Richardson combined in an eminent degree the qualities of a man of action with those of a scholar: his character was strong, vigorous and clear cut. After a brilliant career as a student, and later as a colleague of Professor L N G Filon at the Imperial College, London, he went on active service with the British Expeditionary Force, 1914-19, receiving the D.S.O. for his part in the Battle of Bullecourt, 1917. A college friend who fought by his side wrote of him: "By common consent he was the most distinguished combatant officer of the London O.T.C. before World War I : in that war he became a legendary figure. His bravery was outstanding." In 1918 he was seriously wounded; and as a result of an incredibly grave operation carried out by Sauerbruck in Munich for the removal of a bullet from the lung, he recovered a measure of health, but the strain of the operation left a legacy of leukaemia, a progressive disease which made him an invalid for life. Despite his handicaps he turned his immense vitality to the channels of teaching and research. The main cause of his success in keeping the leukaemia at bay so long lies in the devoted nursing by his wife. In 1922 he married Dr Margaret Harris, a member of the Modern Languages Department of Swansea and an expert on German linguistics.
Richardson's earliest publications were in analysis and hydrodynamics, but his bent lay in algebra. At Swansea he created a flourishing school, and two of his students, R Wilson and D E Littlewood now occupy chairs in the University of Wales. Richardson made substantial and effective contributions, alone and also in collaboration with his pupils, by generalising the results of classical algebra and the theory of numbers to non-commutative and non-associative systems. Later, when the Swansea school of algebra was successfully launched, he turned to the abstract theory itself. Typical of his thought is the paper, Some Combinatorial Problems of Finite Abstract Algebra, which he published in our Proceedings (Proc. Edinburgh Math. Soc. (2) 6 (1939-41), 46-50). He was interested not only in the logical structure of an abstract system but also in the strength of a postulate or the extent of a theorem. The paper was a preliminary survey, as he put it, of the extent to which certain theorems are likely to be true in an arbitrary multiplicative system. Later he turned his attention to the arithmetic of forms and particularly to the composition of quadratic and cubic forms, to which he made remarkable contributions.
Many of Richardson's early works were published in the Messenger of Mathematics and the Philosophical Magazine; the later and generally larger works were published in the Proceedings of the London Mathematical Society, the Philosophical Transactions, the Quarterly Journal of Mathematics, the Annals of Mathematics and the Duke Mathematical Journal. He holds an honoured place among algebraists in the succession of Macmahon, Young and Wedderburn.