Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/184

given OAB is compleated) the new ordinate radius.

I ſay, then, that if the point G is placed in a right line given by poſition, the point g will be alſo placed in a right line given by poſition. If the point G is placed in a conic ſection, the point will be likewiſe placed in a conic ſection. And here I underſtand the circle as one of the conic ſections. But farther, if the point G is placed in a line of the third analytical order, the point g will alſo be placed in a line of the third order, and ſo on in curve lines of higher orders. The two lines in which the points G, g, are placed, will be always of the ſame analytical order. For as ad is to OA, ſo are ad to OD, dg to DG, and AB to AD; and therefore AD is equal to $\textstyle {\frac {OA\times AB}{ad}}$ and DG equal to $\textstyle {\frac {OA\times dg}{ad}}$. Now if the point G is placed in a right line: and therefore, in any equation by which the relation between the abſciſſa AD and the ordinate DG is expreſſed, thoſe in determined lines AD and DG riſe no higher than to one dimenſion, by writing this equation $\textstyle {\frac {OA\times AB}{ad}}$ in place of AD, and $\textstyle {\frac {OA\times dg}{ad}}$ in place of DG, a new equation will be produced, in which the new abſciſſa ad and new ordinate dg riſe only to one dimenſion; and which therefore muſt denote a right line, But if AD and DG (or either of them) had riſen to two dimenſions in the firſt equation, ad and dg would likewiſe have riſen to two dimenſions in the ſecond equation. And ſo on in three or more dimenſions. The indetermined