by Adrian Rice

© Oxford University Press 2004 All rights reserved

**Graves, John Thomas** (1806-1870), jurist and mathematician, born in Dublin on 4 December 1806, was the eldest son of John Crosbie Graves, barrister, grandnephew of Richard Graves DD and cousin of Robert James Graves MD. After attending a school run by the Revd Samuel Field at Westbury-on-Trym, Gloucestershire, he entered Trinity College, Dublin, in 1823. There he distinguished himself in both science and classics, was a fellow student and friend of Sir William Rowan Hamilton, and graduated BA in 1827. He then moved to Oxford and became an incorporated member of Oriel College on 11 November 1830. He proceeded MA at Oxford in 1831, and at Dublin in 1832. Graves was called to the English bar in 1831 as a member of the Inner Temple, having previously (1830) entered the King's Inns, Dublin. He practised for a short time until 1839, when he was appointed professor of jurisprudence at University College, London, in succession to John Austin, who had retired in 1835. Soon after he was also elected an examiner in laws for the University of London.

With two of his professorial colleagues, the classicist Henry Malden and the mathematician Augustus De Morgan, Graves established enduring friendships, serving with the latter on the committee of the Society for the Diffusion of Useful Knowledge. He was elected a fellow of the Royal Society in 1839, and subsequently sat upon its council. He was also a member of the Philological Society and of the Royal Society of Literature. In 1846 Graves was appointed an assistant poor-law commissioner, and in the next year, under the new Poor Law Act, one of the poor-law inspectors of England and Wales. He married Amelia, a daughter of the politician William Tooke, on 24 March 1846. They had no children.

Graves's principal works as a jurist are twelve lectures on the law of nations, reported in the *Law Times,* commencing 25 April 1845, and two elaborate articles contributed to the *Encyclopaedia metropolitana* on Roman law and canon law. He was also a contributor to Smith's *Dictionary of Greek and Roman Biography,* his articles including very full lives of the jurists Cato, Crassus, Drusus, and Gaius, and one on the legislation of Justinian.

Graves also had a high reputation among his contemporaries as a mathematician. In 1826, aged only nineteen, he began research into exponential functions. The results were published in the *Philosophical Transactions* for 1829 under the title 'An attempt to rectify the inaccuracy of some logarithmic formulae'. His principal discovery was the existence of two arbitrary and independent integers in the complete expression of an imaginary logarithm. He thus considered that he had elucidated the subject of the logarithms of negative and imaginary quantities, which at different periods had caused disagreements between mathematicians such as Leibniz and Johann Bernoulli, Euler, and D'Alembert.

However, Graves's conclusions also occasioned some initial controversy since they were not at first unanimously accepted by the British mathematical community, the most prominent objectors being George Peacock and Sir John Herschel. Graves accordingly communicated to the British Association for the Advancement of Science a defence and explanation of his discovery, which was printed in its *Report* for 1834. The same report contained a paper by Hamilton, in which he fully confirmed his friend's conclusions. Hamilton explicitly acknowledged that it was 'in reflecting on the important symbolical results of Mr. Graves respecting imaginary logarithms, and in attempting to explain to himself the theoretical meaning of those remarkable symbolisms' that he was drawn to 'the theory of conjugate functions, which, leading on to a theory of triplets and sets of moments, steps, and numbers' *(Notes and Abstracts of ... the British Association for the Advancement of Science,* 1834, section 2) was to be the foundation of his future remarkable contributions to algebra, culminating in the discovery of quaternions. When this occurred in 1843, it was to Graves that Hamilton made his first written communication, on 17 October. Further acknowledgements of his obligation to Graves for stimulus and suggestion can be found in Hamilton's preface to his *Lectures on Quaternions* and in a prefatory letter to a communication in the *Philosophical Magazine* for December 1844. Graves modestly disclaimed the credit of suggestion.

For many years Graves had been Hamilton's sympathetic friend and mathematical confidant, and the two men maintained an active correspondence, in which they competed with each other in their attempts to produce a full and coherent interpretation of imaginaries. Graves worked at perfecting algebraic language; Hamilton had the higher object of arriving at the meaning of the science and its operations.

Soon after Hamilton's discovery of quaternions Graves concentrated on extending to eight squares Euler's theorem that the sum of four squares multiplied by the sum of four squares gives a product which is also the sum of four squares. He went on to conceive a theory of octaves analogous to Hamilton's theory of quaternions, introducing four imaginaries, additional to Hamilton's i, j, k, and conforming to 'the law of the modulus'. This was of great interest to Hamilton, but turned out to be less effective than quaternions as a practical algebraic system. Further mathematical papers followed, principally on complex numbers and the theory of equations, but these were largely of less significance than his earlier output and declined in number after the mid-1840s, owing to increasing pressure of work.

For many years Graves's principal recreational interest was mathematical bibliography, and his collection of mathematical works of all ages and countries was generally described as one of the most complete and valuable private libraries of the kind ever formed. It was bequeathed in his will to University College, London, in remembrance of their former association. Comprising over ten thousand books, 4577 pamphlets, and a considerable number of manuscripts, covering all aspects of the mathematical sciences over a period of five centuries, the current value of the Graves Library is inestimable, many of the items being exceedingly rare, some probably unique.

Three days after making this bequest, on 29 March 1870, Graves died at Thirlestaine Lodge, his home in Cheltenham. He was buried beside his mother in the graveyard of Swindon church, near Cheltenham. His wife survived him.

ADRIAN RICE

**Sources **

*University College Gazette,* 1 (1886-7), 189-90

*PRS,* 19 (1870-71), xxvii-xxviii

R. P. Graves, *Life of Sir William Rowan Hamilton,* 3 vols. (1882-9)

A. R. Dorling, 'The Graves mathematical collection in University College London', *Annals of Science,* 33 (1976), 307-9

University College London, *Proceedings at the Annual General Meeting of the Members of the College* (1870-71), 19

University College London, *Catalogue of books in the general library and in the south library at University College London,* 1 (1879), iii

*CGPLA Eng. & Wales* (1870)

**Archives **

TCD, corresp. and papers

UCL, collections, lecture notes | BL, letters to Charles Babbage, Add. MSS 37189-37200, *passim*

Bodl. Oxf., corresp. with Sir Thomas Phillipps

TCD, letters to Sir W. R. Hamilton

UCL, letters to Society for the Diffusion of Useful Knowledge

**Likenesses **

photograph (after portrait), UCL

**Wealth at death **

under £25,000: probate, 26 April 1870, *CGPLA Eng. & Wales*

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