GEOMETRICAL AXIOMS. 47 



dation of any system of geometry, is the expression 

 that it gives for the distance between two points lying 

 in any direction towards one another, beginning with 

 the infinitesimal interval. He took from analytical 

 geometry the most general form for this expression, 

 that, namely, which leaves altogether open the kind of 

 measurements by which the position of any point is 

 given. 1 Then he showed that the kind of free mobi- 

 lity without change of form which belongs to bodies 

 in our space can only exist when certain quantities 

 yielded by the calculation 2 quantities that coincide 

 with Gauss's measure of surface-curvature when they 

 are expressed for surfaces have everywhere an equal 

 value. For this reason Biemann calls these quantities, 

 when they have the same value in all directions for a 

 particular spot, the measure of curvature of the space 

 at this spot. To prevent misunderstanding, 3 I will 

 once -more observe that this so-called measure of 

 space-curvature is a quantity obtained by purely ana- 

 lytical calculation, and that its introduction involves no 

 suggestion of relations that would have a meaning 

 only for sense-perception. The name is merely taken, 



1 For the square of the distance of two infinitely near points the 

 expression is a homogeneous quadric function of the differentials of 

 their co-ordinates. 



2 .They are algebraical expressions compounded from the co- 

 efficients of the various terms in the expression for the square of the 

 distance of two contiguous points and from their differential quotients. 



3 As occurs, for instance, in the above-mentioned work of Tobias, 

 pp. 70, etc. 



