How did Descartes know what sort of a curve an equation traces?

Question

For instance, it is known that the equation $x^2 + y^2 = r^2$ forms a circle. And the following is the graph of the folium of Descartes defined by $x^3 + y^3 - 3axy = 0$. My question is how did he know that this is the curve the equation will trace? I'm sure he wasn't plotting each point one by one so how did he do it? Curves can be traced relatively easily using calculus but Descartes did not have access to the method (even though Fermat did to a certain extent even before calculus officially came into being). So what other method did Descartes use to trace curves in his analytic geometry?

Answer 1

In the letter to Mersenne [Lettre XCIX, Janvier 1638, Oeuvres de Descartes: Correspondance I, Charles Adam & Paul Tannery eds., 1897, page 490] the curve is described.

The context is Descartes' discusssion of Fermat's Methodus ad disquirendam maximam et minimam compared to his own methods to find the tangent to a curve :

let the curved line BDN such that, taken a point B whatever on the curve and draw the perpendicular BC [(to the horizontal) and being C the intersection of the horizontal with the curve] the cubes of the two segments BC and CD are jointly equal to the parallelepiped built with the said lines and the assumed line P (in a way that : being P=9 and CD=2, we will have BC=4, because the cubes of 2 and 4 are 8 and 64 and 8+64=72 and also the parallellepiped formd by 2, 4 and 9 is 72).

In modern terms, the equation is :

$$x^3+y^3=pxy.$$

The curve has been named galand [ribbon node] by Roberval.

---

On how to draw the locus in general, we can see :

• Jan de Witt’s Elementa Curvarum Linearum Liber Secundus, Albert Grootendorst et alii editors, (2010)

as well as Part II of :

• Henk J.M. Bos, Redefining Geometrical Exactness : Descartes' Transformation of the Early Modern Concept of Construction (2001).