Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/218

 of ovals that are not touched by conjugate figures running out in infinitum.

Hence the area of an ellipis, decribed by a radius drawn from the focus to the moving body, is not to be found from the time given, by a finite equation; and therefore cannot be determined by the decription of curves geometrically rational. Thoe curves I call geometrically rational, all the points whereof may be determined by lengths that are defineable by equations, that is, by the complicated ratio's of lengths. Other curves (uch as pirals, quadratrixes, and cycloids) I call geometrically irrational. For the lengths which are or are not as number to number (according to the tenth book of elements) are arithmetically rational or irrational. And therefore I cut off an area of an ellipis proportional to the time in which it is decribed by a curve geometrically irrational, in the following manner.

To find the place of a body moving in a given elliptic trajectory at any aigned time.

Suppoe A (Pl. 14. Fig. 2.) to be the principal vertex, S the focus, and O the centre of the ellipis APB; and let P be the place of the body to be found. Produce OA to G, o as OG may be to OA as OA to OS. Erect the perpendicular GH; and about the centre O, with the interval OG, decribe the circle GEF; and on the ruler GH