Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

DOCTRINAL INTERPRETATIONS 68
space, which we describe by calling them respectively
projective straight line, projective plane and projective
space. The adjective has fine propriety, but that need
not here detain us. You can readily prove, or you may
assume, that the " ideal " points of the projective plane
constitute a straight line-called the "ideal" line, or
line at infinity; and that the locus of the " ideal " points
of projective space is a plane-called the " ideal " plane,
or plane at infinity. I can not pause here to justify the
convention. It is amply justified by its consequences,
for which, if you be interested, you must repair to projective geometry,-invented by the engineer, Desargues,
a contemporary of Descartes and Pascal,-quickly forgotten-reinvented, in France again, about one hundred
years ago-perhaps the most beautiful branch of mathematics.
We may now proceed to the promised new interpretation of our doctrinal functions. As HAF is simpler
than HAF', let us first deal with the former.
Let 7r denote a projective plane. Let a chosen point
O be the vertex of a pencil of lines of 7r; call each line of the
pencil an O-line. Note that every other pencil of i
contains one and but one O-line. Now let us in thought
remove from Tr, once for all, the O-pencil. We thus
remove one and but one line from every other pencil.
We may conveniently call the pencils, thus bereft of a
line, pathopencils as being defective or, so to speak,
pathological. We have taken from T one and but one
pencil of lines. Our field of operation consists of all that
is left. Denote the field by c>. We are going to give
HaF an interpretation in I,; the interpretation, as you
will see, will be a doctrine about certain things in q —
a geometry of the field. The interpretation results from