SECT. in.J STABILITY OF THE SYSTEM. 23 



It is true that, according to theory, the radial disturbing 

 force should permanently alter the dimensions of all the orbits, 

 and the periodic times of all the planets, to a certain degree. 

 For example, the masses of all the planets revolving within 

 the orbit of any one, such as Mars, by adding to the interior 

 mass, increase the attracting force of the sun, which, there- 

 fore, must contract the dimensions of the orbit of that planet, 

 and diminish its periodic time ; whilst the planets exterior to 

 Mars's orbit must 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 certain 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, suf- 

 ficiently to derange the whole order of nature, to alter the 

 relative positions of the planets, to put an end to the vicis- 

 situdes 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 1 No- 

 thing can be known from observation, since the existence of 

 the human race has occupied comparatively but a point in 

 duration, 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 



