Newton demonstrated, in Book III of his *Principia*, that the deviations of the lunar orbit from a Keplerian ellipse are due to the residual
gravitational influence of the Sun, which is sufficiently large that it is not completely negligible compared with the mutual
gravitational attraction of the Earth and the Moon.
However, Newton was not able to give a full account of these
deviations (in the *Principia*), because of the complexity of the equations of motion that arise in a system of three mutually
gravitating bodies. (See Chapter 9.) In fact, Alexis Clairaut (1713–1765) is generally credited with the first reasonably accurate and complete theoretical explanation of the Moon's orbit to be published.
His method of calculation makes use of an expansion of the lunar equations of motion in terms of various small parameters.
Clairaut, however, initially experienced difficulty in accounting for the precession of the lunar perigee. Indeed, his first calculation
overestimated the period of this precession by a factor of about two, leading him to question Newton's
inverse-square law of gravitation. Later, he realized that he could account for the precession in terms of
standard Newtonian dynamics by
continuing his expansion in small parameters to higher order. (See Section 11.17.) After Clairaut, the theory of lunar motion was further elaborated in major works by
D'Alembert (1717–1783), Euler (1707–1783), Laplace (1749–1827), Damoiseau (1768–1846), Plana (1781–1864), Poisson (1781–1840), Hansen (1795–1874),
De Pontécoulant (1795–1874), J. Herschel (1792–1871), Airy (1801–1892), Delaunay (1816–1872), G.W. Hill (1836–1914), and E.W. Brown (1836–1938) (Brown 1896). The fact that so many celebrated mathematicians and astronomers devoted so much time and effort to lunar
theory is a tribute to its inherent difficulty, as well as its great theoretical and practical interest. Indeed, for a period
of about one hundred years (between 1767 and about 1850) the
so-called *method of lunar distance* was the principal means used by mariners to determine
terrestrial longitude at sea (Cotter 1968). This method depends crucially on a precise knowledge of the position of the
Moon in the sky as a function of time. Consequently, astronomers and mathematicians during the
period in question were spurred to make ever more accurate observations of the Moon's orbit, and to develop lunar theory
to greater and greater precision. An important outcome of these activities were various tables of lunar motion (e.g., those
of Mayer, Damoiseau, Plana, Hansen, and Brown), the majority of which were
published at public expense.

This chapter contains an introduction to lunar theory in which, amongst other things, approximate expressions for evection, variation, the precession of the perigee, and the regression of the ascending node, are derived from the laws of Newtonian mechanics. Further information on lunar theory can be obtained from Godfray (1853), Brown (1896), Adams (1900), and Cook (1988).