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The Four Curves of Alexis Clairaut: Clairaut’s Work

Taner Kiral (Wabash College), Jonathan Murdock (Wabash College), and Colin B. P. McKinney (Wabash College)

The power of the mesolabe compass, then, is not that it solves the specific case of two mean proportionals. Rather, it is able to solve for any number of means. Similarly, Clairaut’s curves started from a desire to solve a specific problem, but he quickly generalized them to produce a family of curves able to solve the general problem of finding any number of means. Clairaut’s work differs from Descartes’ in a number of ways:

  • Clairaut’s first family of curves is a proper super-set of the curves described by Descartes’ mesolabe compass. Clairaut placed an intermediate curve between each of Descartes’.

  • Clairaut gave three other families of curves, similarly motivated, but totally different in construction.

  • Clairaut also investigated properties of his curves such as their tangents and inflection points. This point especially should not be seen as a criticism of Descartes: after all, a considerable amount of time transpired between the first edition of La Géométrie and Clairaut’s paper.

The first point is illustrated in the following figure. The construction given by Descartes is shown, with his curves traced in red. The curves unique to Clairaut are shown in blue.

The equations for each of these curves is listed below, indexed under both Clairaut’s scheme and Descartes’. In the figure, the curves move from left to right from \(Y\) towards \(X\), alternating between red and blue.

Taking \(a = 1\), the general form for Clairaut’s equation is

\(x^{\frac{n + 1}{n}} = \sqrt{x^2 + y^2}\).

When \(n\) is even, raising both sides to the \(n\)th power clears the fractional exponents, yielding the polynomial equation

\(x^{n+1} = (x^2 + y^2)^{\frac{n}{2}}\) (for n even).

If, however, \(n\) is odd, then it is necessary to raise both sides to the \(2n\)th power, yielding

\(x^{2n+2} = (x^2 + y^2)^n\) (for \(n\) odd).

This yields the following list of equations, whose graphs are shown in Figure 3:

  • C1/D1: \(x^4 = x^2 + y^2\)
  • C2: \(x^3 = x^2 + y^2\)
  • C3/D2: \(x^8 = (x^2 + y^2)^3\)
  • C4: x5 = \((x^2 + y^2)^2\)
  • C5/D3: \(x^{12} = (x^2 + y^2)^5\)

Figure 2: Descartes’ and Clairaut’s Curves.

The cases where we take \(n\) odd in Clairaut’s general equation yield precisely the curves given by Descartes.

Given the degree to which Clairaut’s work relates to Descartes’, it is surprising that Clairaut never mentioned Descartes or La Géométrie. In fact, Clairaut never mentioned any other mathematician or work, not even Guisnée, despite using almost precisely the same assignment of points and quantities as Guisnée. Guisnée himself referenced Descartes by name four times. We have a hard time believing that Clairaut had not read Descartes. Even if he had failed to notice the similarity of his curves with those of Descartes when he presented this work, we would expect that someone at the Royal Academy would have. We also would find it strange that, even if he had not noticed the similarity when he presented to the Academy, he would not have subsequently noticed this in the roughly eight-year period between when he presented to the Academy and when his paper was published in the Miscellanea Berolinensia.

Taner Kiral (Wabash College), Jonathan Murdock (Wabash College), and Colin B. P. McKinney (Wabash College), "The Four Curves of Alexis Clairaut: Clairaut’s Work," Convergence (November 2020)