Septembee 23, 1910] 



SCIENCE 



387 



some efifort is required to realize that an 

 apparently so obvious idea embodies a 

 great invention; one to which the Greeks, 

 with their unsurpassed capacity for ab- 

 stract thinking, never attained. An at- 

 tempt to do a multiplication sum in Roman 

 numerals is perhaps the readiest road to an 

 appreciation of the advantages of this 

 great invention. In a large group of sci- 

 ences, the formal element, the common 

 language, so to speak, is supplied by mathe- 

 matics; the range of the application of 

 mathematical methods and symbolism is 

 ever increasing. Without taking too liter- 

 ally the celebrated dictum of the great 

 philosopher Kant, that the amount of real 

 science to be found in any special subject 

 is the amount of mathematics contained 

 therein, it must be admitted that each 

 branch of science which is concerned with 

 natural phenomena, when it has reached a 

 certain stage of development, becomes ac- 

 cessible to, and has need of, mathematical 

 methods and language ; this stage has, for 

 example, been reached in our time by parts 

 of the science of chemistry. Even biology 

 and economics have begun to require 

 mathematical methods, at least on their 

 statistical side. As a science emerges from 

 the stages in which it consists solely of 

 ■more or less systematized descriptions of 

 the phenomena with which it is concerned 

 ii) their more superficial aspect; when the 

 intensive magnitudes discerned in the phe- 

 nomena become representable as extensive 

 magnitudes — then is the beginning of the 

 application of mathematical modes of 

 thought: at a still later stage, when the 

 phenomena become accessible to dynamical 

 treatment, mathematics is applicable to the 

 subject to a still greater extent. 



Mathematics shares with the closely al- 

 lied subject of astronomy the honor of be- 

 ing the oldest of the sciences. When we 

 consider that it embodies, in an abstract 



form, some of the more obvious, and yet 

 fundamental, aspects of our experience of 

 the external world, this is not altogether 

 surprising. The comparatively high de- 

 gree of development which, as recent his- 

 torical discoveries have disclosed, it had 

 attained amongst the Babylonians more 

 than five thousand years B.C., may well as- 

 tonish us. These times must have been 

 preceded by still earlier ages in which the 

 mental evolution of man led him to the use 

 of the tally, and of simple modes of meas- 

 urement, long before the notions of niunber 

 and of magnitude appeared in an explicit 

 form. 



I have said that mathematics is the old- 

 est of the sciences; a glance at its more 

 recent history will show that it has the 

 energy of perpetual youth. The output of 

 contributions to the advance of the science 

 during the last century and more has been 

 so enormous that it is difficult to say 

 whether pride in the greatness of achieve- 

 ment in his subject, or despair at his ina- 

 bility to cope with the multiplicity of its 

 detailed developments, should be the dom- 

 inant feeling of the mathematician. Pew 

 people outside the small circle of mathe- 

 matical specialists have any idea of the 

 vast growth of mathematical literature. 

 The Royal Society Catalogue contains a 

 list of nearly thirty-nine thousand papers 

 on subjects of pure mathematics alone, 

 which have appeared in seven hundred 

 serials during the nineteenth century. 

 This represents only a portion of the total 

 output ; the very large number of treatises, 

 dissertations and monographs published 

 during the century being omitted. During 

 the first decade of the twentieth century 

 this activity has proceeded at an acceler- 

 ated rate. Mathematical contributions to 

 mechanics, physics and astronomy would 

 greatly swell the total. A notion of the 

 range of the literature relating not only to 



