essing, _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 1-1/2 deg. per annum,
whereas observation gives 3 deg., 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
---- + -----,
r^2 r^4

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.

But the strangest part of this tale is to come. For only a few years
ago, Prof. Adams, of Cambridge (Neptune Adams, as he is called), was
editing various old papers of Newton's, now in the possession of the
Duke of Portland, and he found manuscripts bearing on this very point,
and discovered that Newton had reworked out the calculations himself,
had found the cause of the error, had taken into account the terms
hitherto neglected, and so, fifty years before Clairaut, had completely,
though not publicly, solved this long outstanding problem of the
progression of the apses.

(_g_) and (_h_) Two other inequalities he calculated out and predicted,
viz. variation in the motions of the apses and the nodes. Neither of
these had then been observed, but they were afterwards detected and
verified.

A good many other minor irregularities are now known--some thirty, I
believe; and altogeth