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

Lemma XI.

The evaneſcent ſubtenſe 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 ſubtenſe of the conterminate arc. Pl. 2. Fig 4.

Case 1. Let AB be that arc, AD its tangent, BD the ſubtenſe of the angle of contact perpendicular on the tangent, AB the ſubtenſe of the arc. Draw BG perpendicular to the ſubtenſe 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 interſection of the lines BG, AG, when the points D, B, have come to A. It is evident that the distance GJ may be leſs than any aſſignable. 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 aſſum'd of leſs length than any aſſignable, the ratio of equality of AG to Ag may be ſuch as to differ from the ratio of equality by leſs than any aſſignable difference; and therefore the ratio $\scriptstyle {AB^{2}}$ to $\scriptstyle {Ab^{2}}$ may be ſuch as to differ from the ratio of BD to bd by leſs than any aſſignable difference. Therefore by Lem. 1, the ultimate ratio of $\scriptstyle {AB^{2}}$ to $\scriptstyle {Ab^{2}}$ is the ſame with the ultimate ratio of BD to bd. Q.E.D.

Case 2. Now let BD be inclined to AD in any given angle, and the ultimate ratio of BD to bd will