with three, but the existence of a space of four dimensions to correspond to it does not therefore follow. When the geometer speaks of the two real or imaginary intersections of a straight line with a conic, he is really speaking the language of algebra. _Apart from the algebra involved_, it is the height of absurdity to try to distinguish between the two points in which a line _fails to meet a conic!_ *182. The work of Poncelet.* But Poncelet’s right to the title "The Father of Modern Geometry" does not stand or fall with the principle of contingent relations. In spite of the fact that he considered this principle the most important of all his discoveries, his reputation rests on more solid foundations. He was the first to study figures _in homology_, which is, in effect, the collineation described in § 175, where corresponding points lie on straight lines through a fixed point. He was the first to give, by means of the theory of poles and polars, a transformation by which an element is transformed into another of a different sort. Point-to-point transformations will sometimes generalize a theorem, but the transformation discovered by Poncelet may throw a theorem into one of an entirely different aspect. The principle of duality, first stated in definite form by Gergonne,(19) the editor of the mathematical journal in which Poncelet published his researches, was based by Poncelet on his theory of poles and polars. He also put into definite form the notions of the infinitely distant elements in space as all lying on a plane at infinity. *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 Plücker. 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.