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# Title: [Philosophiae Naturalis Principia Mathematica] Philosophiæ Naturalis Principia Mathematica (shelfmark Leibn. Marg. 166)
# Folio 54, page id 1041163592:0088
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54
PTTTLOSODMLSE NAIDRALIS
DEMOrD Corol. Hinc Fllipseos area toras eique proportionale rectanou-
Conronon lum sub axibus, est in ratione compolica ex lubduplicata ratione
laceris recti ex ratione temporis periodici. Namque arca tota eAt
uE area QI X SP ducta in tempus periodicum.
PROPOSITIO ZV. PHEOREMA VII.
Tisdem positis, dico quod Tempora periodica in P. lixsibus sunt in ratione
sesquiplicat a majorum axium.
Namque axis minor est medius proportionalis inter axem majo-
rem ex latus rectum, acque adeo rectangulum sub axibus eAt in ra-
tione composita ex subduplicata ratione lateris recti et selquiplicata
ratione axis majoris. Sed hoc rectangulum, per C orollarium Prov.
XTV. est in ratione composita ex lubduplicata ratione laceris recti
et ratione periodici temporis. Dematur uerobique subduplicata
ratio laceris rectis e manebit selquiplicata ratio majoris axis aequa-
lis rationi periodici temporis. 9. F. D.
Corol. Sune igitur cempora periodica in Ellipsibus eadem ac in
Circulis, quorum diametri aequantur majoribus azibus FIliplcon.
PROPOSITIO XVI TMEOREMA VIII.
Tisdem positis, es actis ad corpora lineis rectis, quae ibidem tangant Or-
bitas, demissisque ab umbilico communi ad hastangentes perpendi-
cularibus: dico quod Velocitates corporum sunt in ratione composi
ta ex ratione perpendiculorum inverse es sub duplicata ratione la-
terum rectorum principalium directe.
Ab umbilico sad tansentem P R demitte perpendiculum SY
ec relocitas corporis P crit reciproce in subduplicata ratione quan-
citaris -2. Nam relocitas illa cA ut arcus quam minimus P B
in data cemporis particula descriptus, hoc CA (per LC. VTL.) ut
tansens P R, id est (ob proportionales P R ad _ F e S = ad SY) ur
 X
=, EIVC VE SY RCCIPTOCE et SPX GF directe; eAque
S~ X∞+
MINII