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

 another in two points, one of thoe interections is not to be found but by an equation of two dimenions, by which the other interections may alo found. Becaue there may be four interections of two conic ections, any one of them is not to be found univerally but by an equation of four dimenions, by which they may be all found together. For if thoe interection everally ought, becaue the law and condition of all is the ame. the calculus will be the ame in every cae, and therefore the concluion always the ame, which mut therefore comprehend interections at once within itelf, and exhibit them all indifferently. Hence it is that the interections of the conic ections with the curves of the third order, becaue they may amount to nine, come out together by equations of ix dimenions and the interections of two curves of third order, becaue they may amount to nine, come out together by equations of nine dimenions this did not necearily happen, we might reduce all olid to plane problems and thoe higher than olid to olid problems. But here I peak 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 ingle curve, but compoed of two or more, whoe interections may be everally found by different calculues. After the ame manner the two interections of right lines with the conic ections come out always by equations of two dimenions; the three interactions of right lines with the irreducible curves of the third order by equations of dimenions; the four interections of right lines with the irreducible curves of the fourth order,