Newton, Isaac - Philosophia naturalis principia mathematica

Newton, Isaac, Philosophia naturalis principia mathematica, 1713

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1

DE MOTU
CORPORUM

PROPOSITIO X. PROBLEMA. V.

Gyretur corpus in Ellipſi: requiritur lex vis centripetæ tendentis ad
centrum Ellipſeos.

Sunto CA, CBſemiaxes Ellipſeos; GP, DKdiametri conju
gatæ; PF, Qtperpendicula ad diametros; Qvordinatim appli
cata ad diametrum

20[Figure 20]
GP; & ſi compleatur
parallelogrammum
QvPR,erit (ex CoNI
cis) PvGad Qv quad.
ut PC quad.ad CD
quad.& (ob ſimilia
triangula Qvt, PCF)
Qv quad.eſt ad Qt
quad.ut PC quad.ad
PF quad.& conjun
ctis rationibus, PvG
ad Qt quad.ut PC
quad.ad CD quad.
& PC quad.ad PF
quad.id eſt, vGad
(Qt quad./Pv) ut PC quad.
ad (CDqXPFq/PCq). Scribe QRpro Pv,& (per Lemma XII.) BCXCA
pro CDXPF,nec non, punctis P& Q coeuntibus, 2PCpro
vG,& ductis extremis & mediis in ſe mutuo, fiet (Qt quad.XPCq/QR)
æquale (2BCqXCAq/PC). Eſt ergo (per Corol. 5 Prop. VI.) vis centri
peta reciproce ut (2BCqXGAq;/PC) id eſt (ob datum 2BCqXCAq)
reciproce ut (1/PC); hoc eſt, directe ut diſtantia PC. que E. I.

In PGab altera parte puncti t poſita intelligatur tu æqualis ipſi
tv; deinde cape uVquæ ſit ad vGut eſt DC quad.ad PC quad.
Et quoniam ex Conicis est Qv quad.ad PvG,ut DC quad.ad
PC quad:erit Qv quad.æquale PvXuV.Unde quadratum chor-

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