GEOMETEICAL AXIOMS. 55 



imagine such spaces of other sorts, it cannot be main- 

 tained that the axioms of geometry are necessary con- 

 sequences of an a priori transcendental form of intui- 

 tion, as Kant thought. 



The distinction between spherical, pseudospherical, 

 and Euclid's geometry depends, as was above observed, 

 on the value of a certain constant called, by Eiemann, 

 the measure of curvature of the space in question. 

 The value must be zero for Euclid's axioms to hold 

 good. If it were not zero, the sum of the angles of 

 a large triangle would differ from that of the angles of 

 a small one, being larger in spherical, smaller in pseu- 

 dospherical, space. Again, the geometrical similarity 

 of large and small solids or figures is possible only in 

 Euclid's space. All systems of practical mensuration 

 that have been used for the angles of large rectilinear 

 triangles, and especially all systems of astronomical 

 measurement which make the parallax of the im- 

 measurably distant fixed stars equal to zero (in pseudo- 

 spherical space the parallax even of infinitely distant 

 points would be positive), confirm empirically the 

 axiom of parallels, and show the measure of curvature 

 of our space thus far to be indistinguishable from zero 

 It remains, however, a question, as Biemann observed, 

 whether the result might not be different if we could 

 use other than our limited base-lines, the greatest oi 

 which is the major axis of the earth's orbit. 



