NATURALIS
PRINCIPIA
MATHEMATICA.
ISAACO NEWTONO,
EQUITE A RATO.
SOCIETATI REGALI,
A
SERENISSIMO REGE
CAROLO II
AD PHILOSOPHIAM PROMOVENDAM
FUNDATÆ,
ET
AUSPICIIS
AUGUSTISSIMÆ REGINÆ
ANNÆ
FLORENTI,
VIRI PRÆSTANTISSIMI
ISAACI NEWTONI
OPUS HOCCE
MATHEMATICO PHYSICUM
EN tibi norma Poli, & divæ libramina Molis,
Computus en Jovis; & quas, dum primordia rerum.
Conderet, omnipotens &longs;ibi Leges ip&longs;e Creator
Dixerit, atque operum quæ fundamenta locarit.
Intima panduntur victi penetralia Cæli,
Nec latet, extremos quæ Vis circumrotet Orbes.
Sol &longs;olio re&longs;idens ad &longs;e jubet omnia prono
Tendere de&longs;cen&longs;u, nec recto tramite currus
Sidereos patitur va&longs;tum per inane moveri;
Sed rapit immotis, &longs;e centro, &longs;ingula gyris.
Hinc patet, horrificis qua &longs;it via flexa Cometis:
Di&longs;cimus hinc tandem, qua cau&longs;a argentea Phœbe
Pa&longs;&longs;ibus haud æquis eat, & cur &longs;ubdita nulli
Hactenus A&longs;tronomo numerorum fræna recu&longs;et:
Cur remeent Nodi, curque Auges progrediantur.
Di&longs;cimus, & quantis refluum vaga Cynthia Pontum
Viribus impellat; fe&longs;&longs;is dum fluctibus ulvam
De&longs;erit, ac nautis &longs;u&longs;pectas nudat arenas;
Alterni&longs;ve ruens &longs;pumantia littora pul&longs;at.
Quæque Scholas hodie rauco certamine vexant,
Obvia con&longs;picimus; nubem pellente Mathe&longs;i:
Quæ &longs;uperas penetrare domos, atque ardua Cæli,
NEWTONI au&longs;piciis, jam dat contingere Templa.
Surgite Mortales, terrenas mittite curas;
Atque hinc cæligenæ vites cogno&longs;cite Mentis,
A pecudum vita longe longeque remotæ.
Qui &longs;criptis primus Tabulis compe&longs;cere Cædes,
Furta & Adulteria, & perjuræ crimina Fraudis;
Quive vagis populis circumdare mœnibus Urbes
Auctor erat; Cereri&longs;ve beavit munere gentes;
Vel qui curarum lenimen pre&longs;&longs;it ab Uva;
Vel qui Niliaca mon&longs;travit arundine pictos
Con&longs;ociare &longs;onos, oculi&longs;que exponere Voces;
Humanam &longs;ortem minus extulit; utpote pauca
In commune ferens mi&longs;eræ &longs;olatia vitæ.
Jam vero Superis convivæ admittimur, alti
Jura poli tractare licet, jamque abdita diæ
Clau&longs;tra patent Naturæ, & rerum immobilis ordo;
Et quæ præteritis latuere incognita &longs;æclis.
Talia mon&longs;trantem ju&longs;tis celebrate Camænis,
Vos qui cæle&longs;ti gaudetis nectare ve&longs;ci,
NEWTONUM clau&longs;i re&longs;erantem &longs;crinia Veri,
NEWTONUM Mu&longs;is carum, cui pectore puro
Phœbus ade&longs;t, totoQ.E.I.ce&longs;&longs;it Numine mentem:
Nec fas e&longs;t propius Mortali attingere Divos.
PRÆFATIO
AD
LECTOREM.
Naturalium inve&longs;tigatione maximi fecerint; & Recentiores,
mi&longs;&longs;is formis &longs;ub&longs;tantialibus & qualitatibus occultis, Phænomena
Naturæ ad leges Mathematicas revocare aggre&longs;&longs;i fint: Vi&longs;um e&longs;t
in hoc Tractatu
tionalem
cam.
utiqueCum autem Artifices pa
rum accurate operari &longs;oleant, fit ut
tria
errores non &longs;unt Artis &longs;ed Artificum. Qui minus accurate ope
ratur, imperfectior e&longs;t Mechanicus, & &longs;i quis accurati&longs;&longs;ime ope
rari po&longs;&longs;et, hic foret Mechanicus omnium perfecti&longs;&longs;imus. Nam &
Linearum rectarum & Circulorum de&longs;criptiones in quibus
metria Has lineas de&longs;cri
berePo&longs;tulat enim ut Tyro
ea&longs;dem accurate de&longs;cribere prius didicerit quam linen attingat
Geometriæ;
uantur, docet. Rectas & Circulos de&longs;cribere Problemata &longs;unt,
Ex
Geometria Ac gloriatur
Fun
datur igitur
quam
curate proponit ac demon&longs;trat. Cum autem artes Manuales in
corporibus movendis præcipue ver&longs;entur, fit ut
nitudinem,Quo &longs;en&longs;u
chanica rationalis
cunque re&longs;ultant, & Virium quæ ad motus quo&longs;cunque requirun
tur, accurate propo&longs;ita ac demon&longs;trata. Pars hæc
Veteribus in
exculta fuit, qui Gravitatem (cum potentia manualis non &longs;it) vix
aliter quam in ponderibus per potentias illas movendis con&longs;iderarunt.
Nos autem non Artibus &longs;ed Philo&longs;ophiæ con&longs;ulentes, deque poten
tiis non manualibus &longs;ed naturalibus &longs;cribentes, ea maxime tracta
mus quæ ad Gravitatem, Levitatem, vim Ela&longs;ticam, re&longs;i&longs;tentiam
Fluidorum & eju&longs;modi vires &longs;eu attractivas &longs;eu impul&longs;ivas &longs;pe
ctant: Et ea propter, hæc no&longs;tra tanquam Philo&longs;ophiæ principia
Mathematica proponimus. Omnis enim Philo&longs;ophiæ difficultas in
eo ver&longs;ari videtur, ut a Phænomenis motuum inve&longs;tigemus vires
Naturæ, deinde ab his viribus demon&longs;tremus phænomena reliqua.
Et huc &longs;pectant Propo&longs;itiones generales quas Libro primo & &longs;ecundo
pertractavimus. In Libro autem tertio Exemplum hujus rei propo
&longs;uimus per explicationem Sy&longs;tematis mundani. Ibi enim, ex phæ
nomenis cæle&longs;tibus, per Propo&longs;itiones in Libris prioribus Mathe
matice demon&longs;tratas, derivantur vires Gravitatis quibus corpora
ad Solem & Planetas &longs;ingulos tendunt. Deinde ex his viribus
per Propo&longs;itiones etiam Mathematicas, deducuntur motus Planeta
rum, Cometarum, Lunæ & Maris. Utinam cætera Naturæ phæ
nomena ex principiis Mechanicis eodem argumentandi genere deri
vare liceret. Nam multa me movent ut nonnihil &longs;u&longs;picer ea om
per cau&longs;as nondum cognitas vel in &longs;e mutuo impelluntur & &longs;e
cundum figuras regulares cohærent, vel ab invicem fugantur &
recedunt: quibus viribus ignotis, Philo&longs;ophi hactenus Naturam fru
&longs;tra tentarunt. Spero autem quod vel huic Philo&longs;ophandi modo,
vel veriori alicui, Principia hic po&longs;ita lucem aliquam præbebunt.
eruditi&longs;&longs;imus
Typothetarum Sphalmata correxit & Schemata incidi curavit, &longs;ed
etiam Auctor fuit ut horum editionem aggrederer. Quippe cum
demon&longs;tratam a me Figuram Orbium cæle&longs;tium impetraverat, ro
gare non de&longs;titit ut eandem cum
Quæ deinde hortatibus & benignis &longs;uis au&longs;piciis effecit ut de ea
dem in lucem emittenda cogitare inciperem. At po&longs;tquam Mo
tuum Lunarium inæqualitates aggre&longs;&longs;us e&longs;&longs;em, deinde etiam ælia
tentare cæpi&longs;&longs;em quæ ad leges & men&longs;uras Gravitatis & aliarum
virium, & Figuras a corporibus &longs;ecundum datas qua&longs;cunque leges
attractis de&longs;cribendas, ad motus corporum plurium inter &longs;e, ad
motus corporum in Mediis re&longs;i&longs;tentibus, ad vires, den&longs;itates &
motus Mediorum, ad Orbes Cometarum & &longs;imilia &longs;pectant, edi
tionem in aliud tempus differendam e&longs;&longs;e putavi, ut cætera rima
rer & una in publicum darem. Quæ ad motus Lunares &longs;pectant,
(imperfecta cum &longs;int,) in Corollariis Propo&longs;itionis
complexus &longs;um, ne &longs;ingula methodo prolixiore quam pro rei digNI
tate proponere, & &longs;igillatim demon&longs;trare tenerer, & &longs;eriem reli
quarum Propo&longs;itionum interrumpere. Nonnulla &longs;ero inventa lo
cis minus idoneis in&longs;erere malui, quam numerum Propo&longs;itionum
& citationes mutare. Ut omnia candide legantur, & defectus,
in materia tam difficili non tam reprehendantur, quam novis Le
ctorum conatibus inve&longs;tigentur, & benigne &longs;uppleantur, enixe rogo.
Dabam
dantur & nonnulla adjiciuntur. In Libri primi Sectione
facilior redditur & amplior. In Libri &longs;ecundi Sectione
Experimentis confirmatur. In Libro tertio Theoria Lunæ & Præ
ce&longs;&longs;io Æquinoctiorum ex Principiis &longs;uis plenius deducuntur, &
Theoria Cometarum pluribus & accuratius computatis Orbium
exemplis confirmatur.
Dabam
Mar. 28. 1713.
PRÆFATIO.
NEWTONIANÆ Philo&longs;ophiæ novam tibi, Lector benevole,
diuQ.E.D.&longs;ideratam Editionem, plurimum nunc emenda
tam atque auctiorem exhibemus. Quæ poti&longs;&longs;imum conti
neantur in hoc Opere celeberrimo, intelligere potes ex Indicibus
adjectis: quæ vel addantur vel immutentur, ip&longs;a te fere docebit
Auctoris Præfatio. Reliquum e&longs;t, ut adjiciantur nonnulla de Me
thodo hujus Philo&longs;ophiæ.
Qui Phy&longs;icam tractandam &longs;u&longs;ceperunt, ad tres fere cla&longs;&longs;es re
vocari po&longs;&longs;unt. Extiterunt enim, qui &longs;ingulis rerum &longs;peciebus Quali
tates &longs;pecificas & occultas tribuerint; ex quibus deinde corporum
&longs;ingulorum operationes, ignota quadam ratione, pendere volue
runt. In hoc po&longs;ita e&longs;t &longs;umma doctrinæ Schola&longs;ticæ, ab
& Peripateticis derivatæ: Affirmant utique &longs;ingulos Effectus ex
corporum &longs;ingularibus Naturis oriri; at unde &longs;int illæ Naturæ
non docent; nihil itaQ.E.D.cent. Cumque toti &longs;int in rerum no
minibus, non in ip&longs;is rebus; Sermonem Q.E.D.m Philo&longs;ophicum
cen&longs;endi &longs;unt adinveni&longs;&longs;e, Philo&longs;ophiam tradidi&longs;&longs;e non &longs;unt cen
&longs;endi.
Alii ergo melioris diligentiæ laudem con&longs;equi &longs;perarunt, rejecta
Vocabulorum inutili farragine. Statuerunt itaque Materiam uNI
ver&longs;am homogeneam e&longs;&longs;e, omnem vero Formarum varietatem, quæ
in corporibus cernitur, ex particularum componentium &longs;implici&longs;&longs;i
mis quibu&longs;dam & intellectu facillimis affectionibus oriri. Et recte
quidem progre&longs;&longs;io in&longs;tituitur a &longs;implicioribus ad magis compo&longs;ita,
&longs;i particularum primariis illis affectionibus non alios tribuunt mo
dos, quam quos ip&longs;a tribuit Natura. Verum ubi licentiam &longs;ibi
a&longs;&longs;umunt, ponendi qua&longs;cunque libet ignotas partium figuras &
magnitudines, incerto&longs;que &longs;itus & motus; quin & fingendi Fluida
quædam occulta, quæ corporum poros liberrime permeent, omNI
potente prædita &longs;ubtilitate, motibu&longs;que occultis agitata; jam ad
&longs;omnia delabuntur, neglecta rerum con&longs;titutione vera: quæ fane
fru&longs;tra petenda e&longs;t ex fallacibus conjecturis, cum vix etiam per
certi&longs;&longs;imas Ob&longs;ervationes inve&longs;tigari po&longs;&longs;it. Qui &longs;peculationum
&longs;ecundum leges Mechanicas accurati&longs;&longs;ime procedant; Fabulam qui
dem elegantem forte & venu&longs;tam, Fabulam tamen concinnare di
cendi &longs;unt.
Relinquitur adeo tertium genus, qui Philo&longs;ophiam &longs;cilicet Ex
perimentalem profitentur. Hi quidem ex &longs;implici&longs;&longs;imis quibus
po&longs;&longs;unt principiis rerum omnium cau&longs;as derivandas e&longs;&longs;e volunt:
nihil autem Principii loco a&longs;&longs;umunt, quod nondum ex Phænome
nis comprobatum fuerit. Hypothe&longs;es non commini&longs;cuntur, neque
in Phy&longs;icam recipiunt, ni&longs;i ut Quæ&longs;tiones de quarum veritate di&longs;
putetur. Duplici itaque Methodo incedunt, Analytica & Syn
thetica. Naturæ vires lege&longs;que virium &longs;impliciores ex &longs;electis
quibu&longs;dam Phænomenis per Analy&longs;in deducunt, ex quibus deinde
per Synthe&longs;in reliquorum con&longs;titutionem tradunt. Hæc illa e&longs;t
Philo&longs;ophandi ratio longe optima, quam præ ceteris merito am
plectendam cen&longs;uit Celeberrimus Auctor no&longs;ter. Hanc &longs;olam uti
Q.E.D.gnam judicavit, in qua excolenda atque adornanda operam
&longs;uam collocaret. Hujus igitur illu&longs;tri&longs;&longs;imum dedit Exemplum,
Mundani nempe Sy&longs;tematis explicationem e Theoria Gravitatis
felici&longs;&longs;ime deductam. Gravitatis virtutem univer&longs;is corporibus in
e&longs;&longs;e, &longs;u&longs;picati &longs;unt vel finxerunt alii: primus Ille & &longs;olus ex Ap
parentiis demon&longs;trare potuit, & &longs;peculationibus egregiis firmi&longs;&longs;i
mum ponere fundamentum.
Scio equidem nonnullos magni etiam nominis Viros, præjudiciis
quibu&longs;dam plus æquo occupatos, huic novo Principio ægre a&longs;&longs;en
tiri potui&longs;&longs;e, & certis incerta identidem prætuli&longs;&longs;e. Horum famam vel
licare non e&longs;t animus: Tibi potius, Benevole Lector, illa paucis ex
ponere lubet, ex quibus Tute ip&longs;e judicium non iniquum feras.
Igitur ut Argumenti &longs;umatur exordium a &longs;implici&longs;&longs;imis & pro
ximis; de&longs;piciamus pauli&longs;per qualis &longs;it in Terre&longs;tribus natura Gra
vitatis, ut deinde tutius progrediamur ubi ad corpora Cæle&longs;tia, lon
gi&longs;&longs;ime a &longs;edibus no&longs;tris remota, perventum fuerit. Convenit jam
inter omnes Philo&longs;ophos, corpora univer&longs;a circumterre&longs;tria gra
vitare in Terram. Nulla dari corpora vere levia, jamdudum
confirmavit Experientia multiplex. Quæ dicitur Levitas relativa,
non e&longs;t vera Levitas, &longs;ed apparens &longs;olummodo; & oritur a præ
pollente Gravitate corporum contiguorum.
Porro, ut corpora univer&longs;a gravitant in Terram, ita Terra vici&longs;
&longs;im in corpora æqualiter gravitat; Gravitatis enim actionem e&longs;&longs;e
mutuam & utrinque æqualem, &longs;ic o&longs;tenditur. Di&longs;tinguatur Terræ
inæquales: jam &longs;i pondera partium non e&longs;&longs;ent in &longs;e mutuo æqua
lia; cederet pondus minus majori, & partes conjunctæ pergerent
recta moveri ad infinitum, ver&longs;us plagam in quam tendit pondus
majus: omnino contra Experientiam. ItaQ.E.D.cendum erit, pon
dera partium in æquilibrio e&longs;&longs;e con&longs;tituta: hoc e&longs;t, Gravitatis
actionem e&longs;&longs;e mutuam & utrinque æqualem.
Pondera corporum, æqualiter a centro Terræ di&longs;tantium, &longs;unt ut
quantitates materiæ in corporibus. Hoc utique colligitur ex
æquali acceleratione corporum omnium, e quiete per ponderum
vires cadentium: nam vires quibus inæqualia corpora æqualiter
accelerantur, debent e&longs;&longs;e proportionales quantitatibus materiæ
movendæ. Jam vero corpora univer&longs;a cadentia æqualiter acce
lerari, ex eo patet, quod in Vacuo
æqualia &longs;patia cadendo de&longs;cribunt, &longs;ublata &longs;cilicet Aeris re&longs;i&longs;tentia:
accuratius autem comprobatur per Experimenta Pendulorum.
Vires attractivæ corporum, in æqualibus di&longs;tantiis, &longs;unt ut
quantitates materiæ in corporibus. Nam cum corpora in Ter
ram & Terra vici&longs;&longs;im in corpora momentis æqualibus gravitent;
Terræ pondus in unumquodque corpus, &longs;eu vis qua corpus Ter
ram attrahit, æquabitur ponderi corporis eju&longs;dem in Terram.
Hoc autem pondus erat ut quantitas materiæ in corpore: itaque
vis qua corpus unumquodque Terram attrahit, &longs;ive corporis vis
ab&longs;oluta, erit ut eadem quantitas materiæ.
Oritur ergo & componitur vis attractiva corporum integrorum
ex viribus attractivis partium: &longs;iquidem aucta vel diminuta mole
materiæ, o&longs;ten&longs;um e&longs;t, proportionaliter augeri vel diminui ejus vir
tutem. Actio itaque Telluris ex conjunctis partium Actionibus
conflari cen&longs;enda erit; atque adeo corpora omnia Terre&longs;tria &longs;e
mutuo trahere oportet viribus ab&longs;olutis, quæ &longs;int in ratione ma
teriæ trahentis. Hæc e&longs;t natura Gravitatis apud Terram: videa
mus jam qualis &longs;it in Cælis.
Corpus omne per&longs;everare in &longs;tatu &longs;uo vel quie&longs;cendi vel movendi
uniformiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur
&longs;tatum illum mutare; Naturæ lex e&longs;t ab omnibus recepta Philo&longs;o
phis. Inde vero &longs;equitur, corpora quæ in Curvis moventur, atque
adeo de lineis rectis Orbitas &longs;uas tangentibus jugiter abeunt, Vi
aliqua perpetuo agente retineri in itinere curvilineo. Planetis
igitur in Orbibus curvis revolventibus nece&longs;&longs;ario aderit Vis aliqua,
per cujus actiones repetitas inde&longs;inenter a Tangentibus deflectantur.
Jam illud concedi æquum e&longs;t, quod Mathematicis rationibus
colligitur & certi&longs;&longs;ime demon&longs;tratur; Corpora nempe omnia, quæ
moventur in linea aliqua curva in plano de&longs;cripta, quæque radio
ducto ad punctum vel quie&longs;cens vel utcunque motum de&longs;cribunt
areas circa punctum illud temporibus proportionales, urgeri a
Viribus quæ ad idem punctum tendunt. Cum igitur in confe&longs;&longs;o
&longs;it apud A&longs;tronomos, Planetas primarios circum Solem, &longs;ecunda
rios vero circum &longs;uos primarios, areas de&longs;cribere temporibus pro
portionales; con&longs;equens e&longs;t ut Vis illa, qua perpetuo detorquen
tur a Tangentibus rectilineis, & in Orbitis curvilineis revolvi co
guntur, ver&longs;us corpora dirigatur quæ &longs;ita &longs;unt in Orbitarum cen
tris. Hæc itaque Vis non inepte vocari pote&longs;t, re&longs;pectu quidem
corporis revolventis, Centripeta; re&longs;pectu autem corporis cen
tralis, Attractiva; a quacunQ.E.D.mum cau&longs;a oriri fingatur.
Quin & hæc quoque concedenda &longs;unt, & Mathematice demon
&longs;trantur: Si corpora plura motu æquabili revolvantur in Circulis
concentricis, & quadrata temporum periodieorum &longs;int ut cubi di
&longs;tantiarum a centro communi; Vires centripetas revolventium
fore reciproce ut quadrata di&longs;tantiarum. Vel, &longs;i corpora revol
vantur in Orbitis quæ &longs;unt Circulis finitimæ, & quie&longs;cant Orbita
rum Ap&longs;ides; Vires centripetas revolventium fore reciproce ut
quadrata di&longs;tantiarum. Obtinere ca&longs;um alterutrum in Planetis
univer&longs;is con&longs;entiunt A&longs;tronomi. Itaque Vires centripetæ Plane
tarum omnium &longs;unt reciproce ut quadrata di&longs;tantiarum ab Or
bium centris. Si quis objiciat Planetarum, & Lunæ præ&longs;ertim,
Ap&longs;ides non penitus quie&longs;cere; &longs;ed motu quodam lento ferri in
con&longs;equentia: re&longs;ponderi pote&longs;t, etiam&longs;i concedamus hunc mo
tum tardi&longs;&longs;imum exinde profectum e&longs;&longs;e quod Vis centripetæ pro
portio aberret aliquantum a duplicata, aberrationem illam per
computum Mathematicum inveniri po&longs;&longs;e & plane in&longs;en&longs;ibilem
e&longs;&longs;e. Ip&longs;a enim ratio Vis centripetæ Lunaris, quæ omnium ma
xime turbari debet, paululum quidem duplicatam &longs;uperabit; ad
hanc vero &longs;exaginta fere vicibus propius accedet quam ad tripli
catam. Sed verior erit re&longs;pon&longs;io, &longs;i dicamus hanc Ap&longs;idum progre&longs;
&longs;ionem, non ex aberratione a duplicata proportione, &longs;ed ex alia
pror&longs;us diver&longs;a cau&longs;a oriri, quemadmodum egregie common&longs;tratur
in hac Philo&longs;ophia. Re&longs;tat ergo ut Vires centripetæ, quibus Pla
netæ primarii tendunt ver&longs;us Solem & &longs;ecundarii ver&longs;us primarios
&longs;uos, &longs;int accurate ut quadrata di&longs;tantiarum reciproce.
Ex iis quæ hactenus dicta &longs;unt, con&longs;tat Planetas in Orbitis &longs;uis
retineri per Vim aliquam in ip&longs;os perpetuo agentem: con&longs;tat
Vim illam dirigi &longs;emper ver&longs;us Orbitarum centra: con&longs;tat hujus
efficaciam augeri in acce&longs;&longs;u ad centrum, diminui in rece&longs;&longs;u ab eo
dem: & augeri quidem in eadem proportione qua diminuitur qua
dratum di&longs;tantiæ, diminui in eadem proportione qua di&longs;tantiæ
quadratum augetur. Videamus jam, comparatione in&longs;tituta inter
Planetarum Vires centripetas & Vim Gravitatis, annon eju&longs;dem
forte &longs;int generis. Eju&longs;dem vero generis erunt, &longs;i deprehendan
tur hinc & inde leges eædem eædemque affectiones. Primo ita
que Lunæ, quæ nobis proxima e&longs;t, Vim centripetam expendamus.
Spatia rectilinea, quæ a corporibus e quiete demi&longs;&longs;is dato tem
pore &longs;ub ip&longs;o motus initio de&longs;eribuntur, ubi a viribus quibu&longs;cun
que urgentur, proportionalia &longs;unt ip&longs;is viribus: Hoc utique con
&longs;equitur ex ratiociniis Mathematicis. Erit igitur Vis centripeta
Lunæ in Orbita &longs;ua revolventis, ad Vim Gravitatis in &longs;uperficie
Terræ, ut &longs;patium quod tempore quam minimo de&longs;criberet Luna
de&longs;cendendo per Vim centripetam ver&longs;us Terram, &longs;i circulari om
ni motu privari fingeretur, ad &longs;patium quod eodem tempore quam
minimo de&longs;cribit grave corpus in vicinia Terræ, per Vim gravita
tis &longs;uæ cadendo. Horum &longs;patiorum prius æquale e&longs;t arcus a Luna
per idem tempus de&longs;cripti &longs;inui ver&longs;o, quippe qui Lunæ tran&longs;la
tionem de Tangente, factam a Vi centripeta, metitur; atque adeo
computari pote&longs;t ex datis tum Lunæ tempore periodico tum di
&longs;tantia ejus a centro Terræ. Spatium po&longs;terius invenitur per Ex
perimenta Pendulorum, quemadmodum docuit
itaque calculo, &longs;patium prius ad &longs;patium pofterius, &longs;eu vis cen
tripeta Lunæ in Orbita &longs;ua revolventis ad vim Gravitatis in &longs;u
perficie Terræ, erit ut quadratum &longs;emidiametri Terræ ad Orbitæ
&longs;emidiametri quadratum. Eandem habet rationem, per ea quæ
&longs;uperius o&longs;tenduntur, vis centripeta Lunæ in Orbita &longs;ua revol
ventis ad vim Lunæ centripetam prope Terræ &longs;uperficiem. Vis
itaque centripeta prope Terræ &longs;uperficiem æqualis e&longs;t vi Gravita
tis. Non ergo diver&longs;æ &longs;unt vires, &longs;ed una atque eadem: &longs;i enim
diver&longs;æ e&longs;&longs;ent, corpora viribus conjunctis duplo celerius in Ter
ram caderent quam ex vi &longs;ola Gravitatis. Con&longs;tat igitur Vim
illam centripetam, qua Luna perpetuo de Tangente vel trahitur
vel impellitur & in Orbita retinetur, ip&longs;am e&longs;&longs;e vim Gravitatis
terre&longs;tris ad Lunam u&longs;que pertingentem. Et rationi quidem con
&longs;entaneum e&longs;t ut ad ingentes di&longs;tantias illa &longs;e&longs;e Virtus extendat,
cacuminibus, ob&longs;ervare licet. Gravitat itaque Luna in Terram:
quin & actione mutua, Terra vici&longs;&longs;im in Lunam æqualiter gravitat:
id quod abunde quidem confirmatur in hac Philo&longs;ophia, ubi agi
tur de Maris æ&longs;tu & Æquinoctiorum præce&longs;&longs;ione, ab actione tum
Lunæ tum Solis in Terram oriundis. Hinc & illud tandem edo
cemur, qua nimirum lege vis Gravitatis decre&longs;cat in majoribus a
Tellure di&longs;tantiis. Nam cum Gravitas non diver&longs;a &longs;it a Vi cen
tripeta Lunari, hæc vero &longs;it reciproce proportionalis quadrato
di&longs;tantiæ; diminuetur & Gravitas in eadem ratione.
Progrediamur jam ad Planetas reliquos.
Quoniam revolu
tiones primariorum circa Solem & &longs;ecundariorum circa Jovem &
Saturnum &longs;unt Phænomena generis eju&longs;dem ac revolutio Lunæ
circa Terram, quoniam porro demon&longs;tratum e&longs;t vires centripetas
primariorum dirigi ver&longs;us centrum Solis, &longs;ecundariorum ver&longs;us
centra Jovis & Saturni, quemadmodum Lunæ vis centripeta ver&longs;us
Terræ centrum dirigitur; adhæc, quoniam omnes illæ vires &longs;unt
reciproce ut quadrata di&longs;tantiarum a centris, quemadmodum vis
Lunæ e&longs;t ut quadratum di&longs;tantiæ a Terra: concludendum erit
eandem e&longs;&longs;e naturam univer&longs;is. Itaque ut Luna gravitat in Ter
ram, & Terra vici&longs;&longs;im in Lunam; &longs;ic etiam gravitabunt omnes
&longs;ecundarii in primarios &longs;uos, & primarii vici&longs;&longs;im in &longs;ecundarios;
&longs;ic & omnes primarii in Solem, & Sol vici&longs;&longs;im in primarios.
Igitur Sol in Planetas univer&longs;os gravitat & univer&longs;i in Solem.
Nam &longs;ecundarii dum primarios &longs;uos comitantur, revolvuntur in
terea circum Solem una cum primariis. Eodem itaque argumento,
utriu&longs;que generis Planetæ gravitant in Solem, & Sol in ip&longs;os.
Secundarios vero Planetas in Solem gravitare abunde in&longs;uper
con&longs;tat ex inæqualitatibus Lunaribus; quarum accurati&longs;&longs;imam
Theoriam, admiranda &longs;agacitate patefactam, in tertio hujus Operis
libro expo&longs;itam habemus.
Solis virtutem attractivam quoquover&longs;um propagari ad ingen
tes u&longs;Q.E.D.&longs;tantias, & &longs;e&longs;e diffundere ad &longs;ingulas circumjecti &longs;pa
tii partes, aperti&longs;&longs;ime colligi pote&longs;t ex motu Cometarum; qui ab
immen&longs;is intervallis profecti feruntur in viciniam Solis, & non
nunquam adeo ad ip&longs;um proxime accedunt ut Globum ejus, in
Periheliis &longs;uis ver&longs;antes, tantum non contingere videantur. Ho
rum Theoriam ab A&longs;tronomis antehac fru&longs;tra quæ&longs;itam, no&longs;tro
tandem &longs;æculo feliciter inventam & per Ob&longs;ervationes certi&longs;
&longs;ime demon&longs;tratam, Præ&longs;tanti&longs;&longs;imo no&longs;tro Auctori debemus. Patet
habentibus moveri, & radiis ad Solem ductis areas temporibus
proportionales de&longs;cribere. Ex hi&longs;ce vero Phænomenis manife
&longs;tum e&longs;t & Mathematice comprobatur, vires illas, quibus Cometæ
retinentur in orbitis &longs;uis, re&longs;picere Solem & e&longs;&longs;e reciproce ut qua
drata di&longs;tantiarum ab ip&longs;ius centro. Gravitant itaque Cometæ
in Solem: atque adeo Solis vis attractiva non tantum ad corpora
Planetarum in datis di&longs;tantiis & in eodem fere plano collocata,
&longs;ed etiam ad Cometas in diver&longs;i&longs;&longs;imis Cælorum regionibus & in
diver&longs;i&longs;&longs;imis di&longs;tantiis po&longs;itos pertingit. Hæc igitur e&longs;t natura
corporum gravitantium, ut vires &longs;uas edant ad omnes di&longs;tantias in
omnia corpora gravitantia. Inde vero &longs;equitur, Planetas & Co
metas univer&longs;os &longs;e mutuo trahere, & in &longs;e mutuo graves e&longs;&longs;e:
quod etiam confirmatur ex perturbatione Jovis & Saturni, A&longs;tro
nomis non incognita, & ab actionibus horum Planetarum in &longs;e in
vicem oriunda; quin & ex motu illo lenti&longs;&longs;imo Ap&longs;idum, qui &longs;u
pra memoratus e&longs;t, quique a cau&longs;a con&longs;imili profici&longs;citur.
Eo demum pervenimus ut dicendum &longs;it, & Terram & Solem &
corpora omnia cæle&longs;tia, quæ Solem comitantur, &longs;e mutuo attrahere.
Singulorum ergo particulæ quæque minimæ vires &longs;uas attractivas
habebunt, pro quantitate materiæ pollentes; quemadmodum &longs;u
pra de Terre&longs;tribus o&longs;ten&longs;um e&longs;t. In diver&longs;is autem di&longs;tantiis,
erunt & harum vires in duplicata ratione di&longs;tantiarum reciproce:
nam ex particulis hac lege trahentibus componi debere Globos
eadem lege trahentes, Mathematice demon&longs;tratur.
Conclu&longs;iones præcedentes huic innituntur Axiomati, quod a
nullis non recipitur Philo&longs;ophis; Effectuum &longs;cilicet eju&longs;dem ge
neris, quorum nempe quæ cogno&longs;cuntur proprietates eædem &longs;unt,
ea&longs;dem e&longs;&longs;e cau&longs;as & ea&longs;dem e&longs;&longs;e proprietates quæ nondum cog
no&longs;cuntur. Quis enim dubitat, &longs;i Gravitas &longs;it cau&longs;a de&longs;cen&longs;us
Lapidis in
Si Gravitas mutua fuerit inter Lapidem & Terram in
quis negabit mutuam e&longs;&longs;e in
& Terræ componatur, in
quis negabit &longs;imilem e&longs;&longs;e compo&longs;itionem in
Terræ ad omnia corporum genera & ad omnes di&longs;tantias propa
getur in
In hac Regula fundatur omnis Philo&longs;ophia: quippe qua &longs;ublata
nihil affirmare po&longs;&longs;imus de Univer&longs;is. Con&longs;titutio rerum &longs;ingula
rum innote&longs;cit per Ob&longs;ervationes & Experimenta: inde vero non
mus.
Jam cum Gravia &longs;int omnia corpora, quæ apud Terram vel in
Cælis reperiuntur, de quibus Experimenta vel Ob&longs;ervationes in
&longs;tituere licet; omnino dicendum erit, Gravitatem corporibus uNI
ver&longs;is competere. Et quemadmodum nulla concipi debent cor
pora, quæ non &longs;int Exten&longs;a, Mobilia, & Impenetrabilia; ita nulla
concipi debere, quæ non &longs;int Gravia. Corporum Exten&longs;io, Mobi
litas, & Impenetrabilitas non ni&longs;i per Experimenta innote&longs;cunt:
eodem plane modo Gravitas innote&longs;cit. Corpora omnia de qui
bus Ob&longs;ervationes habemus, Exten&longs;a &longs;unt & Mobilia & Impene
trabilia: & inde concludimus corpora univer&longs;a, etiam illa de qui
bus Ob&longs;ervationes non habemus, Exten&longs;a e&longs;&longs;e & Mobilia & Im
penetrabilia. Ita corpora omnia &longs;unt Gravia, de quibus Ob&longs;er
vationes habemus: & inde concludimus corpora univer&longs;a, etiam
illa de quibus Ob&longs;ervationes non habemus, Gravia e&longs;&longs;e. Si quis
dicat corpora Stellarum inerrantium non e&longs;&longs;e Gravia, quandoqui
dem eorum Gravitas nondum e&longs;t ob&longs;ervata; eodem argumento
dicere licebit neque Exten&longs;a e&longs;&longs;e, nec Mobilia, nec Impenetrabilia,
cum hæ Fixarum affectiones nondum &longs;int ob&longs;ervatæ. Quid opus
e&longs;t verbis? Inter primarias qualitates corporum univer&longs;orum vel
Gravitas habebit locum; vel Exten&longs;io, Mobilitas, & Impenetra
bilitas non habebunt. Et natura rerum vel recte explicabitur
per corporum Gravitatem, vel non recte explicabitur per corpo
rum Exten&longs;ionem, Mobilitatem, & Impenetrabilitatem.
Audio nonnullos hanc improbare conclu&longs;ionem, & de occultis
qualitatibus ne&longs;cio quid mu&longs;&longs;itare. Gravitatem &longs;cilicet Occultum
e&longs;&longs;e quid, perpetuo argutari &longs;olent; occultas vero cau&longs;as pro
cul e&longs;&longs;e ablegandas a Philo&longs;ophia. His autem facile re&longs;pon
detur; occultas e&longs;&longs;e cau&longs;as, non illas quidem quarum exi&longs;tentia
per Ob&longs;ervationes clari&longs;&longs;ime demon&longs;tratur, &longs;ed has &longs;olum quarum
occulta e&longs;t & ficta exi&longs;tentia nondum vero comprobata. Gravitas
ergo non erit occulta cau&longs;a motuum cæle&longs;tium; &longs;iquidem ex Phæ
nomenis o&longs;ten&longs;um e&longs;t, hanc virtutem revera exi&longs;tere. Hi potius
ad occultas confugiunt cau&longs;as; qui ne&longs;cio quos Vortices, materiæ
cuju&longs;dam pror&longs;us fictitiæ & &longs;en&longs;ibus omnino ignotæ, motibus
ii&longs;dem regendis præficiunt.
Ideone autem Gravitas occulta cau&longs;a dicetur, eoque nomine
rejicietur e Philo&longs;ophia, quod cau&longs;a ip&longs;ius Gravitatis occulta e&longs;t
& nondum inventa? Qui &longs;ic &longs;tatuunt, videant nequid &longs;tatu
lantur. Etenim cau&longs;æ continuo nexu procedere &longs;olent a compo
&longs;itis ad &longs;impliciora: ubi ad cau&longs;am &longs;implici&longs;&longs;imam perveneris, jam
non licebit ulterius progredi. Cau&longs;æ igitur &longs;implici&longs;&longs;imæ nulla
dari pote&longs;t mechanica explicatio: &longs;i daretur enim, cau&longs;a non
dum e&longs;&longs;et &longs;implici&longs;&longs;ima. Has tu proinde cau&longs;as &longs;implici&longs;&longs;imas
appellabis occultas, & exulare jubebis? &longs;imul vero exulabunt
& ab his proxime pendentes & quæ ab illis porro pendent,
u&longs;Q.E.D.m a cau&longs;is omnibus vacua fuerit & probe purgata Phi
lo&longs;ophia.
Sunt qui Gravitatem præter naturam e&longs;&longs;e dicunt, & Miraculum
perpetuum vocant. Itaque rejiciendam e&longs;&longs;e volunt, cum in Phy
&longs;ica præternaturales cau&longs;æ locum non habeant. Huic ineptæ
pror&longs;us objectioni diluendæ, quæ & ip&longs;a Philo&longs;ophiam &longs;ubruit
univer&longs;am, vix operæ pretium e&longs;t immorari. Vel enim Gravita
tem corporibus omnibus inditam e&longs;&longs;e negabunt, quod tamen dici
non pote&longs;t: vel eo nomine præter naturam e&longs;&longs;e affirmabunt, quod
ex aliis corporum affectionibus atque adeo ex cau&longs;is Mechanicis
originem non habeat. Dantur certe primariæ corporum affecti
ones; quæ, quoniam &longs;unt primariæ, non pendent ab aliis. Vide
rint igitur annon & hæ omnes &longs;int pariter præter naturam, eo
que pariter rejiciendæ: viderint vero qualis &longs;it deinde futura
Philo&longs;ophia.
Nonnulli &longs;unt quibus hæc tota Phy&longs;ica cæle&longs;tis vel ideo minus
placet, quod cum
po&longs;&longs;e videatur. His &longs;ua licebit opinione frui; ex æquo autem
agant oportet: non ergo denegabunt aliis eandem libertatem
quam &longs;ibi concedi po&longs;tulant. NEWTONIANAM itaque Philo&longs;ophi
am, quæ nobis verior habetur, retinere & amplecti licebit, & cau&longs;as
&longs;equi per Phænomena comprobatas, potius quam fictas & nondum
comprobatas. Ad veram Philo&longs;ophiam pertinet, rerum naturas
ex cau&longs;is vere exi&longs;tentibus derivare: eas vero leges quærere, qui
bus voluit Summus opifex hunc Mundi pulcherrimum ordinem
&longs;tabilire; non eas quibus potuit, &longs;i ita vi&longs;um fui&longs;&longs;et. Rationi enim
con&longs;onum e&longs;t, ut a pluribus cau&longs;is, ab invicem nonnihil diver&longs;is,
idem po&longs;&longs;it Effectus profici&longs;ci: hæc autem vera erit cau&longs;a, ex qua
vere atque actu profici&longs;citur; reliquæ locum non habent in Philo
&longs;ophia vera. In Horologiis automatis idem Indicis horarii mo
tus vel ab appen&longs;o Pondere vel ab intus conclu&longs;o Elatere oriri po
te&longs;t. Quod &longs;i oblatum Horologium revera &longs;it in&longs;tructum Pondere;
ficta motum Indicis explicare &longs;u&longs;cipiet: oportuit enim internam
Machinæ fabricam penitius per&longs;crutari, ut ita motus propo&longs;iti prin
cipium verum exploratum habere po&longs;&longs;et. Idem vel non ab&longs;imile
feretur judicium de Philo&longs;ophis illis, qui materia quadam &longs;ubti
li&longs;&longs;ima Cælos e&longs;&longs;e repletos, hanc autem in Vortices inde&longs;inenter
agi voluerunt. Nam &longs;i Phænomenis vel accurati&longs;&longs;ime &longs;atisfacere
po&longs;&longs;ent ex Hypothe&longs;ibus &longs;uis; veram tamen Philo&longs;ophiam tradi
di&longs;&longs;e, & veras cau&longs;as motuum cæle&longs;tium inveni&longs;&longs;e nondum di
cendi &longs;unt; ni&longs;i vel has revera exi&longs;tere, vel &longs;altem alias non ex
i&longs;tere demon&longs;traverint. Igitur &longs;i o&longs;ten&longs;um fuerit, univer&longs;orum
corporum Attractionem habere verum locum in rerum natura;
quinetiam o&longs;ten&longs;um fuerit, qua ratione motus omnes cæle&longs;tes ab
inde &longs;olutionem recipiant; vana fuerit & merito deridenda objectio,
&longs;i quis dixerit eo&longs;dem motus per Vortices explicari debere, etiam&longs;i
id fieri po&longs;&longs;e vel maxime conce&longs;&longs;erimus. Non autem concedimus:
Nequeunt enim ullo pacto Phænomena per Vortices explicari;
quod ab Auctore no&longs;tro abunde quidem & clari&longs;&longs;imis rationibus
evincitur; ut &longs;omniis plus æquo indulgeant oporteat, qui inep
ti&longs;&longs;imo figmento re&longs;arciendo, novi&longs;que porro commentis ornando
infelicem operam addicunt.
Si corpora Planetarum & Cometarum circa Solem deferantur
a Vorticibus; oportet corpora delata & Vorticum partes proxime
ambientes eadem velocitate eademque cur&longs;us determinatione mo
veri, & eandem habere den&longs;itatem vel eandem Vim inertiæ pro
mole materiæ. Con&longs;tat vero Planetas & Cometas, dum ver&longs;an
tur in ii&longs;dem regionibus Cælorum, velocitatibus variis variaque
cur&longs;us determinatione moveri. Nece&longs;&longs;ario itaque &longs;equitur, ut
Fluidi cæle&longs;tis partes illæ, quæ &longs;unt ad ea&longs;dem di&longs;tantias a Sole,
revolvantur eodem tempore in plagas diver&longs;as cum diver&longs;is ve
locitatibus: etenim alia opus erit directione & velocitate, ut tran
&longs;ire po&longs;&longs;int Planetæ; alia, ut tran&longs;ire po&longs;&longs;int Cometæ. Quod cum
explicari nequeat; vel fatendum erit, univer&longs;a corpora cæle&longs;tia
non deferri a materia Vorticis; vel dicendum erit, eorundem mo
tus repetendos e&longs;le non ab uno eodemque Vortice, &longs;ed a pluribus
qui ab invicem diver&longs;i &longs;int, idemque &longs;patium Soli circumjectum
pervadant.
Si plures Vortices in eodem &longs;patio contineri, & &longs;e&longs;e mutuo pe
netrare, motibu&longs;Q.E.D.ver&longs;is revolvi ponantur; quoniam hi mo
tus debent e&longs;&longs;e conformes delatorum corporum motibus, qui
valde eccentricis, nunc ad Circulorum proxime formam acceden
tibus; jure quærendum erit, qui fieri po&longs;&longs;it, ut iidem integri con
&longs;erventur, nec ab actionibus materiæ occur&longs;antis per tot &longs;æcula
quicquam perturbentur. Sane &longs;i motus hi fictitii &longs;unt magis com
po&longs;iti & difficilius explicantur, quam veri illi motus Planetarum
& Cometarum; fru&longs;tra mihi videntur in Philo&longs;ophiam recipi:
omnis enim Cau&longs;a debet e&longs;&longs;e Effectu &longs;uo &longs;implicior. Conce&longs;&longs;a
Fabularum licentia, affirmaverit aliquis Planetas omnes & Cometas
circumcingi Atmo&longs;phæris, adin&longs;tar Telluris no&longs;træ; quæ quidem
Hypothe&longs;is rationi magis con&longs;entanea videbitur quam Hypothe
&longs;is Vorticum. Affirmaverit deinde has Atmo&longs;phæras, ex natura
&longs;ua, circa Solem moveri & Sectiones Conicas de&longs;cribere; qui
&longs;ane motus multo facilius concipi pote&longs;t, quam con&longs;imilis motus
Vorticum &longs;e invicem permeantium. Denique Planetas ip&longs;os &
Cometas circa Solem deferri ab Atmo&longs;phæris &longs;uis credendum e&longs;&longs;e
&longs;tatuat, & ob repertas motuum cæle&longs;tium cau&longs;as triumphum agat.
Qui&longs;quis autem hanc Fabulam rejiciendam e&longs;&longs;e putet, idem & alte
ram Fabulam rejiciet: nam ovum non e&longs;t ovo &longs;imilius, quam Hy
pothe&longs;is Atmo&longs;phærarum Hypothe&longs;i Vorticum.
Docuit
nem a cur&longs;u rectilineo oriri a Gravitate lapidis in Terram, ab oc
culta &longs;cilicet qualitate. Fieri tamen pote&longs;t ut alius aliquis, na&longs;i
acutioris, Philo&longs;ophus cau&longs;am aliam commini&longs;catur. Finget igi
tur ille materiam quandam &longs;ubtilem, quæ nec vi&longs;u, nec tactu,
neque ullo &longs;en&longs;u percipitur, ver&longs;ari in regionibus quæ proxime
contingunt Telluris &longs;uperficiem. Hanc autem materiam, in di
ver&longs;as plagas, variis & plerumque contrariis motibus ferri, & li
neas Parabolicas de&longs;cribere contendet. Deinde vero lapidis de
flexionem pulchre &longs;ic expediet, & vulgi plau&longs;um merebitur. La
pis, inquiet, in Fluido illo &longs;ubtili natat; & cur&longs;ui ejus ob&longs;equen
do, non pote&longs;t non eandem una &longs;emitam de&longs;cribere. Fluidum
vero movetur in lineis Parabolicis; ergo lapidem in Parabola
moveri nece&longs;&longs;e e&longs;t. Quis nunc non mirabitur acuti&longs;&longs;imum huju&longs;ce
Philo&longs;ophi ingenium, ex cau&longs;is Mechanicis, materia &longs;cilicet &
motu, phænomena Naturæ ad vulgi etiam captum præclare de
ducentis? Quis vero non &longs;ub&longs;annabit bonum illum
magno molimine Mathematico qualitates occultas, e Philo&longs;ophia
feliciter exclu&longs;as, denuo revocare &longs;u&longs;tinuerit? Sed pudet nugis
diutius immorari.
Summa rei huc tandem redìt: Cometarum ingens e&longs;t numerus;
motus eorum &longs;unt &longs;umme regulares, & ea&longs;dem leges cum Plane
tarum motibus ob&longs;ervant. Moventur in Orbibus Conicis, hi or
bes &longs;unt valde admodum eccentrici. Feruntur undiQ.E.I. omnes
Cælorum partes, & Planetarum regiones liberrime pertran&longs;eunt,
& &longs;æpe contra Signorum ordinem incedunt. Hæc Phænomena
certi&longs;&longs;ime confirmantur ex Ob&longs;ervationibus A&longs;tronomicis: & per
Vortices nequeunt explicari: Imo, ne quidem cum Vorticibus
Planetarum con&longs;i&longs;tere po&longs;&longs;unt. Cometarum motibus omnino lo
cus non erit; ni&longs;i materia illa fictitia penitus e Cælis amo
veatur.
Si enim Planetæ circum Solem a Vorticibus devehuntur; Vor
ticum partes, quæ proxime ambiunt unumquemque Planetam, eju&longs;
dem den&longs;itatis erunt ac Planeta; uti &longs;upra dictum e&longs;t. Itaque
materia illa omnis quæ contigua e&longs;t Orbis magni perimetro, pa
rem habebit ac Tellus den&longs;itatem: quæ vero jacet intra Orbem
magnum atque Orbem Saturni, vel parem vel majorem habebit.
Nam ut con&longs;titutio Vorticis permanere po&longs;&longs;it, debent partes mi
nus den&longs;æ centrum occupare, magis den&longs;æ longius a centro abire.
Cum enim Planetarum tempora periodica &longs;int in ratione &longs;e&longs;qui
plicata di&longs;tantiarum a Sole, oportet partium Vorticis periodos
eandem rationem &longs;ervare. Inde vero &longs;equitur, vires centrifugas
harum partium fore reciproce ut quadrata di&longs;tantiarum. Quæ
igitur majore intervallo di&longs;tant a centro, nituntur ab eodem re
cedere minore vi: unde &longs;i minus den&longs;æ fuerint, nece&longs;&longs;e e&longs;t ut ce
dant vi majori, qua partes centro propiores a&longs;cendere conantur.
