Page:Popular Science Monthly Volume 75.djvu/125

Rh 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 common properties we shall not only see a bond connecting these fund a* 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 the 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 interesting 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 commutative 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 connection 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 elementary 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 trigonometry, yet these operations serve to explain some facts which present themselves in the most elementary arithmetic processes. For instance, the operations of subtracting from 2 and dividing 2 lead, in