THE FOURTH DIMENSION. 



in like manner of the congruence of triangles I and III. It would 

 discover the equality of the three sides and the three angles, but it 

 could never succeed in so superposing the two triangles on each 

 other as to make them coincide. A three-dimensional being, how- 

 ever, can do this very easily. It has simply to turn triangle I about 

 one of its sides and to shove the triangle, thus brought into the po- 

 sition of its reflexion in a mirror, into the position of triangle III. 

 Similarly, triangles II and III may be made to coincide by moving 

 either out of the plane of the paper around one of its sides as axis 

 and turning it until it again falls in the plane of the paper. The 

 triangle thus turned over can then be brought into the position of 

 the other. 



Later on we shall revert to these two kinds of congruence : 

 "congruence by displacement" and "congruence by circumversion." 

 For the present we will start from the fact that it is always possible 

 within the limits of a plane to take a triangle out of one position 

 and bring it into another without altering its sides and angles. The 

 question is, whether this is only possible in the plane, or whether 

 it can also be done on other surfaces. 



We find that there are certain surfaces in which this is possi- 

 ble, and certain others in which it is not. For instance, it is im- 

 possible to move the triangle drawn on the surface of an egg into 

 some other position on the egg's surface without a distension or 

 contraction of some of the triangle's parts. On the other hand, it 

 is quite possible to move the triangle drawn on the surface of a 

 sphere into any other position on the sphere's surface without a 

 distension or contraction of its parts. The mathematical reason of 

 this fact is, that the surface of a sphere, like the plane, has every- 

 where the same curvature, but that the surface of an egg at differ- 

 ent places has different curvatures. Of a plane we say that it has 

 everywhere the curvature zero ; of the surface of a sphere we say it 

 has everywhere a positive curvature, which is greater in proportion 

 as the radius is smaller. There are surfaces also which have a con- 

 stant negative curvature ; these surfaces exhibit at every point in 

 directions proceeding from the same side a partly concave and a 

 partly convex structure, somewhat like the centre of a saddle. 



