THE CONCHOIDS OF NICOMEDES 
239 
length of the slot in Hi? on each side of F, the extremity P of 
the ruler describes the curve which is called a conchoid or 
cochloid. Nicomedes called the straight line AB the ruler 
(Karcor), the fixed point G the pole (ttoXos-), and the constant 
length PD the distance {SidcrT^ga). 
The fundamental property of the curve, which in polar 
coordinates would now be denoted by the equation 
r = a + b sec 6, 
is that, if any radius vector be drawn from C to the curve, as 
CP, the length intercepted on the radius vector between the 
curve and the straight line AB is constant. Thus any veva-is 
in which one of the two given lines (between which the 
straight line of given length is to be placed) is a straight line 
can be solved by means of the intersection of the other line 
with a certain conchoid having as its pole the fixed point 
to which the inserted straight line must verge {veva.v). Pappus 
tells us that in practice the conchoid was not always actually 
drawn but that ‘ some for greater convenience, moved a ruler 
about the fixed point until by trial the intercept was found to 
be equal to the given length. 1 
In the figure above (p. 236) showing the reduction of the 
trisection of an angle to a vevaLS the conchoid to be used 
would have B for its 'pole, AC for the ‘ruler’ or hose, a length 
equal to 2 AB for its distance; and E would be found as the 
intersection of the conchoid with FA produced. 
Proclus says that Nicomedes gave the construction, the 
order, and the properties of the conchoidal lines 2 ; but nothing 
1 Pappus, iv, p. 246. 15. 2 Proclus on Eucl. I, p. 272. 3-7.