Newton, Isaac - Philosophia naturalis principia mathematica

Inveniatur vis quæ tendit ab Hyperbolæ centro C.Prodibit hæc
diſtantiæ CPproportionalis. Inde vero (per Corol. 3 Prop. VII.)
vis ad umbilicum Stendens erit ut (PEcub/SPq), hoc eſt, ob datam PE,
reciproce ut SPque Q.E.I.

Eodem modo demonſtratur quod corpus, hac vi centripeta in
centrifugam verſa, movebitur in Hyperbola conjugata.

LEMMA XIII.

Latus rectum Parabolæ ad verticem quemvis pertinens, eſt quadru
plum diſtantiæ verticis illius ab umbilico figuræ.Patet ex Conicis.

LEMMA XIV.

Perpendiculum quod ab umbilico Parabolæ ad tangentem ejus demitti
tur, medium eſt proportionale inter diſtantias umbilici a puncto con
tactus & a vertice principali figuræ.

Sit enim AQPParabola, Sumbilicus ejus, Avertex principa
lis Ppunctum

23[Figure 23]
contactus, PO
ordinatim ap
plicata ad dia
metrum prin
cipalem, PM
tangens dia
metro princi
pali occurrens
in M,& SN,
linea perpen
dicularis ab umbilico in tangentem. Jungatur AN,& ob æquales
MS& SP, MN& NP, MA& AO,parallelæ erunt rectæ
AN& OP,& inde triangulum SANrectangulum erit ad A&
ſimile triangulis æqualibus SNM, SNP:Ergo PSeſt ad SN,
ut SNad SA. Q.E.D.

Corol.1. PSqueeſt ad SNqueut PSad SA.

Corol.2. Et ob datam SA,eſt SNqueut PS.

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