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-
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 &
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 moctu 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
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
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
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,
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-
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
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
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
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-
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
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
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
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.
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; codem pror&longs;us
modo &longs;oluendæ &longs;unt, quo iam &longs;upra &longs;olutæ &longs;unt aliæ circa motum:
vtrùm verò motus, & quies dicant aliquid di&longs;tinctum à mobili, 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 codem 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
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,
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
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 magnte. e
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 effectus
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, codem modo &longs;e habet, ac &longs;i e&longs;&longs;et cau&longs;a; quippe
certum e&longs;t phy&longs;icè ignem calefacere, Solem illuminare, quod &longs;atis e&longs;t.
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 codem 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;
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
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-
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 codem contradictoria dici
non po&longs;&longs;unt per Ax. 1. n. 3. Igitur impetus non e&longs;t idem cum &longs;ub&longs;tantià
mobilis; igitur di&longs;tinctus; 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.
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;torelem, putantque
non e&longs;&longs;e locutum ex propriâ &longs;ententiâ: Alij dicunt Ari&longs;totelem quidem
tribui&longs;&longs;e aliquam vim extrin&longs;ecam aëri; non tamen nega&longs;&longs;e intrin&longs;ecam
impetus; quidquid &longs;it, ip&longs;a verba Ari&longs;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.
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:
&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 dcor
&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.
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.
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
&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.
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
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-
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;trucretur. Primum dicere ab&longs;urdume&longs;t; quia &longs;i hoc e&longs;&longs;et
multisictibus repetitis tandem moueretur totum mobile; &longs;i verò de
ftrui 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 fubiecti, 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-
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
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, codem in&longs;tanti e&longs;&longs;et in duobus locis, quod dici non
pote&longs;t; & iam diximus in Th. 26. igitur impetus primo in&longs;tanti quo
e&longs;t non habet &longs;uum motum.
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 codem 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.
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
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 codem 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 codem 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.
& efficientis ad extra
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,
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 cnim 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
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
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.
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.
contactu
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
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 codem 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.
tas; cum &longs;cilicet maior globus in minorem impingitur
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; crgo in
qualibet parte &longs;ubiecti &longs;unt plures; &longs;ed hoc e&longs;t e&longs;&longs;e inten&longs;um, vt con&longs;tat,
igitur impetus remi&longs;&longs;us producit inten&longs;um; quod e&longs;t paradoxon egre
gium.
in maiorem incidit
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 æqualem
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, certèvel producent plures vel
pauciores, vel totidem, non plures; cur enim potius 31. quam 32.
nec etiam pauciores; cur enim potius 20. quam 18, &c. Igitur totidem;
quia cum &longs;int plures numeri plurium partium &longs;upra 30. & pauciorum
infra vt patet; &longs;itque tantùm vnicus numerus æqualium; certè quod
vnum e&longs;t, determinatum e&longs;t, per Ax. 5. hæc ratio licèt videatur 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;truiquidem
aliquid impetus in globo impacto, vt videbimus infrà; cum tamen de
&longs;truatur tantùm &longs;equenti po&longs;t ictum in&longs;tanti; certè cum exi&longs;tat adhuc
ip&longs;o in&longs;tanti contactus, nece&longs;&longs;ariò agit, quippe aliquid vltimo in&longs;tanti
pote&longs;t agere; adde quod illud ip&longs;um repugnat manife&longs;tæ experientiæ;
licèt enim aliquando de&longs;truatur totus impetus in globo impacto, quod
&longs;æpè accidit in ludo Tudiculario, nam illicò &longs;i&longs;tit pila eburnea; alius
tamen globus velociter mouetur, cuius effectus rationem infrà 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;pondco, 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, aqitali 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.
immobili
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.
&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;emium,
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
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 ra
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;
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.
& in
in qua vero proportione determinabimus aliàs.
non,
&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
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;uimus
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 mallci F, quia
&longs;cilicet trapezus OD e&longs;t minor &longs;ubduplo alterius NE; quotâ vero parte
&longs;it minor facilè pote&longs;t &longs;ciri opera Geometriæ: &longs;ed hæc omnia determi
nabimus.
in infinitum
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;imilem
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.
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 alliud e&longs;t, vnde po&longs;&longs;it
determinari extentio qualitatum, præter ip&longs;am exten&longs;ionem &longs;ubjecti;
quod maximè in impetu videre e&longs;t, cuius partes in mobili den&longs;o minori
extentioni &longs;ubjacent, quàm in mobili raro; cum ex maiore ictu &longs;eu 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 modo
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 porti mobilis in&longs;unt
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.
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-
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.
pertinet
&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 potest
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 producit
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 motu
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
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.
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,
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.
impetus, vt plures partes ver&longs;us centrum motus producantur in pondere, quod
attollitur
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 huius
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 vectis
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.
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-
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 heterogencos. Secunda e&longs;t in motu circulari, applicata &longs;cilicet
potentia centro; cum propagatur per partes æquales in perfectione, &
inæquales in numero. Tertia e&longs;t in vecte, cum propagatur per partes
æquales in numero, & inæquales in perfectione.
non pote&longs;t e&longs;&longs;e impetus, ni&longs;i exigat motum per Th.14. nec exigere mo-
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.
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.
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.
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
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.
minatu
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 contactus
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 lineam
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 priori
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.
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 reflexi
tio e&longs;t, quia maior globus maius e&longs;t impedimentum, hinc nunquam
quie&longs;cit minor globus impactus.
ctat centra, &longs;eruat
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
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 rata
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 obice
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
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.
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
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 duplum
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.
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.
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 lineæ
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.
flexo grauium; ratio e&longs;t. quia mutatur linea.
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
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.
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.
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;iuc &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-
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
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;
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.
tas e&longs;t ip&longs;e impetus innatus, de qua infrà:
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 deinceps
&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 cuidens; 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 codem 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 langudiorem; 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.
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 6longs;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
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 impedimentum
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 momenta
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 tempus
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 tardiore
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 permutata
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.
&longs;tantia æqualem impetum addant
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 decurritur
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 velocitatis
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 quotientis
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
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
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 aceclera
to
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;iuc 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
(40/25), &longs;i 12. (57/21), &longs;i 14. (77/28), atque ita deinceps.
Ex quo ob&longs;erua mirabilem con&longs;equutionem; quippe &longs;i a&longs;&longs;umantur
tantùm duo termini, & diuidantur bifariam, &longs;umma po&longs;terioris 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 mobili
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.
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
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 motus
&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 æqualis
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
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 minùs
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 codem 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.
dus
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
dictum illud vulgare,
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
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
vibrationes vtriu&longs;que, vbi 100. è minoribus numeratç 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; uod 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 fai&longs;um e&longs;t; quia in eodem plano inclinato &longs;upponitur eadem
inclinatio; &longs;ecus in quadrante, cuius &longs;ingula puncta nouam faciunt 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 nnmerantur; 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 arithinetica 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 diffetentiæ, nec centuplum, nec
millecuplum; &longs;ed de his &longs;atis quæ ex dictis &longs;uprà facilè intelligi po&longs;&longs;unt:
quare veniemus iam ad rationes.
Prima ratio, quam affert Galileus e&longs;t; quia cum natura in &longs;uis 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 qautuor
partes æquales, quæ nobis repre&longs;entent 4. partes temporis æquales;
haud dubiè, cùm acquirantur temporibus æqualibus æqualia velocitatis
momenta; haud dubiè, inquam, his 4. temporibus AB, BC, CD, DE, ac-
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;uniliter 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 tempote
AB; idem dico de trapezo CBIH, qui e&longs;t triplus trianguli IBA; & de
trapezo GDCH, qui e&longs;t quintuplus; igitur triangulum HCA e&longs;t 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.
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
tota velocitas in hypothe&longs;i Galilci acqui&longs;ita in 4.partibus temporis a&longs;
&longs;umptis e&longs;t, vt triangulum AFN; acqui&longs;ita verò in no&longs;tra hypothe&longs;i e&longs;t vt
&longs;umma rectangulorum CB, CI, EK, EN, quæ &longs;umma e&longs;t ad triangulum
AFN, vt 10, ad 8. vel vt 5.ad 4. igitur maior 1/4; nam prima pars 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;eliema hîc habes; &longs;int enim 3.partes temporis &longs;en&longs;ibiles A
CDFE, & &longs;patium vt triangulum AFN, &longs;patia verò acqui&longs;ita in &longs;ingulis
partibus, vt portiones trianguli prædicti, quæ ip&longs;is re&longs;pondent v. g. 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 quabilis, quod erat demon&longs;trandum. Præterea &longs;i diai
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;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-
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 onminò equale in&longs;tanti,
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;titutibus 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-
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,
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 accelerato
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
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.
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.
impetus
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
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;patiorum
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;eruarentur
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.
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
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 codem impetu, quàm duo impetus ad oppo&longs;itas lineas deter
minati; igitur impetus naturalis aduentitius de&longs;truitur, &c.
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, & idco corpora &longs;ur&longs;um ire, quia alia deor&longs;um eunt.
motus
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.
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
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.
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-
pauciores &longs;u&longs;tinent alterius leuioris, &longs;eu rarioris, vt con&longs;tat; v.g. certum
e&longs;t
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.
tat
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.
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.
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
grauitat
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.
ris
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 vacuo
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/
co tempore quo decurreret in vacuo 10. palmos &longs;patij, in aqua decurre
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 codem 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 codem tempore
conficit 48. pedes; igitur velociùs moueretur in aëre quàm in vacuo;
igitur e&longs;t aliquid aliud quod impedit motum; vt enim optimè monet
Mer&longs;ennus, &longs;i grauitas aquæ &longs;it ad grauitatem aëris vt 400 ad 1.& 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 medy
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.
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.
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 & nexum
uendus; igitur ex vtroque capite impeditur motus.
quibu&longs;dam filamentis
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
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 aëre
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
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;ibiliter
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;tentia
den&longs;iore ab codem 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.
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
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;itratio 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.
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.
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 latus
plum lateris maioris, & facies &longs;ubquadrupla; &longs;olidum verò &longs;ub
octuplum.
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 miuuitur quàm facies, vt dictum e&longs;t; igitur
plus minuitur grauitas, quæ e&longs;t cau&longs;a virium quàm hæc re&longs;i&longs;tentia; 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
accipiantur duo cubi maior, & minor eiu&longs;dem materiæ, & dimittantur
ex eadem altitudine codem 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, &
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.
cendunt
&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
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 materiæ
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.
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 grauioris
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?
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.
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;tantibus
&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.
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 codem modo cre&longs;cit velocitas, quo cre&longs;cunt numeri prædicti; &longs;ed
eodem modo cre&longs;cunt &longs;patia, &longs;i dumtaxat accipiantur in &longs;ingulis in&longs;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
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 cunjuncto, 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.
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.
&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.
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.
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.
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.
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.
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 naturalis
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.
&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?
per æqualis; igitur haberet &longs;emper æqualem motum per Ax.3.l.2. igitur
motus e&longs;&longs;et æquabilis, contra Th.15.
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.
&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 muato&longs;i quæ e&longs;t de quæ alias,
ratiocur 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.
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;truatur
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.
ratur
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 erunr æqualia primo in
&longs;tanti motus naturalis, probatur, cum &longs;ingulis in&longs;tantibus æqualibus de
&longs;truatur vnus gradus; certè 4. in&longs;tantibus durat motus.
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 percurritur
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
nora
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,
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.
perfectione
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;t
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 infrá
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 æqualia
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;us
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;eruari
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 aëre
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;patij
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.
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 proijciatur
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.
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;um
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 concurrit
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;t
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 innato
æqualibus vnum gradum; quia pugnat pro rata; ita pror&longs;us impedit, ne
tiùs, quàm tres?
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 destruatur
&longs;tructiua impetus violenti; ac proinde æquabilis e&longs;&longs;et, &longs;emperque dura
ret, de&longs;tructiua inquam &longs;uo modo.
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.
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.
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 codem 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
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;tantis
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.
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 innato
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;equenti
dus impetus innati exigit motum deor&longs;um quàm gradus impetus vio
lenti &longs;ursùm; igitur neuter habebit motum per Th.133.l. 1.
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.
&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
tur quies illa durat tantùm vno in&longs;tanti.
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
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 Galileus
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;cenderat
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;as
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;um
nec in dubium reuocari pote&longs;t.
tum
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.
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.
proportionem ad vtrumque, quæ appellari pote&longs;t, aliquando ad neutrum,
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 Rhombi
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.
alium
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.
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;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.
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
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.
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 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.
mixto
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.
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.
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 codem 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 retardato
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
mo, quia inductione facta non
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;cendit
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 alis
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.
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.
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 EA
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.
lem
in medio libero, cùm in plano duro horizontali per lineam rectam pro
iici po&longs;&longs;it.
nunquam e&longs;t rectus, &longs;ed inflectitur
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 attingat
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-
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.
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.
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
&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;um
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;u
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.
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.
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.
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;itam
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.
Hinc etiam colliges contra Vfanum globum è tormento emi&longs;&longs;um per
inclinatam &longs;ur&longs;um non ferri primò per lineam rectam, quia mouetur
motu mixto, qui rectus e&longs;&longs;e non pote&longs;t in hoc ca&longs;u per Th.54.
Motus mixtus arcus de&longs;cen&longs;us v&longs;que ad centrum terræ durare po&longs;&longs;et
&longs;i producerentur tot partes impetus quot &longs;unt in&longs;tantia illius motus; quia
cùm &longs;emper de&longs;truatur minor impetus, & minor in infinitum, po&longs;t ali
quod &longs;patium de&longs;cen&longs;us tam parùm de&longs;truitur v&longs;que ad centrum terræ vt
non adæquet totus ille impetus primam partem primo in&longs;tanti de&longs;tru
ctam, at tunc linea motus à perpendiculari deor&longs;um di&longs;tingui non
pote&longs;t.
Sed ne Geometriam omninò de&longs;picere videar, in circulo demon&longs;tro
proportiones omnes in quibus decre&longs;cit motus violentus per quamlibet
lineam inclinatam &longs;ur&longs;um, vel deor&longs;um; &longs;it ergo circulus ADGQ cen
tro B; &longs;it motus violentus &longs;ur&longs;um BD coniunctus cum naturali BR, &longs;int
que ex gr. BR. RQ æquales; hand dubiè linea motus erit BC, quia na
turalis BR pugnat pro rata per Th.134.l.1. eritque BC &longs;ubdupla BD;
igitur centro R. &longs;emidiametro RC de&longs;cribatur circulus CLPS, erit
æqualis priori, ducanturque ex centro B infinitæ lineæ BE. BF. BK.
BN, & vt res fit clarior, &longs;int omnes anguli DBE. EBF. FBG, &c. æquales &longs;cilicet grad. 30. & ex punctis E.F.G.K.N.q.
ducantur lineæ
ad circunferentiam circuli CLPS. parallelæ DP.Dico omnes e&longs;&longs;e æqua
les DC; nam primò FH. GL. KM. QP &longs;unt æquales, vt patet: deinde
CE & QO &longs;unt æquales; igitur EV. OX, quod etiam certum e&longs;t; igi
tur &longs;i &longs;upponatur idem motus violentus æqualis BD per omnes inclina
tas BE. BF, &c. coniunctus naturali æquali BR; primum &longs;patium erit
BC, &longs;ecundum BV, tertium BH, quartum BL, quintum BM, &longs;extum
BO
Th.5. &longs;imiliter ex BF. FH. fit BH, ex BG. GL. fit BL; denique ex
nometria facilè intelligi po&longs;&longs;unt.
plane horizontali, vnde incipit iactus
iactus horizontalis habeat arcum a&longs;cen&longs;us, & de&longs;cen&longs;us.
acquirit in eodem plano horizontali, &longs;cilicet in eo à cuius extremitate inci
pit iactus
rat in hotizontali plano per Theorema 60. certè quò propiùs ad illum
iactus inclinatus accedet, minùs acquiret; idem dico de iactu hori
zontali.
zontali, plùs acquiret in eodem plano horizontali
cedit ad vtrumque, minùs acquirit, igitur plùs acquirit, quò plùs re
cedit.
facit angulum
rit in eodem plano horizontali, quàm reliqui omnes
e&longs;t, ratio e&longs;t quia ab horizontali & verticali maximè omnium di&longs;tat;
igitur maximus e&longs;t per Theorema 62. nec e&longs;t vlla alia ratio geome
trica.
probatur, quia qua proportione ad horizontalem &longs;eu verticalem acce
dit iactus, in ea proportione minor e&longs;t; igitur qui æqualiter acce
dunt in proportione æquali, minores &longs;unt; igitur æquales, quod mo
dica figura ob oculos ponet; &longs;it enim quadrans ABF, iactus verti
calis AB, horizontalis AF, medius AD, hic maximus omnium
erit; at verò AC, & AE, qui ab AD æqualiter di&longs;tant, erunt æ
quales.
Ob&longs;eruabis primò, omitti à me multa quæ &longs;uis Parabolis aliqui af
fingunt, quæ nec experimentis, nec vllis rationibus con&longs;en
tiunt.
Secundò rationem i&longs;torum omnium Theorematum; quia quo iactus
ad verticalem propiùs accedit, maior quantitas impetus de&longs;truitur
v.g. in AD plùs quàm in GK; igitur citò deficiunt vires huic iactui;
adde quod acquirit in verticali, quod alius acquirit in horizontali; at
horizontale, tùm quia propior e&longs;t, tum quia citò naturalis impetus
acceleratur; igitur plùs acquirit in perpendiculari deor&longs;um, quàm in
horizontali; quæ omnia ex certis principiis, non fictitiis dedu
cuntur.
