The Four Curves of Alexis Clairaut: Technical Notes

On the Finding of Mean Proportionals

The problem of finding a number of mean proportionals between two given segments has its origins in the Elements of Euclid. The case of one mean is easy. However, many mathematicians in the Greek period toiled over the problem of how to find two means. The case of two means is equivalent to the more famous problem of doubling the cube; namely, given a cube, to construct a new cube double in volume. Hippocrates of Chios (470–410 BCE) realized that this problem could be solved by finding the first of two mean proportionals between the volume of the original cube and the volume of the desired cube (generally double).

What Hippocrates realized is that, assuming \(\mathbf{A}\) and \(\mathbf{D}\) are the given magnitudes, then

\(\mathbf{A}\) : \(\mathbf{D}\) :: cube on \(\mathbf{A}\) : cube on \(\mathbf{B}\) iff \(\mathbf{A}\) : \(\mathbf{B}\) :: \(\mathbf{B}\) : \(\mathbf{C}\) :: \(\mathbf{C}\) : \(\mathbf{D}\).

Hence, if we let \(\mathbf{D}\) = 2 and \(\mathbf{A}\) = 1, then the cube with edge-length \(\mathbf{B}\) will have volume 2. The problem, then, is finding \(\mathbf{B}\). Eutocius of Ascalon (480–540) recorded a number of ancient solutions to this problem, which the reader can find in the original Greek in [5] or [7], or in English in either [6] or [12].

In the Elements, Euclid called this result by the name of triplicate ratio. He first defined duplicate ratio. If three magnitudes \(\mathbf{A}\), \(\mathbf{B}\), and \(\mathbf{C}\) are in continuous proportion, then \(\mathbf{B}\) is the mean proportional between \(\mathbf{A}\) and \(\mathbf{C}\), that is,

\(\mathbf{A}\) : \(\mathbf{B}\) :: \(\mathbf{B}\) : \(\mathbf{C}\).

Then \(\mathbf{A}\) has to \(\mathbf{C}\) the duplicate ratio of that which \(\mathbf{A}\) has to \(\mathbf{B}\). For Clairaut, this can be expressed as

\(\mathbf{A} : \mathbf{C} :: (\mathbf{A} : \mathbf{B})^2 :: \mathbf{A}^2 : \mathbf{B}^2\)

Similarly, if there are four magnitudes \(\mathbf{A}\), \(\mathbf{B}\), \(\mathbf{C}\), and \(\mathbf{D}\), then \(\mathbf{B}\) is the first of two means proportional between \(\mathbf{A}\) and \(\mathbf{D}\) (the second being \(\mathbf{C}\)). Then \(\mathbf{A}\) has to \(\mathbf{D}\) the triplicate ratio of that which \(\mathbf{A}\) has to \(\mathbf{B}\). Symbolically,

\(\mathbf{A} : \mathbf{C} :: (\mathbf{A} : \mathbf{B})^3 :: \mathbf{A}^3 : \mathbf{B}^3\)

Clairaut made use of the generalization of these concepts to any number of means. If there are magnitudes \(\mathbf{X}_0\), ..., \(\mathbf{X}_{n+1}\), then the term \(\mathbf{X}_1\) is the first of \(n\) means proportional between \(\mathbf{X}_0\) and \(\mathbf{X}_{n+1}\), and

\(\mathbf{X}_0 : \mathbf{X}_{n+1} :: \mathbf{X}_0^{n+1} : \mathbf{X}_1^{n+1}\) .

Here, the two given segments are \(\mathbf{CG}\) and \(\mathbf{CM}\); \(\mathbf{CF}\) is the first of \(n\) means proportional. Clairaut used this result to yield the equations for his generalized curves in each of the four problems below.

On the Translation of Genre

In our translation of Clairaut’s article, there was one mathematical term whose meaning was unclear: “genre”. Clairaut’s curves are fourth degree, which he calls “of the third genre”.

Possible translations of “genre” include “gender”, “genre”, and “genus”; we also investigated the possibility of “genre” translating to “degree” or “class”. We decided on “genre” for the following reasons. “Gender” would likely mean that the classification was binary, but this does not make sense given that Clairaut’s curves are, as he says, of the third genre. This can be seen in Problem III, point 1. “Genre” and “degree” are clearly not equivalent in the text, since, as we have said, his curves are of fourth degree but third genre.

“Genus” seemed another possibility to use as a translation. However, the word “genus” has a meaning in modern mathematics, both in topology and projective geometry, and it was not clear if Clairaut’s intended meaning matched the modern concept, despite the fact that his four curves are of the third genus (in the modern sense). This is because each of them is non-singular, and so the genus-degree formula

\(g = \frac{(d - 1)(d - 2)}{2}\)

gives \(g = 3\) when \(d = 4\). As we will see shortly, however, it is a coincidence that the genus of these curves matches the “genre” assigned by Clairaut.

D. E. Smith, in his edition of Descartes’ La Géométrie [3], translated “genre” as “class”. However, Descartes’ use of the word “genre” does not line up with Clairaut’s. Descartes organized curves into classes based on degree, with first and second degree curves (corresponding to lines and conic sections) being of the first class. Third and fourth degree curves are of the second class, etc. This clearly does not match Clairaut’s use, since under Descartes’ system, a fourth degree equation would be of the second class. The meaning of “genre” then must have changed between Descartes and Clairaut.

The answer comes from Guisnée’s book Application de l’algèbre à la géométrie. “Genre” means one less than the degree. We know from our Introduction that Clairaut studied from this very book only a few years prior to preparing the results of his article. We thus conclude that Clairaut’s sense of “genre” is the same as Guisnée’s. As for how to translate the word, we decided to simply use the English “genre”, and not “class”. This allows us to avoid mixing up Clairaut’s sense of “genre” with Descartes’, and our usage also will not tempt the reader into thinking of “class” in the modern sense as in \(C^{\infty}\).

Ultimately the word has no mathematical importance: since it is just one less than the degree, stating the genre does not add any information. It is important, though, in understanding how Clairaut, Guisnée, and other mathematicians of their time thought philosophically about the classification of algebraic curves.