Mathematics at Balliol College, Oxford

Victorian Balliol

William Spottiswoode (1825-1883)

After graduating William Spottiswoode took over the family printing house, but continued to take an active interest in mathematics and physics. His main contributions were to geometry, optics, and electricity (particularly electrical discharge in rarified gases), but an 1861 paper On typical mountain ranges: an application of the calculus of probabilities to physical geography, which attempted to use statistical methods to distinguish between competing geological theories, was apparently the inspiration for Francis Galton's work on statistics. Spottiswoode was also renowned for his interest in linguistics and oriental languages. He became President of the Royal Society in 1878 and still held that office at the time of his death in 1883.

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Henry John Stephen Smith (1826-1883)

Henry Smith was one of the most influential British pure mathematicians of the second half of the nineteenth century. He won the major mathematical prizes of both the German and French Academies of Sciences. He was elected Savilian Professor of Geometry in 1861, and became Keeper of the University Museum in 1874.

Smith graduated with First Class Honours in both Classics and Mathematics after less than two years at the college, and won the major University Prizes in both subjects. He was also President of the Oxford Union. In 1849 the College appointed him its Mathematics Lecturer but also asked him to set up and run the first college chemistry teaching laboratory in Oxford, for which task he equipped himself by studying in Oxford with Neville Story-Maskelyne and at the Royal College of Chemistry with August Hofmann.

His main mathematical work was in number theory, geometry and the theory of elliptic functions. From 1859 onwards he presented a series of Reports on Number Theory to the British Association for the Advancement of Science. In 1861 he proved the existence and uniqueness of what is now called Smith normal form of a matrix with integer entries. This has subsequently been used to prove the cyclic decomposition theorem for modules, but Smith's first application of this result was to determine when linear Diophantine equations admit solutions, settling a longstanding problem first studied by Greek mathematicians. Shortly afterwards he extended the idea to solve another outstanding problem of determining in how many ways a given positive integer could be expressed as the sum of a given number of squares. (The special cases of two, four and six squares had been treated by Jacobi, and three squares by Gauss.) Due to an oversight the French Academy set this as its Prize problem in 1882, unaware that Smith had published the result fifteen years earlier. Smith died before the competition could be judged and was awarded the prize posthumously with the young Prussian mathematician Hermann Minkowski.

Although it was not his main area of interest, it was Smith who first described what is now known as the Cantor set, in 1875, some eight years before Cantor. This was probably the first recursive definition of a fractal set to be published.

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Francis Ysidro Edgeworth (1845-1926)

With Galton and Pearson, Frank Edgeworth is generally regarded as one of the founders of mathematical statistics. Edgeworth studied classics at Balliol, and his interest in statistics was aroused by a course in political economics given by his tutor, Benjamin Jowett. After leaving the College he first studied law, but in his spare time read mathematics and in 1877 published New and Old Methods of Ethics, which clearly showed the breadth of his reading. This was followed four years later by Mathematical Psychics: An Essay on the Application of Mathematics to the Moral Sciences. Edgeworth followed these early works by a statistical analysis of errors. Edgeworth's Theorem giving the correlation coefficients of the multi-dimensional normal distribution, was merely one of many results discovered in the course of a systematic study of the area. In 1891 Edgeworth became Professor of Political Economy.

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Charles Howard Hinton (1853-1907)

Charles Howard Hinton devised methods of visualising the geometry of higher dimensions, which he explained in various books, and also exploited in various science fiction stories, one of which is said to have inspired Edwin Abbot's Flatland. He coined the name tesseract for the four-dimensional analogue of a cube. Added to these modest mathematical contributions were some noteworthy eccentricities. Hinton married Mary Everest Boole, the eldest daughter of George Boole, the founder of mathematical logic. He also married Maud Wheldon, and was tried in the Old Bailey for bigamy. After serving a one day prison sentence for the offence, he fled with his (first) family to Japan, where he taught for some years, before taking up a post at Princeton University. Whilst there, in 1897, he designed a gunpowder powered baseball pitcher, to assist the team in its batting practice.

Leonard James Rogers (1862-1933)

Leonard Rogers is now chiefly remembered for a remarkable set of identities which are special cases of results which he had published in 1894, but which had gone unnoticed until Ramanujan rediscovered them in 1913. Unlike Rogers, Ramanujan could not prove his formulae and sought help from many of the leading mathematicians of the day, but none could supply a proof. Eventually, in 1917, Ramanujan found Rogers' paper and contacted him, whereupon Rogers supplied another simpler proof, which was soon followed by two more. Rogers was described by G.H. Hardy as " a mathematician of great talent" and "a fine analyst... but no-one paid much attention to anything he did". That neglect can be gauged by the fact in 1936 the future Fields Medallist, Atle Selberg, published a "generalisation" of the Rogers-Ramanujan identities which turned out, in fact, to be another special case of Rogers' original result.

Julian Lowell Coolidge (1873-1954)

Julian Lowell Coolidge studied at Balliol from 1895-7. After further studies in Europe (with Study) he returned to his native city of Boston, where he became a Professor at Harvard. Coolidge wrote numerous monographs and texts on geometry, probability and the history of mathematics. His Mathematics of the great amateurs is perhaps the best-known.

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