John, like his younger brother Charles Graves, was educated at a school run by the Rev Samuel Field at Westbury-on-Trym, near Bristol, Gloucestershire in England, where he was prepared for university. Returning to Ireland, he entered Trinity College, Dublin, in 1823. One of his fellow students was William Rowan Hamilton and the two soon became friends. Hamilton had studied Clairaut's Algebra, Laplace's Mécanique Céleste and Newton's Principia before entering Trinity College. He had also come to the attention of John Brinkley, the Andrews Professor of Astronomy at Trinity who was an excellent mathematician having studied the latest continental mathematics. This friendship with Hamilton rapidly brought Graves into contact with the latest mathematics. Graves was also fortunate to benefit from the mathematical reforms that Bartholomew Lloyd had introduced at Trinity almost ten years before he began his studies there. Bartholomew Lloyd had become professor of mathematics at Trinity in 1813 and, taking over a department in which the teaching had been very old-fashioned with no calculus taught, he had quickly introduced the continental approach to calculus teaching from Lacroix's textbook Traité élémentaire de calcul differéntiel et du calcul intégral, from Poisson's Traité de mécanique, and from Laplace's Mécanique Céleste.
As an undergraduate, Graves excelled in science and classics but after graduating from Trinity College, Dublin, with a B.A. in 1827 he decided that he wanted a career in law. However, in 1826, while still an undergraduate, Graves had began to study complex logarithms and produced an extension of Euler's formulas for logarithms given in Introductio in Analysin Infinitorum (1748). Graves felt that he had answered problems that had arisen concerning complex logarithms which had been argued over by Gottfried Leibniz and Johann Bernoulli, Leonhard Euler and Jean d'Alembert. He did not publish this work until 1829 when his paper An Attempt to Rectify the Inaccuracy of some Logarithmic Formulae was published in the Philosophical Transactions of the Royal Society of London. The paper, communicated by John Frederick William Herschel, was read on 18 December 1828. It begins:-
From the recent researches of MM Poisson and Poinsot on angular section, and their discovery of error in trigonometrical formulae usually considered complete, my attention has been drawn to analogous incorrectness in logarithmic series. Accordingly, the end proposed in the present investigation is the exhibition in an amended form of two fundamental developments, as principles employed in their establishment admit of application in expanding by different methods various similar functions, and tend to elucidate other parts of the exponential theory.This paper produced a considerable reaction from a number of mathematicians supporting his results, and a number who objected to them. Alexandre Vincent, a professor at the Royal College at Reims in France, claimed that earlier, in 1825, he had found the same results as Graves but had not published them. Vincent's proof was published in 1832. Martin Ohm also claimed to have arrived at the same results earlier than Graves but he had published them in Versuch eines volkommen consequenten Systems der Mathematik (1823). However, this work only became known in Britain when the second edition appeared in 1829. Graves was disappointed to find that others had found the results earlier but even more disappointed to find that several British mathematicians claimed that his results were incorrect. Both George Peacock and John Herschel expressed doubts concerning the validity of Graves' results and Peacock printed his objections in his Report to the British Association for the Advancement of Science in 1833. Hamilton, however, strongly supported Graves' results and when he spoke at the 1834 meeting of the British Association for the Advancement of Science in Edinburgh, he offered another proof of the results. Hamilton presented the paper On Conjugate Functions or Algebraic Couples, as tending to illustrate generally the Doctrine of Imaginary Quantities, and as confirming the Results of Mr Graves respecting the existence of Two independent Integers in the complete expression of an Imaginary Logarithm. He wrote, rather strangely, in the third person:-
When translated into the language of imaginaries, they agree with the results respecting imaginary exponential functions, direct and inverse, which were published by Mr Graves in the 'Philosophical Transactions' for 1829, and it was in meditating on those results of Mr Graves that Mr Hamilton was led, several years ago, to this theory of conjugate functions, as tending to illustrate and confirm them. For example, Mr Graves had found, for the logarithm of unity to the Napierian base, [an] expression ... which is more general than the usual expression. This result of Mr Graves appeared erroneous to the author of the excellent Report on Algebra, which was lately printed for the Association; but it is confirmed by Mr Hamilton's theory ...Augustus De Morgan expressed doubts about the correctness of the results in 1836 and, in the same year Graves replied to De Morgan by publishing in the Philosophical Magazine an alternative and shorter proof in his paper On the lately proposed logarithms of unity, in reply to Professor De Morgan. De Morgan was convinced by this proof writing:-
I see no further objection to Mr Graves's system, but quite the contrary ... the extension may be highly useful, and I am happy to bear testimony to the ingenuity which suggested it, and the talent with which it has been carried out.However, Peacock was still not convinced and had published a "proof" that Graves was wrong in his Treatise on Algebra of 1830. Duncan Farquharson Gregory pointed out the error in Peacock's "proof" in 1837 so, in the second edition of his Treatise on Algebra in 1845, Peacock admitted his error and gave a proof of Graves' result.
