Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/755

CURVE 719 1829), subsequent to the memoir by Gergonne, " Consider ations philosophiques sur les elemens de la science de 1 eteudue" (Gerg., t. xvi., 1825-26). In this memoir by Gergonne, the theory of duality is very clearly and explicitly stated ; for instance, we find " dans la geometric plane, a chaque theoreme il en repond necessairement un autre qui s en deduit en echangeant simplement entre eux les deux mots points et droites ; tandis que dans la geometric de 1 espace ce sont les mots points et plans qu il faut echanger entre eux pour passer d un theoreme a son correlatif ; " 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 theoremes, <fec.," which is im portant as first introducing the word class. We find in it explicitly the two correlative definitions : " a plane curve is said to be of the ?th degree (order) when it has with a line m real or ideal intersections," and " a plane curve is said to be of the ?th 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 recip rocal 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 corresponds in the other figure a line and point respectively. It is to be understood that the theorem ex tends 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 corre sponding curve must be of the class m. Trilinear coordinates (to be again referred to) were first used by Bobillier in the memoir, " Essai sur un nouveau mode de recherche des proprietes de Fetendue " (Gerg., t. xviii., 1827-28). It is convenient to use these rather than Cartesian coordinates. We represent a curve of the order m by an equation (*x, ?/, z) m Q, the function on the left hand being a homogeneous rational and integral func tion of the order m of the three coordinates (x, y, z) ; clearly the number of constants is the same as for the equa tion (*$x, y, 1 ) = in Cartesian coordinates. The theory of duality is considered and developed, but chiefly in regard to its metrical applications, by Chasles in the " M6moire de g6ome trie sur deux principes generaux de la science, la dualite", et 1 homographie," which forms a sequel to the "Aperu Historique sur Forigine et le developpement des methodes en Ge ome trie" (Mem. de Brux., t. xi., 1837). We now come to Pliicker ; his " six equations " were given in a short memoir in Crelle (1842) preceding his great work, the Theorie der Algebraisclien Curven (1844). Pliicker first gave a scientific dual definition of a curve, viz., " A curve is a locus generated by a point, and en veloped by a line, the point moving continuously along the line, while the line rotates continuously about the point ; " the point is a point (ineunt) of the curve, the line is a tangent of the curve. And, assuming the above theory of geometrical imaginaries, a curve such that m of its points are situate in an arbitrary line is said to be of the order m ; a curve such that n of its tangents pass through an arbitrary point is said to be of the class n ; as already appearing, this notion of the order and class of a curve is, however, due to Gergonne. Thus the line is a curve of the order 1 and class ; and corre sponding dually thereto, we have the point as a curve of the order and class 1. Pliicker moreover imagined a system of line-coordinates (tangential coordinates). The Cartesian coordinates (x, y) and trilinear coordinates (x, y, z) are point-coordinates for determining the position of a point ; tho new coordinates, say (, r), ) are line-coordinates for determining the posi tion of a line. It is possible, and (not so much for any application thereof as in order to more fully establish the analogy between the two kinds of coordinates) important, to give independent quantitative definitions of the two kinds of coordinates ; but we may also derive the notion of line-coordinates from that of point-coordinates; viz., taking r + rjy + tz = to be the equation of a line, we say that (, t], ) are the line-coordinates of this line. A linear relation a+br) + c0 between these coordinates deter mines a point, viz., the point whose point-coordinates are (a, b, c) ; in fact, the equation in question a+br) + c = expresses that the equation r + yy + & = 0, where (x, y, z) are current point-coordinates, is satisfied on writing therein x, y, z = a, b, c ; or that the line in question passes through the point (a, b, c). Thus (, r}, ) are the line-co ordinates of any line whatever ; but when these, instead of being absolutely arbitrary, are subject to the restriction a + by + c = 0, this obliges the line to pass through a point (a, b, c) ; and the last-mentioned equation a+br) + c, = is considered as the line-equation of this point. A line has only a point-equation, and a point has only a line-equation ; but any other curve has a point- equation and also a line-equation ; the point-equation (*&*> 2/> z ) m = is the relation which is satisfied by the point-coordinates (x, y, z) of each point of the curve ; and similarly the line-equation (*$, -rj, ) n = is the relation which is satisfied by the line-coordinates (, 77, ) of each line (tangent) of the curve. There is in analytical geometry little occasion for any explicit use of line-coordinates ; but the theory is very important ; it serves to show that in demonstrating by point-coordinates any purely descriptive theorem whatever, we demonstrate the correlative theorem ; that is, we do not demonstrate the one theorem, and then (as by the method of reciprocal polars) deduce from it the other, but we do at one and the same time demonstrate the two theorems ; our (x, y, z) instead of meaning point-coordinates may mean line-coordinates, and the demonstration is then in every step of it a demonstration of the correlative theorem. The above dual generation explains the nature of the singularities of a plane curve. The ordinary singularities, arranged according to a cross division, are Proper. Improper. Point-singu- ( 1. Tlie stationary point, 2. The double point, larities ( cusp, or spinode ; or node ; Line-singu- 3. The stationary tan- 4. The double tan- larities ( gent, or inflexion ; gent ; arising as follows : 1. The cusp : the point as it travels along the line may come to rest, and then reverse the direction of its motion. 3. The stationary tangent : the line may in the course of its rotation come to rest, and then reverse the direction of its rotation. 2. The node : the point may in the course of its motion come to coincide with a former position of the point, the two positions of the line not in general coinciding. 4. The double tangent : the line may in the course of its motion come to coincide with a former position of the line, the two positions of the point not in general coinciding

It may be remarked that we cannot with a real point and