July 23, 1915] 



SCIENCE 



109 



Helmholtz's Faraday lecture, and certainly 

 the influence of Thomson and Tait's great 

 treatise on "Natural Philosophy," might 

 help to explain the rapid growth of inter- 

 est and discussion. The lesser, though well 

 marked, influx of energy in the years fol- 

 lowing 1895 may be due in part to the dis- 

 covery of radium and the new theories of 

 electrons and atoms. Of course such sur- 

 mises can only be verified or corrected by 

 close examination of the subjects that are 

 grouped together indiscriminately in this 

 division. 



Such fluctuations occur not only in these 

 most concrete branches, but are discernible 

 in all the others. Such variations in the 

 choice of subjects for investigation one is 

 tempted to call changes of fashion, so 

 capricious or accidental do they appear to 

 a superficial observer. On this same dia- 

 gram, in geometry, note the dispropor- 

 tionate breadth of the stream through the 

 '70 's, and its slow narrowing thereafter. 

 See in analysis, too, the breadth near 1880, 

 and later, some expansion soon after 1890. 

 Algebra enjoyed maximal periods not far 

 from 1878, and has overflowed its usual 

 boundaries again ever since 1899. The 

 gradual augmentation in philosophy, his- 

 tory, biography might have been expected, 

 but is also in part traceable to the influ- 

 ence of imitation and to vagaries in classi- 

 fication. 



One may pick flaws in classification, but 

 there are excuses, one of them in this very 

 fact of changing fashions. For a set of 

 categories that answer well enough in 1868 

 do not contain explicitly all the topics that 

 interest even conservative mathematicians 

 in 1880, still less in 1910. No long disquisi- 

 tion on this subject is needed, for nearly 

 every one has some pet grievance against a 

 much more elaborate arrangement, the 

 Dewey system of classification of books. 

 But also a part of the fault, in the case be- 



fore us, lay in the imperfect ofiice arrange- 

 ments even more than in the system 

 adopted. Against this source of error the 

 more recent French system has guarded, 

 by adopting numbers for a designating 

 mark. Of course the Jahrhuch, published 

 in Berlin, ignores this improvement, but 

 it does from time to time add new titles to 

 chapters and new subdivisions under them. 

 My two complaints against the editors were 

 first their lack of discrimination between 

 two subjects called by the same name: 

 theory of forms in the algebra of continu- 

 ous variables, and theory of forms in the 

 domain of whole numbers or integers. 

 When looking for important memoirs on 

 the one subject, one must always search 

 with a careful eye the other chapter also. 

 And in the second place they had made no 

 place for any Theory of Groups except that 

 concerned with permutations (hardly even 

 for that until very recent years), and 

 ignored completely the creation of a great 

 new department of activity by Sophus Lie 

 in 1870, under the same caption with a 

 difi'erence, viz., the "Theory of Continuous 

 Groups. ' ' 



In all this tabulation and construction of 

 graphs, it has been assumed that one article 

 is like another in importance, one idea as 

 fertile as its rival, one name of an investi- 

 gator no more to be noticed than any other. 

 This assumption, however, is harmless, for 

 it deceives nobody. As well try to convince 

 a student of stars that all parts of the 

 nightly sky are of equal brightness, that 

 there are no luminaries of first magni- 

 tude and no Galaxy resplendent with its 

 organized myriads ! But in a first approxi- 

 mation we are at liberty to make errors, 

 provided we do so with the consciousness 

 that they are errors and that they call for 

 subsequent discrimination and revision. 



Look then at a few graphs relating to 

 single divisions or subdivisions. 



