John Thomas Graves (4 December 1806-29 March 1870) was an Irish jurist and mathematician.
He was a friend of William Rowan Hamilton, and is credited both with inspiring Hamilton to discover the quaternions in October 1843 and then discovering their generalization the octonions himself, he called them octaves, later that same year. He was the brother of both the mathematician and bishop Charles Graves and the writer and clergyman Robert Perceval Graves.
Born in Dublin 4 December 1806, he was son of John Crosbie Graves, barrister, grandnephew of Richard Graves, D.D., and cousin of Robert James Graves, M.D. He was sent to school in England, in the parish of Westbury-on-Trym, then a village outside Bristol, under the Rev. Samuel Feild, later vicar of Hatherleigh.
Feild has been described as a prominent second generation evangelical Anglican; he was one of two curates in the parish, under Richard Carrow, the parish priest, as perpetual curate.
Graves was an undergraduate at Trinity College, Dublin, where he distinguished himself in both science and classics, and was a class-fellow and friend of William Rowan Hamilton, graduating B.A. in 1827. He then moved to Oxford, where he became an incorporated member of Oriel College, 11 November 1830.
Graves proceeded M.A. at Oxford in 1831, and at Dublin in 1832.
Having in 1830 entered the King's Inns, Dublin, Graves was called to the English bar in 1831, as a member of the Inner Temple. For a short time he went the Western circuit. In 1839 he was appointed professor of jurisprudence in London University College in a delayed succession to John Austin. Not long after, he was elected an examiner in laws in the University of London.
More information: Math-University of California
Graves was one of the committee of the Society for the Diffusion of Useful Knowledge.
In 1839, he was elected as a Fellow of the Royal Society, and he subsequently sat on 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.
In 1846, Graves married Amelia Tooke, a daughter of William Tooke, and died without issue on 29 March 1870 at Cheltenham.
In his twentieth year (1826) Graves engaged in researches on the exponential function and the complex logarithm; they were printed in the Philosophical Transactions for 1829 under the title An Attempt to Rectify the Inaccuracy of some Logarithmic Formulæ. Alexandre-Joseph-Hidulphe Vincent claimed to have arrived in 1825 at similar results, which, however, were not published by him till 1832. The conclusions announced by Graves were not at first accepted by George Peacock, who referred to them in his Report on Algebra, nor by Sir John Herschel. Graves communicated to the British Association in 1834 (Report for that year) on his discovery.
In the same report is a supporting paper by Hamilton, 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. It was an anticipation, as far as publication was concerned, of an extended memoir, which had been read by Hamilton before the Royal Irish Academy on 24 November 1833, On Conjugate Functions or Algebraic Couples, and subsequently published in the seventeenth volume of the Transactions of the Royal Irish Academy.
For many years Graves and Hamilton maintained a correspondence on the interpretation of imaginaries.
In 1843 Hamilton discovered the quaternions, and it was to Graves that he made on 17 October his first written communication of the discovery. In his preface to the Lectures on Quaternions and in a prefatory letter to a communication to the Philosophical Magazine for December 1844 are acknowledgments of his indebtedness to Graves for stimulus and suggestion. Immediately after the discovery of quaternions, before the end of 1843, Graves successfully extended to eight squares Euler's four-square identity, and went on to conceive a theory of octaves, now called octonions, analogous to Hamilton's theory of quaternions, introducing four imaginaries additional to Hamilton's i, j and k, and conforming to the law of the modulus.
Octonions are a contemporary if abstruse area of contemporary research of the Standard Model of particle physics.
Graves devised also a pure-triplet system founded on the roots of positive unity, simultaneously with his brother Charles Graves, the bishop of Limerick. He afterwards stimulated Hamilton to the study of polyhedra, and was told of the discovery of the icosian calculus.
Graves contributed also to the Philosophical Magazine for April 1836 a paper On the lately proposed Logarithms of Unity in reply to Professor De Morgan, and in the London and Edinburgh Philosophical Magazine for the same year a postscript entitled Explanation of a Remarkable Paradox in the Calculus of Functions, noticed by Mr. Babbage.
More information: nLab
To the same periodical he contributed in September 1838 A New and General Solution of Cubic Equations; in 1839 a paper On the Functional Symmetry exhibited in the Notation of certain Geometrical Porisms, when they are stated merely with reference to the arrangement of points; and in April 1845 a paper on the Connection between the General Theory of Normal Couples and the Theory of Complete Quadratic Functions of Two Variables.
A subsequent number contains a contribution On the Rev. J. G. MacVicar's Experiment on Vision, on the work of John Gibson Macvicar; and the Report of the Cheltenham meeting in 1856 of the British Association contains abstracts of papers communicated by him On the Polyhedron of Forces and On the Congruence nx ≡ n + 1 (mod. p.).
For many years he collected mathematical works. This portion of his library, more than ten thousand books and about five thousand pamphlets he bequeathed to University College, London.
In mathematics, the octonions are a normed division algebra over the real numbers, a kind of hypercomplex number system.
The octonions are usually represented by the capital letter O, using boldface O or blackboard bold O {\displaystyle \mathbb {O} } \mathbb {O}.
Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative. They are also power associative.
Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used.
Octonions are related to exceptional structures in mathematics, among them the exceptional Lie groups.
Octonions have applications in fields such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson construction to the octonions produces the sedenions.
The octonions were discovered in 1843 by John T. Graves, inspired by his friend William Rowan Hamilton's discovery of quaternions. Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 26 December 1843.
He first published his result slightly later than Arthur Cayley's article. The octonions were discovered independently by Cayley and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves' discovery.
More information: Alberto Elduque-Universitat de Zaragoza
You know, people think mathematics is complicated.
Mathematics is the simple bit.
It's the stuff we can understand.
John H. Conway
No comments:
Post a Comment