A&longs;cendent ergo den&longs;iores, de&longs;cendent minus den&longs;æ, & loeorum
fiet invicem permutatio; donec ita fuerit di&longs;po&longs;ita atque ordinata
materia fluida totius Vorticis, ut conquie&longs;cere jam po&longs;&longs;it in æqui
librio con&longs;tituta. Si bina Fluida, quorum diver&longs;a e&longs;t den&longs;itas,
in eodem va&longs;e continentur; utique futurum e&longs;t ut Fluidum, cu
jus major e&longs;t den&longs;itas, majore vi Gravitatis infimum petat locum:
& ratione non ab&longs;imili omnino dicendum e&longs;t, den&longs;iores Vorticis
partes majore vi centrifuga petere &longs;upremum locum. Tota igi
tur illa & multo maxima pars Vorticis, quæ jacet extra Telluris
orbem, den&longs;itatem habebit atque adeo vim inertiæ pro mole ma
teriæ, quæ non minor erit quam den&longs;itas & vis inertiæ Telluris:
inde vero Cometis trajectis orietur ingens re&longs;i&longs;tentia, & valde ad
modum &longs;en&longs;ibilis; ne dicam, quæ motum eorundem penitus &longs;i&longs;tere
atque ab&longs;orbere po&longs;&longs;e merito videatur. Con&longs;tat autem ex motu Co-
minimum &longs;entiri pote&longs;t; atque adeo neutiquam in materiam ul
lam incur&longs;are, cujus aliqua &longs;it vis re&longs;i&longs;tendi, vel proinde cujus ali
qua &longs;it den&longs;itas &longs;eu vis Inertiæ. Nam re&longs;i&longs;tentia Mediorum ori
tur vel ab inertia materiæ fluidæ, vel a defectu lubricitatis. Quæ
oritur a defectu lubricitatis, admodum exigua e&longs;t; & &longs;ane vix
ob&longs;ervari pote&longs;t in Fluidis vulgo notis, ni&longs;i valde tenacia fuerint
adin&longs;tar Olei & Mellis. Re&longs;i&longs;tentia quæ &longs;entitur in Aere, Aqua,
Hydrargyro, & huju&longs;modi Fluidis non tenacibus fere tota e&longs;t
prioris generis; & minui non pote&longs;t per ulteriorem quemcunque
gradum &longs;ubtilitatis, manente Fluidi den&longs;itate vel vi inertiæ, cui
&longs;emper proportionalis e&longs;t hæc re&longs;i&longs;tentia; quemadmodum clari&longs;
&longs;ime demon&longs;tratum e&longs;t ab Auctore no&longs;tro in peregregia Re&longs;i&longs;ten
tiarum Theoria, quæ paulo nunc accuratius exponitur, hac &longs;e
cunda vice, & per Experimenta corporum cadentium plenius
confirmatur.
Corpora progrediendo motum &longs;uum Fluido ambienti paulatim
communicant, & communicando amittunt, amittendo autem re
tardantur. E&longs;t itaque retardatio motui communicato proportio
nalis; motus vero communicatus, ubi datur corporis progredientis
velocitas, e&longs;t ut Fluidi den&longs;itas; ergo retardatio &longs;eu re&longs;i&longs;tentia
erit ut eadem Fluidi den&longs;itas; neque ullo pacto tolli pote&longs;t, ni&longs;i
a Fluido ad partes corporis po&longs;ticas recurrente re&longs;tituatur motus
ami&longs;&longs;us. Hoc autem dici non poterit, ni&longs;i impre&longs;&longs;io Fluidi in cor
pus ad partes po&longs;ticas æqualis fuerit impre&longs;&longs;ioni corporis in Flui
dum ad partes anticas, hoc e&longs;t, ni&longs;i velocitas relativa qua Flui
dum irruit in corpus a tergo, æqualis fuerit velocitati qua cor
pus irruit in Fluidum, id e&longs;t, ni&longs;i velocitas ab&longs;oluta Fluidi re
currentis duplo major fuerit quam velocitas ab&longs;oluta Fluidi pro
pul&longs;i; quod fieri nequit. Nullo igitur modo tolli pote&longs;t Flui
dorum re&longs;i&longs;tentia, quæ oritur ab corundem den&longs;itate & vi in
ertiæ. Itaque concludendum erit; Fluidi cæle&longs;tis nullam e&longs;&longs;e
vim inertiæ, cum nulla &longs;it vis re&longs;i&longs;tendi: nullam e&longs;&longs;e vim qua
motus communicetur, cum nulla &longs;it vis inertiæ: nullam e&longs;&longs;e vim
qua mutatio quælibet vel corporibus &longs;ingulis vel pluribus indu
catur, cum nulla &longs;it vis qua motus communicetur: nullam e&longs;&longs;e
omnino efficaciam, cum nulla &longs;it facultas mutationem quamlibet
inducendi. Quidni ergo hanc Hypothe&longs;in, quæ fundamento
plane de&longs;tituitur, quæque naturæ rerum explicandæ ne minimum
quidem in&longs;ervit, inepti&longs;&longs;imam vocare liceat & Philo&longs;opho pror-Qui Cælos materia fluida repletos e&longs;&longs;e volunt,
hanc vero non inertem e&longs;&longs;e &longs;tatuunt; Hi verbis tollunt Vacuum,
re ponunt. Nam cum huju&longs;modi materia fluida ratione nulla
&longs;ecerni po&longs;&longs;it ab inani Spatio; di&longs;putatio tota fit de rerum no
minibus, non de naturis. Quod &longs;i aliqui &longs;int adeo u&longs;Q.E.D.
diti Materiæ, ut Spatium a corporibus vacuum nullo pacto ad
mittendum credere velint; videamus quo tandem oporteat illos
pervenire.
Vel enim dicent hanc, quam confingunt, Mundi per omnia
pleni con&longs;titutionem ex voluntate Dei profectam e&longs;&longs;e, propter
eum finem, ut operationibus Naturæ &longs;ub&longs;idium præ&longs;ens haberi
po&longs;&longs;et ab Æthere &longs;ubtili&longs;&longs;imo cuncta permeante & implente;
quod tamen dici non pote&longs;t, &longs;iquidem jam o&longs;ten&longs;um e&longs;t ex Co
metarum phænomenis, nullam e&longs;&longs;e hujus Ætheris efficaciam: vel
dicent ex voluntate Dei profectam e&longs;&longs;e, propter finem aliquem
ignotum; quod neQ.E.D.ci debet, &longs;iquidem diver&longs;a Mundi con
&longs;titutio eodem argumento pariter &longs;tabiliri po&longs;&longs;et: vel denique
non dicent ex voluntate Dei profectam e&longs;&longs;e, &longs;ed ex nece&longs;&longs;itate
quadam Naturæ. Tandem igitur delabi oportet in fæces &longs;ordi
das Gregis impuri&longs;&longs;imi. Hi &longs;unt qui &longs;omniant Fato univer&longs;a
regi, non Providentia; Materiam ex nece&longs;&longs;itate &longs;ua &longs;emper & ubi
que extiti&longs;&longs;e, infinitam e&longs;&longs;e & æternam. Quibus po&longs;itis, erit
etiam undiquaque uniformis: nam varietas formarum cum nece&longs;
&longs;itate omnino pugnat. Erit etiam immota: nam &longs;i nece&longs;&longs;ario
moveatur in plagam aliquam determinatam, cum determinata ali
qua velocitate; pari nece&longs;&longs;itate movebitur in plagam diver&longs;am
cum diver&longs;a velocitate; in plagas autem diver&longs;as, cum diver&longs;is
velocitatibus, moveri non pote&longs;t; oportet igitur immotam e&longs;&longs;e.
Neutiquam profecto potuit oriri Mundus, pulcherrima forma
rum & motuum varietate di&longs;tinctus, ni&longs;i ex liberrima voluntate
cuncta providentis & gubernantis Dei.
Ex hoc igitur fonte promanarunt illæ omnes quæ dicuntur
Naturæ leges: in quibus multa &longs;ane &longs;apienti&longs;&longs;imi con&longs;ilii, nulla
nece&longs;&longs;itatis apparent ve&longs;tigia. Has proinde non ab incertis con
jecturis petere, &longs;ed Ob&longs;ervando atque Experiendo addi&longs;cere de
bemus. Qui veræ Phy&longs;icæ principia Lege&longs;que rerum, &longs;ola men
tis vi & interno rationis lumine fretum, invenire &longs;e po&longs;&longs;e confi
dit; hunc oportet vel &longs;tatuere Mundum ex nece&longs;&longs;itate fui&longs;le, Le
ge&longs;que propo&longs;itas ex eadem nece&longs;&longs;itate &longs;equi; vel &longs;i per volun
tatem Dei con&longs;titutus &longs;it ordo Naturæ, &longs;e tamen, homuncionem Sana om
nis & vera Philo&longs;ophia fundatur in Phænomenis rerum: quæ &longs;i
nos vel invitos & reluctantes ad huju&longs;modi principia deducunt,
in quibus clari&longs;&longs;ime cernuntur Con&longs;ilium optimum & Dominium
&longs;ummum &longs;apienti&longs;&longs;imi & potenti&longs;&longs;imi Entis; non erunt hæc ideo
non admittenda principia, quod quibu&longs;dam for&longs;an hominibus
minus grata &longs;int futura. His vel Miracula vel Qualitates occultæ
dicantur, quæ di&longs;plicent: verum nomina malitio&longs;e indita non &longs;unt
ip&longs;is rebus vitio vertenda; ni&longs;i illud fateri tandem velint, utique
debere Philo&longs;ophiam in Athei&longs;mo fundari. Horum hominum
gratia non erit labefactanda Philo&longs;ophia, &longs;iquidem rerum ordo
non vult immutari.
Obtinebit igitur apud probos & æquos Judices præ&longs;tanti&longs;&longs;ima
Philo&longs;ophandi ratio, quæ fundatur in Experimentis & Ob&longs;erva
tionibus. Huic vero, dici vix poterit, quanta lux accedat, quanta
dignitas, ab hoc Opere præclaro Illu&longs;tri&longs;&longs;imi no&longs;tri Auctoris; cujus
eximiam ingenii felicitatem, difficillima quæque Problemata eno
dantis, & ad ea porro pertingentis ad quæ nec &longs;pes erat humanam
mentem a&longs;&longs;urgere potui&longs;&longs;e, merito admirantur & &longs;u&longs;piciunt qui
cunque paulo profundius in hi&longs;ce rebus ver&longs;ati &longs;unt. Clau&longs;tris
ergo referatis, aditum Nobis aperuit ad pulcherrima rerum my
&longs;teria. Sy&longs;tematis Mundani compagem eleganti&longs;&longs;imam ita tan
dem patefecit & penitius per&longs;pectandam dedit; ut nec ip&longs;e, &longs;i
nunc revivi&longs;ceret, Rex
gratiam in ea de&longs;ideraret. Itaque Naturæ maje&longs;tatem propius jam
licet intueri, & dulci&longs;&longs;ima contemplatione frui, Conditorem vero
ac Dominum Univer&longs;orum impen&longs;ius colere & venerari, qui fructus
e&longs;t Philo&longs;ophiæ multo uberrimus. Cæcum e&longs;&longs;e oportet, qui ex
optimis & &longs;apienti&longs;&longs;imis rerum &longs;tructuris non &longs;tatim videat Fabri
catoris Omnipotentis infinitam &longs;apientiam & bonitatem: in&longs;anum,
qui profiteri nolit.
Extabit igitur Eximium NEWTONI Opus adver&longs;us Atheorum
impetus muniti&longs;&longs;imum præ&longs;idium: neque enim alicunde felicius,
quam ex hac pharetra, contra impiam Catervam tela depromp&longs;eris.
Hoc &longs;en&longs;it pridem, & in pereruditis Concionibus Anglice Latineque
editis, primus egregie demon&longs;travit Vir in omni Literarum genere
præclarus idemque bonarum Artium fautor eximius RICHARDUS
BENTLEIUS, Sæculi &longs;ui & Academiæ no&longs;træ magnum Orna
mentum, Collegii no&longs;tri
tegerrimus. Huic ego me pluribus nominibus ob&longs;trictum fateri
denegabis. Is enim, cum a longo tempore Celeberrimi Auctoris
amicitia intima frueretur, (qua etiam apud Po&longs;teros cen&longs;eri non
minoris æ&longs;timat, quam propriis Scriptis quæ literato orbi in de
liciis &longs;unt inclare&longs;cere) Amici &longs;imul famæ & &longs;cientiarum incre
mento con&longs;uluit. Itaque cum Exemplaria prioris Editionis rari&longs;
&longs;ima admodum & immani pretio coemenda &longs;upere&longs;&longs;ent; &longs;ua&longs;it Ille
crebris efflagitationibus & tantum non objurgando perpulit deNI
que Virum Præ&longs;tanti&longs;&longs;imum, nec mode&longs;tia minus quam eruditi
one &longs;umma In&longs;ignem, ut novam hanc Operis Editionem, per om
nia elimatam denuo & egregiis in&longs;uper acce&longs;&longs;ionibus ditatam, &longs;uis
&longs;umptibus & au&longs;piciis prodire pateretur: Mihi vero, pro jure
&longs;uo, pen&longs;um non ingratum demandavit, ut quam po&longs;&longs;et emendate
id fieri curarem.
Maii 12. 1713.
ROGERUS COTES Collegii
A&longs;tronomiæ & Philo&longs;ophiæ Experimentalis
Profe&longs;&longs;or
TOTIUS OPERIS.
PAG.
DEFINITIONES. 1
AXIOMATA, SIVE LEGES MOTUS. 12
SECT. I.
rum.
SECT. II.
SECT. III.
cis.
SECT. IV.
& Hyperbolieorum ex Umbilico dato.
SECT. V.
SECT. VI.
SECT. VII.
SECT. VII.
quibu&longs;cunque centripetis agitata revolvuntur.
SECT. IX.
Motu Ap&longs;idum.
SECT. X.
Funependulorum Motu reciproco.
SECT. XI.
tentium.
SECT. XII.
SECT. XIII.
vis.
SECT. XIV.
tripetis ad &longs;ingulas Magni alicujus corporis partes ten
dentibus agitantur.
SECT. I.
Velocitatis.
SECT. II.
tione Velocitatis.
SECT. III.
Velocitatis, partim in eju&longs;dem ratione duplicata.
SECT. IV.
253
SECT. V.
dro&longs;tatica.
SECT. VI.
272
SECT. VII.
SECT. VIII.
SECT. IX.
REGULÆ PHILOSOPHANDI 357
PHÆNOMENA 359
PROPOSITIONES 362
SCHOLIUM GENERALE. 481
NATURALIS
Principia
MATHEMATICA.
Magnitudine conjunctim.
AER, den&longs;itate duplicata, in &longs;patio etiam duplicato fit qua
druplus; in triplicato &longs;extuplus. Idem intellige de Nive &
Pulveribus per compre&longs;&longs;ionem vel liquefactionem conden
&longs;atis. Et par e&longs;t ratio corporum omnium, quæ per cau&longs;as qua&longs;cun
Q.E.D.ver&longs;imode conden&longs;antur. Medii interea, &longs;i quod fuerit, in
ter&longs;titia partium libere pervadentis, hic nullam rationem habeo.
Hanc autem Quantitatem &longs;ub nomine Corporis vel Ma&longs;&longs;æ in &longs;e
quentibus pa&longs;&longs;im intelligo. Innote&longs;cit ea per corporis cuju&longs;que
Pondus. Nam Ponderi proportionalem e&longs;&longs;e reperi per experi
menta Pendulorum accurati&longs;&longs;ime in&longs;tituta, uti po&longs;thac docebitur.
titate Materiæ conjunctim.
Motus totius e&longs;t &longs;umma motuum in partibus &longs;ingulis; adeoque
in corpore duplo majore æquali cum velocitate duplus e&longs;t, & du
pla cum velocitate quadruplus.
quantum in &longs;e e&longs;t, per&longs;everat in &longs;tatu &longs;uo vel quie&longs;cendi vel
movendi uniformiter in directum.
Hæc &longs;emper proportionalis e&longs;t &longs;uo corpori, neQ.E.D.ffert quic
quam ab Inertia ma&longs;&longs;æ, ni&longs;i in modo concipiendi. Per inertiam
materiæ, fit ut corpus omne de &longs;tatu &longs;uo vel quie&longs;cendi vel moven
di difficulter deturbetur. Unde etiam vis in&longs;ita nomine &longs;ignifican
ti&longs;&longs;imo Vis Inertiæ dici po&longs;&longs;it. Exercet vero corpus hanc vim &longs;olum
modo in mutatione &longs;tatus &longs;ui per vim aliam in &longs;e impre&longs;&longs;am facta;
re&longs;i&longs;tentia, quatenus corpus ad con&longs;ervandum &longs;tatum &longs;uum relucta
tur vi impre&longs;&longs;æ; impetus, quatenus corpus idem, vi re&longs;i&longs;tentis ob
&longs;taculi difficulter cedendo, conatur &longs;tatum ejus mutare. Vulgus
re&longs;i&longs;tentiam quie&longs;centibus & impetum moventibus tribuit: &longs;ed mo
tus & quies, uti vulgo concipiuntur, re&longs;pectu &longs;olo di&longs;tinguuntur
ab invicem;
&longs;centia &longs;pectantur.
vel quie&longs;cendi vel movendi uniformiter in directum.
Con&longs;i&longs;tit hæc vis in actione &longs;ola, neque po&longs;t actionem permanet
in corpore. Per&longs;everat enim corpus in &longs;tatu omni novo per &longs;olam
vim inertiæ. E&longs;t autem vis impre&longs;&longs;a diver&longs;arum originum, ut ex
Ictu, ex Pre&longs;&longs;ione, ex vi Centripeta.
Centrum undique trahuntur, impelluntur, vel utcunque tendunt.
Hujus generis e&longs;t Gravitas, qua corpora tendunt ad centrum ter
ræ; Vis Magnetica, qua ferrum petit magnetem; & Vis illa,
neis, & in lineis curvis revolvi coguntur. Lapis, in funda circum-
di&longs;tendit,
mittitur, avolat. Vim conatui illi contrariam, qua funda lapidem
in manum perpetuò retrahit & in orbe retinet, quoniam in manum
ceu orbis centrum dirigitur, Centripetam appello. Et par e&longs;t ratio
corporum omnium, quæ in gyrum aguntur. Conantur ea omnia a
centris orbium recedere; & ni&longs;i ad&longs;it vis aliqua conatui i&longs;ti contra
ria, qua cohibeantur & in orbibus retineantur, quamQ.E.I.eò Centri
petam appello, abibunt in rectis lineis uniformi cum motu. Pro
jectile, &longs;i vi Gravitatis de&longs;titueretur, non deflecteretur in terram, &longs;ed
in linea recta abiret in cælos; idque uniformi cum motu, &longs;i modo
aeris re&longs;i&longs;tentia tolleretur. Per gravitatem &longs;uam retrahitur a cur&longs;u
rectilineo & in terram perpetuo flectitur, idque magis vel minus
pro gravitate &longs;ua & velocitate motus. Quo minor erit ejus gravitas pro quantitate materiæ vel major &c.
vel major velocitas quacum projicitur, eo minus deviabit a cur&longs;u
rectilineo & longius perget. Si Globus plumbeus, data cum velo
citate &longs;ecundum lineam horizontalem a montis alicujus vertice vi
pulveris tormentarii projectus, pergeret in linea curva ad di&longs;tantiam
duorum milliarium, priu&longs;quam in terram decideret: hic dupla cum
velocitate qua&longs;i duplo longius pergeret, & decupla cum velocitate
qua&longs;i decuplo longius: &longs;i modo aeris re&longs;i&longs;tentia tolleretur. Et augendo
velocitatem augeri po&longs;&longs;et pro lubitu di&longs;tantia in quam projiceretur,
& minui curvatura lineæ quam de&longs;criberet, ita ut tandem caderet
ad di&longs;tantiam graduum decem vel triginta vel nonaginta; vel etiam
ut terram totam circuiret priu&longs;quam caderet; vel denique ut in
terram nunquam caderet, &longs;ed in cælos abiret & motu abeundi per
geret in infinitum. Et eadem ratione, qua Projectile vi gravitatis
in orbem flecti po&longs;&longs;et & terram totam circuire, pote&longs;t & Luna vel
vi gravitatis, &longs;i modo gravis &longs;it, vel alia quacunque vi, qua in ter
ram urgeatur, retrahi &longs;emper a cur&longs;u rectilineo terram ver&longs;us, &
in orbem &longs;uum flecti: & ab&longs;que tali vi Luna in orbe &longs;uo retineri
non pote&longs;t. Hæc vis, &longs;i ju&longs;to minor e&longs;&longs;et, non &longs;atis flecteret Lunam
de cur&longs;u rectilineo: &longs;i ju&longs;to major, plus &longs;atis flecteret, ac de orbe
terram ver&longs;us deduceret. Requiritur quippe, ut &longs;it ju&longs;tæ magnitudinis:
& Mathematieorum e&longs;t invenire Vim, qua corpus in dato quovis
orbe data cum velocitate accurate retineri po&longs;&longs;it; & vici&longs;&longs;im inve
nire Viam curvilineam, in quam corpus e dato quovis loco data
cum velocitate egre&longs;&longs;um a data vi flectatur. E&longs;t autem vis hujus cen
tripetæ Quantitas trium generum, Ab&longs;oluta, Acceleratrix, & Motrix.
ES.
pro Efficacia cau&longs;æ eam propagantis a centro per regiones in circuitu.
Ut vis Magnetica pro mole magnetis vel inten&longs;ione virtutis major
in uno magnete, minor in alio.
proportionalis, quam dato tempore generat.
Uti Virtus magnetis eju&longs;dem major in minori di&longs;tantia, minor
in majori: vel vis Gravitans major in vallibus, minor in cacumiNI
bus præaltorum montium, atque adhuc minor (ut po&longs;thac patebit)
in majoribus di&longs;tantiis a globo terræ; in æqualibus autem di&longs;tan
tiis eadem undique, propterea quod corpora omnia cadentia (gra
via an levia, magna an parva) &longs;ublata Aeris re&longs;i&longs;tentia, æqualiter
accelerat.
Motui, quem dato tempore generat.
Uti Pondus majus in majore corpore, minus in minore; inque
corpore eodem majus prope terram, minus in cælis. Hæc Quantitas
e&longs;t corporis totius centripetentia &longs;eu propen&longs;io in centrum, & (ut ita
dicam) Pondus; & innote&longs;cit &longs;emper per vim ip&longs;i contrariam & æ
qualem, qua de&longs;cen&longs;us corporis impediri pote&longs;t.
Ha&longs;ce virium quantitates brevitatis gratia nominare licet vires
motrices, acceleratrices, & ab&longs;olutas; & di&longs;tinctionis gratia referre ad
Corpora, centrum petentia, ad corporum Loca, & ad Centrum virium:
nimirum vim motricem ad Corpus, tanquam conatum & propen&longs;io
nem totius in centrum ex propen&longs;ionibus omnium partium compo&longs;i
tam; & vim acceleratricem ad Locum corporis, tanquam efficaciam
quandam, de centro per loca &longs;ingula in circuitu diffu&longs;am, ad movenda
corpora quæ in ip&longs;is &longs;unt; vim autem ab&longs;olutam ad Centrum, tan
quam cau&longs;a aliqua præditum, &longs;ine qua vires motrices non propa
gantur per regiones in circuitu; &longs;ive cau&longs;a illa &longs;it corpus aliquod
centrale (quale e&longs;t Magnes in centro vis magneticæ, vel Terra in Mathe
maticus duntaxat e&longs;t hic conceptus. Nam virium cau&longs;as & &longs;edes phy
&longs;icas jam non expendo.
E&longs;t igitur vis acceleratrix ad vim motricem ut celeritas ad mo
tum. Oritur enim quantitas motus ex celeritate ducta in quanti
tatem materiæ, & vis motrix ex vi acceleratrice ducta in quantita
tem eju&longs;dem materiæ. Nam &longs;umma actionum vis acceleratricis in
&longs;ingulas corporis particulas e&longs;t vis motrix totius. Unde juxta
&longs;uperficiem Terræ, ubi gravitas acceleratrix &longs;eu vis gravitans in
corporibus univer&longs;is eadem e&longs;t, gravitas motrix &longs;eu pondus e&longs;t ut
corpus: at &longs;i in regiones a&longs;cendatur ubi gravitas acceleratrix fit mi
nor, pondus pariter minuetur, eritque &longs;emper ut corpus in
gravitatem acceleratricem ductum. Sic in regionibus ubi gravitas
acceleratrix duplo minor e&longs;t, pondus corporis duplo vel triplo
minoris erit quadruplo vel &longs;extuplo minus.
Porro attractiones & impul&longs;us eodem &longs;en&longs;u acceleratrices & mo
trices nomino. Voces autem Attractionis, Impul&longs;us, vel Propen
&longs;ionis cuju&longs;cunQ.E.I. centrum, indifferenter & pro &longs;e mutuo pro
mi&longs;cue u&longs;urpo; has vires non Phy&longs;ice &longs;ed Mathematice tantum con
&longs;iderando. Unde caveat lector, ne per huju&longs;modi voces cogitet me
&longs;peciem vel modum actionis cau&longs;amve aut rationem Phy&longs;icam ali
cubi definire, vel centris (quæ &longs;unt puncta Mathematica) vires
vere & Phy&longs;ice tribuere; &longs;i forte aut centra trahere, aut vires cen
trorum e&longs;&longs;e dixero.
Hactenus voces minus notas, quo &longs;en&longs;u in &longs;equentibus acci
piendæ &longs;int, explicare vi&longs;um e&longs;t. Nam Tempus, Spatium, Locum
& Motum, ut omnibus noti&longs;&longs;ima, non definio. Notandum tamen, quod
vulgus quantitates ha&longs;ce non aliter quam ex relatione ad &longs;en&longs;ibilia
concipiat. Et inde oriuntur præjudicia quædam, quibus tollendis
convenit ea&longs;dem in ab&longs;olutas & relativas, veras & apparentes, ma
thematicas & vulgares di&longs;tingui.
I.
Tempus Ab&longs;olutum, verum, & mathematicum, in &longs;e & natura
&longs;ua
nomine dicitur Duratio: Relativum, apparens, & vulgare e&longs;t &longs;en&longs;ibilis
& externa quævis Durationis per motum men&longs;ura (&longs;eu accurata
&longs;eu inæquabilis) qua vulgus vice veri temporis utitur; ut Hora,
Dies, Men&longs;is, Annus.
II.
Spatium Ab&longs;olutum, natura &longs;ua ab&longs;que relatione ad externum
quodvis, &longs;emper manet &longs;imilare & immobile: Relativum e&longs;t &longs;patii
hujus men&longs;ura &longs;eu dimen&longs;io quælibet mobilis, quæ a &longs;en&longs;ibus no&longs;tris
per &longs;itum &longs;uum ad corpora definitur, & a vulgo pro &longs;patio immo
bili u&longs;urpatur: uti dimen&longs;io &longs;patii &longs;ubterranei, aerei vel cæle&longs;tis
definita per &longs;itum &longs;uum ad Terram. Idem &longs;unt &longs;patium ab&longs;olutum
& relativum, &longs;pecie & magnitudine; &longs;ed non permanent idem &longs;em
per numero. Nam &longs;i Terra, verbi gratia, movetur; &longs;patium Aeris
no&longs;tri, quod relative & re&longs;pectu Terræ &longs;emper manet idem, nunc
erit una pars &longs;patii ab&longs;oluti in quam Aer tran&longs;it, nunc alia pars ejus;
& &longs;ic ab&longs;olute mutabitur perpetuo.
III.
Locus e&longs;t pars &longs;patii quam corpus occupat,
&longs;patii vel Ab&longs;olutus vel Relativus. Pars, inquam, &longs;patii; non Situs
corporis, vel Superficies ambiens. Nam &longs;olidorum æqualium
æquales &longs;emper &longs;unt loci; Superficies autem ob di&longs;&longs;imilitudinem
figurarum ut plurimum inæquales &longs;unt; Situs vero proprie loquen
do quantitatem non habent,
loeorum. Motus totius idem e&longs;t cum &longs;umma motuum partium,
hoc e&longs;t, tran&longs;latio totius de &longs;uo loco eadem e&longs;t cum &longs;umma tran&longs;la
tionum partium de locis &longs;uis;
loeorum partium, & propterea internus & in corpore toto.
IV.
Motus Ab&longs;olutus e&longs;t tran&longs;latio corporis de loco ab&longs;oluto in
locum ab&longs;olutum, Relativus de relativo in relativum. Sic in navi
quæ velis pa&longs;&longs;is fertur, relativus corporis Locus e&longs;t navigii regio illa
in qua corpus ver&longs;atur, &longs;eu cavitatis totius pars illa quam corpus
implet,
perman&longs;io corporis in eadem illa navis regione vel parte cavita
tis. At quies Vera e&longs;t perman&longs;io corporis in eadem parte &longs;patii
illius immoti in qua navis ip&longs;a una cum cavitate &longs;ua & contentis
univer&longs;is movetur. Unde &longs;i Terra vere quie&longs;cit, corpus quod rela
tive quie&longs;cit in navi, movebitur vere & ab&longs;olute ea cum velocitate
qua navis movetur in Terra. Sin Terra etiam movetur; orietur
verus & ab&longs;olutus corporis motus, partim ex Terræ motu vero in
&longs;patio immoto, partim ex navis motu relativo in Terra: & &longs;i cor
pus etiam movetur relative in navi; orietur verus ejus motus, par
tim ex vero motu Terræ in &longs;patio immoto, partim ex relativis mo
tibus tum navis in Terra, tum corporis in navi; & ex his motibus
relativis orietur corporis motus relativus in Terra. Ut &longs;i Terræ pars
illa, ubi navis ver&longs;atur, moveatur vere in orientem cum velocitate
partium 10010; & velis
velocitate partium decem; Nauta autem ambulet in navi ori-
ab&longs;olute in &longs;patio immoto cum velocitatis partibus 10001 in o
rientem, & relative in terra occidentem ver&longs;us cum velocitatis
partibus novem.
Tempus Ab&longs;olutum a relativo di&longs;tinguitur in A&longs;tronomia per Æ
quationem temporis vulgi. Inæquales enim &longs;unt dies Naturales,
qui vulgo tanquam æquales promen&longs;ura temporis habentur. Hanc
inæqualitatem corrigunt A&longs;tronomi, ut ex veriore tempore
motus cæle&longs;tes. Po&longs;&longs;ibile e&longs;t, ut nullus &longs;it motus æquabilis quo
Tempus accurate men&longs;uretur. Accelerari & retardari po&longs;&longs;unt motus
omnes, &longs;ed fluxus temporis Ab&longs;oluti mutari nequit. Eadem e&longs;t du
ratio &longs;eu per&longs;everantia exi&longs;tentiæ rerum; &longs;ive motus &longs;int celeres, &longs;ive
tardi, &longs;ive nulli: proinde hæc a men&longs;uris &longs;uis &longs;en&longs;ibilibus merito
di&longs;tinguitur, & ex ii&longs;dem colligitur per Æquationem A&longs;tronomi
cam. Hujus autem æquationis in determinandis Phænomenis ne
ce&longs;&longs;itas, tum per experimentum Horologii O&longs;cillatorii, tum etiam
per eclip&longs;es Satellitum Jovis evincitur.
Ut partium Temporis ordo e&longs;t immutabilis, &longs;ic etiam ordo par
tium Spatii. Moveantur hæ de locis &longs;uis, & movebuntur (ut ita
dicam) de &longs;eip&longs;is. Nam tempora & &longs;patia &longs;unt &longs;ui ip&longs;orum &
rerum omnium qua&longs;i Loca. In Tempore quoad ordinem &longs;ucce&longs;&longs;i
onis; in Spatio quoad ordinem &longs;itus locantur univer&longs;a. De illo
rum e&longs;&longs;entia e&longs;t ut &longs;int Loca: & loca primaria moveri ab&longs;urdum
e&longs;t. Hæc &longs;unt igitur ab&longs;oluta Loca; & &longs;olæ tran&longs;lationes de his lo
cis &longs;unt ab&longs;oluti Motus.
Verum quoniam hæ Spatii partes videri nequeunt, & ab invi
cem per &longs;en&longs;us no&longs;tros di&longs;tingui; earum vice adhibemus men&longs;uras
&longs;en&longs;ibiles. Ex po&longs;itionibus enim & di&longs;tantiis rerum a corpore ali
quo, quod &longs;pectamus ut immobile, de&longs;inimus loca univer&longs;a: deinde
etiam & omnes motus æ&longs;timamus cum re&longs;pectu ad prædicta loca,
quatenus corpora ab ii&longs;dem transferri concipimus. Sic vice loco
rum & motuum ab&longs;olutorum relativis utimur; nec incommode in
rebus humanis: in Philo&longs;ophicis autem ab&longs;trahendum e&longs;t a &longs;en&longs;ibus.
Fieri etenim pote&longs;t, ut nullum revera quie&longs;cat corpus, ad quod loca
motu&longs;que referantur.
Di&longs;tinguuntur autem Quies & Motus ab&longs;oluti & relativi ab invi
cem per Proprietates &longs;uas & Cau&longs;as & Effectus. Quietis proprietas
e&longs;t, quod corpora vere quie&longs;centia quie&longs;cunt inter &longs;e. Ideoque
cum po&longs;&longs;ibile &longs;it, ut corpus aliquod in regionibus Fixarum, aut longe
ultra, quie&longs;cat ab&longs;olute; &longs;ciri autem non po&longs;&longs;it ex &longs;itu corporum
ad invicem in regionibus no&longs;tris, horumne aliquod ad longin-
&longs;itu inter &longs;e definiri nequit.
Motus proprietas e&longs;t, quod partes, quæ datas &longs;ervant po&longs;itiones
ad tota, participant motus eorundem totorum. Nam Gyrantium
partes omnes conantur recedere ab axe motus, & Progredientium
impetus oritur ex conjuncto impetu partium &longs;ingularum. Motis
igitur corporibus ambientibus, moventur quæ in ambientibus rela
tive quie&longs;cunt. Et propterea motus verus & ab&longs;olutus definiri ne
quit per tran&longs;lationem e vicinia corporum, quæ tanquam quie&longs;cen
tia &longs;pectantur. Debent enim corpora externa non &longs;olum tanquam qui
e&longs;centia &longs;pectari, &longs;ed etiam vere quie&longs;cere. Alioquin inclu&longs;a om
nia, præter tran&longs;lationem e vicinia ambientium, participabunt
etiam ambientium motus veros; & &longs;ublata illa tran&longs;latione non
vere quie&longs;cent, &longs;ed tanquam quie&longs;centia &longs;olummodo &longs;pectabun
tur. Sunt enim ambientia ad inclu&longs;a, ut totius pars exterior ad
partem interiorem, vel ut cortex ad nucleum. Moto autem cor
tice, nucleus etiam,
totius movetur.
Præcedenti proprietati affinis e&longs;t, quod moto Loco movetur una
Locatum: adeoque corpus, quod de loco moto movetur, participat
etiam loci &longs;ui motum. Motus igitur omnes, qui de locis motis
fiunt, &longs;unt partes &longs;olummodo motuum integrorum & ab&longs;olutorum:
& motus omnis integer componitur ex motu corporis de loco &longs;uo
primo, & motu loci hujus de loco &longs;uo, & &longs;ic deinceps; u&longs;Q.E.D.m
perveniatur ad locum immotum, ut in exemplo Nautæ &longs;upra me
morato. Unde motus integri & ab&longs;oluti non ni&longs;i per loca immota
definiri po&longs;&longs;unt: & propterea hos ad loca immota, relativos ad mo
bilia &longs;upra retuli. Loca autem immota non &longs;unt, ni&longs;i quæ omnia
ab infinito in infinitum datas &longs;ervant po&longs;itiones ad invicem; atque
adeo &longs;emper manent immota, &longs;patiumque con&longs;tituunt quod Immo
bile appello.
Cau&longs;æ, quibus motus veri & relativi di&longs;tinguuntur ab invicem,
&longs;unt Vires in corpora impre&longs;&longs;æ ad motum generandum. Motus
verus nec generatur nec mutatur, ni&longs;i per vires in ip&longs;um corpus mo
tum impre&longs;&longs;as: at motus relativus generari & mutari pote&longs;t
viribus impre&longs;&longs;is in hoc corpus. Sufficit enim ut imprimantur in
alia &longs;olum corpora ad quæ fit relatio, ut iis cedentibus mutetur
relatio illa in qua hujus quies vel motus relativus con&longs;i&longs;tit. Rur
&longs;um motus verus a viribus in corpus motum impre&longs;&longs;is &longs;emper muta
tur; at motus relativus ab his viribus non mutatur nece&longs;&longs;ario. Nam
&longs;i eædem vires in alia etiam corpora, ad quæ &longs;it relatio, &longs;ic impri-
motus relativus con&longs;i&longs;tit. Mutari igitur pote&longs;t motus omnis relati
vus ubi verus con&longs;ervatur, & con&longs;ervari ubi verus mutatur; & prop
terea motus verus in eju&longs;modi relationibus minime con&longs;i&longs;tit.
Effectus quibus motus ab&longs;oluti & relativi di&longs;tinguuntur ab invi
cem, &longs;unt vires recedendi ab axe motus circularis. Nam in motu
circulari nude relativo hæ vires nullæ &longs;unt, in vero autem & ab&longs;o
luto majores vel minores pro quantitate motus. Si pendeat &longs;itula
a filo prælongo, agaturque perpetuo in orbem, donec filum a con
tor&longs;ione admodum rige&longs;cat, dein impleatur aqua, & una cum aqua
quie&longs;cat; tum vi aliqua &longs;ubitanea agatur motu contrario in orbem,
& filo &longs;e relaxante, diutius per&longs;everet in hoc motu; &longs;uperficies a
quæ &longs;ub initio plana erit, quemadmodum ante motum va&longs;is: at
po&longs;tquam, vi in aquam paulatim impre&longs;&longs;a, effecit vas, ut hæc quoque
&longs;en&longs;ibiliter revolvi incipiat; recedet ip&longs;a paulatim a medio, a&longs;cen
detque ad latera va&longs;is, figuram concavam induens, (ut ip&longs;e exper
tus &longs;um) & incitatiore &longs;emper motu a&longs;cendet magis & magis, do
nec revolutiones in æqualibus cum va&longs;e temporibus peragendo,
quie&longs;cat in eodem relative. Indicat hic a&longs;cen&longs;us conatum rece
dendi ab axe motus, & per talem conatum innote&longs;cit & men&longs;ura
tur motus aquæ circularis verus & ab&longs;olutus, motuique relativo
hic omnino contrarius. Initio, ubi maximus erat aquæ motus rela
tivus in va&longs;e, motus ille nullum excitabat conatum recedendi ab
axe: aqua non petebat circumferentiam a&longs;cendendo ad latera va
&longs;is, &longs;ed plana manebat, & propterea motus illius circularis verus
nondum inceperat. Po&longs;tea vero, ubi aquæ motus relativus decre
vit, a&longs;cen&longs;us ejus ad latera va&longs;is indicabat conatum recedendi ab
axe; atque hic conatus mon&longs;trabat motum illius circularem verum
perpetuo cre&longs;centem, ac tandem maximum factum ubi aqua quie
&longs;cebat in va&longs;e relative. Igitur conatus i&longs;te non pendet a tran&longs;la
tione aquæ re&longs;pectu corporum ambientium, & propterea motus cir
cularis verus per tales tran&longs;lationes definiri nequit. Unicus e&longs;t cor
poris cuju&longs;que revolventis motus vere circularis, conatui unico tan
quam proprio & adæquato effectui re&longs;pondens: motus autem rela
tivi pro variis relationibus ad externa innumeri &longs;unt; & relationum
in&longs;tar, effectibus veris omnino de&longs;tituuntur, ni&longs;i quatenus verum
illum & unicum motum participant. Unde & in Sy&longs;temate eorum
qui Cælos no&longs;tros infra Cælos Fixarum in orbem revolvi volunt,
& Planetas &longs;ecum deferre; &longs;ingulæ Cælorum partes, & Planetæ
qui relative quidem in Cælis &longs;uis proximis quie&longs;cunt, moven-Mutant enim po&longs;itiones &longs;uas ad invicem (&longs;ecus quam fit
in vere quie&longs;centibus) unaque cum cælis delati participant eorum
motus, & ut partes revolventium totorum, ab eorum axibus rece
dere conantur.
ES.
Igitur quantitates relativæ non &longs;unt eæ ip&longs;æ quantitates, quarum
nomina præ &longs;e ferunt, &longs;ed earum men&longs;uræ illæ &longs;en&longs;ibiles (veræ an
errantes) quibus vulgus loco quantitatum men&longs;uratarum utitur. At
&longs;i ex u&longs;u definiendæ &longs;unt verborum &longs;ignificationes; per nomina il
la Temporis, Spatii, Loci & Motus proprie intelligendæ erunt hæ
men&longs;uræ; & &longs;ermo erit in&longs;olens & pure Mathematicus, &longs;i quantita
tes men&longs;uratæ hic intelligantur. Proinde vim inferunt Sacris
Literis, qui voces ha&longs;ce de quantitatibus men&longs;uratis ibi interpre
tantur. Neque minus contaminant Mathe&longs;in & Philo&longs;ophiam,
qui quantitates veras cum ip&longs;arum relationibus & vulgaribus men
furis confundunt.
Motus quidem veros corporum &longs;ingulorum cogno&longs;cere, & ab
apparentibus actu di&longs;criminare, difficillimum. e&longs;t propterea quod
partes &longs;patii illius immobilis, in quo corpora vere moventur, non
incurrunt in &longs;en&longs;us. Cau&longs;a tamen non e&longs;t pror&longs;us de&longs;perata.
Nam
&longs;uppetunt argumenta, partim ex motibus apparentibus qui &longs;unt
motuum verorum differentiæ, partim ex viribus quæ &longs;unt mo
tuum verorum cau&longs;æ & effectus. Ut &longs;i globi duo, ad datam ab in
vicem di&longs;tantiam filo intercedente connexi, revolverentur circa
commune gravitatis centrum; innote&longs;ceret ex ten&longs;ione fili cona
tus globorum recedendi ab axe motus, & inde quantitas motus
circularis computari po&longs;&longs;et. Deinde &longs;i vires quælibet æquales in
alternas globorum facies ad motum circularem augendum vel mi
nuendum &longs;imul imprimerentur, innote&longs;ceret ex aucta vel diminuta
fili ten&longs;ione augmentum vel decrementum motus; & inde tandem
inveniri po&longs;&longs;ent facies globorum in quas vires imprimi deberent,
ut motus maxime augeretur; id e&longs;t, facies po&longs;ticæ, &longs;ive quæ in mo
tu circulari &longs;equuntur. Cognitis autem faciebus quæ &longs;equuntur,
& faciebus oppo&longs;itis quæ præcedunt, cogno&longs;ceretur determinatio
motus. In hunc modum inveniri po&longs;&longs;et & quantitas & determi
natio motus hujus circularis in vacuo quovis immen&longs;o, ubi nihil
extaret externum & &longs;en&longs;ibile quocum globi conferri po&longs;&longs;ent. Si
jam con&longs;tituerentur in &longs;patio illo corpora aliqua longinqua datam
inter &longs;e po&longs;itionem &longs;ervantia, qualia &longs;unt Stellæ Fixæ in regionibus
no&longs;tris: &longs;ciri quidem non po&longs;&longs;et ex relativa globorum tran&longs;latione
inter corpora, utrum his an illis tribuendus e&longs;&longs;et motus. At &longs;i
e&longs;&longs;e quam motus globorum requireret; concludere liceret mo
tum e&longs;&longs;e globorum, & corpora quie&longs;cere; & tum demum ex
tran&longs;latione globorum inter corpora, determinationem hujus
motus colligere. Motus autem veros ex eorum cau&longs;is, effecti
bus, & apparentibus differentiis colligere; & contra ex motibus
&longs;eu veris &longs;eu apparentibus eorum cau&longs;as & effectus, docebitur
fu&longs;ius in &longs;equentibus. Hunc enim in finem Tractatum &longs;equentem
compo&longs;ui.
SIVE
LEGES MOTUS.
formiter in directum, ni&longs;i quatenus a viribus impre&longs;&longs;is cogitur
&longs;tatum illum mutare.
PRojectilia per&longs;everant in motibus &longs;uis, ni&longs;i quatenus a re&longs;i
&longs;tentia aeris retardantur, & vi gravitatis impelluntur deor&longs;um.
Trochus, cujus partes cohærendo perpetuo retrahunt &longs;e&longs;e a mo
tibus rectilineis, non ce&longs;&longs;at rotari, ni&longs;i quatenus ab aere retardatur.
Majora autem Planetarum & Cometarum corpora motus &longs;uos &
progre&longs;&longs;ivos & circulares in &longs;patiis minus re&longs;i&longs;tentibus factos con
&longs;ervant diutius.
&longs;ecundum lineam rectam qua vis illa imprimitur.
Si vis aliqua motum quemvis generet; dupla duplum, tripla tri
plum generabit, &longs;ive &longs;imul & &longs;emel, &longs;ive gradatim & &longs;ucce&longs;&longs;ive im
pre&longs;&longs;a fuerit. Et hic motus (quoniam in eandem &longs;emper plagam
cum vi generatrice determinatur) &longs;i corpus antea movebatur, mo
tui ejus vel con&longs;piranti additur, vel contrario &longs;ubducitur, vel obli
quo oblique adjicitur, & cum eo &longs;ecundum utriu&longs;Q.E.D.termina
tionem componitur.
porum duorum actiones in &longs;e mutuo &longs;emper e&longs;&longs;e æquales & in par
tes contrarias dirigi.
Quicquid premit vel trahit alterum, tantundem ab eo premitur
vel trahitur. Si quis lapidem digito premit, premitur & hujus
digitus a lapide. Si equus lapidem funi alligatum trahit, retrahe
tur etiam & equus (ut ita dicam) æqualiter in lapidem: nam funis
utrinQ.E.D.&longs;tentus eodem relaxandi &longs;e conatu urgebit equum ver
&longs;us lapidem, ac lapidem ver&longs;us equum; tantumQ.E.I.pediet pro
gre&longs;&longs;um unius quantum promovet progre&longs;&longs;um alterius. Si corpus
aliquod in corpus aliud impingens, motum ejus vi &longs;ua quomodo
cunque mutaverit, idem quoque vici&longs;&longs;im in motu proprio eandem
mutationem in partem contrariam vi alterius ob æqualitatem pre&longs;
&longs;ionis mutuæ) &longs;ubibit. His actionibus æquales fiunt mutationes,
non velocitatum, &longs;ed motuum; &longs;cilicet in corporibus non aliunde
impeditis. Mutationes enim velocitatum, in contrarias itidem
partes factæ, quia motus æqualiter mutantur, &longs;unt corporibus re
ciproce proportionales. Obtinet etiam hæc Lex in Attractionibus,
ut in Scholio proximo probabitur.
pore de&longs;cribere, quo latera &longs;eparatis.
Si corpus dato tempore, vi &longs;ola
formi cum motu ab
&longs;ola
retur ab
rallelogrammum
que feretur id eodem tempore in diagonali ab
niam vis
Legem 11 nihil mutabit velocitatem accedendi ad lineam illam
a vi altera genitam. Accedet igitur corpus eodem tempore ad lineam
ris reperietur alicubi in linea illa
poris eju&longs;dem reperietur alicubi in linea
lineæ concur&longs;u Perget autem motu rectili
neo ab
E
obliquis
AD
& re&longs;olutio abunde confirmatur ex Mechanica.
Ut &longs;i de rotæ alicujus centro
ON
res ponderum ad movendam rotam: Per centrum
intervallorum
de&longs;cribatur circulus occurrens filo
rallela &longs;it
filorum puncta
an non affixa ad planum rotæ; pon
dera idem valebunt, ac &longs;i &longs;u&longs;pende
rentur a punctis
Ponderis autem
ta per lineam
in vires
tro nihil valet ad movendam rotam; vis autem altera
do radium
ter traheret radium
pondus
vim
ad
radii in directum po&longs;iti
&longs;tent in æquilibrio: quæ e&longs;t proprietas noti&longs;&longs;ima Libræ, Vectis, &
Axis in Peritrochio. Sin pondus alterutrum &longs;it majus quam in hac
ratione, erit vis ejus ad movendam rotam tanto major.
Quod &longs;i pondus
partim incumbat plano obliquo
rizonti, po&longs;terior plano
deor&longs;um tendens, exponatur per lineam
vires
lela; & pondas
planum
num
dus jam vicem præ&longs;tat plani &longs;ublati, tendetur illud eadem vi
qua planum antea urgebatur. Unde ten&longs;io fili hujus obliqui erit
ad ten&longs;ionem &longs;ili alterius perpendicularis
eoque &longs;i pondus
ratione reciproca minimarum di&longs;tantiarum &longs;uorum &longs;uorum
AM
valebunt ad rotam movendam, atque adeo &longs;e mutuo &longs;u&longs;tinebunt,
ut quilibet experiri pote&longs;t.
Pondus autem
habet cunei inter corporis fi&longs;&longs;i facies internas: & inde vires cunei
& mallei innote&longs;cunt: utpote cum vis qua pondus
&longs;ecundum lineam ut
urget planum alterum
&longs;imilem virium divi&longs;ionem colligitur; quippe quæ cuneus e&longs;t a ve
cte impul&longs;us. U&longs;us igitur Corollarii hujus lati&longs;&longs;ime patet, & late
patendo veritatem &longs;uam evincit; cum pendeat ex jam dictis Mecha
nica tota ab Auctoribus diver&longs;imode demon&longs;trata. Ex hi&longs;ce enim
facile derivantur vires Machinarum, quæ ex Rotis, Tympanis,
Trochleis, Vectibus, nervis ten&longs;is & ponderibus directe vel obli
que a&longs;cendentibus, cæteri&longs;que potentiis Mechanicis componi &longs;o
lent, ut & vires Tendinum ad animalium o&longs;&longs;a movenda.
ad eandem partem, & differentiam factorum ad contrarias, non
mutatur ab actione corporum inter &longs;e.
Etenim actio eique contraria reactio æquales &longs;unt per Legem 111,
adeoque per Legem 11 æquales in motibus efficiunt mutationes ver
&longs;us contrarias partes. Ergo &longs;i motus fiunt ad eandem partem; quic
quid additur motui corporis fugientis, &longs;ubducetur motui corporis
in&longs;equentis &longs;ic, ut &longs;umma maneat eadem quæ prius. Sin corpora ob
viam eant; æqualis erit &longs;ubductio de motu utriu&longs;que, adeoQ.E.D.ffe
rentia motuum factorum in contrarias partes manebit eadem.
