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Mathematical Principles of Natural Philosophy and His System of the World (2 Volumes)
Mathematical Principles of Natural Philosophy and His System of the World - 2 Volumes Author:Isaac Newton Purchase of this book includes free trial access to www.million-books.com where you can read more than a million books for free. This is an OCR edition with typos. Excerpt from book: SECTION IV. Of the circular motion of bodies tn rejifting mediums. Lemma III. Let P QR be a fpiral cutting all the radii S P, S Q, S R, drc. in equal an... more »gles. 1)raw the right line P T touching the fpiral in any point P, and cutting the radius S Q in T; draw P O, QO perpendicular to the fpiral, and meeting in O, and join S O. 1 fay-, that if the points P and approach andcoincide', the angle P S O will become a right angle, and the ultimate ratio of the rettangle T Qx 2 P S to PQ will be the ratio of equality. PI. 4. Fig. i. For from the right angles O P O, O QR, fubduct the equal angles SP 0, S OR, and there will remain the equal angles O?£, OQS. Therefore a circle which paffes thro' the points O, S, P, will pafs alfo thro' the point (2. Let the points P and O coincide, snd this circle will touch the fpiral in the place of coincidence P O, and will therefore cut the right line O P perpendicularly. Therefore O P will become a diameter of this circle, and the angle OSP, being in a femidrclej becomes a right one. O. E, D. Draw OD SE perpendicular to OP, and the ultimate ratio's of the lines will be as follows ; TO to PD as TS or PS to P E, or t P O to z P S ; and PD to 7J2.as P to 2 PO ; and, fxfquopertnrbate, TOto PCas P]?to 2 PS. Whence PCP becomes equal to 7" ? x zPS. Q.E.D. Proposition XV- Theorem XII. If the denfity of a medium in each place thereof be reciprocally as the diflance of the places from an immoveable centre, and the centripetal force be in the duplicate ratio of the den- jit y : I fay, that a bcdy may revoke in a Jjpiral which cuts all the radii drawn from that centre in a given angle. PI. 4. Fig. 2. Suppofe every thing to be as in the foregoing Lemma, and produce SO to ", fo that 5F"may be equal to S P. In any time let a body, in a refitting med...« less