We now come to Julius Plücker; his “six equations” were given in a short memoir in Crelle (1842) preceding his great work, the Theorie der algebraischen Curven (1844). Plücker first gave a scientific dual definition of a curve, viz.; “A curve is a locus generated by a point, and enveloped 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 0; and corresponding dually thereto, we have the point as a curve of the order 0 and class 1.
Plücker, moreover, imagined a system of line-co-ordinates (tangential co-ordinates). (See [Geometry]: Analytical.) The Cartesian co-ordinates (x, y) and trilinear co-ordinates (x, y, z) are point-co-ordinates for determining the position of a point; the new co-ordinates, say (ξ, η, ζ) are line-co-ordinates for determining the position 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 co-ordinates) important, to give independent quantitative definitions of the two kinds of co-ordinates; but we may also derive the notion of line-co-ordinates from that of point-co-ordinates; viz. taking ξx + ηy + ζz = 0 to be the equation of a line, we say that (ξ, η, ζ) are the line-co-ordinates of this line. A linear relation aξ + bη + cζ = 0 between these co-ordinate determines a point, viz. the point whose point-co-ordinates are (a, b, c); in fact, the equation in question aξ + bη + cζ = 0 expresses that the equation ξx + ηy + ζz = 0, where (x, y, z) are current point-co-ordinates, 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 (ξ, η, ζ) are the line-co-ordinates of any line whatever; but when these, instead of being absolutely arbitrary, are subject to the restriction aξ + bη + cζ = 0, this obliges the line to pass through a point (a, b, c); and the last-mentioned equation aξ + bη + cζ = 0 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 (*
x, y, z)m = 0 is the relation which is satisfied by the point-co-ordinates (x, y, z) of each point of the curve; and similarly the line-equation (*
ξ, η, ζ)n = 0 is the relation which is satisfied by the line-co-ordinates (ξ, η, ζ) of each line (tangent) of the curve.
There is in analytical geometry little occasion for any explicit use of line-co-ordinates; but the theory is very important; it serves to show that in demonstrating by point-co-ordinates 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-co-ordinates may mean line-co-ordinates, and the demonstration is then in every step of it a demonstration of the correlative theorem.
7. Singularities of a Curve. Plücker’s Equations.—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-singularities— | 1. The stationary point, cusp or spinode; | 2. The double point or node; |
| Line-singularities— | 3. The stationary tangent or inflection; | 4. The double tangent; |