262 HERMANN VON HELMHOLTZ 



as independent of direction, in the form laid down by Riemann, 

 and he shortly sums up the conditions required by the latter, 

 in saying that a point of an n-fo\d complex is determined by 

 n co-ordinates, that there is further an equation between the 

 2 n co-ordinates of any pair of points infinitely close together, 

 independent of their motion, which is identical for all congruent 

 pairs of points, and that, lastly, with otherwise perfectly free 

 mobility of the solid body, the property of monodromy of space 

 must be fulfilled, whereby when a solid body of n dimensions 

 rotates round n i fixed points the rotation shall bring it back 

 without reversal to its original position. And in applying these 

 conditions to the case of three independent variables, he is able 

 to show on purely analytical grounds that a homogeneous 

 function of the second degree exists between the increments of 

 the same, which persists unaltered during rotation, and which 

 accordingly gives a measure of the linear elements, independent 

 of direction. 



In his development of these considerations an error crept in 

 owing to Helmholtz's statement that if infinite extension of 

 space be required, no geometry other than the Euclidean is 

 possible, whereas Beltrami showed that the geometry of 

 Lobatschewsky is also admissible, by which in a space extended 

 infinitely in all directions, figures congruent with a given 

 figure can be constructed in all parts of the same, while, further, 

 only one shortest line is possible between any two points ; but 

 the axiom of parallel lines no longer holds. It is only when 

 the measure of spatial curvature is everywhere at zero value 

 that such a space corresponds with Euclid's axioms, and this 

 space is then termed by Helmholtz a plane space. If the 

 measure of curvature is constant and positive, we arrive at 

 spherical space, in which the straightest lines return upon 

 themselves, and there are no parallels; such a space is like 

 the surface of a sphere, unlimited but not of infinite magnitude. 

 If, lastly, the measure of curvature is constant and negative, 

 then in such pseudo-spherical surfaces the straightest lines 

 proceed to infinity, and in each planest surface a bundle of the 

 straightest lines can be drawn through every point, which do 

 not intersect any other given straightest line of the same 

 surface. In a space of which the measure of curvature is other 

 than zero, triangles of large superficies will have a different 



