Projective geometry, by Oswald Veblen and John Wesley Young.

CHAPTER V*

CONIC SECTIONS
41. Definitions. Pascal's and Brianchon's theorems.
DEFINITION. The set of all points of intersection of homologous
lines of two projective, nonperspective flat pencils which are on the
same plane but not on the same point is called a point conic (fig. 49).
The plane dual of a point conic is called a line conic (fig. 50). The
space dual of a point conic is called a cone of planes; the space dual
FIG. 49 FIG. 50
of a line conic is called a cone of lines. The point through which
pass all the lines (or planes) of a cone of lines (or planes) is called
the vertex of the cone. The point conic, line conic, cone of planes,
and cone of lines are called one-dimensionalforns of the second degree.t
The following theorem is an immediate consequence of this definition.
THEOREM 1. The section of a cone of lines by a plane not on the
vertex of the cone is a point conic. The section of a cdne of planes by
a plane not on the vertex is a line conic.
Now let Al and B. be the centers of two flat pencils defining a
point conic. They are themselves, evidently, points of the conic, for the
line A1B1 regarded as a line of the pencil on A1 corresponds to some
other line through B1 (since the pencils are, by hypothesis, projective
* All the developments of this chapter are on the basis of Assumptions A, E, P,
and Ho.
t A fifth one-dimensional form -a self-dual form of lines in space called the
regulus -will be defined in Chap. XI. This definition of the first four one-dimensional forms of the second degree is due to Jacob Steiner (1796-1863). Attention
will be called to other methods of definition in the sequel.
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