GEOMETRICAL AXIOMS. 39 



every kind of surface. The condition under which a 

 surface possesses this important property was pointed 

 out by Gauss in his celebrated treatise on the cur- 

 vature of surfaces. 1 The ' measure of curvature,' as he 

 called it, i.e. the reciprocal of the product of the 

 greatest and least radii of curvature, must be every- 

 where equal over the whole extent of the surface. 



Gauss showed at the same time that this measure 

 of curvature is not changed if the surface is bent with- 

 out distension or contraction of any part of it. Thus 

 we can roll up a flat sheet of paper into the form of 

 a cylinder, or of a cone, without any change in the 

 dimensions of the figures taken along the surface of 

 the sheet. Or the hemispherical fundus of a bladder 

 may be rolled into a spindle-shape without altering the 

 dimensions on the surface. Geometry on a plane will 

 therefore be the same as on a cylindrical surface ; only 

 in the latter case we must imagine that any number of 

 layers of this surface, like the layers of a rolled sheet 

 of paper, lie one upon another, and that after each 

 entire revolution round the cylinder a new layer is 

 reached different from the previous ones. 



These observations are necessary to give the reader a 

 notion of a kind of surface the geometry of which is on 

 the whole similar to that of the plane, but in which 



1 Gauss, Werke, Bd. IV. p. 215, first published in Commentatianes 

 Soc. Reg. Scientt. Gottengensis recentiorcs, vol. vi., 1828. 



