The Precession of the Equinoxes

Proposition 39 Problem 20: Find the precession of the equinoxes.

The middle horary motion of the moon’s nodes in a circular orbit when the nodes are in quadratures was 16" 35’" 16 36

The half of which, 8″ 17‴ 38iv.18v. (for the reasons above explained) is the mean horary motion of the nodes in such an orbit, which motion in a whole sidereal year becomes 20° 11′ 46″.

Because, therefore, the nodes of the moon in such an orbit would be yearly transferred 20° 11′ 46″ in antecedentia; and, if there were more moons, the motion of the nodes of every one (by Cor. 16, Prop. LXVI. Book 1) would be as its periodic time; if upon the surface of the earth a moon was revolved in the time of a sidereal day, the annual motion of the nodes of this moon would be to 20° 11′ 46″ as 23h.56′, the sidereal day, to 27d.7h.43′, the periodic time of our moon, that is, as 1436 to 39343. And the same thing would happen to the nodes of a ring of moons encompassing the earth, whether these moons did not mutually touch each the other, or whether they were molten, and formed into a continued ring, or whether that ring should become rigid and inflexible.

Let us, then, suppose that this ring is in quantity of matter equal to the whole exterior earth PapAPepE, which lies without the sphere Pape (see fig. Lem. II); and because this sphere is to that exterior earth as aC² to AC² - aC², that is (seeing PC or aC the least semi-diameter of the earth is to AC the greatest semi-diameter of the same as 229 to 230), as 52441 to 459.

If this ring encompassed the earth round the equator, and both together were revolved about the diameter of the ring, the motion of the ring (by Lem. III) would be to the motion of the inner sphere as 459 to 52441 and 1000000 to 925275 conjunctly, that is, as 4590 to 485223.

Therefore, the motion of the ring would be to the sum of the motions of both ring and sphere as 4590 to 489813. Wherefore if the ring adheres to the sphere, and communicates its motion to the sphere, by which its nodes or equinoctial points recede, the motion remaining in the ring will be to its former motion as 4590 to 489813; upon which account the motion of the equinoctial points will be diminished in the same proportion.

Wherefore the annual motion of the equinoctial points of the body, composed of both ring and sphere, will be to the motion 20° 11′ 46″ as 1436 to 39343 and 4590 to 489813 conjunctly, that is, as 100 to 292369. But the forces by which the nodes of a number of moons (as we explained above).

Therefore, by which the equinoctial points of the ring recede (that is, the forces 3IT, in fig. Prop. XXX), are in the several particles as the distances of those particles from the plane QR;

By these forces the particles recede from that plane: and therefore (by Lem. II) if the matter of the ring was spread all over the surface of the sphere, after the fashion of the figure PapAPepE, in order to make up that exterior part of the earth, the total force or power of all the particles to wheel about the earth round any diameter of the equator, and therefore to move the equinoctial points, would become less than before in the proportion of 2 to 5.

Wherefore the annual regress of the equinoxes now would be to 20° 11′ 46″ as 10 to 73092; that is, would be 9″ 56‴ 50iv.

But because the plane of the equator is inclined to that of the ecliptic, this motion is to be diminished in the proportion of the sine 91706 (which is the co-sine of 23½ deg.) to the radius 100000; and the remaining motion will now be 9″ 7‴ 20iv. which is the annual precession of the equinoxes arising from the force of the sun.

But the force of the moon to move the sea was to the force of the sun nearly as 4,4815 to 1; and the force of the moon to move the equinoxes is to that of the sun in the same proportion. Whence the annual precession of the equinoxes proceeding from the force of the moon comes out 40″ 52‴ 52iv. and the total annual precession arising from the united forces of both will be 50″ 00‴ 12iv. the quantity of which motion agrees with the phaenomena; for the precession of the equinoxes, by astronomical observations, is about 50″ yearly.

If the height of the earth at the equator exceeds its height at the poles by more than 171⁄6 miles, the matter thereof will be more rare near the surface than at the centre; and the precession of the equinoxes will be augmented by the excess of height, and diminished by the greater rarity.