THE FUTURE OF MATHEMATICS 121 



amples to elementary subjects. Our first effort will be directed towards 

 exhibiting some territory which is common to each of the four subjects 

 — arithemtic, geometry, algebra and trigonometry. By observing com- 

 mon properties we shall not only see a bond connecting these funda- 

 mental subjects, but we shall also be led to general methods which 

 make it unnecessary to study the same properties in different forms. 

 The thing to be emphasized is that these four elementary subjects have 

 in common fundamental notions which not only connect them, but 

 also establish contact between them and many other subjects. Such a 

 fundamental notion is a group of order 8, known as the octic group. 

 Some of the properties of this group may be easily seen by considering 

 the possible movements of space which transform a square into itself. 



The period or order of a movement represents the number of times 

 th« movement must be made in order to arrive at the identity, or at 

 the original position. It is clear that the eight movements of the 

 square include two of period four, five of period two, and the identity 

 A profound study of these eight movements would disclose many in- 

 teresting facts. For instance, it would be seen that only two of them 

 (the square of these of period four and the identity) are commutative 

 with each one of others, while each one of the remaining six is com- 

 mutative with only four of the possible eight movements. Although 

 a profound study of this group of eight movements would be necessary 

 to exhibit the fundamental role which it plays in the various subjects, 

 it is not necessary to enter deeply into its properties in order to see 

 that it is common to the four subjects mentioned above. 



At a first thought it might appear as if these eight movements had 

 nothing in common with trigonometry, but a very fundamental con- 

 nection may be seen as follows : If the vertex of the angle A is the 

 center of a square and the initial line of A coincides with a line of 

 symmetry of the square, the operations of taking the complement and 

 the supplement of A correspond to movements transforming the square 

 into itself. Hence the eight angles which may be obtained from a given 

 angle by a repetition of finding supplement and complement may be 

 placed in a (1, 1) correspondence with the eight movements of the 

 square. As these eight angles play such a fundamental role in ele- 

 mentary trigonometry, it has been suggested that our ordinary school 

 trigonometry might appropriately be called the trigonometry of the 

 octic group, or the trigonometry of the group of movements of the 

 square. 



Although the eight operations of the octic group do not occupy such 

 an important place in elementary arithmetic as in geometry and trig- 

 onometry, yet these operations serve to explain some facts which pre- 

 sent themselves in the most elementary arithmetic processes. For 

 instance, the operations of subtracting from 2 and dividing 2 lead, in 



vol. lxxv. — 9. 



