32o POPULAR SCIENCE MONTHLY 



literature places determinants under algebra. If one were inclined to 

 adopt the common definition that arithmetic is the science of the rela- 

 tions existing between numbers, one would be perplexed by the fact 

 that the theory of groups of finite order is classed with arithmetic in 

 the encyclopedia mentioned above, while it might be difficult to name 

 any other mathematical subject which makes less direct use of numbers 

 than this theory does. 



Although these conflicting uses of the term arithmetic preclude the 

 possibility of formulating a definition which is in accord with the usage 

 of all of the prominent mathematicians, yet this term presents very 

 much less serious difficulties than that of algebra from the standpoint 

 of giving an acceptable definition. All are agreed that the four 

 fundamental operations with natural numbers constitute a part of 

 arithmetic. In fact, all that is generally studied in the elementary 

 schools under the title of arithmetic is now universally regarded as a 

 part of this subject, even if the Greeks called it logistica and dignified 

 what is now generally known as higher arithmetic, or number theory, 

 by the term arithmetic. While it might be difficult to find anything 

 which was included under the term arithmetic during the entire his- 

 toric period of mathematics, it is not difficult to find things which are 

 now universally accepted as parts of this subject. 



When we come to the term algebra, on the contrary, it seems im- 

 possible to find any common ground. If we think of algebra as a 

 generalized arithmetic in which numbers are replaced by symbols which 

 may have any numerical value, we are perplexed by such statements 

 as " In arithmetic it is customary to represent any number whatever 

 by a letter, it being understood that this letter represents the same 

 number as long as the same subject is under consideration." 2 On the 

 other hand, if one were inclined to consider the elements of the theory 

 of equations as the peculiar sphere of algebra, the recent standard 

 encyclopedia of elementary mathematics by Weber and Wellstein, 3 in 

 which simple and quadratic equations are classed under arithmetic, 

 would imply that such usage was not universal among eminent authori- 

 ties. 



Coming to the term geometry, we encounter scarcely less trouble. 

 On the one hand, we find it advocated that geometry should be recog- 

 nized as a science independent of mathematics, just as psychology is 

 gradually being recognized as an independent science and not as a 

 branch of philosophy, 4 while, on the other, we find that the Paris 

 Academy of Sciences uses the term geometry as a synonym for pure 

 mathematics. In the one case, the term geometry is used for what is 



2 " Encyclopedic des sciences niath§matiques " (1904), p. 22. 

 1 Published by B. G. Teubner, Leipzig, Germany. 



4 Bocher, Bulletin of the American Mathematical Society, Vol. 11 (1904), 

 p. 124. 



