Henry John Stephen Smith (1826–1883) FRS was Savilian Professor of Geometry at Oxford University from 1861 to 1883 (Figure 1). He distinguished himself as a superb lecturer and researcher who brought international recognition to Oxford mathematics. His unique and caring personality ensured he was held in widespread affection and admiration by his students and the University community. The two volumes of Henry Smith’s collected mathematical papers, first published in 1894, are little read today. They include interesting biographical material, but it may not be immediately obvious why he was regarded, and not just at his alma mater, as being in the front rank of European mathematicians. To understand the reasons behind this it will be helpful to look more closely at Henry Smith’s life in mathematics. His mathematical writings, on which his reputation chiefly rests, were on the theory of numbers and elliptic functions, topics in which European mathematicians were pre-eminent. He prepared a series of Reports on the Theory of Numbers, commissioned by the British Association for the Advancement of Science, from 1859 to 1865. These were followed by his memoirs on the arithmetical theory of integral quadratic forms leading eventually to his crowning memoir to the French Académie des Sciences for its Grand Prix des Sciences Mathématiques of 1882. He advanced the theory of quadratic forms by returning to original sources and remaining true to arithmetic, confirming Henry Smith’s reputation as a brilliant arithmetician for whom fine arithmetical details and presentation mattered. The principle theme of these memoirs was the classification of quadratic forms. In addition to considering his early education and life in mathematics, this thesis will include a careful distillation of some of the mathematical techniques contained in these memoirs. It will reveal that Henry Smith’s mathematical techniques and presentation style was guided and influenced by the earlier writings of Gauss, Eisenstein, and Dirichlet. In any attempt to assess Henry Smith’s reputation in his day, and since, such considerations will be important.