Tertiò, ob&longs;eruabis talem e&longs;&longs;e hypothe&longs;im illam Paraboli&longs;tarum, de
qua &longs;uprà; &longs;it enim iactus verticalis EA; medius EB; certè ex eorum
etiam principio eo tempore, quo motu æquabili percurreret mobile &longs;pa
tium EA, motu naturaliter retardato percurreret &longs;patium EG &longs;ubdu
plum; atqui percurrit EG eo tempore, quo idem percurreret GE motu
naturaliter accelerato; &longs;ed percurret inclinatam EC eo tempore quo
percurret EA, &longs;cilicet motu æquabili; &longs;unt enim æquales: Volunt autem
FE diuidi in 16. partes, & ED in 8. ducique parallelas HQ IP, &c. & ac
cipi VR (1/16) FE, ita vt RQ &longs;it ad RH vt 9.ad 7. & PS (4/16) & NT (9/16), vel O
T (1/16) PS (4/16) PR (9/16); igitur eo tempore, quo mobile e&longs;&longs;et in IX, erit in M;
igitur motus naturalis acqui&longs;iuit XM, id e&longs;t 1/4 AE; igitur eo tempore quo
e&longs;&longs;et in B erit in D; igitur motus naturalis acqui&longs;iuit BD quadruplum X
M; nam &longs;i vno tempore motu æquabili conficit EX, duobus conficit E
D & &longs;i motu naturaliter accelerato conficit vno tempore XM, duobus
conficit BD iuxta proportionem Galilei, in qua &longs;patia &longs;unt vt temporum
quadrata; & quo tempore motu æquabili conficeret EA, vel EB naturali
conficeret GE vel CZ æqualem GE; ducatur igitur linea per puncta E.
RS, OM, hæc e&longs;t &longs;emiparabola cui &longs;i addas MZD, habebis totam ampli
tudinem Parabolæ ED, hoc e&longs;t totum &longs;patium, quod acquirit in plano
horizontali ED iactus medius EB.
Si verò &longs;it inclinata EY; vt habeatur iuxta hanc hypothe&longs;im amplitu
do horizontalis; fiat &longs;emicirculus centro G, &longs;emidiametro GE; &longs;it per
pendicularis YK, erit &longs;ubdupla amplitudo; &longs;icut perpendicularis XL de
finit &longs;ubduplam amplitudinem LE iactus EB; &longs;imiliter YK definit &longs;ubdu
plam amplitudinem iactus E 4.3. nam arcus YX e&longs;t æqualis arcui X 4.
igitur anguli YEC, CE. 3. &longs;unt æquales; hinc iactus &longs;unt æquales &longs;upra, &
infra grad.45. vt autem habeatur altitudo Parabolæ &longs;ubdupla XL e&longs;t al
titudo Parabolæ iactus EC, &longs;ubdupla YX e&longs;t altitudo iactus EY, &longs;ubdu
pla 4.K e&longs;t altitudo iactus E 3.
Ex his facilè iuxta hypethe&longs;im tabulæ omnium iactuum, cuiu&longs;libet
eleuationis con&longs;trui po&longs;&longs;unt; de quibus habes plura apud Galileum in
dialogis, & plurima apud Mer&longs;ennum in Bali&longs;tica; quare ab illis ab&longs;ti
neo: præ&longs;ertim cum &longs;it fal&longs;a illa hypothe&longs;is, eiu&longs;que &longs;ectatores vltrò fa
teantur tabulas illas non parum à vero abe&longs;&longs;e, de quo vide Mer&longs;ennum
prop. 30. Bali&longs;t.
Quartò, po&longs;&longs;unt iuxta no&longs;tram hypothe&longs;im tabulæ nouæ con&longs;trui, quod
& ego præ&longs;tarem, ni&longs;i pror&longs;us inutiles e&longs;&longs;ent; quare prudenter omi&longs;&longs;as
e&longs;&longs;e prudentes omnes cen&longs;ebunt, cum hîc calculatorem non
lo&longs;ophum; id certè tolerari potuit in analyticis, quæ &longs;ine calculationibus
intelligi non po&longs;&longs;unt; &longs;ed minimè ferendum in Phy&longs;ica, quæ &longs;ucculen-
Praxis Theoricæ in his omninò præferenda e&longs;t; quamquam huic etiam
parti dee&longs;&longs;e nolumus, &longs;ed in &longs;ingularem libellum omnes i&longs;tas tabulas &
alias huiu&longs;modi remittimus; cum hic tantùm rerum phy&longs;icarum cau&longs;as
explicemus.
biè iactus omnium maximus erit horizontalis in vtraque hypothe&longs;i. Primo in
hypothe&longs;i Galilci, in qua Parabola GD figurâ &longs;uperiore habet maximum
omnium amplitudinem; licèt iactus per GX; ex quo &longs;equitur, non ha
beat impetum maiorem, quâm iactus per EY, vel EX; in no&longs;tra verò, ia
ctus per BG primo tempore plùs acquirit in horizontali BG, quàm ia
ctus per BF; igitur plùs etiam &longs;ecundo tempore; nam BF acquirit tantùm
primo tempore BH, at verò BG acquirit RL; adde quod minùs perit ex
iactu BG; quippe a&longs;&longs;umatur BL in B 2. & GL in 2. 3. detrahitur tantùm
G. 3.ex BG; at verò a&longs;&longs;umatur BH in B 4. & FH in 4.5. detrahitur F 5.ex
BF; igitur plùs ex BF quàm ex BG; quæ omnia ex &longs;uperioribus regulis
iu&longs;ta no&longs;tram hypothe&longs;im præ&longs;criptis con&longs;equuntur.
æqualiter ab horizontali, vnde incipit iactus, distantes; æquales inquam in ali
quo plano horizontali, inferiore
BP nihil acquiritur in horizontali, vt con&longs;tat; &longs;i verò iactus &longs;it per BG
maximum &longs;patium acquirunt in horizontali plano inferiore; igitur qua
proportione propiùs accedent lineæ &longs;eu iactus ad BD, PP minùs acqui
rent; qua verò proportione propiùs accedent ad RG plùs acquirent; igi
tur æqualiter plùs, & minùs hinc inde, &longs;i æqualiter hinc inde di&longs;tent; im
mò hoc ip&longs;um præ&longs;entibus oculis intueri licèt; &longs;i enim iactus BF compa
retur cum iactu BK; certè BK acquirit RK, BF acquirit BH æqualem B
K; &longs;ed BF & BK æqualiter di&longs;tant ab horizontali BG; nam arcus GF, &
GK &longs;unt æquales, vt con&longs;tat: idem dico de iactu BE, & BX, qui acquirunt
æquale &longs;patium in horizontali æquale &longs;cilicet BZ.
Ob&longs;eruabis hoc omninò licèt mirum cuiquam fortè videatur, certè
in&longs;titutum e&longs;&longs;e à natura; &longs;i enim comparentur omnes iactus &longs;uprà hori
zontalem BG, haud dubiè cum duo extremi &longs;cilicet BD, & BG nihil
pror&longs;us acquirant, vt con&longs;tat ex dictis, iactus medius &longs;cilicet ad gradum
45.erit omnium maximus, quia æqualiter ab vtraque extremitate di&longs;tat,
vt demon&longs;trauimus &longs;uprà; &longs;i verò comparentur omnes iactus, qui po&longs;
&longs;unt fieri à centro B per totum &longs;emicirculum
tremi BD, BQ nihil pror&longs;us acquirant, vt con&longs;tat, iactus medius, &longs;cilicet
ad gradum 90.qui e&longs;t BG erit omnium maximus, quia æqualiter ab vtra-
dio iactu 45. di&longs;tantes æqualem amplitudinem acquirunt in horizontali
BG, ita qui æqualiter di&longs;tant à medio iactu 90.vel horizontali BG æqua
lem amplitudinem acquirunt in aliquo plano horizontali, &longs;cilicet in eo
vnde vterque iactus de&longs;init in perpendicularem deor&longs;um.
Ob&longs;eruabis &longs;ecundo, omnes perpendiculares deor&longs;um perinde accipi,
atque &longs;i e&longs;&longs;ent parallelæ propter in&longs;en&longs;ibilem differentium; quod certè
ab omnibus admittitur; quomodo verò per diuer&longs;a plana deor&longs;um cor
pus tendere po&longs;&longs;it, v&longs;que ad centrum terræ, Libro &longs;equenti explica
bimus.
violento, quàm in iactu per inclinatam &longs;ur&longs;um
&longs;emidiametro AG; &longs;itque AG horizontalis, & AO perpendiculatis deor
&longs;um; &longs;it iactus per inclinatam &longs;ur&longs;um AD, &longs;itque impetus violentus vt A
D, & naturalis deor&longs;um vt DE; linea motus erit DAE; igitur a&longs;&longs;umatur A
E in AC, & DE in CB, ex impetu AD detrahitur DB, vt con&longs;tat ex dictis
quia totius ille fru&longs;trà e&longs;t; &longs;it autem inclinata deor&longs;um cum impetu vio
lento æquali AI æqualis AD, &longs;itque naturalis deor&longs;um acceleratus pro
rata plani inclinati vt IL, linea motus erit AL; a&longs;&longs;umatur AK, vt AL, &
KH vt IL, detrahitur tantùm IH, &longs;ed IH e&longs;t minor DB; igitur tempore
&longs;equenti æquali impetus violentus inclinatæ &longs;ur&longs;um erit vt EF æqualis
AB inclinatæ deor&longs;um, vt LM, quæ maior e&longs;t EF, quia e&longs;t æqua
lis AH.
Ratio à priori e&longs;t, quia cum inclinata deor&longs;um faciat acutum angu
lum cum perpendiculari deor&longs;um, cum quo obtu&longs;um facit inclinata &longs;ur
&longs;um, maior e&longs;t in illa linea motus; e&longs;t enim maior diagonalis, in hac ve
rò minor, igitur in illa minùs impetus e&longs;t fru&longs;trà, in i&longs;ta verò plùs, igitur
minùs impetus in illa de&longs;truitur, plùs in i&longs;ta; quæ omnia con&longs;tant ex
Th. 110. & 139. & 140. l.1. habes etiam in qua proportione decre&longs;cat
impetus.
qui fit per inclinationem &longs;ur&longs;um plùs detrahitur, in perpendiculari deor
&longs;um nihil detrahitur, in perpendiculari &longs;ur&longs;um totus detrahitur qui po
te&longs;t extrahi, id e&longs;t ex collectione vtriu&longs;que naturalis, & violenti dupli
naturalis in prima linea motus; hæc omnia &longs;equuntur ex dictis.
Obiici pote&longs;t vnum &longs;atis difficile; quia &longs;i in perpendiculari deor&longs;um
purà in AP nihil detrahitur impetus violenti, igitur cre&longs;cit &longs;emper vis
ictus, quod videtur e&longs;&longs;e contra experientiam.
Re&longs;p. me aliquando fui&longs;&longs;e in ea &longs;ententiâ, vt reuerâ exi&longs;timarem de
cre&longs;cere impetum violentum in iactu perpendiculari deor&longs;um; cum
etiam exi&longs;timarem decre&longs;cere vim ictus; &longs;ed re melius con&longs;iderata, cum
nunquam id experiri potuerim; nam &longs;emper &longs;entio vim ictus maiorem,
dine de&longs;cendit; certè ni fallor cum ratio demon&longs;tratiua pro hac &longs;en
tentia faciat, non dubitaui ampliùs priorem &longs;ententiam immutare.
Porrò ratio, quæ pro hac &longs;ententia facit, remque ip&longs;am euincit, talis
e&longs;t; certum e&longs;t impetum violentum de&longs;trui à naturali aliquando in ma
iori, aliquando in minori proportione, vt con&longs;tat ex dictis; illa autem,
&longs;eu maior, &longs;eu minor proportio aliam regulam non habet præter illam
quam toties inculcauimus, id e&longs;t impetum de&longs;trui pro rata, id e&longs;t qua
proportione e&longs;t fru&longs;trà, id e&longs;t qua proportione e&longs;t minor motus eo, qui
e&longs;&longs;et ab vtroque impetu &longs;i ad
atqui cum proiicitur mobile deor&longs;um, vterque impetus ad
neam e&longs;t determinatus; igitur nihil motus dee&longs;t per Th.138.l.1. igitur
nihil impetus e&longs;t fru&longs;trà; igitur nihil impetus illius de&longs;truitur.
Quod dictum e&longs;&longs;e velim non con&longs;iderata medij re&longs;i&longs;tentiâ, quæ certè
aliquid impetus de&longs;truit, quod tamen in&longs;en&longs;ibile e&longs;t in medio libero, pu
tà in aëre; &longs;i enim in&longs;en&longs;ibilis e&longs;t hæc re&longs;i&longs;tentia in motu naturali; dum
mobile &longs;it eius &longs;oliditatis, quæ &longs;uperet facilè vim aëris; certè etiam in
&longs;en&longs;ibilis e&longs;t in motu proiectorum, præ&longs;ertim in mediocri &longs;patio, e&longs;t
enim par vtrobique ratio.
Equidem fateor in longi&longs;&longs;imo &longs;patio po&longs;&longs;e tandem de&longs;trui totum im
petum violentum; nam &longs;i aliquid in dato &longs;patio de&longs;truitur; igitur in ma
iore piùs de&longs;truitur; atque ita deinceps, donec tandem totus de&longs;tructus
&longs;it; at verò in iis altitudinibus, ex quibus corpus deor&longs;um proiicere po&longs;
&longs;umus, vix quidquam facit prædicta re&longs;i&longs;tentia.
Nec e&longs;t quod aliquis dicat ab hac re&longs;i&longs;tentia non de&longs;trui impetum
naturalem in motu naturaliter accelerato, vt dictum e&longs;t in &longs;ecundo lib.
Igitur nec de&longs;trui violentum; nam qua proportione cre&longs;cit medij re&longs;i
&longs;tentia, cre&longs;cunt vires impetus, qui perpetuò augetur; vnde cum
remaneat &longs;emper eadem re&longs;i&longs;tentiæ proportio &longs;icut primo tempore mo
tus impedit hæc re&longs;i&longs;tentia, ne tantillùm impetus producatur; ita &longs;ecun
do tempore impedit ne tantillùm æquale producatur; igitur nihil pro
ducti impetus ab illa de&longs;truitur propter augmentum continuum: at ve
rò cum impetus violentus non intendatur; certè &longs;i tantillùm illus perit,
primo vel &longs;ecundo in&longs;tanti motus, propter medij re&longs;i&longs;tentis, tantillùm
æquale &longs;ingulis temporibus æqualibus de&longs;truitur; igitur cum infinitus
non &longs;it po&longs;t longi&longs;&longs;imum &longs;patij tractum totus tandem de&longs;truetur vio
lentus &longs;olo &longs;uper&longs;tite naturali.
Hinc fortè &longs;agitta ex notabili altitudine minùs ferit; quia materia illa
lignea, & plumea, ex qua con&longs;tat, multùm ab aëre re&longs;i&longs;tente accipit de
trimenti: adde quod licèt initio deor&longs;um rectà emittatur; attamen mini
mo aëris flatu declinat tantillùm obliqua; hæc verò obliquitas maximam
ictus vim infringit, & conflictus impetuum qua&longs;i ip&longs;um ictum di&longs;trahit,
quod facilè probabis, &longs;i modico ferè tactu cadentem perpendiculariter
&longs;agittam à &longs;uo tramite deturbes.
