&longs;t, totam rem
i&longs;tam paulò fu&longs;iùs e&longs;&longs;e explicandam iudico. Primò itaque certum e&longs;t
partes medij, quæ prius in fronte erant, retroire; hoc ip&longs;um videmus in
naui quæ &longs;ulcat aquas, hoc ip&longs;um accidit in omni corpore natante etiam
immobili, quippe partes aquæ retinentur ab illa membranula, de qua &longs;u
prà; &longs;ic enim &longs;æpè a&longs;&longs;urgunt, & intume&longs;cunt &longs;upra labra va&longs;is;
cur verò
continui penè circulares limbi dilatentur: Re&longs;p. nullo flante vento
vix aliquem circulum huiu&longs;modi in &longs;uperficie aquæ apparere à fronte,
&longs;ed tantùm à tergo, & lateribus, qua&longs;i ad in&longs;tar pyramidis; &longs;ed de his aliàs
fusè.
Secundò certum e&longs;t numerum partium, quas impellit maior cubus A
E; e&longs;&longs;e quadruplum numeri partium, quas impellit cubus BG:
&longs;int autem
v.g.8. partes re&longs;i&longs;tentes cubo maiori, &longs;unt duæ re&longs;i&longs;tentes cubo minoris;
&longs;ed vires cubi maioris &longs;unt ad vires cubi minoris vt 8. ad 1. igitur vires
vt 8. &longs;uperabunt faciliùs re&longs;i&longs;tentiam vt 8. quam vires vt 1. re&longs;i&longs;tentiam
vt 2.vnde duplò velociùs moueretur, ni&longs;i aër duplò velociori motu amo
uendus e&longs;&longs;et, quod vt clarius explicetur;
Sit cubus maior AF octuplus cubi GI, vt iam dictum e&longs;t;
haud
dubiè aër qui &longs;ub&longs;tat cubo AF e&longs;t quadruplus aëris, qui &longs;ub&longs;tat cubo GI,
vnde &longs;i vires cubi AF e&longs;&longs;ent quadruplæ virium cubi GI, e&longs;&longs;et æqualis
proportio in vtroque virium, & re&longs;i&longs;tentiæ; &longs;ed &longs;unt octuplæ;
igitur faci
liùs vincetur re&longs;i&longs;tentia; igitur amouebitur aër faciliùs; &longs;it autem aër
expre&longs;&longs;us in globulis EFB, &c. cuius &longs;uperficies cum relinquatur retrò
ver&longs;us AB, & occupetur illa quæ e&longs;t in fronte EF; haud dubiè partes
hinc inde diuiduntur in D, & &longs;egmentum NB tran&longs;it in locum relicti
loci BC, FN tran&longs;it in NB, & DF, in FN; idem dico de &longs;egmentis oppo
&longs;itis; idem pror&longs;us dico de minori globo;
nam MH tran&longs;it in HQ, & H
Q in QG, & QG in GL, idem dico de &longs;egmentis oppo&longs;itis; igitur hæc
e&longs;t circuitio partium medij, quàm &longs;uprà indicauimus; hinc aër, qui amo
uetur à corpore graui de&longs;cendente moueri debet nece&longs;&longs;ariò velociùs
quàm ip&longs;um corpus graue, quod de&longs;cendit.
In hoc porrò ob&longs;erua &longs;egmentum MH moueri tardiùs quàm DF;
quia
conficit &longs;ubduplum &longs;patium, eo tempore, quo DF conficit duplum;
e&longs;t; &longs;ed vires cubi AF &longs;unt ad vires cubi GI, vt 8. ad 1. partes verò aëris,
quas impellit AF, &longs;unt ad partes aëris, quas impellit GI, vt 4.ad 1. igitur
&longs;i partes aëris mouerentur æquali motu cum ip&longs;is cubis, à quibus mo
uentur; certè maior moueretur motu velociori;
vt autem moueantur par
tes DF duplò velociore motu, quàm partes MH; debent vires, quæ mo
nent DF, e&longs;&longs;e in ratione dupla ad illas, quæ mouent MH, id e&longs;t eo tem
pore, quo vires vt 8.mouebunt mobile vt 4. motu vt 2. vires vt 1.moue
bunt mobile vt 1. motu vt 1. licèt enim &longs;uperficies aëris EF moueatur
deor&longs;um; attamen ab alio aëere inferiore ita repertitur, vt &longs;ur&longs;um ver&longs;us
FN repellatur.
Equidem tota &longs;uperficies aëris DF, cum pluribus partibus con&longs;tet,
non pote&longs;t &longs;imul tran&longs;ire in FN; quia pars D antequam perueniat ad F
tran&longs;it per medium DF; igitur &longs;ucce&longs;&longs;iuè per mea ad illud &longs;patium DF,
quo tempore quie&longs;ceret globus AF, quod ridiculum e&longs;t.
Quare fit nece&longs;&longs;ariò aliqua circuitio, & partium aëris commixtio,
&longs;eu conflictus; ita vt retroeant pul&longs;æ prius & repercu&longs;&longs;æ;
non quidem
tramite recto, &longs;ed cum aliqua circuitione; quod certè facilè concipi po
te&longs;t, quæ circuitio eò maior e&longs;t, quo latera cuborum &longs;unt maiora; ita
que cum hæc &longs;atis fusè videantur e&longs;&longs;e explicata, &longs;it.
cenduntprobatur, quia licèt &longs;it maior proportio actiuitatis minus ad
&longs;uam re&longs;i&longs;tentiam, quàm alterius; illud tamen compen&longs;atur;
eóque par
tes aëris velociùs moueri debeant iuxta rationem laterum, vt patet ex
dictis; vnde nece&longs;&longs;ariò &longs;equitur motus æqualis in vtroque cubo;
igitur
licèt maioris cubi vires habeant maiorem proportionem ad molem,
quæ præcipuum illius motus retardat; tum tamen aër, qui re&longs;i&longs;tit maiori
cubo debeat amoueri, vt dictum e&longs;t velociore motu quam aër, qui re&longs;i
&longs;tit minori, &longs;itque eadem proportio re&longs;i&longs;tentiæ ratione motus minoris
ad maiorem, quæ e&longs;t ratione molis maioris ad minorem; certè ratio
compo&longs;ita vtriu&longs;què erit eadem in vtroque cubo; igitur æquè velociter
vterque de&longs;cendet: hinc &longs;atís facilè &longs;oluitur ratio Galilei, quam multi
parum cauti pro demon&longs;tratione venditarunt, ad aliam verò rationem,
quam ex minuto puluere ducere videtur, etiam facilè re&longs;ponderi pote&longs;t;
ideo corpu&longs;cula illa diu fluitare in aëre, tùm quòd minimo ferè tenuis
auræ flatu agitentur; &longs;ic pulueris nubes medius ventus agit;
quis enim
ne&longs;cit aëris partes agitari perpetuò; immò & aquæ inter &longs;e mi&longs;ceri;
igi
tur ab agitationis veluti impre&longs;&longs;ione fluitant illa corpu&longs;cula, cum mini
mus ferè impetus extrin&longs;ecus illa commouere po&longs;&longs;it; tùm etiam quòd à
filamentis illis, quibus partes aëris implicantur facilè detineantur; ana
logiam habes in lapillo, qui ab araneæ tela intercipitur.
motu propter immò e&longs;t perfectior æqualitas in globis,
quàm in cubis; quia perfectior fit circuitio, vt con&longs;ideranti patebit;
hinc globus eiu&longs;dem materiæ, & grauitatis cum cubo de&longs;cendit velociùs
quia &longs;cilicet aër in de&longs;cen&longs;u globi faciliùs agitur retrò, vt con&longs;tat.
quâm globus eiu&longs;dem materiæquia breuiore circuitu partes re
troeunt; quippe tunc maxima e&longs;t facilitas in pellendo aëre, qui e&longs;t à fron
te mobilis, cum velociùs moueri non debet ip&longs;o mobili; atqui hoc ip
&longs;um e&longs;t quod accidit mobili vtrimque aucto; nam linea curua DBA,
quam percurrit de&longs;criptum mobile, non e&longs;t multò longior; at verò in
quadrato &longs;uperiori AF maiori e&longs;t duplò; in circulo quidem minor dia
meter &longs;emiperipheriæ, &longs;ed non duplò.
ctangulum BF &longs;it in fronte tardiùs de&longs;cendet, quàm &longs;i in fronte &longs;it re
ctangulum CE, vel rectangulum FH; hinc tribus motibus diuer&longs;is de&longs;
cendere pote&longs;t idem parallipedum, modò habeat &longs;emper alteram facie
rum horizonti parallelam; hinc cylindrus eiu&longs;dem grauitatis de&longs;cendet
velociùs quàm parallelipedum, vt patet ex dictis; ex quibus facilè intel
ligi pote&longs;t, quænam corpora faciliùs quàm alia de&longs;cendant; quippe illa
regula e&longs;t certi&longs;&longs;ima quàm &longs;uprà attulimus. Porrò ob&longs;eruabis omne
corpus difficiliùs pelli per lineam perpendicularem quàm per obliquam;
hinc globus pellit tantùm vnicum punctum perpendiculariter; idem di
co de cono; cylindrus verò vnam lineam, cubus integrum planum.
in fronte, habet maiorem, inæquali motu de&longs;cendunt
vtroque &longs;unt æquales vires, &longs;ed diuer&longs;a re&longs;i&longs;tentia.
aëre fluitant, vel aquis innatant
vires.
Ob&longs;eruabis primam &longs;uperficiem aquæ habere maiorem quamdam re
&longs;i&longs;tentiam propter illam, qua&longs;i membranulam, de qua &longs;uprà; vnde a&longs;&longs;ur
git quiddam lymbus in margine bracteæ ferri, vel auri innatantis; vel
etiam globuli paulò grauioris aquâ, igitur vt immergatur corpus debet
e&longs;&longs;e grauius totâ illâ aquâ, quæ capacitatem illam non cauam occu
paret.
quia &longs;cilicet alte
rum e&longs;t grauius, quod &longs;uppono; igitur æqualis e&longs;t re&longs;i&longs;tentia, & vires
inæquales; igitur non e&longs;t eadem proportio actiuitatis: & re&longs;i&longs;tentiæ; igi
tur non e&longs;t æqualis motus per Ax.5.
mon&longs;tro; quia globi eiu&longs;dem materiæ inæqualiter de&longs;cendunt per Th.
113. &longs;ed duo globi æquales diuer&longs;æ materiæ de&longs;cendunt inæqualiter per
Th.118. igitur, & inæquales; quod dico de globis', dicatur de cubis, &
aliis figuris &longs;imilibus.
dum grauioriscon&longs;tat experientia; ratio e&longs;t, quia cum globus ferreus de&longs;
cendat velociùs, quàm ligneus per Th. 118. in data ratione, putà (1/100)
haud dubiè bractea ferri non modo (1/100) tardiùs de&longs;cendet, verùm etiam
(20/100) in quo non e&longs;t difficultas.
vel grauius, vel leuius, vel grauioris materiæ, vel leuioris velociùs de&longs;cendat
vt con&longs;tat ex regulis præ&longs;criptis.
inæqualiter iuxta proportionem grauitatis, & re&longs;i&longs;tentiæ medij compa
ratæ cum vtroque, v.g. plumbo detrahitur (1/4800); ligno verò (8/300) v. g. &longs;i
grauitas ligni &longs;it ad grauitatem aëris vt 300.ad 1. & plumbi vt 4800. ad
1. &longs;it enim altitudo 33. pedum 4. digit. reducantur in digitos erunt 400.
in lineas 4800. igitur detrahetur vna linea &longs;patij plumbeo globo; ligneo
verò vnus digitus cum 4. lineis; &longs;ed quis hoc ob&longs;eruet?
quia aër in perexigua
illa foramina inten&longs;us frangitur, re&longs;ilit, ac proinde motum impedit; talis
e&longs;t medulla &longs;ambuci, &longs;pongia, &longs;tupa, &c. immò a&longs;perum corpus tardiùs
de&longs;cendit, quòd &longs;cilicet aër ab a&longs;perioribus illis &longs;alebris re&longs;iliens mo
tum retardet, hinc &longs;ibilus ille auditur &c.
Ex his con&longs;tat quid dicendum &longs;it de motu corporum grauium in
medio, &longs;iue &longs;int eiu&longs;dem materiæ, & &longs;imilis figuræ, maioris vel minoris,
vel æqualis; tunc enim de&longs;cendunt æqualiter contra Galileum, &longs;iue
&longs;int diuer&longs;æ materiæ, & &longs;imilis figuræ, æqualis, vel inæqualis,
guræ; tunc enim de&longs;cendunt modò æqualiter, modò inæqualiter;
æquali
ter certè, cum figura compen&longs;at materiam; cum verò non compen&longs;at,
inæqualiter pro rata; denique &longs;i comparentur duo corpora cum diuer&longs;is
mediis; primo inuenienda e&longs;t proportio motuum vtriu&longs;que in eodem
tùm &longs;ingulorum in diuer&longs;is mediis, vt &longs;uprà dictum e&longs;t.
Probatur, quia tota
diuer&longs;itas vel inæqualitas mediorum petitur à diuer&longs;a proportione acti
uitatis cum re&longs;i&longs;tentia medij per Ax. 5. &longs;ed in vacuo nulla e&longs;t re&longs;i&longs;ten
tia; igitur nulla proportio; igitur nulla ratio motus inæqualis.
&longs;tantibusprobatur, quia &longs;ingulis in&longs;tantibus plures partes medij &longs;unt
&longs;uperandæ; cre&longs;cunt enim &longs;patia, vt con&longs;tat ex dictis; igitur cre&longs;cit re&longs;i
&longs;tentia &longs;ingulis in&longs;tantibus.
quia cre&longs;cit iux
ta rationem plurium partium medij, quæ temporibus æqualibus percur
runtur; &longs;ed eæ cre&longs;cunt iuxta rationem &longs;patiorum, vt con&longs;tat.
quæ ratio &longs;equitur progre&longs;&longs;ionem arithmeticam &longs;implicem numerorum
1.2.3.4.5.6. ex &longs;uppo&longs;itione quòd tempus con&longs;tet ex partibus finitis actu;
nam eodem modo cre&longs;cit velocitas, quo cre&longs;cunt numeri prædicti; &longs;ed
eodem modo cre&longs;cunt &longs;patia, &longs;i dumtaxat accipiantur in &longs;ingulis in&longs;tan
tibus; re&longs;i&longs;tentia cre&longs;cit iuxta rationem &longs;patiorum; igitur iuxta ratio
nem velocitatum.
Ob&longs;eruabis, &longs;i tempus con&longs;tet ex infinitis actu partibus, ita vt &longs;ingu
læ partes motus &longs;ingulis partibus temporis & infinitæ infinitis re&longs;pon
deant; non pote&longs;t e&longs;&longs;e alia progre&longs;&longs;io, in qua fiat acceleratio motus na
turalis, quàm illa Galilei iuxta hos numeros 1. 3. 5. 7. vt con&longs;tat ex dictis
per illud Principium;
tis momenta&longs;i verò tempus con&longs;tat ex finitis in&longs;tantibus æqualibus, nul
la datur progre&longs;&longs;io motus naturaliter accelerati; quia motus accelerari
non pote&longs;t; ne &longs;cilicet eodem in&longs;tanti mobile &longs;it in pluribus locis adæ
quatis; denique &longs;i tempus con&longs;tat ex finitis in&longs;tantibus actu, & infinitis
potentiâ, non pote&longs;t e&longs;&longs;e alia progre&longs;&longs;io huius accelerationis, quam hæc
no&longs;tra iuxta numeros toties repetitos 1.2.3.4.5. attamen quia illa finita
in&longs;tantia &longs;unt ferè innumera in qualibet parte &longs;en&longs;ibili temporis, in
praxi &longs;ine &longs;en&longs;ibili errore in partibus temporis &longs;en&longs;ibilibus po&longs;&longs;umus
tur, meo iudicio, propo&longs;itæ quæ&longs;tionis difficultas.
