*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 *l'Index du répertoire bibliographique des sciences mathématiques.* In order that a method of classification should give promise of usefulness for a long period of time, it must therefore be so constructed as to admit readily of indefinite extensions. This is a characteristic property of the two important methods of classification which have been adopted after international conferences, viz., *l'Index du répertoire* 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 remembered 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 developments, 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