*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 two given lines. Draw through _P_ a line parallel to this line. This is the required line. *3.* Given a parallelogram in the same plane with a given segment _AC_, to construct linearly the middle point of _AC_. *4.* Given four harmonic lines, of which one pair are at right angles to each other, show that the other pair make equal angles with them. This is a theorem of which frequent use will be made. *5.* Given the middle point of a line segment, to draw a line parallel to the segment and passing through a given point. *6.* A line is drawn cutting the sides of a triangle _ABC_ in the points _A’_, _B’_, _C’_ the point _A’_ lying on the side _BC_, etc. The harmonic conjugate of _A’_ with respect to _B_ and _C_ is then constructed and called _A"_. Similarly, _B"_ and _C"_ are constructed. Show that _A"B"C"_ lie on a straight line. Find other sets of three points on a line in the figure. Find also sets of three lines through a point. CHAPTER III - COMBINATION OF TWO PROJECTIVELY RELATED FUNDAMENTAL FORMS [Figure 9] FIG. 9