Page:Newton's Principia (1846).djvu/429

PROPOSITION XXVIII. PROBLEM IX.

To find the diameters of the orbit, in which, without eccentricity, the moon would move.

The curvature of the orbit which a body describes, if attracted in lines perpendicular to the orbit, is as the force of attraction directly, and the square of the velocity inversely. I estimate the curvatures of lines compared one with another according to the evanescent proportion of the sines or tangents of their angles of contact to equal radii, supposing those radii to be infinitely diminished. But the attraction of the moon towards the earth in the syzygies is the excess of its gravity towards the earth above the force of the sun 2PK (see Fig. Prop. XXV), by which force the accelerative gravity of the moon towards the sun exceeds the accelerative gravity of the earth towards the sun, or is exceeded by it. But in the quadratures that attraction is the sum of the gravity of the moon towards the earth, and the sun's force KT, by which the moon is attracted towards the earth. And these attractions, putting N for $\scriptstyle {\frac {AT+CT}{2}}$, are nearly as $\scriptstyle {\frac {178725}{AT^{2}}}-{\frac {2000}{CT\times N}}$ and $\scriptstyle {\frac {178725}{CT^{2}}}+{\frac {1000}{AT\times N}}$, or as 178725N $\scriptstyle \times$ CT² - 2000AT² $\scriptstyle \times$ CT, and 178725N $\scriptstyle \times$ AT² + 1000CT² $\scriptstyle \times$ AT. For if the accelerative gravity of the moon towards the earth be represented by the number 178725, the mean force ML, which in the quadratures is PT or TK, and draws the moon towards the earth, will be 1000, and the mean force TM in the syzygies will be 3000; from which, if we subtract the mean force ML, there will remain 2000, the force by which the moon in the syzygies is drawn from the earth, and which we above called 2PK. But the velocity of the moon in the syzygies A and B is to its velocity in the quadratures C and D as CT to AT, and the moment of the area, which the moon by a radius drawn to the earth describes in the syzygies, to the moment of that area described in the quadratures conjunctly; that is, as 11073CT to 10973AT. Take this ratio twice inversely, and the former ratio once directly, and the curvature of the orb of the moon in the syzygies will be to the curvature thereof in the quadratures as 120406729 $\scriptstyle \times$ 178725AT² $\scriptstyle \times$ CT² $\scriptstyle \times$ N - 120406729 $\scriptstyle \times$ 2000AT⁴ $\scriptstyle \times$ CT to 122611329 $\scriptstyle \times$ 178725AT² $\scriptstyle \times$ CT² $\scriptstyle \times$ N + 122611329 $\scriptstyle \times$ 1000CT⁴ $\scriptstyle \times$ AT, that is, as 2151969AT $\scriptstyle \times$ CT $\scriptstyle \times$ N - 24081AT³ to 2191371AT $\scriptstyle \times$ CT $\scriptstyle \times$ N + 12261CT³.

Because the figure of the moon's orbit is unknown, let us, in its stead, assume the ellipsis DBCA, in the centre of which we suppose the earth to be situated, and the greater axis DC to lie between the quadratures as the lesser AB between the syzygies. But since the plane of this ellipsis is revolved about the earth by an angular motion, and the orbit, whose curvature we now examine, should be described in a plane void of such motion