May 13, 1892.] 



SCIENCE. 



273 



ceed to make models of the faces as follows (Fig. 1). The 

 side ab of the original square abed traces the square abfe, 

 which he places, as in the figure, in the only position known 

 to him subject to the condition that ab is one of its sides. 

 The three other squares are similarly placed as in the figure, 

 and now five of the six squares are shown in positions which 

 are correct with reference to their generating lines. But the 

 corner a is in this figure represented as the generator of two 

 lines ae, which is evidently incorrect. The outer squares 



are therefore to be turned through 90° about their generating 

 lines until the two lines ae become one and the four spaces 

 between ee, ff, gg, hh. disappear. He cannot imagine how 

 this is to be done, but he can suppose the central square to 

 move away and disappear in the to him unknown direc- 

 tion, carrying with it the outer squares which would then 

 appear to sink into the centre and disappear as they reached 

 their generating lines until at last the lines e/, fg, gh, he 

 reach the position now occupied by the sides of the square 

 abed and become in the picture, what they are really, the 

 sides of the sixth square efgh. Supposing, in the next place, 

 that the square abed as it moves away is still visible, but 

 smaller by perspective, the plane being could construct a 

 model which is to us a perspective view of a cube and which 

 would represent to him fairly well the relations of the bojin- 

 daries of a cube (Fig. 2). 



Let us proceed in the same way. A cube moving in a 

 direction perdendicular to all of its faces traces out a rectangu- 

 lar tessaract. Each face traces a new cube, each line a 

 square, each point a line. Counting up we find the tessaract 



/ 



a i 

 rf c 



is bounded by 8 cubes; has 24 squares, which do not enclose 

 the tessaract, but appear here and there as lines do upon a 

 cube, interfaces, not surfaces; 32 lines or edges, and 16 

 angular points. A little calculation now shows that each 

 face is common to two cubes, each line to three faces, and 

 each point to four lines. All this seems very abstract, but it 

 becomes real and evident when we make a model. Placing 

 a cube on each face of the original cube, after the analogy 

 of the plane being's squares, we have these six cubes in the 

 only positions known to us which satisfy their genetic con- 



ditions (see Fig. 3). The eighth cube is represented by the 

 outer faces of the six cubes, and it is evident that the three 

 lines marked cC are really one, the two faces bC are one 

 face, and so on. We may now imagine the central cube to 

 move away in the fourth dimension and the others sink in- 

 ward and disappear as they reach the present boundaries of 

 the central cube, where they turn at a right-angle into the 

 new direction. Finally all the outer faces will meet as the 

 boundaries of the eighth cube DF. Supposing the cubes 

 elastic, we may stretch their outer faces and diminish the 

 inner until we obtain the perspective view of a tessaract, as 

 shown in Fig. 4, where the relations of the various boun- 

 daries of the tessaract are more easily studied. Incidentally 

 we have learned also that a solid section of the tessaract, 

 when taken parallel to a cube-boundary, is a cube. 



The Second Lesson. 



Turning again to our imaginary plane being for sugges- 

 tions, let us see how motion in the third dimension would 

 appear to him. If a cylinder were passing perpendicularly 

 through his plane he would see only a stationary circle, or 

 if it were oblique, a moving ellipse. A cone would appear 

 as a growing or diminishing circle, a beaded rod as an oscil- 

 lating circle, a corkscrew as an ellipse moving in a circular 

 orbit, and so forth. The stem of a dichotoraous tree would 

 be to him a wooden circle which, as the branches approach. 



widens out, becomes constricted, and finally divides into two 

 circles, which repeat the process indefinitely. We may im- 

 agine a plane philosopher who, after watching this process 

 for some time, constructs a theory of the evolution of circles. 

 But his idea that all these circles have been developed from 

 one is hardly more than a caricature of the truth. 



Every person who has watched the self-division of in- 

 fusoria under the microscope must be struck with the analogy 

 of these two processes. A little reflection enables us to see 

 that race-unity may be more than a figure of speech or a 

 creation of the fancy; that the organic forms that existed for 

 us yesterday and those that will exist for us to-morrow may 

 be but parts of larger units of which the forms we see to-day 

 are only solid sections. True, this is only a suggestion; but 

 it is a suggestion that carries with it an unavoidable sense of 

 freedom, of fetters loosed, of largeness, and of reality, to any- 

 one who will for a time yield himself to its influence. It is 

 a step toward the poet's view, 



" All are but parts of one stupendous whole. 

 Whose body Nature is, and God the soul." 



If four-fold space exists, it is evident that it must contain 

 an infinite variety of three- fold spaces, of which we know 

 only one. It must also be everywhere possible for a four^ 

 fold being to step out of our space at any point and re-enter 