Dices, etiam in glande è tormento explo&longs;a hoc ip&longs;um cernitur
Re&longs;p. e&longs;t minor vis ictus inflicti à glande deor&longs;um, quàm &longs;ur&longs;um vt
aliqui putant; id autem ex duplici capite procedere; primum e&longs;t, cum fe
ratur glans ab igne per aliquod tempus, non e&longs;t dubium, quin vis ignis
&longs;ur&longs;um maior &longs;it quàm deor&longs;um; cum &longs;ur&longs;um gemino qua&longs;i impetu fera
tur, deor&longs;um verò impetu tantùm explo&longs;ionis; &longs;ecundum e&longs;t, quia cum
glans iam deor&longs;um &longs;ua &longs;ponte de&longs;cendat, haud dubiè ab igne minus eò
impelli pote&longs;t, vt &longs;æpè diximus &longs;uprà; quidquid &longs;it, &longs;i proiiciatur deor&longs;um
globus plumbeus vel arcu, vel manu, ob&longs;eruabitur maiorem ab eo ictum
infligi, quàm &longs;i &longs;ua &longs;ponte de&longs;cenderet.
motu naturali acquireret illum impetum quem habet motu violento, acquirit
triplum illius &longs;patium
deor&longs;um A E, in qua motu naturali dato tempore acquiratur AB, & &longs;e
cundo tempore æquali BC; &longs;itque impetus violentus vt AC: Dico quod
æquali tempore prioribus acquireret AE triplum AC, quia motu ve
loci vt AC acquirit CE eo tempore, quo motu veloci vt AB acquirit A
B, & veloci vt BC acquirit BC; nam eo tempore, quo acquirit AB acqui
rit CD, & eo tempore, quo acquirit BC acquirit DE; ergo eo tempore,
quo acquirit AC acquirit CE; ergo &longs;i iungatur motus naturalis violento,
eo tempore, quo motu naturali acquiretur tantùm AC, motu mixto ex
naturali & tali violento acquiretur AE, id e&longs;t triplum: &longs;i verò moueatur
duobus temporibus, ita vt primò acquirat AC, & altero triplum AC,
&longs;itque coniunctus impetus violentus vt AC; certè duobus temporibus
acquiretur motu mixto octuplum AC, &longs;ed hæc &longs;unt facilia.
tandem de&longs;truatur impetus violentus, vbi totus de&longs;tructus e&longs;t, minor e&longs;t ictus
quàm e&longs;&longs;et. &longs;i corpus graue &longs;olo impetu natur ali eò de&longs;cendi&longs;&longs;et
&longs;tro, &longs;it enim &longs;patium AD, quod percurrit motu mixto eo tempore, quo
motu naturali puro &longs;patium BC idem mobile percurreret, &longs;itque de&longs;tru
ctus in puncto D totus impetus violentus; certè remanet tantùm natu
ralis acqui&longs;itus eo tempore, quo mobile percurrit BC; &longs;ed temporibus æ
qualibus acquiruntur æqualia velocitatis momenta; igitur æqualis im
petus; igitur in C tantùm ille impetus, qui e&longs;&longs;et in E vel in D; &longs;ed dum
percurreret ED motu puro naturali, augetur impetus; igitur maior e&longs;&longs;et
impetus in D &longs;ub finem motus naturalis per AD, quam motus mixti per
eamdem AD; igitur maior ictus &longs;ub finem naturalis; igitur minus &longs;ub fi
nem violenti.
quàm &longs;ua &longs;ponte demi&longs;&longs;um
Ob&longs;eruabis &longs;crupulum adhuc fortè hærere, cur &longs;cilicet impetus
gationem; &longs;ed profectò nullo modo impetus ille violentus impedit effe
ctum impetus naturalis innati vel addititij; quia vterque totum &longs;uum ef
fectum &longs;ortitur; quod autem &longs;pectat ad propagationem; certè ita propa
gatur, vt temporibus æqualibus æqualis impetus accedat.
Dices, debes quidem nouus impetus accedere, &longs;ed non tali
modo.
Re&longs;p. non e&longs;&longs;e alium modum à natura in&longs;titutum, ni&longs;i vt temporibus
æqualibus æqualia velocitatis momenta acquirantur.
Dices præterea, fru&longs;trà accedit nouus impetus naturalis, cum iam ad
&longs;it violentus, qui eius munere defungi pote&longs;t.
Re&longs;p. cau&longs;am nece&longs;&longs;ariam nece&longs;&longs;ariò agere; igitur corpus graue perpe
tuò in medio libero &longs;uum motum intendit.
enim perpendicularis deor&longs;um AB &longs;it iactus per inclinatam AF; &longs;itque
impetus violentus vt AE naturalis vt EC, linea motus erit AC; a&longs;&longs;umatur
AF æqualis AC, & DF æqualis EC, &longs;itque CH vt AD, & impetus natu
ralis auctus vt HK, linea motus erit CK; &longs;it CI æqualis DK, & IG æqua
lis HK, & KL æqualis CG; &longs;it que impetus naturalis &longs;ecundò auctus vt L
M; linea motus erit KM; igitur connectantur puncta AC, KM per lineam
curuam, hæc e&longs;t linea quæ&longs;ita, vt con&longs;tat ex dictis &longs;uprà.
ducitur naturalis
lis e&longs;&longs;et ictus, & æquabilis motus.
tam altitudinem AR, non habet totum impetum naturalem, quem acquireret
motu naturali per totam AR, &longs;ed tantùm illum, quem acquireret in compo&longs;ita
ex &longs;egmentis NO, PB, QR
concurrit impetus naturalis, &longs;ed etiam violentus vt con&longs;tat.
quiri
quireretur.
naturalis deor&longs;um, & horizontalis impre&longs;&longs;us à naui, vt con&longs;tat ex defini
tione 1.hyp.2. & Ax.1.
retardato
li proiecto per horizontalem Th.30. e&longs;t enim pror&longs;us eadem, cum à na
ui reuera imprimatur impetus iis omnibus, quæ motu nauis fe
runtur.
ip&longs;am ex parte
proportione, in qua acceleratur per lineam perpendicularem deor
&longs;um per Th. 29.&longs;ed iuxta rationem planorum inclinatorum per Theo
rema 31. nec etiam violentus de&longs;truitur vniformiter, &longs;ed pro rata per
Th. 39.
fine plùs additur naturalis & minùs detrahitur violenti
ictus in fine ni&longs;i malus nauis ad eam altitudinem a&longs;cenderet, ad quam
profectò nullus a&longs;cendit, quæ omnia con&longs;tant per Theorema 34.
35. 36.
e&longs;t Parabola vt volunt aliqui; hinc non eo tempore de&longs;cendit in nauim
prædictus globus, quo de&longs;cenderet per ip&longs;am perpendicularem motu
purè naturali ex eadem altitudine, &longs;ed maiore tempore; quia motu mix
to non acceleratur iuxta proportionem motus naturalis puri per Th.
77. quod confirmatur illis omnibus experimentis, quæ &longs;uprà adduxi
Th. 46.
dit, e&longs;&longs;etque aptata &longs;agitta, & directa horizontaliter in naui; haud
dubiè &longs;i po&longs;t aliquod tempus &longs;taret illicò immota nauis: emitteretur &longs;a
gita, non minore certè vi quàm ab ip&longs;o arcu
nauis appellitur ad littus, &longs;i &longs;tatim &longs;ub&longs;i&longs;tat; omnia quæ &longs;unt in
naui &longs;uccutiuntur &
matur hic impetus à naui impre&longs;&longs;us, per quem Galileus ex hypothe&longs;i mo
tus æ&longs;tum maris explicat exemplo appul&longs;arum nauium ad littus, quæ
aquam vehunt.
infringat è &longs;ummo malo nauis ad imum ferè malum de&longs;cendit,
perientia à Galileo producta, non tamen adinuenta, à Ga&longs;&longs;endo do
cti&longs;&longs;imè & eleganti&longs;&longs;imè explicata, ab omnibus Copernici &longs;ectatoribus
toties decantata, quæ vulgus ignobile ad admirationem adducit; imò
plures è Philo&longs;ophis fuere, qui eam in dubium adducerent, cum cam &longs;uis
principiis, ne dicam fortè &longs;omniis aduer&longs;ari putarent; certi&longs;&longs;imum tamen
e&longs;t illud experimentum centies, imò millies comprobatum, totis etiam
vrbibus &longs;pectantibus. Nec ratio huius experimenti adeo ab&longs;tru&longs;a e&longs;t,
vel recondita, quin à vulgari, ne dicam triobolari Philo&longs;opho &longs;tatim ex
plicari po&longs;&longs;it; cum enim imprimatur à naui mobili impetus pendulo
globo per horizontalem, & alius ab ip&longs;a grauitate deor&longs;um per Th. 71.
certè mouetur globus demi&longs;&longs;us re&longs;ecto funiculo motu mixto ex hori
zontali nauis, naturali corporis grauis; igitur per lineam curuam, quæ
ferè ad imum malum terminatur &longs;ed modicum figuræ adhibendum e&longs;t;
&longs;it planum aquæ
dicularis altus 48. pedes; diuidatur in 4. partes æquales; corpus graue
conficiat &longs;patium illud duobus &longs;ecundis, v.g.igitur AK vno &longs;ecundo; e&longs;t
autem VK 12. pedum; iam verò moueatur nauis per horizontalem IH,
vel AL maxima qua&longs;i velocitate qua triremis moueri pote&longs;t; ita vt vna
hora faciat 16. milliaria Germanica, & 15′.4. milliaria, 3′ 800. pa&longs;&longs;us,
1′ 266. 1″ 4. pa&longs;&longs;us & (13/30); &longs;upponamus 1″ conficere 18. pedes, &longs;itque AC
18. & AK vel CE 12. haud dubiè motu mixto faciet lineam AE, & &longs;e
cundo tempore lineam EH, donec tandem cadat in punctum H nauis,
quò ferè peruenit punctum I; nam eodem modo retardatur motus
nauis; immò plùs quàm motus globi; quod &longs;cilicet partes aquæ, quæ à
naui diuiduntur multum re&longs;i&longs;tant; vnde fit compen&longs;atio; nam initio
motus violentus, qua&longs;i &longs;ecum rapit motum naturalem initio tardi&longs;&longs;i
mum; præ&longs;ertim cum non acceleretur, ni&longs;i iuxta rationem plani incli
nati, vt &longs;uprà dictum e&longs;t, & in fine naturalis rapit violentum.
Dixi ad imum ferè malum; nam reuera aliquid dee&longs;t quod tamen in
&longs;en&longs;ibile e&longs;t; &longs;ed quia modico tempore globus de&longs;cendit; &longs;it enim malus
108. pedum altitudinis, de&longs;cendit globus tempore 3″; &longs;it 192.4; &longs;it &longs;i
fieri pote&longs;t 432. de&longs;cendet 6″, &longs;ed nunquam accedit ad tantam altitudi
nem, igitur duobus vel tribus &longs;ecundis de&longs;cendit; igitur modico tem
pore; igitur violentus motus cen&longs;eri debet eo tempore æquabilis &longs;en&longs;i
biliter; & cum motus nauis nunquam &longs;it eiu&longs;dem velocitatis cum illa
quæ acquiritur tempore 2″ in de&longs;cen&longs;u, quia cum in de&longs;cen&longs;u acquiran
tur, hoc dato tempore ferè 48. pedes &longs;patij; certè motu æquabili cuius
ctum e&longs;t &longs;uprà; igitur vix nauis percurrit in horizontali æqualem lineam
longitudini mali eo tempore, quo globus nauim attingit &longs;it enim
altitudo mali FA 48. pedum; &longs;it amplitudo &longs;patij horizontalis æqualis
FA; haud dubiè 1″ percurret AD, id e&longs;t 12.pedes ferè, quo tempore per
currat FG. 24. pedes & 20″ percurret DF, & GI. &longs;i motus &longs;umatur vt
æquabilis, vel GH, &longs;i retardatur, igitur 1°ree;″ mobile percurrit &longs;egmentum
curuæ AE & 2°ree; EH.
Et licèt videatur tantùm acquirere MI, quæ e&longs;t minor DF 15. per
pendiculari deor&longs;um, acquirit totam EH, quæ non modo e&longs;t à motu na
turali, verùm etiam à motu violento; nec enim motu naturali dum mi
&longs;cetur cum alio, tantùm acquiritur deor&longs;um, quantùm reuerâ acquiritur
motu naturali puro, vt &longs;uprà monuimus; quia tamen etiam deor&longs;um mo
tus violentus deflectitur, etiam aliquid &longs;patij ratione violenti deor&longs;um
acquiritur; &longs;i enim vbi peruenit in E vterque impetus intactus remane
ret &longs;ine acce&longs;&longs;ione, &longs;ine imminutione; haud dubiè per
&longs;it tangens huius curuæ AEH &longs;uum cur&longs;um pro&longs;equeretur; igitur ac
quireret deor&longs;um totam DN, vel EO propter impetum naturalem præ
uium; &longs;i verò aliquid naturalis accedat, quid mirum &longs;i ratione illius ac
quiratur MI, vel NF?
Dices non de&longs;cendit tam citò motu naturali accelerato, mixto cum
violento, quàm motu puro naturali.
Re&longs;pondeo concedo; vnde nunquam ex A in H 2″ de&longs;cendit; &longs;ed
tardiùs, licèt FA &longs;it 48. ped. &longs;ed parùm abe&longs;t tùm propter minorem re&longs;i
&longs;tentiam huius impetus violenti, qui facilè detorquetur, & con&longs;equen
tur minùs illius perit, tùm quia etiam de&longs;truitur aliquid violenti; igitur
paulò plùs temporis collocat in GI, quàm in FG.
Ob&longs;eruabis primò, &longs;i nouus impetus accedat, non e&longs;&longs;e expectandum
hunc effectum; quippe nihil accipit à naui globus deinceps, vbi &longs;emel
re&longs;ecto fune ab ea qua&longs;i &longs;eparatur.
Secundò, &longs;i &longs;tatim &longs;i&longs;tat nauis demi&longs;&longs;o globo ad vnum malum nullo
modo de&longs;cendet, vt patet, &longs;ed antè.
Tertiò, &longs;i demittatur globus dum &longs;i&longs;tit nauis, tùm deinde, vbi
demi&longs;&longs;us e&longs;t, impellatur nauis; non de&longs;cendet etiam ad radicem, &longs;ed
retrò.
Quartò, motus nauis non e&longs;t æquabilis, quidquid dicat Galileus; alio
quin vna remorum impul&longs;ione opus e&longs;&longs;et, vt &longs;emper eodem motu moue
retur, aut certè &longs;i continua remigatione impellatur; cre&longs;ceret in infini
tum velocitas motus, &longs;i nihil de priori, velocitate detraheretur; retarda
tur igitur ille nauis motus propter re&longs;i&longs;tentiam aquæ, cuius partes & im
pellendæ & &longs;ulcandæ, &longs;eu diuidendæ &longs;unt; hinc fiunt ro&longs;tratæ naues
vel cu&longs;pidatæ vt faciliùs aquam findere po&longs;&longs;int; igitur ille motus nauis
non e&longs;t æquabilis; Idem pror&longs;us dicendum e&longs;t de impetu impre&longs;&longs;o in
jecto per horizontalem vel inclinatam luculenter demon&longs;trauimus.
Quintò &longs;i demittatur ex alia naui proxima immobili perpendiculari
ter omninò de&longs;cendet; Vnde valde hallucinantur ij, qui exi&longs;timant hunc
motum e&longs;&longs;e ab aëre quem nauis commouet, quod fal&longs;i&longs;&longs;imum e&longs;t, quia
pertica ad in&longs;tar mali parùm aëris commouet; adde quod aër retrò agi
tur, vt patet in aqua; præterea &longs;i è curru immobili demittatur globus eo
tempore, quo alius currus præteruolat, de&longs;cendit perpendiculariter; &longs;i ve
rò è curru mobili etiam in maiori di&longs;tantia porrecta &longs;cilicet maximè
extra currum demittente dextera; globus ab ip&longs;o curru capietur; hîc
etiam ob&longs;eruabis idem pror&longs;us accidere in curru mobili, quod in naui; &longs;i
enim è fene&longs;tra currus mobilis demittas pilam, &longs;emper cadet ex aduer&longs;o;
idem dico de currente equo, cui in&longs;idens demittat globum, imò &longs;i locus
&longs;it planus & politus, pila per aliquod tempus currum, vel equitem in&longs;e
quetur, quod qui&longs;que probare poterit, vt reuerâ centies probatum
fuit.
Sextò ad rationem Galilei, qui contendit motum circularem circa
centrum terræ e&longs;&longs;e æquabilem, quia &longs;cilicet mobile non recedit à centro:
leuis e&longs;t omninò ratio; quia globus in medio aëre motu mixto mouetur,
id e&longs;t habet impetum partim deor&longs;um, partim per tangentem, & nullo
modo per circularem, vt certum e&longs;t; nec enim rotata alium impetum im
primunt, igitur violentus e&longs;t; igitur de&longs;trui debet etiam iuxta commu
nia principia: adde quod motus mixtus fit per Diagonalem quod etiam
ip&longs;e admittit; igitur totus impetus æqualem motum non habet; nec enim
Diagonalis æqualis e&longs;t vnquam duobus lateribus; igitur aliquid illius
fru&longs;trà e&longs;t; igitur de&longs;trui debet; præterea licèt motus circularis &longs;it peren
nis circa centrum mundi; nam de illo tantùm e&longs;t quæ&longs;tio, hoc ip&longs;um
&longs;upponit primò motum illum e&longs;&longs;e &longs;implicem; &longs;ecundò, nullam pror&longs;us
e&longs;&longs;e re&longs;i&longs;tentiam; atqui in hoc ca&longs;u vtrumque deficit; nam motus ille
circularis non e&longs;t &longs;implex &longs;ed mixtus, & obe&longs;t re&longs;i&longs;tentia aquæ, vt &longs;uprà
dicat e&longs;&longs;e circularem? adde quod nauis imprimit tantùm rectum per
tangentem, vt iam &longs;uprà dictum e&longs;t; porrò ad illud, quod dicit non de
&longs;trui motum circularem à naturali, cui non e&longs;t contrarius, cum non re
moueat longiùs à centro; videtur omninò di&longs;&longs;imulare cau&longs;am impetus
vt &longs;uprà dictum e&longs;t; ex qua &longs;equitur aliquid impetus fru&longs;trà e&longs;&longs;e; ac pro
inde de&longs;trui per Axioma illud toties decantatum,
Præterea non video quomodo hanc rationem proponat magnus Gali
leus, qui nullum alium impetum violentum de&longs;trui putat, nî&longs;i tantùm il
lum, qui e&longs;t per lineam verticalem &longs;ur&longs;um; nam ex motu illo impre&longs;&longs;o
æquabili, & naturali accelerato &longs;uas Parabolas ad&longs;truit.
