Mathematicians in History: Who was Henry John Stephen Smith?

February 11, 2010
Henry John Stephen Smith (1826-1883) is now ranked as one of the great British mathematicians of the 19th century. However, during the 19th century, his name was not well known and he was not compared with contemporary British mathematical giants George Stokes, Lord Kelvin, James Sylvester, George Boole, William Rowan Hamilton, George Green, or Arthur Cayley. A natural question to ask is.....why not? When Smith's mathematical achievements are now surveyed and taken into account, it is extraordinary to consider that Smith was, as Oxford mathematician Keith Hannabus recently labeled him, "the mathematician the world forgot".

Henry Smith was born in Dublin, Ireland on November 2, 1826 and the family soon thereafter moved to England. Smith entered Oxford University in 1844 after winning the
top scholarship to Balliol College. In 1848, he gained first class honors in both classics
and mathematics at Oxford and was named a Fellow and then a Tutor at Balliol.
In 1860, Smith was appointed Savilian Chair of Geometry at Oxford; it is noteworthy that some of his eventual successors to this position were G. H. Hardy, E. C. Titchmarsh and Sir Michael Atiyah. Moreover, Sir Edmond Halley, of Halley's Comet fame, was a predecessor to Smith in this Chair. In 1861, Smith was elected as a Fellow of the Royal Society of London (along with the physicist James Clerk Maxwell). He also served as President of the London Mathematical Society from 1874-1876.

Smith's mathematical work was extensive and bridged many diverse areas of mathematics. His earlier work, in number theory, is remarkable and shows his panoramic view of the subject; in fact, his "Report on the Theory of Numbers" is described as the 'most complete and elegant monument ever erected to the theory of numbers.' Smith's own work in number theory and algebra was highly regarded. For example, in 1861, Smith proved the existence and uniqueness of what is now called the Smith normal form of a matrix with integer entries. Smith's first application of this result was to determine when linear Diophantine equations have solutions, settling a longstanding problem first studied by the early Greeks.

In 1855, Smith published an ingenious new existence proof of Pierre de Fermat's famous two-squares problem; that is, he showed that any prime number p that is congruent to 1 modulo 4 can be written as the sum of two integer squares; remarkably, Smith wrote this two-page paper in Latin as an homage to his mathematical hero, Carl Friedrich Gauss. It should be noted that the first existence proof of this result was given by the great Swiss mathematician Leonhard Euler (who spent seven years working on the problem) in 1749. Interestingly, Smith's proof is quite easy to understand and seems to refute a claim made by Hardy in his "A Mathematician's Apology" that there is "no proof within the comprehension of anybody but a fairly expert mathematician."

Smith's contributions to mathematical analysis are also outstanding...but, surprisingly, not nearly as well known. Only in recent years has Smith's work in this subject become more fully appreciated. In an 1875 paper "On the integration of discontinuous functions," Smith essentially constructs the middle-thirds Cantor set (some eight years before Cantor did!) along with the Sierpinski gasket and the Koch snowflake. In this paper, Smith also seems to have been the first mathematician to perceive the connection between measure and integral. In fact, he corrected a mistake in Georg Riemann's work on the relatively new (at the time) theory of Riemann integration; in fact, Smith showed that the scope of the Riemann integral is not quite as extensive as some had claimed. It was unfortunate for Smith - in fact for all of mathematics - that his 1875 paper received far less attention than it rightfully deserved. The American mathematical historian, Thomas Hawkins, recently remarked that "Probably the development of a measure-theoretic viewpoint within integration theory would have been accelerated had the contents of Smith's paper been known to mathematicians.....".

Henry Smith died on February 9, 1883. His funeral procession in Oxford was a quarter of a mile long. This display of final respect was due mostly to Smith's immense personal charm and popularity with his contemporaries. Even Smith's closest friends did not know of his mathematical achievements at the time of his death. Indeed, it now seems that a combination of his reluctance to self promote together with a serious miscalculation by mathematicians of his era led to his anonymity. Fortunately, mathematicians today are more aware of his manifold contributions to mathematics!
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