Explanatory Notes in 



slightly to increase; at speeds expressed in thousands of 

 miles per second it begins to increase faster; and at the 

 speed of light it suddenly appears to become infinite 

 whatever that may mean. Hence there are some who 

 think that bodies can never move through the ether faster 

 than the velocity of light. 



Page 12 



In Euclidian Geometry only one straight line can be 

 drawn through a given point parallel to another straight 

 line. That no more than one is possible has never been 

 proven: it was a definite postulate or axiom made by 

 Euclid, but it seemed incapable of proof. Within recent 

 times pure mathematicians have found it possible to devise 

 other more general systems of Geometry, in which two or 

 many such lines can be drawn. Thus Euclidian Ge- 

 ometry, which still appears to suit our own spatial expe- 

 rience very well, can be regarded as a special case of more 

 generalised and comprehensive theoretical systems. 



Abstract propositions may be absolutely and completely 

 true: practical experience can approximate to them more 

 or less precisely in Geometry more precisely than in any 

 other subject. The relation between our systems of 

 thought on the one hand, and our actual experience on the 

 other, is well illustrated by the following quotation from 

 Poincare": 



"The principles of geometry are not experimental facts. 

 . . . Euclid's postulate cannot be proved by experiment. 

 ... If Lobatschewsky's geometry is true, the parallax of a 

 very distant star will be finite. If Riemann's is true, it will 

 be negative. These are results which seem within the 

 reach of experiment, and some have hoped that astronomi- 

 cal observations may enable us to decide between the 

 geometries. But what we call a straight line in astronomy 

 is simply the path of a ray of light. If, therefore, we were 



