Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/217

 by equations of four dimenions, and o on in infinitum. Wherefore the innumerable interections of a right line with a piral, ince this is but one imple curve. and not reducible to more curves, require equations infinite in number of dimenions and roots, by which they may be all exhibited together. For the law and calculus of all is the ame. For if a perpendicular is let fall from the pole upon that interecting right line, and that perpendicular together with the interecting line revolves about the pole, the interections of the piral will mutually pas the one into the other; and that which was firt or nearet, after one revolution, will be the econd, after two, the third, and o on; nor will the equation in the mean time be changed, but as the magnitudes of thoe quantities are changed, by which the poition of the interecting line is determined. Wherefore ince thoe quantities after every revolution return to their firt magnitudes, the equation will return to its firt form, and conequently one and the ame equation will exhibit all the interections, and will therefore have an infinite number of roots, by which they may be all exhibited. And therefore the interection of a right line with a piral cannot be univerally found by any finite equation; and of conequence there is no oval figure whoe area, cut off by right lines at pleaure, can be univerally exhibited by any uch equation.

By the ame argument, if the interval of the pole and point by which the piral is decribed, 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 univerally exhibited by any finite equation. But here I peak