&longs;cant impetus in ratione velocitatum, vt con&longs;tat, & cre&longs;cat re&longs;i&longs;tentia
medij in eadem ratione per Theor. 127. cre&longs;cit etiam in ratione im
petuum.
lisquia cre&longs;cunt vires, vt cre&longs;cit impetus; nam impetus e&longs;t
vis illa, quâ mobile &longs;uperat re&longs;i&longs;tentiam medij vt con&longs;tat, &longs;ed re&longs;i&longs;ten
tia cre&longs;cit vt impetus per Th. 128. igitur cre&longs;cit in ratione virium.
progre&longs;&longs;io effectuum.
in quo e&longs;t re&longs;i&longs;tentia, vt 1. Secundo in&longs;tanti &longs;it impetus vt 2. re&longs;i&longs;tentia in
2. &longs;patiis vt 2. haud dubiè &longs;i vno in&longs;tanti vnus gradus impetus &longs;uperat
re&longs;i&longs;tentiam vt 1. dum percurrit 1.&longs;patium; certè 2. gradus impetus vno
in&longs;tanti &longs;uperabunt re&longs;i&longs;tentiam vt 2. dum conficit mobile 2. &longs;patia; at
que ita deinceps.
po&longs;t aliquod &longs;patium decur&longs;um ex naturaliter accelerato non fieri æquabilem,
quia in tantum fieret æquabilis in quantum tanta e&longs;&longs;et re&longs;i&longs;tentia, vt no
uam accelerationem impediret; &longs;ed hæc ratio nulla e&longs;t;
quia in eadem
ratione cre&longs;cit re&longs;i&longs;tentia, in qua cre&longs;cunt vires per Th. 129. igitur non
mutatur progre&longs;&longs;io motuum per Th. 130. igitur nec acceleratio; igitur
motus naturalis ex accelerato non fit æquabilis: Equidem, vt iam &longs;uprà
dictum e&longs;t, in minori &longs;emper ratione cre&longs;cit velocitas, itémque ip&longs;a re&longs;i
&longs;tentia quod in omni progre&longs;&longs;ione arithmetica iuxta numeros 1.2.3.4.5.
Ob&longs;eruabis remitti à nobis motum leuium &longs;ur&longs;um in 5. Tomum, in cu
ius tertio libro agemus de graui, & leui; quia ideo corpus a&longs;cendit, quia
ab alio de&longs;cendente truditur &longs;ur&longs;um.
&longs;ur&longs;um Perpendiculariter.
OMnis certè motus, qui e&longs;t à principio ex
trin&longs;eco, violentus appellari pote&longs;t, attamen
hîc non ago de omni violento, &longs;ed dumta
xat de illo, qui fit &longs;ursùm per lineam verticalem; quia
&longs;cilicet ex diametro opponitur motui naturali, qui
fit deorsùm perpendiculariter; igitur cum de hoc
ip&longs;o in &longs;ecundo Libro egerimus, de illo in hoc non
agemus.
neam verticalem à principio extrin&longs;eco mediatè, vel immediatè vt
plurimùm.
Dixi à principio extrin&longs;eco, &longs;iue conjuncto, vt cum manu attollo &longs;ur
&longs;um corpus graue, &longs;iue non conjuncto, vt cum quis proiicit lapidem &longs;ur
sùm, &longs;iue &longs;it verum principium effectiuum, vt cum impetus, quem poten
tia motrix producit in manu, producit alium in mobili; &longs;iue non &longs;it
principium effectiuum, &longs;ed tantùm determinans, vt cum mobile quod
cadit deor&longs;um, &longs;ur&longs;um deinde repercutitur; nec enim corpus repercu
tiens producit impetum nouum, vt dicemus cum de motu reflexo; quin
potiùs producti partem de&longs;truit per accidens, & quidquid illius &longs;upere&longs;t,
ad nouam lineam determinat; quod quomodo fiat fusè &longs;uo loco expli
cabimus, igitur licèt corpus reflectens &longs;it tantùm principium nouæ de
terminationis, non verò alicuius impetus producti, dici pote&longs;t princi
pium huius motus violenti.
Dixi vt plurimùm, nam &longs;i terra ducto per centrum foramine e&longs;&longs;et
peruia, haud dubiè lapis demi&longs;&longs;us versùs centrum iret motu naturaliter
oppo&longs;itum circumferentiæ punctum iret, motu certè violento, qui ta
men ab extrin&longs;eco non e&longs;&longs;et.
& nemo e&longs;t qui eam in dubium vocet.
pe de&longs;trui hoc tantùm dicitur, quod de&longs;init e&longs;&longs;e.
turæ. Hoc Axioma idem e&longs;t cum Axiom. 14. l. 1. n.
2. vnde alia expli
catione minimè indiget; hoc ip&longs;um etiam demon&longs;traui in Th.147.149.
150,&c. l. 1.
ticalem per hyp. 1. &longs;ed hic motus e&longs;t à principio extrin&longs;eco, igitur e&longs;t
violentus per def.1. probatur minor; Primò, quia illud e&longs;t principium,
&longs;eu cau&longs;a motus, ex cuius applicatione &longs;emper &longs;equitur motus per Ax.11.
l. 1.n. 1. &longs;ed ex applicatione potentiæ extrin&longs;ecæ v. g. arcus, manus, &c.
ad lineam &longs;ur&longs;um &longs;emper &longs;equitur motus &longs;ur&longs;um; igitur e&longs;t illius cau&longs;a.
Secundò probatur, quia mobile projectum &longs;ursùm mouetur adhuc &longs;epa
ratum à potentia motrice per hyp. 6.l.1. igitur potentia motrix impre&longs;
&longs;it aliquid mobili, vi cuius deinde mouetur, igitur hic motus e&longs;t à prin
cipio extrin&longs;eco.
Diceret fortè aliquis produci hunc motum ab ip&longs;o mobili; &longs;ed con
trà; igitur &longs;emper produceret, quod ab&longs;urdum e&longs;t:
dicet, ad hoc vt pro
ducat determinari debere ab aliquo, &longs;ed contrà; illud à quo determina
tur vel e&longs;t extrin&longs;ecum, vel intrin&longs;ecum, &longs;i primum, ergo hic motus e&longs;t
&longs;emper à principio extrin&longs;eco, quod &longs;atis e&longs;t e&longs;&longs;e determinans per def.1.
&longs;i verò e&longs;t intrin&longs;ecum; igitur &longs;emper e&longs;&longs;et hic motus, quamdiu e&longs;&longs;et
ip&longs;um mobile, quod e&longs;t contra hyp. 1. nam reuera non &longs;emper mo
uetur.
Diceret fortè alius excitari quædam corpu&longs;cula, à quibus mouetur
corpus graue &longs;ursùm; &longs;ed contrà; nam vel &longs;unt in ip&longs;o mobili illa cor
pu&longs;cula, vel extra mobile; &longs;i primum; igitur hic motus &longs;emper erit ab
extrin&longs;eco mediatè, cum ab extrin&longs;eco excitentur; &longs;ed hoc &longs;ufficit ad
hoc; vt motus dicatur violentus per def. 1. &longs;i verò &longs;unt extra mobile;
igitur motus ille e&longs;t &longs;emper ab extrin&longs;eco, idque duplici nomine.
Denique diceret alius ex &longs;uppo&longs;itione, quod terra moueatur non po&longs;
&longs;e corpus graue proiici &longs;ursùm per lineam verticalem, ni&longs;i tantùm ad
&longs;peciem; vt &longs;i quis è naui mobili &longs;ur&longs;um proiiceret pilam rectà omni
nò, quoad eius fieri po&longs;&longs;it; videbitur enim iis, qui vehuntur eadem naui
ri&longs;&longs;imè ferri per lineam nouam inclinatam.
Re&longs;pondeo etiam admi&longs;&longs;a &longs;uppo&longs;itione dici à me motum illum &longs;ur
&longs;um e&longs;&longs;e per lineam verticalem, quando eadem linea recta connectit
&longs;emper hæc tria puncta; &longs;cilicet centrum terræ, idem punctum &longs;uperfi
ciei terræ, & ip&longs;am pilam; ad illud verò quod dicitur de naui, non diffi
teor verum e&longs;&longs;e; &longs;ed dico non e&longs;&longs;e propriè motum violentum, de quo hîc
tantùm e&longs;t quæ&longs;tio, &longs;ed e&longs;&longs;e motum mixtum, de quo fusè &longs;uo loco. Ob&longs;er
uabis autem hîc me ab&longs;tinere à refellendis ab&longs;urdis illis &longs;uppo&longs;itioni
bus, quibus præmi&longs;&longs;æ objectiones innituntur; nam, cui quæ&longs;o in men
tem venire pote&longs;t ab ip&longs;a entitate corporis grauis produci motum in &longs;e?
quis credat produci frigus ab igne? calorem à niue?
lucem à tenebris?
quæ porrò fabulæ, quæ commenta, quæ &longs;omnia excogitari po&longs;&longs;unt, quæ
non vile&longs;cant &longs;i cum his comparentur.
Illa quoque corpu&longs;cula excitata leuiora &longs;unt, quàm vt aliquod præfe
rant rationis momentum; cum mera &longs;int philo&longs;ophiæ ludibria.
Denique illa hypothe&longs;is de terræ motu nullis demon&longs;trationibus fir
mata e&longs;t, vt videbimus &longs;uo loco.
Vnum fortè e&longs;t, quod difficilius obiici pote&longs;t;
&longs;it enim linea vertica
lis AC, &longs;itque globus in A æqualiter impul&longs;us per lineas AD & AB;
haud dubiè &longs;i anguli DAC, BAC &longs;int æquales: certè mobile feretur
per lineam verticalem AC, vt con&longs;tat ex dictis. Re&longs;pondeo motum illum
e&longs;&longs;e violentum; e&longs;t enim à principio extrin&longs;eco, coque gemino, &longs;eu mix
to, in quo non e&longs;t difficultas.
hypoth. 1. igitur habet cau&longs;am per Ax.8.l.1.
liter motus per Th.18.l.1.
quia ibi e&longs;t cau&longs;a, vbi
e&longs;t effectus formalis, &longs;ed motus e&longs;t effectus formalis &longs;ecundarius impe
tus per Th.15.l.1. igitur cum motus &longs;it in projecto &longs;ur&longs;um, in eo e&longs;t etiam
impetus: præterea &longs;ecunda pars motus non ponitur à potentia motrice;
quia illa non e&longs;t applicata mobili cum ponitur noua pars motus, igitur
ab alia cau&longs;a applicata, &longs;ed nulla e&longs;t extrin&longs;eca, vt patet, nulla intrin&longs;eca
præter impetum.
Diceret aliquis ab aëre extrin&longs;ecùs ambiente mobile ip&longs;um propelli;
&longs;ed contra, nam aër, & omne aliud medium re&longs;i&longs;tit potiùs quàm iuuet, vt
demon&longs;trauimus l. &longs;ecundo Th. 1. Nec dicas fui&longs;&longs;e mentem Ari&longs;totelis,
cum nobiles Peripatetici contrâ &longs;entiant; Albertus Magnus, Toletus,
Scaliger, Suarius, & recentiores; neque hoc negauit vnquam Ari&longs;tote-
violentum e&longs;&longs;e à principio intrin&longs;eco contra def. 1. nam e&longs;t quidem à
principio intrin&longs;eco formali, non tamen à principio intrin&longs;eco mouen
te vel agente; nec enim impetus e&longs;t cau&longs;a efficiens motus &longs;ui &longs;ubjecti;
&longs;ed cau&longs;a formalis vt &longs;æpè explicuimus.
Diceret fortè alius primam partem motus produci à potentiâ motri
ce, &longs;ecundam verò ab entitate ip&longs;ius corporis; &longs;ed contrà; igitur corpus
e&longs;&longs;et cau&longs;a nece&longs;&longs;aria; igitur &longs;emper produceret. Dices &longs;emper producere
&longs;i determinetur, &longs;ed contrà; à quo determinatur ad producendam &longs;ecun
dam partem? nihil e&longs;t enim applicatum, à quo determinari po&longs;&longs;it;
Dices
accepi&longs;&longs;e determinationem; &longs;ed contrà; quid e&longs;t illa determinatio?
Dices e&longs;&longs;e modum; igitur permanentem; igitur e&longs;t cau&longs;a motus per Ax.
1. l. 1. n. 1. igitur e&longs;t impetus per def.
3. l. 1. Dices determinari à priori
parte motus; &longs;ed contrà primò, nam reuerâ non e&longs;t illa pars cum deter
minatur corpus. Secundò, quid e&longs;t illa prima pars motus, ni&longs;i migratio è
loco in locum, quæ reuerâ à potentia motrice produci propriè non po
te&longs;t per Th.2. l. 1. &longs;ed de his iam fusè actum e&longs;t in toto ferè libro primo,
&longs;ed præ&longs;ertim in Th.6.
hoc iam &longs;uprà demon
&longs;tratum e&longs;t, &longs;cilicet phy&longs;icè; immò ex motu violento maximè probatur
dari impetum, & vix quidquam e&longs;t in rerum naturâ, quod clariùs euin
cat aliquid de nouo produci.
Probatur, quia e&longs;t de nouo;
igi
tur non e&longs;t à &longs;e per Ax. 8. l. 1. igitur e&longs;t ab alio; igitur ab aliqua
cau&longs;a.
Probatur primò, quia aliquis
motus violentus e&longs;t à cau&longs;a extrin&longs;eca per def.1. Secundò, e&longs;t ab aliqua
cau&longs;a applicata, &longs;ed e&longs;t tantùm applicata potentia motrix; igitur e&longs;t cau
&longs;a, per Ax. 11. l. 1. nec enim producitur hic impetus ab entitate corpo
ris projecti, quod plu&longs;quàm certum e&longs;t ex dictis; hîc enim tantùm
e&longs;t quæ&longs;tio de illo motu, qui extrin&longs;ecùs aduenit, non vero de reflexo
&longs;ursùm, &c.
quia potentia motrix non agit ad extra ni&longs;i
per impetum productum in organo, vt patet; præterea &longs;i e&longs;t cau&longs;a vni
uoca &longs;ufficiens applicata, non e&longs;t ponenda æquiuoca per Ax.11.l.1. adde
quod impetus producitur &longs;emper ad extra ab alio impetu per Th. 42.
l.1.nec in his hactenus propo&longs;itis vlla e&longs;t difficultas.
Probatur,
igitur ille motus habet cau&longs;am, vt &longs;æpè dictum e&longs;t; non aliam, quàm im
petum per Th.4. non productum de nouo, quippe nulla e&longs;t cau&longs;a mobili
applicata per Th. 7. & 8. igitur iam antè productam; igitur con&longs;er
uatur.
vt patet ex dictis, non
ab aëre; igitur à nullo corpore;
igitur ab alia causâ in&longs;en&longs;ibili;
igitur
illam e&longs;&longs;e oportet, & no&longs;&longs;e rerum omnium exigentias, & po&longs;&longs;e cuncta
producere; quippe con&longs;eruatio e&longs;t repetita productio;
immò con&longs;erua
re per actionem, per quam &longs;it res in tali loco, & tali tempore; illa porrò
cau&longs;a in&longs;en&longs;ibilis incorporea, quæ vbique e&longs;t, & &longs;emper, Deus e&longs;t: Nec
puta po&longs;&longs;e exi&longs;tentiam cau&longs;æ primæ probari &longs;en&longs;ibiliori, vt &longs;ic loquar,
argumento, quàm eo, quod petitur ex motu projectorum, quorum motus
durat etiam&longs;i à potentia motrice mobile ip&longs;um &longs;it &longs;eparatum.
