Newton demonstrated, in Book III of his Principia, that the deviations of the lunar orbit from a Keplerian ellipse are due to the
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 complete account of these
deviations, due to the complexity of the equations of motion which arise in a system of three mutually
gravitating bodies (see Chapter 13). In fact, Clairaut (1713-1765) is generally credited with the first reasonably accurate and complete theoretical explanation of the Moon's orbit.
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 on, he realized that he could indeed account for the precession, in terms of
standard Newtonian dynamics, by
continuing his expansion in small parameters to higher order. 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 (1795-1824), J. Herschel (1792-1871), Airy (1801-1892), Delaunay (1816-1872), G.W. Hill (1836-1914), and E.W. Brown (1836-1938). 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
longitude at sea. 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), most of which were
published at public expense. Finally, it is worth noting that the development of lunar theory gave rise to many
significant advances in mathematics and dynamics, and was, in many ways, a precursor to the discovery of
chaotic motion in the 20th century (see Chapter 15).

This chapter contains an introduction to lunar theory in which approximate expressions for evection, variation,
the precession of the perigee, and the regression of the ascending node are derived from the laws of Newtonian dynamics.^{}