PROFESSOR AT HEIDELBERG 259 



magnitudes must be assumed in analytical geometry, in order 

 to establish its propositions completely from the outset. He 

 was then able to consider the possibility of the logical formula- 

 tion of a different system of axioms, since the necessary cal- 

 culation of analytical geometry is a purely logical operation, 

 incapable of yielding any relation between the magnitudes in- 

 volved in it, other than those already contained in the equations 

 proposed for the calculation. 



It has been shown by Gauss that while the square of the 

 length of a linear element in a plane is expressed by the sum 

 of the squares of the increments of the two rectangular co- 

 ordinates, the square of a linear element upon any given surface 

 appears as a homogeneous function of the second degree of the 

 increments of two general co-ordinates, which determine the 

 situation of a point upon a surface. If figures of finite magni- 

 tude are to be movable towards all parts of such a surface 

 without alteration in their measurements as made upon the 

 surface itself, and capable of rotation round any given point, 

 then further the surfaces must have a constant measure of 

 curvature at every point, the measure of curvature of the 

 surface at any point being defined by Gauss as the reciprocal 

 ratio of an infinitely small part of the surface surrounding this 

 point to that part of the surface which is drawn through spherical 

 radii parallel to the normals, upon the unit-sphere. But even 

 upon surfaces with a constant measure of curvature, where free 

 mobility of the figures is possible, geometry would assume a 

 form wholly different from our geometry. 



Helmholtz starts with the assumption that as inhabitants 

 of tri-dimensional space it is possible for us to conceive 

 the various ways in which beings living in a surface would 

 form their conceptions of space, and to picture to ourselves 

 their sensory impressions ; spaces of more than three dimen- 

 sions are, however, inconceivable to us, since all our means 

 of sensory perception extend only to tri-dimensional space; 

 and then he goes on to consider geometry as it would appear 

 to intelligent beings of only two dimensions. 



He propounds the question what would become of the axioms of 

 our geometry, as, that there is only one shortest distance between 

 any two points in space, the straight line ; that, further, through 

 any three points that do not lie in a straight line, a plane surface 



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