Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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ut (
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mkXms/mt
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) ad (
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rkq/2kC
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), vel ut
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mkXms
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ad
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rk
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quadratum; hoc eſt, ſi
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capiantur datæ quantitates F, G in ea ratione ad invicem quam
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habet angulus
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VCP
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ad angulum
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VCp,
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ut GG-FF ad FF. </
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<
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>Et
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propterea, ſi centro
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C
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intervallo quovis
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CP
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vel
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Cp
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deſcribatur
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Sector circularis æqualis areæ toti
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VPC,
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quam corpus
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P
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tempore
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quovis in Orbe immobili revolvens radio ad centrum ducto de
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ſcrip ſit: differentia virium, quibus corpus
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P
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in Orbe immobili &
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corpus
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p
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in Orbe mobili revolvuntur, erit ad vim centripetam qua
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corpus aliquod radio ad centrum ducto Sectorem illum, eodem tem
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pore quo deſcripta ſit area
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VPC
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uniformiter deſeribere potuiſſet,
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ut GG-FF ad FF. </
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<
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>Namque Sector ille & area
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pCk
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ſunt ad in
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vicem ut tempora quibus deſcribuntur. </
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DE MOTU
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CORPORUM</
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Corol.
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2. Si Orbis
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VPK
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Ellipſis ſit umbilicum habens
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C
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& Ap
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ſidem ſummam
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V;
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eique ſimilis & æqualis ponatur Ellipſis
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upk,
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ita ut ſit ſemper
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pC
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æqualis
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PC,
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& angulus
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VCp
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ſit ad angulum
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VCP
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in data ratione G ad F; pro altitudine autem
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PC
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vel
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pC
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ſcribatur A, & pro Ellipſeos latere recto ponatur 2 R: erit vis qua
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corpus in Ellipſi mobili revolvi poteſt, ut (FF/AA)+(RGG-RFF/A
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cub.
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)
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& contra. </
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<
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>Exponatur enim vis qua corpus revolvatur in immota
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Ellipſi per quantitatem (FF/AA), & vis in
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V
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erit (FF/
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CV quad.
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). Vis au
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tem qua corpus in Circulo ad diſtantiam
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CV
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ea cum velocitate
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revolvi poſſet quam corpus in Ellipſi revolvens habet in
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V,
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eſt ad vim qua corpus in Ellipſi revolvens urgetur in Apſide
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V,
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ut dimidium lateris recti Ellipſeos. </
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<
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>ad Circuli ſemidiametrum
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CV,
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adeoque valet (RFF/
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CV cub.
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): & vis quæ ſit ad hanc ut GG-FF ad
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FF, valet (RGG-RFF/
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CV cub.
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): eſtque hæc vis (per hujus Corol. </
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>1.)
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differentia virium in
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V
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quibus corpus
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P
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in Ellipſi immota
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VPK,
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& corpus
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p
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in Ellipſi mobili
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upk
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revolvuntur. </
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<
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>Unde cum (per
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hanc Prop.) differentia illa in alia quavis altitudine A ſit ad ſe
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ipſam in altitudine
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CV
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ut (1/A
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cub.
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) ad (1/
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CV cub.
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), eadem differentia
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in omni altitudine. </
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<
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>A valebit (RGG-RFF/A
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cub.
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). Igitur ad vim (FF/AA)
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qua corpus revolvi poteſt in Ellipſi immobili
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VPK,
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addatur ex
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ceſſus (RGG-RFF/A
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cub.
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) & componetur vis tota (FF/AA)+(RGG-RFF/A
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cub.
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) </
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