40
which will generate the locus._
*69. Pascal's theorem.* The points A, B, C, D, S, and _S'_ may thus be considered as chosen arbitrarily on the locus, and the following remarkable theorem follows at once.
_Given six points, 1, 2, 3, 4, 5, 6, on the point-row of the second order, if we call_
_L the intersection of 12 with 45,_
_M the intersection of 23 with 56,_
_N the intersection of 34 with 61,_
_then __L__, __M__, and __N__ are on a straight line._
[Figure 13]
FIG. 13
*70.* To get the notation to correspond to the figure, we may take (Fig. 13) _A = 1_, _B = 2_, _S' = 3_, _D = 4_, _S = 5_, and _C = 6_. If we make _A = 1_, _C=2_, _S=3_, _D = 4_, _S'=5_, and. _B = 6_, the points L and N are interchanged, but the line is left unchanged. It is clear that one point may be named arbitrarily and the other five named in _5! = 120_ different ways, but since, as we have seen, two different assignments of names give the same line, it follows that there cannot be more than 60 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 th