# 1911 Encyclopædia Britannica/Conchoid

**CONCHOID** (Gr. κόγχη, shell, and εἶδος, form), a plane curve
invented by the Greek mathematician Nicomedes, who devised
a mechanical construction for it and applied it to the problem
of the duplication of the cube, the construction of two
mean proportionals between two given quantities, and possibly
to the trisection of an angle as in the 8th lemma of Archimedes.
Proclus grants Nicomedes the credit of this last application, but
it is disputed by Pappus, who claims that his own discovery was
original. The conchoid has been employed by later mathematicians,
notably Sir Isaac Newton, in the construction of
various cubic curves.

The conchoid is generated as follows:—Let O be a fixed point
and BC a fixed straight line; draw any line through O intersecting
BC in P and take on the line PO two points X, X′, such
that PX = PX′ = a constant quantity.
Then the locus of X and X′ is the
conchoid. The conchoid is also the
locus of any point on a rod which
is constrained to move so that it
always passes through a fixed point,
while a fixed point on the rod travels
along a straight line. To obtain the
equation to the curve, draw AO
perpendicular to BC, and let AO = *a*; let the constant quantity
PX = PX′ = *b*. Then taking O as pole and a line through O
parallel to BC as the initial line, the polar equation is *r* = *a* cosec θ
± *b*, the upper sign referring to the branch more distant from
O. The cartesian equation with A as origin and BC as axis of
*x* is *x*^{2}*y*^{2} = (*a* + *y*)^{2} (*b*^{2} − *y*^{2}). Both branches belong to the same
curve and are included in this equation. Three forms of the
curve have to be distinguished according to the ratio of *a* to *b*.
If *a* be less than *b*, there will be a node at O and a loop below the
initial point (curve 1 in the figure); if *a* equals *b* there will be
a cusp at O (curve 2); if *a* be greater than *b* the curve will not
pass through O, but from the cartesian equation it is obvious
that O is a conjugate point (curve 3). The curve is symmetrical
about the axis of *y* and has the axis of *x* for its asymptote.