This theorem of the generation of a conic by means of projec tive pencils makes it possible to deduce all the properties of conic sections from the theory of one dimensional projectivities. For example, the fact that there is one and only one conic through five coplanar points, no three of which are collinear, is an immediate consequence of the fact that a projectivity between two pencils of lines is fully determined by the correspondence of three lines of the one pencil with three lines of the other. The theorem is a good example of the unifying power of projective geometry, for it includes as special cases a host of theorems about the construction of conic sections when the given five points are in special position from the point of view of ele mentary geometry. An equal number of additional theorems is obtained by applying the principle of duality in the plane; and a corresponding group of theorems about cones is obtained by space duality.
The Ruled Quadric.—Let A 1,A2,A3,A4 be four points of a line a and four points of a line b which is not in the same plane with a. In case then any straight line c which meets the three lines A2B2, will also meet A4B4. Hence there is a whole family of straight lines which meet the lines A1B1, A2B2, A3B3, A4B4. The lines of this family fill up a surface which is called a quadric sur face because its equation is of the second degree. It is called a ruled quadric because of the straight lines which lie upon it. In deed, the surface may be described as follows : let a, b and c be the three lines already described under these names. There is a family of lines each of which meets a, b and c and there is just one line of this family through each point of a, b or c. This fam ily may be called the first ruling or regulus of the quadric. Any line which meets three lines of the first ruling meets all lines of this ruling, and the family of all such lines is a regulus just like the first ruling. The quadric consists of the points on the two rulings. There is one line of each ruling through each point of the quadric, and the plane containing these two lines is tangent to the quadric.
But for further discussion of this question the reader must be referred to the article on surfaces (see SURFACE) and to the appro priate chapters in books on projective and analytic geometry. The ruled quadric from the point of view of Euclid is either a hyper bolic paraboloid or a hyperboloid of one sheet, according as the plane at infinity is or is not tangent to it.
One-dimensional Forms.—This by no means exhausts the list of seemingly complicated geometric figures whose theory can be deduced from the simple propositions about one-dimensional projectivities. The essential element in any such deduction is to find a family of geometric figures which are in such a correspond ence with the points of a line that the theory of projectivities can be applied. Any such family of geometric figures is called a one dimensional form. Examples are : a range of points, a pencil of lines, the set of all planes having a line in common, the points of a conic section, the lines of a regulus, the generating lines of a cone, the points of a rational cubic curve in space, the tangents to a conic section, the set of all conic sections having four points in common, and so on. When the general ideas about projectivities of one-dimensional forms are combined with simple theorems about the particular figures to which they are being applied, they often give us flashes of insight into unexpected branches of mathe matics, such as, for example, quaternions (q.v.) and biquaternions.
Cross Ratio.—If two points, A,B, go by projection to two points, A',B', there is no general relation between the distance AB and the distance A'B'. The same remains true of the ratios elements of a one-dimensional form (e.g., three points of a line) any fourth element C is uniquely determined by its cross ratio ABCD. That is, if C is given, the cross ratio is a unique number x, and if a number x is given, there is one and only one element C such that x is the cross ratio ABCD (if C=A, x=o; if C=B, x= oo; if C = D, x= 1). The number x is what we call the co-ordi nate of the element C of the one-dimensional form. A co-ordinate is a number which serves as a name for the element ; and these names have been assigned in such a way that any two distinct ele ments have distinct names. It can be proved that any projectivity by which each element of a one-dimensional form corresponds to an element of the same form, can be represented by an equation That is to say, for any pro jectivity of a one-dimensional form, there are four numbers a,b,c,d such that every element x is carried by the projectivity to that element whose name x is related to x by the equation written above.