Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/117

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will be as the verſed line directly and the ſquare of the time inverſely. Q. E. D.

And the ſame thing may alſo be eaſily demonſtrated by corol. 4. lem. 10.

Plate 3, Figure 2
Plate 3, Figure 2

Cor. 1. If a body P revolving about the centre S, (Pl. 3. Fig. 2.) deſcribes a curve line APQ which a right line ZPR touches in any point P; and from any other point Q of the curve. QR is drawn parallel to the diſtance SP, meeting the tangent in R; and QT is drawn perpendicular to the diſtance SP: the centripetal force will be reciprocally as the ſolid , if the ſolid be taken of that magnitude which it ultimately acquires when the points P and Q coincide. For QR is equal to the verſed ſine of double the arc QP, whoſe middle is P: and double the triangle SQP, or SP x QT is proportional to the time, in which that double arc is deſcribed; and therefore may be uſed for the exponent of the time.

Cor. 2. By a like reaſoning, the centripetal force is reciprocally as the ſolid if ST is a perpendicular from the centre of force on PR the tangent of the orbit. For the rectangles ST x QP and SP x QT are equal.

Cor. 3. If the orbit is either a circle, or touches or cuts a circle concentrically, that is contains with a circle the leaſt angle of contact or ſection, having the ſame curvature and the ſame radius of curvature at the point P; and if P, V be a chord of this circle, drawn from the body through the centre of force; the centripetal force will be reciprocally as the ſolid . For PV is