Induction by enumeration has a clear interpretation within the numerical paradigm of inductive discovery (i.e., the one pioneered by [Gold, 1967]). The concept is less easily interpreted within the first-order paradigm discussed by [Kelly, 1996, Martin & Osherson, 1998], in which the scientist's data amount to the basic diagram of a structure. We formulate two kinds of enumerative induction that are appropriate to the first-order paradigm, and analyze their potential for discovery. Among other results, it is shown that one form of enumerative induction achieves maximum inductive competence. * We offer warm thanks to two generous referees for their constructive remarks on an earlier draft, particularly, for spotting an error in the original proofs of Propositions (28) and (72).