John Wallis, discussing this fraction in his _Arithmetica Infinitorum_ (1656), gives many of the elementary properties of the convergents to the general continued fraction, including the rule for their formation. Huygens (_Descriptio automati planetarii_, 1703) uses the simple continued fraction for the purpose of approximation when designing the toothed wheels of his _Planetarium_. Nicol Saunderson (1682-1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler's _Algebra_ (1795). Moritz A. Stern wrote at length on the subject in _Crelle's Journal_ (x., 1833; xi., 1834; xviii., 1838). The theory of the convergence of continued fractions is due to Oscar Schloemilch, P. F. Arndt, P. L. Seidel and Stern. O. Stolz, A. Pringsheim and E. B. van Vleck have written on the convergence of infinite continued fractions with complex elements. REFERENCES.--For the further history of continued fractions we may refer the reader to two papers by Gunther and A. N. Favaro, _Bulletins di bibliographia e di storia delle scienze mathematische e fisicke_, t. vii., and to M. Cantor, _Geschichte der Mathematik_, 2nd Bd. For text-books treating the subject in great detail there are those of G. Chrystal in English; Serret's _Cours d`algebre superieure_ in French; and in German those of Stern, Schloemilch, Hatterdorff and Stolz. For the application of continued fractions to the theory of irrational numbers there is P. Bachmann's _Vorlesungen ueber die Natur der Irrationalzahnen_ (1892). For the application of continued fractions to the theory of lenses, see R. S. Heath's _Geometrical Optics_, chaps. iv. and v. For an exhaustive summary of all that has been written on the subject the reader may consult Bd. 1 of the _Encyklopaedie der mathematischen Wissenschaften_ (Leipzig). (A. E. J.)