403] 
on pascal’s theorem. 
133 
Considei now the six planes a, b, c, f g, li, and taking in the first instance an 
arbitrary point of projection, and a plane of projection which is also arbitrary—the 
line of intersection ab of the planes a and b will be projected into a line ab, and 
the point of intersection of the planes a, b, c into a point abc; and so in other cases. 
We have thus a plane figure, consisting of the fifteen lines ab, ac, ...gh, and of the 
twenty points abc, abf, ...fgli\ and which is such, that on each of the lines there lie 
four of the points, and through each of the points there pass three of the lines, viz. 
the points abc, abf abg, abh lie on the line ab; and the lines be, ca, ab meet in the 
point abc, and so in other cases. If now the point of projection instead of being 
arbitrary, be one of the above-mentioned four points 0, then the projections of the 
lines af ’ bg, ch meet in a point, and the like for each of the six triads of lines; 
that is in the plane figure we have six points 1, 2, 3, 4, 5, 6, each of them the 
intersection of three lines as shown in the diagram, 
1 = af. bg . ch, 
2 = ag .bh. cf 
3 = ah .bf . eg, 
4 = af .bh. eg, 
5 = ag .bf. ch, 
6 = ah .bg . cf, 
and these six points lie in a conic. It is clear that the lines af, ag, ah; bf, bg, bh; 
cf, eg, ch are the lines 14, 25, 36; 35, 16, 24; 26, 34, 15 respectively. 
Conversely, starting from the points 1, 2, 3, 4, 5, 6 on a conic, and denoting the 
lines 14, 25, 36; 35, 16, 24 ; 26, 34, 15 (being, it may be noticed, the sides and 
diagonals of the hexagon 162435) in the manner just referred to, then it is possible 
to complete the figure of the fifteen lines ab, ac,...gh and of the twenty points 
abc, abf ...fgh, such that each line contains upon it four points, and that through each 
point there pass three lines, in the manner already mentioned. 
Of the fifteen lines, nine, viz. the lines af, ag, ah; bf, bg, bh ; cf, eg, ch are, as has 
been seen, lines through two of the six points 1, 2, 3, 4, 5, 6; the remaining lines are 
be, ca, ab; gh, hf, fg. These are Pascalian lines, 
be of the hexagon 162435, 
ca „ 152634, 
ab „ 142536, 
gh „ 152436, 
hf „ 142635, 
bg „ 162534,