Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/94

 If a right line AE and a curve line ABC, both given by poition, cut each other in a given angle; and to that right line, in another given angle, BD, CE are ordinately applied, meeting the curve in B, C; and the points B and C together approach towards, and meet in, the point A: I ay that the areas of the triangles ABD, ACE, will ultimately be one to the other in the duplicate ratio of the ides.

For while the points B, C approach towards the point A, uppose always AD to be produced to the remote points d and e, o as Ad, Ae may be proportional to AD, AE; and the ordinates db, ec, to be drawn parallel to the ordinates DB and EC, and meeting AB and AC produced in b and c. Let the curve Abc be imilar to the curve ABC, and draw the right line Ag o as to touch both curves in A, and cut the ordinates DB, EC, db, ec, in F, G, f, g. Then upposing the length Ae to remain the ame, let the points B and C meet the point A; and the angle cAg vanihing, the curvilinear areas Abd, Ace will coincide with the rectilinear areas Afd, Age; and therefore (by Lem 5) will be one to other in the duplicate ratio of the sides Ad, Ae. But the areas ABD, ACE are always proportional to these areas; and so the sides AD, AE are to these sides. And therefore the areas ABD, ACE are ultimately