After studying at the Universities of Chicago and Harvard, Birkhoff returned to Chicago as a graduate student. His first research was on the asymptotic properties of certain ordinary differential equations, for which he was awarded the degree of Ph.D. in 1907. In the same year he was appointed Instructor at the University of Wisconsin. It was here, under the influence of Professor E B van Vleck, that he started to work at the theory of linear difference equations. In 1909 Birkhoff was elected to an Assistant Professorship at Princeton, where he remained until 1919 when he became Professor of Mathematics at Harvard.
The paper which first brought Birkhoff into prominence was concerned with the theory of systems of linear difference equations (Trans. American Math. Soc., XII (1911)). Before this time the only rigorous solutions of such systems had been obtained either by using asymptotic series or by means of the Laplace Transform. Birkhoff used a new direct matrix method. He showed that, in the plane of the complex variable z, a system of linear difference equations of the first order with rational coefficients possesses two complete sets of solutions with remarkable properties; these he termed principal solutions. The principal solutions have singularities which are simply related to the poles of the coefficients and have a completely specified asymptotic form. Moreover, he proved that such a system has a purely Riemannian theory in which certain rational functions of exp (2πiz) play a part like that of the monodromic group constants in the theory of ordinary linear differential equations. By this justly celebrated memoir, Birkhoff made a striking advance in the subject, to which he continued to make further valuable contributions for many years. At the same time he was continuing his researches in the subject of his doctorate thesis, and published many important papers concerned mainly with the problem of constructing a system of linear differential equations with prescribed singularities and monodromic group.
The period from 1912 to 1927 Birkhoff devoted mainly to a study of the theory of dynamical systems. His work in this field was of great value, and for it he was elected the first holder of the Bôcher Prize of the American Mathematical Society in 1923. A connected account of his contributions to dynamical theory will be found in his book Dynamical Systems, which appeared in 1927 as one of the American Mathematical Society's Colloquium Publications.
One of Birkhoff's earliest dynamical papers dealt with Poincaré's Last Geometrical Theorem (Trans. American Math. Soc., XIV (1913)). Poincaré had shown, in 1912, that the existence of an infinite number of periodic orbits in the restricted problem of three bodies would follow at once from the following simple geometrical theorem: "Given a ring O < a ≤ r ≤ b in the plane with polar co-ordinates (r, 0) and a one-to-one continuous area preserving transformation T of the ring into itself which advances points on r = a and regresses points on r = b, then there will be at least two points of the ring which are invariant under T." Birkhoff was the first to give a proof of this theorem, which he subsequently extended and applied to various dynamical problems.
Birkhoff's greatest paper on dynamics (Trans. American Math. Soc., XVIII (1917)) was concerned with dynamical systems with two degrees of freedom. He showed that all problems relating to such systems could be reduced to the problem of determining the orbits of a particle moving on a smooth surface which rotates about a fixed axis with constant angular velocity and carries with it a conservative field of force. This simplification of the problem enabled him to show how the existence of periodic orbits may be proved and their form determined. This memoir was followed by others dealing with the theory of periodic orbits and the problem of three bodies. Many fellows of the Society will remember the inspiring course of lectures on his own work in dynamics which Professor Birkhoff gave at the St Andrews Mathematical Colloquium in 1926.
After 1927 Birkhoff published numerous papers on both pure and applied mathematics, notably on the Theory of Relativity and on the Ergodic Theorem, to which he made substantial contributions. His main interest at this time was, however, in the development of a mathematical theory of aesthetics. The fundamental problem with which he was concerned was to define, within a given class of aesthetic objects, the order 0 and the complexity C so that the ratio M = O/C is the aesthetic measure of any object in the class. The results he obtained were described in his book Aesthetic Measure, published in 1933. This is not the place to discuss whether Birkhoff had not embarked on a vain task in attempting to found an analytical theory of aesthetics. Many would admit that he was on fairly safe ground when discussing the aesthetic measure of the more elementary forms of decorative art, and yet would claim that in the higher arts, such as music, poetry, sculpture, and painting, there are elements which must defy analytical treatment. Whatever one's views, one cannot but admire the patience, the thoroughness, and the perseverance with which Birkhoff mastered many highly technical matters in this interesting study.
Professor Birkhoff was a prominent member of the International Congresses of Mathematicians held between the two wars, and formed a valuable link between American men of science and their colleagues in Europe. He received many honours, was president of the American Mathematical Society during 1924-26, president of the American Association for the Advancement of Science during 1936-37, and was an honorary member of the Edinburgh and London Mathematical Societies and of many European academies (including the Institut de France, the Lincei, and the Pontifical Academy). He was a frequent visitor to this country; on his last visit, in 1938, he was one of the American delegates at the British Association Meeting in Cambridge and at the Gregory Tercentenary Celebrations, when he received the honorary degree of LL.D. of the University of St Andrews.
He was elected an Honorary Fellow of the Society in 1943.