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TEE POPULAR SCIENCE MONTHLY 



any curvature. It is well to note that in an exactly similar way our 

 space may be curved without our being conscious of it. So he might 

 feel just as certain that his space is "straight-line" space as we, the 

 high and mighty 3-space beings, do that our space is Euclidean (or 

 space of zero curvature). 



Again, suppose the tube to be bent in an egg-shaped curve where 

 the curvature is not constant. Here the unodim's world would still 

 be one-dimensional, but as his body would be bent a little more in one 

 part of his world than in another, it is possible that he may feel that 

 there is some variety in his space. He may walk a little straighter at 

 times and less straight at others. Whether his 1-space is straight or 

 curved, and, if curved, whatever may be the variety of its convolu- 

 tions, the unodim can not know of the existence of a world of 2-space or 

 3-space. 



If a 2-space body, say a square, passes through his 1-space world, 

 he sees only the 1-space section of the square. 



In Fig. 3, xy, the 1-space world, is represented as being in the same 



Fig. 3. 



plane with the square mn. The square may cut xy at right angles 

 or obliquely. In any case the unodim sees at any moment only the 

 part of the square common to his world and is not conscious that there 

 is any more to the square. 



Two-space 



Nest let us consider 2-space. 



Assume a 2-space being, which we shall call a duodim. that is, a 

 flat being (theoretically with no thickness) with length and breadth 

 and confined to a surface having length and breadth but no thickness. 

 Such a being could move to the right or left or forward or baclcward, 

 we will say, but neither up nor down from the surface. In fact, he 

 knows neither vp nor down : the surface is his world. 



His position in his world is easily located by the Cartesian system 

 of coordinates, that is, with reference to its distance from, say, two 

 straight lines at right angles to each other. For illustration, define his 

 world as the geometrical plane formed by the two lines xx', yy' inter- 

 secting eacli other at right angles. Enijiloying the usual notation, we 

 consider distances measured perpendicular to the F-axis as positive 

 if measured to the right, and negative if to the left of the F-axis. Such 

 a distance is called the abscissa of the given point. Similarly, distances 



