LUNAR THEORY Historical.—In ancient times the moon was a very impor tant body for night illumination and a great deal of attention was given to its motions. The observations made were of an exacti tude far in advance of those in other physical sciences. Hip parchus (2nd century B.c.) discovered the eccentricity of the moon's orbit and the motion of the apse. The inclination of the orbit to the ecliptic and the motion of the node were also deter mined by him. He explained the motion of the moon as uniform in a circular orbit with the earth placed eccentrically and this was amply sufficient for the observations of that time. The numer ical values were derived from observations of eclipses. For fixing the quantities which require the lapse of long intervals of time for their accurate evaluation he made use of the oldest eclipse observations available. At the times of eclipses the inequality in longitude due to eccentricity cannot be separated from an impor tant solar perturbation called the evection. Consequently, he got a value of the eccentricity in longitude (true value 6° •29) erro neous by the amount of the evection (0.27). The latter inequal ity was discovered by Ptolemy (1 A.D. 140). No great improve ment in the knowledge of the moon's motion was made, except for slightly improved accuracy in the mean motions till Tycho Brahe (1546-1601) discovered another inequality due to solar attraction, called the variation, which has greatest effect when the moon is or distant from the sun on either side. The coefficient of this term is o°.66. The explanation of all these inequalities and the construction of an adequate theory of the moon's motion had to wait the discovery of universal gravitation.
the mean motion of the perigee was only about half the observed value, but it appears from manuscripts that are extant that he succeeded later in explaining the observed motions within 8%. Newton's results were given in a geometrical form but he probably first obtained them by the method of fluctions. No substantial advance was made on Newton's work till Clairaut developed his analytical theory 6o years later. Clairaut's work was followed by that of d'Alembert and Euler in the eighteenth century. A new epoch in the theory was then opened by the work of Laplace.
The analytical methods which have been developed may be divided into three classes.
(I) Laplace and his immediate successors, especially Plana, effected the integration by expressing the time in terms of the moon's true longitude. Then, by inverting the series, the longi tude was expressed in terms of the time.
(2) By the second general method the moon's co-ordinates are obtained in terms of the time by the direct integration of the differential equations of motions, retaining as algebraic symbols the values of the various elements. Most of the elements are small numerical fractions : e the eccentricity of the moon's orbit, about 0.055; e', the eccentricity of the earth's orbit, about 0.017; y, the sine of half the inclination of the moon's orbit, about 0.046; m, the ratio of the mean motions of the earth and moon, about 0.075. The expressions for the longitude, latitude and paral lax appear as infinite trigonometric series, in which the coefficients of the sines and cosines are themselves infinite series proceeding according to the powers of the above small quantities. This method was applied with success by Pontecoulant and Sir John Lubbock and afterwards by Delaunay.