234. Newton’s lunar theory may be said to have given a _qualitative_ account of the lunar inequalities known by observation at the time when the _Principia_ was published, and to have indicated others which had not yet been observed. But his attempts to explain these irregularities _quantitatively_ were only partially successful. Euler, Clairaut, and D’Alembert threw the lunar theory into an entirely new form by using analytical methods instead of geometrical; one advantage of this was that by the expenditure of the necessary labour calculations could in general be carried further when required and lead to a higher degree of accuracy. The result of their more elaborate development was that—with one exception—the inequalities known from observation were explained with a considerable degree of accuracy quantitatively as well as qualitatively; and thus tables, such as those of Clairaut, based on theory, represented the lunar motions very closely. The one exception was the secular acceleration: we have just seen that Euler failed to explain it; D’Alembert was equally unsuccessful, and Clairaut does not appear to have considered the question. 235. The chief inequalities in planetary motion which observation had revealed up to Newton’s time were the forward motion of the apses of the earth’s orbit and a very slow diminution in the obliquity of the ecliptic. To these may be added the alterations in the rates of motion of Jupiter and Saturn discovered by Halley (chapter X., § 204). Newton had shewn generally that the perturbing effect of another planet would cause displacements in the apses of any planetary orbit, and an alteration in the relative positions of the planes in which the disturbing and disturbed planet moved; but he had made no detailed calculations. Some effects of this general nature, in addition to those already known, were, however, indicated with more or less distinctness as the result of observation in various planetary tables published between the date of the _Principia_ and the middle of the 18th century. The irregularities in the motion of the earth, shewing themselves as irregularities in the apparent motion of the sun, and those of Jupiter and Saturn, were the most interesting and important of the planetary inequalities, and prizes for essays on one or another subject were offered several times by the Paris Academy. The perturbations of the moon necessarily involved—by the principle of action and reaction—corresponding though smaller perturbations of the earth; these were discussed on various occasions by Clairaut and Euler, and still more fully by D’Alembert. In Clairaut’s paper of 1747 (§ 233) he made some attempt to apply his solution of the problem of three bodies to the case of the sun, earth, and Saturn, which on account of Saturn’s great distance from the sun (nearly ten times that of the earth) is the planetary case most like that of the earth, moon, and sun (cf. § 228).