Poncelet’s two memoirs Sur les centres des moyennes harmoniques and Sur la théorie générale des polaires réciproques, although presented to the Paris Academy in 1824, were only published (Crelle, t. iii. and iv., 1828, 1829) subsequent to the memoir by Gergonne, Considérations philosophiques sur les élémens de la science de l’étendue (Gerg. t. xvi., 1825-1826). In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated; for instance, we find “dans la géométrie plane, à chaque théorème il en répond nécessairement un autre qui s’en déduit en échangeant simplement entre eux les deux mots points et droites; tandis que dans la géométrie de l’espace ce sont les mots points et plans qu’il faut échanger entre eux pour passer d’un théorème à son corrélatif”; and the plan is introduced of printing correlative theorems, opposite to each other, in two columns. There was a reclamation as to priority by Poncelet in the Bulletin universel reprinted with remarks by Gergonne (Gerg. t. xix., 1827), and followed by a short paper by Gergonne, Rectifications de quelques théorèmes, &c., which is important as first introducing the word class. We find in it explicitly the two correlative definitions: “a plane curve is said to be of the mth degree (order) when it has with a line m real or ideal intersections,” and “a plane curve is said to be of the mth class when from any point of its plane there can be drawn to it m real or ideal tangents.”
It may be remarked that in Poncelet’s memoir on reciprocal polars, above referred to, we have the theorem that the number of tangents from a point to a curve of the order m, or say the class of the curve, is in general and at most = m(m − 1), and that he mentions that this number is subject to reduction when the curve has double points or cusps.
The theorem of duality as regards plane figures may be thus stated: two figures may correspond to each other in such manner that to each point and line in either figure there correspond in the other figure a line and point respectively. It is to be understood that the theorem extends to all points or lines, drawn or not drawn; thus if in the first figure there are any number of points on a line drawn or not drawn, the corresponding lines in the second figure, produced if necessary, must meet in a point. And we thus see how the theorem extends to curves, their points and tangents; if there is in the first figure a curve of the order m, any line meets it in m points; and hence from the corresponding point in the second figure there must be to the corresponding curve m tangents; that is, the corresponding curve must be of the class m.
Trilinear co-ordinates (see [Geometry]: Analytical) were first used by E. E. Bobillier in the memoir Essai sur un nouveau mode de recherche des propriétés de l’étendue (Gerg. t. xviii., 1827-1828). It is convenient to use these rather than Cartesian co-ordinates. We represent a curve of the order m by an equation (*
x, y, z)m = 0, the function on the left hand being a homogeneous rational and integral function of the order m of the three co-ordinates (x, y, z); clearly the number of constants is the same as for the equation (*
x, y, 1)m = 0 in Cartesian co-ordinates.
The theorem of duality is considered and developed, but chiefly in regard to its metrical applications, by Michel Chasles in the Mémoire de géométrie sur deux principes généraux de la science, la dualité et l’homographie, which forms a sequel to the Aperçu historique sur l’origine et le développement des méthodes en géométrie (Mém. de Brux. t. xi., 1837).