i22 THE POPULAR SCIENCE MONTHLY 



general, to eight distinct numbers. Starting with 5, these eight num- 

 bers are 



5, - 3, %, - %, %, %, %, %. 



ISTo 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 alge- 

 bra is furnished by the three-valued function xy -f- zw, which is funda- 

 mental 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 solu- 

 tion 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. Stich 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 fur- 

 nish the most active fields of future investigations in accord with the 

 predictions of Poincare, 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 Mathe- 

 matical Association, which is as follows : " I believe Professor Perry 

 will get some very good material for applications oxit 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 appli- 

 cations now." 3 



3 The Mathematical Gazette, January, 1909, p. 17. 