Ut &longs;i corpus &longs;phæricum
beatQ.E.D.as velocitatis partes; &
tium decem, & &longs;umma erit partium &longs;exdecim. In corporum igitur
concur&longs;u, &longs;i corpus
quinque, corpus
&
ma partium &longs;exdecim ut prius. Si corpus
vel decem vel undecim vel duodecim, adeoque progrediatur po&longs;t
concur&longs;um cum partibus quindecim vel &longs;exdecim vel &longs;eptendecim
vel octodecim; corpus
vel cum una parte progredietur ami&longs;&longs;is partibus novem, vel qui
e&longs;cet ami&longs;&longs;o motu &longs;uo progre&longs;&longs;ivo partium decem, vel cum una par
te regredietur ami&longs;&longs;o motu &longs;uo & (ut ita dicam) una parte amplius,
vel regredietur cum partibus duabus ob detractum motum progre&longs;
&longs;ivum partium duodecim. AtQ.E.I.a &longs;ummæ motuum con&longs;pirantium
15+1 vel 16+c, & differentiæ contrariorum 17-1 & 18-2 &longs;emper
erunt partium &longs;exdecim, ut ante concur&longs;um & reflexionem. CogNI
tis autem motibus quibu&longs;cum corpora po&longs;t reflexionem pergent, in
venietur cuju&longs;que velocitas, ponendo eam e&longs;&longs;e ad velocitatem ante
reflexionem, ut motus po&longs;t e&longs;t ad motum ante. Ut in ca&longs;u ultimo, ubi
corporis
decim po&longs;tea, & velocitas partium duarum ante reflexionem; in
venietur ejus velocitas partium &longs;ex po&longs;t reflexionem, dicendo, ut
motus partes &longs;ex ante reflexionem ad motus partes octodecim po&longs;t
ea, ita velocitatis partes duæ ante reflexionem ad velocitatis partes
&longs;ex po&longs;tea.
Quod &longs;i corpora vel non Sphærica vel diver&longs;is in rectis moventia
incidant in &longs;e mutuo oblique, & requirantur eorum motus po&longs;t refle
xionem; cogno&longs;cendus e&longs;t &longs;itus plani a quo corpora concurrentia tan
guntur in puncto concur&longs;us: dein corporis utriu&longs;que motus (per
Corol.11.) di&longs;tinguendus e&longs;t in duos, unum huic plano perpendicu
larem, alterum eidem parallelum: motus autem paralleli, propter
ea quod corpora agant in &longs;e invicem &longs;ecundum lineam huic plano
perpendicularem, retinendi &longs;unt iidem po&longs;t reflexionem atque an
tea; & motibus perpendicularibus mutationes æquales in partes con
trarias tribuendæ &longs;unt &longs;ic, ut &longs;umma con&longs;pirantium & differentia
contrariorum maneat eadem quæ prius. Ex huju&longs;modi reflexio
nibus oriri etiam &longs;olent motus circulares corporum circa centra pro
pria. Sed hos ca&longs;us in &longs;equentibus non con&longs;idero, & nimis longum
e&longs;&longs;et omnia huc &longs;pectantia demon&longs;trare.COROLLARIUM IV.
nibus corporum inter &longs;e non mutat &longs;tatum &longs;uum vel motus vel quie
tis; & propterea corporum omnium in &longs; mutuo agentium (exclu&longs;is
actionibus & impedimentis externis) commune Centrum gravitatis
vel quie&longs;cit vel movetur uniformiter in directum.
Nam &longs;i puncta duo progrediantur uniformi cum motu in lineis
rectis, & di&longs;tantia eorum dividatur in ratione data, punctum divi
dens vel quie&longs;cit vel progreditur uniformiter in linea recta. Hoc
po&longs;tea in Lemmate XXIII demon&longs;tratur, &longs;i corpora quotcunque moventur uNI
formiter in lineis rectis, commune centrum gravitatis duorum quorumvis vel quie&longs;cit vel progreditur uniformiter in linea recta; propterea quod linea, horum corporum centra in recta uniformiter
progredientia jungens, dividitur ab hoc centro communis corporum duo
rum & centri communis tertii in data ratione.Eodem modo &
commune centrum horum trium & quarti cuju&longs;vis vel quie&longs;cit vel
progreditur uniformiter in linea recta; propterea quod ab eo divi
ditur di&longs;tantia inter centrum commune trium & centrum quarti in
data ratione, & &longs;ic in infinitum.Igitur in &longs;y&longs;temate corporum quæ
actionibus in &longs;e invicem alii&longs;que omnibus in &longs;e extrin&longs;ecus impre&longs;
&longs;is omnino vacant, adeoque moventur &longs;ingula uniformiter in rectis
&longs;ingulis, commune omnium centrum gravitatis vel quie&longs;cit vel mo
vetur uniformiter in directum.
Porro in &longs;y&longs;temate duorum corporum in &longs;e invicem agentium,
cum distantiæ centrorum utriusque a communi gravitatis centro &longs;int
reciproce ut corpora; erunt motus relativi corporum eorundem, vel
accedendi ad centrum illud vel ab eodem recedendi, æqualibus mutationibus in
partes contrarias factis, atque adeo ab actionibus horum corpo
rum inter &longs;e, nec promovetur nec retardatur nec mutationem pa
titur in &longs;tatu &longs;uo quoad motum vel quietem.In &longs;y&longs;temate autem
corporum plurimum, quoniam duorum quorumvis in &longs;e mutuo agen
tium commune gravitatis centrum ob actionem illam nullatenus
tercedit, commune gravitatis centrum nihil inde patitur; di&longs;tantia
autem horum duorum centrorum dividitur a communi corporum
omnium centro in partes &longs;ummis totalibus corporum quorum
&longs;unt centra reciproce proportionales; adeoque centris illis duobus
&longs;tatum &longs;uum movendi vel quie&longs;cendi &longs;ervantibus, commune omNI
um centrum &longs;ervat etiam &longs;tatum &longs;uum: manife&longs;tum e&longs;t quod com
mune illud omnium centrum ob actiones binorum corporum inter
&longs;e nunquam mutat &longs;tatum &longs;uum quoad motum & quietem. In tali
autem &longs;y&longs;temate actiones omnes corporum inter &longs;e, vel inter bina
&longs;unt corpora, vel ab actionibus inter bina compo&longs;itæ; & propterea
communi omnium centro mutationem in &longs;tatu motus ejus vel quie
tis nunquam inducunt. Quare cum centrum illud ubi corpora non
agunt in &longs;e invicem, vel quie&longs;cit, vel in recta aliqua progreditur uNI
formiter; perget idem, non ob&longs;tantibus corporum actionibus inter
&longs;e, vel &longs;emper quie&longs;cere, vel &longs;emper progredi uniformiter in dire
ctum; ni&longs;i a viribus in &longs;y&longs;tema extrin&longs;ecus impre&longs;&longs;is deturbetur de hoc
&longs;tatu. E&longs;t igitur &longs;y&longs;tematis corporum plurium Lex eadem quæ cor
poris &longs;olitarii, quoad per&longs;everantiam in &longs;tatu motus vel quietis. Mo
tus enim progre&longs;&longs;ivus &longs;eu corporis &longs;olitarii &longs;eu &longs;y&longs;tematis corporum
ex motu centri gravitatis æ&longs;timari &longs;emper debet.
VF.
tium illud quie&longs;cat, &longs;ive moveatur idem uniformiter in directum
ab&longs;que motu circulari.
Nam differentiæ motuum tendentium ad eandem partem, & &longs;um
mæ tendentium ad contrarias, eædem &longs;unt &longs;ub initio in
hypothe&longs;i) & ex his &longs;ummis vel differentiis oriuntur congre&longs;&longs;us & im
petus quibus corpora &longs;e mutuo feriunt. Ergo per Legem 11 æquales e
runt congre&longs;&longs;uum effectus in
tus inter &longs;e in uno ca&longs;u æquales motibus inter &longs;e in altero. Idem com
probatur experimento luculento. Motus omnes eodem modo &longs;e ha
bent in Navi, &longs;ive ea quie&longs;cat, &longs;ive moveatur uniformiter in directum.
bus æqualibus &longs;ecundum lineas parallelas urgeantur; pergent omnia
eodem modo moveri inter &longs;e, ac &longs;i viribus illis non e&longs;&longs;ent incitata.
Nam vires illæ æqualiter (pro quantitatibus movendorum corpo-
ter (quoad velocitatem) movebunt per Legem 11. adeoque nunquam
mutabunt po&longs;itiones & motus eorum inter &longs;e.
Hactenus principia tradidi a Mathematicis recepta & experien
tia multiplici confirmata. Per Leges duas primas & Corollaria duo
prima
temporis, & motum Projectilium fieri in Parabola; con&longs;pirante ex
perientia, ni&longs;i quatenus motus illi per aeris re&longs;i&longs;tentiam aliquantu
lum retardantur. Ab ii&longs;dem Legibus & Corollariis pendent de
mon&longs;trata de temporibus o&longs;cillantium Pendulorum, &longs;uffragante Ho
rologiorum experientia quotidiana. Ex his ii&longs;dem & Lege tertia
&
cipes, regulas congre&longs;&longs;uum & reflexionum duorum corporum &longs;e
or&longs;im invenerunt, & eodem fere tempore cum
communicarunt, inter &longs;e (quoad has leges) omnino con&longs;pirantes:
& primus quidem
tum prodiderunt. Sed & veritas comprobata e&longs;t a
ram Ve
rum, ut hoc experimentum cum Theoriis ad amu&longs;&longs;im congruat, ha
benda e&longs;t ratio cum re&longs;i&longs;tentiæ aeris, tum etiam vis Ela&longs;ticæ con
currentium corporum. Pendeant corpora
æqualibus
&longs;cribantur &longs;emicirculi Tra
hatur corpus
corpore
ad punctum
tardatio ex re&longs;i&longs;tentia aeris.
Hujus
ta &longs;ita in medio, ita &longs;cilicet
ut
que
Et i&longs;ta
tionem in de&longs;cen&longs;u ab
quam proxime. Re&longs;tituatur
corpus Cadat corpus
citas ejus in loco reflexionis
fimo e&longs;&longs;e ut chordam arcus quem cadendo de&longs;crip&longs;it, Propo&longs;itio e&longs;t
e&longs;t Geometris noti&longs;&longs;ima. Po&longs;t reflexionem perveniat corpus
locum
tur locus
nem redeat ad locum
ita videlicet ut
ponatur velocitas quam corpus
in loco
pus
thodo corrigendus erit locus
veniendus locus
cuo. Hoc pacto experiri licet omnia perinde ac &longs;i in vacuo con
&longs;tituti e&longs;&longs;emus. Tandem ducendum erit corpus
cus
loco
habeatur motus ejus in loco Et &longs;ic
corpus
ejus proxime po&longs;t reflexionem. Et &longs;imili methodo, ubi corpora duo
&longs;imul demittuntur de locis diver&longs;is, inveniendi &longs;unt motus
tam ante, quam po&longs;t reflexionem; & tum demum conferendi &longs;unt
motus inter &longs;e & colligendi effectus reflexionis. Hoc modo in
Pendulis pedum decem rem tentando, idQ.E.I. corporibus tam
inæqualibus quam æqualibus, & faciendo ut corpora de intervallis
ampli&longs;&longs;imis, puta pedum octo vel duodecim vel &longs;exdecim, concurre
rent; reperi &longs;emper &longs;ine errore trium digitorum in men&longs;uris, ubi
corpora &longs;ibi mutuo directe occurrebant, quod æquales erant muta
tiones motuum corporibus in partes contrarias illatæ, atque adeo
quod actio & reactio &longs;emper
erant æquales. Ut &longs;i corpus
novem partibus motus, & a
mi&longs;&longs;is &longs;eptem partibus perge
bat po&longs;t reflexionem cum du
abus; corpus
partibus i&longs;tis &longs;eptem. Si cor
pora obviam ibant
duodecim partibus &
bat
que. De motu ip&longs;ius
in plagam contrariam: & &longs;ic de motu corporis
ducendo partes quatuordecim, fient partes octo in plagam contra
riam. Quod &longs;i corpora ibant ad eandam plagam,
partibus quatuordecim, &
reflexionem pergebat
tuordecim, facta tran&longs;latione partium novem de
in reliquis. A congre&longs;&longs;u & colli&longs;ione corporum nunquam muta
batur quantitas motus, quæ ex &longs;umma motuum con&longs;pirantium &
differentia contrariorum colligebatur. Nam errorem digiti unius
& alterius in men&longs;uris tribuerim difficultati peragendi &longs;ingula
&longs;atis accurate. Difficile erat, tum pendula &longs;imul demittere fic, ut
corpora in &longs;e mutuo impingerent in loco infimo
kSed & in
ip&longs;is pilis inæqualis partium den&longs;itas, & textura aliis de cau&longs;is irre
gularis, errores inducebant.
MOTUS
Porro nequis objiciat Regulam, ad quam probandam inventum
e&longs;t hoc experimentum, præ&longs;upponere corpora vel ab&longs;olute dura
e&longs;&longs;e, vel &longs;altem perfecte ela&longs;tica, cuju&longs;modi nulla reperiuntur in
compo&longs;itionibus naturalibus; addo quod Experimenta jam de&longs;crip
ta &longs;uccedunt in corporibus mollibus æque ac in duris, nimirum a
conditione duritiei neutiquam pendentia. Nam &longs;i Regula illa in
corporibus non perfecte duris tentanda e&longs;t, debebit &longs;olummodo
reflexio minui in certa proportione pro quantitate vis Ela&longs;ticæ. In
Theoria
cem cum velocitate congre&longs;&longs;us. Certius id affirmabitur de perfecte
Ela&longs;ticis. In imperfecte Ela&longs;ticis velocitas reditus minuenda e&longs;t &longs;i
mul cum vi Ela&longs;tica; propterea quod vis illa; (ni&longs;i ubi partes cor
porum ex congre&longs;&longs;u læduntur, vel exten&longs;ionem aliqualem qua&longs;i &longs;ub
malleo patiuntur,) certa ac determinata &longs;it (quantum &longs;entio) faci
atque corpora redire ab invicem cum velocitate relativa, quæ &longs;it ad
relativam velocitatem concur&longs;us in data ratione. Id in pilis ex lana
arcte conglomerata & fortiter con&longs;tricta &longs;ic tentavi. Primum demit
tendo Pendula & men&longs;urando reflexionem, inveni quantitatem vis
Ela&longs;ticæ; deinde per hanc vim determinavi reflexiones in aliis ca
&longs;ibus concur&longs;uum, & re&longs;pondebant Experimenta. Redibant &longs;emper
pilæ ab invicem cum velocitate relativa, quæ e&longs;&longs;et ad velocitatem
relativam concur&longs;us ut 5 ad 9 circiter. Eadem fere cum velocitate
redibant pilæ ex chalybe: aliæ ex &longs;ubere cum paulo minore: in vi
treis autem proportio erat 15 ad 16 circiter. Atque hoc pacto Lex
tertia quoad ictus & reflexiones per Theoriam comprobata e&longs;t, quæ
cum experientia plane congruit.
SIVE
In Attractionibus rem &longs;ic breviter o&longs;tendo.
Corporibus duobus
quibu&longs;vis
interponi quo congre&longs;&longs;us eorum impediatur. Si corpus alterutrum
in prius
pre&longs;&longs;ione corporis Præ
valebit pre&longs;&longs;io fortior, facietque ut &longs;y&longs;tema corporum duorum &
ob&longs;taculi moveatur in directum in partes ver&longs;us
liberis &longs;emper accelerato abeat in infinitum. Quod e&longs;t ab&longs;urdum &
Legi primæ contrarium. Nam per Legem primam debebit &longs;y&longs;tema
per&longs;everare in &longs;tatu &longs;uo quie&longs;cendi vel movendi uniformiter in di
rectum, proindeque corpora æqualiter urgebunt ob&longs;taculum, & id
circo æqualiter trahentur in invicem. Tentavi hoc in Magnete &
Ferro. Si hæc in va&longs;culis propriis &longs;e&longs;e contingentibus &longs;eor&longs;im po
&longs;ita, in aqua &longs;tagnante juxta fluitent; neutrum propellet alterum,
&longs;ed æqualitate attractionis utrinque &longs;u&longs;tinebunt conatus in &longs;e mu
tuos, ac tandem in æquilibrio con&longs;tituta quie&longs;cent.
Sic etiam gravitas inter Terram & ejus partes, mutua e&longs;t.
Se
cetur Terra
& æqualia erunt harum pondera in &longs;e mu
tuo. Nam &longs;i plano alio
cetur in partes duas
quarum
&longs;ci&longs;&longs;æ
media
tram partium extremarum propendebit,
&longs;ed inter utramQ.E.I. æquilibrio, ut ita
dicam, &longs;u&longs;pendetur, & quie&longs;cet. Pars autem extrema
&longs;uo pondere incumbet in partem mediam, & urgebit illam in
partem alteram extremam
tiæ
in &longs;e mutuo &longs;unt æqualia, uti volui o&longs;tendere. Et ni&longs;i pondera illa
æqualia e&longs;&longs;ent, Terra tota in libero æthere fluitans ponderi majori
cederet, & ab eo fugiendo abiret in infinitum.
Ut corpora in concur&longs;u & reflexione idem pollent, quorum ve
locitates &longs;unt reciproce ut vires in&longs;itæ: &longs;ic in movendis In&longs;tru
mentis Mechanicis agentia idem pollent & conatibus contrariis &longs;e
mutuo &longs;u&longs;tinent, quorum velocitates &longs;ecundum determinationem Sie pondera æquipollent
ad movenda brachia Libræ, quæ o&longs;cillante Libra &longs;unt reciproce ut
eorum velocitates &longs;ur&longs;um & deor&longs;um: hoc e&longs;t, pondera, &longs;i recta
a&longs;cendunt & de&longs;cendunt, æquipollent, quæ &longs;unt reciproce ut pun
ctorum a quibus &longs;u&longs;penduntur di&longs;tantiæ ab axe Libræ; &longs;in planis
obliquis alii&longs;ve admotis ob&longs;taculis impedita a&longs;cendunt vel de&longs;cen
dunt oblique, æquipollent quæ &longs;unt reciproce ut a&longs;cen&longs;us & de&longs;cen
&longs;us, quatenus facti &longs;ecundum perpendiculum: id adeo ob determi
nationem gravitatis deor&longs;um. Similiter in Trochlea &longs;eu Poly&longs;pa&longs;to
vis manus funem directe trahentis, quæ &longs;it ad pondus vel directe
vel oblique a&longs;cendens ut velocitas a&longs;cen&longs;us perpendicularis ad ve
locitatem manus funem trahentis, &longs;u&longs;tinebit pondus. In Horolo
giis & &longs;imilibus in&longs;trumentis, quæ ex rotulis commi&longs;&longs;is con&longs;tructa
&longs;unt, vires contrariæ ad motum rotularum promovendum & impe
diendum, &longs;i &longs;unt reciproce ut velocitates partium rotularum in quas
imprimuntur, &longs;u&longs;tinebunt &longs;e mutuo. Vis Cochleæ ad premendum
corpus e&longs;t ad vim manus manubrium circumagentis, ut circularis
velocitas manubrii ea in parte ubi a manu urgetur, ad velocitatem
progre&longs;&longs;ivam cochleæ ver&longs;us corpus pre&longs;&longs;um. Vires quibus Cu
neus urget partes duas ligni fi&longs;&longs;i &longs;unt ad vim mallei in cuneum, ut
progre&longs;&longs;us cunei &longs;ecundum determinationem vis a malleo in ip&longs;um
impre&longs;&longs;æ, ad velocitatem qua partes ligni cedunt cuneo, &longs;ecundum
lineas faciebus cunei perpendiculares. Et par e&longs;t ratio Machina
rum omnium.
Harum efficacia & u&longs;us in eo &longs;olo con&longs;i&longs;tit, ut diminuendo velo
citatem augeamus vim, & contra: Unde &longs;olvitur in omni aptorum
in&longs;trumentorum genere Problema,
di,Nam &longs;i Ma
chinæ ita formentur, ut velocitates Agentis & Re&longs;i&longs;tentis &longs;ine reci
proce ut vires; Agens re&longs;i&longs;tentiam &longs;u&longs;tinebit: & majori cum veloci
tatum di&longs;paritate eandem vincet. Certe &longs;i tanta &longs;ic velocitatum
di&longs;paritas, ut vincatur etiam re&longs;i&longs;tentia omnis, quæ tam ex conti
guorum & inter &longs;e labentium corporum attritione, quam ex con
tinuorum & ab invicem &longs;eparandorum cohæ&longs;ione & elevandorum
ponderibus orirj &longs;olet; &longs;uperata omni ea re&longs;i&longs;tentia, vis redun
dans accelerationem motus &longs;ibi proportionalem, partim in parti
bus machinæ, partim in corpore re&longs;i&longs;tente producet. Ceterum
Mechanicam tractare non e&longs;t hujus in&longs;tituti. Hi&longs;ce volui tan
tum o&longs;tendere, quam late pateat quamque certa &longs;it Lex tertia
Motus. Nam &longs;i æ&longs;timetur Agentis actio ex ejus vi & veloci-
ctim ex ejus partium &longs;ingularum velocitatibus & viribus re&longs;i&longs;tendi
ab earum attritione, cohæ&longs;ione, pondere, & acceleratione ori
undis; erunt actio & reactio, in omni in&longs;trumentorum u&longs;u,
&longs;ibi invicem &longs;emper æquales. Et quatenus actio propagatur per
in&longs;trumentum & ultimo imprimitur in corpus omne re&longs;i&longs;tens,
ejus ultima determinatio determinationi reactionis &longs;emper erit
contraria.
CORPORUM
MOTU CORPORUM
LIBER PRIMUS.
demon&longs;trantur.
tempore quovis finito con&longs;tanter tendunt, & ante finem tempo
ris illius propius ad invicem accedunt quam pro data quavis diffe
tia, fiunt ultimo æquales.
Si negas; fiant ultimò inequales, & &longs;it earum ultima differentia
data differentia
prehen&longs;a, in&longs;cribantur parallelogramma quotcunque
&c.
Bb, Cc, Dd, &c.
rallelis contenta; & compleantur paral-
tudo minuatur, & numerus augeatur
in infinitum: dico quod ultimæ rationes,
quas habent ad &longs;e invicem Figura in
&longs;cripta
AalbmcndoE,
Nam Figuræ in&longs;criptæ & circum&longs;criptæ differentia e&longs;t &longs;umma pa
rallelogrammorum
nium ba&longs;es) rectangulum &longs;ub unius ba&longs;i
latitudo ejus Er
go (per Lemma 1) Figura in&longs;cripta & circum&longs;cripta & multo magis
Figura curvilinea intermedia fiunt ultimo æquales.
lelogrammorum latitudines
& omnes minuuntur in infinitum.
Sit enim
logrammum
&longs;criptæ & Figuræ circum&longs;criptæ; at latitudine &longs;ua
diminuta, minus fiet quam datum quodvis rectangulum.
coincidit omni ex parte cum Figura curvilinea.
cum Figura curvilinea.
CORPORUM
eorundem arcuum comprehenditur.
non &longs;unt rectilineæ, &longs;ed rectilinearum limites curvilinei.
parallelogrammorum &longs;eries, &longs;itQ.E.I.em amborum numerus, & ubi
latitudines in infinitum diminuuntur, rationes ultimæ parallelo
grammorum in una Figura ad parallelogramma in altera, &longs;ingulorum
ad fingula, &longs;int eædem; dico quod Figuræ duæ
Etenim ut &longs;unt parallelogramma &longs;ingula ad &longs;ingula, ita (compo
nendo) fit &longs;umma omnium ad &longs;ummam omnium, & ita Figura ad
Figuram; exi&longs;tente nimirum Figura priore (per Lemma 111) ad &longs;um
mam priorem, & Figura po&longs;teriore ad &longs;ummam po&longs;teriorem in ra
tione æqualitatis.
partium numerum utcunQ.E.D.vidantur; & partes illæ, ubi numerus
earum augetur & magnitudo diminuitur in infinitum, datam obti
neant rationem ad invicem, prima ad primam, &longs;ecunda ad &longs;ecundam,
cæteræque &longs;uo ordine ad cæteras: erunt tota ad invicem in eadem
illa data ratione. Nam &longs;i in Lemmatis hujus Figuris &longs;umantur pa-
&longs;ummæ parallelogrammorum; atque adeo, ubi partium & paralle
logrammorum numerus augetur & magnitudo diminuitur in infiNI
tum, in ultima ratione parallelogrammi ad parallelogrammum, id
e&longs;t (per hypothe&longs;in) in ultima ratione partis ad partem.
proportionalia, tam curvilinea quam rectilinea; & areæ &longs;unt in
duplicata ratione laterum.
aliquo
tangatur a recta utrinque producta
AD;
accedant & coëant; dico quod angulus
BAD,
tus, minuetur in infinitum & ultimo e
vane&longs;cet.
Nam &longs;i angulus ille non evane&longs;cit, continebit arcus
gente
ad punctum
ad invicem est ratio æqualitatis.
Nam dum punctum
parallela agatur
Et punctis
evane&longs;cet; adeoque rectæ &longs;emper &longs;initæ
dius Unde & hi&longs;ce
&longs;emper proportionales rectæ
CORPORUM
quamvis
untem perpetuo &longs;ecans in
hæc
vane&longs;centem
habebit æqualitatis, eo quod
completo parallelogrammo
litatis ad
AG,
ma ab&longs;ci&longs;&longs;arum omnium
cus
timis argumentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt.
AD,
puncta
triangulorum evane&longs;centium est &longs;imilitudinis, & ultima ratio
æqualitatis.
Nam dum punctum
accedit,
ad puncta longinqua
ip&longs;ique
untibus punctis
ne&longs;cet, & propterea triangula tria &longs;emper
finita
que eo nomine &longs;imilia & æqualia. Unde
& hi&longs;ce &longs;emper &longs;imilia & proportionalia
invicem &longs;imilia & æqualia.
mentatione, pro &longs;e invicem u&longs;urpari po&longs;&longs;unt.
angulo dato
tim applicentur
puncta
angulorum
ratione laterum.
Etenim dum puncta
dunt ad punctum
&longs;emper
ginqua
&longs;is
rigantur ordinatæ
tis
rant ip&longs;is
in
tas
Tum manente longitudine
& angulo
cum rectilineis
cata ratione laterum
&longs;unt areæ
areæ
AE.
illa determinata & immutabilis &longs;it, five eadem continuo auge
atur vel continuo diminuatur, &longs;unt ip&longs;o motus initio in duplica
ta ratione Temporum.
Exponantur tempora per lineas
per ordinatas
ut areæ
initio (per Lemma IX) in duplicata ratione temporum
CORPORUM
lium Figurarum partes temporibus proportionalibus de&longs;cribentium
Errores, qui viribus quibu&longs;vis æqualibus ad corpora &longs;imiliter ap
plicatis generantur, & men&longs;urantur per di&longs;tantias corporum a Fi
gurarum &longs;imilium locis illis ad quæ corpora eadem temporibus ii&longs;
dem proportionalibus ab&longs;que viribus i&longs;tis pervenirent, &longs;unt ut qua
drata temporum in quibus generantur quam proxime.
Figurarum &longs;imilium partes &longs;imiliter applicatis generantur, &longs;unt ut
vires & quadrata temporum conjunctim.
ra urgentibus diver&longs;is viribus de&longs;cribunt. Hæc &longs;unt, ip&longs;o motus iNI
tio, ut vires & quadrata temporum conjunctim.
directe & quadrata temporum inver&longs;e.
& vires inver&longs;e.
Si quantitates indeterminatæ diver&longs;orum generum conferantur
inter &longs;e, & earum aliqua dicatur e&longs;&longs;e ut e&longs;t alia quævis directe vel
inver&longs;e: &longs;en&longs;us e&longs;t, quod prior augetur vel diminuitur in eadem
ratione cum po&longs;teriore, vel cum ejus reciproca. Et &longs;i earum aliqua
dicatur e&longs;&longs;e ut &longs;unt aliæ duæ vel plures directe vel inver&longs;e: &longs;en&longs;us
e&longs;t, quod prima augetur vel diminuitur in ratione quæ componitur
ex rationibus in quibus aliæ vel aliarum reciprocæ augentur vel di
minuuntur. Ut &longs;i A dicatur e&longs;&longs;e ut B directe & C directe & D in
ver&longs;e: &longs;en&longs;us e&longs;t, quod A augetur vel diminuitur in eadem ratione
cum BXCX1/D, hoc e&longs;t, quod A & (BC/D) &longs;unt ad invicem in ratio
ne data.
ram finitam ad punctum contactus habentibus, est ultimo in ra
tione duplicata &longs;ubten&longs;æ arcus contermini.
tactus ad tangentem perpendicularis
&longs;ubten&longs;æ
&longs;itque
accedunt u&longs;que ad
e&longs;&longs;e pote&longs;t quam a&longs;&longs;ignata quævis. E&longs;t autem (ex natura circulorum
per puncta
æquale
adeoque ratio
nitur ex rationibus
Sed quoniam
dine quavis a&longs;&longs;ignata, fieri pote&longs;t ut ratio
ad
pro differentia quavis a&longs;&longs;ignata, adeoque ut ra
tio
tione
a&longs;&longs;ignata. E&longs;t ergo, per Lemma 1, ratio ultima
quovis dato, & eadem &longs;emper erit ratio ultima
prius, adeoque eadem ae
tum punctum convergente, vel alia quacunque lege con&longs;tituatur;
tamen anguli
vergent & propius accedent ad invicem quam pro differentia qua
vis a&longs;&longs;ignata, adeoque ultimo æquales erunt, per Lem. I & prop
terea lineæ
rum &longs;inus
etiam illorum quadrata ultimo ut &longs;ubten&longs;æ
&longs;agittæ quæ chordas bi&longs;ecant & ad datum punctum convergunt.
Nam &longs;agittæ illæ &longs;unt ut &longs;ubten&longs;æ
corpus data velocitate de&longs;cribit arcum.
cata ratione laterum
db
exi&longs;tentia. Sic & triangula
ratione laterum
plicatæ &longs;ubduplicatam, quæ nempe ex &longs;implici & &longs;ubduplicata com
ponitur, quamque alias Se&longs;quialteram dicunt.
CORPORUM
cata ratione ip&longs;arum
Adb
neorum
rundem triangulorum. Et inde hæ areæ & hæc &longs;egmenta erunt in
triplicata ratione tum tangentium
arcuum
Cæterum in his omnibus &longs;upponimus angulum contactus nec in
finite majorem e&longs;&longs;e angulis contactuum, quos Circuli continent cum
tangentibus &longs;uis, nec ii&longs;dem infinite minorem; hoc e&longs;t, curvaturam
ad punctum
intervallum Capi enim pote&longs;t
ut
tem
erit infinite minor Circularibus. Et &longs;imili argumento &longs;i fiat
&longs;ucce&longs;&longs;ive ut habebitur &longs;eries an
gulorum contactus pergens in infinitum, quorum quilibet po&longs;te
rior e&longs;t infinite minor priore. Et &longs;i fiat
habebitur alia &longs;eries infinita
angulorum contactus, quorum primus e&longs;t eju&longs;dem generis cum Cir
cularibus, &longs;ecundus infinite major, & quilibet po&longs;terior infinite ma
jor priore. Sed & inter duos quo&longs;vis ex his angulis pote&longs;t &longs;eries
utrinQ.E.I. infinitum pergens angulorum intermediorum in&longs;eri,
quorum quilibet po&longs;terior erit infinite major minorve priore. Ut
&longs;i inter terminos Et rur
&longs;us inter binos quo&longs;vis angulos hujus &longs;eriei in&longs;eri pote&longs;t &longs;eries no
va angulorum intermediorum ab invicem infinitis intervallis diffe
rentium. Neque novit natura limitem.
Quæ de curvis lineis deque &longs;uperficiebus comprehen&longs;is demon
&longs;trata &longs;unt, facile applicantur ad &longs;olidorum &longs;uperficies curvas &
contenta. Præmi&longs;i vero hæc Lemmata, ut effugerem tædium dedu
cendi perplexas demon&longs;trationes, more veterum Geometrarum, ad
ab&longs;urdum. Contractiores enim redduntur demon&longs;trationes per me
thodum Indivi&longs;ibilium. Sed quoniam durior e&longs;t Indivi&longs;ibilium hy
pothe&longs;is, & propterea methodus illa minus Geometrica cen&longs;etur;
malui demon&longs;trationes rerum &longs;equentium ad ultimas quantitatum
ad limites &longs;ummarum & rationum deducere; & propterea limitum
illorum demon&longs;trationes qua potui brevitate præmittere. His enim
idem præ&longs;tatur quod per methodum Indivi&longs;ibilium; & principiis de
mon&longs;tratis jam tutius utemur. Proinde in &longs;equentibus, &longs;iquando
quantitates tanquam ex particulis con&longs;tantes con&longs;ideravero, vel &longs;i
pro rectis u&longs;urpavero lineolas curvas; nolim indivi&longs;ibilia, &longs;ed eva
ne&longs;centia divi&longs;ibilia, non &longs;ummas & rationes partium determinata
rum, &longs;ed &longs;ummarum & rationum limites &longs;emper intelligi; vimque
talium demon&longs;trationum ad methodum præcedentium Lemmatum
&longs;emper revocari.
Objectio e&longs;t, quod quantitatum evane&longs;centium nulla &longs;it ultima
proportio; quippe quæ, antequam evanuerunt, non e&longs;t ultima, ubi
evanuerunt, nulla e&longs;t. Sed & eodem argumento æque contendi po&longs;&longs;et
nullam e&longs;&longs;e corporis ad certum locum pervenientis velocitatem ul
timam: hanc enim, antequam corpus attingit locum, non e&longs;&longs;e ulti
mam, ubi attingit, nullam e&longs;&longs;e. Et re&longs;pon&longs;io facilis e&longs;t: Per velocita
tem ultimam intelligi eam, qua corpus movetur neque antequam
attingit locum ultimum & motus ce&longs;&longs;at, neque po&longs;tea, &longs;ed tunc
cum attingit; id e&longs;t, illam ip&longs;am velocitatem quacum corpus attin
git locum ultimum & quacum motus ce&longs;&longs;at. Et &longs;imiliter per ulti
mam rationem quantitatum evane&longs;centium, intelligendam e&longs;&longs;e ratio
nem quantitatum non antequam evane&longs;cunt, non po&longs;tea, &longs;ed qua
cum evane&longs;cunt. Pariter & ratio prima na&longs;centium e&longs;t ratio qua
cum na&longs;cuntur. Et &longs;umma prima & ultima e&longs;t quacum e&longs;&longs;e (vel
augeri & minui) incipiunt & ce&longs;&longs;ant. Extat limes quem velocitas
in fine motus attingere pote&longs;t, non autem tran&longs;gredi. Hæc e&longs;t
velocitas ultima. Et par e&longs;t ratio limitis quantitatum & propor
tionum omnium incipientium & ce&longs;&longs;antium. Cumque hic limes
&longs;it certus & definitus, Problema e&longs;t vere Geometricum eundem de
terminare. Geometrica vero omnia in aliis Geometricis determi
nandis ac demon&longs;trandis legitime u&longs;urpantur.
Contendi etiam pote&longs;t, quod &longs;i dentur ultimæ quantitatum eva
ne&longs;centium rationes, dabuntur & ultimæ magnitudines: & &longs;ic quan
titas omnis con&longs;tabit ex Indivi&longs;ibilibus, contra quam
Incommen&longs;urabilibus, in libro decimo Elementorum, demon&longs;travit.
Verum hæc Objectio fal&longs;æ innititur hypothe&longs;i. Ultimæ rationes
illæ quibu&longs;cum quantitates evane&longs;cunt, revera non &longs;unt rationes
quantitatum ultimarum, &longs;ed limites ad quos quantitatum &longs;ine limi
te decre&longs;centium rationes &longs;emper appropinquant; & quas propius
a&longs;&longs;equi po&longs;&longs;unt quam pro data quavis differentia, nunquam vero
infinitum. Res clarius intelligetur in infinite magnis.
Si quantitates
duæ quarum data e&longs;t differentia auges ntur in infinitum, dabitur
harum ultima ratio, nimirum ratio æqualitatis, nec tamen ideo da
buntur quantitates ultimæ &longs;eu maximæ quarum i&longs;ta e&longs;t ratio. Igitur
in &longs;equentibus, &longs;iquando facili rerum conceptui con&longs;ulens dixero
quantitates quam minimas, vel evane&longs;centes, vel ultimas; cave in
telligas quantitates magnitudine determinatas, &longs;ed cogita &longs;emper
diminuendas &longs;ine limite.
CORPORUM
ductis de&longs;cribunt, & in planis immobilibus con&longs;i&longs;tere, & e&longs;&longs;e tem
poribus proportionales.
Dividatur tempus in partes æquales, & prima temporis parte de
&longs;cribat corpus vi in&longs;ita rectam
nil impediret, recta
pergeret ad
Leg. 1.) de&longs;cribens
lineam
ip&longs;i
diis
centrum actis, con
fectæ forent æqua
les areæ
Verum ubi corpus
venit ad
centripeta impul
&longs;u unico &longs;ed mag
no, efficiatque ut
corpus de recta
declinet & pergat
in recta
temporis parte, corpus (per Legum Corol. 1.) reperietur in
ob parallelas
triangulo
in faciens ut corpus &longs;ingulis temporis particulis &longs;in
gulas de&longs;eribat rectas jacebunt hæ omnes in
eodem plano; & triangulum Æqualibus igitur tempori
bus æquales areæ in plano immoto de&longs;cribuntur: & componendo,
&longs;unt arearum &longs;ummæ quævis
pora de&longs;criptionum. Augeatur jam numerus & minuatur latitudo
triangulorum in infinitum; & eorum ultima perimeter
Corollarium quartum Lemmatis tertii) erit linea curva: adeoque vis
centripeta, qua corpus a tangente hujus curvæ perpetuo retrahitur,
aget inde&longs;inenter; areæ vero quævis de&longs;criptæ
temporibus de&longs;criptionum &longs;emper proportionales, erunt ii&longs;dem tem
poribus in hoc ca&longs;u proportionales.
&longs;patiis non re&longs;i&longs;tentibus reciproce ut perpendiculum a centro illo in
Orbis tangentem rectilineam demi&longs;&longs;um. E&longs;t enim velocitas in locis
illis
CD, DE, EF
demi&longs;&longs;a.
re&longs;i&longs;tentibus ab eodem corpore &longs;ucce&longs;&longs;ive de&longs;criptorum chordæ
BC
lis
tum diminuuntur, producatur utrinque; tran&longs;ibit eadem per cen
trum virium.
tibus de&longs;criptorum chordæ
parallelogramma
cem in ultima ratione diagonalium
nitum diminuuntur. Nam corporis motus
tur (per Legum Corol. 1.) ex motibus
qui
po&longs;itionis hujus generabantur ab impul&longs;ibus vis centripetæ in B &
bus a motibus rectilineis retrahuntur ac detorquentur in orbes cur
vos &longs;unt inter &longs;e ut arcuum æqualibus temporibus de&longs;criptorum &longs;a
gittæ illæ quæ convergunt ad centrum virium, & chordas bi&longs;ecant Nam hæ &longs;agittæ &longs;unt &longs;e
mi&longs;&longs;es diagonalium de quibus egimus in Corollario tertio.
CORPORUM
gittæ ad &longs;agittas horizonti perpendiculares arcuum Parabolieorum
quos projectilia eodem tempore de&longs;cribunt.
IV, ubi plana
in quibus corpora moventur, una cum centris virium quæ in ip&longs;is
fita &longs;unt, non quie&longs;cunt, &longs;ed moventur uniformiter in directum.
&longs;cripta, & radio ducto ad punctum vel immobile, vel motu rectili
neo uniformiter progrediens, de&longs;cribit areas circa punctum illud
temporibus proportionales, urgetur a vi centripeta tendente ad idem
punctum.
quetur de cur&longs;u rectilineo per vim aliquam in ip&longs;um agentem (per
Leg. 1.) Et vis illa qua corpus de cur&longs;u rectilineo detorquetur, &
cogitur triangula quam minima circa
punctum immobile
git in loco XL,
Lib. 1 Elem.
& Leg.
11.) hoc e&longs;t, &longs;ecundum lineam
Agit ergo &longs;emper &longs;ecundum lineas tendentes ad punctum
illud immobile
quie&longs;cat &longs;uperficies in qua corpus de&longs;cribit figuram curvilineam,
&longs;ive moveatur eadem una cum corpore, figura de&longs;cripta, & puncto
&longs;uo
temporibus proportionales, vires non tendunt ad concur&longs;um radio
rum; &longs;ed inde declinant in con&longs;equentia &longs;eu ver&longs;us plagam in quam
fit motus, &longs;i modo arearum de&longs;criptio acceleratur: &longs;in retardatur, de
clinant in antecedentia.
ratur, virium directiones declinant a concur&longs;u radiorum ver&longs;us plagam
in quam &longs;it motus.
Urgeri pote&longs;t corpus a vi centripeta compo&longs;ita ex pluribus viri
bus. In hoc ca&longs;u &longs;en&longs;us Propo&longs;itionis e&longs;t, quod vis illa quæ ex om
nibus componitur, tendit ad punctum
perpetuo &longs;ecundum lineam &longs;uperficiei de&longs;criptæ perpendicularem;
hæc faciet ut corpus deflectatur a plano &longs;ui motus: &longs;ed quantita
tem &longs;uperficiei de&longs;criptæ nec augebit nec minuet, & propterea in
compo&longs;itione virium negligenda e&longs;t.
ducto de&longs;cribit areas circa centrum illud temporibus proportiona
les, urgetur vi compo&longs;ita ex vi centripeta tendente ad corpus il
lud alterum, & ex vi omni acceleratrice qua corpus illud alterum
urgetur.
Sit corpus primum
VI.) &longs;i vi nova, quæ æqualis & contraria &longs;it illi qua corpus alterum
perget corpus primum
ea&longs;dem ac prius: vis autem, qua corpus alterum
de&longs;truetur per vim &longs;ibi æqualem & contrariam; & propterea (per
Leg. 1.) corpus illud alterum
e&longs;cet vel movebitur uniformiter in directum: & corpus primum
urgente differentia virium, id e&longs;t, urgente vi reliqua perget areas
temporibus proportionales circa corpus alterum Ten
dit igitur (per Theor. 11.) differentia virium ad corpus illud alte
rum
&longs;cribit areas temporibus proportionales; atQ.E.D. vi tota (&longs;ive &longs;im
plici, &longs;ive ex viribus pluribus, juxta Legum Corollarium &longs;ecundum,
compo&longs;ita,) qua corpus prius
gum Corollarium) vis tota acceleratrix qua corpus alterum urgetur:
vis omnis reliqua qua corpus prius urgetur tendet ad corpus alte
rum
tionales, vis reliqua tendet ad corpus alterum
portionales.
CORPORUM
bit areas quæ, cum temporibus collatæ, &longs;unt valde inæquales; &
corpus illud alterum
rectum: actio vis centripetæ ad corpus illud alterum
vel nulla e&longs;t, vel mi&longs;cetur & componitur cum actionibus admodum
potentibus aliarum virium: Vi&longs;que tota ex omnibus, &longs;i plures &longs;unt
vires, compo&longs;ita, ad aliud (&longs;ive immobile &longs;ive mobile) centrum
dirigitur. Idem obtinet, ubi corpus alterum motu quocunque mo
vetur; &longs;i modo vis centripeta &longs;umatur, quæ re&longs;tat po&longs;t &longs;ubductio
nem vis totius in corpus illud alterum
Quoniam æquabilis arearum de&longs;criptio Index e&longs;t Centri, quod
vis illa re&longs;picit qua corpus maxime afficitur, quaque retrahitur a mo
tu rectilineo & in orbita &longs;ua retinetur: quidni u&longs;urpemus in &longs;equen
tibus æquabilem arearum de&longs;criptionem, ut Indicem Centri circum
quod motus omnis circularis in &longs;patiis liberis peragitur?
tripetas ad centra eorundem circulorum tendere; & e&longs;&longs;e inter &longs;e,
ut &longs;unt arcuum &longs;imul de&longs;criptorum quadrata applicata ad circulo
rum radios.
Tendunt hæ vires ad centra circulorum per Prop.II. & Corol.
II.
Prop. 1; & &longs;unt inter &longs;e ut arcuum æqualibus temporibus quam miNI
mis de&longs;criptorum &longs;inus ver&longs;i per Corol. IV. Prop.
I; hoc e&longs;t, ut qua
drata arcuum eorundem ad diametros circulorum applicata per
Lem. VII: & propterea, cum hi arcus &longs;int ut arcus temporibus
quibu&longs;vis æqualibus de&longs;cripti, & diametri &longs;int ut eorum radii; vi
res erunt ut arcuum quorumvis &longs;imul de&longs;criptorum quadrata ap
plicata ad radios circulorum.
res centripetæ &longs;unt ut velocitatum quadrata applicata ad radios
circulorum: hoc e&longs;t, ut cum Geometris loquar, vires &longs;unt in ra
tione compo&longs;ita ex duplicata ratione velocitatum directe & ratione
&longs;implici radiorum inver&longs;e.
ratione radiorum directe & ratione velocitatum inver&longs;e, vires cen
tripetæ &longs;unt reciproce ut quadrata temporum periodieorum appli
cata ad circulorum radios; hoc e&longs;t, in ratione compo&longs;ita ex ratione
radiorum directe & ratione duplicata temporum periodieorum in
ver&longs;e.
locitates &longs;int ut radii; erunt etiam vires centripetæ ut radii: &
contra.
duplicata radiorum; æquales erunt vires centripetæ inter &longs;e: &
contra.
tates æquales; vires centriperæ erunt reciproce ut radii: & contra.
rum & propterea velocitates reciproce in radiorum ratione &longs;ubdu
plicata; vires centripetæ erunt reciproce ut quadrata radiorum:
& contra.
pote&longs;tas quælibet
pote&longs;tas
& contra.
bus corpora &longs;imiles figurarum quarumcunque &longs;imilium, centraque
in figuris illis &longs;imiliter po&longs;ita habentium, partes de&longs;cribunt, con&longs;e
quuntur ex Demon&longs;tratione præcedentium ad ho&longs;ce ca&longs;us applicata.
Applicatur autem &longs;ub&longs;tituendo æquabilem arearum de&longs;criptionem
pro æquabili motu, & di&longs;tantias corporum a centris pro radiis u&longs;ur
pando.
cus, quem corpus in circulo data vi centripeta uniformiter revolven
do tempore quovis de&longs;cribit, medius e&longs;t proportionalis inter dia
metrum circuli, & de&longs;cen&longs;um corporis eadem data vi eodem que tem
pore cadendo confectum.
Ca&longs;us Corollarii &longs;exti obtinet in corporibus cæle&longs;tibus, (ut &longs;eor
&longs;um collegerunt etiam no&longs;trates
propterea quæ &longs;pectant ad vim centripetam decre&longs;centem in dupli
cata ratione di&longs;tantiarum a centris, decrevi fu&longs;ius in &longs;equentibus
exponere.
CORPORUM
Porro præcedentis propo&longs;itionis & corollariorum ejus beneficio,
colligitur etiam proportio vis centripetæ ad vim quamlibet notam,
qualis e&longs;t ea Gravitatis. Nam &longs;i corpus in circulo Terræ concen
trico vi gravitatis &longs;uæ revolvatur, hæc gravitas e&longs;t ip&longs;ius vis centri
peta. Datur autem, ex de&longs;cen&longs;u gravium, & tempus revolutionis
unius, & arcus dato quovis tempore de&longs;criptus, per hujus Corol.
IX. Et huju&longs;modi propo&longs;itionibus
tu
ribus centrifugis contulit.
Demon&longs;trari etiam po&longs;&longs;unt præcedentia in hunc modum.
In cir
culo quovis de&longs;cribi intelligatur Polygonum laterum quotcunque.
Et &longs;i corpus, in polygoni lateribus data cum velocitate movendo,
ad ejus angulos &longs;ingulos a circulo reflectatur; vis qua &longs;ingulis re
flexionibus impingit in circulum erit ut ejus velocitas: adeoque
&longs;umma virium in dato tempore erit ut velocitas illa & numerus re
flexionum conjunctim: hoc e&longs;t (&longs;i polygonum detur &longs;pecie) ut longi
tudo dato illo tempore de&longs;cripta & longitudo eadem applicata ad
Radium circuli; id e&longs;t, ut quadratum longitudinis illius applicatum
ad Radium: adeoque, &longs;i polygonum lateribus infinite diminutis co
incidat cum circulo, ut quadratum arcus dato tempore de&longs;cripti ap
plicatum ad radium. Hæc e&longs;t vis centrifuga, qua corpus urget cir
culum: & huic æqualis e&longs;t vis contraria, qua circulus continuo re
pellit corpus centrum ver&longs;us.
ribus ad commune aliquod centrum tendentibus de&longs;cribit, centrum
illud invenire.
Figuram de&longs;criptam tangant rectæ tres
punctis totidem
erigantur perpendicula
punctis illis
id e&longs;t, ita ut &longs;it
perpendiculorum terminos
DBE, EC
rent in centro qæ&longs;ito
Nam perpendicula a centro
in tangentes
Corol. 1. Prop.I.) &longs;unt reciproce
ut velocitates corporis in punctis adeoque per con&longs;tructio
nem ut perpendicula
recte, id e&longs;t ut perpendicula a pun
cto Un
de facile colligitur quod puncta Et &longs;imili
argumento puncta
am in una recta; & propterea centrum
ver&longs;atur.
que revolvatur, & arcum quemvis jamjam na&longs;centem tempore quàm
minimo de&longs;cribat, & &longs;agitta arcus duci intelligatur quæ chordam bi
&longs;ecet, & producta tran&longs;eat per centrum virium: erit vis centripeta
in medio arcus, ut &longs;agitta directe & tempus bis inver&longs;e.
Nam &longs;agitta dato tempore e&longs;t ut vis (per Corol.4 Prop.I,) & augen
do tempus in ratione quavis, ob auctum arcum in eadem ratione &longs;a
gitta augetur in ratione illa duplicata (per Corol. 2 & 3, Lem.
XI,) ad
eoque e&longs;t ut vis &longs;emel & tempus bis. Subducatur duplicata ratio tempo
ris utrinque, & fiet vis ut &longs;agitta directe & tempus bis inver&longs;e.
Idem facile demon&longs;tratur etiam per Corol.
4 Lem.
X.
circa centrum
curvam
quovis
quovis Curvæ puncto
demittatur
ad di&longs;tantiam illam
tripeta erit reciproce ut &longs;olidum
(
titas, quæ ultimò fit ubi coeunt puncta
guli
proportionale e&longs;t, ideoque pro temporis exponente &longs;cribi pote&longs;t.
CORPORUM
(
bis tangentem Nam rectangula
æquantur.
culo quam minimum continet, eandem habens curvaturam eundem
que radium curvaturæ ad punctum contactus
&longs;it circuli hujus a corpore per centrum virium acta: erit vis centri
peta reciproce ut &longs;olidum
& chorda illa inver&longs;e. Nam velocitas e&longs;t reciproce ut perpendicu
lum I Prop.
