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§§ 234, 235] motions of the moon far more accurately than those of D'Alembert, and were even superior in some points to those based on Euler's very much more elaborate second theory; Clairaut's last tables were seldom in error more than 1$1⁄2$', and would hence serve to determine the longitude to within about $3⁄4$°. Clairaut's tables were, however, never, much used, since Tobias Mayer's as improved by Bradley were found in practice to be a good deal more accurate; but Mayer borrowed so extensively from observation that his formulae cannot be regarded as true deductions from gravitation in the same sense in which Clairaut's were. Mathematically Euler's second theory is the most interesting and was of the greatest importance as a basis for later developments. The most modern lunar theory is in some sense a return to Euler's methods.

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