84 LAMBERT— THE STRAIGHT LINE CONCEPT. [April i 4 , 



THE EXPRESSION FOR DISTANCE. 



Much of the apparent mystery of hyperbolic and elliptic geom- 

 etry vanishes when precision is given to the term distance. Dis- 

 tance is the result of measurement, and the measurement of a straight 

 line requires that any part of the straight line may be applied any- 

 where along the straight line. If A, B, Care any three points in 

 a straight line, and B is between A and C, the expression for dis- 

 tance must satisfy the equation 



distance AB + distance BC= distance AC. 



In a system of measurement introduced by Cayley in the Sixth 

 Memoir on Quantics and developed by Klein, the expression for 

 the distance between two points on a straight line is a function ot 

 the cross-ratio of these two points and two fixed points on the 

 straight line. Let the fixed points be X, YandA, B, Cany three 

 points taken in order on the straight line. By definition the 

 cross-ratio of the four points A, B, X, Fis 



(AX)/(AY)+(BX)/(BY). 



It follows from this definition that : 



cross-ratio ABXY X cross-ratio B CX Y= cross-ratio ACXY. 

 Applying logarithms to this equation 



log cross-ratio ABXY ' -\- log cross-ratio BCXY 



= log-cross-ratio A CYX. 

 The expression 



k log cross -ratio ABXY 



where k denotes any constant may therefore be taken as the expres- 

 sion for the distance between the points A, B} 



The pair of fixed points X, Fis called the absolute of linear 

 measurement. When one of the points A, B coincides with a 

 point of the absolute the distance AB becomes infinite. Hence 

 when the absolute consists of two distinct real points, the straight 

 line has two points at infinity ; when the absolute consists of two 

 coincident points, the straight line has one point at infinity ; when 



1 The constant k must be so determined that the expression for distance has a 

 real value. Since the logarithm is a many-valued function for which the series 

 of values differ by multiples of 2k\/— I, when k is imaginary the expression for 

 distance is a many-valued function for which the series of values differs by multi- 

 ples of some real constant. 



