general, to eight distinct numbers. Starting with 5, these eight numbers are
5,—3, 2⁄5,—2⁄3, 8⁄5, 8⁄3, 5⁄4, 3⁄4.
No new number is obtained by dividing 2 by any of these numbers or by subtracting any of them from 2. The proof of the fact that the eight operations by means of which each one of these eight numbers may be derived from any one of them have the same properties in relation to each other as the eight movements of the square is not difficult, but it involves details which may be omitted in a popular exposition.
An instance where the octic group plays an important role in algebra is furnished by the three-valued function xy + zw, which is fundamental in the theory of the general equation of the fourth degree. On account of the existence of this function the solution of the general equation of the fourth degree may be made to depend upon the solution of the general equation of the third degree. This function is transformed into itself by eight substitutions, and we may arrange its letters separately on the vertices of a square in such a way that the eight substitutions transforming the function into itself correspond to the eight movements which transform the square into itself. Such an arrangement exhibits the intimate relations between this function and the movements of a square, and the preceding examples illustrate the fact that the octic group finds application in each of the elementary subjects—arithmetic, algebra, geometry and trigonometry, and that it forms a part of the domain common to all of these disciplines.
In a similar manner other groups could be traced through these elementary subjects of mathematics and it could be shown that the theory of these groups may be used to clarify many fundamental points and to exhibit deep-seated contact. If the common domains will furnish the most active fields of future investigations in accord with the predictions of Poincaré, and if we may expect the greatest future progress to be based upon the modeling of the less advanced science upon the one which has made the more progress, it is reasonable to expect that a subject like group theory will grow in favor, and that some of the elements of this subject will become a part of the ordinary courses in secondary mathematics. In support of this view we may quote a recent statement by Professor Bryan, President of the Mathematical Association, which is as follows: "I believe Professor Perry will get some very good material for applications out of the theory of groups, when explorers have first made their discoveries, and when the colonists have been over it and surveyed it, and discovered means for cultivating it. We do not know anything about its practical applications now."[1]
- ↑ The Mathematical Gazette, January, 1909, p. 17.
