  We present a self-contained elementary and detailed exposition of Mertens'
own proof of his theorem on the divergence of the series of the reciprocals of
the primes and compare it with the modern proofs. His proof contains explicit
numerical error estimates and computations of the Mertens "constant," both of
which are missing from modern expositions. In as much as his strategy and
techniques are quite different from today's methods, his original methods
deserve to be better known.
