36 OEIGIN AND SIGNIFICANCE OF 



of my non-mathematical hearers without giving rise 

 to misconception. 



Now if beings of this kind lived on an infinite 

 plane, their geometry would be exactly the same as 

 our planimetry. They would affirm that only one 

 straight line is possible between two points ; that 

 through a third point lying without this line only one 

 line can be drawn parallel to it ; that the ends of a 

 straight line never meet though it is produced to 

 infinity, and soon. Their space might be infinitely ex- 

 tended, but even if there were limits to their move- 

 ment and perception, they would be able to represent 

 to themselves a continuation beyond these limits ; and 

 thus their space would appear to them infinitely ex- 

 tended, just as ours does to us, although our bodies 

 cannot leave the earth, and our sight only reaches as 

 far as the visible fixed stars. 



But intelligent beings of the kind supposed might 

 also live on the surface of a sphere. Their shortest or 

 straightest line between two points would then be an 

 arc of the great circle passing through them. Every 

 great circle, passing through two points, is by these 

 divided into two parts ; and if they are unequal, the 

 shorter is certainly the shortest line on the sphere be- 

 tween the two points, but also the other or larger arc 

 of the same great circle is a geodetic or straightest 

 line, i.e. every smaller part of it is the shortest line 



