23 6] Planetary Theory 301 



improvement on Newton's estimate from tides (chapter ix., 

 189), and the second, which was entirely new, previous 

 estimates having been merely conjectural, is in tolerable 

 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 Jerome Le Francois Lalande 

 (1732-1807), who is chiefly known as an admirable popu- 

 lariser 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 treat- 

 ment of planetary perturbations in the second volume of 

 bis 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 satis- 

 factorily. 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 per- 

 turbations 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 



* They give about 78 for the mass of Venus compared to that of 

 the earth. 



