stone of the whole problem was stubborn. He was compelled to go away again to perform military duty, and his mind was full of other things. But one day while crossing the boulevard the solution of the last difficulty suddenly appeared and upon verification was found to be correct.
In this account of the birth and growth of mathematical development, which he assures us is practically the same as for all such developments, it is obvious that the central notion is that of generalization. Elliptic, abelian and theta functions are in turn generalized into a new class of transcendents. Inversion of differentials is generalized into inversion of differential equations. This notion of generalization we need to inspect a little closely. Mathematical generalization consists of two types of thought, often not discriminated, and often scarcely to be discriminated from each other. One type consists in so stating a known theorem that it will be true of a wider class than in its first statement, and the predicate asserted has a wider significance. In such generalization the first statement of the theorem becomes a mere particular case of the second statement. Examples will occur readily to every one. There are two forms of this type: in one, many known cases are brought together under one law; in the other form, the law thus found is made to apply to other known cases, perhaps never before suspected to be related to the first set. It is the guiding threads of analogy that usually bring about these forms of generalization. This kind of generalizing power Poincare had in high degree. In his memoir on "Partial Differential Equations of Physics "[1] he says:
If one looks at the different problems of the integral calculus which arise naturally when he wishes to go deep into the different parts of physics, it is impossible not to be struck by the analogies existing. Whether it be electrostatics, or electrodynamics, the propagation of heat, optics, elasticity or hydrodynamics, we are led always to differential equations of the same family; and the boundary conditions though different, are not without some resemblances. . . . One should therefore expect to find in these problems a large number of common properties.
Also in his "Nouvelles Méthodes de la Mécanique Céleste" he says:
The ultimate aim of celestial mechanics is to solve the great question whether Newton's law alone will explain all astronomical phenomena.
In his address awarding Poincare the gold medal of the Royal Astronomical Society, G. H. Darwin[2] said:
The leading characteristic of M. Poincare 's work appears to be the immense wideness of the generalizations, so that the abundance of possible illustrations is sometimes almost bewildering. This power of grasping abstract principles is the mark of the intellect of the true mathematician; but to one accustomed rather to deal with the concrete the difficulty of completely mastering the argument is sometimes great.
