181.  Poncelet and Cauchy. The efforts of Poncelet to compel the acceptance of this principle independent of analysis resulted in a bitter and perhaps fruitless controversy between him and the great analyst Cauchy.  In his review of Poncelet’s great work on the projective properties of figures(18) Cauchy says, “In his preliminary discourse the author insists once more on the necessity of admitting into geometry what he calls the ‘principle of continuity.’  We have already discussed that principle ... and we have found that that principle is, properly speaking, only a strong induction, which cannot be indiscriminately applied to all sorts of questions in geometry, nor even in analysis.  The reasons which we have given as the basis of our opinion are not affected by the considerations which the author has developed in his Traite des Proprietes Projectives des Figures.”  Although this principle is constantly made use of at the present day in all sorts of investigations, careful geometricians are in agreement with Cauchy in this matter, and use it only as a convenient working tool for purposes of exploration.  The one-to-one correspondence between geometric forms and algebraic analysis is subject to many and important exceptions.  The field of analysis is much more general than the field of geometry, and while there may be a clear notion in analysis to, correspond to every notion in geometry, the opposite is not true.  Thus, in analysis we can deal with four cooerdinates as well as 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.