I.
detur etiam punctum
inveniri pote&longs;t lex vis centripetæ, qua corpus quodvis
rectilineo perpetuò retractum in figuræ illius perimetro detinebitur
eamque revolvendo de&longs;cribet. Nimirum computandum e&longs;t vel &longs;o
lidum (
portionale. Ejus rei dabimus exempla in Problematis &longs;equentibus.
petæ tendentis ad punctum quodcunQ.E.D.tum.
E&longs;to Circuli circumferentia
quod vis ceu ad
dit
latum
movebitur
ad locum priorem
punctum
& acta circuli diametro
gatur
perpendiculum
&longs;it & occurrat tum circulo in
ob &longs;imilia triangula
e&longs;t
(
(
Sic fiet (
Corol.1 & 5 Prop.VI.) vis centripeta e&longs;t reciproce ut (
id e&longs;t, (ob datum
altitudinis
Ad tangentem
& ob &longs;imilia triangula
peta e&longs;t reciproce ut (
proce ut
per tendit, locetur in circumferentia hujus circuli, puta ad
vis centripeta reciproce ut quadrato cubus altitudinis
culo
idem
tempore periodico circum aliud quod
vis virium centrum
ut
quæ a primo virium centro
bis tangentem
tiæ corporis a &longs;ecundo virium centro
parallela e&longs;t. Nam, per con&longs;tructionem hujus Propo&longs;itionis, vis
prior e&longs;t ad vim po&longs;teriorem, ut
triangula
CORPORUM
centrum
orbe eodemque tempore periodico circum aliud quodvis virium
centrum
&longs;tantia corporis a primo virium centro
a &longs;ecundo virium centro
rium centro
cundo virium centro di&longs;tantiæ Nam vires in
hoc Orbe, ad ejus punctum quodvis
eju&longs;dem curvaturæ.
vis centripetæ tendentis ad punctum adeo longinquum
omnes
A Circuli centro
perpendiculariter &longs;ecans in
triangula
e&longs;t
æquale e&longs;t rectangulo &longs;ive coeuntibus punctis
angulo
ad
æquale (2
E&longs;t ergo (per Corol. 1 & 5 Prop.
VI.) vis centripeta reciproce ut
(2
reciproce ut
Idem facile colligitur etiam ex Propo&longs;itione præcedente.
Et &longs;imili argumento corpus movebitur in Ellip&longs;i vel etiam in
Hyperbola vel Parabola, vi centripeta quæ &longs;it reciproce ut cu
bus ordinatim applicatæ ad centrum virium maxime longinquum
tendentis.
vis centripetæ tendentis ad
centrum Spiralis.
Detur angulus indefinite par
vus
angulos dabitur &longs;pecie figura
(
& recta
Lemma XI.) in duplicata ratione ip&longs;ius
(
e&longs;t ut 1 & 5 Prop.
VI.) vis centripeta e&longs;t
reciproce ut cubus di&longs;tantiæ
Perpendiculum
tangentis chorda
ideoque 3 & 5 Prop.VI.)
reciproce ut vis centripeta.
qua&longs;vis conjugatas de&longs;cripta, e&longs;&longs;e inter &longs;e æqualia.
Con&longs;tat ex Conicis.
CORPORUM
centrum Ellip&longs;eos.
Sunto
gatæ;
cata ad diametrum
parallelogrammum
cis)
ut
quad.
triangula
quad.
ctis rationibus,
ad
quad.
&
quad.
(
ad (
pro
æquale (25 Prop.
VI.) vis centri
peta reciproce ut (2
reciproce ut (1/
In
Et quoniam ex Conicis est
etiam per punctum
eju&longs;dem erit curvaturæ cum &longs;ectione conica in
(2
proce ut (23 Prop.
VI.) hoc e&longs;t (ob
datum 2
PRIMUS.
vici&longs;&longs;im, &longs;i vis &longs;it ut di&longs;tantia, movebitur corpus in Ellip&longs;i centrum
habente in centro virium, aut forte in Circulo, in quem utique
Ellip&longs;is migrare pote&longs;t.
cum centrum idem factarum periodica tempora. Nam tempora
illa in Ellip&longs;ibus &longs;imilibus æqualia &longs;unt per Corol. 3 & 8, Prop.
IV:
in Ellip&longs;ibus autem communem habentibus axem majorem, &longs;unt ad
invicem ut Ellip&longs;eon areæ totæ directe & arearum particulæ &longs;imul
de&longs;criptæ inver&longs;e; id e&longs;t, ut axes minores directe & corporum ve
locitates in verticibus principalibus inver&longs;e; hoc e&longs;t, ut axes illi mi
nores directe & ordinatim applicatæ ad axes alteros inver&longs;e; & prop
terea (ob æqualitatem rationum directarum & inver&longs;arum) in ra
tione æqualitatis.
Si Ellip&longs;is, centro in infinitum abeunte vertatur in Parabolam,
corpus movebitur in hac Parabola; & vis ad centrum infinite di
&longs;tans jam tendens evadet æquabilis. Hoc e&longs;t Theorema
Et &longs;i coni &longs;ectio Parabolica, inclinatione plani ad conum &longs;ectum
mutata, vertatur in Hyperbolam, movebitur corpus in hujus pe
rimetro, vi centripeta in centrifugam ver&longs;a. Et quemadmo
dum in Circulo vel Ellip&longs;i, &longs;i vires tendunt ad centrum figuræ
in Ab&longs;ci&longs;&longs;a po&longs;itum, hæ vires augendo vel diminuendo Ordinatas in
ratione quacunQ.E.D.ta, vel etiam mutando angulum inclinationis
Ordinatarum ad Ab&longs;ci&longs;&longs;am, &longs;emper augentur vel diminuuntur in
ratione di&longs;tantiarum a centro, &longs;i modo tempora periodica maneant
æqualia: &longs;ic etiam in figuris univer&longs;is, &longs;i Ordinatæ augeantur vel di
minuantur in ratione quacunQ.E.D.ta, vel angulus ordinationis ut
cunque mutetur, manente tempore periodico; vires ad centrum
quodcunQ.E.I. Ab&longs;ci&longs;&longs;a po&longs;itum tendentes a binis quibu&longs;vis figurarum locis, ad quæ termi
nantur Ordinatæ corre&longs;pondentibus Ab&longs;ci&longs;&longs;arum punctis in&longs;i&longs;tentes, augentur vel &c. augentur vel diminuun
tur in ratione di&longs;tantiarum a centro.
CORPORUM
tis ad umbilicum Ellip&longs;eos.
E&longs;to Ellip&longs;eos umbilicus
tum diametrum
lem e&longs;&longs;e &longs;emiaxi ma
jori
acta ab altero Ellip
&longs;eos umbilico
nea
rallela, (ob æquales
&longs;emi&longs;umma &longs;it ip&longs;a
rum
(ob parallelas
PR
les
ip&longs;arum
quæ
totum 2
quant. Ad
mittatur perpendicularis
(&longs;eu (2
2 Lem. VII.)
e&longs;t ratio æqualitatis; &
ut
Lem XII.) ut
nibus, Ducantur hæc æqualia in
(1
& 5 Prop. VI.) vis centripeta reciproce e&longs;t ut
proce in ratione duplicata di&longs;tantiæ
PRIMUS.
Cum vis ad centrum Ellip&longs;eos tendens, qua corpus
illa revolvi pote&longs;t, &longs;it (per Corol. I Prop.
X) ut
poris ab Ellip&longs;eos centro
genti
&longs;eos punctum
(3 Prop.
VII,) hoc e&longs;t, &longs;i punctum
cus Ellip&longs;eos, adeoque
Eadem brevitate qua traduximus Problema quintum ad Parabo
lam, & Hyperbolam, liceret idem hic facere: verum ob dignita
tem Problematis & u&longs;um ejus in &longs;equentibus, non pigebit ca&longs;us ce
teros demon&longs;tratione confirmare.
dentis ad umbilicum figuræ.
Sunto
jugatæ;
applicata ad diametrum
rallelogrammum
ver&longs;o
adeo ut
rallelas
PH,Ad
mittatur perpendicularis
pali (&longs;eu (2
id e&longs;t, ut 2.
Lem. VII.)
ratio æqualitatis; &
ad XII.) ut
ad
tionalia Ducantur hæc æqualia in
(I
& 5 Prop. VI.) vis centripeta reciproce e&longs;t ut
reciproce in ratione duplicata di&longs;tantiæ
CORPORUM
PRIMUS.
Inveniatur vis quæ tendit ab Hyperbolæ centro
di&longs;tantiæ Inde vero (per Corol.
3 Prop.
VII.)
vis ad umbilicum
reciproce ut
Eodem modo demon&longs;tratur quod corpus, hac vi centripeta in
centrifugam ver&longs;a, movebitur in Hyperbola conjugata.
plum di&longs;tantiæ verticis illius ab umbilico figuræ.
tur, medium e&longs;t proportionale inter di&longs;tantias umbilici a puncto con
tactus & a vertice principali figuræ.
Sit enim
lis
contactus,
ordinatim ap
plicata ad dia
metrum prin
cipalem,
tangens dia
metro princi
pali occurrens
in
linea perpen
dicularis ab umbilico in tangentem. Jungatur
&longs;imile triangulis æqualibus
ut
CORPORUM
quæ ab umbilico in ip&longs;am perpendicularis e&longs;t, incidit in rectam
quæ Parabolam tangit in vertice principali.
petæ tendentis ad umbilicum hujus figuræ.
Maneat con&longs;tructio Lemmatis, &longs;itque
rabolæ, & a loco
parallelam
rallelam & occurrentem tum diametro
latera
Sed, ex Conicis, quadratum ordinatæ
latere recto & &longs;egmento diametri XIII.) rectangu
lo 4
tio 2 Lem.
VII.) fit ratio æqualitatis.
Er
go
in ca&longs;u, æquale
e&longs;t rectangu
lo 4
E&longs;t autem (ob
&longs;imilia trian
gula
SPN)
ad
hoc e&longs;t (per
Corol. 1. Lem.
XIV.) ut
ad 4IX. Lib.
v.
Elem.)
4Ducantur hæc æqualia in (
(1 & 5
Prop. VI.) vis centripeta e&longs;t reciproce ut
tam 4
cunque cum velocitate exeat de loco
ciproce proportionalis quadrato di&longs;tantiæ loeorum a centro, &longs;imul
agitetur; movebitur hoc corpus in aliqua &longs;ectionum Conicarum
umbilicum habente in centro virium; & contra. Nam datis umbi
lico & puncto contactus & po&longs;itione tangentis, de&longs;cribi pote&longs;t &longs;ectio
Conica quæ curvaturam datam ad punctum illud habebit. Datur
autem curvatura ex data vi centripeta: & Orbes duo &longs;e mutuo tan
gentes, eadem vi centripeta de&longs;cribi non po&longs;&longs;unt.
PRIMUS.
&longs;it, qua lineola
po&longs;&longs;it, & vis centripeta potis &longs;it eodem tempore corpus idem mo
vere per &longs;patium
ctione, cujus latus rectum principale e&longs;t quantitas illa (
ultimo fit ubi lineolæ Circu
lum in his Corollariis refero ad Ellip&longs;in, & ca&longs;um excipio ubi cor
pus recta de&longs;cendit ad centrum.
peta &longs;it reciproce in duplicata ratione di&longs;tantiæ loeorum a centro;
dico quod Orbium Latera recta principalia &longs;unt in duplicata ratio
one arearum quas corpora, radiis ad centrum ductis, eodem tempore
de&longs;cribunt.
Nam, per Corol.
2. Prop.
XIII, Latus rectum
titati (
minima
e&longs;t (per Hypothe&longs;in) reciproce ut
CORPORUM
lum &longs;ub axibus, e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione
lateris recti & ratione temporis periodici. Namque area tota e&longs;t
ut area ducta in tempus periodicum.
&longs;e&longs;quiplicata majorum axium.
Namque axis minor e&longs;t medius proportionalis inter axem majo
rem & latus rectum, atque adeo rectangulum &longs;ub axibus e&longs;t in ra
tione compo&longs;ita ex &longs;ubduplicata ratione lateris recti & &longs;e&longs;quiplicata
ratione axis majoris. Sed hoc rectangulum, per Corollarium Prop.
XIV. e&longs;t in ratione compo&longs;ita ex &longs;ubduplicata ratione lateris recti
& ratione periodici temporis. Dematur utrobique &longs;ubduplicata
ratio lateris recti, & manebit &longs;e&longs;quiplicata ratio majoris axis æqua
lis rationi periodici temporis.
Circulis, quorum diametri æquantur majoribus axibus Ellip&longs;eon.
bitas, demi&longs;&longs;i&longs;que ab umbilico communi ad has tangentes perpendi
cularibus: dico quod Velocitates corporum &longs;unt in ratione compo&longs;i
ta ex ratione perpendiculorum inver&longs;e & &longs;ubduplicata ratione la
terum rectorum principalium directe.
Ab umbilico
& velocitas corporis
titatis (
in data temporis particula de&longs;criptus, hoc e&longs;t (per Lem. VII.) ut
tangens
(XIV.
PRIMUS.
duplicata ratione perpendiculorum & duplicata ratione veloci
tatum.
lico communi di&longs;tantiis, &longs;unt in ratione compo&longs;ita ex ratione di
&longs;tantiarum inver&longs;e & &longs;ubduplicata ratione laterum rectorum princi
palium directe. Nam perpendicula jam &longs;unt ip&longs;æ di&longs;tantiæ.
minima ab umbilico di&longs;tantia, e&longs;t ad velocitatem in Circulo in ea
dem à centro di&longs;tantia, in &longs;ubduplicata ratione lateris recti princi
palis ad duplam illam di&longs;tantiam.
ocribus di&longs;tantiis ab umbilico communi &longs;unt eædem quæ corporum
gyrantium in Circulis ad ea&longs;dem di&longs;tantias; hoc e&longs;t (per Corol 6.
Prop. IV.) reciproce in &longs;ubduplicata ratione di&longs;tantiarum.
Nam
perpendicula jam &longs;unt &longs;emi-axes minores; & hi &longs;unt ut mediæ
proportionales inter di&longs;tantias & latera recta. Componatur hæc
ratio inver&longs;e cum &longs;ubduplicata ratione laterum rectorum directe, &
fiet ratio &longs;ubduplicata di&longs;tantiarum inver&longs;e.
ut perpendiculum demi&longs;&longs;um ab umbilico ad tangentem.
CORPORUM
tione di&longs;tantiæ corporis ab umbilico figuræ; in Ellip&longs;i magis varia
tur, in Hyperbola minus, quam in hac ratione. Nam (per Corol.
2. Lem. XIV.) perpendiculum demi&longs;&longs;um ab umbilico ad tangentem
Parabolæ e&longs;t in &longs;ubduplicata ratione di&longs;tantiæ. In Hyperbola per
pendiculum minus variatur, in Ellip&longs;i magis.
co di&longs;tantiam, e&longs;t ad velocitatem corporis revolventis in Circulo
ad eandem a centro di&longs;tantiam, in &longs;ubduplicata ratione numeri bi
narii ad unitatem; in Ellip&longs;i minor e&longs;t, in Hyperbola major quam
in hac ratione. Nam per hujus Corollarium &longs;ecundum, velocitas
in vertice Parabolæ e&longs;t in hac ratione, & per Corollaria &longs;exta hu
jus & Propo&longs;itionis quartæ, &longs;ervatur eadem proportio in omnibus
di&longs;tantiis. Hinc etiam in Parabola velocitas ubique æqualis e&longs;t ve
locitati corporis revolventis in Circulo ad dimidiam di&longs;tantiam, in
Ellip&longs;i minor e&longs;t, in Hyperbola major.
locitatem gyrantis in Circulo in di&longs;tantia dimidii lateris recti princi
palis Sectionis, ut di&longs;tantia illa ad perpendiculum ab umbilico in
tangentem Sectionis demi&longs;&longs;um. Patet per Corollarium quintum.
6. Prop.
IV.) velocitas gyrantis
in hoc Circulo &longs;it ad velocitatem gyrantis in Circulo quovis alio,
reciproce in &longs;ubduplicata ratione di&longs;tantiarum; fiet ex æquo velo
citas gyrantis in Conica &longs;ectione ad velocitatem gyrantis in Circulo
in eadem di&longs;tantia, ut media proportionalis inter di&longs;tantiam illam
communem & &longs;emi&longs;&longs;em principalis lateris recti &longs;ectionis, ad per
pendiculum ab umbilico communi in tangentem &longs;ectionis de
mi&longs;&longs;um.
&longs;tantiæ loeorum a centro, & quod vis illius quantitas ab&longs;oluta &longs;it
cognita; requiritur Linea quam corpus de&longs;cribit, de loco dato, cum
data velocitate, &longs;ecundum datam rectam egrediens.
Vis centripeta tendens ad punctum
bita quavis data
&longs;ectionem
recta aliqua
intelligantur perpendicula, erit (per Corol. 1. Prop.
XVI.) latus re
ctum principale Coni&longs;ectionis ad latus rectum principale Orbitæ, in
ratione compo&longs;ita ex duplicata ratione perpendiculorum & dupli
cata ratione velocitatum, atque adeo datur. Sit i&longs;tud
tur præterea Coni&longs;e
ctionis umbilicus
Anguli
plementum ad du
os rectos fiat angu
lus
tur po&longs;itione linea
alter De
mi&longs;&longs;o ad
diculo
―SP+PH: quad. -LX―SP+PH=SPq.+2SPH+PHq.
-LX―SP+PH.
+LX―SP+PH,
&longs;eu
tam longitudine quam po&longs;itione. Nimirum &longs;i ea fit corporis &c.
in
velocitas, ut latus rectum
jacebit
adeoque figura erit Ellip&longs;is, & ex datis umbilicis
principali
latus rectum
nita erit, & propterea figura erit Parabola axem habens
lelum lineæ Quod &longs;i corpus majori adhuc
cum velocitate de loco &longs;uo
ad alteram partem tangentis, adeoque tangente inter umbilicos per
gente, figura erit Hyperbola axem habens principalem æqualem dif
ferentiæ linearum
PRIMUS.
latere recto
ad
4
divi&longs;im
CORPORUM
invenietur Orbita expedite, capiendo &longs;cilicet latus rectum ejus, ad
duplam di&longs;tantiam
ad velocitatem corporis in Circulo, ad di&longs;tantiam
Corol. 3. Prop.
XVI.) dein
tiam inter latus rectum & 4
Conica, & ex Orbe &longs;uo impul&longs;u quocunque exturbetur; cogno&longs;ci
pote&longs;t Orbis in quo po&longs;tea cur&longs;um &longs;uum peraget. Nam componen
do proprium corporis motum cum motu illo quem impul&longs;us &longs;olus
generaret, habebitur motus quocum corpus de dato impul&longs;us loco,
&longs;ecundum rectam po&longs;itione datam, exibit.
nuo perturbetur, innote&longs;cet cur&longs;us quam proxime, colligendo mu
tationes quas vis illa in punctis quibu&longs;dam inducit, & ex &longs;eriei ana
logia mutationes continuas in locis intermediis æ&longs;timando.
Si corpus
punctum quodcunQ.E.D.tum
tendente moveatur in perimetro
datæ cuju&longs;cunque Sectionis co
nicæ cujus centrum &longs;it
quiratur Lex vis centripetæ: du
catur
la, & Orbis tangenti
currens in
Corol. 1 & Schol.
Prop.
X, & Corol.
3 Prop.
VII.) erit ut
(
PRIMUS.
rum ex umbilico dato.
punctum quodvis tertium
perpendiculo
TR
get: & contra, &longs;i tangit, erit
Secet enim perpendiculum
ctam
in
Unde punctum
cas & Hyperbolicas, quæ tran&longs;ibunt per puncta data, & rectas po
&longs;itione datas contingent.
Sit
cipalis Trajectoriæ cuju&longs;vis;
bet tran&longs;ire; & Centro
vallo
perbola, de&longs;cribatur circulus
perpendiculum
tr,
Sit
munis, & umbilicis
dato de&longs;cribatur Trajectoria.
Dico factum. Nam Trajecto
ctoria de&longs;cripta (eo quod
+SP
in Hyperbola æquatur axi)
tran&longs;ibit per punctum
(per Lemma &longs;uperius) tanget
rectam
mento vel tran&longs;ibit eadem per
puncta duo
ctas duas q.E.F.
CORPORUM
tran&longs;ibit per puncta data, & rectas po&longs;itione datas continget.
Sit
bendæ. Centro
culum
te perpendicularem
ut &longs;it
bendus e&longs;t alter circulus
punctum
cta
dantur duo puncta
puncta
tangat circulum
&longs;i datur punctum
rem
&longs;cribatur Parabola. Dico factum.
Nam Parabola, ob æquales
matis XIV. Corol. 3.) ob æquales
PRIMUS.
re, quæ per data puncta tran&longs;ibit & rectas tanget pofitione datas.
puncta duo
tio axis principalis ad di&longs;tantiam
umbilieorum. In ea ratione cape
tris
&longs;cribe circulos duos, & ad rectam Dico factum.
Sit enim
alter Figuræ de&longs;criptæ, & cum &longs;it
vi&longs;im ratione,
adeoQ.E.I. ratione quam habet axis principalis Figuræ de&longs;cribendæ
ad di&longs;tantiam umbilieorum ejus; & propterea Figura de&longs;cripta e&longs;t
eju&longs;dem &longs;peciei cum de&longs;cribenda. Cumque &longs;int
ad
ex Conicis manife&longs;tum e&longs;t.
duas Ab umbilico in tangentes demitte
perpendicula
dem ad
quales
& erige perpendiculum infinitum
ductam &longs;eca in
Trajectoriæ de&longs;cribendæ axis prin
cipalis ad umbilieorum di&longs;tantiam.
Super diametro
circulus &longs;ecans Dico factum.
Nam bi&longs;eca
ut
2
angula
adeoque ut
principalis
quam habet Trajectoriæ de&longs;cribendæ axis principalis ad ip&longs;ius um
bilieorum di&longs;tantiam, & propterea eju&longs;dem e&longs;t &longs;peciei. In&longs;uper cum
perpendiculariter bi&longs;ecentur, liquet, ex Lemmate XV, rectas illas
Trajectoriam de&longs;criptam tangere.
CORPORUM
am
dicularem
ad di&longs;tantiam umbilieorum; circuloque &longs;uper diametro
&longs;cripto, &longs;ecetur producta recta
principali rectam Dico fa
ctum. Namque
ut axis principalis Trajectoriæ
de&longs;cribendæ ad di&longs;tantiam um
bilieorum ejus, patet ex demon
&longs;tratis in Ca&longs;u &longs;ecundo, & prop
terea Trajectoriam de&longs;criptam
eju&longs;dem e&longs;&longs;e &longs;peciei cum de&longs;cri
benda; rectam vero
gulus
Conicis.
quæ tangat rectam
tangentem datum, quæque &longs;imilis &longs;it Figuræ In tangentem
pendiculum
gulis autem
troque
lum &longs;ecantem Figuram Denique umbilicis
& axe principali
Conica. Dico factum.
Nam &longs;i agatur
terea
illius umbilieorum intervallum, ut axis
vallum
gulum
axi &
rectam
PRIMUS.
quarum differentiæ vel dantur vel nullæ &longs;unt.
quod invenire oportet; Ob datam differentiam linearum
locabitur punctum
principalis axis differentia illa data. Sit axis ille
demi&longs;&longs;aque
perbolæ,
palis axis differentia inter
perpendicularis, ad quam &longs;i ab Hyperbolæ hujus puncto quovis
demittatur normalis
ad
dem
ideoque &longs;i rectæ
rant in
catur, dabitur po&longs;itione. Eadem
methodo per Hyperbolam ter
tiam, cujus umbilici &longs;unt
& axis principalis differentia re
ctarum
alia recta in qua
Habitis autem duobus Locis recti
lineis, habetur punctum quæ&longs;itum
CORPORUM
ctum
cus alius rectilineus invenietur ut &longs;upra.
Circuli per puncta
Solvitur etiam hoc Lemma problematicum per Librum Tactio
num
puncta data & rectas po&longs;itione datas continget.
Detur umbilicus
dus &longs;it umbilicus alter
qualis axi principali. Junge
Hoc modo &longs;i dentur plures tangen-
æquantur axibus, vel datis longitu
dinibus
que adeo quæ vel æquantur &longs;ibi invi
cem, vel datas habent differentias; &
inde, per Lemma &longs;uperius, datur umbi
licus ille alter
bilicis una cum axis longitudine (quæ
vel e&longs;t
perbola,
PRIMUS.
Ca&longs;us ubi dantur tria puncta &longs;ic &longs;olvitur expeditius.
Dentur
puncta
& productam demitte normales
producta cape
major, æqualis, vel minor fuerit quam
vel Hyperbola; pun
cto
cadente ad eandem
partem lineæ
cum puncto
&longs;ecundo ca&longs;u abeunte
in infinitum; in tertio
cadente ad contrari
am partem lineæ
Nam &longs;i demittantur
ad
ci&longs;&longs;im
probabitur e&longs;&longs;e Jacent ergo puncta
C, D
ab umbilico
a punctis ii&longs;dem ad rectam
Methodo haud multum di&longs;&longs;imili hujus problematis &longs;olutionem
tradit Clari&longs;&longs;imus Geometra
VIII. Prop. XXV.
CORPORUM
ABDC,
producta
tarum ad oppo&longs;ita duo latera
tarum ad alia duo latera oppo&longs;ita
oppo&longs;ita latera ductas parallelas e&longs;
&longs;e alterutri reliquorum laterum,
puta
ac
latera duo ex oppo&longs;itis, puta
&
la. Et recta quæ bi&longs;ecat paralle
la illa latera erit una ex diametris
Conicæ &longs;ectionis, & bi&longs;ecabit eti
am
Produc
applicata ad contrarias partes diametri. Cum igitur puncta
P
gulo, erit (per Prop.17 & 18 Lib. III Conieorum
lum Sed
æquales &longs;unt, utpote æqualium
& inde etiam rectangula
que adeo rectangulum
e&longs;t ad rectangulum
Age
& ip&longs;i
&longs;ecantem
Jam ob &longs;imilia triangula
DBN
Ergo, ducendo antecedentes in
antecedentes & con&longs;equentes in
con&longs;equentes, ut rectangulum
in
ad rectangulum
ad rectangulum
e&longs;t ad rectangulum
PRIMUS.
quatuor
e&longs;&longs;e parallelas lateribus
&longs;ed ad ea utcunQ.E.I.clinatas. Ea
rum vice age
ip&longs;i
ip&longs;i
los triangulorum
PSs, PTt,
ad
ad
mon&longs;trata, ratio
pezii
ra
tanget Conicam &longs;ectionem circa Trapezium de&longs;criptam.
Per puncta
&longs;emper tangere. Si negas,
junge
Conicam &longs;ectionem alibi
quam in
puta in
punctis
datis angulis ad latera Tra
pezii rectæ
&
ut
(per Lem. XVII)
ad
Hypoth.)
ter &longs;imilitudinem Trapeziorum
terminos corre&longs;pondentes hujus, erit
go Trapezia æquiangula
diagonales Incidit itaque
inter&longs;ectionem rectarum
Conicam &longs;ectionem.
CORPORUM
ad alias totidem po&longs;itione datas rectas
&longs;ingulas, in datis angulis ducantur, &longs;itque rectangulum &longs;ub duabus
ductis
punctum
quæ tangit lineas Nam coeat linea
in coeat etiam linea
evadet
C
plius &longs;ecare po&longs;&longs;unt &longs;ed tantum tangent.
Nomen Conicæ &longs;ectionis in hoc Lemmate late &longs;umitur, ita ut
&longs;ectio tam Rectilinea per verticem Coni tran&longs;iens, quam Circularis
ba&longs;i parallela includatur. Nam &longs;i punctum
quævis ex punctis quatuor
guntur. Si Trapezii anguli duo oppo&longs;iti &longs;imul &longs;umpti æquentur
duobus rectis, & lineæ quatuor
latera ejus vel perpendiculariter vel in angulis quibu&longs;vis æqualibus,
&longs;itque rectangulum &longs;ub duabus ductis
lo &longs;ub duabus aliis Idem
fiet &longs;i lineæ quatuor ducantur in angulis quibu&longs;vis & rectangulum
&longs;ub duabus ductis
duæ ultimæ
lorum Cæteris
in ca&longs;ibus Locus puncti
nominantur Sectiones Conicæ. Vice autem Trapezii
&longs;titui pote&longs;t Quadrilaterum cujus latera duo oppo&longs;ita &longs;e mutuo in
&longs;tar diagonalium decu&longs;&longs;ant. Sed & e punctis quatuor
po&longs;&longs;unt unum vel duo abire ad infinitum, eoque pacto latera fi
guræ quæ ad puncta illa convergunt, evadere parallela: quo in
ca&longs;u Sectio Conica tran&longs;ibit per cætera puncta, & in plagas paralle
larum abibit in infinitum.
PRIMUS.
tas rectas
angulis ducantur,
&longs;ub duabus ductis,
duabus,
tione.
Lineæ
gulorum continentes ducuntur, conveniant cum aliis duabus po&longs;i
tione datis lineis in punctis
rectam quamlibet Secet ea
lineas oppo&longs;itas
datos omnes angulos figuræ, dabuntur rationes
tione
dabitur ratio
addendo datas rationes
ratio
punctum
CORPORUM
punctorum infinitorum
ctum quodvis
pote&longs;t. Nam chorda
puncta
e&longs;t, ubi Quo in ca&longs;u,
ultima ratio evane&longs;centium Ip&longs;i
igitur
tima ratione &longs;ectam in
& evane&longs;cens
Per quodvis punctorum
occurrentem Loco in
tur autem punctum XIX.
Bi&longs;eca
Hæc
erit
ver&longs;um, ad quod latus rectum erit
ut
occurrit Loco, linea
infinita, Locus erit Parabola & la
rum rectum ejus ad diametrum
pertinens erit (
ubi puncta
ubi
AtQ.E.I.a Problematis Veterum de quatuor lineis ab
ti & ab
ca, qualem Veteres quærebant, in hoc Corollario exhibetur.
PRIMUS.
P
bus unius angulorum illorum infinite productis
eidem &longs;ectioni Conicæ in
C
tæ duæ
rallelogrammi lateribus
laterum partesEt contra, &longs;i
partes illæ ab&longs;ci&longs;&longs;æ &longs;unt ad invicem in data ratione, punctum
get Sectionem Conicam per puncta quatuor
occurrat
lela &longs;it ipfi
& erit (per Lemma XVII.) re
ctangulum
ctangulum
tione data. Sed e&longs;t
adeoque ut
vici&longs;&longs;im
ad
ut
ad rectangulum
gulum Sed dantur
&
cem, tum &longs;imili ratiocinio regrediendo, &longs;equetur e&longs;&longs;e rectangulum
punctum
tran&longs;euntem per puncta
CORPORUM
Conicæ &longs;ectionis ad punctum
cum puncto
dat; &
vis Conicæ &longs;ectionis punc
tum
CD
ut
&longs;i &longs;it
ad Conicæ Sectionis punc
um aliquod
non &longs;ecat Conicam &longs;ectio
nem in punctis pluribus quam quatuor. Nam, &longs;i fieri pote&longs;t, tran&longs;
eant duæ Conicæ &longs;ectiones per quinque puncta
que &longs;ecet recta
in r. Ergo
invicem æquantur, contra Hypothe&longs;in.
polos ductæ, concur&longs;u &longs;uo
tam rectam
prioribus duabus ad puncta illa data
MBD, MCD
CD
B, C Et vice ver&longs;a, &longs;i rectæ
&longs;uo
ABC,
PRIMUS.
Nam in recta
tum
occurrentes ip&longs;is
CD
cientes angulum
æqualem angulo dato
gulo dato
ergo (ex Hypothe&longs;i)
æquales &longs;int anguli
anguli
aufer communes
&
æquales
NCM
& triangula
immobilia. Ergo
indeQ.E.D.tam rationem inter &longs;e; atque adeo, per Lemma xx,
punctum
contingit &longs;ectionem Conicam, per puncta
Et contra, &longs;i punctum mobile
tran&longs;euntem per data puncta
æqualis angulo dato
lo dato
ctionis puncta immobilia
in puncta duo immobilia
& hæc erit Locus perpetuus puncti illius mobilis
pote&longs;t, ver&longs;etur punctum Tanget ergo
punctum
tran&longs;euntem, ubi punctum Sed
& ex jam demon&longs;tratis tanget etiam punctum
cam per eadem quinque puncta
Ergo duæ &longs;ectiones Co
nicæ tran&longs;ibunt per eadem quinque puncta, contra Corol. 3. Lem.
xx. Igitur punctum
CORPORUM
Dentur puncta quinque
alia duo quævis
hi&longs;que parallelas
inde a polis duobus
tas duas
rem priori & po&longs;teriorem po&longs;teriori) occurrentes in
niQ.E.D. rectis
&longs;cinde qua&longs;vis
earum terminos Nam punc
tum illud
puncta quatuor
&longs;centibus, coit punctum
nica per puncta quinque
PRIMUS.
E punctis datis junge tria quævis
ACB,
ra
punctum
ad punctum
tentur puncta
quibus altera crura
&longs;e decu&longs;&longs;ant. Agatur
recta infinita
rotentur anguli illi mo
biles circum polos &longs;uos
rum
quæ jam &longs;it
&longs;emper in rectam illam
infinitam
rum
riam quæ&longs;itam Nam punctum
XXI, continget &longs;ectionem Conicam per puncta
ubi punctum
&longs;tructionem) accedet ad puncta
tio Conica tran&longs;iens per puncta quinque
&longs;itam, in puncto quovis dato Accedat punctum
punctum
recta inveniri po&longs;&longs;unt, ut in Corollario &longs;ecundo Lemmatis XIX.
Con&longs;tructio prior evadet paulo &longs;implicior jungendo
&longs;i opus e&longs;t, producta capiendo
per
capiendo &longs;emper
currentes in
pHac methodo puncta Trajectoriæ inveniuntur expediti&longs;&longs;ime,
chanice.
CORPORUM
tam continget po&longs;itione datam.
puncta
&
Age
nique, agendo quamvis
ab&longs;cinde
actarum xx) incidet &longs;emper in
Trajectoriam de&longs;cribendam.
PRIMUS.
Revolvatur tum angulus magnitudine datus
ca polum
&longs;ecat radium illum ubi crus alterum
dio in punctis
currant perpetuo radius ille
cruris alterius
quæ&longs;itam.
Nam &longs;i in con&longs;tructionibus Problematis &longs;uperioris accedat punc
tum
ultimo &longs;uo &longs;itu fiet tangens
&longs;itæ evadent eædem cum con&longs;tructionibus hic de&longs;criptis. Delinea
bit igitur cruris
puncta
Junge bina lineis
gulum &longs;ub media proportio
nali inter
dia proportionali inter
dia proportionali inter
ter
tum contactus. Nam &longs;i rectæ
riam &longs;ecet in punctis quibu&longs;
vis
punctum
tione compo&longs;ita ex ratione rectanguli
&longs;eu rectanguli
guli
puncto
Capi autem pote&longs;t punctum
& perinde Trajectoria dupliciter de&longs;cribi.
CORPORUM
duas po&longs;itione datas continget.
Dentur tangentes
puncta
duo quævis
finitam
rentem in punctis
etiam per alia duo quævis
age infinitam
currentem in punctis
ita &longs;eca in
inter
proportionalem inter
&
portionalis inter
diam proportionalem inter
& Nam &longs;i
&longs;upponantur e&longs;&longs;e puncta contactuum alicubi in tangentibus &longs;i
ta; & per punctorum
terutra
lela, quæ occurrat curvæ in
dia proportionalis inter
gulum
ta Porro tangentibus concurrentibus in
rit (ex Conicis) rectangulum
ergo puncta
&longs;unt in una recta. Et eodem argumento probabitur quod puncta
Jacent igitur puncta contactuum
&
ut in ca&longs;u primo Problematis &longs;uperioris.
PRIMUS.
Tran&longs;mutanda &longs;it figura quævis
rectæ duæ parallelæ
& a figuræ puncto quo
vis
ducatur quævis
ip&longs;i De
inde a puncto aliquo
in linea
punctum
recta
currens in
occur&longs;us erigatur recta
habens rationem ad
tum in figura nova Eadem ratione
puncta &longs;ingula figuræ primæ dabunt puncta totidem figura novæ.
nia figuræ primæ, & punctum
puncta omnia figuræ novæ & eandem de&longs;cribet. Di&longs;tinctionis gra
tia nominemus
dium ab&longs;cidentem,
parallelogrammum
CORPORUM
Dico jam quod, &longs;i punctum
tam, punctum Si
punctum
Conicam &longs;ectionem. Conicis &longs;ectionibus hic Circulum annumero.
Porro &longs;i punctum
git Lineam tertii ordinis
Analytici, punctum
tanget Lineam tertii iti
dem ordinis; & &longs;ic de
curvis lineis &longs;uperiorum
ordinum. Lineæ duæ e
runt eju&longs;dem &longs;emper or
dinis Analytici quas pun
cta Et
enim ut e&longs;t
ita &longs;unt
æqualis e&longs;t (
tum
qua relatio inter ab&longs;ci&longs;&longs;am
determinatæ illæ
a&longs;cendunt, &longs;cribendo in hac æquatione (
(
va
dent, atque adeo quæ de&longs;ignat Lineam rectam. Sin
(vel earum alterutra) a&longs;cendebant ad duas dimen&longs;iones in æquati
one prima, a&longs;cendent itidem
da. Et &longs;ic de tribus vel pluribus dimen&longs;ionibus.
Indeterminatæ
per ad eundem dimen&longs;ionum numerum, & propterea Lineæ, quas
puncta
Dico præterea quod &longs;i recta aliqua tangat lineam curvam in fi
tran&longs;lata tanget lineam illam curvam in figura nova: & contra. Nam
&longs;i Curvæ puncta quævis duo accedunt ad invicem & coeunt in fi
gura prima, puncta eadem tran&longs;lata accedent ad invicem & coibunt
in figura nova, atque adeo rectæ, quibus hæc puncta junguntur, &longs;i
mul evadent curvarum tangentes in figura utraque. Componi po&longs;
&longs;ent harum a&longs;&longs;ertionum Demon&longs;trationes more magis Geometrico.
Sed brevitati con&longs;ulo.
PRIMUS.
Igitur &longs;i figura rectilinea in aliam tran&longs;mutanda e&longs;t, &longs;ufficit rec
tarum a quibus conflatur inter&longs;ectiones transferre, & per ea&longs;dem
in figura nova lineas rectas ducere. Sin curvilineam tran&longs;mutare
oportet, transferenda &longs;unt puncta, tangentes & aliæ rectæ quarum
ope curva linea definitur. In&longs;ervit autem hoc Lemma &longs;olutioni
difficiliorum Problematum, tran&longs;mutando figuras propo&longs;itas in &longs;im
pliciores. Nam rectæ quævis convergentes tran&longs;mutantur in pa
rallelas, adhibendo pro radio ordinato primo, lineam quam
vis rectam quæ per concur&longs;um convergentium tran&longs;it: id adeo quia
concur&longs;us ille hoc pacto abit in infinitum, lineæ autem parallelæ
&longs;unt quæ ad punctum infinite di&longs;tans tendunt. Po&longs;tquam autem
Problema &longs;olvitur in figura nova, &longs;i per inver&longs;as operationes tran&longs;
mutetur hæc figura in figuram primam, habebitur &longs;olutio quæ&longs;ita.
Utile e&longs;t etiam hoc Lemma in &longs;olutione Solidorum Problema
tum. Nam quoties duæ &longs;ectiones Conicæ obvenerint, quarum in
ter&longs;ectione Problema &longs;olvi pote&longs;t, tran&longs;mutare licet earum alter
utram, &longs;i Hyperbola &longs;it vel Parabola, in Ellip&longs;in: deinde Ellip&longs;is
facile mutatur in Circulum. Recta item & &longs;ectio Conica, in con
&longs;tructione Planorum Problematum, vertuntur in Rectam & Cir
culum.
tres continget po&longs;itione datas.
Per concur&longs;um tangentium quarumvis duarum cum &longs;e invicem, &
concur&longs;um tangentis tertiæ cum recta illa, quæ per puncta duo data
tran&longs;it, age rectam infinitam; eaque adhibita pro radio ordinato pri
mo, tran&longs;mutetur figura, per Lemma &longs;uperius, in figuram novam. In
gens tertia fiet parallela rectæ per
puncta duo data tran&longs;eunti. Sunto
parallela tran&longs;iens per puncta illa
figura nova tran&longs;ire debet, & pa
rallelogrammum
Secentur rectæ
ita ut &longs;it
rectanguli
ad
&
rum quarum prima e&longs;t recta
rectangulorum Et
enim, ex Conicis, &longs;unt
&
ea
latus quadratum ip&longs;ius
compo&longs;ite, in data ratione omnium antecedentium
omnes con&longs;equentes, quæ &longs;unt latus quadratum rectanguli
recta
data illa ratione puncta contactuum Per
inver&longs;as operationes Lemmatis novi&longs;&longs;imi transferantur hæc pun
cta in figuram primam & ibi, per Probl. XIV, de&longs;cribetur
Trajectoria.
cent vel inter puncta
inter puncta Si punctorum
terutrum cadit inter puncta
po&longs;&longs;ibile e&longs;t.
CORPORUM
quatuor po&longs;itione datas continget.
Ab inter&longs;ectione communi duarum quarumlibet tangentium ad
inter&longs;ectionem communem reliquarum duarum agatur recta infini-XXII) in figuram novam, & tangentes binæ, quæ ad
radium ordinatum primum concurrebant, jam evadent parallelæ. Sun
to illæ
que
re&longs;pondens. Per figuræ centrum
quali
figura nova tran&longs;ire debet. Per Lemmatis XXII operationem in
ver&longs;am transferatur hoc punctum in figuram primam, & ibi habe
buntur puncta duo per quæ Trajectoria de&longs;cribenda e&longs;t. Per ea
dem vero de&longs;cribi pote&longs;t Trajectoria illa per Prob. XVII.
PRIMUS.
minentur, datamque habeant rationem ad invicem, & recta
CD,
tione data in
tione data.
Concurrant enim rectæ
ad
que
data, & propterea dabitur &longs;pecie
triangulum
in
ne
lam rationem, dabitur etiam &longs;pecie triangulum
punctum Junge
&longs;imilia erunt triangula
rationem
& erit &longs;emper Locatur igitur punc
tum
CORPORUM
rallelæ &longs;int ac dentur po&longs;itione; dico quod Sectionis &longs;emidia
meter hi&longs;ce duabus parallela, &longs;it media proportionalis inter ha
rum &longs;egmenta, punctis contactuum & tangenti tertiæ inter
jecta.
Sunto
rallelæ duæ Coni&longs;ec
tionem
gentes in
recta tertia Coni&longs;ec
tionem tangens in
& occurrens prioribus
tangentibus in
&longs;itque
ter Figuræ tangenti
bus parallela: Dico
quod
&longs;unt continue proportionales.
Nam &longs;i diametri conjugatæ
in
mum
ELI, ECH, EBG) AF
ex natura Sectionum Conicarum,
Q.E.D.
erit (ex æquo perturbate)
ut
F
puncta contactuum
PRIMUS.
nem quamcunque Conicam, & ab&longs;cindantur ad tangentem quamvis
quintam; &longs;umantur autem laterum quorumvis duorum contermi
norum ab&longs;ci&longs;&longs;æ terminatæ ad angulos oppo&longs;itos parallelogrammi:
dico quod ab&longs;ci&longs;&longs;a alterutra &longs;it ad latus illud a quo est ab&longs;ci&longs;&longs;a, ut
pars lateris alterius contermini inter punctum contactus & latus
tertium, est ad ab&longs;ci&longs;&longs;arum alteram.
Tangant parallelogrammi
MI
hæc latera in
&
laterum
&longs;ci&longs;&longs;æ
laterum
&longs;ci&longs;&longs;æ
co quod &longs;it
&
per Corollarium &longs;e
cundum Lemmatis &longs;uperioris, e&longs;t
Item
tionem Conicam de&longs;eriptum, dabitur rectangulum
& huic æquale rectangulum
occurrens in
gulo
per puncta bi&longs;ectionum agatur, tran&longs;ibit hæc per centrum Sectio
nis Conicæ. Nam cum &longs;it
XXIII)
& medium rectæ
CORPORUM
Dentur pofitione tangentes
Figuræ quadrilateræ &longs;ub quatuor quibu&longs;vis contentæ
gonales 3. Lem.
XXV) recta
per puncta bi&longs;ectionum acta tran&longs;ibit per centrum Trajectoriæ.
Rur&longs;us Figuræ quadrilateræ
tangentibus contentæ, diagonales (ut ita dicam)
&longs;eca in
ibit per centrum Trajectoriæ. Dabitur ergo centrum in concur&longs;u bi
&longs;ecantium. Sit illud
ad eam di&longs;tantiam ut centrum
& acta Secet hæc tan-
concur&longs;us
ta
punctis contactuum. Patet hoc per Corol.
2. Lem.
XXIV.
Ea
dem methodo invenire licet alia contactuum puncta, & tum de
mum per Probl. XIV. &c.
Trajectoriam de&longs;cribere.
PRIMUS.
Problemata, ubi dantur Trajectoriarum vel centra vel A&longs;ymp
toti, includuntnr in præcedentibus. Nam datis punctis & tangen
tibus una cum centro, dantur alia totidem puncta aliæque tangen
tes a centro ex altera ejus parte æqualiter di&longs;tantes. A&longs;ymptotos
autem pro tangente habenda e&longs;t, & ejus terminus infinite di&longs;tans
(&longs;i ita loqui fas &longs;it) pro puncto contactus. Concipe tangentis cu
ju&longs;vis punctum contactus abire in infinitum, & tangens vertetur in
A&longs;ymptoton, atque con&longs;tructiones Problematis XIV & Ca&longs;us pri
mi Problematis XV vertentur in con&longs;tructiones Problematum ubi
A&longs;ymptoti dantur.
Po&longs;tquam Trajectoria de&longs;cripta e&longs;t, invenire licet axes & umbi
licos ejus hac methodo. In con&longs;tructione & figura Lemmatis XXI,
fac ut angulorum mobi
lium
ra
concur&longs;u Trajectoria de
&longs;cribebatur, &longs;int &longs;ibi invi
cem parallela, eumque
&longs;ervantia &longs;itum revolvan
tur circa polos &longs;uos
in figura illa. Interea ve
ro de&longs;cribant altera an
gulorum illorum crura
&longs;uo
hujus centrum
hoc centro ad Regulam
currentem in
runt ad punctum illud
& contrarium eveniet &longs;i crura eadem concurrunt ad punctum remo
tius Hi&longs;ce
autem datis, umbilici &longs;unt in promptu.
CORPORUM
Axium vero quadrata &longs;unt ad invicem ut
facile e&longs;t Trajectoriam
&longs;pecie datam per data
quatuor puncta de&longs;cri
bere. Nam &longs;i duo ex
punctis datis con&longs;titu
antur poli
dabit angulos mobiles
tem datis de&longs;cribi pote&longs;t
Circulus
Tum ob datam &longs;pecie
Trajectoriam, dabitur
ratio
eoQ.E.I.&longs;a
tro
de&longs;cribe alium circulum,
& recta quæ tangit hunc circulum, & tran&longs;it per concur&longs;um crurum
tum punctum erit Regula illa
betur. Unde etiam vici&longs;&longs;im Trapezium &longs;pecie datum (&longs;i ca&longs;us qui
dam impo&longs;&longs;ibiles excipiantur) in data quavis Sectione Conica in
&longs;cribi pote&longs;t.
Sunt & alia Lemmata quorum ope Trajectoriæ &longs;pecie datæ,
datis punctis & tangentibus, de&longs;cribi po&longs;&longs;unt. Ejus generis
e&longs;t quod, &longs;i recta linea per punctum quodvis po&longs;itione datum
ducatur, quæ datam Coni&longs;ectionem in punctis duobus inter&longs;e
cet, & inter&longs;ectionum intervallum bi&longs;ecetur, punctum bi&longs;ectionis
tanget aliam Coni&longs;ectionem eju&longs;dem &longs;peciei cum priore, atque
axes habentem prioris axibus parallelos. Sed propero ad magis
utilia.
PRIMUS.
idem po&longs;itione datas, quæ non &longs;unt omnes parallelæ, &longs;ingulos ad
&longs;ingulas ponere.
Dantur po&longs;itione tres rectæ infinitæ
tet triangulum
angulus
& angulus Super
DF
tria circulorum &longs;eg
menta
EMF,
angulos angulis
ABC, ACB
re&longs;pective. De&longs;criban
tur autem hæc &longs;egmen
ta ad eas partes linea
rum
literæ
ordine cum literis
eodem cum literis
literis
redeant; deinde com
pleantur hæc &longs;egmenta
in circulos integros. Se
cent circuli duo prio
res &longs;e mutuo in
que centra eorum
cape
tro
de&longs;cribe circulum, qui &longs;ecet circulum primum
tum
lis & æqualis Figuræ
CORPORUM
Agatur enim
QG, QD, PD.
gulo
gulum
Ergo angulus
adeoque angulo
æqualis e&longs;t; & propter
ea punctum
punctum
gulus
midius e&longs;t anguli ad
centrum
lis e&longs;t angulo ad cir
cumferentiam
& angulus
dimidius e&longs;t anguli ad
centrum
qualis e&longs;t complemen
to ad duos rectos an
guli ad circumferenti
am
qualis angulo
funtQ.E.I.eo triangu
la
lia; &
ut
(ex con&longs;tructione) ut
tur itaque
do &longs;imilia e&longs;&longs;e probavimus, &longs;unt etiam æqualia. Unde, cum tan
gant in&longs;uper trianguli
blema.
Concipe Triangulum
puncto
rectum po&longs;itis, mutari in lineam rectam, cujus pars data
tis po&longs;itione datis
tis
cedentem ad hunc ca&longs;um &longs;olvetur Problema.
PRIMUS.
tæ rectis tribus po&longs;itione datis interjacebunt.
De&longs;cribenda &longs;it Trajectoria quæ &longs;it &longs;imilis & æqualis Lineæ cur
væ
partes datis hujus partibus
bitur.
