44 ORIGIN AND SIGNIFICANCE OF 



can represent to ourselves the various cases in which 

 beings on a surface might have to develop their per- 

 ception of space ; for we have only to limit our own 

 perceptions to a narrower field. It is easy to think 

 away perceptions that we have ; but it is very difficult 

 to imagine perceptions to which there is nothing ana- 

 logous in our experience. When, therefore, we pass to 

 space of three dimensions, we are stopped in our power 

 of representation, by the structure of our organs and 

 the experiences got through them which correspond 

 only to the space in which we live. 



There is however another way of treating geometry 

 scientifically. All known space-relations are measur- 

 able, that is, they may be brought to determination of 

 magnitudes (lines, angles, surfaces, volumes). Problems 

 in geometry can therefore be solved, by finding methods 

 of calculation for arriving at unknown magnitudes from 

 known ones. This is done in analytical geometry, where 

 all forms of space are treated only as quantities and 

 determined by means of other quantities. Even the 

 axioms themselves make reference to magnitudes. The 

 straight line is defined as the shortest between two 

 points, which is a determination of quantity. The 

 axiom of parallels declares that if two straight lines in 

 a plane do not intersect (are parallel), the alternate 

 angles, or the corresponding angles, made by a third 

 line intersecting them, are equal; or it may be laid 



