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Encyclopedia Britannica



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
 , 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
'
Double
  and triple concertos are concertos with two or three solo players. A concerto for several solo players is called a concertante.
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
  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
  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
  of y and has the axis of x for its asymptote.


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