Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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<
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>Nam perpendicula a centro
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S
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in tangentes
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PT, QT
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demiſſa (per
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Corol. </
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>1. Prop.I.) ſunt reciproce
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ut velocitates corporis in punctis
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P
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&
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V
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; &c. </
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>adeoque per conſtructio
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nem ut perpendicula
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AP, BQ
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di
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recte, id eſt ut perpendicula a pun
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cto
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D
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in tangentes demiſſa. </
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<
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>Un
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de facile colligitur quod puncta
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S, D, T,
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ſunt in una recta. </
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>Et ſimili
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argumento puncta
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S, E, V
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ſunt eti
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am in una recta; & propterea centrum
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S
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in concurſu rectarum
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TD, VE
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verſatur.
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Q.E.D.
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PROPOSITIO VI. THEOREMA V.
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Si corpus in ſpatio non reſiſtente circa centrum immobile in Orbe quocun
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que revolvatur, & arcum quemvis jamjam naſcentem tempore quàm
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minimo deſcribat, & ſagitta arcus duci intelligatur quæ chordam bi
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ſecet, & producta tranſeat per centrum virium: erit vis centripeta
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in medio arcus, ut ſagitta directe & tempus bis inverſe.
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<
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>Nam ſagitta dato tempore eſt ut vis (per Corol.4 Prop.I,) & augen
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do tempus in ratione quavis, ob auctum arcum in eadem ratione ſa
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gitta augetur in ratione illa duplicata (per Corol. </
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<
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>2 & 3, Lem. </
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<
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>XI,) ad
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eoque eſt ut vis ſemel & tempus bis. </
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<
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>Subducatur duplicata ratio tempo
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ris utrinque, & fiet vis ut ſagitta directe & tempus bis inverſe.
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Q.E.D.
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<
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>Idem facile demonſtratur etiam per Corol. </
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<
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>4 Lem. </
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<
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>X. </
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Corol.
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1. Si corpus
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P
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revolvendo
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<
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circa centrum
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S
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deſcribat lineam
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curvam
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APQ,
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tangat verò recta
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ZPR
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curvam illam in puncto
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quovis
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P,
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& ad tangentem ab alio
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quovis Curvæ puncto
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Q
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agatur
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QR
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diſtantiæ
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SP
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parallela, ac
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demittatur
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QT
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perpendicularis
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ad diſtantiam illam
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SP:
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vis cen
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tripeta erit reciproce ut ſolidum
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(
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SP quad.XQT quad./QR
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) ſi modo ſolidi illius ea ſemper ſumatur quan
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titas, quæ ultimò fit ubi coeunt puncta
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P
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&
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Nam
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QR
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æqualis </
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