42 THE ANATOMY OF SCIENCE 



ducible from the material used by Euclid, with 

 the single elimination of this one postulate. 



For our purpose we may express the postu- 

 lates of Euclid in simpler form. 



1. Through two points there is a unique line 

 and this may be called the straight line. 



2. About any point circles may be described. 



3. Through any point outside a line one and 

 only one parallel can be drawn. 



It is the last postulate which is abandoned in 

 the non-Euclidean geometries of which I have 

 spoken. In the geometry of Lobachevski more 

 than one parallel can be drawn ; in that of Rie- 

 mann none can be drawn. The geometry of 

 navigation which we discussed in our allegory 

 belongs to the last-named class. In it the 

 straight line is what we call a great circle, and, 

 as you know, no two great circles are parallel 

 to each other. 



The geometry of navigation would be differ- 

 ent on a small planet and on a large one. It 

 would depend upon what we call the curvature 

 of the surface. And all of these non-Euclidean 

 geometries of which I have spoken involve a 

 certain absolute magnitude, which by analogy 

 is called curvature. So also these are called 

 "curved" geometries as distinguished from the 



