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POPULAR SCIENCE MONTHLY

literature places determinants under algebra. If one were inclined to adopt the common definition that arithmetic is the science of the relations 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 historic 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 impossible 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."[1] 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,[2] in which simple and quadratic equations are classed under arithmetic, would imply that such usage was not universal among eminent authorities.

Coming to the term geometry, we encounter scarcely less trouble. On the one hand, we find it advocated that geometry should be recognized as a science independent of mathematics, just as psychology is gradually being recognized as an independent science and not as a branch of philosophy,[3] 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

  1. "Encyclopédie des sciences mathématiques" (1904), p. 22.
  2. Published by B. G. Teubner, Leipzig, Germany.
  3. Bôcher, Bulletin of the American Mathematical Society, Vol. 11 (1904), p. 124.