then L, M, and N are on a straight line.
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 [pg 42] 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.
72. Determination of the locus. It is now clear that five points, arbitrarily chosen in the plane, are sufficient to determine a point-row of the second order through them. Two of the points may be taken as centers of two projective pencils, and the three others will determine three pairs of corresponding rays of the pencils, and therefore all pairs. If four points of the locus are [pg 43] given, together with the tangent at one of them, the locus is likewise completely determined. For if the point at which the tangent is given be taken as the center S of one pencil, and any other of the points for S', then, besides the two pairs of corresponding rays determined by the remaining two points, we have one more pair, consisting of the tangent at S and the ray SS'. Similarly, the curve is determined by three points and the tangents at two of them.