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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 229, page id 1041163592:0263
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PRINCIPTA MATHEMATICA.
229
tatem illam datam in subduplicata ratione, quam habet vis Grari. LIVEN
catis ad Medii relistenciam illam cogniram.
PROPOSITIO II. THEOREMA VII.
Positis jam demonstratis, dico quod si Tancentes anculorum seco
ris Circularis es (ectoris Kyperbolici sumantur velocitatibus
proportionales, existente radio justae maenitudinis: erit tempus
omne ascensus futuri ut sector circulu e tempus omne descen-
sus praeteriti ut sector Hyperbolae.
ReCtK AC, qua ris graritatis exxonitur, perpendicularis et 2-
SFCVNDBS.
qualis dueaeur A D. Centro D Temidiamerro  D describatur cum
Circuli quadrans LF F, cum EIyperbola rectangula AV Z axcm
habens IX, verricem principalem A ex ACYMPEOTON DC. Jun-
SANEUF D p, D P, et erit sector Circularis A+ D ut tempus ascensys
omnis Futuri; ec sector LIyperbolicus ATD ue tempus descensus
omnis praeterici. Si modo sectorum Tangentes Ap, AP sInC ut
velocitatCS.
Cas. T. Agatur enim Dvg abscindens seccoris AD+ et trian
suli ADp momenta, seu particulas quam minimas simul delcrip-
caS + D V_ pDg. Cum particulae illae, ob angulum commu-
nem D, Cune in duplicata ratione lacerum, crit particula tV
UE
3Dauad Sed pD quad. CA A D quad + Ap quad. id est,
AD quad + AD X Ak Ccu AD XCk; e qD p CA = A DXP9.
Ergo sectoris particula + D V Cst uF =, id eA, uc relocitaris de-
crementum quam minimum pa directe ec ris illa Ck quae velo-
citarem diminuie inverse, atque adco ut parcicula temporis decre.
mento respondens. EF componendo tc lumma particularum om-
nium + D V in seCtOrC ADF, ut summa particularum cemporis
lingulis velocitaris decrescentis Ap parciculis amistis pg respon-
dentium, ulque dum relocitas illa in nihilum diminuta cva-
nuerie; hoc eAt, Sector torus ADF eA ue alcensus totiuS FuTuEI
TemDuS. Q. F. D.
Ca5.