The Conchoid Family of Curves - National Curve Bank

"The invention 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."

E. H. Lockwood

Historical Sketch

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 proportions between given lengths in order to solve the famous Delian problem.

The Delian problem is also known as duplication of the cube; in other words, finding the edge of a cube having a volume exactly twice that of a given cube. This is one of the three famous constructions 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 Blaise Pascal
Conchoid of Külp
and of course, the oldest, the Conchoid of Nicomedes.

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.

Colson's Conchoid in Instituzioni analitche

Reproduced with permission from the Rare Books Division, Dept. of Rare Books and Special Collections, Princeton University Library.

Please observe that Agnesi first sketched a circle to generate both her Witch and the conchoid before proceeding to give a proof.

For the modern graphing calculator . . . .
Replay the animation.