Petersburg Academy, and was published in 1752, with the title _Théorie de la Lune_. Two years later he published a set of lunar tables, and just before his death (1765) he brought out a revised edition of the _Théorie de la Lune_ in which he embodied a new set of tables. D’Alembert followed his paper of 1747 by a complete lunar theory (with a moderately good set of tables), which, though substantially finished in 1751, was only published in 1754 as the first volume of his _Recherches sur différens points importans du système du Monde_. In 1756 he published an improved set of tables, and a few months afterward a third volume of _Recherches_ with some fresh developments of the theory. The second volume of his _Opuscules Mathématiques_ (1762) contained another memoir on the subject with a third set of tables, which were a slight improvement on the earlier ones. Euler’s first lunar theory (_Theoria Motuum Lunae_) was published in 1753, though it had been sent to the St. Petersburg Academy a year or two earlier. In an appendix[137] he points out with characteristic frankness the defects from which his treatment seems to him to suffer, and suggests a new method of dealing with the subject. It was on this theory that Tobias Mayer based his tables, referred to in the preceding chapter (§ 226). Many years later Euler devised an entirely new way of attacking the subject, and after some preliminary papers dealing generally with the method and with special parts of the problem, he worked out the lunar theory in great detail, with the help of one of his sons and two other assistants, and published the whole, together with tables, in 1772. He attempted, but without success, to deal in this theory with the secular acceleration of the mean motion which Halley had detected (chapter X., § 201). In any mathematical treatment of an astronomical problem some data have to be borrowed from observation, and of the three astronomers Clairaut seems to have been the most skilful in utilising observations, many of which he obtained from Lacaille. Hence his tables represented the actual 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[138] is in some sense a return to Euler’s methods.