An Elementary Course in Synthetic Projective Geometry | Page 6

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 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 equations in three variables; etc. The equation of a line in a plane involves two constants, either of which may take an infinite number of values. From this it follows that there is an infinity of lines in the plane which is of the second order if the infinity of points on a line is