An Elementary Course in Synthetic Projective Geometry | Page 5

we have set the lines through a
point and the points on a line into one-to-one correspondence. This
correspondence is of such fundamental importance in the study of
projective geometry that a special name is given to it. Calling the
totality of points on a line a _point-row_, and the totality of lines
through a point a pencil of rays, we say that the point-row and the
pencil related as above are in perspective position, or that they are
perspectively related.

*7. Axial pencil; fundamental forms.* A similar correspondence may
be set up between the points on a line and the planes through another
line which does not meet the first. Such a system of planes is called an
axial pencil, and the three assemblages--the point-row, the pencil of
rays, and the axial pencil--are called fundamental forms. The fact that
they may all be set into one-to-one correspondence with each other is
expressed by saying that they are of the same order. It is usual also to
speak of them as of the first order. We shall see presently that there are
other assemblages which cannot be put into this sort of one-to-one
correspondence with the points on a line, and that they will very
reasonably be said to be of a higher order.

*8. Perspective position.* We have said that a point-row and a pencil of
rays are in perspective position if each ray of the pencil goes through
the point of the point-row which corresponds to it. Two pencils of rays
are also said to be in perspective position if corresponding rays meet on
a straight line which is called the axis of perspectivity. Also, two
point-rows are said to be in perspective position if corresponding points
lie on straight lines through a point which is called the center of

perspectivity. A point-row and an axial pencil are in perspective
position if each plane of the pencil goes through the point on the
point-row which corresponds to it, and an axial pencil and a pencil of
rays are in perspective position if each ray lies in the plane which
corresponds to it; and, finally, two axial pencils are perspectively
related if corresponding planes meet in a plane.

*9. Projective relation.* It is easy to imagine a more general
correspondence between the points of two point-rows than the one just
described. If we take two perspective pencils, A and S, then a point-row
a perspective to A will be in one-to-one correspondence with a
point-row b perspective to B, but corresponding points will not, in
general, lie on lines which all pass through a point. Two such
point-rows are said to be projectively related, or simply projective to
each other. Similarly, two pencils of rays, or of planes, are projectively
related to each other if they are perspective to two perspective
point-rows. This idea will be generalized later on. It is important to
note that between the elements of two projective fundamental forms
there is a one-to-one correspondence, and also that this correspondence
is in general _continuous_; that is, by taking two elements of one form
sufficiently close to each other, the two corresponding elements in the
other form may be made to approach each other arbitrarily close. In the
case of point-rows this continuity is subject to exception in the
neighborhood of the point "at infinity."

*10. Infinity-to-one correspondence.* It might be inferred that any
infinite assemblage could be put into one-to-one correspondence with
any other. Such is not the case, however, if the correspondence is to be
continuous, between the points on a line and the points on a plane.
Consider two lines which lie in different planes, and take m points on
one and n points on the other. The number of lines joining the m points
of one to the n points jof the other is clearly mn. If we symbolize the
totality of points on a line by [infinity], then a reasonable symbol for
the totality of lines drawn to cut two lines would be [infinity]2. Clearly,
for every point on one line there are [infinity] lines cutting across the

other, so that the correspondence might be called [infinity]-to-one.
Thus the assemblage of lines cutting across two lines is of higher order
than the assemblage of points on a line; and as we have called the
point-row an assemblage of the first order, the system of lines cutting
across two lines ought to be called of the second order.

*11. Infinitudes of different orders.* Now it is easy to set up a
one-to-one correspondence between the points in a plane and the
system of lines cutting across two lines which lie in different planes. In
fact, each line of the system of lines meets the plane in one point, and
each