Page:Newton's Principia (1846).djvu/106

. 4. And therefore the forces are as the spaces described in the very beginning of the motion directly, and the squares of the times inversely.

. 5. And the squares of the times are as the spaces described directly, and the forces inversely.

If in comparing indetermined quantities of different sorts one with another, any one is said to be as any other directly or inversely, the meaning is, that the former is augmented or diminished in the same ratio with the latter, or with its reciprocal. And if any one is said to be as any other two or more directly or inversely, the meaning is, that the first is augmented or diminished in the ratio compounded of the ratios in which the others, or the reciprocals of the others, are augmented or diminished. As if A is said to be as B directly, and C directly, and D inversely, the meaning is, that A is augmented or diminished in the same ratio with $$\scriptstyle \mathrm{B\times C\times\frac{1}{D}}$$, that is to say, that A and $$\scriptstyle \mathrm{\frac{BC}{D}}$$ are one to the other in a given ratio.

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) $$\scriptstyle \mathrm{AB^{2}=AG\times BD}$$, and $$\scriptstyle \mathrm{A}b^{2}=\mathrm{A}g\times bd$$; and therefore the ratio of AB² to Ab² is compounded of the ratios of AG to Ag, and of Bd to bd. But because GJ may be assumed of less length than any assignable, the ratio 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 of AB² to Ab² may be such as to differ from the ratio of BD to bd by less than any assignable difference. There fore, by Lem. I, the ultimate ratio of AB² to Ab² is the same with the ultimate ratio of BD to bd. Q.E.D.

2. Now let BD be inclined to AD in any given angle, and the ultimate ratio of BD to bd will always be the same as before, and therefore the same with the ratio of AB² to Ab². Q.E.D.