The same investigation revealed to Laplace the existence of alterations of a similar character, and due to the same cause, of other elements in the moon’s orbit, which, though not previously noticed, were found to be indicated by ancient eclipse observations. 241. The third volume of the _Mécanique Céleste_ contains a general treatment of the lunar theory, based on a method entirely different from any that had been employed before, and worked out in great detail. “My object,” says Laplace, “in this book is to exhibit in the one law of universal gravitation the source of all the inequalities of the motion of the moon, and then to employ this law as a means of discovery, to perfect the theory of this motion and to deduce from it several important elements in the system of the moon.” Laplace himself calculated no lunar tables, but the Viennese astronomer _John Tobias Bürg_ (1766-1834) made considerable use of his formulae, together with an immense number of Greenwich observations, for the construction of lunar tables, which were sent to the Institute of France in 1801 (before the publication of Laplace’s complete lunar theory), and published in a slightly amended form in 1806. A few years later (1812) _John Charles Burckhardt_ (1773-1825), a German who had settled in Paris and worked under Laplace and Lalande, produced a new set of tables based directly on the formulae of the _Mécanique Céleste_. These were generally accepted in lieu of Bürg’s, which had been in their turn an improvement on Mason’s and Mayer’s. Later work on lunar theory may conveniently be regarded as belonging to a new period of astronomy (chapter XIII., § 286). 242. Observation had shewn the existence of inequalities in the planetary and lunar motions which seemed to belong to two different classes. On the one hand were inequalities, such as most of those of the moon, which went through their cycle of changes in a single revolution or a few revolutions of the disturbing body; and on the other such inequalities as the secular acceleration of the moon’s mean motion or the motion of the earth’s apses, in which a continuous disturbance was observed always acting in the same direction, and shewing no signs of going through a periodic cycle of changes. The mathematical treatment of perturbations soon shewed the desirability of adopting different methods of treatment for two classes of inequalities, which corresponded roughly, though not exactly, to those just mentioned, and to which the names of =periodic= and =secular= gradually came to be attached. The distinction plays a considerable part in Euler’s work (§ 236), but it was Lagrange who first recognised its full importance, particularly for planetary theory, and who made a special study of secular inequalities.