Primò, &longs;i nullus e&longs;&longs;et impetus extrin&longs;ecus,
vel acqui&longs;itus, nullus e&longs;&longs;et motus violentus, ni&longs;i tantùm motus reflexus
cadentium deorsùm. Secundò, &longs;i nullus e&longs;&longs;et Deus, nullus e&longs;&longs;et motus
violentus; immò nec vllus naturaliter acceleratus. Tertiò, &longs;i impetus e&longs;
&longs;et fluens vt motus, nullus e&longs;&longs;et motus violentus. Quartò, &longs;i &longs;ingulæ par
tes motus produci debent ab aliquâ causâ efficiente, nullus etiam e&longs;&longs;et
motus violentus.
quàm &longs;int partes impetus naturalisProbatur, quia &longs;i e&longs;&longs;ent plures natura
lis deorsùm, quàm &longs;int violenti &longs;ur&longs;um, corpus tenderet deor&longs;um; &longs;ed
tardiùs per Th.134.l.1. & &longs;i tot e&longs;&longs;ent vnius, quot alterius, mobile ip&longs;um
non moueretur per Th.133.l.1.
certa e&longs;t. Secundò, quia &longs;i &longs;emper cre&longs;ceret, numquam rediret mobile
contra hyp.1. nec enim ab vllo reflectitur; &longs;i enim reflecteretur ab aëre
inten&longs;us, multò magis remi&longs;&longs;us.
ditur effectus per Th.13. igitur nec cau&longs;a per Ax.2.l.2.
quia mobile tandem redit per hyp.1.
&longs;ed numquam rediret, &longs;i e&longs;&longs;et æquabilis; cur enim potiùs hoc in&longs;tanti
quàm alio? cur ab hoc puncto &longs;patij potiùs, quàm ab alio?
quia &longs;i e&longs;&longs;et intactus, e&longs;&longs;et &longs;em
per æqualis; igitur haberet &longs;emper æqualem motum per Ax.3.l.2. igitur
motus e&longs;&longs;et æquabilis, contra Th.15.
cum enim non remaneat inta
ctus, & æqualis; nec fiat maior per Th.14. certè fit minor, igitur detra
ctione aliqua per Ax.1.l.2.
cur potiùs vno quam alio? quippe illa ratio, quæ probat de vno probat
de &longs;ingulis.
probatur, quia æqualis effectus æqualem cau&longs;am &longs;upponit, per Ax.
3. l. 2.
quia licèt re&longs;i
&longs;tat motui, &longs;eu potius mobili, non tamen e&longs;t ea re&longs;i&longs;tentia, quæ po&longs;&longs;it
impetum tam citò de&longs;truere; probatur primò, quia &longs;i hoc e&longs;&longs;et, de&longs;true
retur æquali tempore per omnem lineam &longs;ur&longs;um, quod e&longs;t contra expe
rientiam, vt dicemus infrà; e&longs;&longs;et enim eadem cau&longs;a applicata;
igitur idem
& æqualis effectus; probatur &longs;ecundò, quia non de&longs;truit aër primum il
lum gradum impetus naturalis acqui&longs;iti, vt con&longs;tat in motu deor&longs;um, qui
tamen e&longs;t imperfecti&longs;&longs;imus; igitur non e&longs;t &longs;ufficiens ad de&longs;truendum im
petum violentum, ni&longs;i longo tempore. Tertiò, globus &longs;ursùm projectus
a&longs;cendit, & deinde de&longs;cendit æquali tempore; igitur &longs;altem &longs;ingulis in
&longs;tantibus de&longs;truitur vnus gradus impetus violenti æqualis primo gradui
innato; atqui aër non pote&longs;t vno in&longs;tanti de&longs;truere impetum æqualem
primo innato; alioqui non intenderetur motus naturalis. Quartò, & hæc
e&longs;t ratio à priori, quotie&longs;cumque &longs;unt in eodem mobili duo impetus ad
oppo&longs;itas lineas determinati, pugnant pro rata, vt demon&longs;trauimus l.1.
Th. 149. 150. 152. & in toto Schol. & multis aliis pa&longs;&longs;im; atqui con&longs;er
uatur &longs;emper impetus naturalis innatus per Sch. Th.152.n.6.l.1.per Th.
9. & Schol.Th.14. & Th.73.l.2.
cet ab impetu innato &longs;i quæ e&longs;t de quæ alias,
ratio cur vno in&longs;tanti de&longs;truerentur duo gradus impetus, quàm 3. 4. 5.
quippe grauitas exigeret de&longs;tructionem omnium: præterea omnis impe
tus de&longs;truitur ne &longs;it fru&longs;trà per Schol, Th.152. & Th.162.l.1. denique &longs;i
e&longs;t ponenda alia cau&longs;a de&longs;tructiua.
pro
batur, quia nulla e&longs;t cau&longs;a extrin&longs;eca de&longs;tructiua &longs;altem adæquatè per hT.
20.igitur e&longs;t intrin&longs;eca per Ax.4. l.2. &longs;ed intrin&longs;eca vel e&longs;t mobilis enti
tas, vel grauitas, vel impetus innatus; &longs;ed mobilis entitas non e&longs;t cau&longs;a
de&longs;tructiua; nec etiam ip&longs;a grauitas per Th.21. igitur impetus naturalis
innatus.
in&longs;tantibus e&longs;t cau&longs;a de&longs;tructiua applicata, igitur &longs;ingulis in&longs;tantibus de
&longs;truit per Ax. 12. l. 1.
liter de&longs;truaturquia &longs;ingulis in&longs;tantibus e&longs;t eadem cau&longs;a de&longs;tructiua ap
plicata; igitur &longs;ingulis in&longs;tantibus æqualiter de&longs;truit per Ax.3.l.2.porrò
in tantum de&longs;truit in quantum efficit, vt aliquid &longs;it fru&longs;trà, vt fusè di
ctum e&longs;t lib.1.vel in quantum exigit eius
&longs;truatur vne in&longs;tanti
& per Th.134.l.1.
tur velocitatis momenta in motu violento
jecto applicata æqualibus temporibus æqualem effectum producit per
Ax.3.l.2. &longs;ed impetus innatus e&longs;t cau&longs;a de&longs;tructiua impetus violenti per
Th. 22. igitur æqualibus temporibus, &c.
raturprobatur quia &longs;ingulis in&longs;tantibus æqualibus acquiritur æqualis
gradus impetus, vt &longs;æpè dictum e&longs;t &longs;uprà; atqui &longs;ingulis in&longs;tantibus de
&longs;truitur vnus gradus impetus violenti per Th.24. &longs;ed ille gradus re&longs;pon
det impetui innato per Th. 25. igitur æqualibus temporibus tantùm de
&longs;truitur violenti, quantùm acquiritur naturalis; cum enim primo in
&longs;tanti &longs;it impetus naturalis, & &longs;ecundo tempore æquali acquiratur æqua
lis, item tertio, quarto, &c. certè cum impetus innatus pugnet cum vio
lento pro rata; nec &longs;it potior ratio cur maiorem portionem quàm mino
rem de&longs;truat, æqualem certè de&longs;truit, itemque &longs;ecundo in&longs;tanti æqua
lem, item tertio, quarto; igitur in eadem proportione decre&longs;cit violentus,
&longs;eu retardatur, in qua naturalis acceleratur.
Hinc inuertenda e&longs;t progre&longs;&longs;ionis linea;
quippe linea AE repræ&longs;en
tat nobis progre&longs;&longs;ionem motus accelerati, quæ fit in in&longs;tantibus, & li
nea FK progre&longs;&longs;ionem motus, quæ fit in partibus temporis &longs;en&longs;ibilibus;
in illa primo in&longs;tanti decurritur primum &longs;patium AB, &longs;ecundo tempore
æquali BC, tertio CD, quarto DE: in hac vero prima parte acquiritur
&longs;patium FG &longs;ecunda æquali primæ GH, tertia HI, quarta IK; igitur &longs;i ac
cipiatur linea AE, progrediendo ab A ver&longs;us E, vel linea FK progre
diendo ab F ver&longs;us K habebitur progre&longs;&longs;io motus naturaliter accelerati;
&longs;i verò accipiatur EA, vel KF, progrediendo &longs;cilicet ab E ver&longs;us A, vel à
K ver&longs;us F, erit progre&longs;&longs;io motus violenti naturaliter retardati; vt con
&longs;tat ex præcedèntibus Theorematis; & quemadmodum progre&longs;&longs;io acce
lerationis in in&longs;tantibus finitis fit iuxta &longs;eriem i&longs;torum numerorum 1.2.
3.4. in partibus verò temporis &longs;en&longs;ibilibus iuxta &longs;eriem i&longs;torum 1.3.5.7.
ita fit omninò progre&longs;&longs;io retardationis in in&longs;tantibus iuxta hos nume
ros 4.3.2.1. in partibus temporis &longs;en&longs;ibilibus iuxta hos 7.5. 3. 1.
gradus impetus quibus violentus &longs;uperat innatum,
petus innati; producantur 5. gradus violenti, quorum &longs;inguli &longs;int æqua
les innato etiam
ualenter id e&longs;t 4. temporibus, quorum &longs;ingula erunt æqualia primo in
&longs;tanti motus naturalis, probatur, cum &longs;ingulis in&longs;tantibus æqualibus de
&longs;truatur vnus gradus; certè 4. in&longs;tantibus durat motus.
illius, quàm tribuimus &longs;uprà accelerationi, a&longs;&longs;umptis &longs;cilicet &longs;patiis æqualibus;
tum &longs;i accipiantur &longs;patia æqualia prime &longs;patie quod decurritur prime in&longs;tan
ti metus naturalis, tum &longs;i accipiantur &longs;patia æqualia date &longs;patie quod in par
te temporis &longs;en&longs;ibili percurriturquippe quemadmodum in progre&longs;&longs;ione
accelerationis decre&longs;cunt tempora; &longs;ic in progre&longs;&longs;ione retardationis
cre&longs;cunt, a&longs;&longs;umptis &longs;cilicet &longs;patiis æqualibus; quare ne iam dicta hic re
petam, con&longs;ule quæ diximus lib.2. de hac progre&longs;&longs;ione.
&longs;unt maiora; & &longs;ub finem in motu violente &longs;unt maiora, in naturali &longs;unt mi
noraquia &longs;cilicet hic acceleratur, ille retardatur:
igitur velo
citas accelerati cre&longs;cit; igitur &longs;i accipiantur &longs;patia æqualia, decre&longs;cit tem
pus; at verò velocitas retardati decre&longs;cit, igitur a&longs;&longs;umptis &longs;patiis æquali
bus, cre&longs;cit tempus; igitur &longs;i accipiatur &longs;patium, quod percurritur primo
in&longs;tanti huius motus, & deinde alia huic æqualia; haud dubiè, cum &longs;e
cundo in&longs;tanti motus &longs;it tardior, &longs;itque a&longs;&longs;umptum æquale &longs;patium; haud
dubiè inquam in&longs;tans &longs;ecundum erit maius primo, & tertium &longs;ecundo,
atque ita deinceps.
&longs;ecundo,quia eadem cau&longs;a breuiore tempore minùs agit
per Ax.3.l.2. & Ax. 13.l.1. num.4. igitur minùs impetus de&longs;truitur pri
mo, quàm &longs;ecundo, & minùs &longs;ecundo quàm tertio, atque ita deinceps;
idem enim dici debet de cau&longs;a de&longs;tructiua, quod de productiua.
Dices, igitur idem impetus de&longs;truitur primo in&longs;tanti, quo e&longs;t, &longs;i de&longs;trui
tur primo in&longs;tanti motus. Re&longs;pondeo negando; quia primo in&longs;tanti, quo
e&longs;t impetus, non e&longs;t motus per Th.34.l.1.
Dices, igitur impetus ille e&longs;t fru&longs;trà, quia nullus effectus, &longs;eu motus
ex eo &longs;equitur; Re&longs;pondeo negando; nam omnes gradus impetus qui ei
dem parti mobilis in&longs;unt, communi qua&longs;i actione, vel exigentia indi
ui&longs;ibiliter exigunt motum.
perfectionecum enim &longs;inguli &longs;ingulis in&longs;tantibus de&longs;truantur, vt dictum
e&longs;t; quippe e&longs;t tantùm vnus gradus impetus innati, & cum &longs;ingula in
&longs;tantia &longs;int inæqualia, etiam &longs;inguli gradus illius impetus &longs;unt inæquales
in perfectione.
alioquin minimè afferri pote&longs;timmò, ni&longs;i hoc e&longs;&longs;et, nulla e&longs;&longs;et huiu&longs;modi
naturalis retardatio; nam producantur, &longs;i fieri pote&longs;t, omnes æquales, &longs;int
que v.g.20. nunquid po&longs;&longs;unt e&longs;&longs;e 40. perfectionis &longs;ubduplæ, vel 10. du
plæ, vel 5. quadruplæ &c. cur autem potiùs vnum dices quàm aliud?
at
verò optimam inde reddo rationem quòd cum &longs;int omnes inæquales, cò
plures &longs;unt, quò maior e&longs;t ni&longs;us; pauciores verò, quò minor.
v. g. quà proportione primum in&longs;tans e&longs;t minus &longs;ecundo, & &longs;ecundum
tertio, ita ille gradus impetus, qui de&longs;truitur primo in&longs;tanti, e&longs;t minor
vel imperfectior co, qui de&longs;truitur &longs;ecundo, & qui de&longs;truitur &longs;ecundo
imperfectior co, qui de&longs;truitur tertio, atque ita deinceps.
ti, de quo infráquod &longs;equitur ex dictis nece&longs;&longs;ariò: vtrùm verò ille &longs;it æ
qualis omninò in perfectione impetui naturali innato, dicemus
infrà.
Hic ob&longs;eruabis mirabilem &longs;anæ naturæ prouidentiam, quæ motus
omnes cum ip&longs;o naturali ita compo&longs;uit, vt &longs;it veluti regula omnium mo
tuum, &longs;itque vnum qua&longs;i principium perfectionis totius impetus; tùm in
tùm in violento, in cuius progre&longs;&longs;ione de&longs;truitur &longs;emper perfectior;
producitur imperfectior ab eadem cau&longs;a in minoribus temporibus, &
de&longs;truitur perfectior ab eadem cau&longs;a in maioribus temporibus; & cum
impetus innatus &longs;it cau&longs;a de&longs;tructiua impetus violenti, habet inæqualem
proportionem cum &longs;uo effectu pro temporibus inæqualibus; & cum
idem impetus innatus &longs;it qua&longs;i principium crementi, vel accelerationis,
&longs;icut e&longs;t principium retardationis; certè pro inæqualitate temporum e&longs;t
diuer&longs;a proportio crementorum; quo nihil clarius in hac materia meo
iudicio dici pote&longs;t.
