Pierre de Fermat

Portrait of Pierre de Fermat

The Margin Was Too Narrow · Magistrate & Mathematician · 1607–1665

The Toulouse magistrate who invented coordinate geometry in his spare time, founded probability by letter with Pascal, transformed the theory of numbers, and published almost none of it.

Who was Fermat?

Pierre de Fermat was born near Montauban in 1607, studied law, and in 1631 became a councillor of the parlement of Toulouse, where he served for the rest of his life. Mathematics was what he did when the court rose, and he treated it as a private pleasure: he almost never published, and what survives is largely letters. Around 1636 he circulated a short work, the Introduction to Plane and Solid Loci. In it a curve is studied by taking a fixed line, measuring a distance along it, raising an ordinate, and stating the relation between those two lengths. That is coordinate geometry, and Fermat had it before Descartes published his Géométrie in 1637 — a fact Descartes never conceded and which soured relations between them permanently. Fermat's version is in some ways more limited: his ordinates are often raised obliquely rather than square to the axis, and he admits no quantity less than nothing, so his figures occupy a single quarter. He also had a method for maxima, minima and tangents which anticipates a central move of the calculus. Descartes attacked it, was answered, and eventually accepted it. In 1654 Pascal wrote to him about the division of a stake in an interrupted game — the problem Pacioli had answered wrongly a century and a half earlier. Over a handful of letters the two of them settled it, by reasoning about how many rounds each player still needed rather than how many had been won. That correspondence is generally taken as the beginning of the mathematical theory of probability. His deepest work was on whole numbers, and it is here that his habit of announcing results without proofs is most exasperating. He asserted a great many things about primes, divisors and sums of squares, most of them true and most of them proved by other people over the following two centuries. In the margin of his copy of Diophantus he wrote that no cube is the sum of two cubes, nor any higher power, and that he had found a truly marvellous demonstration which the margin was too narrow to contain. Andrew Wiles proved it in 1994, using mathematics unimaginable in the seventeenth century. Almost nobody believes Fermat had a valid proof.

Major achievements

Life in brief

Explore the life of Fermat — five chapters

  1. The Councillor of Toulouse — Mathematics in the hours the court left over
  2. A Curve as a Relation — Plane and Solid Loci, about 1636
  3. The Letters of 1654 — How a stake should be divided
  4. The Theory of Numbers — Assertions, mostly true, mostly unproved
  5. The Margin Was Too Narrow — A note that took three hundred and fifty years

Read the five-chapter life of Pierre de Fermat →

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