All that is necessary is to place an “s” before each of the names given for a tesseract block. We then obtain apellatives which, like the colour names on page 174, fig. 103, apply to all the points, lines, faces, solids, and to the hyper-solid of the tesseract. These names have the advantage over the colour marks that each point, line, etc., has its own individual name. In the diagrams I give the names corresponding to the positions shown in the coloured plate or described on p. 174. By comparing cubes 1, 2, 3 with the first row of cubes in the coloured plate, the systematic names of each of the points, lines, faces, etc., can be determined. The asterisk shows the origin from which the names run. These point, line, face, etc., names should be used in connection with the corresponding colours. The names should call up coloured images of the parts named in their right connection. [Illustration] It is found that a certain abbreviation adds vividness of distinction to these names. If the final “en” be dropped wherever it occurs the system is improved. Thus instead of “senen,” “seten,” “selen,” it is preferable to abbreviate to “sen,” “set,” “sel,” and also use “san,” “sin” for “sanen,” “sinen.” [Illustration] [Illustration] We can now name any section. Take _e.g._ the line in the first cube from senin to senel, we should call the line running from senin to senel, senin senat senel, a line light yellow in colour with null points. [Illustration] Here senat is the name for all of the line except its ends. Using “senat” in this way does not mean that the line is the whole of senat, but what there is of it is senat. It is a part of the senat region. Thus also the triangle, which has its three vertices in senin, senel, selen, is named thus: Area: setat. Sides: setan, senat, setet. Vertices: senin, senel, sel. The tetrahedron section of the tesseract can be thought of as a series of plane sections in the successive sections of the tesseract shown in fig. 114, p. 191. In b_{0} the section is the one written above. In b_{1} the section is made by a plane which cuts the three edges from sanen intermediate of their lengths and thus will be: Area: satat. Sides: satan, sanat, satet. Vertices: sanan, sanet, sat. The sections in b_{2}, b_{3} will be like the section in b_{1} but smaller. Finally in b_{4} the section plane simply passes through the corner named sin. Hence, putting these sections together in their right relation, from the face setat, surrounded by the lines and points mentioned above, there run: 3 faces: satan, sanat, satet 3 lines: sanan, sanet, sat and these faces and lines run to the point sin. Thus the tetrahedron is completely named. The octahedron section of the tesseract, which can be traced from fig. 72, p. 129 by extending the lines there drawn, is named: Front triangle selin, selat, selel, setal, senil, setit, selin with area setat.