Proposition 9 Problem 4

If a body revolves in a spiral PQS, cutting all the radii SP, SQ, etc., in a given angle; it is proposed to find the law of the centripetal force tending of that spiral.

Suppose the indefinitely small angle PSQ to be given because, then all the angles are given, the shape SPRQT will be given in specie.

Therefore the ratio QT/QR is also given, and QT2/QR is as QT that is as SP.

But if the angle PSQ is any way changed, the right line QR, subtending the angle of contact QPR (by Lemma 11) will be changed in the duplicate ratio of PR or QT.

Therefore the ratio QT2/QR remains the same as before, that is, as SP. And

…

is as SP³, 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 same otherwise.

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 SP³ is as SY² … PV, that is (by Corol. 3 and 5, Prop. VI) reciprocally as the centripetal force.

Lemma 12

All parallelograms circumscribed about any conjugate diameters of a given ellipsis or hyperbola are equal among themselves.

This is demonstrated by the writers on the conic sections.