20 STABILITY OF SYSTEM. SECT. III. 



have the contrary effect. But the mass of the whole of the 

 planets and satellites taken together is so small, when compared 

 with that of the sun, that these effects are quite insensible, and 

 could only have been discovered by theory. And, as it is cer- 

 tain that the length of the major axes and the mean motions are 

 not permanently changed by any other power whatever, it may 

 be concluded that they are invariable. 



With the exception of these two elements, it appears that all 

 the bodies are in motion, and every orbit in a state of perpetual 

 change. Minute as these changes are, they might be supposed 

 to accumulate in the course of ages, sufficiently to derange the 

 whole order of nature, to alter the relative positions of the planets, 

 to put an end to the vicissitudes of the seasons, and to bring 

 about collisions which would involve our whole system, now so 

 harmonious, in chaotic confusion. It is natural to inquire, what 

 proof exists that nature will be preserved from such a catastrophe ? 

 Nothing can be known from observation, since the existence of 

 the human race has occupied comparatively but a point in dura- 

 tion, while these vicissitudes embrace myriads of ages. The 

 proof is simple and conclusive. All the variations of the solar 

 system, secular as well as periodic, are expressed analytically by 

 the sines and cosines of circular arcs (N. 76), which increase 

 with the time ; and, as a sine or cosine can never exceed the 

 radius, but must oscillate between zero and unity, however much 

 the time may increase, it follows that when the variations have 

 accumulated to a maximum by slow changes, in however long a 

 time, they decrease, by the same slow degrees, till they arrive at 

 their smallest value, again to begin a new course ; thus for ever 

 oscillating about a mean value. This circumstance, however, 

 would be insufficient, were it not for the small excentricities of 

 the planetary orbits, their minute inclinations to the plane of the 

 ecliptic, and the revolutions of all the bodies, as well planets as 

 satellites, in the same direction. These secure the perpetual 

 stability of the solar system (N. 77). However, at the time 

 that the stability was proved by La Grange and La Place, the 

 telescopic planets between Mars and Jupiter had not been disco- 

 vered ; but La Grange, having investigated the subject under a 

 very general point of view, showed that, if a planetary system be 

 composed of very unequal masses, the whole of the larger would 

 maintain an unalterable stability with regard to the form and 



