44 NUTATION OF LUNAR ORBIT. [SECT. v. 



both are retarded when the mean motion is anticipated. The 

 secular variations in these three elements are in the ratio of 

 the numbers 3, 0735, and 1 ; whence the three motions of the 

 moon, with regard to the sun, to her perigee, and to her nodes, 

 are continually accelerated, and their secular equations are 

 as the numbers 1, 4'702, and 0*612. A comparison of ancient 

 eclipses observed by the Arabs, Greeks, and Chaldeans, im- 

 perfect as they are, with modern observations, confirms these 

 results of analysis. Future ages will develope these great 

 inequalities, which at some most distant period will amount 

 to many circumferences (K 107). They are, indeed, periodic ; 

 but who shall tell their period 1 Millions of years must 

 elapse before that great cycle is accomplished. 



The moon is so near, that the excess of matter at the 

 earth's equator occasions periodic variations in her longitude, 

 and also that remarkable inequality in her latitude, already 

 mentioned as a nutation in the lunar orbit, which diminishes 

 its inclination to the ecliptic when the moon's ascending node 

 coincides with the equinox of spring, and augments it when 

 that node coincides with the equinox of autumn. As the 

 cause must be proportional to the effect, a comparison of these 

 inequalities, computed from theory, with the same given by 

 observation, shows that the compression of the terrestrial 

 spheroid, or the ratio of the difference between the polar and 

 the equatorial diameters, to the diameter of the equator, 

 ^ s 3oi-o3- It i g proved analytically, that if a fluid mass of 

 homogeneous matter, whose particles attract each other in- 

 versely as the squares of the distance, were to revolve about 

 an axis as the earth does, it would assume the form of a 

 spheroid whose compression is 530- Since that is not the 

 case, the earth cannot be homogeneous, but must decrease in 

 density from its centre to its circumference. Thus the moon's 

 eclipses show the earth to be round ; and her inequalities not 

 only determine the form, but even the internal structure of 

 our planet ; results of analysis which could not have been 

 anticipated. Similar inequalities in the motions of Jupiter's 



