THE FOURTH DIMENSION 



385 



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y 



measured perpendicular to the X-axis are positive if measured above and 

 negative if below the X-axis, such, a distance being called the ordinate 

 of the given point. 



Thus it is evident that a point is fully determined in the plane if 

 its abscissa and ordinate are given. Every school boy has met with this 

 principle in the location of a place on the 

 earth's surface by latitude and longitude, 

 where the axes of reference are great circles 

 of the globe. 



Now our duodim has a far more extended 

 space than the unodim, and can do many 

 things that the unodim is totally ignorant of. 

 His space may not necessarily be one of zero 

 curvature, — for it is perfectly consistent with 

 our definition of 2-space for it to be the 



surface of a sphere, of an ellipsoid, of an egg-shaped figure, or what not. 

 It is to be noticed that if the space has constant curvature (including 

 no curvature), a body may be moved from any place to any other place 

 on the surface without changing its shape. 



If there are (-Fig. 5) two triangles like E and F in which the sides 



Fig. 4. 



Fig. 5. 



are equal each to each, but are arranged in reversed order, it is impos- 

 sible in 2-space for F to be made to take the position of E. Here E 

 is the reflection of F in a mirror. An inhabitant of 3-space has no diffi- 

 cult}^, however, in taking up F, turning it over and putting it on the 

 position E. 



A three-dimensional body as such is of course invisible to a 2-space 

 being. If a 3-space body, say a cube, crosses a 2-space, the 2-space 

 being is conscious only of its section with his world. 



Fig. 6 represents the effect of a 3-space body, a cube, passing through 

 the 2-space pictured by the plane " mn." The section ABCD of the 

 plane mn with the cube G is all that a duodim would be conscious of. 

 As G passes through mn, this section, certainly if G is a homogeneous 

 body, will appear the same until suddenly it vanishes as G passes 

 bevond mn. 