Septimò, non e&longs;t tamen quod diffitear ingeniosè excogitatum ab eo
fui&longs;&longs;e, ideo globum è &longs;ummo malo demi&longs;&longs;um ad imum de&longs;cendere, quod
&longs;cilicet de&longs;cendat motu mixto ex naturali accelerato, & violento æqua-
quæ eo tempore, quo corpus graue de&longs;cendit ab A in D motu naturali,
percurrit FG æquabili motu, & con&longs;equenter GI æqualem FG eo tem
pore, quo idem corpus graue percurrit DF triplam AD; igitur globus
demi&longs;&longs;us ex A &longs;uo motu de&longs;cribit Parabolam AEH; quod etiam accidet
a&longs;&longs;umpta quacunque altitudine mali vel quocunque &longs;patio confecto à
naui mobili eo tempore, quo corpus graue motu naturali accelerato
conficit &longs;patium æquale altitudini mali.
Octauò, non e&longs;t tamen di&longs;&longs;imulandum, quod etiam non di&longs;&longs;imulauit
Mer&longs;ennus, talem non fore de&longs;cen&longs;um, &longs;i nauis v. g. eadem cum emi&longs;&longs;a
&longs;agitta, vel explo&longs;a è tormento glande velocitate moueretur; non quod
aër vel medium ob&longs;i&longs;tat, vt ip&longs;i dicunt; hoc enim iam &longs;uprà rejecimus;
&longs;ed quod major impetus violentus efficiat, vt iam &longs;uprà dictum e&longs;t, ne in
tanta proportione naturalis acceleretur; quod etiam &longs;uo boatu intonant
tormenta maiora, è quibus horizontaliter directis explo&longs;æ pilæ per plu
ra &longs;ecunda in libero aëre moueantur, licèt os tormenti à plano horizon
tis vix tribus pedibus ab&longs;it; igitur non de&longs;cribunt &longs;uo motu Parabolas;
hinc &longs;ub finem minor e&longs;t ictus; hinc etiam fatetur idem Mer&longs;ennus &longs;e
cundum &longs;patium horizontale confici tardiore motu quàm primum &
tertium quàm &longs;ecundum, atque ita deinceps.
impetus qui concurrunt ad illum motum
zontalem LF, è qua &longs;ur&longs;um rectâ per lineam perpendicularem LA pro
iiciatur corpus graue; huic certè ine&longs;t triplus impetus, &longs;cilicet duo vio
lenti, alter per verticalem LA impre&longs;&longs;us à proiiciente; alter per horizon
talem LF impre&longs;&longs;us à naui; tertius denique naturalis per ip&longs;am perpen
dicularem deor&longs;um LP; igitur tres i&longs;ti impetus &longs;uo modo concurrunt
ad motum per Ax.1.certè &longs;i ine&longs;&longs;ent tantùm duo impetus &longs;cilicet LA, &
LF, motus fieret per inclinatam rectam LC; vel &longs;i tantùm duo LP, &
LA fieret per ip&longs;am LA motus retardatus; vel &longs;i LF & LP fieret per
curuam deor&longs;um, vt con&longs;tat ex dictis; igitur per aliam lineam fieri de
bet ad quam tres illi impetus concurrunt.
junctus cum horizontali LF, quàm cum nullus e&longs;t horizontalis,
quia &longs;emper mobile deor&longs;um trahit, vt patet.
talis verò e&longs;t æquabilis &longs;altem æquiualenter
rizontalis, in a&longs;cen&longs;u &longs;altem perinde &longs;e habet; immò cum illo conuenit
ad de&longs;truendum violentum &longs;ur&longs;um, id e&longs;t ad deflectendum deor&longs;um
mobile vt con&longs;tat; igitur hic motus con&longs;tat ex naturali & horizontali
puncto E &longs;cilicet EF, ED, EA; ex EA ED fit mixtus EG, ex EA,
EF, violentus EB; denique ex mixto EG à naturali EF fit EC, quæ
omnia &longs;unt clara.
&longs;ine horizontali
a&longs;cendit in C, &longs;ed DC, & EB &longs;unt eiu&longs;dem altitudinis.
Ob&longs;eruabis, licèt i&longs;te motus non fiat per lineam parabolicam, vt &longs;uprà
demon&longs;trauimus Th. 54. & reliquis; quia tamen &longs;en&longs;ibiliter proximè
accedit, deinceps vtemur Parabola vt in fig. Th. 83. & horizontalem
motum accipiemus pro æquabili; licèt omninò æquabilis non &longs;it; ni&longs;i
tantùm æquiualenter; dixi æquiualenter, quia eodem modo &longs;e habet hic
motus, ac &longs;i per inclinatam &longs;ur&longs;um LC impetu &longs;cilicet LC mobile pro
iiceretur; &longs;ed in hoc ca&longs;u de&longs;trueretur impetus ille per inclinatam &longs;im
plex; igitur & mixtus; quia tamen ille qui remanet partim ex LA, par
tim ex LF eodem modo ferè &longs;e habet ac &longs;i totus LF intactus maneret;
hinc dictum e&longs;t &longs;uprà æquiualenter e&longs;&longs;e æquabilem.
tantum ex verticali & horizontali &longs;ine naturali a&longs;cenderet
ret in C fig. Th.83. &longs;inc impetu naturali, &longs;ed FC & LA æquales &longs;unt;
atqui motu violento puro, ni&longs;i naturalis obe&longs;&longs;et, a&longs;cenderet in A; at ve
rò &longs;i obe&longs;t naturalis; a&longs;cendit tantùm motu violento in K, & mixto in
in D; quia ex K in L motu naturali tot acquireret mobile gradus impe
tus naturalis quot amittit in motu violento ab L in K; &longs;ed cum in impe
tu acqui&longs;ito à K in L motu æquabili a&longs;cenderet ab L in A, quæ e&longs;t dupla
LK vt o&longs;tendimus in &longs;ecundo libro; &longs;ed motu mixto, & verticali, & ho
rizontali a&longs;cenderet in C; &longs;ed FD e&longs;t &longs;ubdupla FE; igitur motu mixto
a&longs;cendit ad &longs;ubduplam altitudinem, &c.
tu mixto ex verticali retardato, horizontali æquabili, & naturali item æqua
bili, de&longs;cendit etiam motu mixto ex horizontali retardato &longs;altem æquiualenter,
& naturali accelerato
remaneat aliquid impetus verticalis qui in communem lineam abit cum
horizontali; res tamen perinde &longs;e habet atque &longs;i totus verticalis de&longs;true
retur, & totus horizontalis intactus permaneret; igitur de&longs;cen&longs;us fit mo
tu mixto ex naturali accelerato & horizontali retardato per Th.30. quia
tamen modico illo tempore parùm retardatur, vt &longs;uprà monui, &longs;en&longs;ibili
ter accipi pote&longs;t pro æquabili.
iiciatur ex L in A, eo tempore, quo nauis mouetur ex L in F, certè &longs;i
tempus illud diuidatur bifariam prima parte mobile percurret LI tri
plam IK in verticali, & LM &longs;ubduplam LF in horizontali; igitur erit
in G; &longs;ecunda verò perte temporis in verticali percurrit IK, & MF in
horizontali; igitur erit in D; præterea &longs;i accipiantur duæ aliæ partes tem
poris æquales; prima in perpendiculari deor&longs;um percurret DE æqua
lem LK, & in horizontali DO; igitur erit in N; &longs;ecunda vero in per
pendiculari percurret NQ triplam NO, & NR in horizontali; igitur
erit in S; &longs;ed hæc e&longs;t Parabola; nam vt &longs;e habent quadrata applicatarum
v.g. EG, FL, ita &longs;agittæ DE, DF; dixi &longs;en&longs;ibiliter, nam vt &longs;uprà mo
nui e&longs;t alia linea, quæ tamen proximè accedit ad Parabolam.
e&longs;t
horizontalis tùm nauis tùm mobilis omninò æquabilis, à quo tamen
tantillùm deficit, &longs;ed in tam breui tempore non e&longs;t &longs;en&longs;ibile.
remittit, tantùm initio remittit horizontali naturalis tardior, & &longs;ub finem ve
locior detrahit
ticali LK, & multùm &longs;ub finem; in de&longs;cen&longs;u verò curua DS accedit
propiùs ad horizontalem DT, à qua multùm recedit &longs;ub finem.
probata centies experientia; idem dico de &longs;agitta, arcu emi&longs;&longs;a, glande
tormento explo&longs;a, &c. &longs;ic dum demittis manu in eadem naui aliquod
graue deor&longs;um, eadem &longs;emper à te di&longs;tantia cadit; &longs;ic in rhodis currenti
bus poma odorifera, &longs;ur&longs;um modica vi projecta eadem &longs;emper excipiun
tur manu, perinde atque &longs;i currus ip&longs;e &longs;taret. Ita pror&longs;us &longs;e res habet
dum in&longs;idens equo etiam pernici&longs;&longs;imè currenti ludis huiu&longs;modi moti
bus; quorum nullum pror&longs;us di&longs;crimen ob&longs;eruabis in naui, &longs;iue &longs;tet &longs;iue
moueatur &longs;olito cur&longs;u; &longs;i enim eadem velocitate, qua vel emi&longs;&longs;a &longs;agitta,
vel glans explo&longs;a moueretur; haud dubiè maximum di&longs;crimen inter
cederet.
rectà ferri iudicabis
pter motum nauis, in eadem &longs;emper e&longs;&longs;e putas, in qua pila &longs;emper
occurrat; licèt reuerâ qui &longs;unt in naui immobili rem aliter e&longs;&longs;e
lem illi, quam di&longs;cus per lineam inclinatam &longs;ur&longs;um proiectus &longs;uo mo
tu de&longs;criberet; neque mirum e&longs;t, cum &longs;int eædem vtriu&longs;que rationes, cum
hac tantum differentia, quòd inclinata di&longs;ci &longs;it motus &longs;implicis, inclina
ta verò pilæ a&longs;cendentis &longs;it motus mixti ex horizontali & verticali, æ
quabili quidem in a&longs;cen&longs;us accelerato in de&longs;cen&longs;u.
tum terræ, &longs;equi &longs;cilicet ex eo lapidem proiectum &longs;ur&longs;um per verticalem longo
interuallo ver&longs;us occa&longs;um retrò de&longs;cen&longs;urum,
terræ &longs;uppo&longs;ito non &longs;equeretur, cum non &longs;equatur ex motu nauis.
Igitur alia ratione impugnari debet hypothe&longs;is illa, quæ terræ motum
de&longs;truit; quod certè &longs;i à me fieri po&longs;&longs;it, in tractatu de corporibus cœle&longs;ti
bus, vel de nouo &longs;y&longs;temate aliquando præ&longs;tabimus; non tamen e&longs;t quod
hîc di&longs;&longs;imulem aliquorum agendi methodum, qui ex hoc phœnome
no con&longs;tanter a&longs;&longs;erunt terram moueri; nam primò, &longs;equeretur tantùm
moueri circa centrum id e&longs;t motu orbis, non verò motu centri; quæ e&longs;t
hypothe&longs;is Origani. Secundò ex quiete terræ hoc idem phœnomenon
&longs;equitur; quippe, &longs;i terra quie&longs;cit, eadem manu cadentem excipio lapi
dem, quæ &longs;ur&longs;um rectà proiicit; igitur quemadmodum ex hoc non infero
terræ quietem, &longs;ed aliunde; ita neque ex hoc inferri pote&longs;t terræ motus;
cum enim duplex hypothe&longs;is codem phœnomeno &longs;tare pote&longs;t, neutra ex
eo euincitur; igitur &longs;icuti fateor ex hoc phœnomeno minimè demon
&longs;trari terræ quietem ita & tu fateri debes ex eo minimè ad&longs;trui po&longs;&longs;e
terræ motum.
Adde quod, haud dubiè &longs;i terra quie&longs;cit citiùs proiectus lapis &longs;ur&longs;um
de&longs;cendit, quàm &longs;i mouetur; nec enim vt dictum e&longs;t &longs;uprà proiecta velo
ci&longs;&longs;imo motu per horizontalem de&longs;cendunt eo tempore, quo ex eadem
altitudine motu purè naturali de&longs;cenderent; quod multis euincitur ex
perimentis, vt vidimus in Th.46. atqui punctum terræ &longs;ub æquatore ve
loci&longs;&longs;imè moueretur, quod vno temporis &longs;ecundo conficeret 1250.pedes
geometricos &longs;i 5. pedes geometrici tribuantur pa&longs;&longs;ui, 4000. pa&longs;&longs;us leucæ
germanicæ, 15. leucæ germanicæ gradui Æquatoris, toti demum Æqua
tori 360. gradus; cum autem iactus medius tormenti validi&longs;&longs;imi &longs;it
15000. pedum, duretque 30″ temporis; certè 30″ temporis con&longs;icit pun
ctum æquatoris 37500. pedes; igitur mouetur velociùs explo&longs;a glande;
igitur &longs;i hæc velocitas glandis impedit, ne tàm citò deor&longs;um cadat, ma
jor velocitas motus terræ potiori iure illud ip&longs;um impediet; igitur &longs;i
terra quie&longs;cit, globus &longs;ur&longs;um proiectus velociùs recidet in terram, et&longs;i
terra moueatur tardiùs.
Ob&longs;eruabis duos tantùm motus in naui mobili fui&longs;&longs;e hactenus explica
tos; primus e&longs;t, quo demittitur plumbea pila è &longs;ummo mali; &longs;ecundus e&longs;t,
quo ex
cet, quot po&longs;&longs;unt duci lineæ è &longs;ummo malo in orbem quoquo ver&longs;um;
quarum hæ &longs;unt præcipuæ. &longs;it apex mali B; circa quem de&longs;cribatur cir
culus ACDE, &longs;itque primò circulus ille verticalis parallelus &longs;cilicet li
neæ directionis nauis BA, quæ &longs;it v. g. ver&longs;us Boream; primò habes li
neam verticalem &longs;ur&longs;um BE; &longs;ecundò perpendicularem deor&longs;um BC;
tertiò lineam directionis ver&longs;us Boream BA; quartò illi oppo&longs;itum
ver&longs;us Au&longs;trum BD; tùm voluatur circulus circa axem immobilem AD
per quadrantem integrum, dum &longs;cilicet BE &longs;it ad Ortum, quæ e&longs;t quinta
linea, & BC ip&longs;i oppo&longs;ita ad Occa&longs;um, quæ e&longs;t &longs;exta. Igitur habes 6. li
neas; &longs;cilicet &longs;ur&longs;um, deor&longs;um, ver&longs;us Boream & Au&longs;trum, ver&longs;us Ortum,
& Occa&longs;um; linea quæ tendit deor&longs;um pote&longs;t dupliciter con&longs;iderari, vel
enim demittitur &longs;ua &longs;ponte, vel proiicitur.
Iam verò inter
horizonti parallelas, quæ vt con&longs;iderentur; cen&longs;eatur prædictus circulus
parallelus horizonti, ita vt ex centro B ducantur ad
lineæ, quot &longs;unt puncta in circumferentia; &longs;ecundò inclinatas &longs;ur&longs;um &
inclinatas deor&longs;um; &longs;imiliter inter Occa&longs;um & Au&longs;trum, inter Au&longs;trum
& Ortum, inter Ortum & Boream; porrò exprimes omnes lineas, &longs;i api
cem mali fingas centrum globi, &longs;eu &longs;i in circulo prædicto verticali à
centro B ad circumferentiam ducantur tot lineæ quot po&longs;&longs;unt duci,
tuncque circa axem EC immobilem voluatur circulus, &c. his po&longs;i
tis &longs;it.
lum
de&longs;cenderet vt con&longs;tat per Th. 69. &longs;ed profectò nihil acquiret in hori
zontali globus, quod non acquirat nauis; igitur imùm ferè malum attin
git &longs;ed opus e&longs;t aliqua figurâ; &longs;it enim apex mali A, de&longs;cendatque pri
mò ex A &longs;ua &longs;ponte in H; haud dubiè &longs;i eo tempore, quo motu na
turali conficit AD, mixto deor&longs;um conficit AF, eo tempore cadet in G
ex A &longs;i hic impetus deor&longs;um adueniat; &longs;ed res e&longs;t clara; hæc porrò figura
non e&longs;t Parabola, licèt &longs;it curua; con&longs;tat autem hîc motus ex naturali
accelerato, ex impre&longs;&longs;o deor&longs;um æquabili per &longs;e, & horizontali &longs;en&longs;i
biliter æquabili; pote&longs;t autem de&longs;ignari hæc linea motus ex &longs;uprà
dictis.
