by equations of four dimenſions, and ſo on in infinitum. Wherefore the innumerable interſections of a right line with a ſpiral, ſince this is but one ſimple curve. and not reducible to more curves, require equations infinite in number of dimenſions and roots, by which they may be all exhibited together. For the law and calculus of all is the ſame. For if a perpendicular is let fall from the pole upon that interſecting right line, and that perpendicular together with the interſecting line revolves about the pole, the interſections of the ſpiral will mutually paſs the one into the other; and that which was firſt or neareſt, after one revolution, will be the ſecond, after two, the third, and ſo on; nor will the equation in the mean time be changed, but as the magnitudes of thoſe quantities are changed, by which the poſition of the interſecting line is determined. Wherefore ſince thoſe quantities after every revolution return to their firſt magnitudes, the equation will return to its firſt form, and conſequently one and the ſame equation will exhibit all the interſections, and will therefore have an infinite number of roots, by which they may be all exhibited. And therefore the interſection of a right line with a ſpiral cannot be univerſally found by any finite equation; and of conſequence there is no oval figure whoſe area, cut off by right lines at pleaſure, can be univerſally exhibited by any ſuch equation.
By the ſame argument, if the interval of the pole and point by which the ſpiral is deſcribed, is taken proportional to that part of the perimeter of the oval which is cut off; it may be proved that the length of the perimeter cannot be univerſally exhibited by any finite equation. But here I ſpeak
