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 (*≬ x, y, 1)2 = 0; or by an equation of the third order, &c.; or what is the same thing, according as the equation is linear, quadric, cubic, &c.
A curve of the first order is a right line; and conversely every right line is a curve of the first order. A curve of the second order is a conic, and is also called a quadric curve; and conversely every conic is a curve of the second order or quadric curve. A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.
A curve of the order m has for its equation (≬ x, y, 1)m = 0; and when the coefficients of the function are arbitrary, the curve is said to be the general curve of the order m. The number of coefficients is ½(m + 1)(m + 2); but there is no loss of generality if the equation be divided by one coefficient so as to reduce the coefficient of the corresponding term to unity, hence the number of coefficients may be reckoned as ½(m + 1)(m + 2) − 1, that is, ½m(m + 3); and a curve of the order m may be made to satisfy this number of conditions; for example, to pass through ½m(m + 3) points.
It is to be remarked that an equation may break up; thus a quadric equation may be (ax + by + c)(a′x + b′y + c′) = 0, breaking up into the two equations ax + by + c = 0, a′x + b′y + c′ = 0, viz. the original equation is satisfied if either of these is satisfied. Each of these last equations represents a curve of the first order, or right line; and the original equation represents this pair of lines, viz. the pair of lines is considered as a quadric curve. But it is an improper quadric curve; and in speaking of curves of the second or any other given order, we frequently imply that the curve is a proper curve represented by an equation which does not break up.
4. Intersections of Curves.—The intersections of two curves are obtained by combining their equations; viz. the elimination from the two equations of y (or x) gives for x (or y) an equation of a certain order, say the resultant equation; and then to each value of x (or y) satisfying this equation there corresponds in general a single value of y (or x), and consequently a single point of intersection; the number of intersections is thus equal to the order of the resultant equation in x (or y).
Supposing that the two curves are of the orders m, n, respectively, then the order of the resultant equation is in general and at most = mn; in particular, if the curve of the order n is an arbitrary line (n = 1), then the order of the resultant equation is = m; and the curve of the order m meets therefore the line in m points. But the resultant equation may have all or any of its roots imaginary, and it is thus not always that there are m real intersections.
The notion of imaginary intersections, thus presenting itself, through algebra, in geometry, must be accepted in geometry—and it in fact plays an all-important part in modern geometry. As in algebra we say that an equation of the mth order has m roots, viz. we state this generally without in the first instance, or it may be without ever, distinguishing whether these are real or imaginary; so in geometry we say that a curve of the mth order is met by an arbitrary line in m points, or rather we thus, through algebra, obtain the proper geometrical definition of a curve of the mth order, as a curve which is met by an arbitrary line in m points (that is, of course, in m, and not more than m, points).
The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m = 2, if the hyperbola xy − 1 = 0 be cut by the line y = β, the resultant equation in x is βx − 1 = 0, and there is apparently only the intersection (x = 1/β, y = β); but the theorem is, in fact, true for every line whatever: a curve of the order m meets every line whatever in precisely m points. We have, in the case just referred to, to take account of a point at infinity on the line y = β; the two intersections are the point (x = 1/β, y = β), and the point at infinity on the line y = β.
It is, moreover, to be noticed that the points at infinity may be all or any of them imaginary, and that the points of intersection, whether finite or at infinity, real or imaginary, may coincide two or more of them together, and have to be counted accordingly; to support the theorem in its universality, it is necessary to take account of these various circumstances.
5. Line at Infinity.—The foregoing notion of a point at infinity is a very important one in modern geometry; and we have also to consider the paradoxical statement that in plane geometry, or say as regards the plane, infinity is a right line. This admits of an easy illustration in solid geometry. If with a given centre of projection, by drawing from it lines to every point of a given line, we project the given line on a given plane, the projection is a line, i.e. this projection is the intersection of the given plane with the plane through the centre and the given line. Say the projection is always a line, then if the figure is such that the two planes are parallel, the projection is the intersection of the given plane by a parallel plane, or it is the system of points at infinity on the given plane, that is, these points at infinity are regarded as situate on a given line, the line infinity of the given plane.[1]
- ↑ In solid geometry infinity is a plane—its intersection with any given plane being the right line which is the infinity of this given plane.
