The Imaginary in Geometry $j 



(o, i) (i, o) = arc (i, o) ( — I, o) = tt/ 2 - Similarly the area 

 of the conic, defined as ffdxdy, is ir. 



As heretofore an angle 6 will have the ratio of a complex sector 

 to the triangle O, (i, o), (01). 



All these statements remain true no matter what sort of dia- 

 gram may be gotten by point projection of this diagram. The 

 ellipse, in particular, might get projected into a hyperbola, the 

 supplementary curves then becoming ellipses. 



The correspondence between the various projections is com- 

 plete, so that when we are proving a theorem concerning any 

 diagram, we are at one and the same time proving a like theorem 

 concerning every projection of that diagram. For example, in 

 the representation of a complex element by an arrow whose butt 

 is at the center of the /-ray and whose barb is at the center of the 

 7-ray through the element, it was found, when the element be- 

 longed to a circle, that butt and barb were inverse points with 

 respect to that circle. In the projection butt and barb are in- 

 verse, in terms of the projected cooordinates, with respect to the 

 projection of the circle, be it what conic it may. 



In all the above, both the centre of projection and the plane 

 receiving the projection are assumed to be real. Before consid- 

 ering projection from an imaginary center or thrown upon an 

 imaginary plane, it will be necessary to consider imaginary ele- 

 ments in space. 



Von Staudt's Representation 1 



In Fig. 33 I and J are the intersections of the line s with 

 the conic C 2 . Now, without changing the conic at all, merely 

 revolving the axes, keeping them conjugate, and always regard- 

 ing z as the projection of a line at co parallel to the F-axis, we 

 get a different pair of intersections and a different pair of sup- 

 plementary curves. Thus, what we are to consider as the imagi- 

 nary intersection of a curve and a line depends upon the choice 

 of axes, depends upon what line through the origin we shall 

 regard as the projection of a parallel to the line at infinity. \\ q 

 can avoid this by saying that the / intersection is the collectivity 



1 Von Staudt, Geometrie der Lagc. Liiroth, Math. Ann., Vol. VIII. 



37 



