In the account of the creation of the fuchsian functions we see this power of finding examples of his generalizations, that is to say, of applying them. By these functions he could solve differential equations, he could express the coordinates of algebraic curves as fuchsian functions of a parameter, he could solve algebraic equations of any order. Humbert put it succinctly thus: "Poincaré handed us the keys of the world of algebra." Again, from the simplification of the theory of algebraic curves he was able to reach results which led to a generalization of the fuchsian functions to the zetafuchsian functions, which he afterward used in differential equations, the starting point of the problem. He applied the theory of continuous groups to hypercomplex numbers and then applied hypercomplex numbers to the periods of abelian integrals and the algebraic integration of differential equations of certain types. He applied fuchsian functions to the theory of arithmetic forms and opened a wide development of the theory of numbers. He applied fundamental functions to the potential theory of surfaces in general, showing how the Green's function could be constructed for any surface, permitting the solution of the problem. He developed' integral invariants, which persist through cycles of space and time. He dared to apply the kinetic theory of gases and the theory of radiant matter to the Milky Way itself, suggesting that probably we are a speck in a spiral nebula. He analyzed mathematically the rings of Saturn into a swarm of satellites, and the spectroscope confirmed his conclusions, a piece of work ranking with the mathematical discovery of Neptune. He found a generalization for figures of equilibrium of the heavenly bodies, discovering an infinity of forms and pointing out the stable transition shapes from one type to another, of which the piriform was quite new; at the same time throwing light on the problems of cosmogony. He applied trigonometric series, divergent series, and even the theory of probabilities, to show that the stability or instability of our universe has never been demonstrated, but that if probability is measured by continuous functions only, the universe is most probably stable.
There is no essential difference between generalizations of this type in whatever realm they appear. It is generalization to see that projective geometry merely states the invariancies of the projective group, and elementary geometry is a collection of statements about the invariants of the orthogonal group. Expansions in sines and cosines, or Legendrian polynomials, or Bessel functions are particular cases of expansions in fundamental functions, and these arise from the inversion of definite integrals. It is also generalization to reduce the phenomena of light to a wave-theory, then the phenomena of light, electricity and magnetism to ether-properties. It is generalization to reduce physics and physical chemistry to the study of quanta of energy, and, I might add, to reduce all the physical sciences to a study of the kinematics of four-dimensional
