Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/294

 les) of two ocillations, will be always as thoe parts, and therefore will caue thoe parts to be decribed together. But bodies which continually decribe together parts proportional to the wholes, will decribe the wholes together alo.  Q. E. D.

Hence if the body T (Pl. 20. Fig. 3.) hanging by a rectilinear thread AT from the centre A, decribe the circular arc STRQ, and in the mean time be acted on by any force tending downwards with parallel directions, which is to the uniform force of gravity as the arc TR to its ine TN, the times of the everal ocillations will be equal. For becaue TZ, AR are parallel. the triangle ATN, ZTY are imilar; and therefore TZ will be to AT as TY to TN; that is, if the uniform force of gravity be expreed by the given length AT the force TZ by which the ocillations become iochronous, will be to the force of gravity AT; as the arc TR equal to TY is to TN the ine of that arc.

And therefore in clocks, if forces were impreed by ome machine upon the pendulum which preerves the motion, and o compounded with the force of gravity, that the whole force tending downwards hould be always as a line produced by applying the rectangle under the arc TR and the radius AR to the line TN, all the ocillations will become iochronous.