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

 given OAB is compleated) the new ordinate radius.

I ay, then, that if the point G is placed in a right line given by poition, the point g will be alo placed in a right line given by poition. If the point G is placed in a conic ection, the point will be likewie placed in a conic ection. And here I undertand the circle as one of the conic ections. But farther, if the point G is placed in a line of the third analytical order, the point g will alo be placed in a line of the third order, and o on in curve lines of higher orders. The two lines in which the points G, g, are placed, will be always of the ame analytical order. For as ad is to OA, o 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 abcia AD and the ordinate DG is expreed, thoe in determined lines AD and DG rie no higher than to one dimenion, 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 abcia ad and new ordinate dg rie only to one dimenion; and which therefore mut denote a right line, But if AD and DG (or either of them) had rien to two dimenions in the firt equation, ad and dg would likewie have rien to two dimenions in the econd equation. And o on in three or more dimenions. The indetermined