We have moved well ahead to complete the story of Graves' 1826 results on complex logarithms which took nearly 20 years before all were happy with it. Let us return to looking at Graves' career after he graduated and had decided to study law. After graduating from Trinity College, Dublin, he moved to the Inner Temple in London to gain legal qualifications. He gave his address as the Inner Temple when he submitted his logarithm paper to the Royal Society of London in 1828. Back in Ireland, he continued his legal training and entered the King's Inns, Dublin in 1830. Moving to Oxford, he became an incorporated member of Oriel College on 11 November 1830 and was awarded an M.A. in the following year. Later in 1831 he was called to the English bar as member of the Inner Temple. In 1832 he was awarded an M.A. by Trinity College, Dublin. For the next few years Graves practiced law.
Graves was not too happy as a lawyer and applied for a position with the Irish Society. He was unsuccessful and wrote a letter to James David Forbes (1809-1868), who was at that time professor of Natural Philosophy at the University of Edinburgh. The letter, dated 11 March 1838, was sent from Graves' chambers at 20 Southampton Buildings, Chancery Lane, London. In the letter, Graves, who was making an application to University College London, wrote:-
I wish to make my escape from the bar, being fonder of literary and scientific than of professional pursuits.In 1839 Graves was appointed as a Professor of Jurisprudence at University College, London. In the same year he was elected a fellow of the Royal Society of London. Soon after taking up his professorship, Graves was elected as an examiner in laws for the University of London. Working at University College London, Graves was now a colleague of Augustus De Morgan and the two became close friends. The Society for the Diffusion of Useful Knowledge had been set up by the same reformers who founded London University and De Morgan was an enthusiast for the Society. Graves and De Morgan served together on the Society's committee.
Graves had certainly not given up mathematics but had continued to correspond with Hamilton. In fact his early work on logarithms was undertaken since he believed that it might lead to the discovery of new imaginary numbers. Graves continued to work on the idea but he thought about such things in a rather different way from Hamilton who wrote to him in 1835 (see for example ):-
... we belong to opposite poles in algebra since you, like Peacock, seem to consider algebra as a 'system of signs and their combinations', somewhat analogous to syllogisms expressed in letters while I am never satisfied unless I think that I can look beyond or through the signs to the things signified.Graves' ideas regarding possible new complex numbers continued and he tried to multiply lines in space given by triples. In corresponding with Hamilton and telling him about his ideas in this area, he kept Hamilton's interest in generalising complex numbers to three dimensions. After Hamilton made his discovery of the quaternions in 1843 the first person he wrote to telling of his discovery was Graves who replied on 26 October complimenting Hamilton on his novel idea and adding (see for example ):-
There is still something in the system which gravels me. I have not yet any clear views as to the extent to which we are at liberty arbitrarily to create imaginaries, and to endow them with supernatural properties.The reason that it was Graves that he turned to first was, as Hamilton explained in a letter to the Philosophical Magazine on 20 November 1844, because his discussions with Graves had played an important role in the discovery. You can read part of Hamilton's letter at THIS LINK.
