lines ad, dg in the ſecond equation, and AD, DG, in the firſt will always riſe to the ſame number of dimenſions; and therefore the lines in which the points G, g, are placed are of the ſame analytical order.
I ſay farther, that if any right line touches the curve line in the firſt figure, the ſame right line tranſferred the ſame way with the curve into the new figure, will touch that curve line in the new figure. and vice verſa. For if any two points of the curve in the firſt figure are ſuppoſed to approach one the other till they come to coincide; the ſame points tranſferred will approach one the other till they come to coincide in the new figure; and therefore the lines with which thoſe points are joined will come together tangents of the curves in both figures. I might have given demonſtrations of theſe aſſertions in a more geometrical form; but I ſtudy to be brief.
Wherefore if one rectilinear figure is to be tranſformed into another we need only tranſfer the interſections of the right lines of which the firſt figure conſiſts, and through the tranſferred interſections to draw right lines in the new figure. But if a curvilinear figure is to be tranſformed we muſt tranſfer the points, the tangents, and other right lines, by means of which the curve line is defined. This lemma is of uſe in the ſolution of the more difficult problems. For thereby we may tranſform the propoſed figures if they are intricate into others that are more ſimple, Thus any right lines converging to a point are tranſformed into parallels; by taking for the firſt ordinate radius any right line that paſſes through the point of concourſe of the converging lines, and that, beauſe their point of concourſe is by this means made