&longs;us Boream, mouebitur globus motu mixto ex duplici horizontali per
lineam ferè æquabili; id e&longs;t &longs;en&longs;ibiliter, licèt geometricè loquendo retardetur,
& naturali accelerato
quam conficiat eo tempore, quo conficit AH motu naturali, motu mixto
perueniet in K; &longs;i verò duplicetur horizontalis, ita vt eo tempore quo
conficit AH, conficiat AD, motu mixto perueniet in L; hæc autem curua
ctus horizontalis e&longs;t veloci&longs;&longs;imus, qualis in arce, vel in tormentis belli
cis, eodem tempore mobile non decidit in terram, quo de&longs;cenderet mo
tu purè naturali ex eadem altitudine.
duplam amplitudinem in plano hòrizontali acquireret, v.g. &longs;i
emi&longs;&longs;a arcu extra nauim ex A perueniret in K, in naui mobili perueniret
in L; &longs;i verò nauis, vt reuerâ fit, tardiùs moueatur, &longs;agitta è naui emi&longs;&longs;a
ver&longs;us Boream &longs;cilicet acquiret pro rata, id e&longs;t &longs;i nauis motus &longs;it tantùm
&longs;ubduplus perueniret in M; &longs;i &longs;ubquadruplus in N &c.
quam &longs;uo motu conficit nauis maiorem iactum habebit, non tamen &longs;en&longs;ibiliter
quia motus nauis parum addit; ob&longs;eruabis tamen non videri maiorem
quàm &longs;i nauis quie&longs;ceret, quia eo tempore, quo &longs;agitta ex A peruenit in
L, nauis ex H peruenit in K; igitur videtur &longs;emper e&longs;&longs;e idem iactus, &longs;iue
moueatur nauis &longs;iue non, quia e&longs;t &longs;emper eadem di&longs;tantia nauis, & ter
mini iactus; cum nauis id totum acquirat &longs;patij, quod motui &longs;agittæ
accedit.
brachium rotat, tùm præuio cur&longs;u impetum auget,
tus brachij imprimitur mobili, &longs;ed etiam impetus totius corporis; hinc
etiam &longs;i præmittatur cur&longs;us longiore &longs;altu
&longs;patium traiicitur; quæ omnia ex ii&longs;dem principiis manife&longs;tè &longs;e
quuntur.
motu mixto ex duobus horizontalibus ad oppo&longs;itas lineas, & ex naturali ac
celerato
tur; certè &longs;i nauis motu &longs;ubduplo in oppo&longs;itam partem feratur, peruenit
tantùm in K, quæ omnia con&longs;tant ex dictis; nam impetus oppo&longs;iti pu
gnant pro rata, vt &longs;æpè diximus; videbitur tamen e&longs;&longs;e æqualis iactus; &longs;i
enim eo tempore, quo &longs;agitta peruenit in K, nauis fertur in oppo&longs;itam
partem &longs;patio æquali KL, haud dubiè di&longs;tantia &longs;emper erit æqualis; tan
tùm enim recedit ver&longs;us Boream nauis, quantùm &longs;agitta à puncto L ad
punctum K reducitur.
lineam GC &longs;eu TA ver&longs;us Boream, & &longs;agitta è &longs;ummo malo emitteretur
per lineam TO ver&longs;us Au&longs;trum, de&longs;cenderet per lineam T.G. nec quidquam
menti etiam ne latum quidem vnguem pertran&longs;iret in horizontali, vide
tur tamen &longs;emper e&longs;&longs;e idem iactus; nam eo tempore, quo &longs;agitta caderet
à T in G, nauis e&longs;&longs;et in C, atqui CG & GM &longs;unt a&longs;&longs;umptæ æquales; hinc
potiùs arcus e&longs;&longs;et emi&longs;&longs;us quàm &longs;agitta, & tormentum explo&longs;um quàm
globus.
Ob&longs;eruabis, &longs;i nauis motus &longs;it ad motum &longs;agittæ v. g. in ratione &longs;ub
dupla, &longs;cilicet vt FG, vel LM ad GM peruenit in L per Parabolam TL; &longs;t
vt EG vel KM ad GL peruenit in K per Parabolam TK; &longs;i vt DG vel I
M ad GM peruenitin I per Parabolam TI, &c. vnde vides Parabolas
i&longs;tas &longs;emper in infinitum contrahi, donec tandem in rectam TG de&longs;i
nant vbi motus nauis e&longs;t æqualis motui &longs;agittæ: Parabolas dixi &longs;en&longs;ibi
liter, &longs;cilicet eo modo, quo &longs;uprà.
partem in quam fertur nauis per &longs;patium æquale differentia illorum motuum,
v.g. &longs;i nauis moueatur per GM & &longs;agitta per TA, &longs;itque motus nauis ad
motum &longs;agittæ, vt GM, ad IM; eo tempore quo nauis attinget M, &longs;agitta
cadet in I, & &longs;i motus &longs;it vt GM ad KM cadet in K vel vt GM ad GL
cadet in L. per Parabolas, quæ omnia con&longs;tant ex dictis, & ex Theore
mate per 134. l.1.
in &longs;agittarium
haud dubiè per Th. &longs;uperius eo tempore, quo nauis peruenit ad M &longs;a
gitta attinget punctum L, & eo tempore quo nauis e&longs;&longs;et in L &longs;agitta e&longs;
&longs;et in puncto Y, &longs;i cum nauis peruenit in L illicò &longs;i&longs;tat &longs;agitta, cadet in
ip&longs;am nauim; nam cadet in L quod clarum e&longs;t: dixi &longs;i nauis &longs;i&longs;tat po&longs;t
emi&longs;&longs;am &longs;agittam, &longs;i enim nauis &longs;emper moueatur, æquabilis &longs;emper e&longs;&longs;e
videbitur &longs;agittæ iactus, &longs;i enim è naui immobili emi&longs;&longs;a fui&longs;&longs;et prædicta
&longs;agitta per horizontalem TO, acqui&longs;iui&longs;&longs;et &longs;patium vel amplitudinem G
L; &longs;ed videtur confeci&longs;&longs;e ML, cum nauis mouetur; atqui ML e&longs;t æqualis
LG, quid clarius?
Hinc &longs;i quis in naui currat per lineam directionis id e&longs;t ver&longs;us eain
partem, in quam mouetur nauis, curret velociùs; immò &longs;i ambulet, ingen
tes faciet pa&longs;&longs;us &longs;eu &longs;altus v.g.&longs;i nauis conficit &longs;patium GM eo tempore
quo aliquis &longs;altat ex G in H; haud dubiè amplitudo eius &longs;altus erit com
po&longs;ita ex tota GM & GH; &longs;i verò in partem oppo&longs;itam ver&longs;us C currat:
vel currit velociùs, vel tardiùs, vel æquali motu: &longs;i primum, aliquid &longs;patij
acquiret ver&longs;us C æqualis &longs;cilicet
ver&longs;us M &longs;patio æquali eidem differentiæ; &longs;i tertium, nec accedet, nec re
cedet, &longs;ed totis viribus currens &longs;eu tentans currere in eodem &longs;emper lo-
bis circa centrum immobile.
trianguli orthogonij, cuius ba&longs;is &longs;it horizontalis & perpendiculum &longs;patium,
quod percurritur motu naturali æquali tempore, idque in naui mobili
in eam
accelerato & inclinato mixto ex horizontali & alio inclinato &longs;it enim
horizontalis AD, perpendicularis AMK, &longs;it AM &longs;patium quod percurri
tur in perpendiculari motu purè naturali, eo tempore, quo percurritur
AC &longs;ubdupla AD, &longs;itque AM &longs;ubdupla AC, & &longs;ecundo tempore æquali
percurratur in horizontali CD, & in perpendiculari MK tripla AM;
erit motus mixtus per lineam parabolicam ANH; nam &longs;uppono hori
zontalem æquabilem, cùm parùm ab eo ab&longs;it, vt &longs;upradictum e&longs;t; præ&longs;er
tim cum &longs;en&longs;ibiliter hæc linea &longs;it parabolica.
Iam verò in eadem naui proiiciatur mobile per inclinatam AP, quæ
&longs;it diagonalis quadrati AP, & impetus perinclinatam AP &longs;it ad impetum
per horizontalem AC, vt AP ad AC; ducatur LPF parallela MN, & CF
parallela AP; denique diagonalis AF: haud dubiè ML e&longs;t æqualis AM, vt
patet; & &longs;i motus e&longs;&longs;et tantum mixtus ex AC & AP fieret per diagona
lem AF, quam mobile eodem tempore percurreret quo vel AC vel AP;
igitur &longs;i dum percurrit AF percurrit AM, motu naturali, certè dum per
currit AN &longs;ubdupla AF, percurret tantùm &longs;ubquadruplam AM; a&longs;&longs;uma
tur ergo NO æqualis AS, & FG æqualis AM;
e&longs;t linea quç&longs;ita.
Itaque idem dicendum e&longs;t de his inclinatis, quod de aliis &longs;uprà di
ctum e&longs;t Th.72. ni&longs;i quod accipitur inclinata mixta ex horizontali & da
ta inclinata, v.g. ANF ex AC & AP; hæc autem linea non e&longs;t Parabolica,
quia quadratum MN, vel VO e&longs;t ad quadratum RG vt 1.ad 4.at verò &longs;a
gitta AV e&longs;t ad &longs;agittam AP, vt 5.ad 12.porrò hæc linea &longs;ecat Parabolam
vt patet; &longs;i verò accipiatur inclinatata AI, mixta inclinata erit AH igitur
a&longs;&longs;umatur HX æqualis AM, & PZ æqualis AS ducetur linea huius mo
tus per AZX. quænam verò &longs;int hç lineæ, dicemus aliàs Tomo &longs;equenti.
erit etiam mixtus ex naturali, & inclinato ex horizontali, & data inclinata
vnde idem pror&longs;us
clinata, de qua multa &longs;uprà dicta &longs;unt à Th.47. &longs;uppo&longs;ito tamen motu na
turali accelerato, ad quem proximè accedit propter mutationem perpe
tuam lineæ. &longs;it enim inclinata &longs;ur&longs;um AB, quæ percurratur motu
æquabili eo tempore, quo horizontalis AE, vel quo motu naturali LA;
diuidatur AE bifariam in D; ducatur DG, tùm DC, AC, hæc e&longs;t linea mo
tus mixti ex inclinata AG, & horizontali AD; &longs;equitur deinde Parabola;
nam &longs;ico tempore quo percurritur AD, percurritur AG, & LM vel FA;
igitur FO, quæ e&longs;t &longs;ubquadrupla FA; igitur a&longs;&longs;umatur NH æqualis FO, &
CK æqualis FA, & ducatur curua per puncta AHK; hæc e&longs;t &longs;emiparabo
la, nam KI e&longs;t ad KE vt quadratum IH ad quadratum EA.
Vnde vides omnes inclinatas &longs;ur&longs;um v&longs;que ab horizontali DB ad
verticalem DA inclu&longs;iuè e&longs;&longs;e Parabolas; omnes verò inclinatas ab ca
dem horizontali DB ad perpendicularem DC inclu&longs;iuè non e&longs;&longs;e Para
bolas, &longs;ed propiùs accedere ad rectam, vnde aliquis &longs;u&longs;picari po&longs;&longs;et e&longs;&longs;e
Hyperbolas.
tam directionis nauis,
neam rectam, &longs;ur&longs;um vel deor&longs;um, v.g.
deor&longs;um KB, mixta erit KL; &longs;it etiam inclinata KL, & horizontalis
CH; mixta erit KH, cui addatur in eadem KF portio &longs;patij, quod motu
naturali percurritur; idem dico de aliis inclinatis.
Præterea &longs;it horizontalis VX, inclinata
ex VOVX fiet VS detracta &longs;cilicet portioni &longs;patij, quod detrahitur à
motu naturali; &longs;i verò &longs;it vel major motus horizontalis, vel minor eo,
quem a&longs;&longs;ump&longs;imus, non percurrit mobile lineam rectam &longs;ed vel Para
bolam &longs;i &longs;ur&longs;um proiiciatur, vel &longs;i deor&longs;um aliam nouam, quam ad Hy
perbolam accedere &longs;uprà diximus.
Hinc certè, quod mirabile dictu e&longs;t, &longs;i è puncto nauis V &longs;ur&longs;um per
inclinatam VO proiiciatur, &longs;tatimque po&longs;t proiectionem &longs;i&longs;tat nauis, in
ip&longs;am nauim de&longs;cendet mobile; atque ita ex his habeo omnes motus cir
culi verticalis paralleli lineæ directionis; quare &longs;upere&longs;t vt explicemus
alios motus; ac primò quidem per circulum horizontalem, cuius habeo
quoque duas lineas, &longs;cilicet communes &longs;ectiones horizontalis & prio
ris verticalis, id e&longs;t lineam directionis ver&longs;us Boream, & oppo&longs;itam ver
&longs;us Au&longs;trum.
monebitur motu mixto ex duplici horizontali, & naturali deor&longs;um
enim horizontalis ver&longs;us Boream AC, & alia horizontalis AH ver&longs;us
ortum in eodem plano horizontali; certè ex vtraque fit mixta AK, quæ
&longs;i percurratur æquali tempore cum AC, & eius &longs;ubdupla cum AB, AC
verò æquali tempore cum AF; quamquàm &longs;uppono iam e&longs;&longs;e perpendi
cularem deor&longs;um AB; denique cum AG &longs;ubquadrupla AF a&longs;&longs;umatur
ED æqualis AG perpendiculariter ducta in AD, & KL æqualis AF
parallela ED, & per puncta AEL ducatur curua, hæc e&longs;t linea motus
quæ&longs;ita; voluatur autem triangulum AKL, donec &longs;it parallelum circulo
verticali vel alteri, ACO erit in proprio &longs;itu; vnde eo tempore, quo e&longs;
&longs;et in DE punctum nauis A e&longs;&longs;et in B, & co, quo e&longs;&longs;et in KL, punctum A
e&longs;&longs;et in C; hoc e&longs;t &longs;ingula puncta AK, è regione AC ductis parallelis
ver&longs;us ortum.
Hinc &longs;i ex A dirigas
no&longs;&longs;e debent rei tormentariæ præfecti; quippe &longs;agitta aberrabit à &longs;copo
ver&longs;us Boream declinans toto eo &longs;patio, quod conficit nauis codem tem
pore, quo mouetur &longs;agitta; ita pror&longs;us &longs;i moueatur H ver&longs;us K, vt attin
gas ex puncto immobili A debes dirigere ictum in K, &longs;i quo tempore
&longs;agitta conficit AK &longs;copus H percurrit HK.Idem pror&longs;us dicendum e&longs;t
de iaculatione per lineam oppo&longs;itam ver&longs;us occa&longs;um.
Si verò proiiciatur mobile per lineam inter Boream, & Ortum, linea
motus erit Parabola cuius Tangens erit mixta ex horizontali ver&longs;us
Boream, & declinante ver&longs;us Ortum, v. g. &longs;it horizontalis ver&longs;us Boream
AF, quam hactenus a&longs;&longs;ump&longs;i pro linea directionis; &longs;it linea ver&longs;us
Ortum AC; &longs;it declinans ver&longs;us Boream AL; &longs;itque impetus AL, ad
AE vt AL ad AE, quod hactenus &longs;uppo&longs;ui; &longs;it LG æqualis AE, AG
e&longs;t mixta ex AE, AL; a&longs;&longs;umatur KI, & GH vt iam diximus; fiatque
Parabola AIH, quæ circa axem AE ita voluatur, vt &longs;it perpendicularis
plano horizontali LF.
Idem dico de omni alia declinante vel à Borea ad Ortum, vel ad Oc
ca&longs;um.
&longs;it vt linea; conficit lineam parabolicam, cuius tangens vel amplitudo e&longs;t re
sta ad Ortum
tum, ND ad Occa&longs;um; &longs;it NL declinans ab au&longs;tro ad Ortum, &longs;itque im
petus per NL ad impetum per NF, vt NL ad NF; mixta ex NF NL
e&longs;t HK; &longs;it autem KH æqualis &longs;patio, quod conficitur motu naturali eo
tempore, quo percurritur NF, &longs;it KI æqualis NK, & IG quadrupla KH;
Parabola NHG e&longs;t linea motus quæ&longs;ita dum voluatur NIG circa axem
NI, dum IG pendeat perpendicularitur ex plano horizontali ON.
Idem fiet, &longs;i proiiciatur per declinantem NB ab Au&longs;tro &longs;cilicet ad
Occa&longs;um.
ctio cum horizontali tendit ad Ortum, conficit lineam parabolicam, cuius am
plitudo e&longs;t mixta ex horizontali ver&longs;us Boream, & horizontali ver&longs;us Ortum,
&longs;it linea ver&longs;us Boream AB, ver&longs;us Ortum AK, mixta ex vtraque AF,
linea inclinata &longs;ur&longs;um AP, Parabola AMN, quæ vertatur circa A do
nec incubet AFG, denique AFG circa FA voluatur, donec incubet
perpendiculariter plano; porrò perinde e&longs;t, &longs;iue proiiciatur per inclina
tam &longs;ur&longs;um ver&longs;us Ortum, &longs;iue ver&longs;us Occa&longs;um.
