The invention of continued fractions is ascribed generally to Pietro Antonia Cataldi, an Italian mathematician who died in 1626. He used them to represent square roots, but only for particular numerical examples, and appears to have had no theory on the subject. A previous writer, Rafaello Bombelli, had used them in his treatise on Algebra (about 1579), and it is quite possible that Cataldi may have got his ideas from him. His chief advance on Bombelli was in his notation. They next appear to have been used by Daniel Schwenter (1585-1636) in a Geometrica Practica published in 1618. He uses them for approximations. The theory, however, starts with the publication in 1655 by Lord Brouncker of the continued fraction
| 1 | 1² | 3² | 5² | ||||
| 1 | + | 2 | + | 2 | + | 2 | + ... |
as an equivalent of π ⁄ 4. This he is supposed to have deduced, no one knows how, from Wallis’ formula for 4 ⁄ π viz.
| 3 . 3 . 5 . 5 . 7 . 7 ... |
| 2 . 4 . 4 . 6 . 6 . 8 ... |
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 Schlömilch, 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`algèbre supérieure in French; and in German those of Stern, Schlömilch, Hatterdorff and Stolz. For the application of continued fractions to the theory of irrational numbers there is P. Bachmann’s Vorlesungen über 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 Encyklopädie der mathematischen Wissenschaften (Leipzig).
(A. E. J.)
CONTOUR, CONTOUR-LINE (a French word meaning generally “outline,” from the Med. Lat. contornare, to round off), in physical geography a line drawn upon a map through all the points upon the surface represented that are of equal height above sea-level. These points lie, therefore, upon a horizontal plane at a given elevation passing through the land shown on the map, and the contour-line is the intersection of that horizontal plane with the surface of the ground. The contour-line of 0, or datum level, is the coastal boundary of any land form. If the sea be imagined as rising 100 ft., a new coast-line, with bays and estuaries indented in the valleys, would appear at the new sea-level. If the sea sank once more to its former level, the 100-ft. contour-line with all its irregularities would be represented by the beach mark made by the sea when 100 ft. higher. If instead of receding the sea rose continuously at the rate of 100 ft. per day, a series of levels 100 ft. above one another would be marked daily upon the land until at last the highest mountain peaks appeared as islands less than 100 ft. high. A record of this series of advances marked upon a flat map of the original country would give a series of concentric contour-lines narrowing towards the mountain-tops, which they would at last completely surround. Contour-lines of this character are marked upon most modern maps of small areas and upon all government survey and military maps at varying intervals according to the scale of the map.