CONCHOID (Gr. oyXn, shell, and ethos, form)

This article appears in Volume V06, Page 827 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: COM-COR

CONCHOID (Gr. oyXn, shell, and ethos, 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

CUBE (Gr. K46os, a cube)

, the construction of two mean proportionals between two given quantities, and possibly to the trisection of an

angle

ANGLE (from the Lat. angulus, a corner, a diminutive, of which the primitive form,
angus
- ANGUS, EARLS OF
, does not occur in Latin; cognate are the Lat. angere, to compress into a bend or to strangle, and the Gr. ayKOS, a bend; both connected with the Aryan root ank-, to

as in the 8th lemma of Archimedes.

Proclus

PROCLUS, or PROCULUS (A.D. 410-485)

grants Nicomedes the credit of this last application, but it is disputed by Pappus, who claims that his own discovery was
'

Double

and triple concertos are concertos with two or three solo players. A concerto for several solo players is called a concertante.

original

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 0 be a fixed point and BC a fixed straight line; draw any line through 0 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
i A, is constrained to move so that it
& C 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 A0=a; let the constant quantity PX = PX' = b. Then taking 0 as

pole

and a line through 0 parallel to BC as the initial line, the polar equation is r =a cosec B b, the upper sign referring to the branch more distant from O. The cartesian equation with A as origin and BC as

axis

AXIS (Lat. for " axle ")

of x is x2y2=(a+y)2 (b2y2). 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 0 and a loop below the initial point (curve i in the figure); if a equals b there will be a

cusp

CUSP (Lat. cuspis, a spear, point)

at 0 (curve 2); if a be greater than b the curve will not pass through 0, but from the cartesian equation it is obvious that 0 is a conjugate point (curve 3). The curve is symmetrical about the

axis

AXIS (Lat. for " axle ")

of y and has the axis of x for its asymptote.

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