98 FUNDAMENTAL PRINCIPLES OP MATHEMATICS. 



characteristics of our space. Thus it may be shown that such peculiar- 

 ities do indeed exist which depend on the completely free motion of 

 solid bodies without change of form, and on the particular value of the 

 measure of curvature. In the space under consideration this must be 

 put equal to zero — as it is, among surfaces, for the plane alone and 

 surfaces derived from it in order to give rise to the axioms of Euclid 

 concerning the singularity of the shortest line and the essential con- 

 ditions to parallelism. If the curvature was different from zero, triangles 

 of great area would, it is true, have for the sum of their angles a value 

 different from smaller ones; but the result of geometrical and astro- 

 nomical measurements, which always gives the sum of the angles of a 

 triangle as very near but never exactly equal to two right angles, 

 warrants us only in the conclusion that the measure of curvature for 

 our space is extremely small. It can not be proved that its value is 

 zero — it is an axiom. Helmholtz went, however, still further. He 

 showed that the consideration of a spherical or pseudospherical world 

 developed by analogy from the plane might be extended in all direc- 

 tions, so that the axioms of our geometry throughout can not be fixed 

 in their present form by our intuitive faculty. And he even made it 

 appear plausible that if our eyes were provided with suitable convex 

 glasses we might come to look upon the pseudospherical space as quite 

 natural, and that we would be deceived in our estimations of size and 

 distance only in the first few instances. 



These researches formed in part the subjects of some lectures deliv- 

 ered in Heidelberg in 1868. Twenty years later he referred again in 

 his article on "Shortest lines in the color system" to the results 

 obtained by himself and Eiemann. They found that all the character- 

 istics of our particular kind of space may be derived from the fact that 

 one may express the distance between two neighboring points in terms 

 of the corresponding increments of their coordinates. To know the 

 distance between two points of a solid it requires that these end points 

 shall be completely given, and the distance shall remain constant 

 through whatever movements and displacements the solid body be 

 subjected. Analogously, Helmholtz proposed to determine colors by the 

 quantities of three suitably chosen primary colors required to produce 

 them, these primary colors taking the place of coordinates. In this 

 analogy the conception of difference between two colors nearly alike 

 corresponded to the distance between points in space. He deduced a 

 very simple analytical expression which he hoped would play the same 

 role in expressing the difference in hue that the formula for the length 

 of the linear element does in geometry. This expression determined 

 the variation in brightness and hue which occurs corresponding to a 

 simultaneous change in the quantities of the three primary colors 

 uniting to produce it. Analogously to the shortest line between 

 two points he defined as the shortest series of colors that series of 

 intermediate shades between given end colors of different brightness 



