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Let S be any curve and let A
be a fixed point. If a straight line is drawn through A to meet
curve at Q, and if P and P' are points on this line such that PQ = P'Q
a constant term, the locus of the points P and P' is called a conchoid
the curve with respect to the fixed point A.
The Conchoid family of curves is easily
entered and modified on a graphing calculator.
Be sure to use (1/cos) for the secant term.
of the conchoid ( mussel-shell shape ) is ascribed to Nicomedes
by Pappus and other classical authors; it was a favourite with
the mathematicians of the seventeenth century as a specimen for the new
method of analytical geometry and calculus. It could be used (as
was the purpose of the invention) to solve the two problems of doubling
the cube and of trisecting an angle; and hence for every
cubic or quartic problem. For this reason Newton suggested that
it should be treated as a standard curve."
Nicomedes (ca. 225 BC) is credited with being the first
to investigate the conchoid, a name from the Greek meaning
"shell-like" or "shell form." He sought to find two mean
given lengths in order to solve the famous Delian problem.
The Delian problem is also known as duplication of the
in other words, finding the edge of a cube having a volume exactly
that of a given cube. This is one of the three famous
dating from antiquity. If, contrary to Euclidean assumptions, we
permit ourselves to mark a straight edge, the conchoid may, in fact be
applied to these famous problems.
With adjustments in various constant terms, the conchoid
is modified into a circle, spiral, limaçon, or cardiod.
Moreover, we find equations for the conchleoid and conchal as
well as focal conchoids of conic sections in the literature. But
the most famous are the . . . . . .
- Conchoid of Dürer
- Conchoid of de Sluze
- Limaçon of Pascal, father of
- Conchoid of Külp
- and of course, the oldest, the Conchoid of
To underscore the historical importance of the conchoid,
we have selected Figure #139 of John Colson's translation of Maria
Gaetana Agnesi's Instituzioni analitche, now generally
recognized as the first book of mathematics written by a woman.
Colson held the prestigious Lucasian chair at
Cambridge. In 1801 he chose to translate Agnesi's widely
acclaimed treatise on Calculus written one-half century earlier (1748).
Unfortunately, his translation of the name of the curve now known in
English speaking countries as the Witch of Agnesi was a
slight, but lasting error. However, his Conchoid of Nicomedes
was accurate and an indication of the enduring fame of this popular
curve. Colson's EXAMPLE IV speaks for itself.
permission from the Rare Books Division, Dept. of Rare Books and
Collections, Princeton University Library.
Please observe that
Agnesi first sketched a circle to generate both her Witch
and the conchoid
before proceeding to give
Links and Books
|Eves, Howard, An Introduction
to the History of Mathematics, 6th ed,. The Saunders College
|Gray, Alfred, Modern
Differential Geometry of Curves and Surfaces with MATHEMATICA®, 2nd ed., CRC Press,
1998, p. 898.
|Lockwood, E. H., A Book of
Curves, Cambridge University Press, 1961.
|Shikin, Eugene V., Handbook
and Atlas of Curves, CRC Press, 1995.
|Yates, Robert, CURVES AND
THEIR PROPERTIES, The National Council of Teachers of Mathematics,
Code and animation contributed by
Gustavo Gordillo, 2006.