[1] Neville's principal areas of expertise were geometrical, with differential geometry dominating much of his early work.
During his time in Cambridge, he had been greatly influenced by Bertrand Russell's work on the logical foundations of mathematics and in 1922 he published his Prolegomena to Analytical Geometry.
In 1914, as a visiting lecturer, he travelled to India, where, in response to a request from Hardy, he managed to persuade the Indian mathematician Ramanujan to accompany him back to England, thus playing a vital role in the initiation of one of the most celebrated mathematical collaborations of the last hundred years.
Neville had a keen interest in elliptic functions, having taught the subject to postgraduate students at Reading since the 1920s.
He believed that the subject's recent decline in popularity was due to its dependence on a mass of complicated formulae, a variety of differing and confusing notations, and an artificial definition relying on a familiarity with theta functions.
A period of recuperation from an illness in 1940 gave him the opportunity to put several years of lecture notes into publishable form.
He regularly attended meetings of the British Association for the Advancement of Science, being President of Section A (Mathematics and Physics) in 1950.