i9°5-] 



LAMBERT— THE STRAIGHT LINE CONCEPT. 85 



the absolute consists of a pair of imaginary points, the straight 

 line has no point at infinity. 



In the geometry of two dimensions the absolute must be the locus 

 of the point pairs which are the absolute of all the lines in the 

 plane. It follows that the points on the absolute are the points at 

 infinity in the plane. By substituting in the equation of the abso- 

 lute f(x, y) = o for x and y respectively (x x + Xx 2 )/(i + '•) an d 

 0'i + l )\)/( l + '•) there will be found two values of X, say A, and 

 /.,, to which correspond the points of intersection of the straight 

 line through the points (.\\, j\) and (x 2 , v 2 ) with the absolute, and 

 the cross-ratio of the points (.v,,/,), (x v y 3 ) and the points of 

 intersection with the absolute is Xjk % . This cross-ratio is there- 

 fore readily calculated whether the points of intersection are real or 

 imaginary. 



If the equation of the absolute in homogeneous coordinates is 

 v 2 + v 2 — 4a 2 /" = o, in order that the distance between two points 

 within the absolute shall be real the constant k must be real. 

 Every straight line determined by two points within the absolute 

 has two points at infinity and we have the hyperbolic geometry of 

 two dimensions. Points without the absolute are non-existent in 

 this geometry. 



If the equation of the absolute in homogeneous coordinates is 

 x 2 +y + ^z 2 / 2 = o the constant k must be assumed imaginary in 

 order that the distance between two points of the plane shall be 

 real. The straight line has no point at infinity and we have the 

 elliptic geometry. In this geometry the straight line has a finite 

 length and must return into itself. The distance between two 

 points has a series of values differing by multiples of the length of 

 the entire straight line. 



The points whose homogeneous coordinates are x = 1, y = 

 y/ — 1, t= o) x = 1, y = — s/ — 1, /= o satisfy the equations of 

 the absolute in both the hyberbolic and elliptic geometries. These 

 two points, named the imaginary circular points at infinity, consti- 

 tute the absolute of plane parabolic geometry. The parabolic 

 geometry is therefore a common limiting case of the hyperbolic 

 and elliptic geometries. By a suitable choice of the constant k 

 the parabolic geometry becomes the geometry of Euclid. 



In the geometry of three dimensions the absolute of hyperbolic 

 geometry may be written v 2 4- r 2 + z 1 — 4a 2 / 2 = o ; the absolute 



