Lemma IX.
If a right line AE and a curve line ABC, both given by poſition, 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 ſay that the areas of the triangles ABD, ACE, will ultimately be one to the other in the duplicate ratio of the ſides.
For while the points B, C approach towards the point A, ſuppose always AD to be produced to the remote points d and e, ſo 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 ſimilar to the curve ABC, and draw the right line Ag ſo as to touch both curves in A, and cut the ordinates DB, EC, db, ec, in F, G, f, g. Then ſupposing the length Ae to remain the ſame, let the points B and C meet the point A; and the angle cAg vaniſhing, 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
