The Four Curves of Alexis Clairaut

Alexis Clairaut, born in 1713 to mathematician and teacher Jean-Baptiste Clairaut and mother Catherine, was a mathematician who showed promise from a very young age. In 1726 he presented on four new families of curves and their properties to the French Royal Academy of Sciences. Clairaut published these findings in 1734 in “Quatre Problèmes sur de Nouvelles Courbes” (“Four Problems on New Curves”) in the fourth volume of the journal of the Royal Prussian Academy of Sciences, Miscellanea berolinensia ad incremental scientiarum. Each of the four families of algebraic curves that he investigated was partly motivated by the classical Greek problem of finding mean proportionals between two given line segments. Clairaut also investigated the analytic properties of his curves by finding tangents, inflection points, and quadratures.

Clairaut’s 1734 paper, written and published in French, has not yet been translated to English. We present a dual language edition—French (pdf) and English (pdf)—to make Clairaut’s paper accessible by a modern audience.

In this companion article, we provide an overview of the historical and mathematical contexts of Clairaut's paper, relating his work in particular to that of Descartes. We then describe the provenance of Clairaut's text and the editorial conventions that we adopted in preparing its French transcription and English translation. This is followed by some brief technical notes to help a modern reader better understand the problem of finding mean proportionals that played the central role in Clairaut's findings, as well as his use of the French mathematical term genre. We close with a few comments about what Clairaut's paper might offer today's students, based on our own experience as a faculty-student research-and-translation team.