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Newton Principia, Book I, Corollary 2 to Proposition 45. Waff (1976) gives a detailed discussion of Newton's work on the motion of the lunar apogee. The advance of the apsidal line and the retrogression of the nodes are simple consequences of the differing lengths of the sidereal, anomalistic, and draconitic months (Ts, Ta, and Td, for example). We have Ta/ Ts — 1 ^ 0.008 53, so each sidereal month the perigee of the Moon advances by about 0.008 53 x 360° ^ 3° 4'. However, Td/Ts — 1 ^ —0.00402, so the nodes move backwards by about 1° 27' per sidereal month, or equivalently by about 19° 21' per sidereal year.

parameters that were used justified using the theory of gravity. Newton's lunar theory was thus far from perfect, and he was well aware of the fact. But in view of the immense difficulty of the problem he was considering, and the influence his ideas and methods had on those who followed in his footsteps, Newton's lunar theory should be considered as one of the great achievements of the Principia.

The lunar theory actually appears in the Principia in two quite different ways. In Book I, Section 11 (and notably in Proposition 66 and its twenty-two corollaries) Newton treats different aspects of the general problem of three mutually interacting bodies qualitatively. The Moon is not mentioned specifically, but it is clear from the choice of letters in his diagrams that it was the Moon for which the theory was intended. The general theory was then applied to the specific case of the Moon's motion in Book III, some of the calculations having been described above.

But Newton used also his perturbation theory to tackle three other problems, always starting from a diagram similar to that shown in Figure 8.8. By taking E to be the Sun, and S and M to be two different planets, Newton was able to consider the effect of one planet on another's orbit. The relative masses of the bodies that make up the Solar System showed that, apart from in the Sun-Earth-Moon system, the perturbing effects would be small, except perhaps in the case of Jupiter and Saturn. Newton did attempt to quantify this latter effect, but with little success.

Another application of the technique was to explain the tides. Let M be part of a belt of matter encircling the Earth E and then let this ring shrink until it is contiguous with the Earth. If the belt is fluid, it can be thought of as representing the seas, and Newton assumed that the motion of the fluid would be governed by the same laws as governed the motion of a solid particle. The theory then showed that the basic tidal phenomena were the combined effect of the perturbing effect of the Sun and the Moon. Notwithstanding the fact that the dynamical theory Newton developed was the first that could predict quantitatively the rise and fall of the oceans with tolerable accuracy, it is fundamentally wrong. Newton's analysis was based solely on the component of the perturbing force that was parallel to the gravitational attraction (the vertical component), but the dynamics of tidal motion are in fact determined mainly by the horizontal motion of the seas, which in turn is determined by the horizontal component of the tide-generating force.80

80 See Palmieri (1998) for more details.

Finally, and most remarkably, Newton used his theory to explain the precession of the equinoxes, a phenomenon for which no one had previously suggested anything close to a correct explanation. He realized that the gravitational attraction of the Sun and Moon on the extra material around the equator of the Earth would result in its axis rotating in a small conical motion, like a top. He computed its effect as 50" per year, agreeing with the accepted value of the time. Newton's solution to this problem was imperfect - hardly surprising, since no theory of rigid body dynamics existed81 - but represented one of the more unexpected successes of gravitation theory. George Airy later said:

... if at this time we might presume to select the part of the Principia which probably astonished and delighted and satisfied its readers more than any other, we should fix, without hesitation, on the explanation of the precession of the

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