& finis huius cum principio illius
tur; &longs;iue temporum æqualium in &longs;patiis inæqualibus; &longs;iue &longs;patio
rum æqualium in temporibus inæqualibus, &longs;iue a&longs;&longs;umantur in&longs;tan
tia in progre&longs;&longs;ione arithmetica &longs;implici iuxta hos numeros 1.2.3.4. &longs;iue
a&longs;&longs;umantur temporis partes &longs;en&longs;ibiles in progre&longs;&longs;ione Galilei iuxta hos
numeros 1.3.5.7. quæ omnia ex dictis nece&longs;&longs;ariò con&longs;equuntur.
etiam aliæ partes motus in di&longs;tantiis æqualibus
percurrit mobile demi&longs;&longs;um ex puncto A deor&longs;um motu naturaliter ac
celerato, & moueatur per 6. in&longs;tantia, &longs;eu 6. tempora æqualia: Primo
in&longs;tanti, quo percurrit &longs;patium AB; haud dubiè, quando peruenit ad pun
ctum G, habet 7. gradus impetus æquales, quia ante motum AB habebat
innatum; &longs;ed in motu illo fluunt 6. tempora æqualia, vt dictum e&longs;t;
igitur
6. acquirit gradus impetus, quorum quidem vltimò acqui&longs;itus nullum
adhuc habuit motum; &longs;ed haud dubiè haberet, &longs;i vlteriùs hic motus pro
pagaretur: his po&longs;itis imprimantur mobili in O 7.gradus impetus æqua
les prioribus &longs;ursùm motu violento, per lineam OH; certè primo in&longs;tan
ti motus, &longs;eu tempore æquali prioribus percurret ON, id e&longs;t 6. &longs;patiola;
quia licèt &longs;int 7.gradus; attamen impetus innatus corporis grauis detra
hit vnum &longs;patium, &longs;imulque de&longs;truit vnum gradum, &longs;ecundo tempore
percurret NM 5. tertio ML 4. quarto LK 3. quinto KI 2. &longs;exto IH 1.
igitur primum violenti ON re&longs;pondet vltimo naturali FG &longs;eu &longs;ecun
dum illius quinto huius, tertium illius quarto huius, quartum tertio,
quintum &longs;ecundo &longs;extum primo, & vici&longs;&longs;im; idem pror&longs;us in progre&longs;&longs;ione
Galilei accidit, a&longs;&longs;umptis &longs;cilicet partibus temporis &longs;en&longs;ibilibus.
tus, quos habuit ab eadem altitudine decidens motu naturali
dictis.
vtriu&longs;que erunt æqualiacon&longs;tat etiam ex dictis v.g. corpus graue, motu
naturali in libero aëre tempore duorum &longs;ecundorum percurrit 48. pe
des, igitur &longs;i moueatur &longs;ur&longs;um æquali tempore percurret 48. pedes per
&longs;e, dico per &longs;e; quippe ratione figuræ corporis &longs;ecus accidere pote&longs;t, vt
plurimùm etiam accedit ratione motus mixti ex motu centri recto, &
motu orbis circulari, de quo infrà.
curruntur &longs;unt æqualia,
nato e&longs;t æqualis impetui producta in principio violenti.
diu durat eiu&longs;dem a&longs;cen&longs;us, & tot habet gradus impetus in fine de&longs;cen&longs;us,
quot habet in principio a&longs;cen&longs;use&longs;t enim æquale &longs;patium;
igitur æquale
tempus; igitur æqualis vtrobique impetus. Sed hîc duo obiici po&longs;&longs;unt,
primò &longs;agittam per lineam verticalem vibratam po&longs;ui&longs;&longs;e tantùm in a&longs;
cen&longs;u 3. &longs;ecunda, in de&longs;cen&longs;u verò 5. vt &longs;æpiùs ob&longs;eruatum e&longs;t, te&longs;te Mer
&longs;enno; &longs;ecundò, &longs;i eodem tempore corpus graue &longs;ursùm proiectum motu
violento a&longs;cenderet, quo deinde de&longs;cendit, in fine de&longs;cen&longs;us æqualis
e&longs;&longs;et ictus, &longs;eu percu&longs;&longs;io vtriu&longs;que; cum tamen illa &longs;it maior, quæ infli
gitur motu violento, vt con&longs;tat multis experimentis.
Re&longs;pondeo ad primum etiam te&longs;te Mer&longs;enno globum ferreum trium
aut 4. librarum &longs;ur&longs;um explo&longs;um è breuiore tormento &longs;ed latiore, æqua
le tempus in a&longs;cen&longs;u, & in de&longs;cen&longs;u in&longs;ump&longs;i&longs;&longs;e; quod reuerâ &longs;ecùs acci
dit &longs;agittæ, cuius differentia a&longs;cen&longs;us, & de&longs;cen&longs;us &longs;en&longs;u etiam percipi
pote&longs;t; tùm quia lignea materia multò leuior e&longs;t ferro, tùm quia leui&longs;&longs;i
mæ illæ pennæ, quibus in&longs;truitur, motum retardant in de&longs;cen&longs;u; quod
maximè confirmatur ex eo quod pluma facilè anhelitu &longs;ur&longs;um pellatur
&longs;atis veloci motu, quæ deinde tardi&longs;&longs;imo &longs;ua &longs;ponte de&longs;cendit: præterea
mucro ferreus, quo &longs;agitta armatur, &longs;emper præire debet, cuius rei ratio
nem afferemus infrà; igitur cum in a&longs;cen&longs;u præeat, vt præeat in de&longs;cen
&longs;u, altera extremitas &longs;emicirculum &longs;uo motu facere debet, qui certè ad
naturalem motum pertinet, altera tamen extremitas, quæ mouetur mo
tu contrario alterius motum retardat; ad &longs;ecundam obiectionem
re&longs;pondebo Th.44.
percurritur motu naturaliter retardato, a&longs;&longs;umptis &longs;cilicet
cum enim motu æquabili compo&longs;ito ex &longs;ubdupla velocitate maximæ, &
minimæ motus accelerati æquali tempore percurratur æquale &longs;patium,
&longs;ubduplum minimæ pro nihilo ferè habetur; igitur pote&longs;t tantùm a&longs;&longs;u-
dubiè &longs;patium duplum percurretur.
con&longs;eruarialioqui violentus e&longs;&longs;et æquabilis, igitur nunquam de&longs;ineret:
quantum ab&longs;urdum! quale incommodum &c.
&longs;patiorum quæ &longs;cilicet toto motu percurruntur in a&longs;cen&longs;u & de&longs;cen&longs;u,
corpus graue cadens ex data altitudine 48 pedum æqualem ictum infli
git in fine de&longs;cen&longs;us, & in principio a&longs;cen&longs;us, quo &longs;cilicet ad
altitudinem a&longs;cenderet; probatur, quia æqualis acquiritur impetus in
de&longs;cen&longs;u alteri, qui de&longs;truitur in a&longs;cen&longs;u, a&longs;&longs;umptis dumtaxat &longs;patiis illis
æqualibus; igitur æqualis e&longs;t in fine de&longs;cen&longs;us, in quo e&longs;t totus acqui&longs;i
tus, atque in principio a&longs;cen&longs;us, in quo nullus e&longs;t de&longs;tructus: ad id verò,
quod dicebatur &longs;uprà de &longs;agitta, cuius ictus maior e&longs;t initio a&longs;cen&longs;us,
quàm in fine de&longs;cen&longs;us non diffiteor; quia materia &longs;agittæ, tùm lignea
tùm plumea motum &longs;atis &longs;uperque retardat, vt differentia ictuum &longs;en&longs;u
ip&longs;o percipi po&longs;&longs;it; quæ tamen nulla perciperetur in a&longs;cen&longs;u de&longs;cen&longs;u
que globi ferrei.
impre&longs;&longs;um de&longs;trui tantùm ab aërequod plu&longs;quàm fal&longs;um e&longs;&longs;e comper
tum e&longs;t, vt demon&longs;trauimus &longs;uprà Th. 20. qua&longs;i verò non ad&longs;it aliqua
cau&longs;a nece&longs;&longs;aria de&longs;tructiua, &longs;cilicet impetus innatus; hinc etiam eum
dem reiicies, qui vult numquam fieri po&longs;&longs;e, vt motu naturaliter accelera
to tanta acquiratur velocitas, quanta imprimitur in motu violento; vult
enim motum acceleratum tran&longs;ire in æquabilem, cuius contrarium de
mon&longs;trauimus &longs;uprà Th. 131, l. 2. igitur cum cre&longs;cat &longs;emper velocitas,
nullus e&longs;t finitus gradus, quem tandem non a&longs;&longs;equatur; immò vt dictum
e&longs;t in præcedenti Th. a&longs;&longs;umptis æqualibus &longs;patiis, impetus, qui e&longs;t in
principio a&longs;cen&longs;us, æqualis e&longs;t cum eo, qui e&longs;t in fine de&longs;cen&longs;us.
Diceret fortè aliquis cadentem globum ex alti&longs;&longs;imæ turris apice de
clinare à perpendiculari antequam terram feriat, vt con&longs;tat ex multis
experimentis; igitur præualet tandem re&longs;i&longs;tentia aëris:
&longs;ed re&longs;pondeo id
tantùm accidere propter currentem illac aëris tractum; alioquin non
e&longs;&longs;et potiùs ratio, cur in vnam partem declinaret, quàm in aliam.
tegri &longs;patijv.g. in &longs;ubduplo &longs;patij &longs;egmento non e&longs;t &longs;ubduplus ictus, &longs;it
enim &longs;patium integrum motus vîolenti OH, & principium motus &longs;it
in O, finis in H; accipiatur &longs;egmentum OM, quod e&longs;t qua&longs;i &longs;ubduplum O
H, ictus in M non e&longs;t profectò &longs;ubduplus ictus in O, &longs;ed tantùm in L, vt
torum integri &longs;patij.
a&longs;&longs;umatur progre&longs;&longs;io Gali
lei in AF, ita vt &longs;i prima parte temporis &longs;en&longs;ibili percurratur &longs;patium
FE 9 partium æqualium; &longs;ecunda percurratur ED. 7. partium, tertia
DC 5. quarta CB 3; quinta BA 1. hoc po&longs;ito facilè erit determinare
rationes ictuum; nam in de&longs;cen&longs;u ictus &longs;unt vt velocitates, & hæ vt tem
pora; igitur &longs;i AB percurritur in dato tempore, & AC in duobus prio
ri æqualibus; certè ictus in de&longs;cen&longs;u AC e&longs;t duplus ictus in de&longs;cen&longs;u
AB; in AD triplus, &c. Igitur in a&longs;cen&longs;u ictus in F erit quintuplus,
ictus in E quadruplus in D triplus, &c. igitur ictus &longs;unt in ratione dupli
cata &longs;patiorum facto &longs;patij initio à &longs;ummo puncto A.
nem, cogno&longs;ci po&longs;&longs;unt vires, quibus ad aliam quamcumque proijciaturv. g.
proiiciatur corpus graue ad altitudinem 48. pedum; vires &longs;unt iis æqua
les, quas acquirit in de&longs;cen&longs;u eiu&longs;dem altitudinis 48. pedum; &longs;it alia di
&longs;tantia 100. pedum; haud dubiè vires nece&longs;&longs;ariæ ad motum hunc violen
tum &longs;unt æquales iis, quas acquireret in de&longs;cen&longs;u 100. pedum per Th.
40. atqui ita &longs;e habent vires acqui&longs;itæ in de&longs;cen&longs;u 48. pedum ad vires
acqui&longs;itas in de&longs;cen&longs;u 100. vt v.g. 48. ad v.g. 100. id e&longs;t ferè vt 7.
ad 10.
&longs;it enim decur&longs;um idem &longs;pa
tium 48. pedum motu violento &longs;ur&longs;um; idque v. g. tempore 2. &longs;ecundo
rum, quod ferè cum experientia con&longs;entit; &longs;it aliud &longs;patium 100. tempus
primi motus e&longs;t ad tempus &longs;ecundi vt v. g. 48. ad v. g. 100. quia &longs;patia
&longs;unt vt quadrata temporum; igitur tempora vt radices 4. hinc vires &longs;unt
in ratione temporum; quia vt temporibus æqualibus acquiruntur æqua
lia velocitatis momenta in motu naturali, ita & de&longs;truuntur æqualia in
motu violento, quæ omnia con&longs;tant; igitur ictus &longs;unt vt vires, vires vt
tempora, tempora denique, vt radices
&longs;ur&longs;umprobatur
quia non tendit vlteriùs; igitur non impeditur eius motus à &longs;uperficie
corporis terminati ad punctum A; igitur nullum impetum in eo produ
cit, qui tantùm producitur ad tollendum impedimentum per Th.44.l.1.
igitur nullum ictum infligit, qui tantùm infligitur per impetum, vt
con&longs;tat.
impetu corporis projecti, tùm ex grauitatione corporis grauis incum
bentis, quæ omnia hîc fu&longs;iùs e&longs;&longs;ent tractanda, ni&longs;i locum proprium infrà
&longs;ibi vendicarent.
petus ad lineas oppo&longs;itas ex diametro determinati ad communem li
neam determinari non po&longs;&longs;unt, cur enim potiùs dextror&longs;um quam &longs;ini
stror&longs;um? igitur non concurrunt ad communem motum, ni&longs;i dicatur
impetus innatus valeo nomine concurrere ad violentum, quod eius li
neam &longs;ingulis temporibus qua&longs;i ca&longs;tiget, vltróque, vel vlteriùs currentem
contineat.
tantùm concurritpatet, cum enim in mobili projecto &longs;ur&longs;um &longs;it tantùm
ille impetus præter innatum, nec innatus concurrat per Th. 52. illum
tantùm concurrere nece&longs;&longs;e e&longs;t: excipe &longs;emper impetum acqui&longs;itum, de
quo iam &longs;uprà.
diante &longs;cilicet impetu producto in organo proprio, non e&longs;t motus
quia primo in&longs;tanti, quo e&longs;t impetus, non e&longs;t motus, per Th.34.l.1.
mo in&longs;tanti, quo e&longs;tprobatur, quia &longs;ecundo in&longs;tanti exigit motum &longs;ui &longs;ub
jecti; igitur tolli etiam impedimentum;
igitur per motum medij;
igitur
priori in&longs;tanti in eodem mobili debet e&longs;&longs;e impetus; igitur produci ab
impetu organi; igitur & in organo ab impetu manus.
dus de&longs;truitur
ciperet, e&longs;&longs;e de&longs;ineret, quod dici non pote&longs;t.
probatur, quia certè tàm impedit primam productionem, quàm con&longs;er
uationem, vt patet; e&longs;t enim par vtrobique ratio; præterea agit in ip&longs;am
manum.
du aquali ip&longs;i impetui innatoquippe &longs;icut de&longs;truit &longs;ingulis in&longs;tantibus
æqualibus vnum gradum; quia pugnat pro rata;
ita pror&longs;us impedit, ne
tiùs, quàm tres?
quippe e&longs;t cau&longs;a ne
ce&longs;&longs;aria; igitur &longs;tatim primo in&longs;tanti exigit de&longs;tructionem;
non certè
pro primo in&longs;tanti per Th.56.igitur pro &longs;ecundo, atque ita pro aliis dein
ceps; de&longs;truitur autem, ne &longs;it fru&longs;trà eo modo, quo diximus &longs;uprà.
numquam destruaturne &longs;i aliquando de&longs;trueretur, nulla e&longs;&longs;et cau&longs;a de
&longs;tructiua impetus violenti; ac proinde æquabilis e&longs;&longs;et, &longs;emperque dura
ret, de&longs;tructiua inquam &longs;uo modo.