Age rectas
los XXVI) Dein
circa triangulum de&longs;cribe Trajectoriam Curvæ
æqualem.
CORPORUM
&longs;itione datas, quæ neque omnes parallelæ &longs;unt, neque ad commune
punctum convergunt, &longs;inguli ad &longs;ingulas con&longs;i&longs;tent.
Dentur po&longs;itione rectæ quatuor
rum prima &longs;ecet &longs;ecundam in
& de&longs;cribendum &longs;it Trapezium
&longs;imile, & cujus angulus
les, tangant cæteras lineas Jungatur
menta
angulo
nearum
ris qui literarum
literis
bem redeant. Compleantur &longs;egmenta in circulos integros, &longs;itque
centrum circuli primi
& utrinque producatur
puncti
circulum primum in
Figuræ
Trapezium
PRIMUS.
Secent enim circuli duo primi
Jungantur
Anguli ad circumferentias
gulorum
illorum dimidiis E&longs;t ergo Figura
ad
FbH, FcIEr
go Figuræ
Quo facto Trapezium
& angulis &longs;uis
q.E.F.
tione datis dato ordine interjectæ, datam habebunt proportionem
ad invicem. Augeantur anguli
GH, HI
ma, ducetur recta
&longs;itione datis
runt ad invicem ut lineæ
dinem inter &longs;e. Idem vero &longs;ic fit expeditius.
CORPORUM
Producantur
occurrens rectæ
ut
factum.
Secet enim
agatur
erunt Secetur
ut
eoque ut
in
ut
&longs;it etiam
neas
In con&longs;tructione Corollarii hujus po&longs;tquam ducitur
& agere
tervallo
PRIMUS.
Problematis hujus &longs;olutiones alias
cogitarunt.
datis in partes &longs;ecabitur, ordine, &longs;pecie & proportione datas.
De&longs;cribenda &longs;it Trajectoria
illius partibus
miles & proportionales, rectis
&
mis, &longs;ecunda &longs;ecundis, tertia ter
tiis interjaceant. Actis rectis
GH, HI, FI,
Lem. XXVII.) Trapezium
quod &longs;it Trapezio
rectas illas po&longs;itione datas
dicto ordine. Dein circa hoc Trapezium de&longs;cribatur Trajectoria
curvæ Lineæ
CORPORUM
Con&longs;trui etiam pote&longs;t hoc Problema ut &longs;equitur.
Junctis
GH, HI, FIConcurrant
quarum
&longs;itque ad
cantur autem
AK, AL,
ordine cum literis
currat rectæ
&longs;itque
cum recta
rem
& &longs;i &longs;uper linea
Trapezium
jectoria &longs;pecie data, &longs;olvetur Problema.
Hactenus de Orbibus inveniendis.
Supere&longs;t ut Motus corpo
rum in Orbibus inventis determinemus.
PRIMUS.
tempus a&longs;&longs;ignatum.
Sit
lis Parabolæ, &longs;itque 4
areæ Parabolicæ ab&longs;cindendæ
quæ radio
poris de vertice de&longs;cripta fuit, vel an
te appul&longs;um ejus ad verticem de&longs;cri
benda e&longs;t. Innote&longs;cit quantitas areæ il
lius ab&longs;cindendæ ex tempore ip&longs;i pro
portionali. Bi&longs;eca
perpendiculum
Circulus centro
de&longs;criptus &longs;ecabit Parabolam in loco
quæ&longs;ito
perpendiculari
AOq+POq-2
2
Pro
3
=(AO+3AS/6)XPO=(4AO-3SO/6)XPO
=areæ
Ergo area ab&longs;ci&longs;&longs;a
&longs;it arcum
cem
eunte, velocitas puncti
ad lineam rectam quam corpus tempore motus &longs;ui ab
cum velocitate quam habuit in vertice
CORPORUM
pus de&longs;crip&longs;it arcum quemvis a&longs;&longs;ignatum
medium ejus punctum erige perpendiculum rectæ
rens in
per æquationes numero terminorum ac dimen&longs;ionum finitas genera
liter inveniri.
Intra Ovalem detur punctum quodvis, circa quod ceu polum re
volvatur perpetuo linea recta, uniformi cum motu, & interea in rec
ta illa exeat punctum mobile de polo, pergatque &longs;emper ea cum
velocitate, quæ &longs;it ut rectæ illius intra Ovalem quadratum. Hoc
motu punctum illud de&longs;cribet Spiralem gyris infinitis. Jam &longs;i areæ
Ovalis a recta illa ab&longs;ci&longs;&longs;æ incrementum per finitam æquationem
inveniri pote&longs;t, invenietur etiam per eandem æquationem di&longs;tantia
puncti a polo, quæ huic areæ proportionalis e&longs;t, adeoque om
nia Spiralis puncta per æquationem finitam inveniri po&longs;&longs;unt: &
propterea rectæ cuju&longs;vis po&longs;itione datæ inter&longs;ectio cum Spirali in
veniri etiam pote&longs;t per æquationem finitam. Atqui recta omnis
infinite producta Spiralem &longs;ecat in punctis numero infinitis, & æqua
tio, qua inter&longs;ectio aliqua duarum linearum invenitur, exhibet ea
rum inter&longs;ectiones omnes radicibus totidem, adeoque a&longs;cendit ad
rot dimen&longs;iones quot &longs;unt inter&longs;ectiones. Quoniam Circuli duo &longs;e
mutuo &longs;ecant in punctis duobus, inter&longs;ectio una non invenietur
ni&longs;i per æquationem duarum dimen&longs;ionum, qua inter&longs;ectio altera
etiam inveniatur. Quoniam duarum &longs;ectionum Conicarum quatuor
e&longs;&longs;e po&longs;&longs;unt inter&longs;ectiones, non pote&longs;t aliqua earum generaliter in
veniri ni&longs;i per æquationem quatuor dimen&longs;ionum, qua omnes &longs;i
mul inveniantur. Nam &longs;i inter&longs;ectiones illæ &longs;eor&longs;im quærantur, quo
niam eadem e&longs;t omnium lex & conditio, idem erit calculus in ca&longs;u
unoquoque & propterea eadem &longs;emper conclu&longs;io, quæ igitur de
bet omnes inter&longs;ectiones &longs;imul complecti & indifferenter exhibere.
tiones &longs;ex dimen&longs;ionum, & inter&longs;ectiones duarum Curvarum tertiæ
pote&longs;tatis, quia novem e&longs;&longs;e po&longs;&longs;unt, &longs;imul prodeunt per æqua
tiones dimen&longs;ionum novem. Id ni&longs;i nece&longs;&longs;ario fieret, reducere licc
ret Problemata omnia Solida ad Plana, & plu&longs;quam Solida ad Soli
da. Loquor hic de Curvis pote&longs;tate irreducibilibus.
Nam &longs;i æqua
tio per quam Curva definitur, ad inferiorem pote&longs;tatem reduci
po&longs;&longs;it: Curva non erit unica, &longs;ed ex duabus vel pluribus compo&longs;i
ta, quarum inter&longs;ectiones per calculos diver&longs;os &longs;eor&longs;im inveniri
po&longs;&longs;unt. Ad eundem modum inter&longs;ectiones binæ rectarum & &longs;ecti
onum Conicarum prodeunt &longs;emper per æquationes duarum dimen
&longs;ionum; ternæ rectarum & Curvarum irreducibilium tertiæ pote&longs;tatis
per æquationes trium, quaternæ rectarum & Curvarvm irreducibi
lium quartæ pote&longs;tatis per æquationes dimen&longs;ionum quatuor, & &longs;ic
in infinitum. Ergo rectæ & Spiralis inter&longs;ectiones numero infinitæ, cum
Curva hæc &longs;it &longs;implex & in Curvas plures irreducibilis, requirunt æ
quationes numero dimen&longs;ionum & radicum infinitas, quibus omnes
po&longs;&longs;unt &longs;imul exhiberi. E&longs;t enim eadem omnium lex & idem calculus.
Nam &longs;i a polo in rectam illam &longs;ecantem demittatur perpendiculum,
& perpendiculum illud una cum &longs;ecante revolvatur circa polum, in
ter&longs;ectiones Spiralis tran&longs;ibunt in &longs;e mutuo, quæque prima erat &longs;eu
proxima, po&longs;t unam revolutionem &longs;ecunda erit, po&longs;t duas tertia,
& &longs;ic deinceps: nec interea mutabitur æquatio ni&longs;i pro mutata mag
nitudine quantitatum per quas po&longs;itio &longs;ecantis determinatur. Unde
cum quantitates illæ po&longs;t &longs;ingulas revolutiones redeunt ad magNI
tudines primas, æquatio redibit ad formam primam, adeoque una
eademque exhibebit inter&longs;ectiones omnes, & propterea radices ha
bebit numero infinitas, quibus omnes exhiberi po&longs;&longs;unt. Nequit
ergo inter&longs;ectio rectæ & Spiralis per æquationem finitam generali
ter inveniri, & idcirco nulla extat Ovalis cujus area, rectis impe
ratis ab&longs;ci&longs;&longs;a, po&longs;&longs;it per talem æquationem generaliter exhiberi.
PRIMUS.
Eodem argumento, &longs;i intervallum poli & puncti, quo Spiralis de
&longs;cribitur, capiatur Ovalis perimetro ab&longs;ci&longs;&longs;æ proportionale, pro
bari pote&longs;t quod longitudo perimetri nequit per finitam æquatio
nem generaliter exhiberi. De Ovalibus autem hic loquor quæ non
tanguntur a figuris conjugatis in infinitum pergentibus.
CORPORUM
Hinc area Ellip&longs;eos, quæ radio ab umbilico ad corpus mobile
ducto de&longs;cribitur, non prodit ex dato tempore, per æquationem
finitam; & propterea per de&longs;criptionem Curvarum Geometrice ra
tionalium determinari nequit. Curvas Geometrice rationales ap
pello quarum puncta omnia per longitudines æquationibus defiNI
tas, id e&longs;t, per longitudinum rationes complicatas, determinari
po&longs;&longs;unt; cætera&longs;que (ut Spirales, Quadratrices, Trochoides) Geo
metrice irrationales. Nam longitudines quæ &longs;unt vel non &longs;unt ut
numerus ad numerum (quemadmodum in decimo Elementorum)
&longs;unt Arithmetice rationales vel irrationales. Aream igitur Ellip&longs;eos
tempori proportionalem ab&longs;cindo per Curvam Geometrice irratio
nalem ut &longs;equitur.
tempus a&longs;&longs;ignatum.
Ellip&longs;eos
centrum, &longs;itque Produc
ut &longs;it
ceu fundo, progrediatur Rota
&longs;uum, & interea puncto &longs;uo
tempus revolutionis unius in Ellip&longs;i. Erigatur perpendiculum
occurrens Trochoidi in
Ellip&longs;i in corporis loco quæ&longs;ito
PRIMUS.
Nam centro
& arcui
Arcui
pendiculum
rentia inter &longs;ectorem
tia rectangulorum 1/2
1/2
æqualitatem datarum rationum
OAarc.
ut
Cæterum, cum difficilis &longs;it hujus Curvæ de&longs;criptio, præ&longs;tat &longs;olu
tionem vero proximam adhibere. Inveniatur tum angulus quidam
B, qui &longs;it ad angulum graduum 57,29578, quem arcus radio æqualis
&longs;ubtendit, ut e&longs;t umbilieorum di&longs;tantia
trum
eadem ratione inver&longs;e. Quibus &longs;emel inventis, Problema deinceps
confit per &longs;equentem Analy&longs;in. Per con&longs;tructionem quamvis (vel.
utcunque conjec
turam faciendo)
cogno&longs;catur cor
poris locus
ximus vero ejus lo
co
axem Ellip&longs;eos or
dinatim applicata
tione diametrorum
Ellip&longs;eos, dabitur
Circuli circum&longs;cri
pti
&longs;tente Sufficit angulum illum rudi calculo in numeris
proximis invenire. Cogno&longs;catur etiam angulus tempori propor-
de&longs;crip&longs;it arcum Sit
angulus i&longs;te N. Tum capiatur & angulus D ad angulum B, ut
e&longs;t &longs;inus i&longs;te anguli
N-
co&longs;inu anguli
auctam ubi major. Po&longs;tea capiatur tum angulus F ad angulum B,
ut e&longs;t &longs;inus anguli
lum N-
dem co&longs;inu anguli
nor e&longs;t, auctam ubi major. Tertia vice capiatur angulus H ad an
gulum B, ut e&longs;t &longs;inus anguli
lus I ad angulum N-
eandem longitudinem co&longs;inu anguli
ubi angulus i&longs;te re
cto minor e&longs;t, auc
tam ubi major. Et
&longs;ic pergere licet in
infinitum. DeNI
que capiatur angu
lus
angulo
+G+I+&c. e t
ex co&longs;inu ejus
& ordinata
e&longs;t ad &longs;inum ejus
locus correctus
e&longs;t, debet &longs;ignum+ip&longs;ius E ubique mutari in-, & &longs;ignum-in+.
Idem intelligendum e&longs;t de &longs;ignis ip&longs;orum G & I, ubi anguli
N-
Convergit autem &longs;eries infinita quam
celerrime, adeo ut vix unquam opus fuerit ultra progredi quam
ad terminum &longs;ecundum E. Et fundatur calculus in hoc Theore
mate, quod area
rectam ab umbilico
mi&longs;&longs;am.
CORPORUM
Non di&longs;&longs;imili calculo conficitur Problema in Hyperbola.
Sit
ejus Centrum Sit ea
ab&longs;cindat veræ proximam. Jun
gatur
A&longs;ymptoton agantur
A&longs;ymptoto alteri parallelæ, & per
Tabulam Logarithmorum dabi
tur Area
area
angulo
&longs;ci&longs;&longs;am
ab&longs;cindendæ A & ab&longs;ci&longs;&longs;æ
differentiam duplam 2
vel 2 A-2
batur autem chorda illa
major &longs;it area ab&longs;cindenda A, &longs;ecus ad puncti
& punctum Et computatione
repetita invenietur idem accuratior in perpetuum.
PRIMUS.
Atque his calculis Problema generaliter confit Analytice.
Ve
rum u&longs;ibus A&longs;tronomicis accommodatior e&longs;t calculus particularis
qui &longs;equitur. Exi&longs;tentibus
L ip&longs;ius latere recto, ac D differentia inter &longs;emiaxem minorem
& lateris recti &longs;emi&longs;&longs;em 1/2 L; quære tum angulum Y, cujus &longs;inus
&longs;it ad Radium ut e&longs;t rectangu
lum &longs;ub differentia illa D, &
&longs;emi&longs;umma axium
ad quadratum axis majoris
tum angulum Z, cujus &longs;inus
&longs;it ad Radium ut e&longs;t duplum
rectangulum &longs;ub umbilieorum
di&longs;tantia
illa D ad triplum quadratum
&longs;emiaxis majoris
angulis &longs;emel inventis; locus corporis &longs;ic deinceps determinabitur.
Sume angulum T proportionalem tempori quo arcus
tus e&longs;t, &longs;cu motui medio (ut loquuntur) æqualem; & angulum
V (primam medii motus æquationem) ad angulum Y (æquatio
nem maximam primam) ut e&longs;t &longs;inus dupli anguli T ad Radium;
tionem maximam &longs;ecundam) ut e&longs;t cubus &longs;inus anguli T ad cubum
Radii. Angulorum T, V, X vel &longs;ummæ T+X+V, &longs;i angulus
T recto minor e&longs;t, vel differentiæ T+X-V, &longs;i is recto major e&longs;t
recti&longs;Q.E.D.obus minor, æqualem cape angulum
medium æquatum;) &, &longs;i
&longs;cindet aream Hæc
Praxis &longs;atis expedita videtur,
propterea quod angulorum per
exiguorum V & X (in minutis
&longs;ecundis, &longs;i placet, po&longs;itorum)
figuras duas ter&longs;ve primas in
venire &longs;ufficit. Sed & &longs;atis ac
curata e&longs;t ad Theoriam Planeta
rum. Nam in Orbe vel Martis
ip&longs;ius, cujus Æquatio centri ma
xima e&longs;t graduum decem, error
vix &longs;uperabit minutum unum
&longs;ecundum. Invento autem angulo motus medii æquati
gulus veri motus
CORPORUM
Hactenus de Motu corporum in lineis Curvis.
Fieri autem po
te&longs;t ut mobile recta de&longs;cendat vel recta a&longs;cendat, & quæ ad i&longs;tiu&longs;
modi Motus &longs;pectant, pergo jam exponere.
PRIMUS.
&longs;tantiæ loeorum a centro, Spatia definire quæ corpus recta cadendo
datis temporibus de&longs;cribit.
lariter de&longs;cribet id, per Corol. 1. Prop.
XIII,
Sectionem aliquam Conicam cujus umbili
cus congruit cum centro virium. Sit Sec
tio illa Conica
Et primo &longs;i Figura Ellip&longs;is e&longs;t, &longs;uper hu
jus axe majore
ta
que adeo etiam tempori proportionalis. Ma
nente axe
Ellip&longs;eos, & &longs;emper manebit area
tempori proportionalis. Minuatur latitudo
illa in infinitum: &, Orbe
cidente cum axe
axis termino
portionalis. Dabitur itaque Spatium
quod corpus de loco
cadendo tempore dato de&longs;cribit, &longs;i modo tempori proportiona
lis capiatur area
tatur perpendicularis
CORPORUM
dem diametrum principalem
& quoniam areæ
CBED, SDEB,
num
proportionalis e&longs;t tempori quo
corpus
dem tempori proportionalis.
Minuatur latus rectum Hyper
bolæ
nente latere tran&longs;ver&longs;o, & coibit
arcus
bilicus
rea
tempori quo corpus
de&longs;cen&longs;u de&longs;cribit lineam
Figura
eodem vertice principali
&longs;cribatur alia Parabola
quæ &longs;emper maneat data interea
dum Parabola prior in cujus perimetro corpus
nuto & in nihilum redacto ejus latere recto, conveniat cum linea
quo corpus illud
vis
lum de&longs;cribentis, in &longs;ubduplicata ratione quam
poris a Circuli vel Hyperbolæ rect angulæ vertice ulteriore
ad Figuræ &longs;emidiametrum principalem
Bi&longs;ecetur
meter, in
hanc diametrum perpendicularis, atque Figu
ræ Con&longs;tat
per Cor. 9. Prop.
XVI, quod corporis in
linea
citas in loco quovis
poris intervallo
culum de&longs;cribentis in &longs;ubduplicata ratione rec
tanguli 1/2 LXE&longs;t au
tem ex Conicis
adeoque (2Ergo ve
locitates illæ &longs;unt ad invicem in &longs;ubduplicata
ratione (
ro ex Conicis e&longs;t
& compo&longs;ite vel divi&longs;im ut
Unde vel dividendo vel componendo fit
(
do
puncto
& corporis jam recta de&longs;cendentis in linea
velocitatem corporis centro
in &longs;ubduplicata ratione ip&longs;ius (
lectis æqualitatis rationibus
duplicata ratione
PRIMUS.
ad
vis revolvens, motu &longs;uo &longs;ur&longs;um ver&longs;o a&longs;cendet ad duplam &longs;uam a
centro di&longs;tantiam.
CORPORUM
tas in loco quovis
velocitati qua corpus centro
culum uniformiter de&longs;cribere
potest.
Nam corporis Parabolam
bentis velocitas in loco quovis 7. Prop.
XVI) æ
qualis e&longs;t velocitati corporis di
midio intervalli
ca idem centrum
de&longs;cribentis. Minuatur Parabolæ
latitudo
arcus Parabolicus
ta
& intervallum
po&longs;itio.
&longs;cripta, æqualis &longs;it areæ quam corpus, radio dimidium lateris recti
Figuræ
dem tempore de&longs;cribere potest.
Nam concipe corpus
lam
ter in Circulo
bere. Erigantur perpendicula
in
rens in
Et quoniam e&longs;t
1. Prop.
XXXIII, e&longs;t
in coitu punctorum
rationes ultimæ. Ergo
ut
de&longs;cendentis velocitas in
corporis Circulum intervallo
trum XXXIII.) Et
hæc velocitas ad velocitatem corporis de&longs;cri
bentis Circulum 6. Prop.
IV, & ex æquo velo
citas prima ad ultimam, hoc e&longs;t lineola
arcum
id e&longs;t in ratione
æquale
le
id e&longs;t area
igitur temporis particulis generantur arearum
duarum particulæ
nitudo earum minuatur & numerus augeatur in infinitum, ratio
nem obtinent æqualitatis, & propterea (per Corollarium Lem
matis IV) areæ totæ &longs;imul genitæ &longs;unt &longs;emper æquales,
PRIMUS.
pra
eoque 1/4
tis velocitas in
uniformiter de&longs;cribi po&longs;&longs;it (per Prop. XXXIV) Et hæc velocitas ad ve
locitatem qua Circulus radio 6. Prop.
IV) e&longs;t in &longs;ubduplicata ratione
æquale 1/4
æqualis areæ
CORPORUM
pora de&longs;cen&longs;us.
Super diametro
tro &longs;ub initio) de&longs;cribe Semicirculum
huic æqualem Semicirculum
plicatam
lem con&longs;titue &longs;ectorem
XXXV, quod corpus cadendo de&longs;cribet &longs;patium
eodem Tempore quo corpus aliud uniformiter cir
ca centrum
a&longs;cen&longs;us vel de&longs;cen&longs;us.
Exeat corpus de loco dato
lineam
In duplicata ratione hujus velocitatis ad
uniformem in Circulo velocitatem, qua cor
pus ad intervallum datum
Si ratio illa e&longs;t numeri binarii ad unita
tem, punctum
&longs;u Parabola vertice
vis recto de&longs;cribenda e&longs;t. Patet hoc per
Prop. XXXIV.
Sin ratio illa minor vel ma
jor e&longs;t quam 2 ad 1, priore ca&longs;u Circulus,
po&longs;teriore Hyperbola rectangula &longs;uper di
ametro Patet per
Prop. XXXIII.
Tum centro
æquante dimidium lateris recti, de&longs;cribatur
Circulus
tis vel de&longs;cendentis loca duo quævis
erigantur perpendicula
tia Conicæ Sectioni vel Circulo in XXXV, corpus
bet &longs;patium
te&longs;t arcum
PRIMUS.
eorum a centro, dico quod cadentium Tempora, Velocitates & Spa
tia de&longs;cripta &longs;unt arcubus, arcuumque finibus rectis & &longs;inibus
ver&longs;is re&longs;pective proportionalia.
Cadat corpus de loco quovis
dum rectam
tervallo
vis
dendo de&longs;cribet Spatium
Demon&longs;tratur eodem modo ex Propo&longs;i
tione X, quo Propo&longs;itio XXXII, ex Propo
&longs;itione XI demon&longs;trata fuit.
&longs;cribit arcum quadrantalem
locis quibu&longs;vis ad u&longs;que centrum cadunt. Nam revolventium tem
pora omnia periodica (per Corol. 3. Prop.
IV.) æquantur.
CORPORUM
curvilinearum quadraturis, requiritu, corporis recta a&longs;cenden
tis vel de&longs;cendentis tum Velocitas in locis &longs;ingulis, tum Tempus
quo corpus ad locum quemvis perveniet: Et contra.
De loco quovis
ejus
illo ad centrum
nalis: Sitque
punctum Coinci
dat autem
perpendiculari
locitas in loco quovis
vilineæ
In
to areæ
nalis, & &longs;it
punctum
pus quo corpus cadendo de&longs;cribit li
neam
Etenim in recta
quam minima
&longs;itque
corpus ver&longs;abatur in
latus quadratum &longs;it ut de&longs;cendentis velocitas, erit area ip&longs;a in du
plicata ratione velocitatis, id e&longs;t, &longs;i pro velocitatibus in
&longs;cribantur V & V+I, erit area
VV+2 VI+II, & divi&longs;im area
(
rationes &longs;umantur, longitudo
tiam ut quantitatis hujus dimidium (IXV/
& tempus inver&longs;e, adeoque &longs;i primæ na&longs;centium rationes &longs;uman
tur, ut (IXV/
proportionalis facit ut corpus ea cum Velocitate de&longs;cendat quæ &longs;it
ut areæ
PRIMUS.
Porro cum tempus, quo quælibet longitudinis datæ lineola
de&longs;cribatur, &longs;it ut velocitas inver&longs;e adeoque ut areæ
quadratum inver&longs;e; &longs;itque
ut idem latus quadratum inver&longs;e: erit tempus ut area
&longs;umma omnium temporum ut &longs;umma omnium arearum, hoc e&longs;t
(per Corol. Lem.
IV) Tempus totum quo linea
area tota
te aliqua uniformi vi centripeta nota (qualis vulgo &longs;upponitur
Gravitas) velocitatem acquirat in loco
quam corpus aliud vi quacunque cadens acqui&longs;ivit eodem loco
& in perpendiculari
uniformis ad vim alteram in loco
de quo corpus alterum cecidit. Namque completo rectangulo
2VI, adeoque ut 1/2 V ad I, id e&longs;t, ut &longs;emi&longs;&longs;is velocitatis totius
ad incrementum velocitatis corporis vi inæquabili cadentis; & &longs;i
militer area
tius ad incrementum velocitatis corporis uniformi vi cadentis;
&longs;intQ.E.I.crementa illa (ob æqualitatem temporum na&longs;centium)
ut vires generatrices, id e&longs;t, ut ordinatim applicatæ
adeoque ut areæ na&longs;centes
areæ totæ
locitatum, & propterea (ob æqualitatem velocitatum) æquantur.
cum velocitate vel &longs;ur&longs;um vel deor&longs;um projiciatur, & detur lex vis
centripetæ, invenietur velocitas ejus in alio quovis loco
do ordinatam
loco
nea
&longs;i is &longs;uperior e&longs;t, ad latus quadratum rectanguli &longs;olius
e&longs;t, ut √
CORPORUM
ciproce proportionalem lateri quadrato ex
& capiendo tempus quo corpus de&longs;crip&longs;it lineam
quo corpus alterum vi uniformi cecidit a
que tempus quo corpus vi uniformi de&longs;cendens de&longs;crip&longs;it lineam
duplicata ratione
te) in ratione
& divi&longs;im, ad tempus quo corpus idem de&longs;crip&longs;it lineolam
ut 2
lam
&longs;it lineam
tempus primum ad tempus ultimum ut rectangulum 2
ad aream
tripetis agitata revolvuntur.
corpus aliud recta a&longs;cendat vel de&longs;cendat, &longs;intque eorum Velocita
tes in aliquo æqualium altitudinum ca&longs;u æquales, Velocitates eorum
in omnibus æqualibus altitudinibus erunt æquales.
De&longs;cendat corpus aliquod ab
moveatur corpus aliud a
tervallis quibu&longs;vis de&longs;cribantur circuli concentrici Junga
tur
culum
vel Quoniam di&longs;tantiæ
Exponantur hæ vires per æ
quales lineolas 2.)
re&longs;olvatur in duas
tem corporis in cur&longs;u illo, &longs;ed retrahet &longs;olummodo corpus a cur
&longs;u rectilineo, facietQ.E.I.&longs;um de Orbis tangente perpetuo deflecte
re, inque via curvilinea In hoc effectu produ
cendo vis illa tota con&longs;umetur: vis autem altera
corporis cur&longs;um agendo, tota accelerabit illud, ac dato tem
pore quam minimo accelerationem generabit &longs;ibi ip&longs;i proportiona
lem. Proinde corporum in
poribus factæ (&longs;i &longs;umantur linearum na&longs;centium
IT, NT
tem inæqualibus ut lineæ illæ & tempora conjunctim. Tempora
autem quibus
tum &longs;unt ut viæ de&longs;criptæ
cur&longs;u corporum per lineas Æquales igitur &longs;unt cor-
entur æquales in &longs;ub&longs;equentibus æqualibus di&longs;tantiis.
PRIMUS.
CORPORUM
Sed & eodem argumento corpora æquivelocia & æqualiter a cen
tro di&longs;tantia, in a&longs;cen&longs;u ad æquales di&longs;tantias æqualiter retarda
buntur.
pedimento quovis politi&longs;&longs;imo & perfecte lubrico cogatur in li
nea curva moveri, & corpus aliud recta a&longs;cendat vel de&longs;cendat,
&longs;intque velocitates eorum in eadem quacunque altitudine æquales:
erunt velocitates eorum in aliis quibu&longs;cunque æqualibus altitudi
nibus æquales. NamQ.E.I.pedimento va&longs;is ab&longs;olute lubrici idem
præ&longs;tatur quod vi tran&longs;ver&longs;a
non acceleratur, &longs;ed tantum cogitur de cur&longs;u rectilineo di&longs;cedere.
tia, ad quam corpus vel o&longs;cillans vel in Trajectoria quacunque re
volvens, deque quovis Trajectoriæ puncto, ea quam ibi habet
velocitate &longs;ur&longs;um projectum a&longs;cendere po&longs;&longs;it; &longs;itque quantitas A
di&longs;tantia corporis a centro in alio quovis Orbitæ puncto, & vis
centripeta &longs;emper &longs;it ut ip&longs;ius A dignitas quælibet A
Index
corporis in omni altitudine A erit ut √P
tur. Namque velocitas recta a&longs;cendentis ac de&longs;cendentis (per Prop.
XXXIX) e&longs;t in hac ip&longs;a ratione.
PRIMUS.
curvilinearum quadraturis, requiruntur tum Trajectoriæ in qui
bus corpora movebuntur, tum Tempora motuum in Trajectoriis
inventis.
Tendat vis quælibet ad centrum
de&longs;criptus, centroque eodem de&longs;cribantur alii quivis circuli
KE
Age tum rectam
tum rectam
puncta
cadere debet ut in loco
tati corporis prioris in
lineola
locitas atque adeo ut latus quadratum areæ
lum
proce ut altitudo
tudo Hanc quantitatem Q/A nominemus Z,
& ponamus eam e&longs;&longs;e magnitudinem ip&longs;ius Q ut &longs;it in aliquo
ca&longs;u √
√
& divi&longs;im
eoque √
AX
AX
(QX
&longs;emper
æquales re&longs;pective, & de&longs;cribantur curvæ lineæ
gatur perpendiculum
VDcd,
gulum
AX
&longs;unt na&longs;centes particulæ
VCX
erit area genita
pori proportionalis, & area genita
nito
&longs;it de loco
dabitur corporis altitudo
æqualis Sector
angulo
pleto illo tempore reperietur.
CORPORUM
Ap&longs;ides Trajectoriarum expedite inveniri po&longs;&longs;unt. Sunt enim
Ap&longs;ides puncta illa in quibus recta
perpendiculariter in Trajectoriam
&
lineam illam
tur; nimirum capiendo &longs;inum ejus ad radium ut
e&longs;t, ut Z ad latus quadratum areæ
libet Conica
occurrens axi infinite producto
ducatur recta
Sectori
Vis centripeta Cubo di&longs;tantiæ loeorum a centro reciproce propor
tionalis, & exeat corpus de loco
lineam rectæ
Trajectoria quam punctum
&longs;ectio
ea Ellip&longs;is &longs;it, a&longs;cendet illud perpetuo & abibit in infinitum. Et con
tra, &longs;i corpus quacunque cum Velocitate exeat de loco
de ut incæperit vel obliQ.E.D.&longs;cendere ad centrum, vel ab eo ob-
in data aliqua ratione. Sed &, Vi centripeta in centrifugam ver&longs;a,
a&longs;cendet corpus obliQ.E.I. Trajectoria
endo angulum
longitudinem Con&longs;equun
tur hæc omnia ex Propo&longs;itione præcedente, per Curvæ cuju&longs;dam
quadraturam, cujus inventionem, ut &longs;atis facilem, brevitatis gratia
mi&longs;&longs;am facio.
PRIMUS.
data cum Velocitate &longs;ecundum datam rectam egre&longs;&longs;i.
Stantibus quæ in tribus Propo&longs;itionibus præcedentibus: exeat
corpus de loco
corpus aliud, vi aliqua uniformi centripeta, de loco
quirere po&longs;&longs;et in
rectæ
ordinatim applicatæ
dataque lege vis centripetæ qua corpus primum agitatur, dantur cur
væ lineæ
& ejus Corol. 1. Deinde ex dato angulo
tium XXVIII, datur
quantitas Q, una cum curvis lineis
pleto tempore quovis
tum area
locus
CORPORUM
Supponimus autem in his Propo&longs;itionibus Vim centripetam in
rece&longs;&longs;u quidem a centro variari &longs;ecundum legem quamcunque quam
quis imaginari pote&longs;t, in æqualibus autem a centro di&longs;tantiis e&longs;&longs;e
undeque eandem. Atque hactenus Motum corporum in Orbibus
immobilibus con&longs;ideravimus. Supere&longs;t ut de Motu eorum in Orbi
bus qui circa centrum virium revolvuntur adjiciamus pauca.
PRIMUS.
Virium revolvente perinde moveri po&longs;&longs;it, atque corpus aliud in
eadem Trajectoria quie&longs;cente.
In Orbe
&longs;itione dato revolvatur
corpus
quæ &longs;it ip&longs;i
angulumque
gulo
nalem con&longs;tituat; & a
rea quam linea
&longs;cribit erit ad aream
&longs;imul de&longs;cribit, ut velo
citas lineæ de&longs;cribentis
neæ de&longs;cribentis
hoc e&longs;t, ut angulus
tione, & propterea tempori proportionalis. Cum area tempori
proportionalis &longs;it quam linea
nife&longs;tum e&longs;t quod corpus, cogente ju&longs;tæ quantitatis Vi centripeta,
revolvi po&longs;&longs;it una cum puncto
idem Fiat angu
lus
guræ
arcum ejus Quæratur igi
tur, per Corollarium quintum propo&longs;itionis VI, Vis centripeta qua
corpus revolvi po&longs;&longs;it in Curva illa linea quam punctum
in plano immobili, & &longs;olvetur Problema.
CORPORUM
liud in eodem Orbe revolvente æqualiter moveri po&longs;&longs;unt, est
in triplicata ratione communis altitudinis inver&longs;e.
Partibus Orbis quie
&longs;centis
&longs;imiles & æquales Or
bis revolventis partes
gatur e&longs;&longs;e quam miNI
ma. A puncto
ctam
pendiculum
que produc ad
Quoniam corporum al
titudines
&
tur, manife&longs;tum e&longs;t quod linearum
decrementa &longs;emper &longs;int æqualia, ideoque &longs;i corporum in locis
Corol. 2.) in binos, quorum hi ver&longs;us centrum, &longs;ive &longs;ecundum
lineas
& &longs;ecundum lineas ip&longs;is
habeant; motus ver&longs;us centrum erunt æquales, & motus tran&longs;
ver&longs;us corporis
tus angularis lineæ
quali in centrum motu æqualiter movebitur a
completo illo tempore reperietur alicubi in linea
punctum
acquiret di&longs;tantiam a linea
pus alterum
poris
tran&longs;ver&longs;us corporis
&longs;tum e&longs;t quod corpus
lineas
lineas illas urgentur. Capiatur autem angulum
eoque Vi majore urgetur quam corpus
angulo
&longs;equentia, vel movetur in antecedentia majore celeritate quam
&longs;it dupla ejus qua linea
re &longs;i Orbis tardius movetur in antecedentia. E&longs;tque Virium dif
ferentia ut loeorum intervallum
ip&longs;ius actione, dato illo temporis &longs;patio, transferri debet. Centro
lineas
quale rectangulo
autem triangula
earumQ.E.D.fferentia
adeoque rectangulum
nis
&longs;unt primæ rationes linearum na&longs;centium; & hinc fit (
e&longs;t lineola na&longs;cens
proce ut cubus altitudinis
PRIMUS.
ad vim qua corpus motu Circulari revolvi po&longs;&longs;it ab
tempore quo corpus
lineola na&longs;cens
capiantur datæ quantitates F, G in ea ratione ad invicem quam
habet angulus Et
propterea, &longs;i centro
Sector circularis æqualis areæ toti
quovis in Orbe immobili revolvens radio ad centrum ducto de
&longs;crip &longs;it: differentia virium, quibus corpus
corpus
corpus aliquod radio ad centrum ducto Sectorem illum, eodem tem
pore quo de&longs;cripta &longs;it area
ut GG-FF ad FF. Namque Sector ille & area
vicem ut tempora quibus de&longs;cribuntur.
CORPORUM
&longs;idem &longs;ummam
ita ut &longs;it &longs;emper
&longs;cribatur A, & pro Ellip&longs;eos latere recto ponatur 2 R: erit vis qua
corpus in Ellip&longs;i mobili revolvi pote&longs;t, ut (FF/AA)+(RGG-RFF/A
& contra. Exponatur enim vis qua corpus revolvatur in immota
Ellip&longs;i per quantitatem (FF/AA), & vis in
tem qua corpus in Circulo ad di&longs;tantiam
revolvi po&longs;&longs;et quam corpus in Ellip&longs;i revolvens habet in
e&longs;t ad vim qua corpus in Ellip&longs;i revolvens urgetur in Ap&longs;ide
ut dimidium lateris recti Ellip&longs;eos. ad Circuli &longs;emidiametrum
adeoque valet (RFF/
FF, valet (RGG-RFF/1.)
differentia virium in
& corpus Unde cum (per
hanc Prop.) differentia illa in alia quavis altitudine A &longs;it ad &longs;e
ip&longs;am in altitudine
in omni altitudine. A valebit (RGG-RFF/A
qua corpus revolvi pote&longs;t in Ellip&longs;i immobili
ce&longs;&longs;us (RGG-RFF/A
PRIMUS.
bilis
que &longs;imilis, æqualis & concentrica ponatur Ellip&longs;is mobilis
&longs;itque 2 R Ellip&longs;eos hujus latus rectum principale, & 2T latus
tran&longs;ver&longs;um &longs;ive axis major, atque angulus
angulum
mobili & mobili temporibus æqualibus revolvi po&longs;&longs;unt, erunt ut
(FFA/T
minetur T, & radius curvaturæ quam Orbis
e&longs;t radius Circuli æqualiter curvi, nominetur R, & vis centripeta
qua corpus in Trajectoria quacunQ.E.I.mobili
te&longs;t, in loco
tur X, altitudine
ratione anguli
corpus idem eo&longs;dem motus in eadem Trajectoria
riter mota temporibus ii&longs;dem peragere pote&longs;t, ut &longs;umma virium
X+(VRGG-VRFF/A
bili, augeri vel minui pote&longs;t ejus motus angularis circa centrum
virium in ratione data, & inde inveniri novi Orbes immobiles in
quibus corpora novis viribus centripetis gyrentur.
&longs;itione datam erigatur perpendiculum
gaturque
ad angulum
vis qua corpus gyrari pote&longs;t in Curva
illa
tangit, erit reciproce ut cubus altitu
dinis
uniformiter progredi pote&longs;t in recta
jam demon&longs;trata) detorQ.E.I.ur motus ille rectilineus in lineam 3. Prop.
XLI inventa, in qua ibi diximus corpora
huju&longs;modi viribus attracta oblique a&longs;cendere.
CORPORUM
&longs;idum.
Problema &longs;olvitur Arithmetice faciendo ut Orbis, quem corpus
in Ellip&longs;i mobili (ut in Propo&longs;itionis &longs;uperioris Corol. 2, vel 3)
revolvens de&longs;cribit in plano immobili, accedat ad formam Orbis
cujus Ap&longs;ides requiruatur, & quærendo Ap&longs;ides Orbis quem cor
pus illud in plano immobili de&longs;cribit. Orbes autem eandem ac
quirent formam, &longs;i vires centripetæ quibus de&longs;cribuntur, inter &longs;e
collatæ, in æqualibus altitudinibus reddantur proportionales. Sit
punctum
titudinum differentia
circa umbilicum &longs;uum
tur, quæQ.E.I. Corollario 2. erat ut (FF/AA)+(RGG-RFF/A
ut (FFA+RGG-RFF/A
(RGG-RFF+TFF-FFX/A
quævis centripeta ad fractionem cujus denominator &longs;it A
numeratores, facta homologorum terminorum collatione, &longs;tatuendi
&longs;unt analogi. Res Exemplis patebit.
ut (A
(T
minis corre&longs;pondentibus, nimirum datis cum datis & non datis
cum non datis, fiet RGG-RFF+TFF ad T
-3TTX+3TXX-X
-XX. Jam cum Orbis ponatur Circulo quam maxime finitimus,
coeat Orbis cum Circulo; & ob factas R, T æquales, atque X in infi-
FF ut TT ad 3 TT id e&longs;t, ut 1 ad 3; adeoque G ad F,
hoc e&longs;t angulus
go cum corpus in Ellip&longs;i immobili, ab Ap&longs;ide &longs;umma ad Ap
&longs;idem imam de&longs;cendendo conficiat angulum
cam) gradum 180; corpus aliud in Ellip&longs;i mobili, atque adeo in
Orbe immobili de quo agimus, ab Ap&longs;ide &longs;umma ad Ap&longs;idem
imam de&longs;cendendo conficiet angulum
adeo ob &longs;imilitudinem Orbis hujus, quem corpus agente uniformi
vi centripeta de&longs;cribit, & Orbis illius quem corpus in Ellip&longs;i re
volvente gyros peragens de&longs;cribit in plano quie&longs;cente. Per &longs;u
periorem terminorum collationem &longs;imiles redduntur hi Orbes, non
univer&longs;aliter, &longs;ed tunc cum ad formam circularem quam maxime
appropinquant. Corpus igitur uniformi cum vi centripeta in
Orbe propemodum circulari revolvens, inter Ap&longs;idem &longs;ummam
& Ap&longs;idem imam conficiet &longs;emper angulum (180/√3) graduum, &longs;eu
103
Ap&longs;idem imam ubi &longs;emel confecit hunc angulum, & inde ad Ap&longs;i
dem &longs;ummam rediens ubi iterum confecit eundem angulum; &
&longs;ic deinceps in infinitum.
PRIMUS.
nitas quælibet A
tatum indices quo&longs;cunQ.E.I.tegros vel fractos, rationales vel irratio
nales, affirmativos vel negativos. Numerator ille A
in &longs;eriem indeterminatam per Methodum no&longs;tram Serierum conver
gentium reducta, evadit T
Et collatis hujus terminis cum terminis Numeratoris alterius
RGG-RFF+TFF-FFX, fit RGG-RFF+TFF ad T
ut-FF ad-Et &longs;umendo ratio
nes ultimas ubi Orbes ad formam circularem accedunt, fit RGG
ad T
& vici&longs;&longs;im GG ad FF ut T
adeoque G ad F, id e&longs;t angulus
ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam in Ellip&longs;i confectus, &longs;it
graduum 180; conficietur angulus
ab Ap&longs;ide &longs;umma ad Ap&longs;idem imam, in Orbe propemodum Cir
culari quem corpus quodvis vi centripeta dignitati A
portionali de&longs;cribit, æqualis angulo graduum (180/√
repetito corpus redibit ab Ap&longs;ide ima ad Ap&longs;idem &longs;ummam, &
&longs;ic deinceps in infinitum. Ut &longs;i vis centripeta &longs;it ut di&longs;tantia cor
poris a centro, id e&longs;t, ut A &longs;eu (A
adeoque angulus inter Ap&longs;idem &longs;ummam & Ap&longs;idem imam æ
qualis (180/2)
nis unius corpus perveniet ad Ap&longs;idem imam, & completa alia
quarta parte ad Ap&longs;idem &longs;ummam, & &longs;ic deinceps per vices in
infinitum. Id quod etiam ex Propo&longs;itione x.
manife&longs;tum e&longs;t.
Nam
corpus urgente hac vi centripeta revolvetur in Ellip&longs;i immobili,
cujus centrum e&longs;t in centro virium. Quod &longs;i vis centripeta &longs;it reci
proce ut di&longs;tantia, id e&longs;t directe ut 1/A &longs;eu (A
eoQ.E.I.ter Ap&longs;idem &longs;ummam & imam angulus erit graduum (180/√2)
&longs;eu 127
tua anguli hujus repetitione, vicibus alternis ab Ap&longs;ide &longs;umma ad
imam & ab ima ad &longs;ummam perveniet in æternum. Porro &longs;i vis
centripeta &longs;it reciproce ut latus quadrato-quadratum undecimæ
dignitatis altitudinis, id e&longs;t reciproce ut A (11/4), adeoQ.E.D.recte ut
(1/A
terea corpus de Ap&longs;ide &longs;umma di&longs;cedens & &longs;ubinde perpetuo de
&longs;cendens, perveniet ad Ap&longs;idem imam ubi complevit revolutionem
integram, dein perpetuo a&longs;cen&longs;u complendo aliam revolutionem in
regram, redibit ad Ap&longs;idem &longs;ummam: & &longs;ic per vices in æternum.
CORPORUM
Altitudinis, &
tripetam e&longs;&longs;e ut (
&longs;eu (per eandem Methodum no&longs;tram Serierum convergentium) ut
(
+(Et &longs;umendo rationes ulti
mas quæ prodeunt ubi Orbes ad formam circularem accedunt, fit
GG ad
vici&longs;&longs;im GG ad FF ut
Quæ proportio, exponendo altitudinem maximam
metice per Unitatem, fit GG ad FF ut
1 ad (
Ap&longs;idem &longs;ummam & Ap&longs;idem imam in Ellip&longs;i immobili &longs;it 180
erit angulus
centripeta quantitati (
lis angulo graduum 180 √(
tripeta &longs;it ut (
180 √(
cilioribus. Quantitas cui vis centripeta proportionalis e&longs;t, re
&longs;olvi &longs;emper debet in Series convergentes denominatorem ha
bentes A
provenit ad ip&longs;ius partem alteram non datam, & pars data nu
meratoris hujus RGG-RFF+TFF-FFX ad ip&longs;ius partem
alteram non datam in eadem ratione ponendæ &longs;unt: Et quantitates
&longs;uperfluas delendo, &longs;cribendoque Unitatem pro T, obtinebitur
proportio G ad F.
PRIMUS.
tas, inveniri pote&longs;t dignitas illa ex motu Ap&longs;idum; & contra.
Nimirum &longs;i motus totus angularis, quo corpus redit ad Ap&longs;idem
eandem, &longs;it ad motum angularem revolutionis unius, &longs;eu graduum
360, ut numerus aliquis
minetur A: erit vis ut altitudinis dignitas illa A
Unde liquet vim illam in majore quam triplicata altitudinis ratione,
in rece&longs;&longs;u a centro, decre&longs;cere non po&longs;&longs;e: Corpus tali vi revolvens
deque Ap&longs;ide di&longs;cedens, &longs;i cæperit de&longs;cendere nunquam perveniet
ad Ap&longs;idem imam &longs;eu altitudinem minimam, &longs;ed de&longs;cendet u&longs;que ad
centrum, de&longs;cribens Curvam illam lineam de qua egimus in Cor. 3.
Prop. XLI.
Sin cæperit illud, de Ap&longs;ide di&longs;cedens, vel minimum
a&longs;cendere; a&longs;cendet in infinitum, neque unquam perveniet ad Ap
&longs;idem &longs;ummam. De&longs;cribet enim Curvam illam lineam de qua ac
tum e&longs;t in eodem Corol. & in Corol.
6, Prop.
XLIV.
Sic & ubi
vis, in rece&longs;&longs;u a centro, decre&longs;cit in majore quam triplicata ratione
altitudinis, corpus de Ap&longs;ide di&longs;cedens, perinde ut cæperit de&longs;cen
dere vel a&longs;cendere, vel de&longs;cendet ad centrum u&longs;que vel a&longs;cendet
in infinitum. At &longs;i vis, in rece&longs;&longs;u a centro, vel decre&longs;cat in minore
quam triplicata ratione altitudinis, vel cre&longs;cat in altitudinis ratione
quacunque; corpus nunquam de&longs;cendet ad centrum u&longs;que, &longs;ed ad
Ap&longs;idem imam aliquando perveniet: & contra, &longs;i corpus de Ap&longs;i
de ad Ap&longs;idem alternis vicibus de&longs;cendens & a&longs;cendens nunquam
appellat ad centrum; vis in rece&longs;&longs;u a centro aut augebitur, aut in
minore quam triplicata altitudinis ratione decre&longs;cet: & quo ci
tius corpus de Ap&longs;ide ad Ap&longs;idem redierit, eo longius ratio virium
recedet a ratione illa triplicata. Ut &longs;i corpus revolutionibus 8 vel
4 vel 2 vel 1 1/2 de Ap&longs;ide &longs;umma ad Ap&longs;idem &longs;ummam alterno de
&longs;cen&longs;u & a&longs;cen&longs;u redierit; hoc e&longs;t, &longs;i fuerit
2 vel 1 1/2 ad 1, adeoque (
vel 4/9-3: erit vis ut A
id e&longs;t, reciproce ut A
Si corpus &longs;ingulis revolutionibus redierit ad Ap&longs;idem eandem immo
tam; erit
& propterea decrementum virium in ratione duplicata altitudinis,
ut in præcedentibus demon&longs;tratum e&longs;t. Si corpus partibus revo
lutionis unius vel tribus quartis, vel duabus tertiis, vel una ter
tia, vel una quarta, ad Ap&longs;idem eandem redierit; erit
1/4 vel 2/3 vel 1/3 vel 1/4 ad 1, adeoque A(
ADenique &longs;i corpus pergendo
tegram, & præterea gradus tres, adeoque Ap&longs;is illa &longs;ingulis corporis
revolutionibus confecerit in con&longs;equentia gradus tres; erit
363
A
ciproce ut ADecre&longs;cit igitur vis centripeta in ratio
ne paulo majore quam duplicata, &longs;ed quæ vicibus 59 3/4 propius ad
duplicatam quam ad triplicatam accedit.
