M. Lagrange, on applying the method of continued fractions to the resolution of numerical equations, discovered that, for those of the second degree, the quotients recur periodically. From this, combined with the previously well known fact that all periodic chain fractions belong to quadratics, he inferred that periodicity is exclusively confined to equations of this order.

In January 1858, I showed to the Royal Society that the series of approximating fractions obtained by M. Lagrange can be continued in the opposite direction, and that the convergence then is to the other root; and enunciated the general theorem, that if any two fractions be assumed, and if a progression be formed from them by combining fixed multiples of their members, this progression, which I called duserr or two-headed, may be continued in either way, and gives on the one hand the one, on the other hand the other root of a quadratic.