Page 48
Prop. X. Prob. V.
Gyretur corpus in Ellipsi: requiritur lex vis centripetae tendentis ad centrum Ellipseos.
Sunto CA, CB
[illustration]
Philosophiæ naturalis principia mathematica autore Js. Newton ...
About this Item
- Title
- Philosophiæ naturalis principia mathematica autore Js. Newton ...
- Author
- Newton, Isaac, Sir, 1642-1727.
- Publication
- Londini :: Jussu Societatis Regiae ac Typis Josephi Streater ...,
- 1687.
- Rights/Permissions
-
To the extent possible under law, the Text Creation Partnership has waived all copyright and related or neighboring rights to this keyboarded and encoded edition of the work described above, according to the terms of the CC0 1.0 Public Domain Dedication (http://creativecommons.org/publicdomain/zero/1.0/). This waiver does not extend to any page images or other supplementary files associated with this work, which may be protected by copyright or other license restrictions. Please go to http://www.textcreationpartnership.org/ for more information.
- Subject terms
- Mechanics -- Early works to 1800.
- Celestial mechanics -- Early works to 1800.
- Link to this Item
-
http://name.umdl.umich.edu/A52251.0001.001
- Cite this Item
-
"Philosophiæ naturalis principia mathematica autore Js. Newton ..." In the digital collection Early English Books Online. https://name.umdl.umich.edu/A52251.0001.001. University of Michigan Library Digital Collections. Accessed June 4, 2024.
Pages
semiaxes Ellipseos; GP, DK diame∣tri conjugatae; PF, Qt perpendicula ad diametros; Qv ordinatim applica∣ta ad diametrum GP; & si complea∣tur parallelogram∣mum QvRP, erit (ex Conicis) PvG ad Qv quad. ut PC quad. ad CD quad. & (ob simi∣lia triangula Qvt, PCF) Qv quad. est ad Qt quad. ut PC quad. ad PF quad. & conjunctis rationibus, PvG ad Qt quad. ut PC quad. ad CD quad. & PC quad. ad PF quad. id est vG ad Qt quad./Pv ut PC quad. ad CDq×PFq/PCq. Scribe QR pro Pv, & (per Lemma xii.) BC×CA pro CD×PF, nec non (punctis P & Q coeun∣tibus) 2 PC pro vG, & ductis extremis & medijs in se mutuo, fiet Qtq×PCq / QR aequale 2 BCq×CAq/PC Est ergo (per Corol. Theor. V.) vis centripeta reciproce ut 2 BCq×CAq/PC, id est
Page 49
(ob datum 2 BCq.×CAq.) ut 1••PC, hoc est, directe ut distantia PC.Q.E.I.
Corol. 1. Unde vicissim si vis sit ut distantia, movebitur corpus in Ellipsi centrum habente in centro virium, aut forte in circulo, in quem Ellipsis migrare potest.
Corol. 2. Et aequalia erunt revolutionum in Figuris universis circa centrum idem factarum periodica tempora. Nam tempora illa in Ellipsibus similibus aequalia sunt per Corol. 3 & 7 Prop. IV: In Ellipsibus autem communem habentibus axem majorem, sunt ad invicem ut Ellipseon areae totae directe & arearum parti∣culae simul descriptae inverse; id est ut axes minores directe & corporum velocitates in verticibus principalibus inverse, hoc est ut axes illi directe & ordinatim applicatae ad axes alteros inverse, & propterea (ob aequalitatem rationum directarum & inversa∣rum) in ratione aequalitatis.
Scholium.
Si Ellipsis, centro in infinitum abeunte, vertatur in Parabo∣lam, corpus movebitur in hac Parabola, & vis ad centrum infini∣te distans jam tendens, evadet aequabilis. Hoc est Theorema Galilei. Et si Conisectio Parabolica, inclinatione plani ad conum sectum mutata, vertatur in Hyperbolam, movebitur corpus in hujus perimetro, vi centripeta in centrifugam versa.