This paper presents a major unpublished memoir written by Charles Hermite on elliptic functions, and dating from 1849: the text is appended. Looking at this contribution in the overall context of that mathematician's work, its import for the development of the elementary theory of elliptic functions is examined. Along with Liouville, in the 1840's Hermite laid the foundations of a general theory of doubly periodic meromorphic functions, bringing to bear in this context the general approach propounded by Cauchy for the theory of complex variable functions. In his lectures from 1868 on at the École polytechnique, and equally at the Paris science faculty, he approached this theory along the lines set by Jacobi, with theta functions as a starting point. These lectures probably provide the best presentation of that theory, prior to the widespread acceptance of the Weierstrass approach.