In this short article he explains the history of the solution of Pell's equation. It was proposed by Fermat in his letter to British mathematicians, and solved by Wallis and Brouncker, jointly, in their correspondence. They explained their solution on a number of special cases. Fermat criticized this solution asserting that they did not give a general proof. Then Weil speculates whether Fermat himself had a "general proof" himself and what this proof could be. He also analyses the "British solution", and establishes its connection with the Euclidean algorithm and a continued fraction algorithm.
The first published complete proof is due to Lagrange. Weil also gives some detail about pre-history of this solution, and mentions Bachet and Pascal.
In the beginning he notices that there is no known connection of this equation with John Pell, who was famous as a mathematician during his life time but published nothing.