Next to the circle we have the conic sections, the invention of them attributed to Plato (who lived 430-347 B.C.); the original definition of them as the sections of a cone was by the Greek geometers who studied them soon replaced by a proper definition in plano like that for the circle, viz. a conic section (or as we now say a “conic”) is the locus of a point such that its distance from a given point, the focus, is in a given ratio to its (perpendicular) distance from a given line, the directrix; or it is the locus of a point which moves so as always to satisfy the foregoing condition. Similarly any other property might be used as a definition; an ellipse is the locus of a point such that the sum of its distances from two fixed points (the foci) is constant, &c., &c.
The Greek geometers invented other curves; in particular, the conchoid (q.v.), which is the locus of a point such that its distance from a given line, measured along the line drawn through it to a fixed point, is constant; and the cissoid (q.v.), which is the locus of a point such that its distance from a fixed point is always equal to the intercept (on the line through the fixed point) between a circle passing through the fixed point and the tangent to the circle at the point opposite to the fixed point. Obviously the number of such geometrical or kinematical definitions is infinite. In a machine of any kind, each point describes a curve; a simple but important instance is the “three-bar curve,” or locus of a point in or rigidly connected with a bar pivoted on to two other bars which rotate about fixed centres respectively. Every curve thus arbitrarily defined has its own properties; and there was not any principle of classification.
2. Cartesian Co-ordinates.—The principle of classification first presented itself in the Géometrie of Descartes (1637). The idea was to represent any curve whatever by means of a relation between the co-ordinates (x, y) of a point of the curve, or say to represent the curve by means of its equation. (See [Geometry]: Analytical.)
Any relation whatever between (x, y) determines a curve, and conversely every curve whatever is determined by a relation between (x, y).
Observe that the distinctive feature is in the exclusive use of such determination of a curve by means of its equation. The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation is x²/a² + y²/b² = 1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.
3. Order of a Curve.—We obtain from the equation the notion of an algebraical as opposed to a transcendental curve, viz. an algebraical curve is a curve having an equation F(x, y) = 0 where F(x, y) is a rational and integral function of the co-ordinates (x, y); and in what follows we attend throughout (unless the contrary is stated) only to such curves. The equation is sometimes given, and may conveniently be used, in an irrational form, but we always imagine it reduced to the foregoing rational and integral form, and regard this as the equation of the curve. And we have hence the notion of a curve of a given order, viz. the order of the curve is equal to that of the term or terms of highest order in the co-ordinates (x, y) conjointly in the equation of the curve; for instance, xy − 1 = 0 is a curve of the second order.
It is to be noticed here that the axes of co-ordinates may be any two lines at right angles to each other whatever; and that the equation of a curve will be different according to the selection of the axes of co-ordinates; but the order is independent of the axes, and has a determinate value for any given curve.
We hence divide curves according to their order, viz. a curve is of the first order, second order, third order, &c., according as it is represented by an equation of the first order, ax + by + c = 0, or say (*
x, y, 1) = 0; or by an equation of the second order, ax² + 2hxy + by² + 2fy + 2gx + c = 0, say (*