122 NEWTON S PBINCIPIA. 



that the squares of the masses, and the fourth powers of 

 the eccentricities, and the angles of the orbits, are neg 

 lected in the calculus.* But Lagrange afterwards showed, 

 that the theorem holds true, even if these quantities be 

 taken into the account. The examination of the moon s 

 motion demonstrates the same important fact, with respect 

 to the permanency of the greater axis and mean motion 

 of the planets ; for if the solar day were now F ^ of a 

 second longer than it was in the age of Hipparchus, the 

 moon s secular equation would be augmented above 42 per 

 cent, or would be in that large proportion greater than it 

 now is known to be. Therefore there has not even 

 been the smallest change of the mean movement of the 

 planets. 



The other changes which take place in the orbits and 

 motions of the heavenly bodies, were found by these great 

 geometricians to follow a law of periodicity which secures 

 the eternal stability of the system. The motion of the 

 earth s orbit we have already seen is so slow, that its axis 

 takes above 109,060 years to perform a complete revolution ; 

 but after that time it occupies precisely the same position 

 in space as it did when this vast period of time began to 

 run. So the eccentricity of the earth s orbit has been for 

 ages slowly decreasing, and the decrease will go on, or 

 the orbit will approach nearer and nearer to a circle, until 

 it reaches a limit which it never can pass. The eccen 

 tricity will then begin slowly to increase until it again 

 reaches its greatest point, beyond which the orbit never 

 can depart from the circular form. The same principle 

 extends itself to all the planets. Thus, the time of the 

 secular variation of Jupiter s eccentricity is 70,400 years. 



All these deductions are the strict analytical conse- 



* M6c. Cel. liv, ii. eh. 7. and 8. (sects. 54 and 63.) 



