  We recall Charles Babbage's 1819 criterion for primality, based on
simultaneous congruences for binomial coefficients, and extend it to a
least-prime-factor test. We also prove a partial converse of his non-primality
test, based on a single congruence. Two problems are posed. Along the way we
encounter Bachet, Bernoulli, Bezout, Euler, Fermat, Kummer, Lagrange, Lucas,
Vandermonde, Waring, Wilson, Wolstenholme, and several contemporary
mathematicians.
