  It was a real breakthrough when Newton discovered that the gravitational forces of the Sun and Moon acting on an oblately spheroidal Earth would produce a wobble of the Earth that, at least qualitatively, could account for the precession of the Equinoxes.  No physical explanation for this phenomenon had been proposed before. Newton faced a problem, however, in trying to carry out a quantitative derivation of the precession: he knew the magnitude of the gravitational action of the Sun on the Earth but not that of the Moon, for he could not obtain the mass of the Moon in the way he had for the Sun, Jupiter, and Saturn insofar as no bodies orbit the Moon. Propositions 36 and 37 endeavor to infer the force of the Moon on the Earth from the difference in the heights of the tides when the Sun and Moon are in conjunction and in opposition. In the first edition Newton managed to derive a value for the rate of the precession in good agreement with the known value, but during the quarter century between the first and second editions he concluded that the value he had used for the Moon's force (6 and 1/3 times the Sun's force) was much too large. The derivation of the precession was therefore extensively revised in the second edition, using a new value for the Moon's force (4.4815 times the Sun's force, still more than a factor of 2 greater than the correct value). In all editions the derivation in Proposition 39 treated the wobble not directly as the motion of a rigid body, but by analogy with the motion of the lunar nodes. By the standards of our present physics, no part of Book 3 is further off-base than Newton's solution for the precession. The phenomenon, however, subsequently provided important evidence for Newton's theory of gravity when d'Alembert in 1749 carried out a successful derivation based on rigid body motion and a correct value of the Moon's force derived from the then recently discovered phenomenon of the nutation of the Earth.