8FC. Vl.J OF NATURAL PHILOSOPHY. 155 terscctions is not to be found but by an equation of two dimensions, fo which the other intersection may be also found. Because there may b(four intersections of two conic sections, any one of them is not to be found universally, but by an equation of four dimensions, by which they may bi> all found together. For if those intersections are severally sought, be cause the law and condition of all is the same, the calculus will be the same in every case, and therefore the conclusion always the same, which must therefore comprehend all those intersections at once within itself, and exhibit them all indifferently. Hence it is that the intersections of the conic se"f ions with the curves of the third order, because they may amount to six, (\,me out together by equations of six dimensions ; and the inter sections of two curves of the third order, because they may amount to nine, come out together by equations of nine dimensions. If this did not ne cessarily happen, we might reduce all solid to plane Problems, and those higher than solid to solid Problems. But here i speak of curves irreduci ble in power. For if the equation by which the curve is defined may bo reduced to a lower power, the curve will not be one single curve, but com posed of two, or more, whose intersections may be severally found by different calculusses. After the same manner the two intersections of right lines with the conic sections come out always by equations of two dimensions ; the three intersections of right lines with the irreducible curves of the third urder by equations of three dimensions ; the four intersections of right lines with the irreducible curves of the fourth order, by equations of four dimensions ; and so on in iitfinitum. Wherefore the innumerable inter sections of a right line with a spiral, since this is but one simple curve and not reducible to more curves, require equations infinite in r- .imber 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 intersec tions 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 equa tion will return to its first form ; and consequently one and the same equation will exhibit all the intersections, and will therefore have an infi nite 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 an^ such equation.