Si verò proiiciatur per inclinatam deor&longs;um ver&longs;us Ortum, de&longs;cribit
lineam, quæ non e&longs;t Parabola, &longs;ed propiùs accedit ad Hyperbolam, cuius
&longs;it enim AC ver&longs;us Boream, AB ver&longs;us Ortum, AD inclinata deor
&longs;um &longs;ub horizontali AB, AG quæ e&longs;t in eodem plano cum AD DG,
mixta ex AD, & AC; a&longs;&longs;umatur EF æqualis &longs;patio, quod conficitur
motu naturali eo tempore, quo conficitur AE, & GH æqualis &longs;patio,
quod conficitur motu naturali eo tempore, quo percurritur AG; duca
tur curua AFH, cuius &longs;itus vt habeatur &longs;it AB ver&longs;us Ortum, ex qua
pendeat perpendiculariter deor&longs;um triangulum ABH, tùm circa axem
AD voluatur triangulum ADH, donec HD &longs;it parallela horizonti; tùm
circa axem AG voluatur triangulum AGH, dum GH &longs;it perpendicu
laris deor&longs;um, tunc enim linea motus AFH habebit proprium &longs;itum;
idem fiet &longs;i proiiciatur per inclinatam deor&longs;um ver&longs;us Occa&longs;um.
tus erit Parabola, cuius amplitudo erit mixta ex declinante horizontali, &
horizontali ver&longs;us Boream,
horizontalis ver&longs;us Ortum AR, declinans à Borea in Ortum AD, mixta
ex AD, AK &longs;it AI, &longs;itque Rhomboides AE parallelus horizonti; &longs;it
EG perpendicularis &longs;ur&longs;um, &longs;it HD parallela GE; differentia &longs;patij,
quod acquiritur motu naturali eo tempore, quo percurritur AI, & FC,
quæ &longs;it &longs;ubdupla EG. Dico lineam motus AHF e&longs;&longs;e parabolicam, quæ
omnia con&longs;tant ex dictis; idemque dictum e&longs;to de omni alia inclinata
&longs;ur&longs;um &longs;imul, & declinante, &longs;eu ver&longs;us Ortum &longs;eu ver&longs;us Occa&longs;um; porrò
triangulum AEG incubat
&longs;i verò proiiciatur per inclinatam deor&longs;um voluatur AKE, dum KO
&longs;it perpendicularis deor&longs;um; &longs;it planum RK horizontale, voluatur
AKE circa A, ita vt KO &longs;it &longs;emper perpendicularis deor&longs;um, donec
AE &longs;ecet planum RK in AD &longs;int IO. & EA vt EF, GH in &longs;uperio
re figura, & per puncta AOM ducatur curua; hæc e&longs;t linea motus
quæ&longs;ita.
de&longs;cribet Parabolam, cuius amplitudo erit mixta ex horizontali ver&longs;us Bo
ream & declinante horizontali ab Au&longs;tro ad Ortum
ver&longs;us Boream, AG ver&longs;us Ortum, AI declinans ab Au&longs;tro ad Ortum,
AG mixta ex AF AI AL inclinata, ANK Parabola; &longs;it enim planum
FI horizontale cui triangulum ALI incubet perpendiculariter in &longs;e
ctione AG, reliqua &longs;unt facilia; idem dico de inclinata &longs;ur&longs;um &longs;imul, &
declinante ab Au&longs;tro ad Occa&longs;um; &longs;i verò &longs;it inclinata deor&longs;um, &longs;it pla
num ACB horizontale, AB &longs;it declinans, AC &longs;it mixta ex AB & ho
rizontali ver&longs;us Boream AF; &longs;it AD inclinata deor&longs;um, fiatque cur
ua AQE more &longs;olito, ita vt triangulum ACE perpendiculariter
deor&longs;um pendeat ex plano horizontali ACB, reliqua &longs;unt facilia.
Ob&longs;eruabis a&longs;&longs;umptam e&longs;&longs;e à me hactenus Parabolam, licèt accurate
non &longs;int parabolicæ lineæ, quia proximè ad Parabolas accedunt;
certè Phy&longs;icè loquendo & &longs;en&longs;ibiliter pro Parabolis a&longs;&longs;umi po&longs;&longs;e ni
hil vetat.
Ex his colligis mirabilium motuum rationem.
Primò mobile proje
ctum per lineam declinantem ab Ortu ferri po&longs;&longs;e rectà ad Ortum.
Secundò projectum per inclinatam deor&longs;um, ferri po&longs;&longs;e per ip&longs;am
perpendicularem deor&longs;um.
Tertiò projectum per inclinatam &longs;ur&longs;um, ferri po&longs;&longs;e per verti
calem.
Quartò, rationem à priori habes, cur &longs;i ex equo vel &longs;puas, vel ali
quid demittas deor&longs;um, rectà perpendiculariter non cadat, &longs;ed &longs;emper
è regione, quod maximè videre e&longs;t cum purgatur nauis mobilis, eiecta
&longs;cilicet aquâ, quæ &longs;emper nauim in&longs;equi videtur, imò & cum quis pe
dem effert in naui hunc motum quoque ob&longs;eruat.
Quintò non erit etiam iniucundum inde elicere quomodo in maiore
naui, di&longs;co ludere vel pila quis po&longs;&longs;it, licèt nauis motus nullo modo lu
dum impediat; quæ omnia ex iis, quæ diximus nece&longs;&longs;ariò con&longs;equuntur,
& quæ manife&longs;tum probat experimentum.
Sextò, inde etiam eruuntur rationes motuum mixtorum ex pluribus
motibus v.g.4.5.6.7.&c.in infinitum &longs;iue in eodem plano, &longs;iue in diuer
&longs;is; In diuer&longs;is vt hactenus explicuimus; in eodem vero &longs;iv.g.per BC,
BE, BA &longs;imul imprimantur impetus eidem mobili qui &longs;int vt ip&longs;æ li
neæ; primò fiat ex BA BC mixta BD, & ex BD BE, mixta BF, vel ex
BE BC mixta BG, & ex BG BA mixta BF, vel ex BE BA mixta
BH, & ex BH BC mixta BF; vides &longs;emper e&longs;&longs;e
mixtam in diuer&longs;is planis; iam o&longs;tendimus e&longs;&longs;e plures &longs;uprà in naui
mobili v.g. per planum verticale, horizontale, & inclinatum.
Septimò, &longs;i in naui mobili curreret equus, vel currus, e&longs;&longs;et motus mix
tus ex quatuor aliis, & &longs;i terra moueretur in naui mobili e&longs;&longs;ent quatuor
motus, &longs;i ex ea aliquod mobile proiiceretur; inuenitur autem linea mix
ta in diuer&longs;is planis per quamdam planorum circuitionem, de qua
&longs;uprà.
Octauò, po&longs;&longs;et facilè in eodem plano motus mixtus conflari ex qua
tuor aliis vel etiam pluribus, &longs;int enim quatuor in eodem plano AD
AE. AF. AH. ex AD AE fit AB, ex AB, A fi fit AC, ex AC AH
fit AG, quæ e&longs;t longior AC, & AC longior AB: po&longs;&longs;es etiam compo
nere ex AH AF, atque ita deinceps eodem ordine, & &longs;emper vltima
linea erit AG, quod certè mirabile e&longs;t, & à Geometris demon&longs;trari
pote&longs;t.
Nonò, ex his motibus mixtis educi po&longs;&longs;unt rationes multorum effe-
nubium, vaporum, ventorumque motus, qui &longs;æpè turbinatim procellas
agunt, quorum turbinum ratio referri non debet, vt videbimus &longs;uo loco,
in repercu&longs;&longs;ionem aliquam, quæ fiat à concauis montibus, qui longi&longs;&longs;i
mo interuallo &longs;æpiùs ab&longs;unt; &longs;ed potiùs petenda e&longs;t ab ip&longs;a mixti motus
naturâ; quippè rara materies venti facilè recipit omnem impetum; ita
que ex prægnantibus &longs;æpè nubibus conferta tenui&longs;&longs;imorum halituum
examina fractis qua&longs;i carceribus quacumque linea erumpunt;
hinc infiniti propemodum motus, hinc turbines illi, &c.
atque hæc de motu mixto ex pluribus
rectis &longs;int &longs;atis.
Planis.
HACTENVS con&longs;iderauimus motum in libe
ro medio; iam verò con&longs;iderabimus in planis
durioribus, in quibus mobilè feratur vel &longs;ua
&longs;ponte vel ab extrin&longs;eco impul&longs;um.
piam moueri po&longs;&longs;it, quod nec &longs;it verticale &longs;ur&longs;um, nec perpendiculare deor
&longs;um,
zontale e&longs;t etiam decliue, vt &longs;uo loco videbimus.
planum, quod minùs recedit à perpendiculari, tardiùs verò per illud, quod plùs
recedit.
tardiore motu de&longs;cendit, quàm in perpendiculari deor&longs;um.
Vtraque hypothe&longs;is certa e&longs;t, & de vtraque &longs;upponimus tantùm, quòd
&longs;it, nam demon&longs;trabimus infrà propter quid &longs;it.
Axioma con&longs;tat ex iis, quæ fusè demon&longs;traui &longs;ecundò lib. adde quod,
deor&longs;um tendere, & corpus graue &longs;ua &longs;ponte moueri idem pror&longs;us &longs;onare
videntur; nec enim loquor de potentiâ motrice animantium, vel de alia
quacumque magneticâ, &longs;ed de potentiâ motrice grauium; graue autem
illud appello, quod in medio rariore po&longs;itum deor&longs;um tendit, ni&longs;i impe
diatur, denique hîc &longs;uppono dari motum naturalem grauium deor&longs;um
probetur hoc Axioma per hypothe&longs;im primam; nam reuerâ &longs;uppono
quòd omnibus experimentis comprobatur, &longs;cilicet corpus graue per pla
num Inclinatum deor&longs;um &longs;ua &longs;ponte de&longs;cendere, non verò a&longs;cendere ni&longs;i
propter aliquam reflexionem.
qui imminuitur
impedimentum?
per &longs;uam virtutem
tur modica altitudo
interuallo ad &longs;e&longs;e inuicem accedunt, pro parallelis accipiuntur.
tur, quia tardiore motu de&longs;cendit mobile per hyp. 2. igitur impeditur
per Axio.2.
innatus
Ax.1. cur enim potiùs ad vnam lineam quàm ad aliam? atqui id tan
tùm planum inclinatum efficit, vel impedit, ne deor&longs;um rectà tendere
po&longs;&longs;it; igitur ex eo tantùm capite impedit.
nullus e&longs;&longs;et omninò motus &longs;uper eodem plano, &longs;ed per planum inclina
tum mobile deor&longs;um mouetur per hyp.1.igitur totus motus non impedi
tur; hinc ratio à priori primæ hypothe&longs;eos.
Axioma 2.cum enim motus imminuatur, quia impeditur per idem Axio
ma; certè quò plùs impeditur, plùs imminuitur; &longs;ed quò plùs imminui
tur, minor e&longs;t, ergo quò plùs impeditur, minor e&longs;t.
quirendam
quod conficitur in perpendiculari deor&longs;um
demon&longs;tretur, aliquid figuræ tribuendum e&longs;t. &longs;it perpendicularis deor-
peruenit in E per planum AE, di&longs;tat æquè à centro, ac &longs;i e&longs;&longs;et in B; &longs;up
pono enim perpendiculares omnes deor&longs;um e&longs;&longs;e parallelas per po&longs;tula
tum; igitur non acce&longs;&longs;it propiùs ad centrum confecto &longs;patio AE, quàm
confecto AB; igitur impeditur in plano AE in ea proportione, in qua
AB e&longs;t minor AE, nam haud dubiè AE e&longs;t maior AB, &longs;it autem dupla v.g.
igitur impeditur non quidem totus motus &longs;ed &longs;ubduplus; in plano verò
AD impeditur iuxta cam proportionem in qua AB e&longs;t minor AD, nec
enim aliunde pote&longs;t impediri, cum &longs;cilicet impediatur tantùm, quia im
peditur linea ad quam ab ip&longs;a natura determinatus e&longs;t per Th.2. v. g.li
nea deor&longs;um AB; quippè lineæ comparantur inter &longs;e v.g. AE cum AB,
nam impedimentum lineæ AE in eo tantùm po&longs;itum e&longs;t, quòd difficiliùs
per illam quàm per AB ad
imperimentum petatur a difficultate; atqui difficultas motus, qui fit per
lineam AE in eo tantùm e&longs;t, quòd &longs;it maius &longs;patium conficiendum, igi
tur quò maius &longs;patium e&longs;t, maior difficultas e&longs;t; igitur quò maior linea
e&longs;t, maius impedimentum e&longs;t.
Adde quod vel impedimenti proportio petitur ab angulis vel à Tan
gentibus, vel à &longs;ecantibus; nihil enim aliud ade&longs;&longs;e pote&longs;t; igitur per Ax.
3. pote&longs;t tantùm impediri ab his; &longs;ed proportio impedimenti non pote&longs;t
e&longs;&longs;e ab angulis; quod probatur primò, quia &longs;i ego quæram à te in qua
proportione motus per AE e&longs;t tardior motu per AB; dices in ea, in qua
angulus EAB e&longs;t maior nullo angulo, quod e&longs;t ridiculum: Equidem di
ceres motum per AD e&longs;&longs;e velociorem motu per AE in ea proportione,
in qua angulus EAB e&longs;t maior angulo BAD, quod tamen fal&longs;um e&longs;t; e&longs;&longs;et
enim ferè duplò maior, quod repugnat
angulum BA, qui &longs;it tantùm vnius gradus &longs;eu minuti, &longs;itque EAB angu
lus 60. grad. &longs;i velocitas motus per AI e&longs;&longs;et ad velocitatem motus per
AE vt angulus EAB ad angulum BAI, motus per AI e&longs;&longs;et &longs;exagecuplò
velocior, quàm per AE, quod e&longs;t ab&longs;urdum: Diceret fortè aliquis in to
to angulo 90. GAB di&longs;tribui huius impedimenti motum v.g. &longs;i angulus
BAI &longs;it 1.grad. motus per AI amittit tantùm (1/90) &longs;ui motus; &longs;i angulus D
AB circiter 40.grad. motus per AD amittit tantùm (40/90), & per AE (60/90); cum
&longs;it angulus BAE 60. grad. igitur motus per AB e&longs;t ad motum per AE
vt 3.ad 1. quod omnibus experimentis repugnat.
Secundò probatur, quia &longs;i fiat inclinata proximè accedens ad AG v.
g.4′.& a&longs;&longs;umatur alia accedens 3′. differentia anguli erit tantùm 2′.
cum
tamen differentia longitudinis plani &longs;eu &longs;ecantis huius, & illius, &longs;it ma
xima, vt con&longs;tat ex canone &longs;inuum, igitur non imminueretur motus in
plano inclinato ratione impedimenti contra Th.4. quis enim neget e&longs;&longs;e
maximum impedimentum motus tantum &longs;patium, quod
Tertiò, omnia experimenta con&longs;entiunt huic Theoremati, & repu
gnant huic propo&longs;itioni quæ petitur ab angulis; adde quod angulus ni
hil pror&longs;us facit ad motum, &longs;ed linea &longs;eu &longs;patium; denique hoc ip&longs;um e&longs;t
quod ab omnibus Mechanicis vulgò &longs;upponitur perinde qua&longs;i prima
Equidem explicari pote&longs;t hæc demon&longs;tratio operâ libræ; &longs;it enim
libra CG cuius centrum immobile e&longs;t A; &longs;it autem diameter libræ CG,
pondus in C &longs;e habet ad pondus in D, tran&longs;lata &longs;cilicet diametro in DH
vt CA, ad BA; igitur pondus in D grauitaret minùs in planum inclina
tum DA, quàm in horizontali CAI; nam pondus in D idem præ&longs;tat, quod
præ&longs;taret appen&longs;um in D fune DE; igitur grauitatio in C e&longs;t ad grauita
tionem in D, vt CA, vel DA ad BA; &longs;ed quâ proportione decre&longs;cit graui
tatio in planum, cre&longs;cit motus in plano inclinato, quia minùs impeditur
per Th.4. igitur in perpendiculari ea nulla e&longs;t gtauitatio in planum; nec
impeditur vllo modo motus, igitur ab E ver&longs;us C ita impeditur motus, vt
AC ver&longs;us C impeditur grauitatio in planum, &longs;ed impeditur grauitatio
in D v.g. in ratione totius CA ad EA, vel DA ad DI; igitur impeditur
motus in eadem proportione v.g. in plano DA ad DB vel AI, igitur in
ratione plani inclinati ad perpendicularem.
Hæc omnia veri&longs;&longs;ima &longs;unt; &longs;upere&longs;t tamen vt &longs;ciatur ratio phy&longs;ica cur
pondus in D æquiualeat ponderi in B quod &longs;upponunt quidem omnes
Mechanici, & omnibus experimentis congruit: Equidem pondus pendu
lum ex D fune DB, vel longiore, e&longs;t eiu&longs;dem momenti, cuius e&longs;t affixum
in D, ita vt linea directionis, quæ ducitur ab eius centro re&longs;pondeat fu
ni DB; vnde rectè concluditur ab Archimede idem pondus affixum bra
chio BA eiu&longs;dem e&longs;&longs;e momenti cum pendulo DB, vel affixo puncto D,
quod certè veri&longs;&longs;umum e&longs;t, nondum tamen rationem phy&longs;icam video;
verum quidem e&longs;t idem pondus pendulum fune DB minoris e&longs;&longs;e
momenti, quàm &longs;i e&longs;&longs;et affixum puncto C; nam &longs;uppono CG e&longs;&longs;e libram
in &longs;itu horizontali; tum quia pondus illud DB trahit deor&longs;um extremum
libræ D per arcum DC longo circuitu, maximè declinante à &longs;ua linea
directionis DB; tùm quia ex hoc &longs;equitur nece&longs;&longs;ariò pondus B deflecti
à &longs;ua perpendiculari curua linea; tùm quia linea DA, quæ rigida &longs;uppo
nitur, re&longs;i&longs;tit motui DB & patet; in qua verò proportione, dictum e&longs;t
certè hactenus, &longs;ed phy&longs;icè non demon&longs;tratum.
