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SCIENCE 



[N. S. Vol. XXXII. No. 821 



problems probably of a character which 

 we can not at present forecast, it is essential 

 that mathematics should be allowed to de- 

 velop itself freely on its own lines. Even 

 if much of our present mathematical theo- 

 rizing turns out to be useless for external 

 purposes, it is wiser, for a well-known rea- 

 son, to allow the wheat and the tares to 

 grow together. It would be easy to estab- 

 lish in detail that many of the applications 

 which have been actually made of mathe- 

 matics were wholly unforeseen by those 

 who first developed the methods and ideas 

 on which they rest. Recently, the more 

 refined mathematical methods which have 

 been applied to gravitational astronomy by 

 Delaunay, G. W. Hill, Poincare, E. W. 

 Brown and others^ have thrown much 

 light on questions relating to the solar 

 system, and have • much increased the ac- 

 curacy of our knowledge of the motions of 

 the moon and the planets. Who knows 

 what weapons forged by the theories of 

 functions, of differential equations, or of 

 groups, may be required when the time 

 comes for such an empirical law as Men- 

 deleeff's periodic law of the elements to 

 receive its dynamical explanation by means 

 of an analysis of the detailed possibilities 

 of relatively stable types of motion, the 

 general schematic character of which will 

 have been indicated by the physicist? It 

 is undoubtedly true that the cleft between 

 pure mathematics and physical science is 

 at the present time wider than formerly. 

 That is, however, a result of the natural 

 development, on their own lines, of both 

 subjects. In the classical period of the 

 eigliteenth century, the time of Lagrange 

 and Laplace, the nature of the physical 

 investigations, consisting largely of the de- 

 tailed working out of problems of gravita- 

 tional astronomy in accordance with New- 

 ton's law, was such that the passage was 

 easy from the concrete problems to the cor- 



responding abstract mathematical ones. 

 Later on, mathematical physicists were 

 much occupied with problems which lent 

 themselves readily to treatment by means 

 of continuous analysis. In our own time 

 the effect of recent developments of physics 

 has been to present problems of molecular 

 and sub-molecular mechanics to which con- 

 tinuous analysis is not at least directly ap- 

 plicable, and can only be made applicable 

 by a process of averaging the effects of 

 great swarms of discrete entities. The 

 speculative and incomplete character of 

 our conceptions of the structure of the ob- 

 jects of investigation has made the appli- 

 cations of dynamics to their detailed eluci- 

 dation tentative and partial. The gener- 

 alized dynamical scheme developed by 

 Lagrange and Hamilton, with its power of 

 dealing with systems, the detailed struc- 

 ture of which is partially unknown, has, 

 however, proved a powerful weapon of at- 

 tack, and affords a striking instance of the 

 deep-rooted significance of mathematical 

 form. The wonderful and perhaps un- 

 precedentedly rapid discoveries in physics 

 which have been made in the last two de- 

 cades have given rise to many questions 

 which are as yet hardly sufficiently definite 

 in form to be ripe for mathematical treat- 

 ment; a necessary condition of which 

 treatment consists in a certain kind of pre- 

 cision in the data of the problems to be 

 solved. 



The difficulty of obtaining an adequate 

 notion of the general scope and aims of 

 mathematics, or even of special branches 

 of it, is perhaps greater than in the case 

 of any other science. Many persons, even 

 such as have made a serious and prolonged 

 study of the subject, feel the difficulty of 

 seeing the wood for trees. The severe de- 

 mands made upon students by the labor 

 of acquiring a difficult technique largely 

 accounts for this; but teachers might do 



