Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/215



There is no oval figure whoe area, cut off by right lines at pleaure, can be univerally found by means of equations of any number of finite terms and dimenions.

Suppoe 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 quare of that right line within the oval. By this motion that point will decribe 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 ditance of the point from the pole, which is proportional to this area, might be found by the ame equation, and therefore all the points of the piral might be found by a finite equation alo; and therefore the interection of a right line given in poition with the piral might alo be found by a finite equation. But every right line infinitely produced cuts a piral in an infinite number of points; and the equation by which any one intersection of two lines is found, at the ame time exhibits all their interections by as many roots, and therefore ries to as many dimenions as there are interactions. Becaue two circles mutually cut one