if such a circle meets the line in the points CC’, then, by the theorem in the geometry of the circle which says that if any chord is _ drawn through a fixed point within a circle, the product of its segments is constant in whatever direction the chord is drawn, and if a secant line be drawn from a fixed point without a circle, the product of the secant and its external segment is constant in whatever direction the secant line is drawn_, we have OC .  OC’ = OG .  OG’ = constant.  So that for all such points OA .  OA’ = OB .  OB’ = OC .  OC’.  Further, the line GG’ meets AA’ in the center of the involution.  To find the double points, if they exist, we draw a tangent from O to any of the circles through GG’.  Let T be the point of contact.  Then lay off on the line OA a line OF equal to OT.  Then, since by the above theorem of elementary geometry OA .  OA’ = OT_2__ = OF__2_, we have one double point F.  The other is at an equal distance on the other side of O.  This simple and effective method of constructing an involution of points is often taken as the basis for the theory of involution.  In projective geometry, however, the circle, which is not a figure that remains unaltered by projection, and is essentially a metrical notion, ought not to be used to build up the purely projective part of the theory.

146. It ought to be mentioned that the theory of analytic geometry indicates that the circle is a special conic section that happens to pass through two particular imaginary points on the line at infinity, called the circular points and usually denoted by I and J.  The above method of obtaining a point-row in involution is, then, nothing but a special case of the general theorem of the last chapter (§ 125), which asserted that a system of conics through four points will cut any line in the plane in a point-row in involution.

[Figure 41]

FIG. 41

147.  Pairs in an involution of rays which are at right angles.  Circular involution. In an involution of rays there is no one ray which may be distinguished from all the others as the point at infinity is distinguished from all other points on a line.  There is one pair of rays, however, which does differ from all the others in that for this particular pair the angle is a right angle.  This is most easily shown by using the construction that employs circles, as indicated above.  The centers of all the circles through G and G’ lie on the perpendicular bisector of the line GG’.  Let this line meet the line AA’ in the point C (Fig. 41), and draw the circle with center C which goes through G and G’.  This circle cuts out two points M and M’ in the involution.  The rays GM and GM’ are clearly at right angles, being inscribed in a semicircle.  If, therefore, the involution of points is projected