322 POPULAR SCIENCE MONTHLY 



a duplicate or an artificial classification. For some purposes, such as 

 arranging books on shelves in a library, the former is not feasible, and 

 hence arises the constant opportunity of complaint on the part of 

 those who use libraries with a view to obtain all available facts along 

 certain lines. This opportunity is inherent in the subject and hence 

 must exist under the most ideal conditions, but it is sometimes made 

 more apparent by the fact that books are not always classified by those 

 who are as familiar with the subject-matter as their authors were 

 at the time of writing. 



Although the present active period in mathematical development 

 has exhibited many relations between subjects which appeared to be 

 unrelated, yet it has been still richer in exhibiting new centers of 

 developments which promise to be useful, and hence it has called for a 

 great extension of classification headings, as may be seen from the 1908 

 edition of VIndex du repertoire bibliographique des sciences mathe- 

 matiques. In order that a method of classification should give promise 

 of usefulness for a long period of time, it must therefore be so con- 

 structed as to admit readily of indefinite extensions. This is a char- 

 acteristic property of the two important methods of classification which 

 have been adopted after international conferences, viz., VIndex du 

 repertoire just mentioned and the International Catalogue mentioned 

 above. The former of these provides for an indefinite extension of its 

 fundamental headings by using the capital letters of the Roman 

 alphabet with various exponents to represent these headings. On the 

 other hand, the International Catalogue divides all mathematics into 

 four parts, in addition to a general heading for history, periodical, 

 general treatises, etc. 



In the first five annual issues of the International Catalogue these 

 four parts into which all mathematics is divided bore the following 

 headings: Fundamental notions, algebra and theory of numbers, 

 analysis, and geometry. In the last issue of this catalogue the first 

 of these headings is replaced by arithmetic and algebra, in accordance 

 with the decision of the international convention of 1905. The term 

 algebra now appears in two of the four headings, and, if it is remem- 

 bered that the theory of numbers is higher arithmetic, this term is 

 implicitly in two of these four headings. This is another evidence of 

 the vagueness of the terms arithmetic and algebra as used in some of 

 the best mathematical literature of the present day, and seems to imply 

 that these terms, especially the former, are more and more devoted to 

 those fundamental notions which are most prominent in the later de- 

 velopments, or have the most frequent uses in related sciences. As the 

 science grows some things which are now classed with analysis or 

 geometry will naturally be put under other headings. 



While it might be impossible to advance good reasons for dividing 



