HERMANN VON HELMHOLTZ 



of a sphere, return on themselves. On such a 

 surface, also, as on a plane, only one shortest line 

 is possible between any two points, but the axiom 

 as to parallels does not hold good. Like the 

 plane and the sphere, it has its measure of curva- 

 ture constant, so that figures constructed on one 

 part of the surface can be transferred to any 

 other place with perfect congruence of form and 

 perfect equality of all dimensions lying on the 

 surface. 



It can also be shown, by a method devised by 

 Riemann, that the space in which we live is three- 

 fold, a surface is two-fold, and a line is an aggregate 

 of points. Time is also an aggregate of one dimen- 

 sion. Colour is an aggregate of three dimensions, 

 because each colour may be represented by a mixture 

 of three primary colours, taken in definite proportions, 

 as may be shown by Clerk Maxwell's colour top. 

 But Helmholtz goes still further. He shows that we 

 can depict the appearance of a spheric or of a pseudo- 

 spheric world, or such a world as we see in a con- 

 vex mirror. Suppose we looked at things through 

 specially-constructed convex glasses, we would at 

 first be confused, more especially by illusions of com- 

 parative size and distance, but, by experience, the 

 space would cease to be strange. It is clear, there- 

 fore, according to this argument, that the axioms of 

 geometry do not all hold good in different varieties 

 of space. Taken by themselves out of all connection 

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