PROPOSITION VIII. PROBLEM III.
From C, the centre of the semi-circle, let the semi-diameter CA be drawn, cutting the parallels at right angles in M and N, and join CP. Because of the similar triangles CPM, PZT, and RZQ, we shall have CP² to PM² as PR² to QT²; and, from the nature of the circle, PR² is equal to the rectangle , or, the points P, Q, coinciding, to the rectangle . Therefore CP² is to PM² as to QT²; and , and . And therefore (by Corol. 1 and 5; Prop. VI.), the centripetal force is reciprocally as ; that is (neglecting the given ratio ), reciprocally as PM³. Q.E.I.
And the same thing is likewise easily inferred from the preceding Proposition.
SCHOLIUM.
And by a like reasoning, a body will be moved in an ellipsis, or even in an hyperbola, or parabola, by a centripetal force which is reciprocally as the cube of the ordinate directed to an infinitely remote centre of force.
PROPOSITION IX. PROBLEM IV.
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