STOKES, George Gabriel. On the Critical Values of the Sums of Periodic Series
Cambridge: John W. Parker, 1847. First edition.

An offprint of the paper in which he inaugurated the fundamental notion of ‘uniform convergence’. “An understanding that the convergence of a series of functions needs to be carefully handled developed only in the late 1840’s, simultaneously but independently in the work of Phillip Seidel, a former pupil of Dirichlet, and George Stokes in Cambridge. Both realized that for an infinite sum of continuous functions to be continuous, convergence must be of a particular kind, which Seidel (in 1848) described as not ‘arbitrarily slow’ and Stokes (in late 1847) as not ‘infinitely slow’. Such ideas foreshadowed the later idea of uniform convergence, where a series of functions not only converges at every point, but does so at a uniform rate over a range of values. Stokes’ paper of 1847 was entitled ‘On the critical values of the sums of periodic series’. His discussion of continuity in §1 shows that even by 1847 the Eulerian definition of a continuous function as one that can be expressed by a single formula had not entirely fallen out of use, but Stokes himself adopted the Bolzano-Cauchy definition: ‘f(x) is called continuous when, for all values of x, the difference between f(x) and f(x&plusminus;h) can be made smaller than any assignable quantity by sufficiently diminishing h’. In the same opening section, Stokes separated convergent series into two classes, either ‘essentially’ or ‘accidentally’ convergent; in modern terms absolutely or conditionally convergent. His definition of ‘infinitely slow’ convergence came only much later in the paper, in §38. … The modern definition of uniform convergence was formulated a few years after the appearance of Stokes’ paper, by Weierstrass in Berlin.” (Stedall: Mathematics Emerging, A Source Book 1540-1900, pp. 527-528).

Large 4to: 286 x 226 mm. Offprint from the Transactions of the Cambridge Philosophical Society, Vol. VIII, Part V, 1847. Dated 1848. Contemporary plain wrappers; upper right corner of title soiled, otherwise fine and unopened. 53 pp.