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§ 236] improvement on Newton's estimate from tides (chapter, § 189), and the second, which was entirely new, previous estimates having been merely conjectural, is in tolerably, agreement with modern measurements. It is worth noticing as a good illustration of the reciprocal influence of observation and mathematical theory that, while Clairaut used Lacaille's observations for his theory, Lacaille in turn used Clairaut's calculations of the perturbations of the earth to improve his tables of the sun published in 1758.

Clairaut's method of solving the problem of three bodies was also applied by Joseph Jérôme Le François Lalande (1732–1807), who is chiefly known as an admirable populariser of astronomy but was also an indefatigable calculator and observer, to the perturbations of Mars by Jupiter, of Venus by the earth, and of the earth by Mars, but with only moderate success.

D'Alembert made some progress with the general treatment of planetary perturbations in the second volume of his Recherches, and applied his methods to Jupiter and Saturn.

236. Euler carried the general theory a good deal further in a series of papers beginning in 1747. He made several attempts to explain the irregularities of Jupiter and Saturn, but never succeeded in representing the observations satisfactorily. He shewed, however, that the perturbations due to the other planets would cause the earth's apse line to advance about 13" annually, and the obliquity of the ecliptic to diminish by about 48" annually, both results being in fair accordance both with observations and with more elaborate calculations made subsequently. He indicated also the existence of various other planetary irregularities, which for the most part had not previously been observed.

In an essay to which the Academy awarded a prize in 1756, but which was first published in 1771, he developed with some completeness a method of dealing with perturbations which he had indicated in his lunar theory of 1753. As this method, known as that of the variation of the elements or parameters, played a very important part