section is a point-row of the second order. It will later appear that a
point-row of the second order is a conic section. In the future,
therefore, we shall refer to a point-row of the second order as a conic.

[Figure 14]

FIG. 14




*74. Conic through five points.* Pascal’s theorem furnishes an elegant
solution of the problem of drawing a conic through five given points. To
construct a sixth point on the conic, draw through the point numbered 1 an
arbitrary line (Fig. 14), and let the desired point 6 be the second point
of intersection of this line with the conic. The point _L = 12-45_ is
obtainable at once; also the point _N = 34-61_. But _L_ and _N_ determine
Pascal’s line, and the intersection of 23 with 56 must be on this line.
Intersect, then, the line _LN_ with 23 and obtain the point _M_. Join _M_
to 5 and intersect with 61 for the desired point 6.

[Figure 15]

FIG. 15




*75. Tangent to a conic.* If two points of Pascal’s hexagon approach
coincidence, then the line joining them approaches as a limiting position