66.  Lines joining four points of the locus to a fifth. Suppose that the points S, S’, B, C, and D are fixed, and that four points, A, A_1_, A_2_, and A_3_, are taken on the locus at the intersection with it of any four harmonic rays through B.  These four harmonic rays give four harmonic points, L, L_1_ etc., on the fixed ray SD.  These, in turn, project through the fixed point M into four harmonic points, N, N_1_ etc., on the fixed line DS’.  These last four harmonic points give four harmonic rays CA, CA_1_, CA_2_, CA_3_.  Therefore the four points A which project to B in four harmonic rays also project to C in four harmonic rays.  But C may be any point on the locus, and so we have the very important theorem,

Four points which are on the locus, and which project to a fifth point of the locus in four harmonic rays, project to any point of the locus in four harmonic rays.

67. The theorem may also be stated thus: 

The locus of points from which, four given points are seen along four harmonic rays is a point-row of the second order through them.

68. A further theorem of prime importance also follows: 

Any two points on the locus may be taken as the centers of two projective pencils 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: