Williams — On the. Fourth Dimension. 



509 



D 



Use has accustomed us to the fact that we may represent 



by projection the figure traced by the square ABCD when 



moved in a direction (parallel 



to AE) perpendicular to all 



the lines contained in itself. 



And in a projection there is 



nothing to hinder us from 



moving the cube AG, thus 



obtained, in a direction re- 

 presented by AK, which shall 



be at right angles to AE, AB, 



and AD. In fact, we might, 



had we been so disposed, 



have moved our square AC 



in that direction, and traced, 



instead of AG, the cube AM, without doing any violence 



to our notions of the propriety 

 of our dealings with the projec- 

 tion. By moving, then, either 

 the cube AG in a direction AK 

 perpendicular to each of its 

 sides, or the cube AM in a 

 direction AE perpendicular to 

 each of its sides, we shall ob- 

 tain a projection of the tessaract 

 AQ which will be found to have 

 all the defining elements which 

 are contained in the table above. 

 There are the eight cubes, AM, 

 EQ, AR, BQ, KF, NG, AG, KQ : 

 the twenty -four squares, AC, 



EG, KM, OQ; AL, EP, DM, HQ ; AN, ER, BM, EQ ; AH, 



KR, BG, LQ; AO, BP, 



DR, CQ; AF, KP, DG, f 



NQ: the thirty-two lines, 



AE, KO, DH, NR, BF, 



LP, CG, MQ, AB, KL, 



EF, OP, HG, RQ, DC, 



NM, AK, EO, DN, HR, 



BL, FP, CM, GQ, AD, 



KN, EH, OR, BO, LM, 



FG, PQ : and the sixteen 



points, A, B, C, D, E, F, 



G, H, K, L, M, N, 0, P, 



Q, R- 



As has been indicated 

 above, any one of the 

 eight cubes here enume- 



D 



N 



M 



