120 THE MATHEMATICAL PRINCIPLES [BOOK I. SP in x. Now. because of the similar triangles Pxv, SPM, and of the equal sides SP, SM of the one, the sides Px or Q,R and Pv of the other will be also equal. But (by the conic sections) the square of the ordinate Q,y is equal to the rectangle under the latus rectum and the segment Pv of the diameter ; that is (by Lem. XIII.), to the rectangle 4PS X Pv, or 4PS X Q,R ; and the points P and Q, coinciding, the ratio of Qv to Q,.r (by Cor. 2, Lem. VII.,) becomes a ratio of equality. And therefore Q,#2, in this case, becomes equal to the rectangle 4PS X Q,R. But (because of the similar triangles Q#T, SPN), Q^'2 is to QT2 as PS2 to SN2, that is (by Cor. 1, Lem. XIV.), as PS to SA ; that is, as 4PS X QR to 4SA x QR, and therefore (by Prop. IX. Lib. V., Elem.) QT* and 4SA X QR are SP2 SP2 X QT2 equal. Multiply these equals by ^-^-, and ^5 — -will become equal to SP2 X 4SA : and therefore (by Cor. 1 and 5, Prop. VL), the centripetal force is reciprocally as SP2 X 4S A ; that is, because 4SA is given, recipro cally in the duplicate ratio of the distance SP. Q.E.I. COR. 1. From the three last Propositions it follows, that if any body P goes from the place P with any velocity in the direction of any right line PR, and at the same time is urged by the action of a centripetal force that is reciprocally proportional to the square of the distance of the places from the centre, the body will move in one of the conic sections, having its fo cus in the centre of force ; and the contrary. For the focus, the point of contact, and the position of the tangent, being given, a conic section may be described, which at that point shall have a given curvature. But the curvature is given from the centripetal force and velocity of the body be ing given ; and two orbits, mutually touching one the other, cannot be de scribed by the same centripetal force and the same velocity. COR. 2. If the velocity with which the body goes from its place P is such, that in any infinitely small moment of time the lineola PR may be thereby describe I: and the centripetal force such as in the same time to move the same body through the space QR ; the body will move in one of QT2. the conic sections, whose principal latus rectum is the quantity Tjfr in its ultimate state, when thelineoke PR, QR are diminished in infinitum. In these Corollaries I consider the circle as an ellipsis ; and I except the case where the body descends to the centre in a right line. PROPOSITION XIV. THEOREM VI. Tf several bodies revolve about one common centre, and the centripetal force is reciprocally in tlie duplicate ratio of the distance of places from the centre ; I say, that the principal latera recta of tfieir orbits are in the duplicate ratio of the areas, which the bodies by radii drawn to the centre describe it\ the same time.