The Folium of Descartes

Find the area of the loop bounded
by the
Folium of Descartes:

Equations

____________

Please click on the thumbnail image below to see an enlarged view of these equations.

"Cogito ergo sum"

( I think, therefore I am.)

Area of the Loop

Many famous curves have names from nature. "Folium" means leaf. "Cissoid" means ivy shaped. "Conchoid" implies the shape of a sea shell. "Petal" and "Rose" are more obvious in English.

This section . . . .

Area of loop

Move the mouse over this line. The slant asymptote is at
t = - 1.
Interactive folium

From Descartes' On the Nature of Curved Lines in
La Géométrie, first published in 1637 as an appendix to
his famous Discours de la Méthode.

Note the illustration on the right: Descartes has joined the arithmetic of multiplication and division to geometry in the style of the ancient Greeks. His next mental step might similarly have been to attempt to unite algebra and geometry. Arguably his most significant contribution to mathematics was to found analytic geometry. Indeed, we call graphing in the plane as "Cartesian" in his honor.

http://www.orst.edu/instruct/phl302/
philosophers/descartes.html

http://hades.ph.tn.tudelft.nl/Internal/PHServices
/Documentation/MathWorld/math/math/f/f234.htm

A Brief Listing of references that should be in most university libraries.

Bell, E. T., Men of Mathematics, Chapter 3 in various editions and publishers. Bell's essay entitled Descartes: Gentleman, Soldier, and Mathematician is a wonderfully written description of a "particularly full and interesting life."

Yates, R. C., Curves and their Properties, NCTM, 1952, pp. 98 - 99. Also in A Handbook on Curves and their Properties, various publishers including the NCTM.