Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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ac denique per punctum
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Q
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agatur
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LR
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quæ ipſi
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SP
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parallela
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ſit & occurrat tum circulo in
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L
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tum tangenti
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PZ
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in
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R.
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Et
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ob ſimilia triangula
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ZQR, ZTP, VPA
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; erit
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RP quad.
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hoc
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eſt
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QRL
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ad
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QT quad.
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ut
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AV quad.
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ad
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PV quad.
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Ideoque
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(
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QRLXPV quad./AV quad.
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) æquatur
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QT quad.
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Ducantur hæc æqualia in
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(
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SP quad./QR
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) &, punctis
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P
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&
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Q
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coeuntibus, ſcribatur
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PV
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pro
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RL.
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Sic fiet (
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SP quad.XPV cub./AV quad.
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) æquale (
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SP quad.XQT quad./QR
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) Ergo (per
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Corol.1 & 5 Prop.VI.) vis centripeta eſt reciproce ut (
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SPqXPV cub./AV quad
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)
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id eſt, (ob datum
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AV quad.
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) reciproce ut quadratum diſtantiæ ſeu
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altitudinis
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SP
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& cubus chordæ
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PV
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conjunctim.
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Q.E.I.
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Idem aliter.
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>Ad tangentem
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PR
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productam demittatur perpendiculum
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SY,
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& ob ſimilia triangula
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SYP, VPA
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; erit
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AV
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ad
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PV
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ut
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SP
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ad
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SY,
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ideoque (
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SPXPV/AV
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) æquale
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SY,
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& (
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SP quad.XPV cub./AV quad.
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) æquale
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SY quad.XPV.
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Et propterea (per Corol.3 & 5 Prop.VI.) vis centri
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peta eſt reciproce ut (
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SPqXPV cub./AVq
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) hoc eſt, ob datam
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AV,
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reci
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proce ut
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SPqXPV cub. </
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<
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E. I.
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Corol.
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1. Hinc ſi punctum datum
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S
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ad quod vis centripeta ſem
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per tendit, locetur in circumferentia hujus circuli, puta ad
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V
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; erit
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vis centripeta reciproce ut quadrato cubus altitudinis
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SP.
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Corol.
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2. Vis qua corpus
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P
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in cir
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<
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culo
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APTV
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circum virium centrum
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<
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S
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revolvitur, eſt ad vim qua corpus
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idem
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P
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in eodem circulo & eodem
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tempore periodico circum aliud quod
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vis virium centrum
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R
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revolvi poteſt,
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ut
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RP quad.XSP
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ad cubum rectæ
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SG
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quæ a primo virium centro
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S
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ad or
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bis tangentem
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PG
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ducitur, & diſtan
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tiæ corporis a ſecundo virium centro
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parallela eſt. </
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<
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>Nam, per conſtructionem hujus Propoſitionis, vis
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prior eſt ad vim poſteriorem, ut
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RPqXPT cub.
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ad
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SPqXPV cub.
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