CURVE. 
461 
785] 
line drawn through it to a fixed point, is constant ; and the “ cissoid ” 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 
pivotted 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. 
The principle of classification first presented itself in the Géométrie of Descartes 
(1637). The idea was to represent any curve whatever by means of a relation between 
the coordinates (x, y) of a point of the curve, or say to represent the curve by means 
of its equation. 
Descartes takes two lines xx, yy', called axes of coordinates, intersecting at a point 
0 called the origin (the axes are usually at right angles to each other, and for the 
V 
N 
X'— 
M 
y 
present they are considered as being so); and he determines the position of a point 
P by means of its distances OM (or NP) = x, and MP (or ON)=y, from these two 
axes respectively; where x is regarded as positive or negative according as it is in 
the sense Ox or Ox from 0; and similarly y as positive or negative according as it 
is in the sense Oy or Oy' from 0; or, what is the same thing, 
X 
y 
In the quadrant 
xy, 
or 
N.E., we have 
+ 
+ 
» 
xy 
N.W. „ 
- 
+ 
» 
xy' 
S.E. 
+ 
- 
xy' 
s.w. „ 
- 
- 
whatever between 
(x, 
y) 
determines a 
curve, 
curve whatever is determined by a relation between (x, y).