An Elementary Course in Synthetic Projective Geometry | Page 6

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 it, is called a plane system and is a
fundamental form of the second order. The point, considered as made
up of all the lines and planes passing through it, is called a point system
and is also a fundamental form of the second order.

*17.* If now we take three lines in space all lying in different planes,
and select l points on the first, m points on the second, and n points on
the third, then the total number of planes passing through one of the
selected points on each line will be lmn. It is reasonable, therefore, to
symbolize the totality of planes that are determined by the [infinity]
points on each of the three lines by [infinity]3, and to call it an
infinitude of the third order. But it is easily seen that every plane in
space is included in this totality, so that _the totality of planes in space
is an infinitude of the third order._

*18.* Consider now the planes perpendicular to these three lines. Every
set of three planes so drawn will determine a point in space, and,

conversely, through every point in space may be drawn one and only
one set of three planes at right angles to the three given lines. It follows,
therefore, that _the totality of points in space is an infinitude of the
third order._

*19. Space system.* Space of three dimensions, considered as made up
of all its planes and points, is then a fundamental form of the third
order, which we shall call a _space system._

*20. Lines in space.* If we join the twofold infinity of points in one
plane with the twofold infinity of points in another plane, we get a
totality of lines of space which is of the fourth order of infinity. _The
totality of lines in space gives, then, a fundamental form of the fourth
order._

*21. Correspondence between points and numbers.* In the theory of
analytic geometry a one-to-one correspondence is assumed to exist
between points on a line and numbers. In order to justify this
assumption a very extended definition of number must be made use of.
A one-to-one correspondence is then set up between points in the plane
and pairs of numbers, and also between points in space and sets of three
numbers. A single constant will serve to define