584 



SCIENCE 



[N. S. Vol. XLHI. No. 1113 



and improving its applications for the wel- 

 fare of all men. 



Every man must do what he can — ^hence 

 comes specialization. Mathematics is and 

 has been a useful kind of work, of value 

 both immediate and prospective. It is of 

 that I am to speak here hriefly. On its 

 utility I need touch but lightly, mentioning 

 a few of the most obvious contributions to 

 other branches of science ; then I must point 

 out more at length certain particular 

 mathematical theories and bodies of rea- 

 soned abstract knowledge developed in the 

 past hundred years. 



These latter take their chance of survival 

 along with the poetry, the art, the philos- 

 ophy of their time — and indeed with much 

 of what is now received as natural science. 

 If it survives, it will be because you, and 

 others like you who are to become intellec- 

 tual leaders in the near future, find in these 

 fields tasks which seem to you worth the 

 doing; things begun which you would 

 gladly finish, errors which must be cleared 

 away to make room for truth ; ideas in germ 

 or questions vaguely hinted at, which can 

 be worthily developed by your arduous 

 labor. 



First, then, let us recall some of those 

 mathematicians 'whose labors have enriched 

 other natural sciences since the time of 

 Lagrange and Laplace. Four physical 

 problems of major importance have de- 

 manded the devotion of mathematicians of 

 the first rank, and have given occasion for 

 the elaboration of theories now generally 

 accepted. These are the problems of the 

 transmission of light, of electrical and mag- 

 netic effects at a distance, of the relation 

 between heat and other forms of energy in 

 the world of force, and the historical ques- 

 tion concerning the origin and growth of 

 the earth on whose surface we live. 



Not that the statement of a physical 

 theory requires a mathematical mind; in- 



deed the observatory and the laboratory 

 are far more likely to be the birthplace of 

 theories than the computing room or the 

 logician's study. But whoever formulates 

 a physical theory with precise terms, defi- 

 nitions and laws, and tests it for consistency 

 of its parts and agreement with a wide 

 range of facts — he is a mathematician ; and 

 if the complexity of his problem drives him 

 to invent new concepts or new short-cuts in 

 argument, he is a creative mathematician. 



Such was Fresnel, who in 1817-19 an- 

 alyzed and pushed to precise formulation 

 the theory of wave-motion in the luminifer- 

 ous ether. The hypothesis of an ether was 

 not unknown at that time, and in acoustics 

 the undulation theory was well established. 

 Authorities, however, seemed overwhelm- 

 ingly in favor of the emission theory of 

 light — Descartes, Newton, Brewster, La- 

 place and Poisson. It required the resolute 

 and unperturfjed mind of a true investi- 

 gator to give due attention to the hypoth- 

 esis, then far from orthodox, of an all-per- 

 vading ether, and to build up a complete 

 theory of the phenomena of diffraction 

 until it brought him to a crucial experi- 

 ment, which even his opponents admitted to 

 be decisive. When its implications were 

 completely analyzed and their consequences 

 demonstrated, doubt and prejudice gave 

 way to clearness and certainty. The con- 

 troversy was practically closed when in 

 1820 Fresnel received the medal of the 

 Paris Academy for his essay on ' ' The Dif- 

 fraction of Light." And this general in- 

 dorsement of the ether hypothesis was most 

 essential for the next pressing problem, 

 that of the transmission of electrical effects. 



The effect of a small closed current of 

 electricity upon a magnetized particle in 

 its field is like that of a magnet, feeble or 

 strong, standing at right angles with the 

 plane of the current. One closed current 

 attracts or repels another, just as one 