&longs;ic na
uis aquis innatans, nubes in aëre liberatæ; halitus, atque adeo ip&longs;æ partes
aquæ, quas perexiguus lapillus in orbes penè innumeros agit, ne quid
dicam de partibus aëris, quæ tam citò & procul mouentur, vt con&longs;tat in
&longs;ono, motu &longs;cilicet ferè æquabili.
tùm
quia plures partes impetus &longs;unt producendæ in &longs;ubjecto grauiore quod
pluribus partibus con&longs;tat, tùm impetus innatus maior e&longs;t, non quidem in
inten&longs;ione &longs;ed in exten&longs;ione, ac proinde impedit ne plures gradus pro
ducantur; quippe maius impedimentum plus impedit, quis hoc neget?
motum &longs;ecundi instantis
dici non debet.
qua&longs;i actione vel exigentia
velocitatem, quæ e&longs;t indiui&longs;ibilis actu.
quia impetus innatus aliquid detrahit, cum exigat motum deor&longs;um per
lineam oppo&longs;itam, igitur imminuitur motus pro rata.
impetus qui destruitur
impetum innatum.
v.g.
motus qui ponitur &longs;ecundo in&longs;tanti non minùs exigitur ab eo gradu im
petus qui de&longs;truitur &longs;ecundò in&longs;tanti, quàm ab aliis, non exigitur qui
dem &longs;ecundo &longs;ed primo pro &longs;ecundo; vnde dixi cau&longs;am partialem, quia
etiam exigitur ab aliis gradibus impetus, qui non de&longs;truuntur exigentiâ
communi; quippe impetus non exigit ni&longs;i pro &longs;ecundo in&longs;tanti;
nec vl
lum ab&longs;urdum e&longs;t eo in&longs;tanti cau&longs;am exigentiæ non exi&longs;tere cum poni
tur eius effectus, &longs;cilicet id quod exigebat priori in&longs;tanti quo erat; nul
lus e&longs;t enim influxus huius cau&longs;æ; præ&longs;ertim cum non &longs;it cau&longs;a
totalis.
Vnde cum effectus qui ponitur &longs;ecundo in&longs;tanti non re&longs;pondeat per
fectioni cau&longs;æ totius propter impedimentum, aliquis gradus cau&longs;æ e&longs;&longs;et
fru&longs;trà; igitur eodem in&longs;tanti &longs;ecundo de&longs;trui debet, alioqui ni&longs;i de&longs;true
retur &longs;ingulis in&longs;tantibus poneretur effectus non re&longs;pondens perfectioni
cau&longs;æ; immò numquam de&longs;trueretur totus motus violentus, vt con&longs;tat;
itaque primo in&longs;tanti omnes gradus impetus qui &longs;unt exigunt motum
pro &longs;ecundo ne aliquis eo in&longs;tanti &longs;it fru&longs;trà &longs;i non exigeret, & &longs;ecundo
in&longs;tanti aliquis gradus impetus de&longs;truitur, ne &longs;it fru&longs;trà eodem in&longs;tanti
&longs;ecundo, cum &longs;cilicet non &longs;int tot gradus motus, quot &longs;unt gradus impe
tus; atque ita deinceps tertio in&longs;tanti de&longs;truitur vnus gradus, vt iam &longs;u
prà dictum e&longs;t.
tertioque, &c. quia talis e&longs;t perfectionis
hoc iam &longs;uprà explicatum e&longs;t; quia
cum motus initio &longs;it velocior, in&longs;tantia &longs;unt minora, igitur minùs im
petus in &longs;ingulis de&longs;truitur, pater ex dictis.
tertij in&longs;tantisquia non pote&longs;t concurrere ad motum ni&longs;i exigendo;
at
qui exigere tantùm pote&longs;t, quando e&longs;t; quod enim non e&longs;t non exigit,
&longs;ed motus tertij in&longs;tantis exigitur &longs;ecundo; &longs;ic enim tota res motus pro
cedit vt impetus primo in&longs;tanti exigat motum pro &longs;ecundo; & &longs;ecundo
pro tertio; & tertio pro quarto, atque ita deinceps;
igitur impetus ille
qui de&longs;truitur; &longs;ecundo in&longs;tanti non exigit motum pro tertio, & qui de
&longs;truitur tertio non exigit pro quarto, atque ita deinceps.
&longs;ed tantùm impedit, immediatè quidem, quia cum exigat motum deor
sùm, facit vt non &longs;it tantus motus &longs;ur&longs;um; mediatè verò, quia cum non
&longs;it tantus motus &longs;ursùm, quantus e&longs;&longs;et, haud dubiè non re&longs;pondet adæ
quatè cau&longs;æ; igitur aliquid cau&longs;æ fru&longs;trà e&longs;t;
igitur de&longs;trui debet; hinc
fru&longs;trà.
probatur, quia hæc
determinat impetum productum in manu vel in organo; hic verò im
petum, quem producit in mobili &longs;ursùm projecto; patet, quia nulla e&longs;t
alia cau&longs;a applicata.
nirent, &longs;i vel vnus gradus violenti e&longs;&longs;et æqualis perfectionis cum innatocum
enim detrahatur &longs;emper pars aliquota alicuius totius, tandem perueni
tur ad vltimam; igitur &longs;int 100. gradus impetus violenti, quorum quili
bet &longs;it æqualis impetui innato; certè cum temporibus æqualibus æqua
lis gradus impetus de&longs;truatur; accipiatur illud tempus, in quo de&longs;trui
tur vnus, haud dubiè 100. æqualibus temporibus de&longs;truentur omnes 100.
igitur 99. in&longs;tantibus de&longs;truentur 99. gradus; igitur &longs;upere&longs;t vnus; igitur
duo illi impetus perueniunt tandem ad æqualitatem.
ueretur &longs;ursùm quàm deor&longs;um in&longs;tanti &longs;equentiprobatur, quia tàm gra
dus impetus innati exigit motum deor&longs;um quàm gradus impetus vio
lenti &longs;ursùm; igitur neuter habebit motum per Th.133.l. 1.
quia in&longs;tanti
immediatè antecedenti erant duo gradus impetus violenti, & vnus in
nati; igitur duo illi præualent pro in&longs;tanti &longs;equenti, in quo e&longs;t æqua
litas.
cedit
Th. 73.
cum enim
&longs;ingulis in&longs;tantibus de&longs;truatur vnus gradus, v. g in&longs;tanti illo, quod &longs;e
quitur po&longs;t in&longs;tans æqualitatis, de&longs;truitur ille gradus, qui &longs;upere&longs;t; nec
pote&longs;t vel plùs, vel minùs de&longs;trui; pugnant enim pro rata; quod certè
cuiquam fortè paradoxor videbitur, &longs;cilicet nullum tune e&longs;&longs;e motum
propter pugnam, cum tamen nulla e&longs;t amplius pugna.
tis &longs;it tantùm impetus innatus per Th. 76. certè non impeditur quomi
nus habeat motum pro in&longs;tanti &longs;equenti, quem reuerà exigit; igitur pro
&longs;ed pro alio antecedente mouebatur; igi
tur quies illa durat tantùm vno in&longs;tanti.
quia nul
la pror&longs;us e&longs;t reflexio, vbi nullum e&longs;t reflectens; atqui nullum e&longs;t refle
ctens, vt patet, quia nullum e&longs;t corpus impediens motus propagationem;
licèt enim medium impediat, non tamen per modum reflectentis pro
priè; immo vt dicemus infrà in puncto reflexionis nulla datur quies; &longs;ed
motus reflexus &longs;ibi vendicat librum &longs;ingularem.
quali motu deor&longs;um cadit,
tus in fine motus violenti, per Th.76. igitur &longs;olus impetus naturalis li
bero motu deorsùm fertur.
accelerari, quia &longs;en&longs;im de&longs;truitur impetus violentus antè impre&longs;&longs;us,
nitus ridiculum e&longs;t; quia lapis deci&longs;us è rupe etiam motu naturaliter
accelerato deor&longs;um cadit, licèt eò nunquam motu violento euectus
fuerit.
Ob&longs;eruabis hanc hypothe&longs;im gradus impetus violenti æqualis perfe
ctionis cum innato e&longs;&longs;e fal&longs;am. Primò, quia commodius e&longs;t potentiæ
motrici producere imperfectiorem impetum, &longs;ic enim plures illius gra
dus producere pote&longs;t. Secundò, quia in reflexo &longs;ur&longs;um vltimus gradus
qui de&longs;truitur e&longs;t imperfectior innato, e&longs;t enim acqui&longs;itus; igitur in omni
alio motu &longs;ursùm. Tertiò, quia violentus e&longs;t cum innato in eadem &longs;ubie
cti parte; &longs;ed idem &longs;ubiectum formas homogeneas non patitur, de quò
aliàs, hinc dicendum &longs;upere&longs;t non quie&longs;cere mobile in fine motus
patet, quia
nihil e&longs;t quod de&longs;truat ip&longs;um impetum; igitur &longs;emper moueretur, ni&longs;i
per accidens ab ip&longs;o medio eius motus retardaretur; vnde dixi
cum ratione medij retardetur; immò quò leuius e&longs;t, faciliùs à medio re
tinetur, vide Th.61.
quia impetus naturalis aduentitius &longs;upponit motum deor&longs;um, ad cuius
inten&longs;ionem à natura fuit in&longs;titutus per re&longs;p. ad quartam obiect.
in di&longs;
&longs;ert.l.2. adde quod tardiùs a&longs;cenderet, quàm de&longs;cenderet; deinde velo
ciùs de&longs;cenderet po&longs;tmotum violentum corpus graue, quàm &longs;i nullo mo
tu violento præuio demitteretur deor&longs;um, quæ omnia experimentis
pa&longs;&longs;im a&longs;&longs;erit GalileusPrimò, quia non &longs;unt infinita in&longs;tantia, &longs;ed retarda
tur tantùm &longs;ingulis in&longs;tantibus; Secundò in medio den&longs;iore minùs du
rat; igitur non tran&longs;it per tot gradus tarditatis;
præterea in plano incli
nato &longs;ur&longs;um în minore proportione retardatur motus, quod etiam in
plano horizontali certi&longs;&longs;imum e&longs;t; quorum omnium rationes &longs;uo loco
videbimus.
Nec e&longs;t quod aliqui dicant infinito tribui non po&longs;&longs;e hæc prædicata
æqualitatis vel inæqualitatis, quod fal&longs;um e&longs;t, loquamur de infinito actu;
&longs;i enim e&longs;&longs;et numerus infinitus hominum, nunquid verum e&longs;&longs;et dicere
numerum oculorum e&longs;&longs;e maiorem numero hominum; nec e&longs;t quod ali
qui confugiant ad di&longs;iunctiones; nos rem i&longs;tam &longs;uo loco fusè tractabi
mus & demon&longs;trabimus, ni fallor, cum Ari&longs;totele, fieri non pò&longs;&longs;e vt &longs;it
aliquod creatum infinitum actu; licèt vltrò concedamus plura e&longs;&longs;e infi
nita potentiâ; & verò certum e&longs;t infinito potentiâ non ine&longs;&longs;e huiu&longs;modi
prædicata æqualitatis, vel inæqualitatis.
nunquam profectò de&longs;cenderatquia cum &longs;ingulis in&longs;tantibus &longs;inguli gra
dus re&longs;pondeant, & duo in&longs;tantia &longs;imul e&longs;&longs;e non po&longs;&longs;int; nunquam certè
verum e&longs;&longs;et dicere fluxi&longs;&longs;e infinita; igitur nec mobile per infinitos tar
ditatis gradus ad quietem perueni&longs;&longs;e; hoc Theorema &longs;upponit e&longs;&longs;e tan
tùm finita in&longs;tantia.
dictis, quia initio motus e&longs;t velocior, igitur plures partes aëris æquali
tempore re&longs;i&longs;tunt; in fine verò è contrario.
quæ cum naturali e&longs;t coniuncta,
verò initio maior, & in fine minor; hinc prima cre&longs;cit cam &longs;uo motu,
&longs;ecunda cum &longs;uo decre&longs;cit.
tet ex dictis; igitur in toto motu eadem e&longs;t re&longs;i&longs;tentiæ proportio.
violento,
bili &longs;cilicet &longs;ur&longs;um repercu&longs;&longs;o; potentia motrix, quæ &longs;equitur ex com
pre&longs;&longs;ione & rarefactione corporum, &longs;ed de his omnibus aliàs.
Ob&longs;eruabis primò &longs;i aliquando accidat, vt aliqui volunt ictum, qui
&longs;tatim initio motus violenti infligitur, non e&longs;&longs;e maximum, &longs;ed minorem
eo, qui po&longs;t aliquod confectum &longs;patium infligitur; quod probant in pila
ex fi&longs;tula ænea &longs;ur&longs;um emi&longs;&longs;a, quæ
quod &longs;anè &longs;i verum e&longs;t, hæc vnica e&longs;t, &longs;eu ratio, &longs;eu cau&longs;a, quòd &longs;cilicet &longs;ur
&longs;um pila pellatur ab igne, qui ab ore fi&longs;tulæ erumpens per aliquod &longs;pa
tium à tergo vrget; igni enim innatum e&longs;t &longs;ur&longs;um euolare.
Ob&longs;eruabis &longs;ecundò, vix po&longs;&longs;e manu mobile &longs;ur&longs;um rectà proiici, quia
&longs;cilicet manus extremitas motu mixto mouetur ex duobus vel pluribus
circularibus, de quo infrà.
Ob&longs;erua tertiò, non tantùm propter grauitationem con&longs;eruari impe
tum naturalem innatum, &longs;ed etiam vt motui violento re&longs;i&longs;tat; at verò
non re&longs;i&longs;teret, ni&longs;i grauitaret.
Ob&longs;erua quartò, reciprocas rationes motus naturalis & violenti; in
quibus mirabile pror&longs;us fuit naturæ in&longs;titutum, cum idem in vtroque il
larum &longs;it principium.
Ob&longs;erua quintò, finem motus violenti e&longs;&longs;e multiplicem, nullum ta
men à natura in&longs;titutum; quippe potentia motrix, quæ agit ex appetitu
elicito, (vt vulgò aiunt,) &longs;eu cum cognitione, finem &longs;ibi proponit ad libi
tùm; illa verò quæ vi compre&longs;&longs;ionis excitatur per accidens &longs;ur&longs;um agit
mobile potiùs, quàm per aliam lineam; repercu&longs;&longs;a &longs;ursùm videntur e&longs;&longs;e
magis iuxta in&longs;titutum naturæ.
duobus, vel pluribus rectis.
