(c) The "annual equation" is a fluctuation introduced into the other perturbations by reason of the varying distance of the disturbing body, the sun, at different seasons of the year. Its magnitude plainly depends simply on the excentricity of the earth's orbit.
Both these perturbations, (b) and (c), Newton worked out completely.
(d) and (e) Next come the retrogression of the nodes and the variation of the inclination, which at the time were being observed at Greenwich by Flamsteed, from whom Newton frequently, but vainly, begged for data that he might complete their theory while he had his mind upon it. Fortunately, Halley succeeded Flamsteed as Astronomer-Royal [see list at end of notes above], and then Newton would have no difficulty in gaining such information as the national Observatory could give.
The "inclination" meant is the angle between the plane of the moon's orbit and that of the earth. The plane of the earth's orbit round the sun is called the ecliptic; the plane of the moon's orbit round the earth is inclined to it at a certain angle, which is slowly changing, though in a periodic manner. Imagine a curtain ring bisected by a sheet of paper, and tilted to a certain angle; it may be likened to the moon's orbit, cutting the plane of the ecliptic. The two points at which the plane is cut by the ring are called "nodes"; and these nodes are not stationary, but are slowly regressing, i.e. travelling in a direction opposite to that of the moon itself. Also the angle of tilt is varying slowly, oscillating up and down in the course of centuries.
(f) The two points in the moon's elliptic orbit where it comes nearest to or farthest from the earth, i.e. the points at the extremity of the long axis of the ellipse, are called separately perigee and apogee, or together "the apses." Now the pull of the sun causes the whole orbit to slowly revolve in its own plane, and consequently these apses "progress," so that the true path is not quite a closed curve, but a sort of spiral with elliptic loops.
But here comes in a striking circumstance. Newton states with reference to this perturbation that theory only accounts for 1½° per annum, whereas observation gives 3°, or just twice as much.
This is published in the Principia as a fact, without comment. It was for long regarded as a very curious thing, and many great mathematicians afterwards tried to find an error in the working. D'Alembert, Clairaut, and others attacked the problem, but were led to just the same result. It constituted the great outstanding difficulty in the way of accepting the theory of gravitation. It was suggested that perhaps the inverse square law was only a first approximation; that perhaps a more complete expression, such as
| A | + | B | , |
| r2 | r4 |
must be given for it; and so on.
Ultimately, Clairaut took into account a whole series of neglected terms, and it came out correct; thus verifying the theory.