of ovals that are not touched by conjugate figures running out in infinitum.
Cor. Hence the area of an ellipſis, deſcribed 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 deſcription of curves geometrically rational. Thoſe 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 (ſuch as ſpirals, 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 ellipſis proportional to the time in which it is deſcribed by a curve geometrically irrational, in the following manner.
Proposition XXXI. Theorem XXIII.
To find the place of a body moving in a given elliptic trajectory at any aſſigned time.
Suppoſe A (Pl. 14. Fig. 2.) to be the principal vertex, S the focus, and O the centre of the ellipſis APB; and let P be the place of the body to be found. Produce OA to G, ſo as OG may be to OA as OA to OS. Erect the perpendicular GH; and about the centre O, with the interval OG, deſcribe the circle GEF; and on the ruler GH