Pater Mer&longs;ennus multis locis ex docti&longs;&longs;imo Roberuallo demon&longs;trat
rem i&longs;tam ingenio&longs;i&longs;&longs;imè; &longs;it enim circulus centro R; &longs;int vectes æqua
les BF horizonti, DN perpendiculari paralleli; tùm CL, FO, æqualiter
inclinati, ducantur CO EL; haud dubiè &longs;i pondera C & L &longs;int æqualia
erit æquilibrium; quod certum e&longs;t, & demon&longs;trabimus cum de libra; e&longs;t
enim quarta propo&longs;itio Vbaldi de libra; &longs;ed pondus in O pendulum &longs;ci
licet filo CO e&longs;t eiu&longs;dem momenti, cuius e&longs;t pondus in P; igitur pon
dus in P æquale ponderi O &longs;u&longs;tineret pondus ML, &longs;ed pondus in P
e&longs;t ad pondus in B vel in F, ad hoc, vt &longs;it æquilibrium, RF ad R
P; igitur pondus in A vel in R, quod erit ad pondus in L, vt P ad R
L, &longs;u&longs;tinebit pondus in L; &longs;ed &longs;i applicetur potentia in C quæ trahat per
tangentem CT, faciet idem momentum quod faceret in B trahens per
tangentem BA; at vicem illius potentiæ gerit pondus B vel A, quod gra
uitat per BA; igitur potentia applicata C per CT, æqualis ponderi A
eadem potentia in L per LG retinebit pondus in L; quæ idem retine
ret applicata in C per CT; cum enim RC & RL &longs;int æquales &longs;i &longs;int ap
plicatæ duæ potentiæ æquales in C quidem per CT, & in L per LG;
haud dubiè erit perfectum æquilibrium; igitur &longs;i pondus A pendeat in
H fune LGH, retinebit pondus L in plano inclinato GLK; e&longs;t autem
pondus H ad pondus LN SR ad RL; &longs;ed triangula RSL, & GKI
&longs;unt proportionalia; igitur pondus in H e&longs;t ad pondus L, vt GI ad G
K; igitur &longs;i vires, quæ retinent pondus in plano inclinato GK &longs;unt ad vi
res, quæ retinent pondus in perpendiculari GI, vt GI ad GK; igitur im
petus &longs;eu motus mobilis in plano GK e&longs;t ad impetum, &longs;eu motum eiu&longs;
dem in perpendiculo GI, vt GI ad GK.
Hæc omnia veri&longs;&longs;ima &longs;unt, &longs;emper tamen de&longs;iderari videtur ratio phy
&longs;ica, cur idem pondus pendulum ex C in O, &longs;it eiu&longs;dem momenti cum
pondere affixo puncto P, &longs;eu brachio libræ horizontalis PS. quod certè
Mechanica Axiomatis, vel hypothe&longs;eos loco iure a&longs;&longs;umere pote&longs;t; at ve
rò phy&longs;ica non &longs;atis habet de re cogno&longs;cere quod &longs;it, ni&longs;i &longs;ciat propter
quid &longs;it; igitur nos aliquam afferre conabimur. Suppono tantùm tunc
e&longs;&longs;e æquilibrium perfectum duorum ponderum æqualium cum
æqualia illa pondera ita &longs;unt appen&longs;a, vt linea directionis vnius æqua
lis &longs;it lineæ directionis alterius, cur enim alterum præualeret &longs;i &longs;int æ
qualia? hoc po&longs;ito.
Dico pondus affixum P æquale ponderi L facere aquilibrium; cum
enim linea directionis &longs;it PO, &longs;i de&longs;cenderet liberè per PO. L eodem
tempore attolleretur per LS, quod certè applicatis planis SL PO facilè
fieri po&longs;&longs;et; &longs;ed eodem modo P grauitat, quo &longs;i de&longs;cenderet per PO; e&longs;t
enim eius linea directionis; atqui tunc faceret æquilibrium, quod o&longs;ten
do; æquale &longs;patium conficeret L, per LS a&longs;cendendo, quod P per PO
de&longs;cendendo; igitur &longs;i attolleret L in S, &longs;imiliter pondus L æquale P in S
attolleret pondus P ex O in P, igitur neutrum præualere pote&longs;t; &longs;ed quia
hæc fu&longs;iùs explicabimus cum de libra, nunc tantùm indica&longs;&longs;e &longs;ufficiat.
Supere&longs;t vt breuiter o&longs;tendamus accipi non po&longs;&longs;e hanc proportio
nem imminutionis motus in plano inclinato à Tangente BE tùm
quia; iam à &longs;ecante accipi o&longs;tendimus, tùm quia &longs;it Tangens BD æqualis
&longs;umi toti &longs;eu perpendiculari AB; &longs;equeretur motum per AD æqualem
e&longs;&longs;e motui per AB; Equidem in maxima di&longs;tantia accedit Tangens ad
&longs;ecantem; igitur eò plùs impeditur motus, quò maius &longs;patium conficien
dum e&longs;t, &c.
pendiculari, vt ip&longs;a perpendicularis ad ip&longs;um planum inclinatum,
citas motus per AE e&longs;t ad velocitatem motus per AB, vt ip&longs;a AB e&longs;t
ad ip&longs;am AE, &longs;it enim AE dupla AB, velocitas per AB e&longs;t dupla veloci
atis per AE.
Ob&longs;erua quæ&longs;o, cum dico motum in plano inclinato e&longs;&longs;e ad motum
in perpendiculo, vt ip&longs;æ lineæ permutando, ita intelligendum e&longs;&longs;e, vt
vel a&longs;&longs;umatur motus in &longs;ingulis in&longs;tantibus, ita vt eo in&longs;tanti, quo datum
&longs;patium in inclinata acquiritur, acquiratur duplum in perpendiculo; quo
po&longs;ito valet certè tantùm illa proportio ratione motus æquabilis, &longs;i &longs;er
uari debet; nam perinde &longs;e habet phy&longs;icè, atque &longs;i e&longs;&longs;et, vt iam fusè ex
plicatum e&longs;t lib.2. in re &longs;imili.
tus non impeditur, cum &longs;it eadem proportio, quæ e&longs;t perpendicularis
ad inclinatam; dixi per &longs;e, nam per accidens in plano &longs;cabro tantillùm
inclinato mobile de&longs;cendit, adde quod corpus graue tamdiu mouetur
quandiu accedere pote&longs;t ad centrum terræ.
dicularis ad inclinatam pote&longs;t e&longs;&longs;e minor in infinitum, quia inclinata
pote&longs;t e&longs;&longs;e longior, & in infinitum.
&longs;ubduplus illius qui fit in perpendiculari
AE e&longs;t dupla AB, &longs;ed motus in AB e&longs;t ad motum in AE vt AE ad AB
per Th.6. igitur e&longs;t duplus.
Ex his reiicies quoque Cardanum, & alios quo&longs;dam, qui diuer&longs;am
proportionem motuum in planis inclinatis deducunt ex diuer&longs;is angu
lis inclinationis; iuxta quam proportionem motus in AE e&longs;&longs;et &longs;ubtri
plus in AB contra experimentum.
eadem cum illa, quam adduximus lib.3. cum de motu naturali, quia &longs;ci
licet prior impetus con&longs;eruatur, & acquiritur nouus, Imò acceleratur
iuxta
lei in partibus temporum &longs;en&longs;ibilibus; vnde a&longs;&longs;umemus deinceps i&longs;tam
Galilei proportionem, quia &longs;cilicet partes temporis &longs;en&longs;ibiles tantùm
a&longs;&longs;umere po&longs;&longs;umus.
in hac habet totum &longs;uum motum, non verò in illa, quia impeditur, ni&longs;i
enim totus e&longs;&longs;et, non grauitaret corpus illud in planum inclinatum;
quippe &longs;uas omnes vires impetus ille exereret circa motum; igitur ali
quid illarum exerit circa motum aliquid circa planum, in quod ex parte
grauitat; igitur idem e&longs;t impetus innatus, adde quod ille e&longs;t in&longs;epa
rabilis.
e&longs;t minor eo, qui producitur in perpendiculari
motus, igitur minor impetus, vt &longs;æpè diximus; &longs;ecundò (hæc e&longs;t ratio
à priori;) quia cum ideo producatur impetus i&longs;te aduentitius, vt motus
acceleretur; certè debet re&longs;pondere motui, qui competit impetui innati;
&longs;i enim nullum habet motum, nullus accedit de nouo impetus, è con
tra verò &longs;i e&longs;t motus, &longs;ed maior, &longs;i maior e&longs;t motus, & minor &longs;i e&longs;t minor;
quia hic impetus tantùm e&longs;t propter motum.
exigit (præ&longs;cindendo à re&longs;i&longs;tentia medij)
tat in planum; alioquin cre&longs;ceret &longs;emper grauitatio; igitur totus exerce
tur circa motum; ratio e&longs;t quia hic impetus addititius non e&longs;t in&longs;titutus
propter grauitationem, &longs;ed tantùm propter motum: adde quod ad om
nem lineam determinari pote&longs;t, &longs;ecùs verò naturalis &longs;altem om
ninò.
quò velociùs mouetur in plano, breuiori tempore &longs;ingulis partibus in
cumbit: &longs;ecundò quia motu illo accelerato qua&longs;i di&longs;trahitur mobile ab
illa linea grauitationis in planum; hinc mobile celeri motu moueretur
in plano illo inclinato, quod eiu&longs;dem &longs;ub&longs;i&longs;tentis grauitationi & ponde
ri vltrò cederet.
deor&longs;um
ctatu &longs;equenti; tamen ratione plani qua&longs;i detorquetur ad lineam plani
ad quam tamen omninò non determinatur, alioquin non grauitaret in
planum: vnde dixi, detorquetur &longs;eu qua&longs;i diuiditur, perinde qua&longs;i e&longs;&longs;et
duplex impetus, quorum alter per lineam perpendicularem deor&longs;um
e&longs;&longs;et determinatus, in quo non e&longs;t difficultas; impetus tamen aduenti
tius determinatur omninò ad lineam plani.
Dubitari pote&longs;t an grauitatio in planum inclinatum &longs;it vt re&longs;iduum
plani, cui detrahitur perpendiculum v.g. &longs;it planum inclinatum CD ad
angulum ACD 60. potentia quæ &longs;u&longs;tinet pondus B per EB e&longs;t ad præ
dictum pondus vt CA ad CD; detrahitur CA ex CD, &longs;upere&longs;t FD æqua
lis &longs;cilicet CA; an fortè grauitatio ponderis B in planum inclinatum C
D e&longs;t ad grauitationem eiu&longs;dem in planum horizontale; quæ e&longs;t graui
tatio tota, id e&longs;t nihil imminuta vt DF ad DC; attollatur enim totum
triangulum CAD in eadem &longs;itu altera manu, & altera filo EB paralle-
pondus trianguli CAD; nunquid fortè altera manus &longs;u&longs;tinebit tantùm
&longs;ubduplum ponderis B? & altera &longs;ubduplum? igitur vt habeatur quod
&longs;u&longs;tinet &longs;uppo&longs;ita dextra v.g. debet &longs;ub&longs;trahi, quod &longs;u&longs;tinet &longs;ini&longs;tra, &longs;ed
quod &longs;u&longs;tinet &longs;ini&longs;tra, e&longs;t vt ip&longs;a potentia, id e&longs;t vt CA ad CD; igitur
tota CD repræ&longs;entat totum pondus, &longs;egmentum CF partem ponderis
quæ competit potentiæ E, FD verò partem quæ &longs;u&longs;tinetur à pla
no CF.
Hinc facilè po&longs;&longs;et determinari quota pars ponderis incubet plano,
&longs;it enim planum inclinatum AC, perpendiculum AB, accipiatur AB
æqualis AB, &longs;itque AC tripla AB, duæ tertiæ ponderis incubant plano
&longs;i verò &longs;it horizontale planum, totum pondus grauitat in illud; nulla e&longs;t
enim perpendicularis, &longs;i &longs;it perpendiculare planum, nihil pror&longs;us gra
uitat; quia nulla e&longs;t inclinata, & quò propiùs accedit planum inclina
tum ad horizontalem plùs grauitat pondus in illud, minùs verò; quò
propiùs accedit ad perpendicularem.
Hinc e&longs;&longs;et oppo&longs;ita ratio grauitationis, & motus, in plano inclinato;
nam quò plùs e&longs;t grauitationis minùs e&longs;t motus, quò plùs motus, minùs
grauitationis; quando verò planum inclinatum e&longs;t duplum perpendicu
culi vt planum CFD, tunc
planum quantùm de motu in eodem plano; ide&longs;t vtrique &longs;ubduplum,
&longs;i verò vt in plano ADC perpendiculum e&longs;t &longs;ubtriplum plani, detrahun
tur de motu 2/3 & de grauitatione 1/3, idem dico de aliis, quæ certè omnia
ex veris principiis phy&longs;icis con&longs;equi videntur, quò enim plus grauitat
mobile in planum, plùs &longs;u&longs;tinetur; quò plùs &longs;u&longs;tinetur, plùs impeditur il
lius motus; &longs;ed hoc repugnat communi Mechanicorum &longs;ententiæ, qui
cen&longs;ent grauitationem in planum inclinatum e&longs;&longs;e ad grauitationem in
horizontale, vt Tangens e&longs;t ad &longs;ecantem, quæ &longs;it linea plani inclinati,
v.g. vt AB ad CD, quod certè omnes &longs;upponunt, &longs;ed minimè
&longs;trant
eo quòd pondus in extremitate libræ affixum habeat diuer&longs;a momenta
iuxta rationem Tangentium ad &longs;ecantes, v.g. in &longs;ecunda figura Th.5.
pondus in D e&longs;t ad pondus in C vt BA ad DA, quod veri&longs;&longs;imum e&longs;t, &
&longs;uprà demon&longs;trauimus; quippe hoc pertinet ad rationem momenti, non
verò grauitationis in planum; adde quod affixum e&longs;t pondus vecti; igi
tur vectis &longs;u&longs;tinet totum illius pondus; vtrùm verò &longs;i pondus in plano
inclinato veluti in vecte moueatur pondus quo grauitat in planum &longs;it
ad pondus quo grauitat in horizontali vt Tangens ad &longs;ecantem, certè
non demon&longs;trant; attamen ita res pror&longs;us &longs;e habet; quare fit.
in planum horizontale, vt Tangens, vel horizontalis ad &longs;ecantem, vel incli
natam,Primò &longs;it planum inclinatum GD, pondus in-
uitationem in horizontalem CD vt CD ad GD; quia pondus F pellit
planum per lineam FE &longs;eu GB Tangentem; quia determinari non po
te&longs;t &longs;eu percu&longs;&longs;io, &longs;eu impre&longs;&longs;io ex alio capite quàm ex linea ducta à
centro grauitatis perpendiculariter in planum, vt demon&longs;trauimus
in Th. 120. l. 1. atqui libræ extremitas G initio de&longs;cendit per Tangen
tem GB, id e&longs;t per minimum arcum, qui ferè concurrit cum Tangente;
&longs;ed ideò de&longs;cendit in AB, quia pellitur deor&longs;um à pondere; igitur men
&longs;ura grauitationis e&longs;t de&longs;cen&longs;us libræ, &longs;ed libra faciliùs de&longs;cendit ex A
deor&longs;um quàm ex G in proportione AD ad CD vel GD ad CD; igitur
grauitatio ponderis in A e&longs;t ad grauitationem eiu&longs;dem in G, vt GD ad
CD; quia rationes cau&longs;arum &longs;unt eædem cum rationibus effectuum.
Præterea &longs;it planum inclinatum GD, &longs;it IF parallela GD; &longs;int IK, I
M & quadrans KFR; punctum I &longs;it centrum libræ immobile; certè &longs;i &longs;it
alterum brachium libræ æquale IF in&longs;tructum æquali pondere F, erit æ
quilibrium; &longs;ed pondus illud in F e&longs;t ad idem in R, vt IM ad IF, &longs;eu vt
CD ad GD, quod erat dem.
Ob&longs;eruabis po&longs;&longs;e facilè ex dictis explicari diuer&longs;as potentias applica
tas ponderi F in eodem plano GD, primò &longs;i accipiatur IHF parallela
GH cum centro immobili I pondus retinebitur, &longs;i potentia in I &longs;it ad
globum vt GC ad GD, vt demon&longs;tratum e&longs;t; &longs;i verò pellat potentia per
lineam IF, globus de&longs;cendet, vt patet.
Hinc &longs;ecundò &longs;u&longs;tinens MF totum pondus F &longs;u&longs;tinet, patet, quia &longs;i
ue planum inclinatum pondus ip&longs;um tangat, &longs;iue perpendiculare, totum
&longs;u&longs;tinet pondus; &longs;ub&longs;tracto enim plano pondus immobile manet, adde
quod non pote&longs;t pondus F &longs;u&longs;tineri in brachio IM, ni&longs;i æquale pondus
ex æquali brachio oppo&longs;ito pendeat.
Tertiò ex puncto T lineâ TFE non pote&longs;t &longs;u&longs;tineri pondus licèt po
tentia in T e&longs;&longs;et infinita, quia ex TE de&longs;cendet in TV, patet; idem
dico de omnibus aliis lineis ductis ab F ad aliquod punctum inter
TM.
Quartò ex puncto X linea XF &longs;u&longs;tinebitur pondus dum potentia ap
plicetur in X, maior quidem potentia applicata in I, &longs;ed minor applica
ta in M; nam potentia M e&longs;t ad potentiam I vt IF ad MF; igitur poten
tia X e&longs;t ad potentiam M vt MF ad XF; ad potentiam verò I vt IF
ad XF.
