The Imaginary in Geometry 35 



can replace the circle by an ellipse and the perpendicular diameters 

 by conjugate diameters. The rectangular hyperbola is replaced 

 by one whose asymptotes pass through the blue points of the 

 elliptic elements at infinity. The argument is the ratio of a real 

 elliptic sector plus an imaginary hyperbolic sector to the triangle 

 having for side the conjugate semidiameters of the ellipse. The 

 cosine and sine ratios have the same denominator but the nu- 

 merator of the first is a triangle whose base is the diameter coinci- 

 dent with the initial line of and whose vertex is the elliptic ele- 

 ment having its blue point on the terminal line of 9. The numera- 

 tor of the cosine ratio has the same vertex but the base is the 

 diameter conjugate to that used in the sine ratio, the diameter in 

 fact parallel to the now elliptic element P. 



A still further generalization can be made if instead of con- 

 fining ourselves to parallel projection we use projection from a 

 point. 1 



The line at infinity then projects into a line z, while parallels to 

 it project into lines meeting it in a certain fixed point Y. The seg- 

 ments representing the circular points at infinity project into un- 

 equal segments ; they are such, namely, that their blue points 

 are harmonically conjugate with respect to their common black 

 point and the fixed point Y. The circular rays through any 

 point of the plane project into two imaginary rays through //, 

 the projected imaginary points. Lines at right angles are pro- 

 jected into lines harmonically conjugate with respect to I and 7. 



Equal consecutive segments along a line project into segments 

 such that of three consecutive division points the middle one is 

 harmonically separated from z by the other two. 



In place of rectangular axes we have axes harmonically conju- 

 gate with regard to /" and /". In particular, we might take a 

 pair of lines one through Y, call it the y-axis, the other through 

 the common point of 7 and / call this the .v-axis and the point X. 

 The axes so chosen intersect in O ; on the .v-axis we take, where 

 we please, a point for (1, o). The coordinates of every point in 

 the plane are then determined. A diagram (fig. 33) makes this 



1 For the analytic theory see Sophus Lie. Vorlesungen iiber contin- 

 uerlichen Gruppen, pp. 13 seq. 



35 



