Newton, Isaac - Philosophia naturalis principia mathematica

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LIBER
PRIMUS.

SECTIO VI.

De Inventione Motuum in Orbibus datis.

PROPOSITIO XXX. PROBLEMA XXII.

Corporis in data Trajectoria Parabolica moti invenire locum ad
tempus aſſignatum.

Sit Sumbilicus & Avertex principa

73[Figure 73]
lis Parabolæ, ſitque 4 ASXMæquale
areæ Parabolicæ abſcindendæ APS,
quæ radio SP,vel poſt exceſſum cor
poris de vertice deſcripta fuit, vel an
te appulſum ejus ad verticem deſcri
benda eſt. Innoteſcit quantitas areæ il
lius abſcindendæ ex tempore ipſi pro
portionali. Biſeca ASin G,erigeque
perpendiculum GHæquale 3 M, &
Circulus centro H,intervallo HS
deſcriptus ſecabit Parabolam in loco
quæſito P.Nam, demiſſa ad axem
perpendiculari PO& ducta PH,eſt
AGq+GHq (=HP q=—AO-AG: quad.+—PO-GH: quad.)=
AOq+POq-2 GAO-2GHXPO+AGq+GHqueUnde
2 GHXPO (=AOq+POq-2GAO)=AOq+1/4 POque
Pro AOqſcribe (AOXPOq/4AS); &, applicatis terminis omnibus ad
3POductiſQ.E.I. 2AS,fiet 4/3 GHXAS(=1/6AOXPO+1/2 ASXPO
=(AO+3AS/6)XPO=(4AO-3SO/6)XPO=areæ —APO-SPO)
=areæ APS.Sed GHerat 3 M, & inde 4/3 GHXASeſt 4 ASXM.
Ergo area abſciſſa APSæqualis eſt abſcindendæ 4ASXM. Q.E.D.

Corol.1. Hinc GHeſt ad AS,ut tempus quo corpùs deſcrip
ſit arcum APad tempus quo corpus deſcripſit arcum inter verti
cem A& perpendiculum ad axem ab umbilico Serectum.

Corol.2. Et Circulo ASPper corpus motum Pperpetuo tranſ
eunte, velocitas puncti Heſt ad velocitatem quam corpus habuit

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