Call the large squares in fig. 2 by the name written in them. It is evident that each can be divided as shown in fig. 1. Then the small square marked 1 will be “en” in “En,” or “Enen.” The square marked 2 will be “et” in “En” or “Enet,” while the square marked 4 will be “en” in “Et” or “Eten.” Thus the square 5 will be called “Ilil.” [Illustration: Fig. 2.] This principle of extension can be applied in any number of dimensions. 2. _Application to Three-Dimensional Space._ To name a three-dimensional collocation of cubes take the upward direction first, secondly the direction towards the observer, thirdly the direction to his right hand. [Illustration] These form a word in which the first letter gives the place of the cube upwards, the second letter its place towards the observer, the third letter its place to the right. We have thus the following scheme, which represents the set of cubes of column 1, fig. 101, page 165. We begin with the remote lowest cube at the left hand, where the asterisk is placed (this proves to be by far the most convenient origin to take for the normal system). Thus “nen” is a “null” cube, “ten” a red cube on it, and “len” a “null” cube above “ten.” By using a more extended sequence of consonants and vowels a larger set of cubes can be named. To name a four-dimensional block of tesseracts it is simply necessary to prefix an “e,” an “a,” or an “i” to the cube names. Thus the tesseract blocks schematically represented on page 165, fig. 101 are named as follows:— [Illustration: 1 2 3] 2. DERIVATION OF POINT, LINE, FACE, ETC., NAMES. [Illustration] The principle of derivation can be shown as follows: Taking the square of squares the number of squares in it can be enlarged and the whole kept the same size. [Illustration] Compare fig. 79, p. 138, for instance, or the bottom layer of fig. 84. Now use an initial “s” to denote the result of carrying this process on to a great extent, and we obtain the limit names, that is the point, line, area names for a square. “Sat” is the whole interior. The corners are “sen,” “sel,” “sin,” “sil,” while the lines are “san,” “sal,” “set,” “sit.” [Illustration] I find that by the use of the initial “s” these names come to be practically entirely disconnected with the systematic names for the square from which they are derived. They are easy to learn, and when learned can be used readily with the axes running in any direction. To derive the limit names for a four-dimensional rectangular figure, like the tesseract, is a simple extension of this process. These point, line, etc., names include those which apply to a cube, as will be evident on inspection of the first cube of the diagrams which follow.