86 LAMBERT— THE STRAIGHT LINE CONCEPT. [April i 4) 



of elliptic geometry x' 2 + y 2 + z' 2 -f 4a 2 /- = o ; the absolute of 

 parabolic geometry, x 2 -f- y 1 -f- r = o, t — o, again a common 

 limiting case of the absolute of hyperbolic and elliptic geometry. 



By taking a in the equation of the absolute sufficiently large the 

 hyperbolic and elliptic geometries approach identity with the para- 

 bolic geometry in finite regions of space, so that experience or 

 experiment can never determine that the space of experience is 

 hyperbolic, elliptic or parabolic. 



The expression for the distance between two points must satisfy 

 the requirement that the distance between two points shall be the 

 same for all positions of the straight line on which the two points 

 are located. A collinear motion of space into itself is represented 

 analytically by a linear transformation which transforms the abso- 

 lute into itself. The cross-ratio is an invariant of linear transforma- 

 tions. Hence the definition of distance k x log cross-ratio satisfies 

 also this requirement of the expression for distance. 



By the calculus of variations it is proved that in the elliptic, 

 hyperbolic and parabolic geometries the straight line is the shortest 

 distance between two points. Hilbert, by taking for absolute a 

 triangle, has proved that the sum of two sides of a triangle may be 

 equal to or less than the third side. 



ANGLE MEASUREMENT. 



In Cayley's system of measurement the measure of an angle is 

 defined as a constant times the logarithm of the cross-ratio of the 

 pencil of four rays formed by the sides of the angle and the 

 tangents to the absolute from the vertex of the angle. If the 

 equation of the absolute in line coordinates is /(//, v) = o, the 

 measurement of angles about the point of intersection of the lines 

 («j, i\) and (u 2 , 7' 2 ) is analytically identical with the measurement 

 of distance on a line through two points. 



It follows from the definition that a right angle is an angle 

 whose sides are harmonic conjugates with respect to the tangents 

 from the angle vertex to the absolute. In the hyperbolic geometry 

 any line through the pole of a given line with respect to the abso- 

 lute intersecting the given line is perpendicular to it ; the angle 

 between lines intersecting on the absolute is zero, hence the two 

 lines drawn from a given point to the intersections of a given line 

 with the absolute are parallel to the given line ; the sum of the 

 angles of a triangle is less than 180 . 



