2 6o HERMANN VON HELMHOLTZ 



can be drawn, which must entirely contain the straight lines 

 joining any two of these points ; and, lastly, that through any 

 point lying outside a straight line only one straight line can be 

 drawn, parallel to it, and never cutting it. The two-dimensional 

 being would indeed be able, as a rule, to draw shortest lines 

 between two points, which he terms ' straightest ' lines, but 

 even in the simplest case of the sphere, an infinite number of 

 straightest lines could be drawn between any two poles ; parallel 

 straightest lines that did not intersect could not be drawn at all, 

 and the sum of the angles of a triangle would always be greater 

 than two right angles, and the more so, the larger the surface 

 of the triangle. The space of these beings would no doubt be 

 unlimited, but it would be found to have finite extension, or 

 at any rate be postulated as having it. Only when the constant 

 measure of curvature is of zero value, i, e. when, according to 

 Gauss, the surface can be spread out on a plane by flexion 

 without extension or disruption, would our geometry hold 

 good. 



Both for Riemann and Helmholtz, however, the question 

 of primary importance was not under what conditions our 

 geometrical axioms might be valid, but under what hitherto 

 not clearly explained conditions we arrived at the knowledge 

 of them. Riemann shows how by a generalization from tri- 

 dimensional space the universal properties of space, its con- 

 tinuity, and the multiplicity of its dimensions, could be expressed, 

 by saying that each particular in the complex which it presents^ 

 i.e. each point, could be determined by measuring n con- 

 tinuously and independently variable magnitudes, which are 

 its co-ordinates, so that space becomes an n-times extended 

 complex, and we ascribe to it n dimensions. Riemann adds 

 as a further necessity that the length of a line must be in- 

 dependent of place and direction, so that every line must be 

 measurable by every other, and since in our actual space the 

 measure of each linear element is the square root of a homo- 

 geneous function of the second degree of the increments of three 

 measurements of whatever kind, he starts in his general 

 investigation from this form of linear element as if it were 

 hypothetical. He finally generalizes the definition of the 

 measure of curvature for w-dimensional space, and shows that, 

 if he adds the final condition that spatial figures shall every- 



