  This is a collection of definitions, notations and proofs for the Bernoulli
numbers $B_n$ appearing in formulas for the sum of integer powers, some of
which can be found scattered in the large related historical literature in
French, English and German. We provide elementary proofs for the original
convention with ${\mathcal B}_1=1/2$ and also for the current convention with
$B_1=-1/2$, using only the binomial theorem and the concise Blissard symbolic
(umbral) notation.
