Additions Chapter VIII — Critique of the Wallis–Brouncker Method for Pell
Summary: Lagrange surveys the historical method (Fermat → Brouncker → Wallis → Euler) for , points out gaps in earlier “demonstrations,” and gives a counterexample showing that taking continued-fraction limits “in plus” rather than “in minus” can prevent the method from ever terminating.
Sources: additions-8 (Articles 85–87).
Last updated: 2026-05-10.
Historical attribution (Art. 85)
The method of ch2.0.7-pell-equation-method in Euler’s main treatise — solving in integers by iterative descent — is not Euler’s invention. Lagrange traces it as follows:
- The problem is due to Fermat, who proposed it as a challenge to English mathematicians (Wallis, Commercium Epistolicum).
- The method was found by Lord Brouncker in response.
- Wallis published it in his Algebra (Chapter 98), and Ozanam later attributed it to Fermat.
- Euler alone noticed that is the key to solving the more general (the subject of add7-integer-quadratic-method).
Lagrange’s own contribution is the first rigid demonstration that is always solvable for non-square, published in the Mélanges de Turin vol. IV — with a shorter, more conceptual proof reproduced in Article 37 of the Additions. He says of Wallis’s claimed proof: “his demonstration, if I may presume to say so, is a mere petitio principii” (Wallis Chapter 99).
Lagrange also points out (Art. 85) that the standard recipe — find one solution of and propagate via — fails when is composite: there can be solutions of the former that are not contained in Euler’s general expressions.
Convergent characterisation (Art. 86)
From Chapter II of the Additions, every integer solution of must arise from a principal convergent of the continued-fraction expansion with all positive (i.e. all approximations taken less than the real values). This positivity restriction is essential to the validity of the method of Problem I (Art. 23).
The substitution chain (Art. 87)
Working with the continued-fraction expansion of , write where the last remainder is and the second-to-last is . Substituting iteratively into (or , neglecting the constant) gives a chain of transformed forms The procedure terminates when the leading coefficient hits unity; setting the second indeterminate to and the first to , and unwinding the substitutions, recovers .
This is the same method as Euler’s Chapter VII, recast.
When the method fails (Art. 87, continued)
Wallis (Ch. 98) and Euler (Art. 102 of the parent Chapter VII) both claim that one may freely choose limits “in plus” or “in minus” — i.e. take the floor or the ceiling of , , etc. — and that this freedom is sometimes useful for shortening the calculation. Lagrange shows this is wrong.
Counterexample: . First step: . Take in minus and substitute :
Branch A — continue in minus. Setting gives , immediately solved by , hence . ✓
Branch B — switch to plus. Setting gives , then , then . Continuing in plus: → , then . Each step keeps producing forms with non-unit leading coefficient ( in cycle); the algorithm never reaches a leading , so never produces a solution.
The same failure occurs whenever the first limit is taken “in minus” and all subsequent in “plus.” Lagrange remarks that the a priori reason can be deduced from the principles of his theory but does not pursue it; the example suffices to show the necessity of “investigating these problems more fully and more rigorously than has hitherto been done.”
Significance
This is one of the earliest documented critiques of an algorithmic method’s completeness in number theory, distinct from criticism of its correctness. The method’s individual steps are valid; what fails is the claim that any choice of approximation direction works. It motivates the careful sign-conventions of continued-fractions and the specific positivity required by the standard convergent expansion.