i8 CONCRETE REPRESENTATIONS OF 



circles in general intersect in two points, which may be real, 

 coincident, or imaginary ; and the point-pair thus determined 

 will not determine uniquely one orthogonal circle, but a pencil 

 of circles. Two such points are inverse points with respect 

 to the fixed circle. We shall see in 18 that the ' distance ' 

 between a pair of inverse points is real or imaginary according 

 as the fundamental circle is imaginary or real. In the former 

 case we may either consider the two points as distinct (so that 

 two straight lines will intersect in two points), or identify 

 them ; and we get the two forms of geometry, Spherical and 

 Elliptic. In the latter case it is necessary to identify the two 

 points, otherwise we should have two real points with an 

 imaginary distance ; thus we get Hyperbolic geometry. 

 Alternatively we may agree to consider only the points in 

 the interior (or exterior) of the fundamental circle. When 

 the fundamental circle reduces to a point 0, one of the points 

 of any point-pair is at O and we need only consider the other 

 point, so that two lines always intersect in just one point. 

 This geometry is Parabolic, and we shall see that it is identical 

 with Euclidean geometry. 



When the fundamental circle is real, two orthogonal 

 circles intersect in two points, real, coincident, or imaginary. 

 This corresponds to the three sorts of line-pairs in Hyperbolic 

 geometry, intersectors, parallels, and non-intersectors. When 

 the fundamental circle is imaginary, two orthogonal circles 

 always intersect in two real points, so that in Elliptic or 

 Spherical geometry parallels and non-intersectors do not exist. 

 When the fundamental circle reduces to a point 0, every 

 orthogonal circle passes through 0, and they cut in pairs in 

 one other real point which may coincide with 0. The latter 

 case corresponds to parallels in Euclidean geometry. 



14. Next, to fix the representation, we have to consider 

 the measurement of distances and angles. 



Let us make the condition that angles are to be the same 

 in the geometry and in its representation, i.e. that the repre- 