In this letter Hamilton mentions some extensions of the quaternions which had been discovered by Graves. In fact in volume 3 of the Proceedings of the Royal Irish Academy (1836-1869) we see precisely how Graves explained his discoveries to Hamilton:-
In a letter dated January 18th, 1844, Mr Graves communicated to Sir William Hamilton his theorem respecting sums of eight squares ... In a letter of somewhat earlier date, but evidently written in haste, upon a journey, and dated December 26th, 1843, analogous expressions had been given, containing, however, some errors in the signs, which were soon afterwards corrected as above. That earlier letter also indicated an expectation that a theory of octaves, including a new and extended system of imaginaries, which had thus been suggested to the writer (J T Graves, Esq.) by Sir William R Hamilton's theory of quaternions, might itself be extended so as to form a theory of what Mr Graves at the time proposed to call 2n-ions: but in a letter written shortly afterwards, doubts were expressed respecting the possibility of this additional extension, from octaves to sets of sixteen.The octaves, which are mentioned in this quote, are today called the octonions or Cayley numbers and not, as one might expect, the Graves numbers. The events which led to this are as follows. Hamilton was too busy to report on Graves' findings until after the same 8-dimensional system had been constructed by Arthur Cayley and published in March 1845. The quote above, which shows Graves' priority, was not published until 14 June 1847. This note contains Graves' Eight Squares Identity but even this remarkable result, which generalised Euler's Four Squares Identity, had been discovered before. The Danish mathematician Ferdinand Degen (1766-1825) seems to have been the first to discover this result in Adumbratio demonstrationis theorematis arithmeticae maxime generalis (1818) which he published in the Mémoires de l'Académie Impériale des Sciences de St Pétersbourg. As an aside, we note that Degen was the mathematician to whom the young Niels Henrik Abel submitted his "solution" of the quintic equation. Degen asked Abel to give an example of his method and at that stage Abel himself discovered his error. Graves seems to be one of those people who made remarkable discoveries but luck was against him and he has almost been forgotten.
For further details of the Eight Squares Identity and Graves-Cayley Numbers, see THIS LINK.
At University College, London, Graves continued to seek systems of algebraic triplets even after Hamilton had discovered the quaternions. He clearly interested his colleague De Morgan in this question as the comments on Charles Graves's publications on algebraic triplets show, see THIS LINK.
On 25 April 1845 the first of twelve lectures by Graves on the law of nations were published in the Law Times. He also wrote :-
... 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.On 24 March 1846 Graves married Amelia Tooke, the daughter of William Tooke and Amelia Shaen. They had no children. William Tooke was a lawyer who become President of the Society of Arts. Before his marriage Graves had resigned his chair of Jurisprudence at University College, London, giving as the reason for his resignation the 'discouragingly low enrolment for his classes'. He stated that his lectures on Roman law, on international law and on general jurisprudence had been poorly attended in contrast with his lectures on practical topics such as the law of equity. In the same year that he married he was appointed as an assistant poor-law commissioner. In 1847 he was promoted to one of the poor-law inspectors of England and Wales, a position which came into existence because of the Poor Laws that came into force in that year.
The botanist Sir Joseph Dalton Hooker (1817-1911) wrote to his fellow botanist John Stevens Henslow in 1858:-
I met a Rev J T Graves at Dublin, a Fellow of Trinity College, Dublin, mathematician, a man of renown in these parts who has been employed by the Government in enquiring on Endowed Schools and other Educational matters. He is immensely strong on your point of teaching the science of Observation to all men, especially the young of all classes, and he has reported the same to the Government in perhaps the very words you would have used.This letter by Hooker does not mean that Graves had changed his profession since we see that Graves continued to write reports as a poor-law inspector up to at least 1866.
There is one aspect of Graves' life which we have not mentioned so far and that was his life-long hobby as a collector of ancient mathematical texts. He bequeathed his collection to University College London, three days before he died at his home, Thirlestaine Lodge in Cheltenham at the age of sixty-three. Several articles describe his amazing collection but perhaps the best would be to quote from  supplied by the library of University College, London:-
The library acquired contains over 10,000 books, 4,600 pamphlets, 51 manuscripts and numerous periodicals, covering mainly mathematics and astronomy. Described as probably the most important private mathematical collection ever made, it contains many rare and possibly unique items. Probably the most important single collection within the Graves material is the Euclid collection, which contains eighty-three of the editions of Euclid's works printed before 1640. The collection includes the 'editio princeps' published by Erhard Ratdolt at Venice in 1482, and amongst the translations are the first into any modern language, the Italian of 1543, the first German translation (1562), the first French (1564), John Day's edition of the first English translation with John Dee's preface of 1570, the first edition in Arabic (1594) and later translations into Turkish, Chinese, Persian, Hebrew, Finnish and many other languages. The Graves collection includes seventy-five of the Library's incunabula and many famous books such as first editions of Copernicus's 'De revolutionibus' of 1543, Newton's 'Principia' and 'Opticks', and Thomas Salusbury's 'Mathematical Collections' of 1661-65. There are first editions of fascinating "association copies" of the works of Priestley, Boyle, Kepler, Galileo, and Napier and important runs of early scientific periodicals. There are also many treasures such as Henry Cavendish's copy of Pascal's 'Traité de l'équilibre des liqueurs' of 1663 and, of special note, a copy of Galileo's 'Il Saggiatore', published in Rome in 1623 and inscribed to Galileo's friend Morandi.
Article by: J J O'Connor and E F Robertson