May 18, 1917] 



SCIENCE 



471 



as a mature science, we are not to think of 

 it as having become dead and unproduc- 

 tive. Like the individual scientific worker 

 on coming to maturity, it has merely 

 reached the period when its deeds become 

 most etfective for the use and satisfaction 

 of mankind. In fact, physics is perhaps at 

 the same time the most mature natural sci- 

 ence at present existing and the one whose 

 recent progress has been the most rapid 

 and the most remarkable. 



In the development of physics the in- 

 finitesimal calculus has persistently played 

 a leading role; its interaction with experi- 

 mental results has been and is fundamental 

 and necessary to the progress we have wit- 

 nessed and yet see to-day. From this crea- 

 tion of the mathematicians and the use 

 made of it by the physicists the world has 

 received a good practically immeasurable 

 in its extent. Sometimes we are tempted to 

 assess the advantages due to each of these 

 elements; but one can hardly expect suc- 

 cess from such a venture. Logically the 

 mathematics is prior; for it could exist of 

 itself, while the physics probably could not. 

 But psychologically and practically they 

 are so bound up that no separation can be 

 made. "Were the mathematics swept away, 

 much of physical theory would likewise 

 have to go ; but on the other hand much of 

 the mathematics would never have existed 

 had it not been called into being by the de- 

 mands of physical science. 



Until recently it was customary to as- 

 sume that nature is essentially continuous 

 in her manifestations. As long as we pro- 

 ceed on that hypothesis the infinitesimal 

 calculus is the natural tool to be employed 

 in the investigation of phenomena; and we 

 should expect to find differential equations 

 and integral equations playing a leading 

 role in the exposition of physical theory. 



That they have done so has furnished a 

 great incentive to some investigators in 

 prosecuting their labors. It is said that 



Poincare was urged on in his studies of dif- 

 ferential equations by the conviction that 

 he was engaged in perfecting the most im- 

 portant tool which could be employed in 

 the investigation of physical phenomena. 

 No doubt it is a similar use of integral equa- 

 tions which drew quickly into that field so 

 large a body of workers and resulted in its 

 so rapid development. The same spur has 

 urged men on in the study of expansion 

 problems in connection with both differen- 

 tial equations and integral equations. It is 

 now a long while since Fourier series were 

 thus introduced; and their properties and 

 availability have been treated in numerous 

 investigations. 



More recently extensive generalizations 

 of these series have made their appearance ; 

 and we have a great class of expansions in 

 the so-called orthogonal and biorthogonal 

 functions arising in the study of differen- 

 tial and integral equations. In the field of 

 differential equations the most important 

 class of these functions was first defined in 

 a general and explicit manner by an Amer- 

 ican mathematician, Professor Birkhoff, of 

 Harvard University; and their leading 

 fundamental properties were developed by 

 him. "We shall doubtless witness a great 

 progress in our knowledge of these func- 

 tions. 



But in the early years of the present cen- 

 tury the world of scientific thought has 

 been unexpectedly confronted with a new 

 situation of a rather astonishing sort. Our 

 unquestioning assumption of the continuity 

 of nature appears now not to have been well 

 founded; and much of the development of 

 theory which has been based on it is con- 

 sequently perhaps to be regarded as only a 

 rough and unsatisfactory first approxima- 

 tion. If certain apparent discontinuities 

 in nature turn out to be real (and it looks 

 now as if they must) then the differential 

 equation will probably lose its place as the 

 most important tool of applied mathematics 



