Page:Popular Science Monthly Volume 64.djvu/373

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N the recent International Catalogue of Scientific Literature, group theory is classed among the fundamental notions of mathematics. The two other subjects which are classed under this heading are 'foundations of arithmetic' and 'universal algebra.' While it might be futile to attempt to popularize those recent advances in mathematics which are based upon a long series of abstract concepts, it does not appear so hopeless to give a popular exposition of fundamental notions. In what follows we shall aim to give such an exposition of some of the notions involved in the theory of groups.

This theory seems to have a special claim on popular appreciation in our country because it is one of the very few subjects of pure mathematics in whose development America has taken a prominent part. The activity of American mathematicians along this line is mainly due to the teachings of Klein and Lie at the universities of Göttingen and Leipzig respectively. During the Chicago exposition, the former held a colloquium at Evanston, in which the fundamental importance of the subject was emphasized and thus brought still more prominently before the American mathematicians.

There is probably no other modern field of mathematics of which so many prominent mathematicians have spoken in such high terms during the last decade. In support of this strong statement we quote the following:

There are two subjects which have become especially important for the latest development of algebra; that is, on the one hand, the ever more dominating theory of groups whose systematizing and clarifying influence can be felt everywhere, and then the deep penetrations of number theory. The theory of groups, which is making itself felt in nearly every part of higher mathematics, occupies the foremost place among the auxiliary theories which are employed in the most recent function theory.

In fine, the principal foundation of Euclid's demonstrations is really the existence of the group and its properties. Unquestionably he appeals to other axioms which it is more difficult to refer to the notion of group. An axiom of this kind is that which some geometers employ when they define a straight line as the shortest distance between two points. But it is precisely such axioms that Euclid enunciates. The others, which are more directly associated with the idea of displacement and with the idea of groups, are the very ones