Philosophiæ naturalis principia mathematica autore Js. Newton ...

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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

LEM. III.

Sit PQRr Spiralis quae secet radios omnes SP, SQ, SR, &c. in aequalibus angulis. Agatur recta PT quaetangat eandem in puncto quovis P, secetque radium SQ in T; & ad Spiralem erectis perpen∣diculis PO, QO concurrentibus in O, jungatur SO. Dico quod fi puncta P & Q accedant ad invicem & coeant, angulus PSO evadet rectus, & ultima ratio rectanguli TQ×PS ad PQ quad. erit ra∣tio aequalitatis.

Etenim de angulis rectis OPQ, OQR subducantur anguli aequales SPQ, SQR, & manebunt anguli aequales OPS, OQS. Ergo circulus qui transit per

[illustration]

puncta O, S, P transibit eti∣am per punctum Q. Coeant puncta P & Q, & hic cir∣culus in loco coitus PQ tan∣get Spiralem, adeoque per∣pendiculariter secabit rectam OP. Fiet igitur OP diame∣ter circuli hujus, & angulus OSP in semicirculo rectus. Q.E.D.

Ad OP demittantur perpendicula QD, SE, & linearum ra∣tiones ultimae erunt hujusmodi: TQ ad PD ut TS vel PS ad PE, seu PO ad PS. Item PD ad PQ ut PQ ad PO. Et ex aequo perturbate TQ ad PQ ut PQ ad PS. Unde fit PQq. aequalis PQ×PS. Q.E.D.