183.  The debt which analytic geometry owes to synthetic geometry. The reaction of pure geometry on analytic geometry is clearly seen in the development of the notion of the class of a curve, which is the number of tangents that may be drawn from a point in a plane to a given curve lying in that plane.  If a point moves along a conic, it is easy to show—­and the student is recommended to furnish the proof—­that the polar line with respect to a conic remains tangent to another conic.  This may be expressed by the statement that the conic is of the second order and also of the second class.  It might be thought that if a point moved along a cubic curve, its polar line with respect to a conic would remain tangent to another cubic curve.  This is not the case, however, and the investigations of Poncelet and others to determine the class of a given curve were afterward completed by Pluecker.  The notion of geometrical transformation led also to the very important developments in the theory of invariants, which, geometrically, are the elements and configurations which are not affected by the transformation.  The anharmonic ratio of four points is such an invariant, since it remains unaltered under all projective transformations.

184.  Steiner and his work. In the work of Poncelet and his contemporaries, Chasles, Brianchon, Hachette, Dupin, Gergonne, and others, the anharmonic ratio enjoyed a fundamental role.  It is made also the basis of the great work of Steiner,(20) who was the first to treat of the conic, not as the projection of a circle, but as the locus of intersection of corresponding rays of two projective pencils.  Steiner not only related to each other, in one-to-one correspondence, point-rows and pencils and all the other fundamental forms, but he set into correspondence even curves and surfaces of higher degrees.  This new and fertile conception gave him an easy and direct route into the most abstract and difficult regions of pure geometry.  Much of his work was given without any indication of the methods by which he had arrived at it, and many of his results have only recently been verified.

185.  Von Staudt and his work. To complete the theory of geometry as we have it to-day it only remained to free it from its dependence on the semimetrical basis of the anharmonic ratio.  This work was accomplished by Von Staudt,(21) who applied himself to the restatement of the theory of geometry in a form independent of analytic and metrical notions.  The method which has been used in Chapter II to develop the notion of four harmonic points by means of the complete quadrilateral is due to Von Staudt.  His work is characterized by a most remarkable generality, in that he is able to discuss real and imaginary forms with equal ease.  Thus he assumes a one-to-one correspondence between the points and lines of a plane, and defines a conic as the locus of points which lie on their corresponding lines, and a pencil of rays of the second order as