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 _linearly_ a line parallel to that
segment. Thus, given that _B_ is the middle point of _AC_, we may draw any
two lines through _A_, and any line through _B_ cutting them in points _N_