The sections between the front and rear triangle, of which one is shown in 1b, another in 2b, are thus named, points and lines, salan, salat, salet, satet, satel, satal, sanal, sanat, sanit, satit, satin, satan, salan. The rear triangle found in 3b by producing lines is sil, sitet, sinel, sinat, sinin, sitan, sil. The assemblage of sections constitute the solid body of the octahedron satat with triangular faces. The one from the line selat to the point sil, for instance, is named selin, selat, selel, salet, salat, salan, sil. The whole interior is salat. Shapes can easily be cut out of cardboard which, when folded together, form not only the tetrahedron and the octahedron, but also samples of all the sections of the tesseract taken as it passes cornerwise through our space. To name and visualise with appropriate colours a series of these sections is an admirable exercise for obtaining familiarity with the subject. EXTENSION AND CONNECTION WITH NUMBERS. By extending the letter sequence it is of course possible to name a larger field. By using the limit names the corners of each square can be named. Thus “en sen,” “an sen,” etc., will be the names of the points nearest the origin in “en” and in “an.” A field of points of which each one is indefinitely small is given by the names written below. [Illustration] The squares are shown in dotted lines, the names denote the points. These points are not mathematical points, but really minute areas. Instead of starting with a set of squares and naming them, we can start with a set of points. By an easily remembered convention we can give names to such a region of points. Let the space names with a final “e” added denote the mathematical points at the corner of each square nearest the origin. We have then for the set of mathematical points indicated. This system is really completely independent of the area system and is connected with it merely for the purpose of facilitating the memory processes. The word “ene” is pronounced like “eny,” with just sufficient attention to the final vowel to distinguish it from the word “en.” [Illustration] Now, connecting the numbers 0, 1, 2 with the sequence e, a, i, and also with the sequence n, t, l, we have a set of points named as with numbers in a co-ordinate system. Thus “ene” is (0, 0) “ate” is (1, 1) “ite” is (2, 1). To pass to the area system the rule is that the name of the square is formed from the name of its point nearest to the origin by dropping the final e. By using a notation analogous to the decimal system a larger field of points can be named. It remains to assign a letter sequence to the numbers from positive 0 to positive 9, and from negative 0 to negative 9, to obtain a system which can be used to denote both the usual co-ordinate system of mapping and a system of named squares. The names denoting the points all end with e. Those that denote squares end with a consonant.