Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/97

 Rh The evanecent ubtene of the angle of contact, in all curves which at the point of contact here have a finite curvature, is ultimately in the duplicate ratio of the ubtene of the conterminate arc. Pl. 2. Fig 4.

Let AB be that arc, AD its tangent, BD the ubtene of the angle of contact perpendicular on the tangent, AB the ubtene of the arc. Draw BG perpendicular to the ubtene AB, and AG to the tangent AD, meeting in G; then let the points D, B and G approach to the points d, b and g, and suppose J to be the ultimate interection of the lines BG, AG, when the points D, B, have come to A. It is evident that the distance GJ may be les than any aignable. But (from the nature of the circles passing through the points A, B, G; A, b, g) $$\scriptstyle{AB^2 = AG \times BD}$$, and $$\scriptstyle{Ab^2 = Ag \times bd}$$; and therefore the ratio $$\scriptstyle{AB^2}$$ to $$\scriptstyle{Ab^2}$$ is compounded of the ratio's of AG to Ag, and of BD to bd. But because GJ may be aum'd of les length than any aignable, the ratio of equality of AG to Ag may be uch as to differ from the ratio of equality by les than any aignable difference; and therefore the ratio $$\scriptstyle{AB^2}$$ to $$\scriptstyle{Ab^2}$$ may be uch as to differ from the ratio of BD to bd by les than any aignable difference. Therefore by Lem. 1, the ultimate ratio of $$\scriptstyle{AB^2}$$ to $$\scriptstyle{Ab^2}$$ is the ame with the ultimate ratio of BD to bd. Q.E.D.

Now let BD be inclined to AD in any given angle, and the ultimate ratio of BD to bd will