Quintò, cùm triangula IF M.HF 4. &longs;int proportionalia, potentia M
e&longs;t ad potentiam I vt HF ad 4. F.
Sextò, &longs;i applicetur potentia, vel in T pellendo per lineam TFE, quæ
cadit perpendiculariter in planum GD, vel &longs;i applicetur in A per lineam
AE trahendo, non poterit retineri globus, quæcunque tandem poten
tia applicetur; quia &longs;emper per GD globus rotari poterit nullo cor
pore impediente; &longs;uppono enim tùm planum tùm globum e&longs;&longs;e perfectè
no nullam e&longs;&longs;e partium compre&longs;&longs;ionem, qua vna pars in aliam qua&longs;i pe
netret; &longs;i enim totus locus datur ad de&longs;cen&longs;um; certè non e&longs;t vlla ratio
propter quam non de&longs;cendat; nec dicas affigi plano GD ab ip&longs;a vi ex
teriùs affigente; quia nullo modo impeditur motus, per datam lineam,
ni&longs;i vel aliquod corpus opponatur, vel alius impetus detrahat ab eadem
linea; atqui nihil horum prorsùs e&longs;t in hoc ca&longs;u.
Si potentia applicetur in N per lineam NF, maior e&longs;&longs;e debet quàm in
I, &longs;ed minor quàm in A; e&longs;t autem ad potentiam in I vt IF ad NF;
quippe re&longs;i&longs;tit planum GD huic potentiæ in N, non tamen re&longs;i&longs;tit in I;
igitur illa maior e&longs;&longs;e debet, quod autem potentia in N &longs;it ad potentiam
in I, vt IF ad NF (po&longs;ito &longs;cilicet quod vtraque pondus E &longs;u&longs;tineat) plùs
quàm certum e&longs;t; quia cùm pondus po&longs;&longs;it tantùm moueri per EG &longs;eu per
lineam FI potentia NF trahit per FN; igitur potentia in N &longs;u&longs;tinens
pondus F e&longs;t ad potentiam in I &longs;u&longs;tinentem idem pondus, vt IF ad NF;
&longs;imiliter potentia in K &longs;u&longs;tinens idem pondus F e&longs;t ad potentiam in I vt
IF ad ZF, nam IZ e&longs;t perpendicularis in KF, donec tandem potentia
&longs;it in A applicata per AF in quam IF cadit perpendiculariter, igitur po
tentia in A debet e&longs;&longs;e infinita.
Octauò, &longs;i pellatur pondus F per omnes lineas contentas &longs;ini&longs;tror&longs;um
inter FT & FA deor&longs;um faciliùs cadet; &longs;i verò trahatur per lineas con
tentas inter TF & FA dextror&longs;um, etiam deor&longs;um cadit; quia perinde
e&longs;t &longs;iue trahatur per lineam IF, &longs;iue pellatur æquali ni&longs;u per lineam VF
quæ concurrit cum FI; & perinde e&longs;t &longs;iue pellatur per IF, &longs;iue trahatur
per FV; idem dictum &longs;it de omnibus aliis lineis, quæ per centrum F
hinc inde ducuntur.
Vnum e&longs;t, quod de&longs;iderari videtur ex quo reliqua ferè omnia depen
dent, quomodo &longs;cilicet potentia in N trahens per FN &longs;it ad potentiam
in I trahentem per FI vt FI e&longs;t ad FN, quod &longs;ic breuiter demon&longs;tro:
&longs;it horizontalis BD, & triangulum ECD; ex centro D ducatur arcus
BE, qui &longs;it v.g. 30.grad. vt CE &longs;it &longs;ubdupla ED; certè potentia in B
e&longs;t ad potentiam in E per EC vt BD, vel ED ad CD; &longs;ed potentia in E
per EA Tangentem e&longs;t æqualis potentiæ in B; &longs;it autem planum EA, &
connectatur AC; triangula AEC & ECD &longs;unt proportionalia; igitur
&longs;it AC verticalis, EC horizontalis, & AE inclinata; &longs;it potentia in A
per AE trahens pondus E; &longs;it potentia C trahens per CE; dico quod
impeditur tractio toto angulo AEC, &longs;icut ante impediebatur grauitatio
toto angulo AEC; igitur vtrobique e&longs;t æquale impedimentum; &longs;ed in
primo ca&longs;u ratione impedimenti ita &longs;e habet potentia in E per EA ad
potentiam in E per EC, vt ED ad CD, vel vt EA ad EC; igitur in &longs;e
cundo in quo e&longs;t idem impedimentum potentia in A per EA e&longs;t ad po
tentiam in C per EC, vt ip&longs;a inclinata AE ad EC.
Nonò denique ob&longs;eruabis, egregium e&longs;&longs;e apud Mer&longs;ennum tractatum
authore docti&longs;&longs;imo Roberuallo &longs;uper hac tota re, in quo certè Geome-
adde quod implicatior illa figura infinitis ferè contexta lineis, quam ha
bet, equidem erudito Geometræ faciet &longs;atis, non tamen rudiori Tyroni,
qui vix in hoc labyrintho tutum &longs;e e&longs;&longs;e putabit.
globus F in plano ED; ducatur FH perpendicularis deor&longs;um; hæc e&longs;t
linea directionis centri grauitatis, vt con&longs;tat; igitur cùm non &longs;u&longs;tinea
tur in prædicta linea, nec enim terminatur ad punctum contactus G, cer
tè debet rotari; adde quod non e&longs;t in æquilibrio, vt patet, ratio autem
inæqualitatis e&longs;t vt GF ad FN, nec vlla e&longs;t difficultas; igitur duplici
qua&longs;i motu de&longs;cendet in prædicto plano ille globus, &longs;cilicet motu centri
propter inclinationem plani, & motu orbis, tùm quia non e&longs;t in æqui
librio, tùm quia in linea directionis FH non &longs;u&longs;tinetur à plano.
centri grauitatis &longs;ecet ip&longs;um planum intra ba&longs;im corpus repit quidem in
prædicto plano &longs;ed non rotatur, &longs;i verò cadat extra ba&longs;im rotatur, non repit
&longs;it enim planum inclinatum BC, cui incubet cubus DL, cuius cen
trum grauitatis &longs;it I; ducatur RG perpendicularis deor&longs;um per cen
trum grauitatis I cadit in punctum G intra ba&longs;im BG; igitur non ro
tabitur, &longs;ed repet; quia &longs;i &longs;u&longs;tinetur in G remoto &longs;en&longs;im plano BC;
haud dubiè portio GD non præponderat portioni GL, vt patet ex
libra.
Sit quoque parallelipedum EK, centrum grauitatis N, perpendicu
laris ducta per centrum HNM cadit intra ba&longs;im; igitur non rotabi
tur, quia &longs;ubmoto plano BC non &longs;u&longs;tinetur quidem in M, &longs;ed minimè
inclinabitur dextror&longs;um; igitur non rotabitur. Si verò cadat extra ba
&longs;im haud dubiè rotabitur, &longs;it enim planum inclinatum AC, cui in
cumbat parallelipedum FN, cuius centrum grauitatis &longs;it L; ducatur L
perpendicularis, cadit in E extra ba&longs;im FD; certè latus DN inclinabi
tur deor&longs;um; igitur rotabitur, quia eodem modo &longs;e habet, quo &longs;e ha
beret, &longs;i &longs;ubmoto plano &longs;u&longs;tineretur in linea DX, &longs;ed trapezus DX
PN triangulo FXD præponderat per regulas libræ, de quibus &longs;uo
loco.
Ob&longs;eruabis autem primò &longs;ciri po&longs;&longs;e data plani inclinatione & ba&longs;i
parallelipedi maximam illius altitudinem, qua po&longs;ita non rotetur;
&longs;ecus verò po&longs;ita quacunque alia maiore; &longs;it enim planum AC, ba
&longs;is parallelipedi FD; erigantur FO, DN perpendiculares in
M: dico FX e&longs;&longs;e maximam altitudinem, vt con&longs;tat ex dictis.
Secundò, quotie&longs;cunque rectangulum, ita e&longs;t &longs;itum, vt eius
diagonalis &longs;it perpendicularis; dico e&longs;&longs;e in perfecto æquilibrio;
&longs;it enim rectangulum BE, cuius diagonalis BE perpendicula
riter cadit in horizontalem AC; certè erit in æqualibrio; &longs;it enim
diui&longs;um per lineam BE ita vt FH vel KI &longs;it libra quæ &longs;u&longs;tineatur in ful
cro BG; &longs;itque totum pondus trianguli BED appen&longs;um brachio GH,
& aliud BET appen&longs;um brachio æquali GF, erit perfectum æquili
brium per regulas libræ, &longs;ed duo triangula eodem modo &longs;e habent
conjuncta, quo &longs;e haberent &longs;eparata & appen&longs;a, vt patet.
Tertiò, omnia rectangula proportionalia in eodem æquilibrio rema
nerent v.g. rectangulum BG cum rectangulo BE, idem dico de Rhom
bo, Rhomboide, &c.
Quartò, inde etiam cogno&longs;citur in qua proportione minuatur pondus.
v. g. &longs;it enim cylindrus AE horizontalis, &longs;u&longs;tineaturque in A immo
biliter, itemque in E; certè qui &longs;u&longs;tinet in E æqualiter &longs;u&longs;tinet; at verò
&longs;i attollatur in AD; certè potentia quæ in D &longs;u&longs;tinet, e&longs;t ad potentiam
quæ &longs;u&longs;tinet in E, vt AF ad AE, quia pondus grauitaret in D & in E in
cadem ratione per Th. 16. &longs;ed potentia &longs;u&longs;tinens adæquat ponderis ra
tionem, &longs;u&longs;tinens inquam, per DH; nam reuerà &longs;u&longs;tinens per DF æqua
lis e&longs;&longs;e debet potentiæ in E: idem dico &longs;i attollatur in AP, nam potentia
trahens in P, per CP, e&longs;t ad potentiam in E, vt QA ad AP, vel AE;
igitur pondus in D e&longs;t ad pondus in P vt FA ad QA.
Quintò, hinc &longs;i duo ferant parallelipedum in &longs;itu inclinato v.g.vt AD,
ferunt inæqualiter, &longs;cilicet in ratione AD FA, itemque &longs;i ferant in &longs;itu
inclinato AP, vel AC, donec tandem AE attollatur in B, nihil amplius
&longs;u&longs;tinet potentia in B, & potentia in A totum &longs;u&longs;tinet.
Sextò, hinc cùm attollitur cylindrus continuò minùs &longs;entitur pondus
& faciliùs attollitur; &longs;ic qui attollunt pontes illos ver&longs;atiles, initio maxi
mo ni&longs;u, & modico &longs;ub finem trahunt.
Septimò ob&longs;eruabis, &longs;i circa centrum immobile A attollatur cylindrus
AE fune BE, potentia po&longs;ita in B, vel fune EO, potentia po&longs;ita in O;
hæc deber e&longs;&longs;e minor quàm po&longs;ita in B, vt autem cogno&longs;catur propor
tio, fiat angulus PAE æqualis angulo OEB; ducatur PQ; dico poten
tiam in O e&longs;&longs;e ad potentiam B, vt AQ ad AP, quia &longs;i anguli OEB &
PAQ &longs;unt æquales etiam anguli APQ & AEB &longs;unt æquales; igitur
perinde e&longs;t &longs;iue trahatur PA circa A per lineam PQ, &longs;iue trahatur EA
circa A per lineam EB. Idem dictum &longs;it de aliis lincis.
Octauò &longs;i attollendum &longs;it rectangulum non quidem circa axem; &longs;ed
circa angulum immobilem, etiam decre&longs;cit proportio ponderis, &longs;it enim
v.g.
tur diagonalis AF, attollatur circa punctum A, ita vt trans&longs;eratur in AG,
ducatur GB perpendicularis: dico potentiam in G e&longs;&longs;e ad potentiam in
in A, vt AB ad AD; quippe res eodem modo &longs;e habet, ac &longs;i AF a&longs;cenderet
potentia in M propter æquilibrium.
Nonò, hinc initio decre&longs;cit in maiori proportione ratione præpon
derantiæ; quia po&longs;ita ba&longs;i KN, angulus KAN e&longs;t omnium maximus; at
verò decre&longs;cit in minori proportione initio ratione &longs;egmenti horizon
talis AD, in quam cadit perpendicularis.
Decimò, &longs;i &longs;it rectangulum oblongum horizontale vt AE diffici
liùs attolletur; quia quadratum AF figuræ prioris debet tantùm attolli
per arcum FM, vt &longs;tatuatur in æquilibro; at verò rectangulum AE fi
guræ huius attolli debet per arcum EC longè maiorem; igitur difficiliùs:
porrò potentia in D e&longs;t ad potentiam in F vt AG ad AF, vt con&longs;tat ex
dictis.
Vndecimò, denique, &longs;i &longs;it rectangulum oblongum, &longs;ed verticale vt
HK longè faciliùs attolletur, quia diagonalis HK debet tantùm percur
rere arcum KM vt &longs;tatuatur in æquilibrio; igitur minorem, igitur longè
faciliùs; porrò hæc omnia omnibus experimentis con&longs;entiunt, & ex
principiis facillimis demon&longs;trantur. Hæc paulò fu&longs;iùs pro&longs;equutus &longs;um,
quia pertinent ad rationem plani inclinati.
in perpendiculari
E, in qua primo tempore &longs;en&longs;ibili percurrat AD; &longs;ecundò DE; certè dato
etiam tempore licèt maiore percurret AB; igitur alio æquali percurret
CB; nam vt &longs;e habet AE ad AG; ita &longs;e habet AD ad AB, & DE ad BC;
quæ omnia &longs;unt certa.
perpendiculari,
AC per Th.6.igitur motus per AC e&longs;t tardior; &longs;ed motu tardiore minùs
&longs;patium conficitur æquali tempore in ca proportione, in qua motus e&longs;t
tardior; &longs;ed proportio velocitatis e&longs;t vt AC ad AE: atqui quâ propor
tione motus e&longs;t tardior alio, maius &longs;patium decurri debet, vt motu acce
lerato per minora crementa acquiratur velocitas alteri æqualis; igitur
eò &longs;patium debet e&longs;&longs;e maius, quò motus erit tardior; igitur debet percur
ri AC in inclinata, & AE in perpendiculari, vt &longs;it æqualis velocitas;
&longs;it autem v.g. AC dupla AE, certè motus per AC e&longs;t &longs;ubduplus motus
pes AE; ducatur EB perpendicularis, certè AB e&longs;t &longs;ubdupla AE; igitur
eo tempore, quo percurret AE, percurret tantùm AB &longs;ubduplum &longs;cili
cet motu &longs;ubduplo; igitur tempore æquali BC triplam AB; &longs;ed tem
poribus æqualibus acquiruntur æqualia velocitatis momenta; igitur ve
locitas in C e&longs;t dupla illius, quæ erat in B; &longs;ed quæ e&longs;t in E e&longs;t dupla il
lius, quæ e&longs;t in B; igitur quæ e&longs;t in E e&longs;t æqualis illi, quæ e&longs;t in C. Adde
quod in ea proportione in qua motus e&longs;t tardior, &longs;patium e&longs;t maius, vt
æqualis velocitas acquiratur; igitur &longs;i quælibet pars &longs;patij motum auget
jus e&longs;t &longs;patio AE in ca proportione in qua motus per AE e&longs;t velocior;
pauciores partes &longs;patij AE augent motum, &longs;ed plùs &longs;ingulæ, & plures
&longs;patij AC augent motum, &longs;ed minùs &longs;ingulæ; &longs;ed cum &longs;int plures in ea
dem proportione, in qua minùs augent; certè plures quarum &longs;ingulæ mi
nùs augent, &longs;imul &longs;umptæ æqualiter augent, v.g. &longs;int AC 4. partes, & AE
2. &longs;ingulæ AE augeant motum vt 4. & &longs;ingulæ AC vt 2. quia in ca pro
portione minùs augent in qua 2. &longs;unt ad 4. certè 2. &longs;imul &longs;umptæ augent
motum vt 8. & 4. &longs;imul &longs;umptæ etiam vt 8. quæ dicta &longs;unt in gratiam
Geometrarum, &longs;ed meliùs adhuc ex dictis patebit.
AG.
quodcertè mirabile e&longs;t.
tum
qualem ictum &longs;ed confecto &longs;patio duplo; accipe AC duplam AE: vis æ
qualem ictum &longs;ed confecto &longs;patio triplo, accipe AG triplam AE.
dines,
quo percurritur perpendicularis AE, vt AC ad AE; probatur, cùm enim
mobile in C & in E habeat æqualem impetum &longs;eu velocitatem per Th.
20. certè cùm motus in AC &longs;it &longs;ubduplus v.g. motus in AE, e&longs;t enim
vt AE ad AC per Th.6. igitur cum &longs;ubduplo motu æquali tempore ac
quiritur &longs;ubduplus impetus; igitur tempore duplo æqualis impetus; at
qui tempus motus per AC e&longs;t ad tempus motus per AE vt AC ad AE,
ide&longs;t duplum; adde quod &longs;i æqualis impetus e&longs;t in C & in E; igitur æqua
lis in D & in B, &longs;ed AB e&longs;t ad BC vt AD ad DE; igitur &longs;i cre&longs;cit impe
tus per partes &longs;ubduplas in AC