Wallis–Brouncker Method

Summary: The historical method for solving Pell’s equation $p^2=A{q}^2+1$ in integers, due to Brouncker (in response to a challenge by Fermat), published by Wallis in his Algebra (1685). Equivalent to expanding $\sqrt{A}$ as a continued fraction and reading off principal convergents — but Wallis claimed any choice of approximation direction works, which Lagrange disproves in add8-pell-method-critique.

Sources: additions-8 (Articles 85–87).

Last updated: 2026-05-10.

---

Origin

• Pierre de Fermat (mid-17th c.) proposed solving $p^2=A{q}^2+1$ as a challenge to English mathematicians, recorded in Wallis’s Commercium Epistolicum.
• Lord Brouncker (William Brouncker, 1620–1684) found the iterative algorithm in response.
• John Wallis published it in his Algebra (1685, Chapter 98).
• Jacques Ozanam later mistakenly credited Fermat.
• Leonhard Euler rediscovered and refined it (Chapter VII of Part II of Elements of Algebra; see ch2.0.7-pell-equation-method) and noticed its broader relevance to $x^2=A{y}^2+B$.

The misnomer “Pell’s equation” is itself a Eulerian misattribution; John Pell had little to do with it.

The algorithm

Starting from $p^2-A{q}^2=1$, observe that for large $q$, $p\approx q\sqrt{A}$. Write the continued fraction expansion of $\sqrt{A}$, $\sqrt{A}=\mu +\frac{1}{{\mu }^{\prime }+\frac{1}{{\mu }^{\prime \prime }+\cdots }}$ and substitute iteratively $p=\mu q+r$, $q={\mu }^{\prime }r+s$, $r={\mu }^{\prime \prime }s+t$, etc. Each substitution transforms the original equation into a new equation in two variables, with smaller coefficients in the same family. When a transformed equation reaches the form $\pm 1\cdot u^2+\cdots =1$ with leading coefficient unity, set the second indeterminate to $0$ and the first to $1$; unwind the substitutions to obtain $(p,q)$.

This is precisely Lagrange’s lagrange-reduction-algorithm specialised to the constant right side $1$.

What’s correct, what’s not

Correct (Lagrange, Art. 86): every solution $(p,q)$ of $p^2-A{q}^2=1$ corresponds to a principal convergent $p/q$ of the continued fraction of $\sqrt{A}$, with all partial quotients positive (i.e., approximations always taken less than the true value).

Wrongly claimed by Wallis and Euler (Art. 87 critique): that one may freely take approximations either “in plus” (ceiling) or “in minus” (floor) at each step, since the remainders decrease either way. Lagrange shows this can prevent the algorithm from terminating: for $p^2-6q^2=1$, taking the first step in minus and all subsequent steps in plus produces a sequence of forms whose leading coefficient never reaches $1$.

Wallis’s “proof” of solvability

Wallis (Ch. 99) attempted to prove that $p^2=A{q}^2+1$ is always solvable for $A>0$ non-square. Lagrange dismisses this as “a mere petitio principii” — i.e., it assumes what it sets out to prove. The first rigorous proof of solvability is Lagrange’s own, in Mélanges de Turin vol. IV, with a more direct version in Article 37 of the Additions.

Relation to modern continued-fraction theory

The Wallis-Brouncker substitution chain is exactly the recurrence for continued-fraction convergents — but historically presented as an algebraic-substitution method without the unifying notion of $p_n/q_n$. The connection is made explicit in add1-continued-fractions (definition) and add2-arithmetic-problems (Pell solvability theorem, Art. 37).

Related pages

• pell-equation
• ch2.0.7-pell-equation-method
• add8-pell-method-critique
• continued-fractions
• convergents
• square-root-continued-fractions
• lagrange-reduction-algorithm
• periodicity-quadratic-irrationals