*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._