39.  Harmonic conjugate of the point at infinity. It is natural to inquire what position of B corresponds to the infinitely distant position of D.  We have proved (§ 27) that the particular quadrangle K, L, M, N employed is of no consequence.  We shall therefore avail ourselves of one that lends itself most readily to the solution of the problem.  We choose the point L so that the triangle ALC is isosceles (Fig. 7).  Since D is supposed to be at infinity, the line KM is parallel to AC.  Therefore the triangles KAC and MAC are equal, and the triangle ANC is also isosceles.  The triangles CNL and ANL are therefore equal, and the line LB bisects the angle ALC. B is therefore the middle point of AC, and we have the theorem

The harmonic conjugate of the middle point of _AC__ is at infinity._

[Figure 7]

FIG. 7

40.  Projective theorems and metrical theorems.  Linear construction. This theorem is the connecting link between the general protective theorems which we have been considering so far and the metrical theorems of ordinary geometry.  Up to this point we have said nothing about measurements, either of line segments or of angles.  Desargues’s theorem and the theory of harmonic elements which depends on it have nothing to do with magnitudes at all.  Not until the notion of an infinitely distant point is brought in is any mention made of distances or directions.  We have been able to make all of our constructions up to this point by means of the straightedge, or ungraduated ruler.  A construction made with such an instrument we shall call a linear construction.  It requires merely that we be able to draw the line joining two points or find the point of intersection of two lines.

41.  Parallels and mid-points. It might be thought that drawing a line through a given point parallel to a given line was only a special case of drawing a line joining two points.  Indeed, it consists only in drawing a line through the given point and through the “infinitely distant point” on the given line.  It must be remembered, however, that the expression “infinitely distant point” must not be taken literally.  When we say that two parallel lines meet “at infinity,” we really mean that they do not meet at all, and the only reason for using the expression is to avoid tedious statement of exceptions and restrictions to our theorems.  We ought therefore to consider the drawing of a line parallel to a given line as a different accomplishment from the drawing of the line joining two given points.  It is a remarkable consequence of the last theorem that a parallel to a given line and the mid-point of a given segment are equivalent data.  For the construction is reversible, and if we are given the middle point of a given segment, we can construct