No celestial body has required as much labor for the study of its motion as the moon. Since Clairault (1747), who indicated a way of constructing a theory containing all the properties of the motion of the Moon, the greatest mathemeticians have dealt with this very difficult problem in celestal mechanics. After d'Alembert, Laplace dealt with a theory of the Moon for more than thirty years. He not only developed a general method for computing the inequalities produced by the attraction of the Sun, in which his predecessors had also been engaged, but he was able to determine for the first time the perturbations caused by the figure of the Earth and the attraction of the planets. One of Laplace's remarkable discoveries was his explanation of the secular acceleration in the motion of the Moon which had been discovered previously by observations. He showed that the cause of the secular acceleration is the secular reduction in the eccentricity of the terrestrial orbit. (Subsequently, this explanation turned out to be insufficient, since it accounted for only 0.8 of the value of the acceleration. The remaining part is explained by the slowing down of the rotation of the earth through tidal friction.) The secular acceleration in the motion of the perigee and in the motion of the node of the lunar orbit which he discovered has the same cause. Laplace's studies, which made up the content of the third volume of his "Mechanique Celeste" (1802) were continued by Damoiseau, who computed the numerical values of the inequalities with great accuracy. The theory of the Moon was developed further by Pontecoulant (1846). Euler (1753) found a new way to to construct a theory of the Moon-- the method of variation of arbitrary constants, or, in other words, the method of an osculating orbit. A considerable development of this idea made it possible for Delaunay to form a very complete letter-symbol theory of the solar inequalities in the motion of the Moon (1846). To give an idea of the amount of work on the theory of the Moon, let us note that it was only after twenty years of research Delaunay was able to obtain a general expression for all the inequalities up to the seventh order relative to the perturbing forces. In the further work of Radeau and Andoillet, Delaunay's theory was improved and used to construct tables of the motion of the moon. Poisson (1835), Puiseux (1864), and M.A. Vil'yev (1919) engaged in the development of Euler's method, namely in the determination of the osculating elements of the lunar orbit. In addition to the method of variation of arbitrary constants, Euler advanced another fruitful idea, which found broad application in subsequent work and which has led as a result to the present extremely accurate tables of the Moon. This idea consists of expanding the coordinates of the Moon into series arranged in powers of the eccentricities of the lunar-solar orbit and of the inclination of the lunar orbit, ie, a series of the type A + +/Be**i + +/Ce**i + +/Dee' + ... + Fi**2 + ... , where the coeffecients are periodic functions. This idea was only used 100 years later by Hill in a number of memoirs starting in 1877. The theory of the motion of the Moon worked out by Hill was then carried by Brown to its present state of completeness and served to set up tables, from which the ephemeris of the Moon has been calculated and published in almanacs since 1923. After 1882, the Nautical Almanac (and, following its example, the greater part of the other almanacs) used Hansen's tables to compute the lunar ephemeris. However, large deviations from them were discovered so that soon after 1883 Newcomb's empirical corrections were introduced into Hansen's tables, first only for the right ascension and declination. After 1897, Newcomb's corrections were also introduced into the parallax and the radius, and, after 1915, into the geocentric elliptic longitude and latitude. However, the mean longitude of the Moon remained without any corrections. Everything came into order only in 1923, when, as was said above, Brown's tables were adopted as the basis of the ephemeris. The chief inequalities of the Moon were discovered already by Ptolemy and Tycho Brahe (about 1580) namely: evection, variation, and annual variation. The terms depending on the eccentricity are called elliptic terms. An idea of the relative size of these inequalities is given by the following equation: True longitude of Moon = mean longitude + 377' sin M + 13' sin 2M + ... + 76' sin (2D - M) + 39' sin 2D - 11' sin M' - 2' sin D + ... where M and M' are the mean anomalies of the Moon and the Sun, M being measured from the mean position of the perigee; and D is the difference between the mean longitudes of the Moon and the Sun. The term with the argument (2D - M) has received the name of "evection". Its period is equal to 31.8 days. The term with argument 2D is called the "variation", its period being equal to 14.7 days. The terms with arguments M, 2M, and 3M are called "elliptic". Their sum gives the "equation of the center". The term with argument M' is called the "annual variation". Its period is 1 year and depends on the ellipticity of the terrestrial orbit. The terms with arguments D, 3D, ... are called the "parabolic inequality", with periods of 29.5306 days, 9.8435 days, etc, respectively. There are analogous terms in the expansions of the radius vector and the latitude of the Moon. These inequalities have a periodical nature. However, in two of the elements of the orbit--the longitude of the perigee and the longitude of the node--there are secular terms in addition to the periodic terms, ie, terms which change gradually. The perigee performs a complete revolution in 8.8503 years, and the line of nodes moves in the opposite direction, performing a complete revolution in 18.5995 years. Brown computed 155 periodic terms with coefficients greater than 0.1" in the expression for the lunar longitude, and more than 500 terms with smaller coefficients. To compute the longitude of the Moon with an accuracy within 0.1", it is necessary to add together 655 terms. To compute the latitude, about 300 terms are sufficient.