239. With the exception of Lagrange's paper on libration, nearly all his and Laplace's important contributions to astronomy were made when Clairaut's and D'Alembert's work was nearly finished, though Euler's activity continued for nearly 20 years more. Lagrange, however, survived him by 30 years and Laplace by more than 40; and together they carried astronomical science to a far higher stage of development than their three predecessors.
240. To the lunar theory Lagrange contributed comparatively little except general methods, applicable to this as to other problems of astronomy but Laplace devoted great attention to it. Of his special discoveries in the subject the most notable was his explanation of the secular acceleration of the moon's mean motion (chapter x., § 201), which had puzzled so many astronomers. Lagrange had attempted to explain it (1774), and had failed so completely that he was inclined to discredit the early observations on which the existence of the phenomenon was based. Laplace, after trying ordinary methods without success, attempted to explain it by supposing that gravitation was an effect not transmitted instantaneously, but that, like light, it took time to travel from the attracting body to the attracted one; but this also failed. Finally he traced it (1787) to an indirect planetary effect. For, as it happens, certain perturbations which the moon experiences owing to the action of the sun depend among other things on the eccentricity of the earth's orbit; this is one of the elements (§ 236) which is being altered by the action of the planets, and has for many centuries been very slowly decreasing; the perturbation in question is therefore being very slightly altered, and the moon's average rate of motion is in consequence very slowly increasing, or the length of the month decreasing. The whole effect is excessively minute, and only becomes perceptible in the course of a long time. Laplace's calculation shewed that the moon would, in the course of a century, or in about 1,300 complete revolutions, gain about 10" (more exactly 10"⋅2) owing to this cause, so that her place in the sky would differ by that amount from what it would be if this disturbing cause did not exist; in two centuries the angle gained would be 40", in three centuries 90", and so on.
