46 



KNOWLEDGE & SCIENTIFIC NEWS. 



[Mar., 1904. 



as a 

 point, line, 

 meaningless 



ward and backward directions which are neither of these' 

 and which are extraspatial ? But, except in this figura- 

 tive sense, the introduction of time as a solution of the 

 fourth-dimensional question is merely a confusion of the 

 problem. Time does not belong to the same category of 

 thought with length, breadth, and thickness. Point, 

 line, surface, solid - these follow each other 

 development in orderly sequence, but 

 surface, solid, time— these terminate in a 

 1:011 sequitiir. 



Again, we may say that as a point is to a line so is a 

 line to a surface, or we may say that as a line is to a 

 surface so is a surface to a solid. This is intelligible, but 

 to add, as a surface is to a solid, so is a solid to time, or 

 to any portion of time, is unintelligible nonsense. More- 

 over, the concept of time does not strike one as being (as 

 the conception of a sphere would be) impossible to 

 Abbott's Flatlanders, whereas it ought to seem even 

 more so if it were actually two dimensions ahead of 

 them. Indeed, it ought to seem impossible to oursehes, 

 unless we are fourth dimensional beings. 



But without confusing the issue by incongruously 

 introducing the concept of Time into the province of 

 Space, let us see what may reasonably be conjectured as 

 to tourth-dimensional existence. 



The elements of the inquiry are strikmgly illustrated 



by Hinton in some such way as the following : 



Two points joined = One line. 

 Two lines joined = One square. 

 Two squares joined = One cube. 

 Two cubes joined = One (?). 

 The series may iie set out in this way: — 



tion of more than three-dimensional form. The figure for 

 the super-cube would, in fact, be like this:— 



Fig. I. 



As the drawing of a cube to a Flatlander would seem 

 to be only two squares on the same surface united by 

 hnes also on that surface, so to us the above figure can at 

 most only convey the idea of two cubes united by lines or 

 perhaps by surfaces. We shall see this better if we draw 

 the figure m perspective stereoscopically and examine the 

 result m the stereoscope, when the two drawings will 

 blend into one apparent solid. Here is such a stereoscopic 

 diagram of the super-cube : — 



No, of Terminal 

 Himensions Points. 



Joining 

 Lines. 



Point 



Line 



S(|uare 



Cube 



Siij)er-rul)e 



n 



16 



Before we proceed to deal with the perspective repre- 

 sentation of the last in this series of figures, the super-cube, 

 it will be well to put ourselves back in imagination into 

 .Abbott's Fldtland, and to consider what would be our 

 b'latlander's impression of the perspective representation 

 (jf the ( ube. '• Here is no third dimension," he would 

 say ; ■' here are but two squares with lines joining them." 

 lo us, who are accustomed mentally to connect such a 

 figure with the similar retinal image which a solid cube 

 forms in our eye, the concept of a third dimension is 

 conveyed by association of ideas, but with the Flat- 

 lander no such association of ideas would exist, because 

 he would have had no experience of " thickness," 



him a Flatland 

 ines. .Similarly, 

 a full-face view of a cube would of course be to 

 him simply a square, and in fact cannot be otherwise 

 rendered on a flat surface. 



Now our relations to fourth-dimensional diagrams must 

 be analogous. It is possible that we miglit make a 

 pictorial rendering of a super-cube on paper, which to a 

 being with senses capable of appreciating fourth-dimen- 

 sional space would be suggestive of a fourth-dimensional 

 super-solid, but to us, with no association of ideas to aid 

 us, the figure must not be expected to afford a representa- 



experience 

 and the figure would remain for 

 one — two squares joined by four " 



Fig. 2. 



This slide, when seen in the stereoscope, shows us a 

 peculiar looking figure, apparently three-dimensional. 

 Now just is the perspective drawing of the cube suggests 

 one tliree-dimensional figure to us but to the Flatlander a 

 pair of united squares, so the above stereogram repre- 

 sents a pair of united solids to us, but to beings with 

 fourth-dimensional perception it might convey the notion 

 of one super solid. 



It becomes e\'ident, therefore, that while our senses are 

 (as at present) limited to three dimensions, we cannot ex- 

 pect in the way thus far indicated to get any nearer to a 

 concept of the super-cube. 



\'et there remains an experiment which carries us just 

 a step further, and brings us to the \'ery verge of a solu- 

 tion of our problem. 



Before we make this experiment it will -again elucidate 

 the matter if once more we imagine ourselves for the 

 moment in Flatland. There the drawing of a cube directly 

 facing us would, as we have seen, be only one square, or 

 more strictly one square exactly behind another. To the 

 Flatlander, who does not know what " behind " means, it 

 would be as though the two squares occupied the same 

 space at the same time. 



Now the analogy from this is obvious, for in the same 

 way under similar < ircumstances the super-cube of fourth- 



