A profoundly influential work on algebra (1842 - 1845)


A treatise on algebra. I. Arithmetical algebra. II. On symbolical algebra, and its applications to the geometry of position.

Published: Cambridge and London, J. and J. J. Deighton and G. F. and J. Rivington, and Whittaker & Co.

Date: 1842 - 1845

Two vols., 8vo (220 × 140 mm), pp. xvi, 399, [1, blank]; x, 455, [1, blank]; occasional ink annotations to margins not affecting text; bookplate of Alwyn Compton to front pastedowns of both volumes; sprinkled edges; very good in half calf over marbled boards, lightly rubbed.

One of the founder members of the Analytical Society in 1815 (along with Woodhouse, Babbage and John Herschel), George Peacock’s early work was an important contribution to the acceptance and use of the Leibnizian notation for differential calculus at Cambridge, best summed up in his Collection of examples of the application of the differential and integral calculus published in 1820. He became Lowndean professor of astronomy and geometry in 1836 in succession to William Lax, having earlier helped to establish the university observatory, his rival for the post being William Whewell; and in 1839 he was made Dean of Ely, not far from Cambridge. In 1830 he wrote a Treatise on algebra, which was followed in 1833 by a Report on the recent progress and present state of certain branches of analysis. Peacock was not a prolific writer, but in preparing a second edition of the 1830 Treatise he so completely re-wrote it, that despite its retaining the same title, it is a substantially new work. ‘The work, the first volume of which is now offered to the public, was designed in the first instance to be a second edition of a Treatise on Algebra, published in 1830. I have found it necessary, in carrying out the principles developed in that work, to present the subject in so novel a form, that I could not with propriety consider it in any other light than as an entirely new treatise.’ (Peacock, 1842, p. iii) ‘Galois’s work on equations solvable by algebraic processes closed a chapter of algebra and, though he introduced ideas such as group and domain of rationality (field) that would bear fruit, the fuller exploitation of these ideas had to await other developments. The next major algebraic creation, initiated by William R. Hamilton, opened up totally new domains while shattering age-old convictions as to how “numbers” must behave … ‘By 1800 the mathematicians were using freely the various types of real numbers and even complex numbers, but the precise definitions of these various types of numbers were not available nor was there any logical justification of the operations with them … It seemed as though the algebra of literal expression possessed a logic of its own, which accounted for its effectiveness and correctness. Hence in the1830s the mathematicians tackled the problem of justifying the operations with literal or symbolic expressions. ‘This problem was first considered by George Peacock … To justify the operations with literal expressions that could stand for negative, irrational, and complex numbers he made the distinction between arithmetical algebra and symbolical algebra. The former dealt with symbols representing the positive integers and so was on solid ground. Here only operations leading to positive integers were permissible. Symbolical algebra adopts the rules of arithmetical algebra but removes the restrictions to positive integers … ‘[In] the second edition of his Treatise on Algebra [Peacock] … introduces a formal science of algebra. In this Treatise Peacock states that algebra like geometry is a deductive science. The processes of algebra have to be based on a compete statement of the body of laws that dictate the operations used in the processes. The symbols for the operations have, at least for the deductive sciences of algebra, no sense other than those given to them by the laws. Thus addition means no more than any process that obeys the laws of addition in algebra … Throughout most of the nineteenth century the view of algebra affirmed by Peacock was accepted.’ (Kline, pp. 772-774)

Very good

Half calf