HENRI POINCARE NORDMANN. 753 



series had been assumed without proof by those who employed them, 

 and that it is probable that in the terms of higher order, t, the tune, 

 enters not only with the sine and cosine, which would lead to periodic 

 compensations of the irregularities, but also outside of the trigono- 

 metric functions, so that certain of the terms, at first negligible, may 

 possibly increase indefinitely with the time. Here with one blow he 

 reduced to naught the conclusions of Laplace and his successors. 



Poincare found later that certain new methods would allow him 

 to express in every case the coordmates of the planets in a purely 

 trigonometric series, avoiding the inconveniences of the former 

 methods, and he proved for the purpose a brilliant series of new 

 theorems of great generality. The rigorous proof of the stability 

 now depended only on knowing whether the new series would be 

 convergent. This was the knot of the problem, for before Poincare 

 all astronomers had supposed a trigonometric series to be absolutely 

 convergent. Pomcare showed that that opinion, despite the fact 

 that it was classic, was erroneous, and indeed that, when we have 

 represented the coordinates of the planets by a convergent series 

 which is not very different from that employed by Laplace, we will 

 not have demonstrated the stability of the solar system. Because 

 of these great results, which are like the crowning of three centuries 

 of incessant research, posterity will certainly place this new treatise 

 on celestial mechanics (Les methodes nouvelles de la mecanique 

 celeste) by the side of the immortal Principia of Newton. All future 

 researches on this subject must be built upon the solid foundations 

 laid by Poincare. 



Celestial mechanics in general considers the planets only as if all 

 their matter were concentrated in mathematical points. It leaves 

 out of consideration the other properties of these objects, evidently 

 generally negligible in comparison with the Newtonian attraction, 

 but whose effects with time may become of importance relative to 

 the stability of the systems. Attacking the question from a new 

 side, Poincare showed that there are three preponderant forces tend- 

 ing to modify the orbits: The resistance, weak though it may be, of 

 the interplanetary medium; the tides which the planets and the sun 

 produce upon each other; and the magnetism of the planets. The 

 accumulated effects of these will fiaally precipitate the planets into 

 the sun. That will be the end of our system of planets. Will that 

 be the end of the human race ? Certainly not, for it is very probable 

 that other changes will have ended terrestrial life long before the day 

 of that final catastrophe; the day ? no, I should not say day, for there 

 will no longer be day and night, for our earth will then forever present 

 the same side toward the sun! Many reasons lead us to believe that 

 in the future as well as in the past the duration of human life upon 

 this globe will be infinitely small compared to the time our earth has 



