216 THE POPULAR SCIENCE MONTHLY 



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 ap- 

 plying them. By these functions he could solve differential equations, 

 he could express the coordinates of algebraic curves as fuchsian func- 

 tions of a parameter, he could solve algebraic equations of any order. 

 Humbert put it succinctly thus : " Poincare 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 generaliza- 

 tion of the 'fuchsian functions to the zetafuchsian functions, which he 

 afterward used in differential equations, the starting point of the prob- 

 lem. He applied the theory of continuous groups to hypercomplex num- 

 bers and then applied hypercomplex numbers to the periods of abelian 

 integrals and the algebraic integration of differential equations of cer- 

 tain 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 gen- 

 eral, 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, dis- 

 covering 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 continu- 

 ous 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 projec- 

 tive 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 



