the earth it is half the
time nearer to the sun than the earth is, and so gets pulled more than
the average, while for the other fortnight it is further from the sun
than the earth is, and so gets pulled less. For the week during which
it is changing from a decreasing half to a new moon it is moving in the
direction of the extra pull, and hence becomes new sooner than would
have been expected. All next week it is moving against the same extra
pull, and so arrives at quadrature (half moon) somewhat late. For the
next fortnight it is in the region of too little pull, the earth gets
pulled more than it does; the effect of this is to hurry it up for the
third week, so that the full moon occurs a little early, and to retard
it for the fourth week, so that the decreasing half moon like the
increasing half occurs behind time again. Thus each syzygy (as new and
full are technically called) is too early; each quadrature is too late;
the maximum hurrying and slackening force being felt at the octants, or
intermediate 45 deg. points.

(_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 regr