*46. Anharmonic ratio.* The corresponding relations between the
trigonometric functions of the angles determined by four harmonic lines
are not difficult to obtain, but as we shall not need them in building up
the theory of projective geometry, we will not discuss them here. Students
who have a slight acquaintance with trigonometry may read in a later
chapter (§ 161) a development of the theory of a more general relation,
called the _anharmonic ratio_, or _cross ratio_, which connects any four
points on a line.




PROBLEMS


*1*. Draw through a given point a line which shall pass through the
inaccessible point of intersection of two given lines. The following
construction may be made to depend upon Desargues’s theorem: Through the
given point _P_ draw any two rays cutting the two lines in the points
_AB’_ and _A’B_, _A_, _B_, lying on one of the given lines and _A’_, _B’_,
on the other. Join _AA’_ and _BB’_, and find their point of intersection
_S_. Through _S_ draw any other ray, cutting the given lines in _CC’_.
Join _BC’_ and _B’C_, and obtain their point of intersection _Q_. _PQ_ is
the desired line. Justify this construction.

*2.* To draw through a given point _P_ a line which shall meet two given
lines in points _A_ and _B_, equally distant from _P_. Justify the
following construction: Join _P_ to the point _S_ of intersection of the
two given lines. Construct the fourth harmonic of _PS_ with respect to the