MOTVM mixtum eum e&longs;&longs;e non dico, qui
ex pluribus aliis motibus componatur;
&longs;eu mi&longs;ceatur; nec enim plures motus
&longs;imul e&longs;&longs;e po&longs;&longs;unt in eodem mobili; cùm
tantùm e&longs;&longs;e po&longs;&longs;it vno dumtaxat in&longs;tan
ti vnica migratio ex loco in locum; nec plura loca
naturali virtute &longs;imul acquiri po&longs;&longs;unt; Igitur nec &longs;i
mul e&longs;&longs;e duo motus; Itaque motus mixtus &longs;implex
e&longs;t, &longs;i con&longs;ideretur ratio, & linea motus; mixtus verò
dicitur, quod ex pluribus re&longs;ultet, qui reuerâ non
&longs;unt, &longs;ed cùm e&longs;&longs;e po&longs;&longs;int, qua&longs;i confluunt in tertium
motum communi &longs;umptu qua&longs;i de vtroque partici
pantem, quod totum fit propter diuer&longs;os impetus,
vel
explicabimus infrà: Porrò in hoc Libro explicamus
tantùm motum mixtum, qui re&longs;ultat ex pluribus re
ctis, vt titulus ip&longs;e præfert.
uer&longs;as lineas determinato, vel eodem ad diuer&longs;ashæc definitio cla
ra e&longs;t; ob&longs;eruabis tantùm ad motum mixtum &longs;ufficere duplicem impe-
nec enim e&longs;t motus purè naturalis, nec etiam violentus, vt con&longs;tat; igi
tur mixtus.
vel deor&longs;um mobile percurrit lineam curuam
di&longs;co ludunt.
lineam perpendicularem deor&longs;um minimè cadit, &longs;ed per curuam inclinatam
hæc hypothe&longs;is mille &longs;altem nititur experimentis; modò &longs;ufficiat quod
&longs;it; nam propter quid &longs;it, demon&longs;trabo.
& proiectum per inclinatam deor&longs;umhæc hypothe&longs;is centies probata fuit;
nec in dubium reuocari pote&longs;t.
tumvel enim retardat, vt impetus innatus retardat violentum, vt &longs;uprà
diximus; vel ad motum vnà cum alio, vel &longs;olus concurrit. Ax.2.
hoc etiam &longs;uprà demon&longs;trauimus, & qui de&longs;truitur, de&longs;truitur quoque
pro rata, ne &longs;it fru&longs;trà qui de&longs;truitur.
rata
quantum pote&longs;t concurrit, igitur pro rata &longs;uæ virtutis.
neæ motus
quabilium, vel alterius æquabilis, & alterius retardati, vel alterius æqua
bilis, & alterius accelerati, vel alterius retardati, & alterius accelera
ti, &c.
fiet per AE, a&longs;&longs;umpta &longs;cilicet DE æquali, & parallela AB, quod probatur
per Th.137.l.1.
qualis, & lineæ determinationum decu&longs;&longs;antur ad angulos rectos
idem Th.137.
alioquin motus e&longs;&longs;et duplus cuiu&longs;li
bet &longs;eor&longs;im &longs;umpti, quod fal&longs;um e&longs;t; nam motus &longs;unt vt lineæ &longs;ed diago
nalis quadrati non e&longs;t dupla lateris; hoc etiam probatur per Th. 141.
& 142.l.1.
enim mobile in A eadem figura &longs;itque impetus per AC, & alter &longs;ubdu
plus prioris per AD, motus fiet per AF ducta DF æquali, & parallela AC,
quod probatur per Th.137.l.1.
decu&longs;&longs;antur ad angulos rectos
ne aliquid &longs;it fru&longs;trà per Ax.2. & &longs;æpè iam probatum e&longs;t.
quales, portio detracta vtrique æqualibus temporibus e&longs;t differentia
diagonalis & compo&longs;itæ ex DA, AB, quod clarum e&longs;t; &longs;i vero impetus
&longs;int inæquales, portio de&longs;tructa erit &longs;emper differentia diagonalis, v.g.
AF & compo&longs;itæ ex AC.AD.
id e&longs;t habet
proportionem ad vtrumque, quæ appellari pote&longs;t, aliquando ad neutrum, ad vtrumque v.g. &longs;i alter impetuum &longs;it 8.alter 6.
haud dubiè linea motus mixti erit 10. ad neutrum vt in diagonali qua
drati, & in multis aliis; ad alterum denique v. g. &longs;i alter &longs;it &longs;ubduplus la
teris æquilateri; alter verò eiu&longs;dem perpendicularis; nam diagonalis, &longs;eu
linea motus mixti erit latus ip&longs;um æquilateri.
les impetus, linea motus mixti erit diagonalis Rhombivt patet per Th.140.
l.1. pote&longs;t autem hæc diagonalis e&longs;&longs;e vel æqualis alteri laterum, vel ma-
minor cùm angulus minor e&longs;t 60. denique e&longs;t maior, cùm maior angu
lus e&longs;t minor 120, quæ omnia con&longs;tant ex Geometria.
les impetus, linea motus mixti erit diagonalis Rhombi
erit, quò angulus erit acutior per Th. 139. l.1. porrò e&longs;t &longs;emper maior
lateribus &longs;eor&longs;im &longs;umptis.
Ob&longs;erua in Rhombo e&longs;&longs;e duas diagonales inæquales, vt con&longs;tat;
igi
tur cùm lineæ determinationum decu&longs;&longs;antur ad angulum obtu&longs;um, linea
motus mixti &longs;emper e&longs;t diagonalis minor; cùm verò decu&longs;&longs;antur ad an
gulum acutum, &longs;emper e&longs;t diagonalis maior.
Hinc quò acutior e&longs;t angulus diagonalis accedit propiùs ad duplum
lateris, donec tandem vtraque linea coëat; tunc enim linea motus e&longs;t du
pla lateris.
Hinc quoque quò angulus e&longs;t obtu&longs;ior diagonalis accedit propiùs ad
nullam, vt &longs;ic loquar, donec tandem vtraque linea concurrat in rectam,
tunc enim nulla e&longs;t diagonalis; igitur nulla linea motus.
nor quidem &longs;i lineæ decu&longs;&longs;entur ad angulum obtu&longs;um; maior verò &longs;i decu&longs;&longs;en
tur ad angulum acutum
proprius ad lineam maioris; hoc est facit angulum acutiorem cum illa
eadem figura &longs;it linea impetus maioris AC, & minoris AD, linea motus
mixti e&longs;t diagonalis AF, quæ accedit propiùs ad AC, quàm ad AD, id e&longs;t
facit angulum acutiorem cum AC, vt patet ex dictis.
cum linea vtriu&longs;que
lineæ determinationum decu&longs;&longs;entur ad angulum obtu&longs;um vel acutum,
vt AC, EG. IM.
aliumcum enim impetus &longs;int vt lineæ, &longs;ub quibus fiunt rectangula vel
Rhomboides; v.g. impetus AC e&longs;t duplus impetus AD, &longs;ed angulus D
AF non e&longs;t duplus anguli FAC, vt con&longs;tat ex Geometria.
Ob&longs;eruabis dari de facto hunc motum mixtum ex duobus æquabilibus
in rerum natura; talis e&longs;t motus nauis, quam geminus ventus impellit in
mari, vel nubis, imò aëris pars in medio aëre, atque adeo ip&longs;ius venti,
&longs;unt enim hi motus æquabiles per &longs;e; quippe retardantur &longs;olummodo
propter re&longs;i&longs;tentiam medij, non verò propter vllam grauitationem.
&longs;it enim duplex impetus
per AE & AH æqualis; ita vt in dato tempore percurrat &longs;eor&longs;im AE mo
tu retardato; item AH iuxta proportionem Galilei;
certè eo tempore quo
percurreret AD in AE, & AI in AH percurrit AG motu mîxto per Th.
5. Similiter eo tempore quo percurreret AE &longs;eor&longs;im, & AH, percurrit
AF per Th.5. Igitur hic motus mixtus e&longs;t rectus, dum &longs;it vterque retar
datus iuxta &longs;imiliter &longs;i alter impetus impetus
&longs;it inæqualis, vt patet in &longs;equenti figura, &longs;it enim impetus per AE, &
alter minor per AH, certè ex AD, AI fit AG, & ex AE, AH fit AF, quam
rectam e&longs;&longs;e con&longs;tat ex Geometria; nec vlla e&longs;t difficultas, quæ ex &longs;upe
rioribus Theorematis facilè &longs;olui non po&longs;&longs;it.
Hinc linea motus mixti ex duobus retardatis &longs;iue æqualibus, &longs;iue
inæqualibus e&longs;t diagonalis parallelogrammatis &longs;ub lineis determina
tionum.
Ob&longs;eruabis dari de facto hunc motum in rerum natura, &longs;i v. g. in pla
no horizontali idem globus, vel &longs;imul gemino ictu impellatur, vel &longs;i iam
impul&longs;um mobile per nouam lineam impellatur.
Proba
tur, quia debet tantùm inuerti linea prioris &longs;cilicet mixti ex duobus re
tardatis; &longs;i enim à puncto F pellatur per FE, FH, motu accelerato, ita
primo, tempori re&longs;pondeat FM, FN, &longs;ecundo NH, ME; haud dubiè li
nea motus mixti erit FA; nam primò tempori re&longs;pondebit FG, & duo
bus FA, vt con&longs;tat ex dictis, &longs;iue vterque impetus &longs;it æqualis, &longs;iue alter
maior altero.
Hinc etiam linea motus mixti ex duobus acceleratis e&longs;t diagonalis,
vt iam &longs;uprà dictum e&longs;t de omnibus aliis.
Ob&longs;eruabis hunc motum dari in rerum natura &longs;altem in corporibus
&longs;ublunaribus; nec enim e&longs;t acceleratus ni&longs;i &longs;it motus naturalis, qui à
duplici impetu e&longs;&longs;e non pote&longs;t.
neam curuam&longs;it enim impetus per AF motu æquabili, & per AC motu
accelerato naturaliter, ita vt eo tempore quo percurritur &longs;eor&longs;im &longs;pa
tium AB percurratur AD triplum; certè ex vtroque primo tempore re
&longs;ultat linea motus mixti AE, &longs;ecundo tempore EG, &longs;ed AEG non e&longs;t
recta; alioquin duo triangula ABE, ACG e&longs;&longs;ent proportionalia, quod
e&longs;t ab&longs;urdum.
quod ip&longs;e Galileus toties in&longs;inuauit, & quiuis
etiam rudior Geometra intelliget; in quo diutiùs non hæreo, præ&longs;ertim
cùm nullus &longs;it motus, qui con&longs;tet ex æquabili, & naturaliter accelerato,
vt demon&longs;trabimus infrà.
curuam
& con&longs;equenter motu naturaliter retardato impellatur per NI motu
æquabili, diuidatur NI in 4. partes æquales v.g. ductis parallelis RD,
NE, PC, &c. a&longs;&longs;umatur NS vel RM, cui affigatur quilibet numerus impar;
putà 7. itaque RM &longs;int 7. ducatur HM parallelæ IN, a&longs;&longs;umatur QL 5.
ducatur GL parallela, accipiatur VK 3. ducatur FK: denique a&longs;&longs;umatur
FAI ducaturque AE parallela IN, & de&longs;cribatur per puncta AKLMN,
linea curua; hæc e&longs;t Parabola, vt con&longs;tat ex Geometria;
nam &longs;i BK e&longs;t 1.
CL erit 4. DM 9. EV 16. &longs;ed æquales &longs;unt AF.AG.AH.AI. prioribus vt
patet; igitur &longs;agittæ &longs;unt vt quadrata
igitur hæc e&longs;t Parabola;
igitur curua, atqui motus mixtus prædictus fieret per hanc lineam, nam
eo tempore quo mobile e&longs;&longs;et in S, erit in M, concurrit enim vterque im
petus pro rata, & eo tempore, quo e&longs;&longs;et in K erit in L, atque ita
deinceps.
Ob&longs;eruabis e&longs;&longs;e pror&longs;us inuer&longs;am prioris, quæ &longs;it ex motu æquabili, &
naturaliter accelerato; &longs;i enim per AE &longs;it æquabilis & æqualis priori
per NI, & per AI &longs;it acceleratus, &longs;i quo tempore peruenit in B motu æ
quabili perueniat in F motu accelerato; haud dubiè perueniet in K, mox
in L, &c. quia eadem proportione, &longs;ed inuer&longs;a quâ retardatur,
acceleratur; igitur &longs;i vltimo tempore retardati acquirit tantùm
YE; primo tempore æquali &longs;cilicet accelerati acquiret AF, atque ita
deinceps &longs;i per NE &longs;it retardatus, & per NI æquabilis linea motus mixti
erit NLA; &longs;i verò &longs;it per AI acceleratus, & per AE æquabilis æqualis
priori per NI, lineamosus mixti erit ALN eadem &longs;cilicet cum priori
mutatis tantùm terminis à quo, & ad quem; vtrùm verò in rerum natu
ra &longs;it huiu&longs;modi motus videbimus infrà.
deor&longs;um motus mixtus, linea per quam fit e&longs;t curua,
per AD, &longs;it acceleratus per AG, a&longs;&longs;umatur AB cum numero impari, putà
5.BC.3. CD.1. accipiatur AE.1. EF.3. ducantur parallelæ BK. CL. DI.
& aliæ EM. FH. GI. & per puncta AM. HI. ducatur linea curua, hæc e&longs;t
linea motus mixti ex retardato & accelerato; hæc porrò non e&longs;t Parabo
la, vt con&longs;tat, quia quadratum AE non e&longs;t ad ad quadratum AF, vt qua
dratum AB, vel EM ad quadratum FH, vel AC.
Ob&longs;eruabis in fine huius motus amplitudinem, &longs;eu &longs;inum rectum li
neæ &longs;cilicet GI, e&longs;&longs;e æqualem altitudini &longs;eu &longs;inui ver&longs;o, vel &longs;agittæ AG;
cùm enim motus naturaliter acceleratus in eadem proportione cre&longs;cat,
quod hic &longs;uppono, in qua retardatus decre&longs;cit; certè AG quæ e&longs;t linea
accelerati e&longs;t æqualis GI, quæ e&longs;t linea retardati: non tamen dicendum
e&longs;t lineam AI e&longs;&longs;e circulum, alioquin GH e&longs;&longs;et æqualis GI, &longs;ed GH e&longs;t, v.
g. 89. cum GI &longs;it radix quadr.81. e&longs;t enim 9. licèt GM &longs;it æqualis GH.
&longs;ed de his lineis infrà. Vtrùm verò &longs;it aliquis motus huiu&longs;modi, videbi
mus in &longs;equentibus Theorematis.
mixtoprobatur, quia &longs;unt duo impetus in eo corpore, &longs;cilicet innatus
deor&longs;um, & impre&longs;&longs;us per horizontalem, vt patet; igitur vterque aliquid
præ&longs;tat ad illum motum per Ax. 1. igitur e&longs;t motus mixtus per def. 1.
Demon&longs;tro; motus mixtus
ex vtroque æquabili e&longs;t rectus per Th.1.& 4. &longs;ed hic motus proiecti per
horizontalem non e&longs;t rectus per hyp.1.
quia nulla e&longs;t cau&longs;a, à qua violentus po&longs;&longs;it accelerari.
Primò, quia
cùm pro tata concurrant po&longs;t integrum quadrantem vix &longs;patium vnius
palmi confeci&longs;&longs;et in perpendiculari deor&longs;um per Th.59.l.2.quod tamen
e&longs;t contra experientiam.Secundò, quia ad aliquod tandem punctum per
ueniretur, in quo mobile haberet tantùm impetum innatun; igitur nul
lus e&longs;&longs;et ictus contra experientiam. Tertiò, quia naturalis impetus in
tenditur in plano inclinato; igitur in motu per inclinatam, e&longs;t enim
motus deor&longs;um; igitur intenditur impetus naturalis, vt patet ex lib. 2.
igitur non e&longs;t mixtus.
accelerato
e&longs;t &longs;uprà, tùm primo, tùm &longs;ecundo libro, nec in hoc e&longs;t vlla diffi
cultas.
demon&longs;tra
tur, primò, quia &longs;ub finem motus e&longs;&longs;et maior impetus; quippè nihil de
traheretur violento, &longs;ed multùm accederet naturali; igitur e&longs;&longs;et maior,
igitur e&longs;&longs;et maior ictus contra hyp. 3. &longs;ecundò, quotie&longs;cunque &longs;unt duo
impetus in eodem mobili ad diuer&longs;as lineas determinati, aliquid illo
rum de&longs;truitur per Th.141.l.1.tertiò &longs;i e&longs;&longs;et vterque æquabilis, aliquid
de&longs;trueretur per Theorema 6. igitur potiori iure, &longs;i impetus naturalis
cre&longs;cat.