CORPORUM
PRIMUS.
ut quadratum altitudinis, revolvatur in Ellip&longs;i umbilicum haben
te in centro virium, & huic vi centripetæ addatur vel auferatur
vis alia quævis extranea; cogno&longs;ci pote&longs;t (per Exempla tertia)
motus Ap&longs;idum qui ex vi illa extranea orietur: & contra. Ut &longs;i
vis qua corpus revolvitur in Ellip&longs;i &longs;it ut (1/AA), & vis extranea ab
lata ut
tiis)
lutionis inter Ap&longs;ides æqualis angulo graduum 180 √(1-
natur vim illam extraneam e&longs;&longs;e 357,
altera qua corpus revolvitur in Ellip&longs;i, id e&longs;t
vel T æquali 1; & 180 √(1-
id e&longs;t, 180
dens, motu angulari 180
imam, & hoc motu duplicato ad Ap&longs;idem &longs;ummam redibit: adeo
que Ap&longs;is &longs;umma &longs;ingulis revolutionibus progrediendo conficiet
1
Hactenus de Motu corporum in Orbibus quorum plana per
centrum Virium tran&longs;eunt. Supere&longs;t ut Motus etiam determine
mus in planis excentricis. Nam Scriptores qui Motum gravium
tractant, con&longs;iderare &longs;olent a&longs;cen&longs;us & de&longs;cen&longs;us ponderum,
tam obliquos in planis quibu&longs;cunQ.E.D.tis, quam perpendicu
lares: & pari jure Motus corporum Viribus quibu&longs;cunque cen-
venit. Plana autem &longs;upponimus e&longs;&longs;e politi&longs;&longs;ima & ab&longs;olute lubrica
ne corpora retardent. Quinimo, in his demon&longs;trationibus, vi
ce planorum quibus corpora incumbunt quæque tangunt incum
bendo, u&longs;urpamus plana his parallela, in quibus centra corpo
rum moventur & Orbitas movendo de&longs;cribunt. Et eadem lege
Motus corporum in &longs;uperficiebus Curvis peractos &longs;ubinde de
terminamus.
CORPORUM
Motu reciproco.
tro tum Plano quocunQ.E.I. quo corpus revolvitur, & conce&longs;
&longs;is Figurarum curvilinearum quadraturis: requiritur Motus cor
poris de loco dato, data cum Velocitate, &longs;ecundum rectam in
Plano illo datam egre&longs;&longs;i.
Sit
dato,
corpus idem in Trajectoria &longs;ua revolvens, &
illa, in Plano dato de&longs;cripta, quam invenire oportet. Jungantur
QS,
corpus trahitur ver&longs;us centrum 2.)
in vires
plano perpendicularem, nil mutat motum ejus in hoc plano. Vis
autem altera
pus directe ver&longs;us punctum
in hoc plano perinde moveri ac &longs;i vis
&longs;ola Data autem
XLII, tum Trajectoria
quam corpus de&longs;cribit, tum locus
vis tempus ver&longs;abitur, tum denique velocitas corporis in loco illo
PRIMUS.
centro; corpora omnia in planis quibu&longs;cunque revolventia de
&longs;cribent Ellip&longs;es, & revolutiones Temporibus æqualibus peragent;
quæque moventur in lineis rectis, ultro citroQ.E.D.&longs;currendo,
&longs;ingulas eundi & redeundi periodos ii&longs;dem Temporibus ab&longs;ol
vent.
Nam, &longs;tantibus quæ
in &longs;uperiore Propo&longs;itio
ne, vis
revolvens trahitur ver
&longs;us centrum
&longs;tantia
ob proportionales
&
tur ver&longs;us punctum
in Orbis plano datum,
e&longs;t ut di&longs;tantia
res igitur, quibus cor
pora in plano
ver&longs;antia trahuntur ver
&longs;us punctum
ratione di&longs;tantiarum æquales viribus quibus corpora undiquaque
trahuntur ver&longs;us centrum
dem Temporibus, in ii&longs;dem Figuris, in plano quovis
punctum
Corol. 2. Prop.
X, & Corol.
2. Prop.
XXXVIII) Temporibus &longs;emper
vel periodos movendi ultro citroQ.E.I. lineis rectis per centrum
in plano illo ductis, complebunt.
CORPORUM
His affines &longs;unt a&longs;cen&longs;us ac de&longs;cen&longs;us corporum in &longs;uperficiebus
curvis. Concipe lineas curvas in plano de&longs;cribi, dein circa axes
quo&longs;vis datos per centrum Virium tran&longs;euntes revolvi, & ea revo
lutione &longs;uperficies curvas de&longs;cribere; tum corpora ita moveri ut
eorum centra in his &longs;uperficiebus perpetuo reperiantur. Si cor
pora illa oblique a&longs;cendendo & de&longs;cendendo currant ultro citroque
peragentur eorum motus in planis per axem tran&longs;euntibus, atque
adeo in lineis curvis quarum revolutione curvæ illæ &longs;uperficies ge
nitæ &longs;unt. I&longs;tis igitur in ca&longs;ibus &longs;ufficit motum in his lineis cur
vis con&longs;iderare.
tarum revolvendo progrediatur in circulo maximo; longitudo
Itineris curvilinei, quod punctum quodvis in Rotæ perimetro da
tum, ex quo Globum tetigit, confecit, (quodque Cycloidem vel
Epicycloidem nominare licet) erit ad duplicatum &longs;inum ver&longs;um
arcus dimidii qui Globum ex eo tempore inter eundum tetigit,
ut &longs;umma diametrorum Globi & Rotæ ad &longs;emidiametrum Globi.
volvendo progrediatur in circulo maximo; longitudo Itineris
curvilinei quod punctum quodvis in Rotæ perimetro datum, ex
quo Globum tetigit, confecit, erit ad duplicatum &longs;inum ver&longs;um
arcus dimidii qui Globum toto hoc tempore inter eundum teti
git, ut differentia diametrorum Globi & Rotæ ad &longs;emidiame
trum Globi.
Sit
rimetro Rotæ. Concipe hanc Rotam pergere in circulo maximo
cus
Globum tetigit in
&longs;inum ver&longs;um arcus 1/2
opus e&longs;t producta) occurrat Rotæ in
EP, VP,
gant
cans rectam in
lineam
lus &longs;ecans
PRIMUS.
CORPORUM
Quoniam Rota eundo &longs;emper revolvitur circa punctum con
tactus
lineam illam curvam
adeo quod recta
rum Cum autem
terea æquales; & angulus
ad
angula
&longs;eu
ita
tum lineæ
tum lineæ curvæ
terea (per Corol. Lem.
IV.) longitudines
crementis illis genitæ, &longs;unt in eadem ratione. Sed, exi&longs;tente
dio, e&longs;t
&longs;inus ver&longs;us eju&longs;dem anguli; & propterea in hac Rota, cujus radius
e&longs;t 1/2
PRIMUS.
Lineam autem
Globum, alteram in po&longs;teriore Cycloidem intra Globum di&longs;tincti
onis gratia nominabimus.
ea in
plus e&longs;t &longs;inus anguli
atque adeo in ratione data.
lineæ rectæ quæ e&longs;t ad Rotæ diametrum
Intra Globum
bi&longs;ecta in
inde occurrens. Agatur
tur ea ad
bum a Rota, cujus diameter &longs;it
teriori occurrant in
AS
Filum parte &longs;ui &longs;uperiore
lum flectatur, parteque reliqua
citur, protendatur in lineam rectam; & pondus
Cycloide data
CORPORUM
Occurrat enim Filum
extremis
ctæ
gurarum
gitudines E&longs;t igi
tur
Proinde, per Corol. 1. Prop.
XLIX, longitudo partis rectæ Fili
æquatur &longs;emper Cycloidis arcui
&longs;emper Cycloidis arcui dimidio 2. Prop.
XLIX) longitudini
per æquale longitudini
data
PRIMUS.
diametrum
clois
&longs;ingulis ut di&longs;tantia loci cuju&longs;que a centro, & hac &longs;ola Vi a
gente corpus
cloidis
æqualia erunt Tempora.
Nam in Cycloidis tangentem
pendiculum
pus
Corol. 2.) re&longs;olvitur in partes
do corpus directe a
tota ce&longs;&longs;at, nullum alium edens effectum; pars autem altera
urgendo corpus tran&longs;ver&longs;im &longs;eu ver&longs;us
ejus in Cycloide; manife&longs;tum e&longs;t quod corporis acceleratio, huic
vi acceleratrici proportionalis, &longs;it &longs;ingulis momentis ut longitudo
ut longitudo 1. Prop.
XLIX,) ut longitudo
arcus Cycloidis
pendiculo
nes eorum &longs;emper erunt ut arcus de&longs;cribendi
tem partes &longs;ub initio de&longs;criptæ ut accelerationes, hoc e&longs;t, ut totæ
&longs;ub initio de&longs;cribendæ, & propterea partes quæ manent de&longs;criben-
etiam ut totæ; & &longs;ic deinceps. Sunt igitur accelerationes atque
adeo velocitates genitæ & partes his velocitatibus de&longs;criptæ par
te&longs;Q.E.D.&longs;cribendæ, &longs;emper ut totæ; & propterea partes de&longs;criben
dæ datam &longs;ervantes rationem ad invicem &longs;imul evane&longs;cent, id e&longs;t,
corpora duo o&longs;cillantia &longs;imul pervenient ad perpendiculum
Cumque vici&longs;&longs;im a&longs;cen&longs;us perpendiculorum de loco in&longs;imo
eo&longs;dem arcus Cycloidales motu retrogrado facti, retardentur in
locis &longs;ingulis a viribus ii&longs;dem a quibus de&longs;cen&longs;us accelerabantur,
patet velocitates a&longs;cen&longs;uum ac de&longs;cen&longs;uum per eo&longs;dem arcus fa
ctorum æquales e&longs;&longs;e, atque adeo temporibus æqualibus fieri; &
propterea, cum Cycloidis partes duæ
pendiculi latus jacentes &longs;int &longs;imiles & æquales, pendula duo o&longs;cil
lationes &longs;uas tam totas quam dimidias ii&longs;dem temporibus &longs;emper
peragent.
CORPORUM
tur in Cycloide, e&longs;t ad totum corporis eju&longs;dem Pondus in loco
alti&longs;&longs;imo
vel
quibus tum o&longs;cillationes totæ, tum &longs;ingulæ o&longs;cillationum partes
peraguntur.
Centro quovis
de&longs;cribe &longs;emicirculum Et
&longs;i vis centripeta, di&longs;tantiis loeorum a centro proportionalis, tendat
ad centrum
in perimetro Globi Prop.
trum tendenti; & eodem tempore quo pendulum
loco &longs;upremo
vires quibus corpora urgentur &longs;unt æquales &longs;ub initio & &longs;patiis
de&longs;cribendis
quantur
de&longs;cribere &longs;patia
gere æqualiter urgeri, & æqualia &longs;patia de&longs;cribere. Quare, per Prop.
XXXVIII, tempus quo corpus de&longs;cribit arcum
veniet ad
locitatem ip&longs;ius in loco infimo
loco
neum lineæ
bus
in o&longs;cillationibus inæqualibus, de&longs;cribantur æqualibus temporibus
arcus totis o&longs;cillationum arcubus proportionales; habentur, ex da
tis temporibus, & velocitates & arcus de&longs;cripti in o&longs;cillationibus
univer&longs;is. Quæ erant primo invenienda.
PRIMUS.
O&longs;cillentur jam Funipendula
corpora in Cycloidibus diver&longs;is
intra Globos diver&longs;os, quorum
diver&longs;æ &longs;unt etiam Vires ab&longs;olu
tæ, de&longs;criptis: &, &longs;i Vis ab&longs;olu
ta Globi cuju&longs;vis
Vis acceleratrix qua
tur in circumferentia hujus Globi,
ubi incipit directe ver&longs;us centrum
ejus moveri, erit ut di&longs;tantia Cor
poris penduli a centro illo & Vis ab&longs;oluta Globi conjunctim, hoc
e&longs;t, ut Itaque lineola
trix
circumferentiæ occurrens in
illud tempus. E&longs;t autem arcus hic na&longs;cens
tione rectanguli Unde Tem
pus o&longs;cillationis integræ in Cycloide
pheria
utque arcus
ut
& Prop.
L) ut √(
Itaque O&longs;cillationes in Globis & Cycloidibus omnibus, quibu&longs;
cunque cum Viribus ab&longs;olutis factæ, &longs;unt in ratione quæ compo
nitur ex &longs;ubduplicata ratione longitudinis Fili directe, & &longs;ubdu
plicata ratione di&longs;tantiæ inter punctum &longs;u&longs;pen&longs;ionis & centrum
inver&longs;e.
CORPORUM
corporum tempora po&longs;&longs;unt inter &longs;e conferri. Nam &longs;i Rotæ, qua Cy
clois intra globum de&longs;cribitur, diameter con&longs;tituatur æqualis &longs;emi
diametro globi, Cyclois evadet Linea recta per centrum globi tran
&longs;iens, & O&longs;cillatio jam erit de&longs;cen&longs;us & &longs;ub&longs;equens a&longs;cen&longs;us in hac
recta. Unde datur tum tempus de&longs;cen&longs;us de loco quovis ad
centrum, tum tempus huic æquale quo corpus uniformiter cir
ca centrum globi ad di&longs;tantiam quamvis revolvendo arcum qua
drantalem de&longs;cribit. E&longs;t enim hoc tempus (per Ca&longs;um &longs;ecun
dum) ad tempus &longs;emio&longs;cillationis in Cycloide quavis
1 ad √(
Cycloide vulgari adinvenerunt. Nam &longs;i Globi diameter augeatur
in infinitum: mutabitur ejus &longs;uperficies &longs;phærica in planum, Vi&longs;que
centripeta aget uniformiter &longs;ecundum lineas huic plano perpendi
culares, & Cyclois no&longs;tra abibit in Cycloidem vulgi. I&longs;to autem
in ca&longs;u longitudo arcus Cycloidis, inter planum illud & punctum
de&longs;cribens, æqualis evadet quadruplicato &longs;inui ver&longs;o dimidii arcus
Rotæ inter idem planum & punctum de&longs;cribens; ut invenit
nus:
quali Cycloide temporibus æqualibus O&longs;cillabitur, ut demon&longs;travit
is erit quem
Aptantur autem Propo&longs;itiones a nobis demon&longs;tratæ ad veram
con&longs;titutionem Terræ, quatenus Rotæ eundo in ejus circulis maxi
mis de&longs;cribunt motu Clavorum, perimetris &longs;uis infixorum, Cycloi
des extra globum; & Pendula inferius in fodinis & cavernis Terra
&longs;u&longs;pen&longs;a, in Cycloidibus intra globos O&longs;cillari debent, ut O&longs;cilla
tiones omnes evadant I&longs;ochronæ. Nam Gravitas (ut in Libro
tertio docebitur) decre&longs;cit in progre&longs;&longs;u a &longs;uperficie Terræ, &longs;ur
&longs;um quidem in duplicata ratione di&longs;tantiarum a centro ejus, de
or&longs;um vero in ratione &longs;implici.
PRIMUS.
bus corpora in datis curvis lineis O&longs;cillationes &longs;emper I&longs;ochro
nas peragent.
O&longs;cilletur corpus
lam in corporis loco quovis
capiatur
gurarum quadraturis (per Methodos vulgares) innote&longs;cit. De pun
cto Agatur
pendiculari illi occurrens in
lis rectæ
CORPORUM
Nam &longs;i vis, qua corpus trahitur de
rectam
Fili
in curva Proinde
cum hæc &longs;it ut via de&longs;cribenda
tardationes in O&longs;cillationum duarum (majoris & minoris) parti
bus proportionalibus de&longs;cribendis, erunt &longs;emper ut partes illæ, &
propterea facient ut partes illæ &longs;imul de&longs;cribantur. Corpora autem
quæ partes totis &longs;emper proportionales &longs;imul de&longs;cribunt, &longs;imul de
&longs;cribent totas.
dens, de&longs;cribat arcum circularem
cundum lineas parallelas deor&longs;um a vi aliqua, quæ &longs;it ad vim uNI
formem Gravitatis, ut arcus
runt O&longs;cillationum &longs;ingularum tempora. Etenim ob parallelas
mis exponatur per longitudinem datam
lationes evadent I&longs;ochronæ, erit ad vim Gravitatis
ad motum con&longs;ervandum impre&longs;&longs;æ ita cum vi Gravitatis componi
po&longs;&longs;int, ut vis tota deor&longs;um &longs;emper &longs;it ut linea quæ oritur appli
cando rectangulum &longs;ub arcu
O&longs;cillationes omnes erunt I&longs;ochronæ.
quibus corpora Vi qualibet centripeta in lineis quibu&longs;cunque cur
vis, in plano per centrum Virium tran&longs;eunte de&longs;criptis, de&longs;cen
dent & a&longs;cendent.
De&longs;cendat corpus de loco quovis
Junga
tur Centro
ex data tum lege vis centripetæ, tum
altitudine
dabitur velocitas corporis in alia qua
vis altitudine XXXIX.
Tempus autem, quo corpus de&longs;cribit
lineolam
gitudo (id e&longs;t ut &longs;ecans anguli
directe, & velocitas inver&longs;e. Tempori
huic proportionalis &longs;it ordinatim appli
cata
erit rectangulum
area
tionale. Ergo &longs;i
nea quam punctum
erit area
pori quo corpus de&longs;cendendo de&longs;crip
&longs;it lineam
PRIMUS.
centrum Virium tran&longs;it, & a corpore in axem demittatur per
pendicularis, eique parallela & æqualis ab axis puncto quovis
dato ducatur: dico quod parallela illa aream tempori proportio
nalem de&longs;cribet.
Sit
Trajectoria quam corpus in eadem de&longs;cribit,
riæ,
perpendicularis,
axe datur educta,
volubilis
re&longs;pondens,
vi centripetæ qua corpus urgetur in centrum
pre&longs;&longs;ionis, qua corpus urget &longs;uperficiem vici&longs;&longs;imque urgetur ver&longs;us
lis;
parallela per corpus tran
&longs;iens, &
a punctis
rallelam illam
perpendiculariter demi&longs;
&longs;æ. Dico jam quod area
tio motus de&longs;cripta, &longs;it
tempori proportionalis.
Nam vis
gum Corol. 2.) re&longs;olvitur
in vires
Vires autem
agendo &longs;ecundum lineam
pendicularem mutant &longs;o
lummodo motum cor
poris quatenus huic plano perpendicularem. Ideoque motus ejus
quatenus &longs;ecundum po&longs;itionem plani factus, hoc e&longs;t, motus pun
cti
bitur, idem e&longs;t ac &longs;i vires
ribus
tur area 1.) tempori proportionalis.
CORPORUM
duo vel plura in eadem quavis recta
beret in &longs;patio libero lineam quamcunque curvam
Vis centripetæ ad centrum datum tendentis, tum &longs;uperficie cur
va cujus axis per centrum illud træn&longs;it; invenieuda est Traje
ctoria quam corpus in eadem &longs;uperficie de&longs;cribet, de loco dato, data
cum Velocitate, ver&longs;us plagam in &longs;uperficie illa datam egre&longs;&longs;um.
Stantibus quæ in &longs;uperiore Propo&longs;itione con&longs;tructa &longs;unt, exeat
ta ejus velocitate in altitudine
quavis altitudine
minimo, de&longs;cribat corpus Trajectoriæ &longs;uæ particulam Jungatur
Circelli centro
ve&longs;tigium Ellipticum in eodem plano Et ob
datum magnitudine & po&longs;itione Circellum, dabitur Ellip&longs;is illa
eo ex dato tempore detur, dabitur
communis ejus & Ellip&longs;eos inter&longs;ectio
in quo Trajectoriæ ve&longs;tigium
tem invenietur Trajectoriæ ve&longs;tigium illud
qua curva linea
inventa fuit. Tum ex &longs;ingulis ve&longs;tigii punctis
num
dabuntur &longs;ingula Trajectoriæ puncta
PRIMUS.
Hactenus expo&longs;ui Motus corporum attractorum ad centrum Im
mobile, quale tamen vix extat in rerum natura. Attractiones enim
fieri &longs;olent ad corpora; & corporum trahentium & attractorum
actiones &longs;emper mutuæ &longs;unt & æquales, per Legem tertiam: ad
eo ut neque attrahens po&longs;&longs;it quie&longs;cere neque attractum, &longs;i duo &longs;int
corpora, &longs;ed ambo (per Legum Corollarium quartum) qua&longs;i at
tractione mutua, circum gravitatis centrum commune revolvantur:
& &longs;i plura &longs;int corpora (quæ vel ab unico attrahantur vel omnia
&longs;e mutuo attrahant) hæc ita inter &longs;e moveri debeant, ut gravitatis
centrum commune vel quie&longs;cat vel uniformiter moveatur in direc
tum. Qua de cau&longs;a jam pergo Motum exponere corporum &longs;e mu
tuo trahentium, con&longs;iderando Vires centripetas tanquam Attractio
nes, quamvis forta&longs;&longs;e, &longs;i phy&longs;ice loquamur, verius dicantur Im
pul&longs;us. In Mathematicis enim jam ver&longs;amur, & propterea mi&longs;&longs;is
di&longs;putationibus Phy&longs;icis, familiari utimur &longs;ermone, quo po&longs;&longs;imus
a Lectoribus Mathematicis facilius intelligi.
CORPORUM
centrum gravitatis, & circum &longs;e mutuo, Figuras &longs;imiles.
Sunt enim di&longs;tantiæ a communi gravitatis centro reciproce pro
portionales corporibus, atque adeo in data ratione ad invicem, &
componendo, in data ratione ad di&longs;tantiam totam inter corpora.
Feruntur autem hæ di&longs;tantiæ circum terminos &longs;uos communi motu
angulari, propterea quod in directum &longs;emper jacentes non mutant
inclinationem ad &longs;e mutuo. Lineæ autem rectæ, quæ &longs;unt in data
ratione ad invicem, & æquali motu angulari circum terminos &longs;uos
feruntur, Figuras circum eo&longs;dem terminos (in planis quæ una cum
his terminis vel quie&longs;cunt vel motu quovis non angulari moven
tur) de&longs;cribunt omnino &longs;imiles. Proinde &longs;imiles &longs;unt Figuræ quæ
his di&longs;tantiis circumactis de&longs;cribuntur.
volvuntur circa gravitatis centrum commune: dico quod Fi
guris, quas corpora &longs;ic mota de&longs;cribunt circum &longs;e mutuo, potest
Figura &longs;imilis & æqualis, circum corpus alterutrum immotum,
Viribus ii&longs;dem de&longs;cribi.
Revolvantur corpora
XX) &longs;imilis Curvis
gravitatis centrum
&
PRIMUS.
rollarium quartum, vel quie&longs;cit vel movetur uniformiter in direc
tum. Ponamus primo quod id quie&longs;cit, inque
pora duo, immobile in
& æqualia. Dein tangant rectæ
nem Figurarum
eoQ.E.I. data ratione. Proinde &longs;i vis qua corpus
pus
ad vim qua corpus
tione data; hæ vires æqualibus temporibus attraherent &longs;emper cor
pora de tangentibus
proportionalia
vis prior efficit ut corpus
poribus complerentur. At quoniam vires illæ non &longs;unt ad invi
cem in ratione
corporum
&longs;ibi mutuo æquales; corpora æqualibus temporibus æqualiter tra
hentur de tangentibus: & propterea, ut corpus po&longs;terius
per intervallum majus
duplicata ratione intervallorum; propterea quod (per Lemma de
cimum) &longs;patia, ip&longs;o motus initio de&longs;cripta, &longs;unt in duplicata ratione
temporum. Ponatur igitur velocitas corporis
tem corporis
&longs;cribantur arcus
quie&longs;centia
mobile
cum &longs;patio in quo corpora moventur inter &longs;e, progreditur unifor
miter in directum; &, per Legum Corollarium &longs;extum, motus
omnes in hoc &longs;patio peragentur ut prius, adeoque corpora de&longs;cri-
CORPORUM
bus &longs;e mutuo trahentia, de&longs;cribunt (per Prop. X,) & circum com
mune gravitatis centrum, & circum &longs;e mutuo, Ellip&longs;es concentri
cas: & vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, &longs;unt Vires di&longs;tan
tiæ proportionales.
ce proportionalibus de&longs;cribunt (per Prop. XI, XII, XIII) & circum
commune gravitatis centrum, & circum &longs;e mutuo, Sectiones conicas
umbilicum habentes in centro circum quod Figuræ de&longs;cribuntur. Et
vice ver&longs;a, &longs;i tales Figuræ de&longs;cribuntur, Vires centripetæ &longs;unt qua
drato di&longs;tantiæ reciproce proportionales.
mune gyrantia, radiis & ad centrum illud & ad &longs;e mutuo ductis,
de&longs;cribunt areas temporibus proportionales.
poris alterutrius
ris quæ corpora circum &longs;e mutuo de&longs;cribunt Figuram &longs;imilem &
æqualem de&longs;cribentis, in &longs;ubduplicata ratione corporis alterins
Namque, ex demon&longs;tratione &longs;uperioris Propo&longs;itionis, tempora
quibus arcus quivis &longs;imiles
duplicata ratione di&longs;tantiarum
duplicata ratione corporis
ponendo, &longs;ummæ temporum quibus arcus omnes &longs;imiles
de&longs;cribuntur, hoc e&longs;t, tempora tota quibus Figuræ totæ &longs;imiles de
&longs;cribuntur, &longs;unt in eadem &longs;ubduplicata ratione.
PRIMUS.
proportionalibus &longs;e mutuo trahentia, revalvuntur circa gravi
tatis centrum commune: dico quod Ellip&longs;eos, quam corpus al
terutrum
lis erit ad Axem principalem Ellip&longs;eos, quam corpus idem
po&longs;&longs;et, ut &longs;umma corporum duorum
medie proportionalium inter hanc &longs;ummam & corpus illud al
terum
Nam &longs;i de&longs;criptæ Ellip&longs;es e&longs;&longs;ent &longs;ibi invicem æquales, tempora
periodica (per Theorema &longs;uperius) forent in &longs;ubduplicata ratione
corporis
tempus periodicum in Ellip&longs;i po&longs;teriore, & tempora periodica eva
dent æqualia; Ellip&longs;eos autem axis principalis (per Prop. XV.) minu
etur in ratione cujus hæc e&longs;t &longs;e&longs;quiplicata, id e&longs;t in ratione, cujus
ratio
Ellip&longs;eos alterius, ut prima duarum medie proportionalium inter
corpus mobile de&longs;criptæ erit ad axem principalem de&longs;criptæ circa
immobile, ut
ter
agitata vel impedita, quomodocunque moveantur; motus eo
rum perinde &longs;e habebunt ac &longs;i non traherent &longs;e mutuo, &longs;ed u
trumque a corpore tertio in communi gravitatis centro con&longs;tituto
Viribus ii&longs;dem traberetur: Et Virium trahentium eadem erit Lex
re&longs;pectu di&longs;tantiæ corporum a centro illo communi atque re&longs;pe
ctu di&longs;tantiæ totius inter corpora.
Nam vires illæ, quibus corpora &longs;e mutuo trahunt, tendendo
ad corpora, tendunt ad commune gravitatis centrum interme-
rent.
CORPORUM
Et quoniam data e&longs;t ratio di&longs;tantiæ corporis utriu&longs;vis a centro
illo communi ad di&longs;tantiam corporis eju&longs;dem a corpore altero, da
bitur ratio cuju&longs;vis pote&longs;tatis di&longs;tantiæ unius ad eandem pote&longs;ta
tem di&longs;tantiæ alterius; ut & ratio quantitatis cuju&longs;vis, quæ ex una
di&longs;tantia & quantitatibus datis utcunQ.E.D.rivatur, ad quantitatem
aliam, quæ ex altera di&longs;tantia & quantitatibus totidem datis da
tamQ.E.I.lam di&longs;tantiarum rationem ad priores habentibus &longs;imiliter
derivatur. Proinde &longs;i vis, qua corpus unum ab altero trahitur, &longs;it
directe vel inver&longs;e ut di&longs;tantia corporum ab invicem; vel ut quæ
libet hujus di&longs;tantiæ pote&longs;tas; vel denique ut quantitas quævis ex
hac di&longs;tantia & quantitatibus datis quomodocunQ.E.D.rivata: erit
eadem vis, qua corpus idem ad commune gravitatis centrum tra
hitur, directe itidem vel inver&longs;e ut corporis attracti di&longs;tantia a cen
tro illo communi, vel ut eadem di&longs;tantiæ hujus pote&longs;tas, vel de
nique ut quantitas ex hac di&longs;tantia & analogis quantitatibus da
tis &longs;imiliter derivata. Hoc e&longs;t, Vis trahentis eadem erit Lex re&longs;pe
ctu di&longs;tantiæ utriu&longs;que.
proportionalibus &longs;e mutuo trahunt, ac de locis datis demittun
tur, determinare Motus.
Corpora (per Theorema novi&longs;&longs;imum) perinde movebuntur ac
&longs;i a corpore tertio, in communi gravitatis centro con&longs;tituto, trahe
rentur; & centrum illud ip&longs;o motus initio quie&longs;cet per Hypothe
&longs;in; & propterea (per Legum Corol. 4.) &longs;emper quie&longs;cet.
Deter
minandi &longs;unt igitur motus corporum (per Prob. XXV,) perinde
ac &longs;i a viribus ad centrum illud tendentibus urgerentur, & habe
buntur motus corporum &longs;e mutuo trahentium.
portionalibus &longs;e mutuo trahunt, deque locis datis, &longs;ecundum datas
rectas, datis cum Velocitatibus exeunt, determinare Motus.
Ex datis corporum motibus &longs;ub initio, datur uniformis motus
centro movetur uniformiter in directum, nec non corporum mo
tus initiales re&longs;pectu hujus &longs;patii. Motus autem &longs;ub&longs;equentes
(per Legum Corollarium quintum, & Theorema novi&longs;&longs;imum)
perinde fiunt in hoc &longs;patio, ac &longs;i &longs;patium ip&longs;um una cum commu
ni illo gravitatis centro quie&longs;ceret, & corpora non traherent &longs;e
mutuo, &longs;ed a corpore tertio &longs;ito in centro illo traherentur. Cor
poris igitur alterutrius in hoc &longs;patio mobili, de loco dato, &longs;ecun
dum datam rectam, data cum velocitate exeuntis, & vi centripeta
ad centrum illud tendente correpti, determinandus e&longs;t motus per
Problema nonum & vice&longs;imum &longs;extum: & habebitur &longs;imul mo
tus corporis alterius e regione. Cum hoc motu componendus
e&longs;t uniformis ille Sy&longs;tematis &longs;patii & corporum in eo gyrantium
motus progre&longs;&longs;ivus &longs;upra inventus, & habebitur motus ab&longs;olutus
corporum in &longs;patio immobili.
PRIMUS.
tione di&longs;tantiarum a centris: requiruntur Motus plurium Cor
porum inter &longs;e.
Ponantur primo corpora duo
tatis centrum
rematis XXI) Ellip&longs;es centra habentes in
Problemate V, innote&longs;cit.
Trahat jam corpus tertium
bus acceleratricibus
& ab ip&longs;is vici&longs;&longs;im trahatur.
Vis 2.)
re&longs;olvitur in vires
& vis
Vires autem
&longs;unt ut ip&longs;arum &longs;umma
atque adeo ut vires accelera
trices quibus corpora
bus corporum
ponunt vires di&longs;tantiis 1. Prop.
X. & Corol.
1 & 8. Prop, IV) efficiunt ut corpora illa de&longs;cribant Ellip&longs;es ut prius,
&longs;ed motu celeriore. Vires reliquæ acceleratrices
nibus motricibus
hendo corpora illa æqualiter & &longs;ecundum lineas
parallelas, nil mutant &longs;itus eorum ad invicem, &longs;ed faciunt ut ip&longs;a
æqualiter accedant ad lineam
dium corporis Impedietur au
tem i&longs;te ad lineam
ex una parte, & corpus
tur circa commune gravitatis centrum
(eo quod &longs;umma virium motricium
tiæ
lip&longs;in circa idem
de&longs;cribet Ellip&longs;in con&longs;imilem e regione. Corpora autem
viribus motricibus
&
po&longs;terius po&longs;teriore) æqua
liter & &longs;ecundum lineas pa
rallelas
tum e&longs;t) attracta, pergent
(per Legum Corollarium
quintum & &longs;extum) circa cen
trum mobile
de&longs;cribere, ut prius.
CORPORUM
Addatur jam corpus quartum
tur hoc & punctum
gravitatis Et eadem
methodo corpora plura adjungere licebit.
Hæc ita &longs;e habent ubi corpora
acceleratricibus majoribus vel minoribus quam quibus trahunt cor
pora reliqua pro ratione di&longs;tantiarum. Sunto mutuæ omnium at
tractiones acceleratrices ad invicem ut di&longs;tantiæ ductæ in corpo
ra trahentia, & ex præcedentibus facile deducetur quod corpora
omnia æqualibus temporibus periodicis Ellip&longs;es varias, circa om
nium commune gravitatis centrum
bunt.
PRIMUS.
&longs;tantiarum ab eorundem centris, moveri po&longs;&longs;e inter &longs;e in El
lip&longs;ibus; & radiis ad umbilicos ductis areas de&longs;cribere tempo
ribus proportionales quam proxime.
In Propo&longs;itione &longs;uperiore demon&longs;tratus e&longs;t ca&longs;us ubi motus plu
res peraguntur in Ellip&longs;ibus accurate. Quo magis recedit Lex vi
rium a Lege ibi po&longs;ita, eo magis corpora perturbabunt mutuos
motus; neque fieri pote&longs;t ut corpora, &longs;ecundum Legem hic po&longs;itam
&longs;e mutuo trahentia, moveantur in Ellip&longs;ibus accurate, ni&longs;i &longs;ervando
certam proportionem di&longs;tantiarum ab invicem. In &longs;equentibus au
tem ca&longs;ibus non multum ab Ellip&longs;ibus errabitur.
varias ab eo di&longs;tantias revolvi, tendantque ad &longs;ingula vires ab&longs;olu
tæ proportionales ii&longs;dem corporibus. Et quoniam omnium com
mune gravitatis centrum (per Legum Corol. quartum) vel quie
&longs;cit vel movetur uniformiter in directum, fingamus corpora mi
nora tam parva e&longs;&longs;e, ut corpus maximum nunquam di&longs;tet &longs;en&longs;ibi
liter ab hoc centro: & maximum illud vel quie&longs;cet vel movebitur
uniformiter in directum, ab&longs;que errore &longs;en&longs;ibili; minora autem re
volventur circa hoc maximum in Ellip&longs;ibus, atque radiis ad idem
ductis de&longs;cribent areas temporibus proportionales; ni&longs;i quatenus
errores inducuntur, vel per errorem maximi a communi illo gravi
tatis centro, vel per actiones minorum corporum in &longs;e mutuo. Di
minui autem po&longs;&longs;unt corpora minora u&longs;Q.E.D.nec error i&longs;te & ac
tiones mutuæ &longs;int datis quibu&longs;vis minores, atque adeo donec Orbes
cum Ellip&longs;ibus quadrent, & areæ re&longs;pondeant temporibus, ab&longs;que
errore qui non &longs;it minor quovis dato.
de&longs;cripto circa maximum revolventium, aliudve quodvis duorum
circum &longs;e mutuo revolventium corporum Sy&longs;tema progredi unifor
miter in directum, & interea vi corporis alterius longe maximi &
ad magnam di&longs;tantiam &longs;iti urgeri ad latus. Et quoniam æquales
vires acceleratrices, quibus corpora &longs;ecundum lineas parallelas ur
gentur, non mutant &longs;itus corporum ad invicem, &longs;ed ut Sy&longs;tema
totum, &longs;ervatis partium motibus inter &longs;e, &longs;imul transferatur effici
unt: manife&longs;tum e&longs;t quod, ex attractionibus in corpus maximum,
ex attractionum acceleratricum inæqualitate, vel ex inclinatione li
nearum ad invicem, &longs;ecundum quas attractiones fiunt. Pone ergo
attractiones omnes acceleratrices in corpus maximum e&longs;&longs;e inter &longs;e
reciproce ut quadrata di&longs;tantiarum; &, augendo corporis maximi
di&longs;tantiam, donec rectarum ab hoc ad reliqua ductarum differen
tiæ re&longs;pectu earum longitudinis & inclinationes ad invicem mino
res &longs;int quam datæ quævis, per&longs;everabunt motus partium Sy&longs;tema
tis inter &longs;e ab&longs;que erroribus qui non &longs;int quibu&longs;vis datis minores.
Et quoniam, ob exiguam partium illarum ab invicem di&longs;tantiam,
Sy&longs;tema totum ad modum corporis unius attrahitur; movebitur
idem hac attractione ad modum corporis unius; hoc e&longs;t, centro
&longs;uo gravitatis de&longs;cribet circa corpus maximum Sectionem aliquam
Conicam (
Ellip&longs;in fortiore,) & Radio ad maximum ducto de&longs;cribet areas
temporibus proportionales, ab&longs;que ullis erroribus, ni&longs;i quas par
tium di&longs;tantiæ (perexiguæ &longs;ane & pro lubitu minuendæ) valeant
efficere.
CORPORUM
Simili argumento pergere licet ad ca&longs;us magis compo&longs;itos in in
finitum.
maximum ad Sy&longs;tema duorum vel plurium, eo magis turbabuntur
motus partium Sy&longs;tematis inter &longs;e; propterea quod linearum a cor
pore maximo ad has ductarum jam major e&longs;t inclinatio ad invicem,
majorque proportionis inæqualitas.
nes acceleratrices partium Sy&longs;tematis ver&longs;us corpus omnium maxi
mum, non &longs;int ad invicem reciproce ut quadrata di&longs;tantiarum a
corpore illo maximo; præ&longs;ertim &longs;i proportionis hujus inæqualitas
major &longs;it quam inæqualitas proportionis di&longs;tantiarum a corpore
maximo: Nam &longs;i vis acceleratrix, æqualiter & &longs;ecundum lineas pa
rallelas agendo, nil perturbat motus inter &longs;e, nece&longs;&longs;e e&longs;t ut ex acti
onis inæqualitate perturbatio oriatur, majorque &longs;it vel minor pro
majore vel minore inæqualitate. Exce&longs;&longs;us impul&longs;uum majorum,
agendo in aliqua corpora & non agendo in alia, nece&longs;&longs;ario muta
bunt &longs;itum eorum inter &longs;e. Et hæc perturbatio, addita perturbatio
ni quæ ex linearum inclinatione & inæqualitate oritur, majorem
reddet perturbationem totam.
culis &longs;ine perturbatione in&longs;igni moveantur; manife&longs;tum e&longs;t, quod
neas parallelas quamproxime.
PRIMUS.
&longs;tantiarum, &longs;e mutuo trahant, & attractiones acceleratrices bi
norum quorumcunQ.E.I. tertium &longs;int inter &longs;e reciproce ut qua
drata di&longs;tantiarum; minora autem circa maximum revolvan
tur: Dico quod interius circa intimum & maximum, radiis
ad ip&longs;um ductis, de&longs;cribet areas temporibus magis proportio
nales, & Figuram ad formam Ellip&longs;eos umbilicum in concur
&longs;u radiorum habentis magis accedentem, &longs;i corpus maximum
his attractionibus agitetur, quam &longs;i maximum illud vel a mi
noribus non attractum quie&longs;cat, vel multo minus vel multo ma
gis attractum aut multo minus aut multo magis agitetur.
Liquet fere ex demon&longs;tratione Corollarii &longs;ecundi Propo&longs;itionis
præcedentis; &longs;ed argumento magis di&longs;tincto & latius cogente &longs;ic
evincitur.
corpora minora
in eodem plano circa
maximum
teriorem
exteriorem
tia corporum
& corporis
andem. In duplicata ratione
rit
& attractio
Hac vi &longs;ola corpus
attractione agitatum, de&longs;cribere deberet & areas, radio
poribus proportionales, & Ellip&longs;in cui umbilicus e&longs;t in centro cor
poris XI. & Corollaria 2 & 3 Theor.
XXI.
Vis
altera e&longs;t attractionis
dita vi priori coincidet cum ip&longs;a, & &longs;ic faciet ut areæ etiamnum tem
poribus proportionales de&longs;cribantur per Corol. 3. Theor.
XXI.
At
quoniam non e&longs;t quadrato di&longs;tantiæ
componet ea cum vi priore vim ab hac proportione aberrantem, id
que eo magis quo major e&longs;t proportio hujus vis ad vim priorem,
cæteris paribus. Proinde cum (per Prop.
XI, & per Corol.
2.
Theor. XXI) vis qua Ellip&longs;is circa umbilicum
debeat ad umbilicum illum, & e&longs;&longs;e quadrato di&longs;tantiæ
proportionalis; vis illa
compo&longs;ita, aberrando
ab hac proportione, fa
ciet ut Orbis
aberret a forma Ellip
&longs;eos umbilicum haben
tis in
gis quo major e&longs;t ab
erratio ab hac propor
tione; atque adeo eti
am quo major e&longs;t proportio vis &longs;ecundæ
teris paribus. Jam vero vis tertia
dum lineam ip&longs;i
vim quæ non amplius dirigitur a
natione tanto magis aberrat, quanto major e&longs;t proportio hujus ter
tiæ vis ad vires priores, cæteris paribus; atque adeo quæ faciet ut
corpus
les de&longs;cribat, atque aberratio ab hac proportionalitate ut tanto ma
jor &longs;it, quanto major e&longs;t proportio vis hujus tertiæ ad vires cæte
ras. Orbis vero
vis tertia duplici de cau&longs;a adaugebit, tum quod non dirigatur a
ad
Quibus intellectis, manife&longs;tum e&longs;t quod areæ temporibus tum max
ime fiunt proportionales, ubi vis tertia, manentibus viribus cæte
ris, fit minima; & quod Orbis
fatam formam Ellipticam, ubi vis tam &longs;ecunda quam tertia, &longs;ed præ
cipue vis tertia, fit minima, vi prima manente.
CORPORUM
Exponatur corporis
trahendo corpora
nil mutarent &longs;itum eorum ad invicem. Iidem jam forent corporum
illorum motus inter &longs;e (per Legum Corol. 6.) ac &longs;i hæ attractio
nes tollerentur. Et pari ratione &longs;i attractio
tractione
neret pars &longs;ola
& Orbitæ forma illa Elliptica perturbaretur. Et &longs;imiliter &longs;i attra
ctio Sic per attractio
nem
ctionem
mutatis: & propterea areæ ac tempora ad proportionalitatem, &
Orbita
dunt, ubi attractio
ma; hoc e&longs;t, ubi corporum
ctæ ver&longs;us corpus
tem; id e&longs;t, ubi attractio
attractionum omnium
maximam & minimam qua&longs;i mediocris, hoc e&longs;t, non multo major
neque multo minor attractione
PRIMUS.
in planis diver&longs;is; & vis
no Orbitæ
corpus At vis altera
agendo &longs;ecundum lineam quæ ip&longs;i
quando corpus
planum Orbitæ
dinem jam ante expo&longs;itam, inducet perturbationem motus in Lati
tudinem, trahendo corpus Et hæc per
turbatio, in dato quovis corporum
vis illa generans
ma, hoc e&longs;t (uti jam expo&longs;ui) ubi attractio
jor, neque multo minor attractione
revolvantur circa maximum
mi
maximum
attrahitur & agitatur atque cætera a &longs;e mutuo.
CORPORUM
ones acceleratrices binorum quorumcunQ.E.I. tertium &longs;int ad invi
cem reciproce ut quadrata di&longs;tantiarum; corpus
am circa corpus
po&longs;itionem
qua corpus
&longs;ecundum lineam
perinde ut ip&longs;a in con&longs;equentia vel in antecedentia dirigitur. Talis
e&longs;t vis
&longs;equentia, motumque accelerat; dein u&longs;que ad
& motum retardat; tum in con&longs;equentia u&longs;que ad
antecedentia tran&longs;eundo a
ribus, velocius movetur in Conjunctione & Oppo&longs;itione quam in
Quadraturis.
draturis quam in Conjunctione & Oppo&longs;itione. Nam corpora ve
lociora minus deflec
tunt a recto tramite. Et
præterea vis
& Oppo&longs;itione, con
traria e&longs;t vi qua cor
pus
adeoque vim illam mi
nuit; corpus autem
minus deflectet a recto
tramite, ubi minus urgetur in corpus
pore Hæc
ita &longs;e habent exclu&longs;o motu Excentricitatis. Nam &longs;i Orbita corpo
ris 9.
o&longs;tendetur) evadet maxima ubi Ap&longs;ides &longs;unt in Syzygiis; indeque
fieri pote&longs;t ut corpus
gius a corpore
pus
nem vis
ob magnitudinem vis
tem vis illa centripeta (per Corol. 2, Prop.
IV.) in ratione compo
&longs;ita ex ratione &longs;implici radii
radius
tripeta diminuitur: auctoque adeo vel diminuto hoc Radio, tem
pus periodicum augeri magis, vel diminui minus quam in Radii hu
jus ratione &longs;e&longs;quiplicata, per Corol. 6. Prop.
IV.
Si vis illa corporis
centralis paulatim langue&longs;ceret, corpus
attractum perpetuo recederet longius a centro
illa augeretur, accederet propius. Ergo &longs;i actio corporis longin
qui
augebitur &longs;imul ac diminuetur Radius
riodicum augebitur ac diminuetur in ratione compo&longs;ita ex ratione
&longs;e&longs;quiplicata Radii & ratione &longs;ubduplicata qua vis illa centripeta
corporis centralis
corporis longinqui
PRIMUS.
pore
rem progreditur & regreditur per vices, &longs;ed magis tamen progre
ditur, & in &longs;ingulis corporis revolutionibus per exce&longs;&longs;um progre&longs;
&longs;ionis fertur in con&longs;equentia. Nam vis qua corpus
corpus
&longs;i augeatur di&longs;tantia
di&longs;tantia, & vis po&longs;terior decre&longs;cit in duplicata illa ratione, adeo
que &longs;umma harum virium decre&longs;cit in minore quam duplicata ra
tione di&longs;tantiæ 1. Prop.
XLV) efficit
ut Aux, &longs;eu Ap&longs;is &longs;umma, regrediatur. In Conjunctione vero &
Oppo&longs;itione, vis qua corpus
inter vim qua corpus
tia illa, propterea quod vis
di&longs;tantiæ
tiæ 1. Prop.XLV) efficit ut Aux progre
diatur. In locis inter Syzygias & Quadraturas pendet motus Au
gis ex cau&longs;a utraque conjunctim, adeo ut pro hujus vel alterius
exce&longs;&longs;u progrediatur ip&longs;a vel regrediatur. Unde cum vis
Syzygiis &longs;it qua&longs;i duplo major quam vis
ce&longs;&longs;us in tota revolutione erit penes vim
gem &longs;ingulis revolutionibus in con&longs;equentia. Veritas autem hujus
& præcedentis Corollarii facilius intelligetur concipiendo Sy&longs;tema
corporum duorum
be Namque horum actioni-
quam duplicata di&longs;tantiæ.
CORPORUM
a decremento vis centripetæ facto in majori vel minori quam du
plicata ratione di&longs;tantiæ
ad Ap&longs;idem &longs;ummam; ut & a &longs;imili incremento in reditu ad Ap
&longs;idem imam; atque adeo maximus &longs;it ubi proportio vis in Ap&longs;ide
&longs;umma ad vim in Ap&longs;ide ima maxime recedit a duplicata ratione
di&longs;tantiarum inver&longs;a: manife&longs;tum e&longs;t quod Ap&longs;ides in Syzygiis
&longs;uis, per vim ablatitiam
locius, inque Quadraturis &longs;uis tardius recedent per vim addititiam
tarditas regre&longs;&longs;us continuatur, fit hæc inæqualitas longe maxima.
di&longs;tantiæ &longs;uæ a centro, revolveretur circa hoc centrum in El
lip&longs;i, & mox, in de&longs;cen&longs;u ab Ap&longs;ide &longs;umma &longs;eu Auge ad Ap&longs;idem
imam, vis illa per acce&longs;&longs;um perpetuum vis novæ augeretur in ra
tione plu&longs;quam dupli
cata di&longs;tantiæ diminu
tæ: manife&longs;tum e&longs;t
quod corpus, perpe
tuo acce&longs;&longs;u vis illius
novæ impul&longs;um &longs;em
per in centrum, magis
vergeret in hoc cen
trum, quam &longs;i urge
retur vi &longs;ola cre&longs;cente
in duplicata ratione di&longs;tantiæ diminutæ, adeoque Orbem de&longs;cri
beret Orbe Elliptico interiorem, & in Ap&longs;ide ima propius acce
deret ad centrum quam prius. Orbis igitur, acce&longs;&longs;u hujus vis no
væ, fiet magis excentricus. Si jam vis, in rece&longs;&longs;u corporis ab
Ap&longs;ide ima ad Ap&longs;idem &longs;ummam, decre&longs;ceret ii&longs;dem gradibus qui
bus ante creverat, rediret corpus ad di&longs;tantiam priorem, adeoque
&longs;i vis decre&longs;cat in majori ratione, corpus jam minus attractum a&longs;
cendet ad di&longs;tantiam majorem & &longs;ic Orbis Excentricitas adhuc ma
gis augebitur. Igitur &longs;i ratio incrementi & decrementi vis centri
petæ &longs;ingulis revolutionibus augeatur, augebitur &longs;emper Excentri
citas; & e contra, diminuetur eadem &longs;i ratio illa decre&longs;cat. Jam
vero in Sy&longs;temate corporum
&longs;unt in Quadraturis, ratio illa incrementi ac decrementi minima e&longs;t, Si Ap&longs;ides con&longs;tituan
gias major quam duplicata di&longs;tantiarum, & ex ratione illa majori
oritur Augis motus veloci&longs;&longs;imus, uti jam dictum e&longs;t. At &longs;i con
&longs;ideretur ratio incrementi vel decrementi totius in progre&longs;&longs;u inter
Ap&longs;ides, hæc minor e&longs;t quam duplicata di&longs;tantiarum. Vis in Ap
&longs;ide ima e&longs;t ad vim in Ap&longs;ide &longs;umma in minore quam duplicata
ratione di&longs;tantiæ Ap&longs;idis &longs;ummæ ab umbilico Ellip&longs;eos ad di
&longs;tantiam Ap&longs;idis imæ ab eodem umbilico: & e contra, ubi
Ap&longs;ides con&longs;tituuntur in Syzygiis, vis in Ap&longs;ide ima e&longs;t ad vim
in Ap&longs;ide &longs;umma in majore quam duplicata ratione di&longs;tantiarum.
Nam vires
nunt vires in ratione minore, & vires
viribus corporis E&longs;t igi
tur ratio decrementi & incrementi totius, in tran&longs;itu inter Ap&longs;ides,
minima in Quadraturis, maxima in Syzygiis: & propterea in tran
&longs;itu Ap&longs;idum a Quadraturis ad Syzygias perpetuo augetur, auget
que Excentricitatem Ellip&longs;eos; inque tran&longs;itu a Syzygiis ad
Quadraturas perpetuo diminuitur, & Excentricitatem diminuit.
