[7] These double rotations of the A and B kinds I should like to call Hamiltons and co-Hamiltons, for it is a singular fact that in his “Quaternions” Sir Wm. Rowan Hamilton has given the theory of either the A or the B kind. They follow the laws of his symbols, I, J, K.
Hamiltons and co-Hamiltons seem to be natural units of geometrical expression. In the paper in the “Proceedings of the Royal Irish Academy,” Nov. 1903, already alluded to, I have shown something of the remarkable facility which is gained in dealing with the composition of three- and four-dimensional rotations by an alteration in Hamilton’s notation, which enables his system to be applied to both the A and B kinds of rotations.
The objection which has been often made to Hamilton’s system, namely, that it is only under special conditions of application that his processes give geometrically interpretable results, can be removed, if we assume that he was really dealing with a four-dimensional motion, and alter his notation to bring this circumstance into explicit recognition.
APPENDIX I
THE MODELS
In Chapter XI. a description has been given which will enable any one to make a set of models illustrative of the tesseract and its properties. The set here supposed to be employed consists of:—
1. Three sets of twenty-seven cubes each.
2. Twenty-seven slabs.
3. Twelve cubes with points, lines, faces, distinguished by colours, which will be called the catalogue cubes.
The preparation of the twelve catalogue cubes involves the expenditure of a considerable amount of time. It is advantageous to use them, but they can be replaced by the drawing of the views of the tesseract or by a reference to figs. [103], [104], [105], [106] of the text.
The slabs are coloured like the twenty-seven cubes of the first cubic block in [fig. 101], the one with red, white, yellow axes.