# Henry John Stephen Smith

Henry John Stephen Smith (November 2 1826 - February 9 1883) was an Irish mathematician, remembered for his work in number theory (elementary divisors, quadratic forms) and matrices. The Smith normal form for matrices are named after him.

## Education

He was born in Dublin, Ireland, the fourth child of John Smith, a barrister, who died when he was two. Mrs. Smith moved to Oxford, and at the age of 15 Henry Smith went to Rugby school (1841). At 19 he won an entrance scholarship at Balliol College, Oxford. For health reasons he went to Paris, and took courses at the Sorbonne and the Collège de France.

In 1849, 23 years old, he finished his undergraduate career with a double-first; that is, in the honors examination for bachelor of arts he took first-class rank in the classics, and also first-class rank in the mathematics. He suffered from poor health, missing his final year at Rugby while convalescing in Italy. In late 1844 he tried for and obtained the Balliol scholarship. While at Oxford his health did not improve, he was struck down with malaria while in Frascati in 1845 and he did not return to Oxford until 1847. He graduated in 1849 with a double first in mathematics and classics. Soon after, he became one of the mathematical tutors of Balliol.

Smith remained at Balliol, becoming a fellow (1850), then working as a tutor before being appointed Savilian Professor of Geometry in 1861. He was elected to the Royal Society and to the Royal Astronomical Society in 1861.

On account of his ability as a man of affairs, Smith was in great demand for University and scientific work of the day. He was made Keeper of the University Museum; he accepted the office of Mathematical Examiner to the University of London; he was a member of a Royal Commission appointed to report on Scientific Education; a member of the Commission appointed to reform the University of Oxford; chairman of the committee of scientists who were given charge of the Meteorological Office, etc. It was not till 1873, when offered a Fellowship by Corpus Christi College, that he gave up his tutorial duties at Balliol.

The London Mathematical Society was founded in 1865. He was for two years president.

## Mathematical publications

His two earliest mathematical papers were on geometrical subjects, but the third concerned the theory of numbers. Following the example of Gauss, he wrote his first paper on the theory of numbers in Latin: "De compositione numerorum primorum formæ ${\displaystyle 4^{n}+1}$ ex duobus quadratis." In it he proves in an original manner the theorem of Fermat---"That every prime number of the form ${\displaystyle 4^{n}+1}$ (${\displaystyle n}$ being an integer number) is the sum of two square numbers." In his second paper he gives an introduction to the theory of numbers.

In 1858 he was selected by the British Association that body to prepare a report upon the Theory of Numbers. It was prepared in five parts, extending over the years 1859-1865. It is neither a history nor a treatise, but something intermediate. The author analyzes with remarkable clearness and order the works of mathematicians for the preceding century upon the theory of congruences, and upon that of binary quadratic forms. He returns to the original sources, indicates the principle and sketches the course of the demonstrations, and states the result, often adding something of his own.

During the preparation of the Report, and as a logical consequence of the researches connected therewith, Smith published several original contributions to the higher arithmetic. Some were in complete form and appeared in the Philosophical Transactions of the Royal Society of London; others were incomplete, giving only the results without the extended demonstrations, and appeared in the Proceedings of that Society. One of the latter, entitled "On the orders and genera of quadratic forms containing more than three indeterminates," enunciates certain general principles by means of which he solves a problem proposed by Eisenstein, namely, the decomposition of integer numbers into the sum of five squares; and further, the analogous problem for seven squares. It was also indicated that the four, six, and eight-square theorems of Jacobi, Eisenstein and Lionville were deducible from the principles set forth.

In 1868 he returned to the geometrical researches which had first occupied his attention. For a memoir on "Certain cubic and biquadratic problems" the Royal Academy of Sciences of Berlin awarded him the Steiner prize.

In February, 1882, he was surprised to see in the Comptes rendus that the subject proposed by the Paris Academy of Science for the Grand prix des sciences mathématiques was the theory of the decomposition of integer numbers into a sum of five squares; and that the attention of competitors was directed to the results announced without demonstration by Eisenstein, whereas nothing was said about his papers dealing with the same subject in the Proceedings of the Royal Society. He wrote to M. Hermite calling his attention to what he had published; in reply he was assured that the members of the commission did not know of the existence of his papers, and he was advised to complete his demonstrations and submit the memoir according to the rules of the competition. According to the rules each manuscript bears a motto, and the corresponding envelope containing the name of the successful author is opened. There were still three months before the closing of the concours (1 June, 1882) and Smith set to work, prepared the memoir and despatched it in time.

Two months after his death the Paris Academy made their award. Two of the three memoirs sent in were judged worthy of the prize. When the envelopes were opened, the authors were found to be Smith and Minkowski, a young mathematician of Koenigsberg, Prussia. No notice was taken of Smith's previous publication on the subject, and M. Hermite on being written to, said that he forgot to bring the matter to the notice of the commission.