NEWTON S PEINCIPIA. 115 



portion would be destroyed. Hence, we arrive through 

 the calculus at this important conclusion, that the law only 

 holds, if the mutual actions of the planets on each other 

 are neglected, and that, therefore, the law is not rigorously 

 true where, as in the case of the earth and others, the 

 actions of the other planets are sensible. 



Again, the inspection of the algebraical expressions shows 

 that the variation in the eccentricity of the earth s orbit 

 produces, likewise, the retardation of the apsides and 

 nodes; and this discovery was also made, apparently, by 

 the mere inspection of the expressions which the calculus 

 had furnished. Thus the expression for the motion of the 

 perigee (or apsides) involves the integral f e f 2 d v (y being 

 the true anomaly);* and this quantity is positive. There 

 fore the decrease of the eccentricity of the earth s orbit, 

 causes a decrease, also, of the perigeal motion of the moon. 

 And one of the terms of the equation to the motion of the 

 nodes contains the same integral f e 2 d v ; consequently 

 the same eccentricity is likewise the cause of the variation 

 in the period of their revolution.! 



Now we have seen how extremely small these irregu 

 larities in the moon s motion are which the theory gives 

 by this analytical process, and that they are hardly sensible 

 in a whole century ; yet it is found that the deductions 

 of the calculus are in a remarkable manner confirmed by 

 actual observation. Practical astronomers, for example, 

 wholly ignorant of Laplace s discoveries, have ascertained 

 that the secular variation in the motion of the moon s 

 apsides, ascertained by comparing the eclipses in the Greek, 

 Arabian, and Chaldean astronomy, with those of the last 



* Angle of the radius vector with the axis of the orbit. 



f Mec. Gel. liv. vii. ch. 1. This wonderful chapter is a mere series 

 of integrations, and contains, from the inspection of the equations, those 

 extraordinary discoveries respecting the laws of the universe. 



I 2 



