Reverting to the purely plane theory, infinity is a line, related like any other right line to the curve, and thus intersecting it in m points, real or imaginary, distinct or coincident.
Descartes in the Géométrie defined and considered the remarkable curves called after him the ovals of Descartes, or simply Cartesians, which will be again referred to. The next important work, founded on the Géométrie, was Sir Isaac Newton’s Enumeratio linearum tertii ordinis (1706), establishing a classification of cubic curves founded chiefly on the nature of their infinite branches, which was in some details completed by James Stirling (1692–1770), Patrick Murdoch (d. 1774) and Gabriel Cramer; the work also contains the remarkable theorem (to be again referred to), that there are five kinds of cubic curves giving by their projections every cubic curve whatever. Various properties of curves in general, and of cubic curves, are established in Colin Maclaurin’s memoir, “De linearum geometricarum proprietatibus generalibus Tractatus” (posthumous, say 1746, published in the 6th edition of his Algebra). We have in it a particular kind of correspondence of two points on a cubic curve, viz. two points correspond to each other when the tangents at the two points again meet the cubic in the same point.
6. Reciprocal Polars. Intersections of Circles. Duality. Trilinear and Tangential Co-ordinates.—The Géométrie descriptive, by Gaspard Monge, was written in the year 1794 or 1795 (7th edition, Paris, 1847), and in it we have stated, in plano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, viz. “Given a surface of the second order and a circumscribed conic surface which touches it ... then if the conic surface moves so that its summit is always in the same plane, the plane of the curve of contact passes always through the same point.” The theorem is here referred to partly on account of its bearing on the theory of imaginaries in geometry. It is in Charles Julian Brianchon’s memoir “Sur les surfaces du second degré” (Jour. Polyt. t. vi. 1806) shown how for any given position of the summit the plane of contact is determined, or reciprocally; say the plane XY is determined when the point P is given, or reciprocally; and it is noticed that when P is situate in the interior of the surface the plane XY does not cut the surface; that is, we have a real plane XY intersecting the surface in the imaginary curve of contact of the imaginary circumscribed cone having for its summit a given real point P inside the surface.
Stating the theorem in regard to a conic, we have a real point P (called the pole) and a real line XY (called the polar), the line joining the two (real or imaginary) points of contact of the (real or imaginary) tangents drawn from the point to the conic; and the theorem is that when the point describes a line the line passes through a point, this line and point being polar and pole to each other. The term “pole” was first used by François Joseph Servois, and “polar” by Joseph Diez Gergonne (Gerg. t. i. and iii., 1810–1813); and from the theorem we have the method of reciprocal polars for the transformation of geometrical theorems, used already by Brianchon (in the memoir above referred to) for the demonstration of the theorem called by his name, and in a similar manner by various writers in the earlier volumes of Gergonne. We are here concerned with the method less in itself than as leading to the general notion of duality.
Bearing in a somewhat similar manner also on the theory of imaginaries in geometry (but the notion presents itself in a more explicit form), there is the memoir by L. Gaultier, on the graphical construction of circles and spheres (Jour. Polyt. t. ix., 1813). The well-known theorem as to radical axes may be stated as follows. Consider two circles partially drawn so that it does not appear whether the circles, if completed, would or would not intersect in real points, say two arcs of circles; then we can, by means of a third circle drawn so as to intersect in two real points each of the two arcs, determine a right line, which, if the complete circles intersect in two real points, passes through the points, and which is on this account regarded as a line passing through two (real or imaginary) points of intersection of the two circles. The construction in fact is, join the two points in which the third circle meets the first arc, and join also the two points in which the third circle meets the second arc, and from the point of intersection of the two joining lines, let fall a perpendicular on the line joining the centre of the two circles; this perpendicular (considered as an indefinite line) is what Gaultier terms the “radical axis of the two circles”; it is a line determined by a real construction and itself always real; and by what precedes it is the line joining two (real or imaginary, as the case may be) intersections of the given circles.
The intersections which lie on the radical axis are two out of the four intersections of the two circles. The question as to the remaining two intersections did not present itself to Gaultier, but it is answered in Jean Victor Poncelet’s Traité des propriétés projectives (1822), where we find (p. 49) the statement, “deux circles placés arbitrairement sur un plan ... ont idéalement deux points imaginaires communs à l’infini”; that is, a circle qua curve of the second order is met by the line infinity in two points; but, more than this, they are the same two points for any circle whatever. The points in question have since been called (it is believed first by Dr George Salmon) the circular points at infinity, or they may be called the circular points; these are also frequently spoken of as the points I, J; and we have thus the circle characterized as a conic which passes through the two circular points at infinity; the number of conditions thus imposed upon the conic is = 2, and there remain three arbitrary constants, which is the right number for the circle. Poncelet throughout his work makes continual use of the foregoing theories of imaginaries and infinity, and also of the before-mentioned theory of reciprocal polars.
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.
