The Mathematical Principles of Natural Philosophy, Book I: Of the Motion of Bodies
Proposition IX. Problem IV
If a body revolves in a spiral PQS, cutting all the radii SP, SQ, &c., in a given angle; it is proposed to find the law of the centripetal force tending to the centre of that spiral.
Suppose the indefinitely small angle PSQ to be given; because, then, all the angles are given, the figure SPRQT will be given in specie. Therefore the ratio is also given, and is as QT, that is (because the figure is given in specie), as SP. But if the angle PSQ is any way changed, the right line QR, subtending the angle of contact QPR (by Lemma XI) will be changed in the duplicate ratio of PR or QT. Therefore the ratio remains the same as before, that is, as SP. And is as, and therefore (by Corol. 1 and 5, Prop. VI) the centripetal force is reciprocally as the cube of the distance SP. Q.E.I.
The perpendicular SY let fall upon the tangent, and the chord PV of the circle concentrically cutting the spiral, are in given ratios to the height SP; and therefore is as, that is (by Corol. 3 and 5, Prop. VI) reciprocally as the centripetal force.