Diceret fortè aliquis impetum de&longs;trui ab aëre, &longs;ed iam &longs;uprà re&longs;pon
&longs;um e&longs;t modicum inde imminui; nec enim vnquam aër in corpore graui
de&longs;truit tantùm impetus, quantùm producitur naturalis &longs;i &longs;it acceleratus;
alioquin motus deor&longs;um non cre&longs;ceret contra experientiam, & &longs;uprà in
toto ferè 2.lib. demon&longs;trauimus.
motus con&longs;tare vel ex naturali æquabili, & violento retardato per Th.
19. vel ex naturali accelerato & violento æquabili per Th. 18. &longs;ed hic
motus neuter e&longs;t, non primum per Th. 25. non &longs;ecundum per Theo
rema 26.
quam probauit, nec probare vnquam potuit; hoc etiam &longs;upponunt
multi Galilei &longs;ectatores, qui cen&longs;ent impetum nunquam de&longs;trui ni&longs;i à
re&longs;i&longs;tentia medij; &longs;ed quæro ab illis quodnam medium de&longs;truat partem
impetus in motu mixto; nec enim linea motus mixti adæquat duas alias
ex quibus qua&longs;i re&longs;ultat; certè hoc non pote&longs;t explicari cum infinitis fetè
aliis, ni&longs;i dicatur impetum de&longs;trui ab alio impetu, eo modo quo &longs;æpè
diximus, hoc e&longs;t ne &longs;it fru&longs;trà; igitur impetus violentus de&longs;truitur ab in
nato, non tamen innatus à violento, vt &longs;æpiùs inculcauimus.
lineam perpendicularem & ex violento retardatoProbatur, &longs;i ita e&longs;t,
additur naturali, quantum detrahitur violento, imò plùs; igitur &longs;emper
e&longs;t in eo mobili æqualis vel maior impetus; igitur æqualis e&longs;t &longs;emper,
vel maior ictus contra hyp. 3. adde quod non minùs impeditur ab im
petu violento naturalis motus, quàm ab inclinato plano; &longs;ed in plano
tenditur in perpendiculari deorsùm; nec enim tam citò de&longs;cendit mobi
le, quod certum e&longs;t, & in lib.de planis inclinatis demon&longs;trabo, cum tan
tùm hîc &longs;upponam ad in&longs;tar phy&longs;icæ hypothe&longs;eos; adde quod idem mo
bile proiectum per horizontalem in data di&longs;tantia minùs ferit, quàm pro
iectum per inclinatam deor&longs;um.
lerato, non eo quidem modo quo acceleratur in perpendiculari, &longs;ed eo quo acce
leratur in plano inclinato, quod hic &longs;ingulis probatur pri
mo, quia inductione facta non &longs;unt enim tantùm
9 combinationes, quia &longs;unt tres differentiæ, &longs;cilicet æquabilibus, retarda
tio, acceleratio; igitur &longs;i 3.ducantur in 3. &longs;unt 9. &longs;unt autem prima ex na
turali, quem deinceps voco primum, æquabili & violento (quem voca
bo &longs;ecundum) æquabili, &longs;ecunda ex prima æquabili & &longs;ecundo accelera
to, tertia ex primo æquabili & &longs;ecundo retardato, quarta ex primo acce
lerato & &longs;ecundo æquabili, quinta ex primo accelerato & &longs;ecundo acce
lerato, &longs;exta ex primo accelerato & &longs;ecundo retardato, &longs;eptima ex primo
retardato & &longs;ecundo æquabili, octaua ex primo retardato & &longs;ecundo ac
celerato, nona ex primo retardato, & &longs;ecundo retardato: non e&longs;t prima
per Th.22. non &longs;ecunda per Th. 21. non tertia per Th. 24. non quarta,
per Th.26. non quinta per T.2h.23. non &longs;exta per Th.29. eo modo quo
diximus, non &longs;eptima per Th. 25. non octaua per Th. 25. non denique
nona per Th.25. igitur debet e&longs;&longs;e alius motus, &longs;ed alius excogitari non
pote&longs;t præter illum quem adduxi. Probatur &longs;ecundò, quia non minùs
impeditur ab impetu violento impetus naturalis acqui&longs;itus quàm à pla
no inclinato vt iam dictum e&longs;t; igitur acceleratur quidem &longs;ed minùs;
nec
enim vterque e&longs;t æquabilis, nam linea e&longs;&longs;et recta per Th.4. & naturalis
cre&longs;cit quia de&longs;cendit deor&longs;um; præterea per Th.24. non pote&longs;t impetus
naturalis e&longs;&longs;e æquabilis, igitur non pote&longs;t violentus e&longs;&longs;e vel æquabilis,
vel acceleratus, igitur retardatus.
proportione quàm fit ea, in qua acceleratur, dum per idem planum inclina
tum de&longs;cenditprobatur, quia &longs;ingulis in&longs;tantibus mutatur inclinatio pla
ni &longs;eu lineæ; igitur &longs;ingulis in&longs;tantibus mutatur proportio accelera
tionis.
tio plani,1. quo magis incur
uatur, accedit propiùs ad perpendicularem, igitur motus magis accele
ratur.
to & retardato, eius linea e&longs;t curua per Th.20. non tamen e&longs;t Parabola,
vt con&longs;tat ex eodem Th.20. Vnde reiicies Galileum, qui vult lineam mo
tus proiecti per horizontalem in aëre libero e&longs;&longs;e Parabolam.
natur alisprobatur, non in minore, quia plùs impetus adderetur quàm de
traheretur; igitur maior e&longs;&longs;et in fine motus quàm initio, igitur maior
ictus contra hyp.;. non in æquali, quia &longs;emper e&longs;&longs;et æqualis ictus con
tra hyp.3.& contra Th.29.
pro rata, vt dicemus infrà; at verò cùm acceleretur tantùm naturalis
iuxta rationem motus, & motus &longs;it iuxta rationem plani, minùs accele
ratur quàm &longs;i caderet mobile perpendiculariter deor&longs;um.
quia &longs;cilicet e&longs;t
minùs impetus, quia plùs detractum e&longs;t quàm additum; nec e&longs;t quod
tribuant hanc retardationem medio; quippe aër non plùs re&longs;i&longs;tit motui
violento quàm naturali; &longs;ed id quod detrahitur ab aëre corpori graui, v.
g. pilæ plumbeæ e&longs;t in&longs;en&longs;ibile, vt fatentur omnes; igitur idem
dum
e&longs;&longs;et rectus contra hyp. 3. prætereà pote&longs;t concurrere;
nec enim &longs;unt li
neæ determinationum oppo&longs;itæ; igitur concurrit per Th.137.l.1.
per lineam horizontalem rectam, vt con&longs;tat, motu æquabili
retardaretur in horizontali, eodem modo moueretur quo in verticali
&longs;ur&longs;um, quæ omnia con&longs;tant ex dictis &longs;uprà.
&longs;it enim mobile pro
jectum ex E per horizontalem EI
&longs;et motu naturaliter accelerato de&longs;cendendo ex A in E;
tium acqui&longs;itum primo in&longs;tanti de&longs;cen&longs;us; BC duplum, CD triplum, &c.
iuxta progre&longs;&longs;ionem arithmeticam, &longs;it EI æqualis EA, diuidatur que eo
dem modo in 4. &longs;patia vt diui&longs;a e&longs;t EA; a&longs;&longs;umpta EO æqualis AB, ducan
tur FN. GM. HL. IK. parallelæ EV; a&longs;&longs;umatur OP æqualis OE, & PQ,
quæ &longs;it ad OE, vt OE ad hypothenu&longs;im &longs;eu planum inclinatum EN, a&longs;-
natum NM; denique a&longs;&longs;umatur ST æqualis OE, tum TV, quæ &longs;it ad OF,
vt QS ad inclinatam ML; ducantur ON. QM. SL. VK. parallelæ EI,
tùm per puncta E.N.M.L.X ducatur curua, hæc e&longs;t linea prædicti motus,
demon&longs;tratur.
Impetus violentus percurrit EF eo tempore, quo naturalis percurrit
EO; igitur linea motus mixti ex vtroque ducitur per punctum N, & licèt
videatur e&longs;&longs;e recta EN, &longs;cilicet diagonalis rectanguli OF, e&longs;t tamen cur
ua, quia mobile non percurrit EF vno in&longs;tanti; igitur nec EO, igitur
motu æqualiter accelerato percurrit EO; igitur EN non e&longs;t recta per
Th.20. Præterea.Secundo tempore impetus innatus remanet; igitur per
curratur OP cui addit ut PQ, quia impetus naturalis minùs cre&longs;cit, vt di
ctum e&longs;t in Th.34. quippe cre&longs;cit iuxta rationem plani inclinati EN.ad
EO permutando, quæ &longs;it v.g. &longs;ubquadrupla; igitur PQ e&longs;t &longs;ubquadrupla
EO; & cùm de&longs;trui &longs;upponatur vnus gradus violenti, v.g. &longs;uper&longs;unt tan
tùm 3. quibus percurritur FG; igitur linea huius motus duci debet per
punctum M, idem dico de punctis L & K, igitur hæc e&longs;t linea motus
mixti, quàm &longs;cilicet corpus graue proiectum per horizontalem &longs;uo fluxu
de&longs;cribit, & cuius alias proprietates demon&longs;trabimus.
v.g. Oq.e&longs;t maior EO, & QS maior OQ atque ita deinceps.
v.g. FG e&longs;t minor EF vno &longs;patio, GH minor EF vno &longs;patio; HI minor
GH vno &longs;patio, quæ omnia con&longs;tant. Vtrùm verò id fiat, dicemus infrà,
& exempli gratia tantùm dictum e&longs;&longs;e volo.
e&longs;t maior
qualis, impetus naturalis deberet cre&longs;cere in eadem proportione, in qua
decre&longs;cit violentus, vt dictum e&longs;t &longs;uprà.
&longs;cit in verticali EAnam EI & EA &longs;unt æquales, &longs;ed EI & VK &longs;unt æqua
les, AE verò e&longs;t linea, vel quam conficit mobile proiectum &longs;ur&longs;um cum
eodem, vel æquali impetu alteri quo proiicitur per horizontalem; &longs;eu
e&longs;t linea quam percurrit corpus graue deor&longs;um, dum acquirit æqualem
impetum alteri impre&longs;&longs;o eidem mobili per horizontalem EI.
lemquippe moueri non pote&longs;t ni&longs;i motu mixto ex naturali accelerato
igitur linea e&longs;t curua; dixi
in medio libero, cùm in plano duro horizontali per lineam rectam pro
iici po&longs;&longs;it.
nunquam e&longs;t rectus, &longs;ed inflectiturratio e&longs;t, quia haud dubiè grauitat, igi
tur incuruatur; vtrùm verò faciat Parabolam hæc linea curua, vt vult
Galileus, examinabimus in libro de lineis motus.
Ob&longs;eruabis funem ten&longs;um &longs;emper incuruari, ni&longs;i fortè ex maxima tra
ctione &longs;uam flexibilitatem amittat, cuius ope tantùm curuatur, imò ita
tendi pote&longs;t, vt ten&longs;ioni cedens frangatur: Equidem po&longs;ito quod vel in
flecti po&longs;&longs;it, vel reduci, nece&longs;&longs;ariò inflectetur in medio, vt benè demon
&longs;trat Galileus in dialogis, no&longs;que infrà ad potentiam vectis reducemus,
ne multiplicemus figuras.
la è tormento explo&longs;a, vel &longs;agitta arcu emi&longs;&longs;a per plura &longs;ecunda minuta mo
ueatur in medio aëre antequam terram attingatquod plu&longs;quàm mille ex
perimentis comprobatum e&longs;t; plura leges apud Mer&longs;ennum, v. g. &longs;it tor
mentum horizonti parallelum extans &longs;upra horizontem tribus pedibus;
certum e&longs;t &longs;patium illud trium pedum confici à globo perpendiculariter
demi&longs;&longs;o tempore 30. tertiorum; cùm tamen explo&longs;us per lineam hori
zontalem terram tantùm attingat po&longs;t 4. &longs;ecunda, ide&longs;t 240. tertia; ita
Mer&longs;ennus l.2. de motu Prop. vltima, imò l. 5. &longs;uæ ver&longs;ionis art.5. con
tra Galileum o&longs;tendit glandem emi&longs;&longs;am è tormento minori conficere
75. exapedas, tempore vnius &longs;ecundi minuti in linea, quæ parùm decli
nat ab horizontali; atqui tempore vnius &longs;ecundi minuti conficit 2.exa
pedas in perpendiculari deor&longs;um; igitur deberet glans infrà &longs;copum de
&longs;cendere notabiliter, id e&longs;t, toto 12. pedum interuallo, cùm tamen vix
tantillùm aberret à &longs;copo 1.Idem Mer&longs;ennus habet in Bali&longs;tica Prop.25.
globum è maiore tormento horizonti parallelo emi&longs;&longs;um in aëre tractu
continuo vola&longs;&longs;e toto tempore 8. &longs;ecundorum, antequam planum hori
zontale attigi&longs;&longs;et, cùm tamen &longs;ex tantùm exapedis tormentum extaret
&longs;upra horizontem; alter globus ex alio tormento explo&longs;us 6. tantum &longs;e
cunda in aëre con&longs;ump&longs;it; imò bombardarum globi aliquando tota 14.
&longs;ecunda po&longs;uerunt; habet idem Mer&longs;ennus alia plura, quorum fides &longs;it
penes authores à quibus accepit; nam vt dicam quod res e&longs;t vix accu
ratè minima illa tempora metiri po&longs;&longs;umus; quidquid &longs;it, ex illis &longs;altem
euinco mobile projectum per horizontalem plùs temporis in&longs;umere in
&longs;uo fluxu, quam &longs;i ex eadem altitudine perpendiculariter demittatur; vt
vult Galileus; cuius ratio alia non e&longs;t ab ea, quàm &longs;uprà indicauimus,
quòd &longs;cilicet motus naturalis minùs cre&longs;cat in motu mixto quàm in na-imò &longs;i cre&longs;ceret vt vult Galileus, ictus;
haud dubiè e&longs;&longs;et maior in fine motus quàm initio, quod omninò expe
rientiæ repugnat.