PRIMUS.
mus planum Orbis
&longs;ita cau&longs;a manife&longs;tum e&longs;t quod, ex viribus
cau&longs;a illa tota, vis
ubi Nodi &longs;unt in Syzygiis, agendo etiam &longs;ecundum idem Orbis
planum, non perturbat hos motus; ubi vero &longs;unt in Quadraturis
eos maxime perturbat, corpu&longs;que
trahendo, minuit inclinationem plani in tran&longs;itu corporis a Qua
draturis ad Syzygias, augetque vici&longs;&longs;im eandem in tran&longs;itu a Syzy
giis ad Quadraturas. Unde fit ut corpore in Syzygiis exi&longs;tente in
clinatio evadat omnium minima, redeatque ad priorem magnitudi
nem circiter, ubi corpus ad Nodum proximum accedit. At &longs;i Nodi
con&longs;tituantur in Octantibus po&longs;t Quadraturas, id e&longs;t, inter
D
ni perpetuo minuitur; deinde in tran&longs;itu per proximos 45 gradus,
u&longs;que ad Quadraturam proximam, inclinatio augetur, & po&longs;tea de
nuo in tran&longs;itu per alios 45 gradus, u&longs;que ad Nodum proximum,
diminuitur. Magis itaQ.E.D.minuitur inclinatio quam augetur, &
propterea minor e&longs;t &longs;emper in Nodo &longs;ub&longs;equente quam in præce-Et &longs;imili ratiocinio, inclinatio magis augetur quam diminui
tur ubi Nodi &longs;unt in Octantibus alteris inter
clinatio igitur ubi Nodi &longs;unt in Syzygiis e&longs;t omnium maxima. In
tran&longs;itu eorum a Syzygiis ad Quadraturas, in &longs;ingulis corporis ad
Nodos appul&longs;ibus, diminuitur, fitque omnium minima ubi Nodi
&longs;unt in Quadraturis & corpus in Syzygiis: dein cre&longs;cit ii&longs;dem gra
dibus quibus antea decreverat, Nodi&longs;que ad Syzygias proximas ap
pul&longs;is ad magnitudinem primam revertitur.
CORPORUM
petuo trahitur de plano Orbis &longs;ui, idQ.E.I. partem ver&longs;us
tran&longs;itu &longs;uo a Nodo
contrariam partem in tran&longs;itu a Nodo
Nodum
perpetuo recedit ab Orbis &longs;ui plano primo
ventum e&longs;t ad Nodum proximum; adeoQ.E.I. hoc Nodo, longi&longs;&longs;i
me di&longs;tans a plano illo primo
non in plani illius Nodo altero
ad partes corporis
teriora vergens. Et &longs;imili argumento pergent Nodi recedere in
tran&longs;itu corporis de hoc Nodo in Nodum proximum. Nodi igi
tur in Quadraturis con&longs;tituti perpetuo recedunt; in Syzygiis (ubi
motus in Latitudinem nil perturbatur) quie&longs;cunt; in locis inter
mediis, conditionis utriu&longs;que participes, recedunt tardius; adeoque,
&longs;emper vel retrogradi vel &longs;tationarii, &longs;ingulis revolutionibus ferun
tur in antecedentia.
lo majores in Conjunctione corporum
po&longs;itione, idque ob majores vires generantes
a magnitudine corporis
rum Et ex aucto corpore
vi centripeta, a qua errores corporis
omnes (paribus di&longs;tantiis) majores in hoc ca&longs;u quam in altero, ubi
corpus
quum e&longs;t, &longs;int quamproxime ut vis
junctim, hoc e&longs;t, &longs;i detur tum di&longs;tantia
ab&longs;oluta, ut
cau&longs;æ errorum & effectuum omnium de quibus actum e&longs;t in præce-
vi ab&longs;oluta corporis
ratione directa vis ab&longs;olutæ corporis
di&longs;tantiæ
ca corpus longinquum
erunt (per Corol. 2. & 6. Prop.
IV.) reciproce in duplicata ratione
temporis periodici. Et inde etiam, &longs;i magnitudo corporis
tionalis &longs;it ip&longs;ius vi ab&longs;olutæ, erunt vires illæ
effectus directe ut cubus diametri apparentis longinqui corporis
corpore Namque hæ rationes eædem &longs;unt
atque ratio &longs;uperior compo&longs;ita.
PRIMUS.
forma, proportionibus & inclinatione ad invicem, mutetur eorum
magnitudo, & &longs;i corporum
in data quavis ratio
ne, hæ vires (hoc e&longs;t,
vis corporis
pus
te in Orbitam
deflectere, & vis cor
poris
idem
deviare cogitur) agunt
&longs;emper eodem mo
do & eadem proportione: nece&longs;&longs;e e&longs;t ut &longs;imiles & proportiona
les &longs;int effectus omnes & proportionalia effectuum tempora; hoc
e&longs;t, ut errores omnes lineares &longs;int ut Orbium diametri, angulares
vero iidem qui prius, & errorum linearium &longs;imilium vel angularium
æqualium tempora ut Orbium tempora periodica.
cem, & mutentur utcunque corporum magnitudines, vires & di
&longs;tantiæ; ex datis erroribus & errorum temporibus in uno Ca&longs;u, col
ligi po&longs;&longs;unt errores & errorum tempora in alio quovis, quam pro
xime: Sed brevius hac Methodo. Vires
tibus, &longs;unt ut Radius
Lem. X) ut vires & quadratum temporis periodici corporis
junctim. Hi &longs;unt errores lineares corporis
gulares e centro
quam omnes in Longitudinem & Latitudinem errores apparentes)
&longs;unt, in qualibet revolutione corporis Conjungantur hæ rationes cum ratio
nibus Corollarii 14, & in quolibet corporum
ubi
volvitur, errores angulares corporis
erunt, in &longs;ingulis revolutionibus corporis illius
temporis periodici corporis
riodici corporis Et inde motus medius Augis erit in da
ta ratione ad motum medium Nodorum; & motus uterque erit ut tempus periodicum corporis &c.
quadratum temporis periodici corporis
temporis periodici corporis Augendo vel minuendo
Excentricitatem & Inclinationem Orbis
tus Augis & Nodorum &longs;en&longs;ibiliter, ni&longs;i ubi eædem &longs;unt nimis
magnæ.
CORPORUM
quam radius
lum
vim mediocrem
vel
nere licet per
longitudo
gitudinem
tem vis mediocris
vel
retinetur in Orbe &longs;uo
circum
corpus
ratione radii
riodici corporis
circum
dem tempore periodico, circum punctum quodvis immobile
di&longs;tantiam
eorum temporum. Datis igitur temporibus periodicis una cum di
&longs;tantia
volvitur, fingamus corpora plura fluida circum idem
les ab ip&longs;o di&longs;tantias moveri; deinde ex his contiguis factis confla
ri Annulum fluidum, rotundum ac corpori
&longs;ingulæ Annuli partes, motus &longs;uos omnes ad legem corporis
Quadraturis. Et Nodi Annuli hujus &longs;eu inter&longs;ectiones ejus cum
plano Orbitæ corporis
gias vero movebuntur in antecedentia, & veloci&longs;&longs;ime quidem in
Quadraturis, tardius aliis in locis. Annuli quoQ.E.I.clinatio varia
bitur, & axis ejus &longs;ingulis revolutionibus o&longs;cillabitur, completaque
revolutione ad pri&longs;tinum &longs;itum redibit, ni&longs;i quatenus per præce&longs;&longs;i
onem Nodorum circumfertur.
PRIMUS.
con&longs;tantem, ampliari & extendi u&longs;que ad hunc Annulum, & alveo
per circuitum excavato continere Aquam, motuque eodem perio
dico circa axem &longs;uum uniformiter revolvi. Hic liquor per vices
acceleratus & retardatus (ut in &longs;uperiore Corollario) in Syzygiis
velocior erit, in Quadraturis tardior quam &longs;uperficies Globi, &
&longs;ic fluet in alveo refluet que ad modum Maris. Aqua revolvendo cir
ca Globi centrum quie&longs;cens, &longs;i tollatur attractio corporis
acquiret motum fluxus & refluxus. Par e&longs;t ratio Globi uniformiter
progredientis in directum & interea revolventis circa centrum
&longs;uum (per Legum Corol. 5.) ut & Globi de cur&longs;u rectilineo uNI
formiter tracti, per Legum Corol. 6. Accedat autem corpus
& ab ip&longs;ius inæquabili attractione mox turbabitur Aqua. Etenim
major erit attractio aquæ propioris, minor ea remotioris. Vis
autem
&longs;am de&longs;cendere u&longs;que ad Syzygias; & vis
&longs;um in Syzygiis, &longs;i&longs;tetQ.E.D.&longs;cen&longs;um ejus & faciet ip&longs;am a&longs;cendere
u&longs;que ad Quadraturas.
bit motus fluendi & refluendi; &longs;ed O&longs;cillatorius ille inclinationis
motus & præce&longs;&longs;io Nodorum manebunt. Habeat Globus eundem
axem cum Annulo, gyro&longs;que compleat ii&longs;dem temporibus, & &longs;uper
ficie &longs;ua contingat ip&longs;um interius, eiQ.E.I.hæreat; & participando
motum ejus, compages utriu&longs;que O&longs;cillabitur & Nodi regredien
tur. Nam Globus, ut mox dicetur, ad &longs;u&longs;cipiendas impre&longs;&longs;iones
omnes indifferens e&longs;t. Annuli Globo orbati maximus inclinationis
angulus e&longs;t ubi Nodi &longs;unt in Syzygiis. Inde in progre&longs;&longs;u Nodo
rum ad Quadraturas conatur is inclinationem &longs;uam minuere, & i&longs;to
conatu motum imprimit Globo toti. Retinet Globus motum im
pre&longs;&longs;um u&longs;Q.E.D.m Annulus conatu contrario motum hunc tollat,
imprimatque motum novum in contrariam partem: Atque hac ra-
Nodorum, & minimus inclinationis angulus in Octantibus po&longs;t
Quadraturas; dein maximus reclinationis motus in Syzygiis, &
maximus angulus in Octantibus proximis. Et eadem e&longs;t ratio Glo
bi Annulo nudati, qui in regionibus æquatoris vel altior e&longs;t paulo
quam juxta polos, vel con&longs;tat ex nateria paulo den&longs;iore. Sup
plet enim vicem Annuli i&longs;te materiæ in æquatoris regionibus exce&longs;
&longs;us. Et quanquam, aucta utcunque Globi hujus vi centripeta,
tendere &longs;upponantur omnes ejus partes deor&longs;um, ad modum gra
vitantium partium telluris, tamen Phænomena hujus & præceden
tis Corollarii vix inde mutabuntur.
CORPORUM
redundans efficit ut Nodi regrediantur, atque adeo per hujus in
crementum augetur i&longs;te regre&longs;&longs;us, per diminutionem vero diminui
tur & per ablationem tollitur; &longs;i materia plu&longs;quam redundans tol
latur, hoc e&longs;t, &longs;i Globus juxta æquatorem vel depre&longs;&longs;ior reddatur
vel rarior quam juxta polos, orietur motus Nodorum in con
&longs;equentia.
tutio Globi. Nimirum &longs;i Globus polos eo&longs;dem con&longs;tanter &longs;ervat,
& motus fit in antecedentia, materia juxta æquatorem redundat;
&longs;i in con&longs;equentia, deficit. Pone Globum uniformem & perfecte
circinatum in &longs;patiis liberis primo quie&longs;cere; dein impetu quocun
que obliQ.E.I. &longs;uperficiem &longs;uam facto propelli, & motum inde
concipere partim circularem, partim in directum. Quoniam Glo
bus i&longs;te ad axes omnes per centrum &longs;uum tran&longs;euntes indifferenter
&longs;e habet, neque propen&longs;ior e&longs;t in unum axem, unumve axis &longs;itum,
quam in alium quemvis; per&longs;picuum e&longs;t quod is axem &longs;uum axi&longs;
Q.E.I.clinationem vi propria nunquam mutabit. Impellatur jam
Globus oblique, in eadem illa &longs;uperficiei parte qua prius, impul&longs;u
quocunque novo; & cum citior vel ferior impul&longs;us effectum nil
mutet, manife&longs;tum e&longs;t quod hi duo impul&longs;us &longs;ucce&longs;&longs;ive impre&longs;&longs;i
eundem producent motum ac &longs;i &longs;imul impre&longs;&longs;i fui&longs;&longs;ent, hoc e&longs;t,
eundem ac &longs;i Globus vi &longs;implici ex utroque (per Legum Corol. 2.)
compo&longs;ita impul&longs;us fui&longs;&longs;et, atque adeo &longs;implicem, circa axem in
clinatione datum. Et par e&longs;t ratio impul&longs;us &longs;ecundi facti in lo
cum alium quemvis in æquatore motus primi; ut & impul&longs;us pri
mi facti in locum quemvis in æquatore motus, quem impul&longs;us &longs;e
cundus ab&longs;que primo generaret; atque adeo impul&longs;uum amborum
factorum in loca quæcunque: Generabunt hi eundem motum cir-
Globus igitur homogeneus & perfectus non retinet motus plures
di&longs;tinctos, &longs;ed impre&longs;&longs;os omnes componit & ad unum reducit, &
quatenus in &longs;e e&longs;t, gyratur &longs;emper motu &longs;implici & uniformi circa
axem unicum, inclinatione &longs;emper invariabili datum. Sed nec vis
centripeta inclinationem axis, aut rotationis velocitatem mutare
pote&longs;t. Si Globus plano quocunque, per centrum &longs;uum & cen
trum in quod vis dirigitur tran&longs;eunte, dividi intelligatur in duo he
mi&longs;phæria; urgebit &longs;emper vis illa utrumque hemi&longs;phærium æqua
liter, & propterea Globum, quoad motum rotationis, nullam in
partem inclinabit. Addatur vero alicubi inter polum & æquato
rem materia nova in formam montis cumulata, & hæc, perpetuo
conatu recedendi a centro &longs;ui motus, turbabit motum Globi, fa
cietque polos ejus errare per ip&longs;ius &longs;uperficiem, & circulos circum
&longs;e punctumque &longs;ibi oppo&longs;itum perpetuo de&longs;cribere. Neque corrige
tur i&longs;ta vagationis enormitas, ni&longs;i locando montem illum vel in polo
alterutro, quo in Ca&longs;u (per Corol. 21) Nodi æquatoris progredien
tur; vel in æquatore, qua ratione (per Corol. 20) Nodi regredi
entur; vel denique ex altera axis parte addendo materiam novam,
qua mons inter movendum libretur, & hoc pacto Nodi vel pro
gredientur, vel recedent, perinde ut mons & hæcce nova materia
&longs;unt vel polo vel æquatori propiores.
PRIMUS.
ad centrum illud ductis, de&longs;cribit areas temporibus magis pro
portionales & Orbem ad formam Ellip&longs;eos umbilicum in centro
eodem habentis magis accedentem, quam circa corpus intimum
& maximum
Nam corporis
tractionem ab&longs;olutam, quæ magis dirigitur in corporum
mune gravitatis centrum
quadrato di&longs;tantiæ
quadrato di&longs;tantiæ
CORPORUM
diis ad centrum illud ductis, de&longs;cribit areas temporibus magis
proportionales, & Orbem ad formam Ellip&longs;eos umbilicum in
centro eodem habentis magis accedentem, &longs;i corpus intimum &
maximum his attractionibus perinde atque cætera agitetur, quam
&longs;i id vel non attractum quie&longs;cat, vel multo magis aut multo
minus attractum aut multo magis aut multo minus agitetur.
Demon&longs;tratur eo
dem fere modo cum
Prop. LXVI, &longs;ed ar
gumento prolixiore,
quod ideo prætereo.
Suffecerit rem &longs;ic æ&longs;ti
mare. Ex demon&longs;tra
tione Propo&longs;itionis
novi&longs;&longs;imæ liquet cen
trum in quod corpus
vitatis duorum illorum. Si coincideret hoc centrum cum centro
illo communi, & quie&longs;ceret commune centrum gravitatis corporum
trium; de&longs;criberent corpus
aliorum duorum ex altera parte, circa commune omnium centrum
quie&longs;cens, Ellip&longs;es accuratas. Liquet hoc per Corollarium &longs;ecun
dum Propo&longs;itionis LVIII collatum cum demon&longs;tratis in Propo&longs;.
LXIV & LXV. Perturbatur i&longs;te motus Ellipticus aliquantulum per
di&longs;tantiam centri duorum a centro in quod tertium
Detur præterea motus communi trium centro, & augebitur per
turbatio. Proinde minima e&longs;t perturbatio ubi commune trium
centrum quie&longs;cit, hoc e&longs;t, ubi corpus intimum & maximum
cæterorum attrahitur: fitque major &longs;emper ubi trium commune il
lud centrum, minuendo motum corporis
gis deinceps magi&longs;que agitatur.
Ellipticas, & arearum de&longs;criptiones fient magis æquabiles, &longs;i cor
pora omnia viribus acceleratricibus, quæ &longs;unt ut eorum vires ab
&longs;olutæ directe & quadrata di&longs;tantiarum inver&longs;e, &longs;e mutuo trahant
agitentque, & Orbitæ cuju&longs;que umbilicus collocetur in communi
centro gravitatis corporum omnium interiorum (nimirum umbi
licus Orbitæ primæ & intimæ in centro gravitatis corporis maxi
mi & intimi; ille Orbitæ &longs;ecundæ, in communi centro gravi
tatis corporum duorum intimorum; i&longs;te tertiæ, in communi cen
tro gravitatis trium interiorum; & &longs;ic deinceps) quam &longs;i corpus
intimum quie&longs;cat & &longs;tatuatur communis umbilicus Orbitarum
omnium.
PRIMUS.
&longs;i corpus aliquod
A viribus acceler atricibus
quæ &longs;unt reciproce ut quadrata di&longs;tantiarum a trahente; &
corpus aliudviribus quæ
&longs;unt reciproce ut quadrata di&longs;tantiarum a trahente: erunt Ab
&longs;olutæ corporum trahentium
ip&longs;a corpora
Nam attractiones acceleratrices corporum omnium
&longs;us
&longs;imiliter attractiones acceleratrices corporum omnium ver&longs;us
paribus di&longs;tantiis, &longs;ibi invicem æquantur. E&longs;t autem ab&longs;oluta vis
attractiva corporis
attractio acceleratrix corporum omnium ver&longs;us
acceleratricem corporum omnium ver&longs;us
ita e&longs;t attractio acceleratrix corporis
acceleratricem corporis
poris
ver&longs;us
quod vires motrices, quæ (per Definitionem &longs;ecundam, &longs;epti
mam & octavam) ex viribus acceleratricibus in corpora attracta
ductis oriuntur, &longs;unt (per motus Legem tertiam) &longs;ibi invicem æqua-Ergo ab&longs;oluta vis attractiva corporis
attractivam corporis
ris
CORPORUM
&longs;eor&longs;im &longs;pectata trahant cætera omnia viribus acceleratricibus quæ
&longs;unt reciproce ut quadrata di&longs;tantiarum a trahente; erunt corpo
rum illorum omnium vires ab&longs;olutæ ad invicem ut &longs;unt ip&longs;a cor
pora.
&longs;eor&longs;im &longs;pectata trahant cætera omnia viribus
acceleratricibus quæ &longs;unt vel reciproce vel directe in ratione dig
nitatis cuju&longs;cunQ.E.D.&longs;tantiarum a trahente, quæve &longs;ecundum Le
gem quamcunque communem ex di&longs;tantiis ab unoquoque trahente
definiuntur; con&longs;tat quod corporum illorum vires ab&longs;olutæ &longs;unt
ut corpora.
ratione duplicata di&longs;tantiarum, &longs;i minora circa maximum in Ellip&longs;i
bus umbilicum communem in maximi illius centro habentibus quam
fieri pote&longs;t accurati&longs;&longs;imis revolvantur, & radiis ad maximum illud
ductis de&longs;cribant areas temporibus quam maxime proportionales:
erunt corporum illorum vires ab&longs;olutæ ad invicem, aut accurate aut
quamproxime in ratione corporum; & contra. Patet per Corol.
Prop. LXVIII collatum cum hujus Corol.
1.
His Propo&longs;itionibus manuducimur ad analogiam inter vires cen
tripetas & corpora centralia, ad quæ vires illæ dirigi &longs;olent. Ra
tioni enim con&longs;entaneum e&longs;t, ut vires quæ ad corpora diriguntur
pendeant ab eorundem natura & quantitate, ut fit in Magneticis.
Et quoties huju&longs;modi ca&longs;us incidunt, æ&longs;timandæ erunt corporum
attractiones, a&longs;&longs;ignando &longs;ingulis eorum particulis vires proprias,
& colligendo &longs;ummas virium. Vocem Attractionis hic generaliter
u&longs;urpo pro corporum conatu quocunque accedendi ad invicem;
&longs;ive conatus i&longs;te fiat ab actione corporum, vel &longs;e mutuo petentium,
vel per Spiritus emi&longs;&longs;os &longs;e invicem agitantium, &longs;ive is ab actione
Ætheris, aut Aeris, Mediive cuju&longs;cunque &longs;eu corporei &longs;eu incorpo
rei oriatur corpora innatantia in &longs;e invicem utcunQ.E.I.pellentis.
Eodem &longs;en&longs;u generali u&longs;urpo vocem Impul&longs;us, non &longs;pecies virium In
Mathe&longs;i inve&longs;tigandæ &longs;unt virium quantitates & rationes illæ, quæ
ex conditionibus quibu&longs;cunque po&longs;itis con&longs;equentur: deinde, ubi
in Phy&longs;icam de&longs;cenditur, conferendæ &longs;unt hæ rationes cum Phæ
nomenis, ut innote&longs;cat quænam virium conditiones &longs;ingulis cor
porum attractivorum generibus competant. Et tum demum de vi
rium &longs;peciebus, cau&longs;is & rationibus Phy&longs;icis tutius di&longs;putare lice
bit. Videamus igitur quibus viribus corpora Sphærica, ex particu
lis modo jam expo&longs;ito attractivis con&longs;tantia, debeant in &longs;e mutuo
agere, & quales motus inde con&longs;equantur.
PRIMUS.
tripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis:
dico quod corpu&longs;culum intra &longs;uperficiem con&longs;titutum his viri
bus nullam in partem attrahitur.
Sit
ca, &
tum. Per
ficiem lineæ duæ
quam minimos
entes; &, ob triangula
(per Corol. 3. Lem.
VII) &longs;imilia, arcus
illi erunt di&longs;tantiis
portionales; & &longs;uperficiei Sphæricæ
particulæ quævis ad
tis per punctum
dique terminatæ, erunt in duplicata
illa ratione. Ergo vires harum particularum in corpus
&longs;unt inter &longs;e æquales. Sunt enim ut particulæ directe & quadrata
di&longs;tantiarum inver&longs;e. Et hæ duæ rationes componunt rationem
Attractiones igitur, in contrarias partes æqualiter fac
tæ, &longs;e mutuo de&longs;truunt. Et &longs;imili argumento, attractiones omnes
per totam Sphæricam &longs;uperficiem a contrariis attractionibus de
&longs;truuntur. Proinde corpus
impellitur.
CORPORUM
con&longs;titutum attrahitur ad centrum Sphæræ, vi reciproce propor
tionali quadrato di&longs;tantiæ &longs;uæ ab eodem centro.
Sint
&longs;ecus in diametris illis productis. Agantur a corpu&longs;culis lineæ
ahb,
mittantur perpendicula
tros perpendicula
(ob æquales
& lineolæ
tio ultima, angulis illis
qualitatis. His itaque con&longs;titutis, erit
&
ut 3. Lem.
VII,) ut arcus
arcum
vel
conjunctis rationibus
ut
&longs;emicirculi Et vires, quibus
hæ &longs;uperficies &longs;ecundum lineas ad &longs;e tendentes attrahunt corpu&longs;cu
la
ad quadrata di&longs;tantiarum &longs;uarum a corporibus, hoc e&longs;t, ut
ad
quæ (facta per Legum Corol. 2. re&longs;olutione virium) &longs;ecundum
lineas
e&longs;t (ob &longs;imilia triangula
ver&longs;us
(
mento vires, quibus &longs;uperficies convolutione arcuum
&longs;criptæ trahunt corpu&longs;cula, erunt ut
eadem ratione erunt vires &longs;uperficierum omnium circularium in quas
utraque &longs;uperficies Sphærica, capiendo &longs;emper Et, per compo&longs;itionem, vires
totarum &longs;uperficierum Sphæricarum in corpu&longs;cula exercitæ erunt
in eadem ratione.
PRIMUS.
tripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis,
ac detur tum Sphæræ den&longs;itas, tum ratio diametri Sphæræ ad
di&longs;tantiam corpu&longs;culi a centro ejus; dico quod vis qua corpu&longs;
culum attrahitur proportionalis erit &longs;emidiametro Sphæræ.
Nam concipe corpu&longs;cula duo &longs;eor&longs;im a Sphæris duabus attrahi,
unum ab una & alterum ab altera, & di&longs;tantias eorum a Sphæra
rum centris proportionales e&longs;&longs;e diametris Sphærarum re&longs;pective,
Sphæras autem re&longs;olvi in particulas &longs;imiles & &longs;imiliter po&longs;itas ad
corpu&longs;cula. Et attractiones corpu&longs;culi unius, factæ ver&longs;us &longs;ingulas
particulas Sphæræ unius, erunt ad attractiones alterius ver&longs;us ana
logas totidem particulas Sphæræ alterius, in ratione compo&longs;ita ex
ratione particularum directe & ratione duplicata di&longs;tantiarum in-Sed particulæ &longs;unt ut Sphæræ, hoc e&longs;t, in ratione triplicata
diametrorum, & di&longs;tantiæ &longs;unt ut diametri, & ratio prior directe
una cum ratione po&longs;teriore bis inver&longs;e e&longs;t ratio diametri ad diame
trum.
CORPORUM
æqualiter attractiva con&longs;tantes, revolvantur; &longs;intQ.E.D.&longs;tantiæ a cen
tris Sphærarum proportionales earundem diametris: Tempora peri
odica erunt æqualia.
di&longs;tantiæ erunt proportionales diametris. Con&longs;tant hæc duo per
Corol. 3. Prop.
IV.
ter den&longs;orum puncta &longs;ingula tendant vires æquales centripetæ de
cre&longs;centes in duplicata ratione di&longs;tantiarum a punctis: vires qui
bus corpu&longs;cula, ad Solida illa duo &longs;imiliter &longs;ita, attrahentur ab ii&longs;
dem, erunt ad invicem ut diametri Solidorum.
centripetæ decre&longs;centes in duplicata ratione di&longs;tantiarum a pun
ctis: dico quod corpu&longs;culum intra Sphæram con&longs;titutum attra
bitur vi proportionali di&longs;tantiæ &longs;uæ ab ip&longs;ius centro.
In Sphæra
locetur corpu&longs;culum
intervallo Manife&longs;tum e&longs;t, per Prop.
LXX, quod Sphæricæ &longs;uperficies concentri
cæ ex quibus Sphærarum differentia
componitur, attractionibus per attractiones
contrarias de&longs;tructis, nil agunt in corpus LXXII, hæc e&longs;t ut
di&longs;tantia
Superficies ex quibus &longs;olida componuntur, hic non &longs;unt pure
Mathematicæ, &longs;ed Orbes adeo tenues ut eorum cra&longs;&longs;itudo in&longs;tar
tur in infinitum. Similiter per Puncta, ex quibus lineæ, &longs;uperficies
& &longs;olida componi dicuntur, intelligendæ &longs;unt particulæ æquales
magnitudinis contemnendæ.
PRIMUS.
attrabitur vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ ab
ip&longs;ius centro.
Nam di&longs;tinguatur Sphæra in &longs;uperficies Sphæricas innumeras
concentricas, & attractiones corpu&longs;culi a &longs;ingulis &longs;uperficiebus
oriundæ erunt reciproce proportionales quadrato di&longs;tantiæ cor
pu&longs;culi a centro, per Prop. LXXI.
Et componendo, fiet &longs;um
ma attractionum, hoc e&longs;t attractio corpu&longs;culi in Sphæram totam, in
eadem ratione.
Sphærarum, attractiones &longs;unt ut Sphæræ. Nam per Prop.
LXXII,
&longs;i di&longs;tantiæ &longs;unt proportionales diametris Sphærarum, vires erunt
ut diametri. Minuatur di&longs;tantia major in illa ratione; &, di&longs;tan
tiis jam factis æqualibus, augebitur attractio in duplicata illa ratio
ne, adeoque erit ad attractionem alteram in triplicata illa ratione,
hoc e&longs;t, in ratione Sphærarum.
plicatæ ad quadrata di&longs;tantiarum.
trahitur vi reciproce proportionali quadrato di&longs;tantiæ &longs;uæ ab ip&longs;ius
centro, con&longs;tet autem Sphæra ex particulis attractivis; decre&longs;cet vis
particulæ cuju&longs;Q.E.I. duplicata ratione di&longs;tantiæ a particula.
tæ, decre&longs;centes in duplicata ratione di&longs;tantiarum a punctis; dico
quod Sphæra quævis alia &longs;imilaris ab eadem attrahitur vi reci
proce proportionali quadrato di&longs;tantiæ centrorum.
Nam particulæ cuju&longs;vis attractio e&longs;t reciproce ut quadratum di
&longs;tantiæ &longs;uæ a centro Sphæræ trahentis, (per Prop. LXXIV) & prop-
co &longs;ito in centro hujus Sphæræ. Hæc autem attractio tanta e&longs;t
quanta foret vici&longs;&longs;im attractio corpu&longs;culi eju&longs;dem, &longs;i modo illud a
&longs;ingulis Sphæræ attractæ particulis eadem vi traheretur qua ip&longs;as
attrahit. Foret autem illa corpu&longs;culi attractio (per Prop.
LXXIV)
reciproce proportionalis quadrato di&longs;tantiæ &longs;uæ a centro Sphæ
ræ; adeoque huic æqualis attractio Sphæræ e&longs;t in eadem ratio
ne.
CORPORUM
neas, &longs;unt ut Sphæræ trahentes applicatæ ad quadrata di&longs;tantiarum
centrorum &longs;uorum a centris earum quas attrahunt.
Nam
que hujus puncta &longs;ingula trahent &longs;ingula alterius, eadem vi qua ab
ip&longs;is vici&longs;&longs;im trahuntur, adeoque cum in omni attractione urgea
tur (per Legem III) tam punctum attrahens, quam punctum at
tractum, geminabitur vis attractionis mutuæ, con&longs;ervatis propor
tionibus.
umbilicum Conicarum Sectionum demon&longs;trata &longs;unt, obtinent ubi
Sphæra attrahens locatur in umbilico & corpora moventur extra
Sphæram.
nicarum Sectionum demon&longs;trantur, obtinent ubi motus peraguntur
intra Sphæram.
riæ den&longs;itatem & vim attractivam) utcunQ.E.D.&longs;&longs;imilares, in
progre&longs;&longs;u vero per circuitum ad datam omnem a centro di&longs;tan
tiam &longs;unt undique &longs;imilares, & vis attractiva puncti cuju&longs;que
decre&longs;cit in duplicata ratione di&longs;tantiæ corporis attracti: dico
quod vis tota qua huju&longs;modi Sphæra una attrahit aliam &longs;it reci
proce proportionalis quadrato di&longs;tantiæ centrorum.
Sunto Sphæræ quotcunque concentricæ &longs;imilares
&c. quarum interiores additæ exterioribus componant materiam
LXXV) trahent Sphæras alias quotcunque concentri
cas &longs;imilares &longs;ingulæ &longs;ingulas, viribus reci
proce proportionalibus quadrato di&longs;tantiæ
vel dividendo, &longs;umma virium illarum omnium, vel exce&longs;&longs;us ali
quarum &longs;upra alias, hoc e&longs;t, vis quas Sphæra tota ex concen
tricis quibu&longs;cunque vel concentricarum differentiis compo&longs;ita
trahit totam ex concentricis quibu&longs;cunque vel concentricarum dif
ferentiis compo&longs;itam Augeatur nu
merus Sphærarum concentricarum in infinitum &longs;ic, ut materiæ den
&longs;itas una cum vi attractiva, in progre&longs;&longs;u a circumferentia ad cen
trum, &longs;ecundum Legem quamcunque cre&longs;cat vel decre&longs;cat: &, ad
dita materia non attractiva, compleatur ubivis den&longs;itas deficiens, eo
ut Sphæræ acquirant formam quamvis optatam; & vis qua harum
una attrahet alteram erit etiamnum (per argumentum &longs;uperius) in
eadem illa di&longs;tantiæ quadratæ ratione inver&longs;a.
PRIMUS.
omnia &longs;imiles, &longs;e mutuo trahant; attractiones acceleratrices &longs;ingula
rum in &longs;ingulas erunt, in æqualibus quibu&longs;vis centrorum di&longs;tantiis,
ut Sphæræ attrahentes.
hentes applicatæ ad quadrata di&longs;tantiarum inter centra.
Sphæras erunt, in æqualibus centrorum di&longs;tantiis, ut Sphæræ attra
hentes & attractæ conjunctim, id e&longs;t, ut contenta &longs;ub Sphæris per
multiplicationem producta.
ad quadrata di&longs;tantiarum inter centra.
CORPORUM
virtute attractiva, mutuo exercita in Sphæram alteram. Nam viri
bus ambabus geminatur attractio, proportione &longs;ervata.
volvantur, &longs;ingulæ circa &longs;ingulas, &longs;intQ.E.D.&longs;tantiæ inter centra re
volventium & quie&longs;centium proportionales quie&longs;centium diame
tris; æqualia erunt Tempora periodica.
tiæ erunt proportionales diametris.
umbilicos Conicarum Sectionum demon&longs;trata &longs;unt, obtinent ubi
Sphæra attrahens, formæ & conditionis cuju&longs;vis jam de&longs;criptæ, lo
catur in umbilico.
ditionis cuju&longs;vis jam de&longs;criptæ.
tionales di&longs;tantiis punctorum a corporibus attractis: dico quod
vis compo&longs;ita, qua Sphæræ duæ &longs;e mutuo trahent, est ut di
&longs;tantia inter centra Sphærarum.
centrum ejus,
tractum,
centrum corpu&longs;culi tran&longs;iens,
ef
catur, huic axi perpendicularia &
hinc inde æqualiter di&longs;tantia a
centro Sphæræ;
nes planorum & axis, &
ctum quodvis in plano
cti
cita, e&longs;t ut di&longs;tantia 2.) &longs;ecundum li
neam
ctorum omnium in plano
culum
ductus in di&longs;tantiam
& di&longs;tantia illa
illam
tum in &longs;ummam di&longs;tantiarum
ductum in duplam centri & corpu&longs;culi di&longs;tantiam
duplum planum
qualium planorum Et &longs;i
mili argumento, vires omnium planorum in Sphæra tota, hinc in
de æqualiter a centro Sphæræ di&longs;tantium, &longs;unt ut &longs;umma planorum
ducta in di&longs;tantiam
tiam centri &longs;ui
PRIMUS.
dem argumento probabitur quod vis, qua Sphæra illa trahitur, erit:
ut di&longs;tantia
ris
e&longs;t ut di&longs;tantia corpu&longs;culi a centro Sphæræ primæ ducta in Sphæ
ram eandem, atque adeo eadem e&longs;t ac &longs;i prodiret tota de corpu&longs;
culo unico in centro Sphæræ; vis tota qua corpu&longs;cula omnia in
Sphæra &longs;ecunda trahuntur, hoc e&longs;t, qua Sphæra illa tota trahitur,
eadem erit ac &longs;i Sphæra illa traheretur vi prodeunte de corpu&longs;culo
unico in centro Sphæræ primæ, & propterea proportionalis e&longs;t di
&longs;tantiæ inter centra Sphærarum.
nem priorem &longs;ervabit.
quoniam vis plani
illo & di&longs;tantia
plano illo & di&longs;tantia
rentia contentorum, hoc e&longs;t, ut &longs;umma æqualium planorum ducta
in &longs;emi&longs;&longs;em differentiæ di&longs;tantiarum, id e&longs;t, ut &longs;umma illa ducta in Et &longs;imili argumento,
attractio planorum omnium
tractio Sphæræ totius, e&longs;t ut &longs;umma planorum omnium, &longs;eu Sphæra
tota, ducta in
nova, intra Sphæram priorem
quod attractio, &longs;ive &longs;implex Sphæræ unius in alteram, &longs;ive mutua
utriu&longs;Q.E.I. &longs;e invicem, erit ut di&longs;tantia centrorum
CORPORUM
di&longs;&longs;imilares & inæquabiles, in progre&longs;&longs;u vero per circuitum ad
datam omnem a centro di&longs;tantiam &longs;int undique &longs;imilares; &
vis attractiva puncti cuju&longs;que &longs;it ut di&longs;tantia corporis attracti:
dico quod vis tota qua huju&longs;modi Sphæræ duæ &longs;e mutuo trahunt
&longs;it proportionalis di&longs;tantiæ inter centra Sphærarum.
Demon&longs;tratur ex Propo&longs;itione præcedente, eodem modo quo
Propo&longs;itio LXXVI ex Propo&longs;itione LXXV demon&longs;trata fuit.
corporum circa centra Conicarum Sectionum demon&longs;trata &longs;unt,
valent ubi attractiones omnes fiunt vi Corporum Sphærieorum
conditionis jam de&longs;criptæ, &longs;untque corpora attracta Sphæræ con
ditionis eju&longs;dem.
Attractionum Ca&longs;us duos in&longs;igniores jam dedi expo&longs;itos; nimi
rum ubi Vires centripetæ decre&longs;cunt in duplicata di&longs;tantiarum ra
tione, vel cre&longs;cunt in di&longs;tantiarum ratione &longs;implici; efficientes
in utroque Ca&longs;u ut corpora gyrentur in Conicis Sectionibus, &
componentes corporum Sphærieorum Vires centripetas eadem Lege,
in rece&longs;&longs;u a centro, decre&longs;centes vel cre&longs;centes cum &longs;eip&longs;is: Quod
e&longs;t notatu dignum. Ca&longs;us cæteros, qui conclu&longs;iones minus ele
gantes exhibent, &longs;igillatim percurrere longum e&longs;&longs;et. Malim
cunctos methodo generali &longs;imul comprehendere ac determinare,
ut &longs;equitur.
culi duo
F, f;
fi di&longs;tantia arcuum
tio ultima lineæ evane&longs;centis
Nam &longs;i linea
& ab VIII, & Corol.
3.
Lem. VII) &longs;imilia, erit
æquo,
PRIMUS.
EF fe,
Sphæricum concavo convexum, ad cujus particulas &longs;ingulas æqua
les tendant æquales vires centripetæ: dico quod Vis, qua &longs;oli
dum illud trahit corpu&longs;culum &longs;itum in
ta ex ratione &longs;olidi
data in loco
Nam &longs;i primo con&longs;ideremus vim &longs;uperficiei Sphæricæ
convolutione arcus
erit &longs;uperficiei pars annularis, convolutione arcus
lineola
chimedesde
neas
hæc ip&longs;a &longs;uperficiei pars annularis; hoc e&longs;t, ut lineola
quod perinde e&longs;t, ut rectangulum &longs;ub dato Sphæræ radio
minor, in ratione
jam intelligatur linea
&longs;ingulæ nominentur
æquales annulos, quorum vires erunt ut &longs;umma omnium
hoc e&longs;t, ut 1/2
jam &longs;uperficies
ercita in corpu&longs;culum
particula aliqua data
CORPORUM
las æquales tendant æquales vires centripetæ, & ad Sphæræ
axem
punctis &longs;ingulis
(DE
&longs;tantiam
Vis tota, qua corpu&longs;culum
area comprehen&longs;a &longs;ub axe Sphæræ
Etenim &longs;tantibus quæ in Lemmate & Theoremate novi&longs;&longs;imo
innumeras æquales
laminas Sphæricas concavo-convexas
diculum
trahit corpu&longs;culum
di&longs;tantiam E&longs;t autem per Lem
ma novi&longs;&longs;imum,
(
ea vis laminæ
di&longs;tantiam
omnium vires in corpus
e&longs;t, Sphæræ vis tota ut area tota
PRIMUS.
eadem &longs;emper maneat in omnibus di&longs;tantiis, & fiat
(
ut area
corpu&longs;culi a &longs;e attracti, & fiat
corpu&longs;culum
&longs;tantiæ corpu&longs;culi a &longs;e attracti, & fiat
vis qua corpu&longs;culum a tota Sphæra attrahitur ut area
particulas tendens ponatur e&longs;&longs;e reciproce ut quantitas V, fiat au
tem
attrahitur ut area
CORPORUM
A puncto
axem
(per Prop. 12, Lib.
2. Elem.)
2
lorum
e&longs;t contento &longs;ub
2
e&longs;t
2
(per Prop. 6, Lib.
2. Elem.) æquatur rectangulo
tur itaque 2
(
præcedentis e&longs;t ut longitudo ordinatim applicatæ
&longs;e&longs;e in tres partes (2
ubi &longs;i pro V &longs;cribatur ratio inver&longs;a vis centripetæ, & pro
dium proportionale inter
ordinatim applicatæ linearum totidem curvarum, quarum areæ per
Methodos vulgatas innote&longs;cunt.
PRIMUS.
dens &longs;it reciproce ut di&longs;tantia; pro V &longs;cribe di&longs;tantiam
2
Pone
pars data 2
gulam 2
eandem longitudinem per motum continuum, ea lege ut inter mo
vendum cre&longs;cendo vel decre&longs;cendo æquetur &longs;emper longitudini
&longs;ubducta de area priore 2
Pars autem tertia (
liter in eandem longitudinem, de&longs;cribet
aream Hyperbolicam; quæ &longs;ubducta de
area
matis con&longs;tructio. Ad puncta
erige perpendicula
A&longs;ymptotis
&longs;cribatur Hyperbola
da
dens &longs;it reciproce ut cubus di&longs;tantiæ, vel (quod perinde e&longs;t) ut cubus
ille applicatus ad planum quodvis datum; &longs;cribe (
dein 2
-(ALBXASq/2PSXLDq),
ut
in longitudinem
am
ducatur &longs;umma &longs;ecundæ & tertiæ, &
manebit area quæ&longs;ita
de talis emergit Problematis con&longs;tru
ctio. Ad puncta
perpendicula
rum
ctum
batur Hyperbola
pendiculis
angulum 2
Hyperbolica
quæ&longs;itam
CORPORUM
tendens, decre&longs;cit in quadruplicata ratione di&longs;tantiæ a particulis;
&longs;cribe (
Cujus tres partes ductæ in longitudinem
idem,
& (
ctionem fiunt
ctis po&longs;terioribus de priore, evadunt (
corpu&longs;culum
reciproce ut
Eadem Methodo determinari pote&longs;t Attractio corpu&longs;culi &longs;iti in
tra Sphæram, &longs;ed expeditius per Theorema &longs;equens.
PRIMUS.
SP
ram in loco quovis
Sphæram in loco
di&longs;tantiarum a centro
centripetarum, in locis illis
Ut &longs;i vires centripetæ particularum Sphæræ &longs;int reciproce ut di
&longs;tantiæ corpu&longs;culi a &longs;e attracti; vis, qua corpu&longs;culum &longs;itum in
trahitur a Sphæra tota, erit ad vim qua trahitur in
compo&longs;ita ex &longs;ubduplicata ratione di&longs;tantiæ
& ratione &longs;ubduplicata vis centripetæ in loco
in centro oriundæ, ad vim centripetam in loco
tro particula oriundam, id e&longs;t, ratione &longs;ubduplicata di&longs;tantiarum Hæ duæ rationes &longs;ubduplicatæ
componunt rationem æqualitatis, & propterea attractiones in
a Sphæra tota factæ æquantur. Simili computo, &longs;i vires particu
larum Sphæræ &longs;unt reciproce in duplicata ratione di&longs;tantiarum, col
ligetur quod attractio in
ad Sphæræ &longs;emidiametrum
plicata ratione di&longs;tantiarum, attractiones in
fuit reciproce ut
&longs;imilis e&longs;t progre&longs;&longs;us in infinitum. Theorema vero &longs;ic demon
&longs;tratur.
CORPORUM
Stantibus jam ante con&longs;tructis, & exi&longs;tente corpore in loco
quovis
Ergo &longs;i agatur
tatis mutandis, evadet ut (
Sphæræ puncto quovis
pote&longs;tatum
(
rationem
tiis
de&longs;cribunt, hi&longs;que proportionales attractiones, &longs;unt in ratione com
po&longs;ita ex &longs;ubduplicatis illis rationibus.
Segmentum quodcunque attrahitur.
Sit
plano Superfi
cie Sphærica
&longs;tinguatur Segmentum in partes
autem &longs;uperficies illa non pure Mathematica, &longs;ed Phy&longs;ica, pro
funditatem habens quam minimam. Nominetur i&longs;ta profundi-
mon&longs;trata
Ponamus præterea vires attractivas par
ticularum Sphæræ e&longs;&longs;e reciproce ut
di&longs;tantiarum dignitas illa cujus Index
e&longs;t
corpus
portionale &longs;it perpendiculum
tum in O; & area curvilinea
quam ordinatim applicata
gitudinem
ducta de&longs;cribit, erit ut vis tota qua
Segmentum totum
PRIMUS.
menti cuju&longs;vis locatum, attrahitur ab eodem Segmento.
A Segmento
Prop.
LXXIX,
LXXX, LXXXI) in ejus axe Centro
lo
Segmentum in partes duas
tis prioris per Prop. LXXXI, & vis partis po&longs;terioris per Prop.
LXXXIII; & &longs;umma virium erit vis Segmenti totius
Explicatis attractionibus corporum Sphærieorum, jam pergere
liceret ad Leges attractionum aliorum quorundam ex particulis at
tractivis &longs;imiliter con&longs;tantium corporum; &longs;ed i&longs;ta particulatim
tractare minus ad in&longs;titutum &longs;pectat. Suffecerit Propo&longs;itiones
qua&longs;dam generaliores de viribus huju&longs;modi corporum, deque mo
tibus inde oriundis, ob earum in rebus Philo&longs;ophicis aliqualem
u&longs;um, &longs;ubjungere.
CORPORUM
fortior &longs;it, quam cum vel minimo intervallo &longs;eparantur ab in
vicem: vires particularum trahentis, in rece&longs;&longs;u corporis attrac
ti, decre&longs;cunt in ratione plu&longs;quam duplicata di&longs;tantiarum a
particulis.
Nam &longs;i vires decre&longs;cunt in ratione duplicata di&longs;tantiarum a par
ticulis; attractio ver&longs;us corpus Sphæricum, propterea quod (per
Prop. LXXIV) &longs;it reciproce ut quadratum di&longs;tantiæ attracti corpo
ris a centro Sphæræ, haud &longs;en&longs;ibiliter augebitur ex contactu; atque
adhuc minus augebitur ex contactu, &longs;i attractio in rece&longs;&longs;u corporis
attracti decre&longs;cat in ratione minore. Patet igitur Propo&longs;itio de
Sphæris attractivis. Et par e&longs;t ratio Orbium Sphærieorum conca
vorum corpora externa trahentium. Et multo magis res con&longs;tat in
Orbibus corpora interius con&longs;tituta trahentibus, cum attractiones
pa&longs;&longs;im per Orbium cavitates ab attractionibus contrariis (per Prop.
LXX) tollantur, ideoque vel in ip&longs;o contactu nullæ &longs;unt. Quod
&longs;i Sphæris hi&longs;ce Orbibu&longs;que Sphæricis partes quælibet a loco con
tactus remotæ auferantur, & partes novæ ubivis addantur: mu
tari po&longs;&longs;unt figuræ horum corporum attractivorum pro lubitu, nec
tamen partes additæ vel &longs;ubductæ, cum &longs;int a loco contactus re
motæ, augebunt notabiliter attractionis exce&longs;&longs;um qui ex contactu
oritur. Con&longs;tat igitur Propo&longs;itio de corporibus Figurarum om
nium.
PRIMUS.
in rece&longs;&longs;u corporis attracti decre&longs;cunt in triplicata vel plu&longs;quam
triplicata ratione di&longs;tantiarum a particulis: attractio longe for
tior erit in contactu, quam cum attrahens & attractum inter
vallo vel minimo &longs;eparantur ab invicem.
Nam attractionem in acce&longs;&longs;u attracti corpu&longs;culi ad huju&longs;modi
Sphæram trahentem augeri in infinitum, con&longs;tat per &longs;olutionem Pro
blematis XLI, in Exemplo &longs;ecundo ac tertio exhibitam. Idem, per
Exempla illa & Theorema XLI inter &longs;e collata, facile colligitur
de attractionibus corporum ver&longs;us Orbes concavo-convexos, &longs;ive
corpora attracta collocentur extra Orbes, &longs;ive intra in eorum cavi
tatibus. Sed & addendo vel auferendo his Sphæris & Orbibus ubi
vis extra locum contactus materiam quamlibet attractivam, eo ut
corpora attractiva induant figuram quamvis a&longs;&longs;ignatam, con&longs;tabit
Propo&longs;itio de corporibus univer&longs;is.
ctiva con&longs;tantia, &longs;eor&longs;im attrahant corpu&longs;cula &longs;ibi ip&longs;is proporti
onalia & ad &longs;e &longs;imiliter po&longs;ita: attractiones acceleratrices cor
pu&longs;culorum in corpora tota erunt ut attractiones acceleratrices
corpu&longs;culorum in eorum particulas totis proportionales & in to
tis &longs;imiliter po&longs;itas.
