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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 49, page id 1041163592:0083
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PRTNOTPTA MATHEMAEICA.
Sed punctis Q _c P cocuntibuS, aquatuE2 P C _ C V. Ergo echiSPEO. LruEN
portionalia L. X Q R et Q E quad. ae quantur. Ducantur taec aequalia in Puruvs
E
2g. XdTG
R _ CEL XSP g. Xquale.
e
Ergo (per Corol L
_C5 PrOD. VI.) vis centripeta reciproce est ut L. Z SP g. id est, reci-
proce in rarione duplicata distancia SP. 9. F. I.
Idem aliter.
Cum vis ad centrum Ellipseos cendens, qua corpus F in F IlipsI
illa revolri pocest, siE (per COrOl. I PFOD. Z) ut C F distantia cOr
Doris. ab Ellipseos CentrO C: duCatUF CF. Parallela FMip(eOS tan-
senti P R: _ vis qua corpus idem P, circum aliud quodris Ellip-
seos punctum S revolri potest, (IC. F dC P SCOnCUTranE In F, Crit uE
 L Cub
e∞  (Der COrOL. 3PFOD. VIT,) hoc eAt, Si punctum S sic umbili-
cus Ellipseos, adeoque P F detUT, UE S P g rCcIpEOCC. Q. F. I.
Eadem brevicace qua craduximus Problema quintum ad Parab0-
lam, ex EIyperbolam, liceret idem hic Facere: verum ob dignita-
tem Problematis ex ulum ejus in sequentibus, non pisebut caus cc-
ceros demonstratione contirmare.
PROPOSITIO XII. PROBLEMA. VII.
Noveatur corpus in Eyperbola: requiritur L ex vis centripeta ten-
dentis ad umbilicum ficurae.
Sunto C A. CB Temi- aXes FIyperboIx; P G, K D diametri con-
JugaCX; P F, Q Derpendicula ad diametros; ex 9u ordinarim
applicata ad diametrum G P. ABatuE SF sccanS cum diametrum
DK in P, cum ordinatim applicatam QV in a, de compleatur pa-
rallelogrammum e R P ∞. PatCE P P aequalem esse semiaxi erans-
verio AC, eo quod, acta ab altero IIy perbolae umbilico sI linca
TTT ipA F C parallela, ob aequales CS, C FL, Xqueneur ES, L I;
adeo uE F F semiditerentia sIC ipTarum F S, PF, id eR (Ob P2
rallelas YFI, P R ex angulos aequales IP R, TYP Z) ipsarum F S
PTI. quarum differentia aZCm tOtum 2 AC adae quat. Ad S F de-
mittatur perpendicularis T. Et Hyperbolae lacere recto princi
Dali (seu 2BC
qC dicto L CrIE L X Q R ad L. Z P V uE Q R ad F V)
idt CA, UE P L. seu AC ad PC: PF L. X X v ad C v P ue L 2d
2