Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/216

another in two points, one of thoſe interſections is not to be found but by an equation of two dimenſions, by which the other interſections may alſo found. Becauſe there may be 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 be all found together. For if thoſe interſection ſeverally ſought, becauſe 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 interſections at once within itſelf, and exhibit them all indifferently. Hence it is that the interſections of the conic ſections with the curves of the third order, becauſe they may amount to nine, come out together by equations of ſix dimenſions and the interſections of two curves of third order, becauſe they may amount to nine, come out together by equations of nine dimenſions this did not neceſſarily happen, we might reduce all ſolid to plane problems and thoſe higher than ſolid to ſolid problems. But here I ſpeak of curves irreducible in power. For if the equations by which the curve is defined may be reduced to a lower power, the curve will not be only ſingle curve, but compoſed 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 interactions of right lines with the irreducible curves of the third order by equations of dimenſions; the four interſections of right lines with the irreducible curves of the fourth order,