Nam &longs;i corpora di&longs;tinguantur in particulas, quæ &longs;int totis pro
portionales & in totis &longs;imiliter &longs;itæ; erit, ut attractio in particulam
quamlibet unius corporis ad attractionem in particulam corre&longs;pon
dentem in corpore altero, ita attractiones in particulas &longs;ingulas
primi corporis ad attractiones in alterius particulas &longs;ingulas corre&longs;
pondentes; & componendo, ita attractio in totum primum corpus
ad attractionem in totum &longs;ecundum.
tias corpu&longs;culorum attractorum, decre&longs;cant in ratione dignitatis
erunt ut corpora directe & di&longs;tantiarum dignitates illæ inver&longs;e. Ut
&longs;i vires particularum decre&longs;cant in ratione duplicata di&longs;tantiarum
a corpu&longs;culis attractis, corpora autem &longs;int ut
eoque tum corporum latera cubica, tum corpu&longs;culorum attracto
rum di&longs;tantiæ a corporibus, ut
ces in corpora erunt ut (
tera illa cubica
tione triplicata di&longs;tantiarum a corpu&longs;culis attractis; attractiones
acceleratrices in corpora tota erunt ut (
les. Si vires decre&longs;cant in ratione quadruplicata; attractiones in
corpora erunt ut (
ca
CORPORUM
hunt corpu&longs;cula ad &longs;e &longs;imiliter po&longs;ita, colligi pote&longs;t ratio decre
menti virium particularum attractivarum in rece&longs;&longs;u corpu&longs;culi at
tracti; &longs;i modo decrementum illud &longs;it directe vel inver&longs;e in ratione
aliqua di&longs;tantiarum.
&longs;int ut di&longs;tantiæ loeorum a particulis: vis corporis totius ten
det ad ip&longs;ius centrum gravitatis; & eadem erit cum vi Globi
ex materia con&longs;imili & æquali con&longs;tantis & centrum habentis
in ejus centro gravitatis.
Corporis
B
quantur inter &longs;e, &longs;int ut di&longs;tan
tiæ
tuantur inæquales, &longs;int ut hæ par
ticulæ in di&longs;tantias &longs;uas
re&longs;pective ductæ. Et exponan
tur hæ vires per contenta illa
& &longs;ecetur ea in
& vis
&
adeoque cum dirigantur in partes contrarias, &longs;e mutuo de&longs;truunt.
Re&longs;tant vires
trum
&longs;i particulæ attractivæ
vitatis centro
PRIMUS.
Eodem argumento, &longs;i adjungatur particula tertia
natur hujus vis cum vi ―
inde oriunda tendet ad commune centrum gravitatis Globi illius
& particulæ
ticularum
&longs;terent in centro illo communi, Globum majorem ibi componentes.
Et &longs;ic pergitur in infinitum. Eadem e&longs;t igitur vis tota particula
rum omnium corporis cuju&longs;cunque
vato gravitatis centro, figuram Globi indueret.
attrahens
attrahens vel quie&longs;cat, vel progrediatur uniformiter in directum;
corpus attractum movebitur in Ellip&longs;i centrum habente in attra
hentis centro gravitatis.
res &longs;unt ut di&longs;tantiæ loeorum a &longs;ingulis: vis ex omnium viri
bus compo&longs;ita, qua corpu&longs;culum quodcunque trahitur, tendet ad
trahentium commune centrum gravitatis, & eadem erit ac &longs;i
trahentia illa, &longs;ervato gravitatis centro communi, coirent & in
Globum formarentur.
Demon&longs;tratur eodem modo, atque Propo&longs;itio &longs;uperior.
hentia, &longs;ervato communi gravitatis centro, coirent & in Globum
formarentur. Ideoque &longs;i corporum trahentium commune gravita
tis centrum vel quie&longs;cit, vel progreditur uniformiter in linea recta:
corpus attractum movebitur in Ellip&longs;i, centrum habente in com
muni illo trahentium centro gravitatis.
CORPORUM
tripetæ, decre&longs;centes in quacunQ.E.D.&longs;tantiarum ratione: inve
nire vim qua corpu&longs;culum attrahitur ubivis po&longs;itum in recta
quæ plano Circuli ad centrum ejus perpendiculariter in&longs;i&longs;tit.
Centro
pendicularis e&longs;t, de&longs;cribi intelligatur Circulus; & invenienda &longs;it vis
qua corpu&longs;culum quodvis A Circuli puncto
quovis
cta
qualis, & erigatur normalis
quæ &longs;it ut vis qua punctum
hit corpu&longs;culum
curva linea quam punctum
petuo tangit. Occurrat eadem Cir
culi plano in
pendiculum
occurrens in
culi
Etenim in
& in Et quoniam vis,
qua annuli punctum quodvis
ut
(
annulus & (
gulum &longs;ub radio
portionales
&longs;eu
&longs;umma virium, quibus annuli omnes in Circulo, qui centro
PRIMUS.
&longs;tantiarum ratione, hoc e&longs;t, &longs;i &longs;it
rea
lum ut (1-
reciproce ut di&longs;tantiarum dignitas quælibet D
ut (1/D
ctio corpu&longs;culi
rus
infinitum erit reciproce ut
ter (
puncta &longs;ingula tendunt vires æquales centripetæ in quacunque
di&longs;tantiarum ratione decre&longs;centes.
In Solidum
hatur corpu&longs;culum
ejus axe
bet
pendiculari &longs;ecetur hoc Solidum,
& in ejus diametro
no aliquo
tran&longs;eunte, capiatur (per Prop.
XC) longitudo
pu&longs;culum
attrahitur proportionalis. Tangat autem punctum
am
CORPORUM
Cylindrus &longs;it, parallelogrammo
luto de&longs;criptus, & vires centri
petæ in &longs;ingula ejus puncta ten
dentes &longs;int reciproce ut quadra
ta di&longs;tantiarum a punctis: erit
attractio corpu&longs;culi
Cylindrum ut
Nam ordinatim applicata
(per Corol. 1. Prop.
XC) erit ut 1-(
gitudinem
in longitudinem
ex curvæ
eadem ducta in longitudinem
ductaQ.E.I. ip&longs;arum
differentiam 1 in ―(
ratur contentum po&longs;tremum 1 in ―(
æqualis 1 in ―(
lis, e&longs;t ut
vis innote&longs;cit qua Sphæ
rois
corpus quodvis
rius in axe &longs;uo
tum. Sit
ctio Conica cujus ordi
natim applicata
quetur &longs;emper longitu
dini
ad punctum illud
quo applicata i&longs;ta Sphæroidem &longs;ecat. A Sphæroidis verticibus
ad ejus axem
æqualia re&longs;pective, & propterea Sectioni Conicæ occurrentia in
&
Sit autem Sphæroidis centrum
ad (
vires &longs;egmentorum Sphæroidis.
PRIMUS.
vis eju&longs;dem diametro, collocetur; attractio erit ut ip&longs;ius di&longs;tantia a
centro. Id quod facilius colligetur hoc argumento.
Sit
Sphærois attrahens, Per
corpus illud
quævis
&
teriorum, exteriori &longs;imilium & concentricarum, quarum prior tran&longs;
eat per corpus
&longs;ecet ea&longs;dem rectas in
omnes axem communem, & erunt rect
arum partes hinc inde interceptæ
&
&
quod rectæ
tur in eodem puncto, ut & rectæ
PC
EPG
gulis verticalibus
nite parvis de&longs;criptos, & lineas etiam
&longs;uperficiebus ab&longs;ci&longs;&longs;æ
corpu&longs;culo
Et pari ratione, &longs;i &longs;uperficiebus Sphæroidum innumerarum &longs;imilium
concentricarum & axem communem habentium dividantur &longs;patia
hent corpus Æquales igitur &longs;unt vires
Coni
mutuo de&longs;truunt. Et par e&longs;t ratio virium materiæ omnis extra Sphæ
roidem intimam
roide intima 3. Prop.
LXXII) at
tractio ejus e&longs;t ad vim, qua corpus
CORPORUM
tripetarum in ejus puncta &longs;ingula tendentium.
E Corpore dato formanda e&longs;t Sphæra vel Cylindrus aliave figu
ra regularis, cujus lex attractionis, cuivis decrementi rationi con
gruens (per Prop. LXXX, LXXXI, & XCI) inveniri pote&longs;t.
Dein fa
ctis experimentis invenienda e&longs;t vis attractionis in diver&longs;is di&longs;tan
tiis, & lex attractionis in totum inde patefacta dabit rationem de
crementi virium partium &longs;ingularum, quam invenire oportuit.
tum, con&longs;tet ex particulis æqualibus æqualiter attractivis, qua
rum vires in rece&longs;&longs;u a Solido decre&longs;cunt in ratione pote&longs;tatis cu
ju&longs;vis di&longs;tantiarum plu&longs;quam quadraticæ, & vi Solidi totius cor
pu&longs;culum ad utramvis plani partem con&longs;titutum trahatur: dico
quod Solidi vis illa attractiva, in rece&longs;&longs;u ab ejus &longs;uperficie pla
na, decre&longs;cet in ratione pote&longs;tatis, cujus latus est di&longs;tantia cor
pu&longs;culi a plano, & Index ternario minor quam Index pote&longs;ta
tis di&longs;tantiarum.
quo Solidum terminatur.
Jaceat Solidum autem ex
parte plani hujus ver&longs;us ip&longs;i
parallela re&longs;olvatur. Et
primo collocetur corpus at
tractum
Agatur autem
nis illis innumeris perpendicularis, & decre&longs;cant vires attractivæ
punctorum Solidi in ratione pote&longs;tatis di&longs;tantiarum, cujus index &longs;it
numerus Ergo (per Corol.
3. Prop.
XC)
ciproce proportionalis, & erit vis illa ut
gulis capiantur longitudines
ip&longs;is reciproce proportionales; & vi
res planorum eorundem erunt ut longitudines captæ, adeoque
&longs;umma virium ut &longs;umma longitudinum, hoc e&longs;t, vis Solidi totius ut
area Sed area illa (per
notas quadraturarum methodos) e&longs;t reciproce ut
terea vis Solidi totius e&longs;t reciproce ut
PRIMUS.
tra Solidum, & capiatur di&longs;tantia
lidi pars
pu&longs;culum
po&longs;itorum punctorum actionibus &longs;e mutuo per æqualitatem tollenti
bus. Proinde corpu&longs;culum
hitur. Hæc autem vis (per Ca&longs;um primum) e&longs;t reciproce ut
hoc e&longs;t (ob æquales
rallelis
ctiva, &longs;ubducendo de vi attractiva Solidi totius infiniti
vim attractivam partis ulterioris
productæ.
jus collata cum attractione partis citerioris nullius pene e&longs;t momen
ti, rejiciatur: attractio partis illius citerioris augendo di&longs;tantiam de
cre&longs;cet quam proxime in ratione pote&longs;tatis
num trahat corpu&longs;culum e regione medii illius plani, & di&longs;tantia
inter corpu&longs;culum & planum collata cum dimen&longs;ionibus corpo
ris attrahentis perexigua &longs;it, con&longs;tet autem corpus attrahens ex
particulis homogeneis, quarum vires attractivæ decre&longs;cunt in
ratione pote&longs;tatis cuju&longs;vis plu&longs;quam quadruplicatæ di&longs;tantiarum;
vis attractiva corporis totius decre&longs;cet quamproxime in ratione
pote&longs;tatis, cujus latus &longs;it di&longs;tantia illa perexigua, & Index terna
rio minor quam Index pote&longs;tatis prioris. De corpore ex particulis
con&longs;tante, quarum vires attractivæ decre&longs;cunt in ratione pote&longs;tatis
triplicatæ di&longs;tantiarum, a&longs;&longs;ertio non valet; propterea quod, in hoc
ca&longs;u, attractio partis illius ulterioris corporis infiniti in Corollario
&longs;ecundo, &longs;emper e&longs;t infinite major quam attractio partis citerioris.
CORPORUM
Si corpus aliquod perpendiculariter ver&longs;us planum datum tra
hatur, & ex data lege attractionis quæratur motus corporis: Sol
vetur Problema quærendo (per Prop. XXXIX) motum corporis recta
de&longs;cendentis ad hoc planum, & (per Legum Corol. 2.) componen
do motum i&longs;tum cum uniformi motu, &longs;ecundum lineas eidem plano
parallelas facto. Et contra, &longs;i quæratur Lex attractionis in planum
&longs;ecundum lineas perpendiculares factæ, ea conditione ut corpus at
tractum in data quacunque curva linea moveatur, &longs;olvetur Proble
ma operando ad exemplum Problematis tertii.
Operationes autem contrahi &longs;olent re&longs;olvendo ordinatim appli
catas in Series convergentes. Ut &longs;i ad ba&longs;em A in angulo quovis
dato ordinatim applicetur longitudo B, quæ &longs;it ut ba&longs;is dignitas
quælibet A
ordinatim applicatæ, vel in ba&longs;em attractum vel a ba&longs;i fugatum,
moveri po&longs;&longs;it in curva linea quam ordinatim applicata termi
no &longs;uo &longs;uperiore &longs;emper attingit: Suppono ba&longs;em augeri parte
quam minima O, & ordinatim applicatam ―(A+O)
Seriem infinitam Aat
que hujus termino in quo O duarum e&longs;t dimen&longs;ionum, id e&longs;t, ter
mino (E&longs;t
igitur vis quæ&longs;ita ut (
(Ut &longs;i ordinatim applicata Parabolam attingat,
exi&longs;tente
tur. Data igitur vi corpus movebitur in Parabola, quemad
modum Quod &longs;i ordinatim applicata
Hyperbolam attingat, exi&longs;tente
2A
corpus movebitur in Hyperbola. Sed mi&longs;&longs;is huju&longs;modi Propo&longs;iti
onibus, pergo ad alias qua&longs;dam de Motu, quas nondum attigi.
PRIMUS.
magni alicujus corporis partes tendentibus agitantur.
di&longs;tinguantur ab invicem, & corpus in tran&longs;itu per hoc &longs;patium
attrahatur vel impellatur perpendiculariter ver&longs;us Medium alter
utrum, neque ulla alia vi agitetur vel impediatur: Sit autem
attractio, in æqualibus ab utroque plano di&longs;tantiis ad eandem
ip&longs;ius partem captis, ubique eadem: dico quod &longs;inus incidentiæ
in planum alterutrum erit ad &longs;inum emergentiæ ex plano altero
in ratione data.
plana duo parallela. Inci
dat corpus in planum pri
us
um intermedium tran&longs;itu
attrahatur vel impellatur
ver&longs;us Medium inciden
tiæ, eaque actione de&longs;cri
bat lineam curvam
emergat &longs;ecundum line
am
gentiæ
pendiculum
rens tum lineæ inciden
tiæ
tum plano incidentiæ
occurrat
&longs;i attractio vel impul&longs;us ponatur uniformis, erit (ex demon&longs;tratis
gulum &longs;ub dato latere recto & linea
&longs;ed & linea
perpendiculum
quales erunt
& additis æqualibus
OI,
lum
lum &longs;ub latere recto &
in data ratione. Sed rect
angulum
e&longs;t rectangulo
e&longs;t, differentiæ quadrato
rum
rationem habet ad &longs;ui ip&longs;ius quartam partem
ratio
ratio dimidiata
angulorum &longs;unt proportionales lateribus oppo&longs;itis. Ergo datur
ratio &longs;inus anguli incidentiæ
tiæ
CORPORUM
lis planis terminata, & agitetur vi quæ &longs;it in
emergentiæ ex plano &longs;ecundo
qui e&longs;t &longs;inus incidentiæ in planum &longs;ecundum
emergentiæ ex plano tertio
&longs;inum emergentiæ ex plano quarto
infinitum: & ex æquo, &longs;inus incidentiæ in planum primum ad &longs;i
num emergentiæ ex plano ultimo in data ratione. Minuantur jam
planorum intervalla & augeatur numerus in infinitum, eo ut attra
ctionis vel impul&longs;us actio, &longs;ecundum legem quamcunque a&longs;&longs;ignatam,
continua reddatur; & ratio &longs;inus incidentiæ in planum primum ad
&longs;inum emergentiæ ex plano ultimo, &longs;emper data exi&longs;tens, etiam
num dabitur.
PRIMUS.
ad ejus velocitatem po&longs;t emergentiam, ut &longs;inus emergentiæ ad
&longs;inum incidentiæ.
Capiantur
occurrentia lineis incidentiæ & emergentiæ
In
normaliter 2) di&longs;tinguatur motus cor
poris in duos, unum planis perpendicularem, al
terum ii&longs;dem parallelum. Vis attractionis vel impul&longs;us, agendo &longs;e
cundum lineas perpendiculares, nil mutat motum &longs;ecundum paralle
las, & propterea corpus hoc motu conficiet æqualibus temporibus
æqualia illa &longs;ecundum parallelas intervalla, quæ &longs;unt inter lineam
æqualibus temporibus de&longs;cribet lineas
citas ante incidentiam e&longs;t ad velocitatem po&longs;t emergentiam, ut
(re&longs;pectu radii
dentiæ.
CORPORUM
po&longs;tea: dico quod corpus, inclinando lineam incidentiæ, refle
ctetur tandem, & angulus reflexionis fiet æqualis angulo inci
dentiæ.
Nam concipe corpus inter parallela plana
de
&longs;cribere arcus Parabolicos, ut &longs;upra; &longs;intque arcus illi
QR,Et &longs;it ea lineæ incidentiæ
mum
in ea ratione quam habet idem &longs;inus incidentiæ ad &longs;inum emer
gentiæ ex plano
tiæ jam factum æqualem radio, angulus emergentiæ erit rectus, ad
eoque linea emergentiæ coincidet cum plano
pus ad hoc planum in puncto
coincidit cum eodem
plano, per&longs;picuum e&longs;t
quod corpus non po
te&longs;t ultra pergere ver
&longs;us planum
nec pote&longs;t idem perge
re in linea emergentiæ
perpetuo attrahitur vel impellitur ver&longs;us Medium incidentiæ. Re
vertetur itaQ.E.I.ter plana
pergendo in arcubus parabolicis arcubus prioribus
angulis in ac prius in
emergetque tandem ea
dem obliquitate in intervalla in infinitum minui & nume
rum augeri, eo ut actio attractionis vel impul&longs;us &longs;ecundum legem
quamcunque a&longs;&longs;ignatam continua reddatur; & angulus emergen
tiæ &longs;emper angulo incidentiæ æqualis exi&longs;tens, eidem etiamnum
manebit æqualis.
PRIMUS.
Harum attractionum haud multum di&longs;&longs;imiles &longs;unt Lucis reflexi
ones & refractiones, factæ &longs;ecundum datam Secantium rationem, ut
invenit
nem, ut expo&longs;uit
& &longs;patio qua&longs;i &longs;eptem vel octo minutorum primorum a Sole ad
Terram venire, jam con&longs;tat per Phænomena Satellitum
&longs;ervationibus diver&longs;orum A&longs;tronomorum confirmata. Radii autem
in aere exi&longs;tentes (uti dudum
nebro&longs;um cubiculum admi&longs;&longs;a, invenit, & ip&longs;e quoque expertus
&longs;um) in tran&longs;itu &longs;uo prope corporum vel opaeorum vel per&longs;picuo
rum angulos (quales &longs;unt nummorum ex auro, argento & ære cu
&longs;orum termini rectanguli circulares, & cultrorum, lapidum aut fra
ctorum vitrorum acies) incurvantur circum corpora, qua&longs;i attracti
in eadem; & ex his radiis, qui in tran&longs;itu illo propius accedunt
ad corpora incurvantur magis, qua
&longs;i magis attracti, ut ip&longs;e etiam dili
genter ob&longs;ervavi. In figura de&longs;ig
nat
dlsld,
nun, mtm, lsl
incurvati; idque magis vel mi
nus pro di&longs;tantia eorum a cultro.
Cum autem talis incurvatio radio
rum fiat in aere extra cultrum, de
bebunt etiam radii, qui incidunt in cultrum, prius incurvari in aere
quam cultrum attingunt. Et par e&longs;t ratio incidentium in vitrum.
Fit igitur refractio, non in puncto incidentiæ, &longs;ed paulatim per
continuam incurvationem radiorum, factam partim in aere ante
quam attingunt vitrum, partim (ni fallor) in vitro, po&longs;tquam illud
ingre&longs;&longs;i &longs;unt: uti in radiis
Igitur ob analogiam quæ e&longs;t inter propagationem radiorum lucis
& progre&longs;&longs;um corporum, vi&longs;um e&longs;t Propo&longs;itiones &longs;equentes in u&longs;us
Opticos &longs;ubjungere; interea de natura radiorum (utrum &longs;int cor
pora necne) nihil omnino di&longs;putans, &longs;ed Trajectorias corporum
Trajectoriis radiorum per&longs;imiles &longs;olummodo determinans.
CORPORUM
mergentiæ in data ratione, quodQ.E.I.curvatio viæ corporum
juxta &longs;uperficiem illam fiat in &longs;patio brevi&longs;&longs;imo, quod ut pun
ctum con&longs;iderari po&longs;&longs;it; determinare &longs;uperficiem quæ corpu&longs;cula
omnia de loco dato &longs;ucce&longs;&longs;ive manantia convergere faciat ad
alium locum datum.
Sit
vergere debent;
de&longs;cribat &longs;uperficiem quæ&longs;itam;
vis; &
Accedat punctum
getur, ad lineam
dem quæ &longs;inus incidentiæ ad &longs;inum emergentiæ. Datur ergo ratio
incrementi lineæ
&longs;i in axe
tran&longs;ire debet, & capiatur ip&longs;ius
tervallis
ubivis tangendo determinabit.
finitum, nunc migret ad alteras partes puncti
guræ illæ omnes quas
ctiones expo&longs;uit. Quarum inventionem cum
fecerit & &longs;tudio&longs;e celaverit, vi&longs;um fuit hac propo&longs;itione expo
nere.
quamvis rectam
& a puncto
telligantur Lineæ curvæ
&longs;emper perpendiculares:
erunt incrementa linea
rum
eo lineæ ip&longs;æ
incrementis i&longs;tis genitæ,
ut &longs;inus incidentiæ & e
mergentiæ ad invicem:
& contra.
PRIMUS.
attractiva
loco dato
cundam attractivam
Juncta
puncto Et po&longs;ito &longs;inu incidentiæ in &longs;uper
ficiem primam ad &longs;inum emergentiæ ex eadem, & &longs;inu emergentiæ
e &longs;uperficie &longs;ecunda ad &longs;inum incidentiæ in eandem, ut quantitas
aliqua data M ad aliam datam N; produc tum
ad
etiam Junge
centro
ductæ in
get Lineam
ciem quæ&longs;itam.
Nam concipe Lineas
neas 2. Prop.
XCVII)
& compo&longs;ite ut
&
CE+BG-FR
proportionales
e&longs;t etiam
divi&longs;im
M ad N, & propterea per
Corol. 2. Prop.
XCVII,
&longs;uperficies
pus, in ip&longs;am &longs;ecundum lineam
ad locum
CORPORUM
Eadem methodo pergere liceret ad &longs;uperficies tres vel plures.
Ad u&longs;us autem Opticos maxime accommodatæ &longs;unt figuræ Sphæ
ricæ. Si Per&longs;picillorum vitra Objectiva ex vitris duobus Sphæri
ce figuratis & Aquam inter &longs;e claudentibus conflentur; fieri pote&longs;t
ut a refractionibus Aquæ errores refractionum, quæ fiunt in vitro
rum &longs;uperficiebus extremis, &longs;atis accurate corrigantur. Talia au
tem vitra Objectiva vitris Ellipticis & Hyperbolicis præferenda
&longs;unt, non &longs;olum quod facilius & accuratius formari po&longs;&longs;int, &longs;ed
etiam quod Penicillos radiorum extra axem vitri &longs;itos accurativs
refringant. Verum tamen diver&longs;a diver&longs;orum radiorum Refrangi
bilitas impedimento e&longs;t, quo minus Optica per Figuras vel Sphæ
ricas vel alias qua&longs;cunque perfici po&longs;&longs;it. Ni&longs;i corrigi po&longs;&longs;int er
rores illinc oriundi, labor omnis in cæteris corrigendis imperite
collocabitur.
SECUNDUS.
MOTU CORPORUM
LIBER SECUNDUS.
Velocitatis.
ami&longs;&longs;us e&longs;t ut &longs;patium movendo confectum.
NAm cum motus &longs;ingulis temporis particulis æqualibus ami&longs;&longs;us
&longs;it ut velocitas, hoc e&longs;t, ut itineris confecti particula: erit,
componendo, motus toto tempore ami&longs;&longs;us ut iter totum.
ris &longs;ola vi in&longs;ita moveatur; ac detur tum motus totus &longs;ub initio, tum
etiam motus reliquus po&longs;t &longs;patium aliquod confectum: dabitur &longs;pa
tium totum quod corpus infinito tempore de&longs;cribere pote&longs;t. Erit
enim &longs;patium illud ad &longs;patium jam de&longs;criptum, ut motus totus &longs;ub
initio ad motus illius partem ami&longs;&longs;am.
tionales.
Sit A ad A-B ut B ad B-C & C ad C-D, &c.
& dividendo
fiet A ad B ut B ad C & C ad D, &c.
CORPORUM
per Medium &longs;imilare moveatur, &longs;umantur autem tempora æqua
lia: velocitates in principiis &longs;ingulorum temporum &longs;unt in pro
gre&longs;&longs;ione Geometrica, & &longs;patia &longs;ingulis temporibus de&longs;cripta
&longs;unt ut velocitates.
cularum initiis agat vis re&longs;i&longs;tentiæ impul&longs;o unico, quæ &longs;it ut velo
citas: erit decrementum velocitatis &longs;ingulis temporis particulis ut
eadem velocitas. Sunt ergo velocitates differentiis &longs;uis proportio
nales, & propterea (per Lem. I. Lib.
II.) continue proportionales.
Proinde &longs;i ex æquali particularum numero componantur tempora
quælibet æqualia, erunt velocitates ip&longs;is temporum initiis, ut ter
mini in progre&longs;&longs;ione continua, qui per &longs;altum capiuntur, omi&longs;&longs;o
pa&longs;&longs;im æquali terminorum intermediorum numero. Componuntur
autem horum terminorum rationes ex æqualibus rationibus termi
norum intermediorum æqualiter repetitis, & propterea &longs;unt æqua
les. Igitur velocitates, his terminis proportionales, &longs;unt in pro
gre&longs;&longs;ione Geometrica. Minuantur jam æquales illæ temporum par
ticulæ, & augeatur earum numerus in infinitum, eo ut re&longs;i&longs;tentiæ
impul&longs;us reddatur continuus; & velocitates in principiis æqualium
temporum, &longs;emper continue proportionales, erunt in hoc etiam
ca&longs;u continue proportionales.
&longs;ingulis temporibus ami&longs;&longs;æ, &longs;unt ut totæ: Spatia autem &longs;ingulis
temporibus de&longs;cripta &longs;unt ut velocitatum partes ami&longs;&longs;æ, (per Prop.
I. Lib II.) & propterea etiam ut totæ.
Hyperbola
diculares, & exponatur tum corporis velocitas tum re&longs;i&longs;tentia Me
dii, ip&longs;o motus initio, per lineam quam
vis datam
quo per lineam indefinitam
pote&longs;t tempus per aream
tium eo tempore de&longs;criptum per lineam
&longs;cent in eadem ratione.
SECUNDUS.
re&longs;i&longs;titur in ratione velocitatis, quodque ab uniformi gravitate
urgetur, definire motum.
Corpore a&longs;cendente, ex
ponatur gravitas per datum
quodvis rectangulum
re&longs;i&longs;tentia Medii initio a&longs;
cen&longs;us per rectangulum
&longs;umptum ad contrarias par
tes. A&longs;ymptotis rectangulis
&longs;cribatur Hyperbola &longs;ecans per
pendicula
corpus a&longs;cendendo, tempore
pore
&longs;patium de&longs;cen&longs;us
de&longs;cen&longs;us 2
proportionales) in horum temporum periodis erunt
ABed,
velocitas, quam corpus de&longs;cendendo pote&longs;t acquirere, erit
Re&longs;olvatur enim rectan
gulum
innumera
&c. quæ &longs;int ut incrementa
velocitatum æqualibus tot
idem temporibus facta; & e
runt nihil,
&c. ut velocitates totæ, at
que adeo (per Hypothe&longs;in)
ut re&longs;i&longs;tentiæ Medii princi
pio &longs;ingulorum temporum
æqualium. Fiat
vitatis ad re&longs;i&longs;tentiam in principio temporis &longs;ecundi, deque vi gravi-
NnHC,ut vires ab&longs;olutæ quibus corpus in principio &longs;ingu
lorum temporum urgetur, atque adeo (per motus Legem 11) ut
incrementa velocitatum, id e&longs;t, ut rectangula
& propterea (per Lem. I. Lib.
II) in progre&longs;&longs;ione Geometrica.
Qua
re &longs;i rectæ productæ occurrant Hyperbolæ
in erunt areæ
æquales, adeoque tum temporibus tum viribus gravitatis &longs;emper
æqualibus analogæ. E&longs;t autem area
3. Lem.
VII,
& Lem. VIII, Lib.
I) ad aream
hoc e&longs;t, ut vis gravitatis ad re&longs;i&longs;tentiam in medio temporis primi.
Et &longs;imili argumento areæ &longs;unt ad areas
smnt,ut vires gravi
tatis ad re&longs;i&longs;tentias in me
dio temporis &longs;ecundi, ter
tii, quarti, &c. Proinde cum
areæ æquales
rLMs, sMNt,&longs;int vi
ribus gravitatis analogæ, e
runt areæ
smnt,re&longs;i&longs;tentiis in mediis &longs;ingulorum temporum, hoc e&longs;t (per
Hypothe&longs;in) velocitatibus, atque adeo de&longs;criptis &longs;patiis analogæ.
Sumantur analogarum &longs;ummæ, & erunt areæ
&c. &longs;patiis totis de&longs;criptis analogæ; necnon areæ
ABsM, ABtN,temporibus.
Corpus igitur inter de&longs;cenden
dum, tempore quovis Q.E.D.
expo&longs;iti in a&longs;cen&longs;u.
CORPORUM
acquirere, e&longs;t ad velocitatem dato quovis tempore acqui&longs;itam, ut
vis data gravitatis qua perpetuo urgetur, ad vim re&longs;i&longs;tentiæ qua in
fine temporis illius impeditur.
velocitatis illius maximæ ac velocitatis in a&longs;cen&longs;u (atque etiam earun
dem differentia in de&longs;cen&longs;u) decre&longs;cit in progre&longs;&longs;ione Geometrica.
rum differentiis de&longs;cribuntur, decre&longs;cunt in eadem progre&longs;&longs;ion
Geometrica.
de&longs;cen&longs;us, & alterum ut velocitas, quæ etiam ip&longs;o de&longs;cen&longs;us initio
æquantur inter &longs;e.
SECUNDUS.
ac tendat perpendiculariter ad planum Horizontis; definire mo
tum Projectilis in eodem, re&longs;i&longs;tentiam velocitati proportionalem
patientis.
Eloco quovis
jectile &longs;ecundum lineam quam
vis rectam
dinem
velocitas &longs;ub initio motus. A
puncto
lem
culum
ut &longs;it
Medii, ex motu in altitudinem
&longs;ub initio orta, ad vim gravi
tatis; vel (quod perinde e&longs;t) ut
&longs;it rectangulum &longs;ub
ad rectangulum &longs;ub
ut re&longs;i&longs;tentia tota &longs;ub initio mo
tus ad vim gravitatis. A&longs;ymptotis
la quævis
dicula
compleatur parallelogrammum
ratione ad
ctum quodvis
diculo
in
in
perveniet ad punctum
punctum
nem
&longs;ymptoton
væ Tangens
CORPORUN
E&longs;t enim N ad
æqualis (
æqualis (
am
Corol. 2.) di&longs;tinguatur motus
corporis in duos, unum a&longs;cen
&longs;us, alterum ad latus. Et cum
re&longs;i&longs;tentia &longs;it ut motus, di&longs;tin
guetur etiam hæc in partes duas
partibus motus proportionales
& contrarias: ideoque longitu
do, a motu ad latus de&longs;cripta, e
rit (per Prop. 11. hujus) ut linea
111. hujus) ut area
-RDGT,
Ip&longs;o autem motus initio area
(&longs;eu (
tunc e&longs;t ad
&longs;eu
ad
in altitudinem ad motum in
longitudinem &longs;ub initio. Cum
igitur
do, ac
tudo, atque
initio ut altitudo ad longitudinem: nece&longs;&longs;e e&longs;t ut
tur in linea
compleatur parallelogrammum
in
lis (
SECUNDUS.
innumeræ Z
progre&longs;&longs;ione Arithmetica. Et hinc Curva
garithmorum facile delineatur.
tere recto quod &longs;it ad 2
ad vim gravitatis, Parabola con&longs;truatur: velocitas quacum corpus
exire debet de loco
formi re&longs;i&longs;tente de&longs;cribat Curvam
ire debet de eodem loco
in &longs;patio non re&longs;i&longs;tente de&longs;cribat Parabolam. Nam Latus re
ctum Parabolæ hujus, ip&longs;o motus initio, e&longs;t (
e&longs;t (
perbolam
(
&
(2
(2
e&longs;t, ut re&longs;i&longs;tentia ad gravitatem.
&longs;ecundum rectam quamvis po&longs;itione datam
&longs;i&longs;tentia Medii ip&longs;o motus initio detur: inveniri pote&longs;t Curva Nam ex data velocitate
notum e&longs;t. Et &longs;umendo 2
ad latus illud rectum, ut e&longs;t vis
gravitatis ad vim re&longs;i&longs;tentiæ,
datur
in
one gravitatis ad re&longs;i&longs;tentiam,
dabitur punctum
datur Curva
CORPORUM
Curva
locitas corporis & re&longs;i&longs;tentia
Medii in locis &longs;ingulis
ex data ratione
tia Medii &longs;ub initio motus, tum
latus rectum Parabolæ: & inde
datur etiam velocitas &longs;ub initio
motus. Deinde ex longitudine
tangentis
proportionalis velocitas, & ve
locitati proportionalis re&longs;i&longs;ten
tia in loco quovis
do 2
Parabolæ ut gravitas ad re&longs;i&longs;tentiam in
augeatur re&longs;i&longs;tentia in eadem ratione, at latus rectum Parabolæ au
geatur in ratione illa duplicata: patet longitudinem 2
in ratione illa &longs;implici, adeoque velocitati &longs;emper proportionalem
e&longs;&longs;e, neque ex angulo
tetur quoque velocitas.
ex Phænomenis quamproxime, & inde colligendi re&longs;i&longs;tentiam &
velocitatem quacum corpus projicitur. Projiciantur corpora duo
&longs;imilia & æqualia eadem cum velocitate, de loco
angulos diver&longs;os
intellectis) & cogno&longs;cantur loca
planum
vel
calculum inventa, auferatur ratio eadem
per experimentum inventa, & exponatur
differentia per perpendiculum
fac iterum ac tertio, a&longs;&longs;umendo &longs;emper
novam re&longs;i&longs;tentiæ ad gravitatem rationem
rectæ
ourva regularis
vera ratio re&longs;i&longs;tentiæ ad gravitatem, quam invenire oportuit. Ex
hac ratione colligenda e&longs;t longitudo
tudo quæ &longs;it ad a&longs;&longs;umptam longitudinem
per experimentum cognita ad longitudinem
erit vera longitudo
tia in locis &longs;ingulis.
SECUNDUS.
Cæterum, re&longs;i&longs;tentiam corporum e&longs;&longs;e in ratione velocitatis, Hy
pothe&longs;is e&longs;t magis Mathematica quam Naturalis. Obtinet hæc ra
tio quamproxime ubi corpora in Mediis rigore aliquo præditis tar
di&longs;&longs;ime moventur. In Mediis antem quæ rigore omni vacant re
&longs;i&longs;tentiæ corporum &longs;unt in duplicata ratione velocitatum. Etenim
actione corporis velocioris communicatur eidem Medii quantitati,
tempore minore, motus major in ratione majoris velocitatis; ad
eoque tempore æquali (ob majorem Medii quantitatem perturba
tam) communicatur motus in duplicata ratione major; e&longs;t que re
&longs;i&longs;tentia (per motus Legem II & III) ut motus communicatus.
Videamus igitur quades oriantur motus ex hac lege Re&longs;i&longs;tentiæ.
CORPORUM
tione Velocitatum.
vi in&longs;ita per Medium &longs;imilare movetur; tempora vero &longs;uman
tur in progre&longs;&longs;ione Geometrica a minoribus terminis ad majores
pergente: dico quod velocitates initio &longs;ingulorum temporum
&longs;unt in eadem progre&longs;&longs;ione Geometrica inver&longs;e, & quod &longs;patia
&longs;unt æqualia quæ &longs;ingulis temporibus de&longs;cribuntur.
Nam quoniam quadrato velocita
tis proportionalis e&longs;t re&longs;i&longs;tentia Me
dii, & re&longs;i&longs;tentiæ proportionale e&longs;t
decrementum velocitatis; &longs;i tempus
in particulas innumeras æquales divi
datur, quadrata velocitatum &longs;ingulis
temporum initiis erunt velocitatum
earundem differentiis proportionalia.
Sunto temporis particulæ illæ
KL, LM,in recta
& erigantur perpendicula
Ll, Mm,Hyperbolæ
centro
in & erit
ut
ubi coeunt
Ll-Mm,ut
Linearum igitur
rum progre&longs;&longs;io. Quo demon&longs;trato, con&longs;equens e&longs;t etiam ut areæ
his lineis de&longs;criptæ &longs;int in progre&longs;&longs;ione con&longs;imili cum &longs;patiis quæ
velocitatibus de&longs;cribuntur. Ergo &longs;i velocitas initio primi tempo
ris
per lineam
&longs;ub&longs;equentes & longitudines de&longs;criptæ per areas
Et compo&longs;ite, &longs;i tempus totum exponatur per &longs;um
mam partium &longs;uarum
&longs;ummam partium &longs;uarum
dividi in partes ut &longs;int
&c. in progre&longs;&longs;ione Geometrica; & erunt partes illæ in eadem pro
gre&longs;&longs;ione, & velocitates in progre&longs;&longs;ione ea
dem inver&longs;a, atque &longs;patia de&longs;cripta æqualia.
SECUNDUS.
partem quamvis
natim applicatam
ordinatam
bolicam adjacentem
quod eodem tempore
re&longs;i&longs;tente de&longs;cribere po&longs;&longs;et, per rectangulum
piendo illud ad &longs;patium quod velocitate uniformi
re&longs;i&longs;tente &longs;imul de&longs;cribi po&longs;&longs;et, ut e&longs;t area Hyperbolica
ad rectangulum
tus initio æqualem e&longs;&longs;e vi uniformi centripetæ, quæ in cadente cor
pore, tempore
citatem
& occurrat A&longs;ymptoto in
tempus exponet quo re&longs;i&longs;tentia prima uniformiter continuata tolle
re po&longs;&longs;et velocitatem totam
gravitatis, aliamve quamvis datam vim centripetam.
quamvis vim centripetam; datur tempus
re&longs;i&longs;tentiæ æqualis generare po&longs;&longs;it velocitatem quamvis
de&longs;cribi debet; ut & &longs;patium
motum &longs;uum cum velocitate illa
dio &longs;imilari re&longs;i&longs;tente de&longs;cribere pote&longs;t.
CORPORUM
ratione velocitatum impedita, & &longs;olis viribus in&longs;itis incitata,
temporibus quæ &longs;unt reciproce ut velocitates &longs;ub initio, de&longs;cri
bunt &longs;emper æqualia &longs;patia, & amittunt partes velocitatum pro
portionales totis.
A&longs;ymptotis rectangulis
CH
vis
exponantur velocitates initi
ales per perpendicula
DE,
ad
perbolæ)
ponendo, ita
areæ
& velocitates primæ
(dividendo) partibus etiam &longs;uis ami&longs;&longs;is
portionales.
temporibus quæ &longs;unt ut motus primi directe & re&longs;i&longs;tentiæ pri
mæ inver&longs;e, amittent partes motuum proportionales totis, &
&longs;patia de&longs;cribent temporibus i&longs;tis in velocitates primas ductis
proportionalia.
Namque motuum partes ami&longs;&longs;æ &longs;unt ut re&longs;i&longs;tentiæ & tempora
Igitur ut partes illæ &longs;int totis proportionales, debe
Proinde tem
pus erit ut motus directe & re&longs;i&longs;tentia inver&longs;e. Quare temporam
particulis in ea ratione &longs;umptis, corpora amittent &longs;emper parti
culas motuum proportionales totis, adeoque retinebunt velocita
tes in ratione prima. Et ob datam velocitatum rationem, de&longs;cri
bent &longs;emper &longs;patia quæ &longs;unt ut velocitates primæ & tempora con
junctim.
SECUNDUS.
ratione diametrorum: Globi homogenei quibu&longs;cunque cum velocita
tibus moti, de&longs;cribendo &longs;patia diametris &longs;uis proportionalia, amit
tent partes motuum proportionales totis. Motus enim Globi cu
ju&longs;que erit ut ejus velocitas & Ma&longs;&longs;a conjunctim, id e&longs;t, ut veloci
tas & cubus diametri; re&longs;i&longs;tentia (per Hypothe&longs;in) erit ut quadra
tum diametri & quadratum velocitatis conjunctim; & tempus (per
hanc Propo&longs;itionem) e&longs;t in ratione priore directe & ratione po&longs;te
riore inver&longs;e, id e&longs;t, ut diameter directe & velocitas inver&longs;e; ad
eoque &longs;patium (tempori & velocitati proportionale) e&longs;t ut dia
meter.
tera diametrorum: Globi homogenei quibu&longs;cunque cum velocitati
bus moti, de&longs;cribendo &longs;patia in &longs;e&longs;quialtera ratione diametrorum,
amittent partes motuum proportionales totis.
ratione dignitatis cuju&longs;cunQ.E.D.ametrorum: &longs;patia quibus Globi
homogenei, quibu&longs;cunque cum velocitatibus moti, amittent partes
motuum proportionales totis, erunt ut cubi diametrorum ad digNI
tatem illam applicati. Sunto diametri D & E; & &longs;i re&longs;i&longs;tentiæ,
ubi velocitates æquales ponuntur, &longs;int ut D
Globi quibu&longs;cunque cum velocitatibus moti, amitteus partes mo
tuum proportionales totis, erunt ut DIgitur de&longs;cri
bendo &longs;patia ip&longs;is D
tates in eadem ratione ad invicem ac &longs;ub initio.
den&longs;iore de&longs;criptum augeri debet in ratione den&longs;itatis. Motus
enim, &longs;ub pari velocitare, major e&longs;t in ratione den&longs;itatis, & tempus
(per hanc Propo&longs;itionem) augetur in ratione motus directe, ac
&longs;patium de&longs;criptum in ratione temporis.
CORPORUM
Medio, quod cæteris paribus magis re&longs;i&longs;tit, diminuendum erit in
ratione majoris re&longs;i&longs;tentiæ. Tempus enim (per hanc Propo&longs;itio
nem) diminuetur in ratione re&longs;i&longs;tentiæ auctæ, & &longs;patium in ra
tione temporis.
rantium in eorundem laterum indices dignitatum & coefficien
tia continue ductis.
Genitam voco quantitatem omnem quæ ex lateribus vel termi
nis quibu&longs;cunque, in Arithmetica per multiplicationem, divi&longs;ionem,
& extractionem radicum; in Geometria per inventionem vel con
tentorum & laterum, vel extremarum & mediarum proportionalium,
ab&longs;que additione & &longs;ubductione generatur. Eju&longs;modi quantita
tes &longs;unt Facti, Quoti, Radices, Rectangula, Quadrata, Cubi, Latera
quadrata, Latera cubica, & &longs;imiles. Has quantitates ut indeterminatas
& in&longs;tabiles, & qua&longs;i motu fluxuve perpetuo cre&longs;centes vel decre
&longs;centes, hic con&longs;idero; & earum incrementa vel decrementa momen
tanea &longs;ub nomine Momentorum intelligo: ita ut incrementa pro
momentis addititiis &longs;eu affirmativis, ac decrementa pro &longs;ubductitiis
&longs;eu negativis habeantur. Cave tamen intellexeris particulas fiNI
tas. Particulæ finitæ non &longs;unt momenta, &longs;ed quantitates ip&longs;æ ex
momentis genitæ. Intelligenda &longs;unt principia jamjam na&longs;centia fi
nitarum magnitudinum. Neque enim &longs;pectatur in hoc Lemmate
magnitudo momentorum, &longs;ed prima na&longs;centium proportio. Eo
dem recidit &longs;i loco momentorum u&longs;urpentur vel velocitates incre
mentorum ac decrementorum, (quas etiam motus, mutationes
& fluxiones quantitatum nominare licet) vel finitæ quævis quanti
tates velocitatibus hi&longs;ce proportionales. Lateris autem cuju&longs;que
generantis Coefficiens e&longs;t quantitas, quæ oritur applicando GeNI
tam ad hoc latus.
Igitur &longs;en&longs;us Lemmatis e&longs;t, ut, &longs;i quantitatum quarumcunque
perpetuo motu cre&longs;centium vel decre&longs;centium A, B, C, &c. mo
menta, vel mutationum velocitates dicantur momentum
vel mutatio geniti rectanguli AB fuerit
tenti ABC momentum fuerit
-2Et generaliter, ut dignitatis
cuju&longs;cunque AItem ut Genitæ
A
tum 3
ve A
Demon&longs;tratur vero Lemma in hunc modum.
SECUNDUS.
ubi de lateribus A & B deerant momentorum dimidia 1/2
fuit A-1/2
mum latera A & B alteris momentorum dimidiis aucta &longs;unt, eva
dit A+1/2
gulo &longs;ubducatur rectangulum prius, & manebit exce&longs;&longs;us
Igitur laterum incrementis totis
mentum
GC momentum (per Cas. 1.) erit
&longs;cribantur AB & Et par e&longs;t ra
tio contenti &longs;ub lateribus quotcunque.
ip&longs;ius A
&longs;ius autem A
+Et eodem argumento momentum dignitatis
cuju&longs;cunque A
in A, una cum 1/A ducto in
hil. Proinde momentum ip&longs;ius 1/A &longs;eu ip&longs;ius A
neraliter cum (1/AEt propterea momentum ip
&longs;ius (1/A
CORPORUM
2A3: ideoque momentum ip&longs;ius A
&longs;ive 1/2Et generaliter &longs;i ponatur A
quale B
le
momento ip&longs;ius A
mentum ip&longs;ius A
cto in A
tum indices
vi vel negativi. Et par e&longs;t ratio contenti &longs;ub pluribus dignitati
bus.
datur, momenta terminorum reliquorum erunt ut iidem termini
multiplicati per numerum intervallorum inter ip&longs;os & terminum
datum. Sunto A, B, C, D, E, F continue proportionales; & &longs;i
detur terminus C, momenta reliquorum terminorum erunt inter
&longs;e ut-2A, -B, D, 2E, 3F.
momenta extremarum erunt ut eædem extremæ. Idem intelligen
dum e&longs;t de lateribus rectanguli cuju&longs;cunQ.E.D.ti.
momenta laterum erunt reciproce ut latera.
In literis quæ mihi cum Geometra periti&longs;&longs;imo
nis abhinc decem intercedebant, cum &longs;ignificarem me compotem
e&longs;&longs;e methodi determinandi Maximas & Minimas, ducendi Tangen
tes, & &longs;imilia peragendi, quæ in terminis &longs;urdis æque ac in ratio
nalibus procederet, & literis tran&longs;po&longs;itis hanc &longs;ententiam involven-
Vir Clari&longs;&longs;imus &longs;e quoQ.E.I. eju&longs;modi methodum incidi&longs;&longs;e, & me
thodum &longs;uam communicavit a mea vix abludentem præterquam in
verborum & notarum formulis, & Idea generationis quantitatum.
Utriu&longs;que fundamentum continetur in hoc Lemmate.
SECUNDUS.
a&longs;cendat vel de&longs;cendat, & &longs;patium totum de&longs;criptum di&longs;tingua
tur in partes æquales, inque principiis &longs;ingularum partium
(addendo re&longs;i&longs;tentiam Medii ad vim gravitatis, quando cor
pus a&longs;cendit, vel &longs;ubducendo ip&longs;am quando corpus de&longs;cendit)
colligantur vires ab&longs;olutæ; dico quod vires illæ ab&longs;olutæ &longs;unt
in progre&longs;&longs;ione Geometrica.
Exponatur enim vis gravitatis per datam lineam
tia per lineam indefinitam
per differentiam
media proportionalis inter
ratione re&longs;i&longs;tentiæ;) incrementum re&longs;i&longs;tentiæ data temporis particu
la factum per lineolam
mentum per lineolam
culis
niam
mentum 2
mentum
Componatur ratio ip&longs;ius
angulum
angulum
ad rectangulum
e&longs;t æqualitatis. Ergo area illa Hyperbolica evane&longs;cens e&longs;t ut
Componitur igitur area tota Hyperbolica
&longs;patio velocitate i&longs;ta de&longs;cripto proportionalis e&longs;t. Dividatur jam
area illa in partes æquales & vi-
erunt in progre&longs;&longs;ione Geo
metrica.
mendo, ad contrariam partem puncti
imnk, knol,con&longs;tabit quod vires ab&longs;olutæ
&longs;unt continue proportionales.
Ideoque &longs;i &longs;patia omnia in a&longs;cen&longs;u &
de&longs;cen&longs;u capiantur æqualia; omnes vires ab&longs;olutæ
IC, KC, LC,erunt continue proportionales.
CORPORUM
perbolicam
poris & re&longs;i&longs;tentia Medii per lineas
& vice ver&longs;a.
dendo pote&longs;t unquam acquirere, exponens e&longs;t linea
tia Medii, invenietur velocitas maxima, &longs;umendo ip&longs;am ad veloci-
SECUMDUS.
ris Circularis & &longs;ectoris Hyperbolici &longs;umantur velocitatibus
proportionales, exi&longs;tente radio ju&longs;tæ magnitudinis: erit tempus
omne a&longs;cen&longs;us futuri ut &longs;ector Circuli, & tempus omne de&longs;cen
&longs;us præteriti ut &longs;ector Hyperbolæ.
Rectæ
qualis ducatur
Circuli quadrans
habens
gantur
omnis futuri; & &longs;ector Hyperbolicus
omnis præteriti. Si modo &longs;ectorum Tangentes
velocitates.
guli
tas
nem
ut (
Ergo &longs;ectoris particula
crementum quam minimum
citatem diminuit inver&longs;e, atque adeo ut particula temporis decre
mento re&longs;pondens. Et componendo fit &longs;umma particularum om
nium
&longs;ingulis velocitatis decre&longs;centis
dentium, u&longs;Q.E.D.m velocitas illa in nihilum diminuta eva
nuerit; hoc e&longs;t, &longs;ector totus