different lines _LMN_ obtained in this way from a given set of six points.
As a matter of fact, the number obtained in this way is in general _60_.
The above theorem, which is of cardinal importance in the theory of the
point-row of the second order, is due to Pascal and was discovered by him
at the age of sixteen. It is, no doubt, the most important contribution to
the theory of these loci since the days of Apollonius. If the six points
be called the vertices of a hexagon inscribed in the curve, then the sides
12 and 45 may be appropriately called a pair of opposite sides. Pascal’s
theorem, then, may be stated as follows:

_The three pairs of opposite sides of a hexagon inscribed in a point-row
of the second order meet in three points on a line._




*71. Harmonic points on a point-row of the second order.* Before
proceeding to develop the consequences of this theorem, we note another
result of the utmost importance for the higher developments of pure
geometry, which follows from the fact that if four points on the locus
project to a fifth in four harmonic rays, they will project to any point
of the locus in four harmonic rays. It is natural to speak of four such
points as four harmonic points on the locus, and to use this notion to
define projective correspondence between point-rows of the second order,
or between a point-row of the second order and any fundamental form of the
first order. Thus, in particular, the point-row of the second order, σ, is
said to be _perspectively related_ to the pencil _S_ when every ray on _S_
goes through the point on σ which corresponds to it.