Lemma XXVIII.
There is no oval figure whoſe area, cut off by right lines at pleaſure, can be univerſally found by means of equations of any number of finite terms and dimenſions.
Suppoſe that within the oval any point is given, about which as a pole a right line is perpetually revolving, with an uniform motion, while in that right line a moveable point going out from the pole, moves always forward with a velocity proportional to the ſquare of that right line within the oval. By this motion that point will deſcribe a spiral with infinite circumgyrations. Now if a portion of the area of the oval cut off by that right line could be found by a finite equation, the diſtance of the point from the pole, which is proportional to this area, might be found by the ſame equation, and therefore all the points of the ſpiral might be found by a finite equation alſo; and therefore the interſection of a right line given in poſition with the ſpiral might alſo be found by a finite equation. But every right line infinitely produced cuts a ſpiral in an infinite number of points; and the equation by which any one intersection of two lines is found, at the ſame time exhibits all their interſections by as many roots, and therefore riſes to as many dimenſions as there are interactions. Becauſe two circles mutually cut one
