Augustus De Morgan

“ Though not elementary algebra, it is within the reach of a student of ordinary books. [674]
Let a continued fraction, such as
a —— b + c —— d + e - f + etc., be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity ”

“ Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. ”

“ Algebra, 1860; Trigonometry, 1854; Optics, 1854, and others.)
678 ↑ This note on Carlyle (1795-1881) is interesting. The translation of Legendre appeared in the same year (1824) as his translation of Goethe's Wilhelm Meister.
679 ↑ Michael Stifel (1487-1567) , also known as Stiefel, Styfel, and Stifelius, was an Augustine monk but became a convert to Lutheranism. He was professor of mathematics at Jena (1559-1567) . His edition of the Coss appeared at Königsberg in 1553, the first edition having been published in 1525. ”