462 
CURVE. 
[785 
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 ^ + p = l, 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 *. 
We obtain from the equation the notion of an algebraical or geometrical as opposed 
to a transcendental curve, viz. an algebraical or geometrical curve is a curve having an 
equation F(x, y) — 0, where F(x, y) is a rational and integral algebraical function of the 
coordinates {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 coordinates {x, y) con 
jointly 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 coordinates 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 coordinates ; 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 (*][#, y, 1) = 0; or by an equation of the second order, 
ax- + 21ixy + by 2 + 2fy + 2gx + c = 0, say (*]£#, y, l) 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. 
* There is no exercise more profitable for a student than that of tracing a curve from its equation, or 
say rather that of so tracing a considerable number of curves. And he should make the equations for him 
self. The equation should be in the first instance a purely numerical one, where y is given or can be 
found as an explicit function of x ; here, by giving different numerical values to x, the corresponding values 
of y may be found ; and a sufficient number of points being thus determined, the curve is traced by drawing 
a continuous line through these points. The next step should be to consider an equation involving literal 
coefficients; thus, after such curves as y=x 3 , y — x (x -1) (x- 2), y = (x - 1) Jx - 2, &c., he should proceed to 
trace such curves as y = (x - a) (x-b) (x - c), y = (x- a) Jx - 0, &c., and endeavour to ascertain for what different 
relations of equality or inequality between the coefficients the curve will assume essentially or notably distinct 
forms. The purely numerical equations will present instances of nodes, cusps, inflexions, double tangents, 
asymptotes, &c.,—specialities which he should be familiar with before he has to consider their general theory. 
And he may then consider an equation such that neither coordinate can be expressed as an explicit function 
of the other of them (practically, an equation such as x 3 + y 3 - 3xy — 0, which requires the solution of a cubic 
equation, belongs to this class) ; the problem of tracing the curve here frequently requires special methods, 
and it may easily be such as to require and serve as an exercise for the powers of an advanced algebraist.