170 ANNUAL REPOET SMITHSONIAN INSTITUTION, 1912. 



mental results may, in spite of their apparent diversity, be deduced 

 from a small number of formulae which are coherent among them- 

 selves. Usually the employment of mathematics in these partial 

 theories is quite independent of the ultimate bases of the theory. 

 And so it is that for many of the phenomena of physical optics the 

 formulae are the same in the mechanical theory of Fresnel and in the 

 electromagnetic theory of Maxwell. In the same way the formulae 

 used by electrical engineers are independent of the diversity of 

 theories concerning the nature of the current. 



If I have been obliged to pomt out, though beyond my subject, 

 this employment of the mathematical tool as an auxiliary to the 

 partial physical theories, it is in order to prevent all misunderstand- 

 ing. It appears certain that for a long time to come, as long, per- 

 haps, as human science shall endure, it will be under this relatively 

 modest form that mathematics will render the greatest service to the 

 physicists. There is no reason why we should be disinterested in 

 the general mathematical theories whereof physics has furnished 

 the model, whether we may be concerned with speculations on 

 partial differential equations suggested by the physics of the conti- 

 nuum or with statistical speculations pertaining to the physics of the 

 discontinuum. But it should be well imderstood that the new mathe- 

 matical theories wliich discontinuity of physical phenomena might 

 suggest can not have the pretention of entirely replacing classical 

 mathematics. These are only new aspects, for which it is proper to 

 make room by the side of older views m such a manner as to augment 

 as much as possible the richness of the abstract world in which we 

 seek models suitable for making us better to comprehend and better 

 to conjecture concrete phenomena. 



III. 



It is frequently a simplification m mathematics to replace a very 

 large finite number by infinity. It is thus that the calculus of defi- 

 nite integrals is frequently more simple than that of summation 

 fornmlae, and that the differential calculus is generally more smiple 

 than that of finite differences. In the same way, we have been 

 led to replace the simultaneous study of a great number of functions 

 of one variable by the study of a continuous infinitude of functions 

 of one variable; that is to say, by the stud}^ of a function of two 

 variables. By a bolder generalization Prof. Vito Volterra has been 

 led to define functions wliich depend upon other functions — that is to 

 say, in the most simple case, functions of lines — in considering them 

 as the limiting cases of fmictions which would depend on a great 

 number of variables or, if one prefers, on a very great number of 

 points of the line. 



