Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/185

 lines ad, dg in the econd equation, and AD, DG, in the firt will always rie to the ame number of dimenions; and therefore the lines in which the points G, g, are placed are of the ame analytical order.

I ay farther, that if any right line touches the curve line in the firt figure, the ame right line tranferred the ame way with the curve into the new figure, will touch that curve line in the new figure. and vice vera. For if any two points of the curve in the firt figure are uppoed to approach one the other till they come to coincide; the ame points tranferred will approach one the other till they come to coincide in the new figure; and therefore the lines with which thoe points are joined will come together tangents of the curves in both figures. I might have given demontrations of thee aertions in a more geometrical form; but I tudy to be brief.

Wherefore if one rectilinear figure is to be tranformed into another we need only tranfer the interections of the right lines of which the firt figure conits, and through the tranferred interections to draw right lines in the new figure. But if a curvilinear figure is to be tranformed we mut tranfer the points, the tangents, and other right lines, by means of which the curve line is defined. This lemma is of ue in the olution of the more difficult problems. For thereby we may tranform the propoed figures if they are intricate into others that are more imple, Thus any right lines converging to a point are tranformed into parallels; by taking for the firt ordinate radius any right line that paes through the point of concoure of the converging lines, and that, beaue their point of concoure is by this means made