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
point in the plane determines one and only one line cutting across the two
given lines--namely, the line of intersection of the two planes determined
by the given point with each of the given lines. The assemblage of points
in the plane is thus of the same order as that of the lines cutting across
two lines which lie in different planes, and ought therefore to be spoken
of as of the second order. We express all these results as follows:

*12.* If the infinitude of points on a line is taken as the infinitude of
the first order, then the infinitude of lines in a pencil of rays and the
infinitude of planes in an axial pencil are also of the first order, while
the infinitude of lines cutting across two "skew" lines, as well as the
infinitude of points in a plane, are of the second order.

*13.* If we join each of the points of a plane to a point not in that
plane, we set up a one-to-one correspondence between the points in a plane
and the lines through a point in space. _Thus the infinitude of lines
through a point in space is of the second order._

*14.* If to each line through a point in space we make correspond that
plane at right angles to it and passing through the same point, we see
that _the infinitude of planes through a point in space is of the second
order._

*15.* If to each plane through a point in space we make correspond the
line in which it intersects a given plane, we see that _the infinitude of
lines in a plane is of the second order._ This may also be seen by setting
up a one-to-one correspondence between the points on a plane and the lines
of that plane. Thus, take a point _S_ not in the plane. Join any point _M_
of the plane to _S_. Through _S_ draw a plane at right angles to _MS_.
This meets the given plane in a line _m_ which may be taken as
corresponding to the point _M_. Another very important method of setting
up a one-to-one correspondence between lines and points in a plane will be
given later, and many weighty consequences will be derived from it.

*16. Plane system and point system.* The plane, considered as made up of
the points and lines in