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. The colours of the three sets of twenty-seven cubes are those of the cubes shown in fig. 101. The slabs are used to form the representation of a cube in a plane, and can well be dispensed with by any one who is accustomed to deal with solid figures. But the whole theory depends on a careful observation of how the cube would be represented by these slabs. In the first step, that of forming a clear idea how a plane being would represent three-dimensional space, only one of the catalogue cubes and one of the three blocks is needed. APPLICATION TO THE STEP FROM PLANE TO SOLID. Look at fig. 1 of the views of the tesseract, or, what comes to the same thing, take catalogue cube No. 1 and place it before you with the red line running up, the white line running to the right, the yellow line running away. The three dimensions of space are then marked out by these lines or axes. Now take a piece of cardboard, or a book, and place it so that it forms a wall extending up and down not opposite to you, but running away parallel to the wall of the room on your left hand. Placing the catalogue cube against this wall we see that it comes into contact with it by the red and yellow lines, and by the included orange face.