752 ANNUAL KEPOET SMITHSONIAN INSTITUTION, 1912. 



from a nebula the process of evolution would stop and become fixed 

 in eternal immobility or perhaps better in invariable mobiUty ? But 

 even great men sometmies make sHps in their logic. There were 

 never men who made none.^ 



Later, two celebrated mathematicians, Lagrange and Poisson, 

 considerably extended the system of Laplace. The indefinite sta- 

 bility of the planetary elements seemed assured forever. The 

 address delivered by an astronomer, and he not one of the least, 

 M. de Pontecoulant, before the Academie des Sciences, when the 

 statue of Poisson was dedicated, shows well the state of belief of the 

 world upon this matter then, and it was scarcely altered at the end of 

 the nineteenth century. 



"For his masterpiece," he said, "Poisson had the honor of solving 

 that most important problem, the stability of the solar system, of 

 which, after the works of Laplace and Lagrange, doubts still existed 

 in the most judicial minds. In the future the harmony of the celestial 

 spheres is assured. Their orbits will never depart from the almost 

 circular form wliich they have to-day and their respective positions 

 will make only slight departures from a mean position in which the 

 succession of centuries will finally see them revolving. The physical 

 universe was therefore built upon indestructible foundations, and 

 God, in order to conserve the human race, will not be obliged, as 

 Newton wrongly believed, to retouch his work." 



So matters stood when Poincare attacked the problem. Soon dis- 

 coveries succeeded discoveries. The problem set is this: Being given 

 several bodies of known masses in given places and with given 

 velocities at some known moment, to determine what these places 

 and velocities will be at any future time, t. For a single planet and 

 the sun the problem is completely solved by the laws of Kepler. But 

 when tivo planets and the sun are considered the reciprocal attraction 

 of the planets upon each other must be considered. Then we have 

 the celebrated problem of three lodies. The difficulties of this latter 

 problem are such that it can be solved only by the method of succes- 

 sive approximations. In the equations which led Laplace and his 

 successors to their conclusions as to the stability, the coordmates of 

 the planets were developed in a series whose terms were arranged in 

 powers of the masses. Poincare first showed that we could not thus 

 obtain an indefinite approximation and that the convergence of the 



' It is related that when Laplace presented his work to Bonaparte, the latter asked whether, as with New- 

 ton, some place had been left to the Creator in the maintonanco of order in the world. Laplace replied: 

 "Citizen, premier consul, I have had no need of such an hypothesis." If that was the response really made, 

 I do not see at all the ground for the irreverontial or atheistical attitude often attributed to Laplace. There 

 may indeed be a very deeply religious sentiment in the belief of a universe so harmoniously constituted 

 that there would be no need of continual retouching for it to preserve its course. "Men," Poincar^ has 

 written, "demand that God continually prove his existence by miracles; but the eternal marvel is that 

 there are not miracles continuously. It is for that very reason that the world is divine because it does 

 work so harmoniously. If it were ruled by caprice, how cpuld wa then prove that chance did not reign 

 supremo?" 



