The operation in use by the ancient geometers for finding the numerical expression for the ratio of two quantities, was to repeat each of them until some multiple of the one agreed with a multiple of the other; the numbers of the repetitions being inversely proportional to the magnitudes.

The modern process, introduced by Lord Brouncker, under the name of continued fractions, is to seek for that submultiple of the one which may be contained exactly in the other; the numbers being then directly proportional to the quantities compared.

On applying this method to the roots of quadratic equations, the integer parts of the denominators were found to recur in periods; and Lagrange showed that, while all irrational roots of quadratics give recurring chain-fractions, every recurring chain-fraction expresses the root of a quadratic; and hence it was argued that this phenomenon of recurrence is exhibited by quadratic equations alone.