by Harvey W. Becher
© Oxford University Press 2004 All rights reserved
Woodhouse, Robert (1773-1827), mathematician, was born at Norwich on 28 April 1773, the son of Robert Woodhouse, a draper, and his wife, Judith, the daughter of the Lowestoft Unitarian minister, James Alderson. Having attended the grammar school at North Walsham, on 20 May 1790 Woodhouse was admitted to Gonville and Caius College, Cambridge, and matriculated in Michaelmas term, 1791. He took a BA as senior wrangler and first Smith's prizeman in 1795, and gained his MA and a fellowship at Caius in 1798. In 1803 he became a fellow of the Royal Society. His younger brother, John Thomas (1780-1845), was also a fellow of Caius.
The historical significance of Woodhouse's mathematics is a function of the backwardness of Cambridge mathematics at the turn into the nineteenth century. Most Cambridge mathematicians continued to labour within the confines of fluxions, a calculus of Newtonian notation and methodology resting upon geometric and physical imagery. By means of a more powerful and abstract calculus expressed in more comprehensive notation, continental mathematicians exhibited that the commitment to fluxions was anachronistic. However, this had little effect at Cambridge, since the primary purpose of a Cambridge education, the core of which was mathematics, was not to produce mathematicians but, rather, liberally educated gentlemen. Euclidean geometry, and Newtonian fluxions, mechanics, and astronomy, seemed best fit for this end, and most Cambridge dons perceived no need to change. Woodhouse did.
Initially Woodhouse focused his reforming efforts on pure mathematics and its foundations. His arguments are given in three articles in the Philosophical Transactions of the Royal Society (1801, 1802) and in a book, The Principles of Analytical Calculation (1803). He demanded that analysis in general and the calculus specifically be placed upon a purely algebraic footing free of geometric and physical encumbrances such as limits or infinitesimals. Moreover he argued that such mathematical entities as complex numbers (for instance the square root of minus one) and divergent series (a series having no single finite sum) were useful, indeed sometimes necessary, for achieving results. For this reason, and because he could not conceive of false premises leading to correct consequences, he objected to their being excluded from analysis.
To justify his contentions, Woodhouse turned to the French doyen of algebraic analysis, J. L. Lagrange. Lagrange had founded the calculus on the expansion of a function in a Taylor series in which the successive coefficients of the terms of the series directly provided the successive derivatives of the function. Finding fault with Lagrange for failing implicitly to avoid limits and for not showing algebraically that each function considered could be developed in a Taylor series, Woodhouse embraced formalism built upon his redefinition of the equal sign. In pure algebra, Woodhouse submitted, '=' did not signify arithmetical equality, but rather, the result of some operation. The analysts' programme, according to Woodhouse, was first to prove algebraically that a function had a Taylor series, then to 'extend' this form (for instance to the complex realm or beyond the radius of convergence), and then to manipulate the algebraic symbols to reach the desired end. Within this framework for algebra, Woodhouse insisted, numerical equality and, therefore, whether or not a series was convergent, were irrelevant. Only in applied mathematics did arithmetic equality become a concern.
Woodhouse's arguments in The Principles, aimed at mathematical virtuosi, were polemical and maladroit, mathematically and grammatically, and as a consequence the book had little immediate impact. However, his views and his notation--the differential, functional, and operator notation of the continent--were imbibed by younger and more bellicose members of the university who, in the second decade of the nineteenth century, stimulated adjustments which would bring Cambridge up to speed in the 1830s. Consequently, the conceptions explicated by Woodhouse became ingrained at Cambridge, thereby facilitating the creation of axiomatic algebra, the development of modern logic, and, correspondingly, the continued scepticism among Cambridge mathematicians toward the convergency considerations which came to dominate continental analysis.
Woodhouse also exercised a less dramatic but more immediate influence. He served as a moderator of the tripos six times between 1799 and 1808, and he turned his attention to astronomy, the apex of the Cambridge curriculum. In the Philosophical Transactions (1804), he reviewed current literature on elliptic integrals, a subject neglected at Cambridge but useful in astronomy. In 1809, using the continental notation with explanations in notes, he provided A Treatise on Plane and Spherical Trigonometry, a textbook treating trigonometry analytically. This was an improvement over the usual geometrically orientated Cambridge texts, especially if astronomy were to be treated analytically. In 1810, in A Treatise on Isoperimetrical Problems and the Calculus of Variations, a work which became influential both as history and as mathematics, he introduced at Cambridge the calculus of variations, a subject crucial to treating astronomy analytically. Finally, between 1812 and 1821, he published textbooks on astronomy which took the student from elementary practical astronomy through physical astronomy. For the physical astronomy, he severed relevant segments from Laplace's Mécanique céleste, the paragon of Newtonian mechanics treated analytically; at times, when employing simpler mathematics, he referenced Laplace's more sophisticated treatment. Woodhouse's publications had an impact: they were cited 147 times as aids for answering questions and solving problems in the tripos for the years 1801 to 1820. It is with justification then, that in 1819 a reviewer wrote that 'No man has done so much to improve the studies of Cambridge as Mr. Woodhouse' (EdinR, 42, 1819, 394).
In 1820 Woodhouse became Lucasian professor of mathematics; the chair was esteemed as that of Newton but brought a niggardly income. In 1822 he attained the more lucrative Plumian professorship of astronomy and experimental philosophy, and as Plumian professor he became the first superintendent of the newly built Cambridge University observatory. Endowed with a decent income, Woodhouse resigned his fellowship and, on 20 February 1823, married Harriet Wilkens, a Norwich architect's daughter, and the sister of the noted architect William Wilkens, sixth wrangler of 1800 and a fellow of Caius. She gave birth to a son, named Robert, in 1825. Two years later, on 23 December 1827, Woodhouse died at Cambridge. He was buried in Caius College chapel.
HARVEY W. BECHER
Sources
H. Becher, 'Woodhouse, Babbage, Peacock, and modern algebra', Historia Mathematica, 7 (1980), 389-400
H. Becher, 'William Whewell and Cambridge mathematics', Historical Studies in the Physical Sciences, 11 (1980-81), 1-48
Venn, Alum. Cant.
N. Guicciardini, The development of Newtonian calculus in Britain, 1700-1800 (1989)
E. Koppelman, 'The calculus of operations and the rise of abstract algebra', Archive for History of Exact Sciences, 8 (1971-2), 155-242
H. Pycior, 'Internalism, externalism, and beyond: 19th-century British algebra', Historia Mathematica, 11 (1984), 424-41
review of Elementary treatise on astronomy, EdinR, 31 (1818-19), 375-94
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