Nec e&longs;t quod aliquis dicat glandem emi&longs;&longs;am per horizontalem tan
tillùm a&longs;cendere; vnde plus temporis in a&longs;cen&longs;u &longs;imul & de&longs;cen&longs;u col
locatur, quàm in &longs;olo de&longs;cen&longs;u; nam primò vix hoc aliquis &longs;ibi per&longs;ua
&longs;erit, cùm experimento percipi non po&longs;&longs;it; Secundò licèt verum e&longs;&longs;et,
non tamen e&longs;t tantus a&longs;cen&longs;us, quin adhuc plùs temporis ponat in a&longs;
cen&longs;u, atqué in de&longs;cen&longs;u, quàm in alti&longs;&longs;ima perpendiculari quadruplæ al
titudinis, vt con&longs;tat; &longs;it enim horizontalis AF, di&longs;tans à plano hori
zontali altitudine BA; &longs;it tormentum directum per lineam AF, & glo
bus percurrat lineam curuam AEF, idque &longs;patio 8.&longs;ecundorum minu
torum; &longs;itque DE 3. pedum;
certè eo tempore quo conficit AE, &longs;i in
perpendiculari conficiat ED, cum ED conficiat tempore 30‴; haud
dubiè AE eodem tempore conficere deberet; &longs;ed conficit AE tempore
4. &longs;ecundorum, vt con&longs;tat ex ip&longs;is multorum ob&longs;eruationibus; igitur to
tam AEF deberet percurrere tempore 1″, id e&longs;t eo tempore quo in per
pendiculari deor&longs;um percurruntur 12. pedes; denique &longs;i verum &longs;it glo
bum a&longs;cendere tantillùm dum emittitur è tormento horizonti paralle
lo; crediderim id e&longs;&longs;e tùm ex aliqua repercu&longs;&longs;ione aëris, tùm eo quod à
flamma &longs;ur&longs;um a&longs;cendente &longs;ur&longs;um etiam aliquantulum inclinetur; quod
verò &longs;pectat ad &longs;agittam, alia cau&longs;a non e&longs;t ni&longs;i modica aëris repercu&longs;&longs;io;
e&longs;t enim leuior &longs;agittæ materia; &longs;ed de repercu&longs;&longs;ione fusè agemus
infrà.
probatur, quia con&longs;tat
ex naturali, & violenti; qui cùm non &longs;int in oppo&longs;itis lineis, ad commu
nem motum concurrunt, vt patet.
linea huius motus e&longs;t curua per hyp. non pertinet etiam hic motus ad
&longs;ecundam combinationem de qua Th. 30. nec ad quintam, nec ad
octauam, nec ad nonam, de aliis videbimus infrà.
probatur,
quia in fine motus e&longs;&longs;et maior impetus, igitur e&longs;&longs;et maior ictus contra ex
perientiam; imò longè maior quàm &longs;i mobile proiiceretur per horizon
talem, quia diutiùs durat ille motus; igitur plures gradus impetus na
turalis acquiruntur; igitur longè maior e&longs;t ictus; prætereà &longs;i impetus
naturalis de&longs;truit impetum &longs;ur&longs;um in verticali, cur non in inclinata? nam
e&longs;t eadem omninò ratio; quippe ideò de&longs;truitur in verticali, quia cor
pus graue &longs;ur&longs;um attollitur; cùm tamen &longs;ua &longs;ponte deor&longs;um ferri debe
ret; &longs;ed non minùs, cùm per inclinatam &longs;ur&longs;um proiicitur, remouetur à
nec ob&longs;tat oppo&longs;itio lineæ verticalis &longs;ur
&longs;um cum perpendiculari deor&longs;um; quia etiam per inclinatam deor&longs;um
fertur in plano inclinato, quæ opponitur ex diametro alteri inclinatæ
&longs;ur&longs;um.
lerato inquam eo modo quo acceleratur in perpendiculari deor&longs;umprobatur
primò, quia motus ille e&longs;&longs;et &longs;emper æqualis, quia tantùm adderetur im
petus quantùm detraheretur, igitur e&longs;&longs;et idem ictus in fine qui in princi
pio; Secundò, quia tempora motuum e&longs;&longs;ent breuiora quàm par &longs;it con
tra experientiam, vt patet ex Th.46.
modo quo diximus in Th.
nem plani inclinati deor&longs;um, vt dictum e&longs;t, &longs;upra horizontalem; nullum
e&longs;t ampliùs planum inclinatum deor&longs;um; igitur nulla acceleratio, imò
impetus naturalis, vt iam &longs;uprà dictum e&longs;t cre&longs;cit tantùm vt motus deor
&longs;um acceleretur; &longs;ed nullus e&longs;t hîc motus deor&longs;um;
modicùm figuræ
rem ob oculos ponit; motus in plano AB e&longs;t ad motum in AC vt
AC ad AB, & in AD, vt AD ad AB, & in AE, vt AE ad AB; igitur immi
nuitur in infinitum; &longs;ed acceleratur in inclinata deor&longs;um iuxta hanc ra
tionem, igitur nulla &longs;upere&longs;t ampliùs proportio, &longs;ecundum quam acce
lerari po&longs;&longs;et in inclinata &longs;ur&longs;um.
&longs;uprobatur, quia nulla alia combinatio præter hanc &longs;upere&longs;t, quam
tertio loco &longs;uprà collocauimus in Th. 30. ratio à priori e&longs;t, quia natura
lis innatus non retardatur; quia nunquam de&longs;truitur, nec acceleratur;
quia &longs;ur&longs;um tendit mobile; igitur &longs;upere&longs;t tantùm quod &longs;it æquabilis,
violentus verò non acceleratur, vt patet, quia nulla e&longs;t cau&longs;a: non e&longs;t
æquabilis, quia coniunctus e&longs;t cum cau&longs;a de&longs;tructiua; igitur e&longs;t re
tardatus.
diximus &longs;uprà in Th.probatur per idem Th.e&longs;t
enim par vtrique motui ratio; quippe hic perinde &longs;e habet, atque &longs;i mo
bile per horizontalem proiiceretur, nam præuius motus
Parabola linea arcus de&longs;cen&longs;us per Th.20.& 27.
&longs;it enim verticalis, AG horizontalis AN,
linea projectionis AD; &longs;itque primum &longs;egmentum AD, quod &longs;cilicet
percurritur eo tempore quo in perpendiculari deor&longs;um percurritur DF,
id e&longs;t, v.g. &longs;exta eius pars, ducatur AFG, &longs;itque FG 5. partium, quarum
&longs;cilicet AD e&longs;t 6. a&longs;&longs;umatur GH æqualis DF, ducaturque FHI; &longs;itque
HI 4. partium, a&longs;&longs;umatur IP æqualis GH, ducaturque HP; accipiatur
PK 3. partium; iam motus naturalis acceleratur eo modo quo &longs;uprà di
ctum e&longs;t iuxta rationem inclinationis deor&longs;um; itaque a&longs;&longs;umatur KL
paulo maior IP; &longs;imiliter ducatur PLM, &longs;itque LM duarum partium,
& MN paulò maior KL, tum &longs;it LNO, &longs;itque NO 1. partis, & OB ma
ior MN, & ducatur curua per puncta A.F.H.P.L.N.B. & habebis
intentum.
Porrò hæc linea non e&longs;t parabolica, vt con&longs;tat ex Geometria & plura
puncta habebis &longs;i minora &longs;patiola a&longs;&longs;umas; &longs;uppono enim DF e&longs;&longs;e tan
tùm id &longs;patij quod primo in&longs;tanti in perpendiculari deor&longs;um à corpore
graui percurritur.
res; vt habes in fig.
AD inclinata 9. partium, FG 7. HI 5. reliqua vt &longs;uprà dictum e&longs;t.
Si verò linea inclinata recedat longiùs ab horizontali, & accedat pro
piùs ad verticalem; vt habeantur puncta, transferantur eadem &longs;patia, &
habebis puncta, per quæ de&longs;cribes prædictam lineam.
Denique &longs;i inclinata accedat propiùs ad horizontalem, transferantur
&longs;imiliter &longs;patia vnius in alteram.
Ob&longs;eruabis autem crementa de&longs;cen&longs;us in GH. IB e&longs;&longs;e iuxta nume
ros impares 1.3.5.7.&c. quandoquidem a&longs;&longs;umitur &longs;patium quod confi
citur in tempore &longs;en&longs;ibili, habita tamen &longs;emper ratione accelerationis,
quæ fit in plano inclinato, vnde cre&longs;cit &longs;emper proportio acceleratio
nis, vt &longs;uprà demon&longs;trauimus; quæ certè proportionum inæqualitas ef
ficit, ne po&longs;&longs;int accuratè de&longs;cribi prædictæ lineæ, &longs;ed tantùm rudi Miner
uâ, cum &longs;ingulis in&longs;tantibus mutetur proportio accelerationis.
Ob&longs;eruabis nondum e&longs;&longs;e à nobis determinatam proportionem illam,
in qua de&longs;truitur impetus violentus in motu mixto, quæ tamen ex dictis
&longs;uprà pote&longs;t colligi; quippe de&longs;truitur pro rata, ide&longs;t qua proportione
linea motus mixti e&longs;t minor linea compo&longs;ita ex vtroque, &longs;it ergo.
probatur, quia natu
ralis non cre&longs;cit, vt patet; con&longs;tat enim arcus a&longs;cen&longs;us ex naturali æqua
bili, &longs;ed aliquis impetus decre&longs;cit, vt con&longs;tat ex dictis, igitur &longs;olus
violentus.
cre&longs;cit, vt dictum e&longs;t &longs;uprà, igitur non decre&longs;cit.
id e&longs;t, qua proportione e&longs;t frustrà;
v.g. &longs;it impetus per AD inclinatam &longs;ur&longs;um, & alius per AB perpendi
cularem deor&longs;um; haud dubiè motus erit per AC;
igitur concurrunt
ad motum AC motus AB & AD, vel potiùs impetus; igitur debet de
&longs;trui impetus in ea proportione, in qua AC e&longs;t minor AG, id e&longs;t com
po&longs;ita ex AD, DC, quod impetus AB non po&longs;&longs;it de&longs;trui; totum id
quod de&longs;truetur detrahetur impetui AD; igitur a&longs;&longs;umatur DF &longs;cilicet
differentia AC, & AG; impetus de&longs;tructus ita &longs;e habet ad impetum
AD, vt DF ad AD, & ad re&longs;iduum impetum ex AD, vt DF ad FA,
quæ omnia con&longs;tant ex Th.7. &longs;it ergo AC fig. 49. perpendicularis &longs;ur
&longs;um, AD inclinata, AB horizontalis; &longs;it impetus violentus re&longs;pondens
AD, & naturalis DG, ducatur AGK, ex AD detrahatur DF, id e&longs;t
differentia AG & compo&longs;itæ ex AD. DG, &longs;upere&longs;t AF, cui a&longs;&longs;umitur
æqualis GK, ex qua detrahitur KH, id e&longs;t differentia GL, & compo&longs;itæ
ex GK, KL, &longs;upere&longs;t GH, cui LO accipitur æqualis, cui detrahitur
OM, id e&longs;t differentia LP & compo&longs;itæ ex LO, OP, &longs;upere&longs;t ML, cui
æqualis accipitur PR, atque ita deinceps. Porrò demon&longs;tratur de&longs;trui
impetum violentum iuxta hanc proportionem; quia de&longs;truitur, qua
proportione e&longs;t fru&longs;trà, pro rata per Ax.2.& Th.7.&longs;ed totus impetus qui
concurrit ad &longs;ecundam lineam AG, e&longs;t compo&longs;itus ex AD, GD; quia &longs;i
naturalis &longs;olus e&longs;&longs;et, percurreret &longs;patium æquale DG; &longs;i verò &longs;olus e&longs;&longs;et
violentus percurreret &longs;patium æquale AD; igitur vterque &longs;imul &longs;umptus
e&longs;t vt
deficit, cùm AG &longs;it motus; certè motus e&longs;t ad impetum, vt AG ad
&longs;itamigitur deficit motus tota DF quæ e&longs;t differentia AG &
igitur impetus e&longs;t fru&longs;trà in ratione DF;
igitur de
bet de&longs;trui in ratione DF; &longs;ed impetus DG &longs;eu naturalis nihil de&longs;trui
tur per Th.57. & 58. igitur ex violento AD de&longs;truitur DF; igitur &longs;u
pere&longs;t tantum AF vel æqualis GK; &longs;imiliter impetui GK & KL re
&longs;pondet motus GL, &longs;ed GL e&longs;t minor compo&longs;ita ex GK & KL &longs;eg
mento KH; igitur e&longs;t fru&longs;trà impetus in ratione KH;
igitur de&longs;truitur
in eadem ratione KH, non ex naturali KL; igitur ex violento GK;
igitur &longs;upere&longs;t tantum GH, vel æqualis LO, in qua &longs;imiliter procedi
tur. & &longs;upere&longs;t LM vel æqualis PR, atque ita deinceps.
Hinc de&longs;truitur impetus initio motus in maiori quantitate, quia
Hinc &longs;ub finem differentia lineæ motus v. g. TB &longs;emper e&longs;t maius
latus trianguli TXB; idem dico de aliis; igitur differentia lineæ motus
& compo&longs;itæ ex duplici impetu e&longs;t &longs;emper minor & minor in in
finitum.
Po&longs;&longs;unt determinari à Geometria omnes anguli triangulorum ADG.
GKL. OLP. nam ADG e&longs;t æqualis CAD, at verò GKL æqualis
KGD, & hic duobus &longs;imul ADG & DAG, igitur determinari facilè
poterunt ex doctrina triangulorum.
Hinc etiam &longs;ciri poterit in quo puncto linea motus v.g. LP cum per
pendiculari OP faciat angulum rectum, quod &longs;atis e&longs;t indica&longs;&longs;e, nam hic
Geometram non ago.
Hinc quoque &longs;ciri pote&longs;t maxima altitudo huius projectionis, quæ
&longs;cilicet in eo puncto e&longs;t, in quo linea motus cum perpendiculari deor
&longs;um facit angulum rectum, v.g. in puncto P, &longs;i angulus LPO e&longs;t
rectus.
Hinc pote&longs;t etiam &longs;ciri altitudo operâ triangulorum productorum
AG 2. GK 3. OLP. quod quiuis Geometra facilè intelliget; hîc quo
que obiter ob&longs;erua vnum, quod &longs;æpè aliàs indicauimus, quanti videlicet
momenti &longs;it Geometria in rebus phy&longs;icis.
Hinc etiam colligo arcum a&longs;cen&longs;us maiorem e&longs;&longs;e arcu de&longs;cen&longs;us &longs;u
pra idem planum horizontale AB; quia in arcu de&longs;cen&longs;us acceleratur
pro ratione diuer&longs;æ inclinationis impetus naturalis; igitur lineam mo
tus addunt propiùs ad perpendicularem, vt vides in TB; igitur minùs
acquirit in horizontali; igitur minor amplitudo horizontalis &longs;ube&longs;t ar
cui de&longs;cen&longs;us projectorum quàm arcui a&longs;cen&longs;us; dixi &longs;uprà idem pla
num, quia arcus de&longs;cen&longs;us infra planum AB propagatur ferè in infi
nitum.
Hinc reiicio Galileum qui nulla pror&longs;us fultus ratione phy&longs;ica vult
vtrumque e&longs;&longs;e æqualem, quod tamen omnibus experimentis repugnat, &
ip&longs;i etiam pueri, qui di&longs;co ludunt ob&longs;eruare po&longs;&longs;unt arcum de&longs;cen&longs;us &longs;ui
di&longs;ci e&longs;&longs;e longè minorem, nec e&longs;t quod ad &longs;uam Parabolam confugiat,
quæ duo fal&longs;a &longs;upponit principia, &longs;cilicet æquabilitatem motus violen
ti, & accelerationem naturalis eo &longs;cilicet modo quo fieret in perpendi
culari; at vtrumque fal&longs;um e&longs;&longs;e &longs;uprà demon&longs;trauimus, adde quod vt iam omnes ob&longs;eruare po&longs;&longs;unt ar
cum a&longs;cen&longs;us maiorem e&longs;&longs;e arcu de&longs;cen&longs;us, quod etiam &longs;upponunt om
nes, qui de re tormentaria &longs;crip&longs;erunt; præ&longs;ertim Vfanus tract. 3.
c. 13.