Newton, Isaac
,
Philosophia naturalis principia mathematica
,
1713
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dicatur Y, atque areæ
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PIGR
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decrementum
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RGgr
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detur, erit
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incrementum areæ Y ut
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PIGR
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-Y. </
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LIBER
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SECUNDUS.</
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>Quod ſi V deſignet vim a gravitate oriundam, arcui deſcribendo
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CD
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proportionalem, qua corpus urgetur in
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D:
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& R pro reſiſten
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tia ponatur: erit V-R vis tota qua corpus urgetur in
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D.
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Eſt
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itaQ.E.I.crementum velocitatis ut V-R & particula illa temporis
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in qua factum eſt conjunctim: Sed & velocitas ipſa eſt ut incre
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mentum contemporaneum ſpatii deſcripti directe & particula ea
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dem temporis inverſe. </
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<
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>Unde, cum reſiſtentia (per Hypotheſin)
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ſit ut quadratum velocitatis, incrementum reſiſtentiæ (per Lem. </
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<
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>II)
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erit ut velocitas & incrementum velocitatis conjunctim, id eſt, ut
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momentum ſpatii & V-R conjunctim; atque adeo, ſi momen
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tum ſpatii detur, ut V-R; id eſt, ſi pro vi V ſeribatur ejus ex
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ponens
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PIGR,
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& reſiſtentia R exponatur per aliam aliquam are
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am Z, ut
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PIGR
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-Z. </
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<
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>Igitur area
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PIGR
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per datorum momentorum ſubductionem
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uniformiter decreſcente, creſcunt area Y in ratione
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PIGR
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-Y,
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& area Z in ratione
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PIGR
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-Z. </
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>Et propterea ſi areæ Y & Z ſi
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mul incipiant & ſub initio æquales ſint, hæ per additionem æqua
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lium momentorum pergent eſſe æquales, & æqualibus itidem mo
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mentis ſubinde decreſcentes ſimul evaneſcent. </
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<
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>Et viciſſim, ſi ſimul
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incipiunt & ſimul evaneſcunt, æqualia habebunt momenta & ſem
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per erunt æquales: id adeo quia ſi reſiſtentia Z augeatur, veloci
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tas una cum arcu illo
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Ca,
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qui in aſcenſu corporis deſcribitur, dimi
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nuetur; & puncto in quo motus omnis una cum reſiſtentia ceſſat
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propius accedente ad punctum
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C,
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reſiſtentia citius evaneſcet quam
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area Y. </
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>Et contrarium eveniet ubi reſiſtentia diminuitur. </
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<
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>Jam vero area Z incipit deſinitque ubi reſiſtentia nulla eſt, hoc
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eſt, in principio & fine motus, ubi arcus
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CD, CD
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arcubus
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CB
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&
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Ca
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æquantur, adeoque ubi recta
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RG
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incidit in rectas
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QE
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&
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CT.
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Et area Y ſeu
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(OR/OQ)IEF-IGH
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incipit deſinitque ubi nulla eſt, ad
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eoque ubi
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(OR/OQ)IEF
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&
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IGH
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æqualia ſunt: hoc eſt (per con
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ſtructionem) ubi recta
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RG
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incidit in rectas
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QE
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&
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CT.
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Proin
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deque areæ illæ ſimul incipiunt & ſimul evaneſcunt, & propterea
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ſemper ſunt æquales. </
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<
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>Igitur area
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(OR/OQ)IEF-IGH
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æqualis eſt
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areæ Z, per quam reſiſtentia exponitur, & propterea eſt ad aream
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PINM
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per quam gravitas exponitur, ut reſiſtentia ad gravita
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tem.
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E. D.
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