TEE CLASSIFICATION OF MATHEMATICS 321 



not regarded as mathematics, and, in the other, it is supposed to com- 

 prise all that is generally included under the term mathematics. With 

 such a wide range of usage among eminent authorities it is evident that 

 an acceptable definition is hopeless. 



These instances appear sufficient to emphasize the fact that the 

 terms arithmetic, algebra and geometry have no definite meanings in 

 mathematical literature. They may be compared with the names of the 

 constellations, which attract the attention of the amateur but are not 

 generally taken very seriously by the professional astronomer since their 

 boundaries are not defined with clearness. Just as it may be difficult to 

 establish a connection between the figures represented by the names of 

 some of the constellations and the arrangement of the brighter stars in 

 them, so it is difficult to see much connection between the meaning of 

 the terms arithmetic, algebra and geometry, and some of the subjects 

 classed under these heads. In a growing science it is very desirable to 

 have some elastic terms — terms to which we assign broader and perhaps 

 even different meanings as our knowledge advances. In fact, the term 

 mathematics is itself preeminently one whose meaning is a matter of 

 slow development, even if we accept such brief definitions as mathe- 

 matics is the science of saving thought, or " mathematics is the science 

 of drawing necessary conclusions." 



The fact that many things which appear unrelated when studied 

 superficially exhibit the most intimate connections when viewed from 

 a higher standpoint has doubtless been a potent cause of the variety of 

 usage as regards general terms of classification. There are no natural 

 lines of division in mathematics. In fact, one of the most attractive 

 phases in the development of mathematics is the discovery of the rela- 

 tions existing between what was supposed to be unrelated. In other 

 words, the unifying of mathematical truths is one of the chief concerns 

 of many of the workers in this domain. Although the elements of 

 arithmetic, algebra and geometry appear sufficiently distinct to the 

 beginner, the marks of distinction vanish one by one as one proceeds 

 in following up the ideas starting from these centers, as is evidenced 

 by the term analytic geometry, since analysis and algebra were syno- 

 nyms for Newton, Euler and Lagrange. 



Notwithstanding the fact that there are no natural lines of division 

 in mathematics, classification is essential and need not be entirely 

 artificial; for, marks of differences which are only superficial are, 

 nevertheless, worthy of note and frequently furnish convenient centers 

 for groups of very closely related ideas. Both subject-matter and 

 method offer many such superficial marks of difference which are 

 utilized for the sake of classification. As we go away from these 

 centers we naturally reach facts which seem equally closely related to 

 more than one center, and in such cases it is necessary to have either 



VOL. LXX, III — 21. 



