I905 j LAMBERT— THE STRAIGHT LINE CONCEPT. 83 



of rational geometry, depends primarily on the fact that a straight 

 line is determined by any two of its points and can be indefinitely 

 extended between any two of its points. Right here arises a ques- 

 tion that can not be answered by experience or experiment. If the 

 straight line is indefinitely extended beyond any two of its points, 

 will there be found on the straight line two points at infinity, one 

 point at infinity, or no point at infinity ? This question, of course, 

 can not be answered until precision has been given to the term 

 distance. 



In the plane determined by a given point and a given straight 

 line, draw a straight line through the given point intersecting the 

 given straight line and revolve the straight line about the given 

 point. When the point of intersection of the revolving line with 

 the given line moves to an infinite distance from the foot of [the 

 perpendicular from the given point to the given line, the revolving 

 line is said to become parallel to the given line. In how many 

 positions does the revolving line become parallel to the given line ? 

 This, again, is a question that can not be answered by experience 

 or experiment. It can be answered only by the axiom of parallels. 



If the axiom of parallels is made to read : Through a given point 

 without a given line one and only one parallel to the line can be 

 drawn — we have a geometry in which the straight line has only 

 one point at infinity. This is the geometry of Euclid. Since the 

 parabola meets the straight line at infinity in only one point, this 

 geometry is also called the parabolic geometry. 



If the axiom of parallels is made to read : Through a given point 

 without a given line two and only two parallels to the line can be 

 drawn — we have a geometry in which the straight line has two 

 and only two points at infinity. Since the hyperbola intersects the 

 straight line at infinity in two points, this geometry is called the 

 hyperbolic geometry. The hyperbolic geometry was developed by 

 Euclidean methods by Lobachevski and Bolyai. 



If the axiom of parallels is made to read : Through a given point 

 without a given line no parallel to the line can be drawn — we have 

 a geometry in which the straight line has no point at infinity. 

 Since the ellipse does not intersect the straight line at infinity in a 

 real point, this geometry is called the elliptic geometry. The 

 elliptic geometry has been discussed by Riemann, Clifford and 

 Newcomb. 



