Quirk and Seymour have shown that a connected simple graph has at least as many spanning trees as circuits. This paper extends and strengthens their result by showing that in a simple binary matroid M the quotient of the number of bases by the number of circuits is at least 2. Moreover, if M has no coloops and rank r, this quotient exceeds 6(r + 1)/19. © 1983, Academic Press Inc. (London) Limited. All rights reserved