Page:Newton's Principia (1846).djvu/449



To a given time to find the inclination of the moon's orbit to the plant of the ecliptic.

Let AD be the sine of the greatest inclination, and AB the sine of the least. Bisect BD in C; and round the centre C, with the interval BC, describe the circle BGD. In AC take CE in the same proportion to EB



as EB to twice BA. And if to the time given we set off the angle AEG equal to double the distance of the nodes from the quadratures, and upon AD let fall the perpendicular GH, AH will be the sine of the inclination required.

For GE² is equal to GH² + HE² = BHD + HE² = HBD + HE² - BH² = HBD + BE² - 2BH $$\scriptstyle \times$$ BE = BE² + 2EC $$\scriptstyle \times$$ BH = 2EC $$\scriptstyle \times$$ AB + 2EC $$\scriptstyle \times$$ BH = 2EC $$\scriptstyle \times$$ AH; wherefore since 2EC is given, GE² will be as AH. Now let AEg represent double the distance of the nodes from the quadratures, in a given moment of time after, and the arc Gg, on account of the given angle GEg, will be as the distance GE. But Hh is to Gg as GH to GC, and, therefore, Hh is as the rectangle GH $$\scriptstyle \times$$ Gg, or GH $$\scriptstyle \times$$ GE, that is, as $$\scriptstyle \frac{GH}{GE}\times$$ GE², or $$\scriptstyle \frac{GH}{GE}\times$$ AH; that is, as AH and the sine of the angle AEG conjunctly. If, therefore, in any one case, AH be the sine of inclination, it will increase by the same increments as the sine of inclination doth, by Cor. 3 of the preceding Prop. and therefore will always continue equal to that sine. But when the point G falls upon either point B or D, AH is equal to this sine, and therefore remains always equal thereto. Q.E.D.

In this demonstration I have supposed that the angle BEG, representing double the distance of the nodes from the quadratures, increaseth uniformly; for I cannot descend to every minute circumstance of inequality. Now suppose that BEG is a right angle, and that Gg is in this case the horary increment of double the distance of the nodes from the sun; then, by Cor. 3 of the last Prop. the horary variation of the inclination in the same case will be to 33″ 10‴ 33iv. as the rectangle of AH, the sine of the inclination, into the sine of the right angle BEG, double the distance of the nodes from the sun, to four times the square of the radius; that is, as AH,