At regular stages during a long and fruitful scientific career he wrote vast treatises, beginning with a well-known textbook on differential equations which led to a many volumed sequel on the general theory. There followed the Theory of Functions of a Complex Variable (1893), and works on the Calculus of Variations and Differential Geometry, besides numerous memoirs and shorter papers on every analytical branch of pure mathematics.
His influence on Cambridge mathematics was revolutionary and profound. As Sadleirian Professor he proved to be a most worthy follower of Cayley, his great and admired teacher, whose collected works he edited with clarity and devotion in a series of fine volumes, and whose algebraic creations in the theory of forms he greatly extended. But already in his Theory of Functions Forsyth had produced a book which, as Professor E T Whittaker tells us, "had perhaps a greater influence on English mathematics than any work since Newton's Principia." For this Forsyth may be regarded as the pioneer in the new British school of analysts, a school which is typified by another great work, Whittaker's Modern Analysis, and later by the Analytical Theory of Numbers of Hardy, Littlewood, Watson and Ramanujan. Forsyth was also influential in the new and freer turn which was taken in school mathematics during the twentieth century.
His writings are architectural and on the grand scale, typical of which is the memoir communicated to this Society in 1922 on "The Concomitants (including Differential Invariants) of Quadratic Differential Forms in Four Variables" (Proc. R.S.E., XLII, 147-212). This arose out of the new impetus given to the study of Riemannian geometry by the discovery of General Relativity. In it Forsyth made his final contribution to his vast programme of work upon the Invariant Theory, wherein he produced systems of forms which were algebraically complete for given ground forms, and thereby handsomely extended and fulfilled the fundamental work of Cayley in higher algebra. No one but a master of analytical technique, involving huge numbers of symbols and identities between them, could hope to do anything effective with these intricate systems of equations. In such work Forsyth was pre-eminent: he used his material strategically and with ease. Behind the baffling prolixity of symbols in his writings lies a great simplicity and the patient working out of a unifying principle. The 1922 memoir, like much of his other work on differential geometry, was based on Lie's theory of groups for infinitesimal transformations.
In 1925 Forsyth communicated a second important memoir to this Society - "A Chapter on the Calculus of Variations: Maxima and Minima, for Weak Variations, of Integrals involving Ordinary Derivatives of the Second Order" (Proc. R.S.E., XLVI, 149-193). This led to the large volume, mostly of original research, which he published on the same subject in 1927. His thoroughness and care for details, the vitality and modesty with which he carried his vast learning, are well remembered among his pupils at Cambridge and London. As a young graduate I owe much to his advice and encouragement, and suggestions for lines of research, and particularly for his insistence that the investigator, young or old, must be prepared to come upon a brick wall, in any problem worth investigating, and that it would fall, but only fall, before determined and systematic blows. I met him last at the Zürich Mathematical Congress, 1932, soon after he had published his Geometry of Four Dimensions - a logical sequel of the great course on Gaussian differential geometry which he used, to deliver at Cambridge. He spoke of the pleasure it gave him to find familiar curves, like the conic and lemniscate of elementary geometry, occupying new and more fundamental roles in the local structure of general (two dimensional) surfaces of fourfold space. He hoped that future geometers, with an analytical leaning, would not leave this fruitful field unexplored.
He was elected an Honorary Fellow in 1900.