460 
[785 
785. 
CURVE. 
[From the Encyclopaedia Britannica, Ninth Edition, vol. vi. (1877), pp. 716—728.] 
This subject is treated here from an historical point of view, for the purpose of 
showing how the different leading ideas in the theory were successively arrived at and 
developed. 
A curve is a line, or continuous singly infinite system of points. We consider in 
the first instance, and chiefly, a plane curve described according to a law. Such a curve 
may be regarded geometrically as actually described, or kinematically as in course of 
description by the motion of a point; in the former point of view, it is the locus 
of all the points which satisfy a given condition; in the latter, it is the locus of a 
point moving subject to a given condition. Thus the most simple and earliest known 
curve, the circle, is the locus of all the points at a given distance from a fixed 
centre, or else the locus of a point moving so as to be always at a given distance 
from a fixed centre. (The straight line and the point are not for the moment regarded 
as curves.) 
Next to the circle we have the conic sections, the invention of them attributed 
to Plato (who lived 430 to 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 piano 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 con 
dition. 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,” which 
is the locus of a point such that its distance from a given line, measured along the