*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