116 NEWTON S PIIINCIPIA. 



century, is about 3.3, or 33 tenths of the moon s mean 

 motion ; and this is the exact result of the calculus. Laplace 

 also discovered, chiefly by similar means, a very small 

 secular inequality in the moon s motion never before sus 

 pected, and produced by the sun s attraction.* It was 

 found by observing, that the divisor of some of the frac 

 tional terms of the equation which shows the inequality 

 is extremely small, and that, consequently, the irregu 

 larity may become sensible. A correction of the tables 

 Avas thus introduced by this great geometrician, in which 

 the theory approaches, on an average, to within J^ of 

 the actual observation. The sign of this inequality being 

 negative, it is a retardation of the mean motion, and is to be 

 set against the secular acceleration. It must be observed, 

 moreover, that the errors of the theory, as compared with 

 the observation, are half of them by excess and half by 

 defect ; so that they may be said to balance each other. 

 The maximum of this inequality is little more than 15&quot;, 

 and its period is 184 years. 



Hitherto of the moon ; but we are, in like manner, con 

 ducted by the same refined, though complicated, analysis 

 to the variations in the orbits, and consequently in the 

 motions of the earth and of the other planets, as well as of 

 the satellites of Jupiter and of Saturn. The most remark 

 able variations produced upon these orbits are the changes 

 in their eccentricity and in their aphelion ; the former 

 being constantly, though slowly, shortened the latter 

 moving round in slow revolutions, as the line of the 

 moon s apsides revolves, but revolves much more swiftly. 



The expressions obtained in the case of any one planet 

 for the eccentricity and perihelion longitude (revolving 

 motion of the axis), are mainly composed of the masses, 



* Mec, Gel. liv. vii. eh. 5. 



