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 the position of a point on a line;
two, a point in the plane; three, a point in space; etc. In the same
theory a one-to-one correspondence is set up between loci in the plane and
equations in two variables; between surfaces in space and equati