  Gottfried Leibniz embarked on a research program to prove all the Aristotelic
categorical syllogisms by diagrammatic and algebraic methods. He succeeded in
proving them by means of Euler diagrams, but didn't produce a manuscript with
their algebraic proofs. We demonstrate how key excerpts scattered across
various Leibniz's drafts on logic contained sufficient ingredients to prove
them by an algebraic method -- which we call the Leibniz-Cayley (LC) system --
without having to make use of the more expressive and complex machinery of
first-order quantificational logic. In addition, we prove the classic
categorical syllogisms again by a relational method -- which we call the
McColl-Ladd (ML) system -- employing categorical relations studied by Hugh
McColl and Christine Ladd. Finally, we show the connection of ML and LC with
Boolean algebra, proving that ML is a consequence of LC, and that LC is a
consequence of the Boolean lattice axioms, thus establishing Leibniz's
historical priority over George Boole in characterizing and applying (a
sufficient fragment of) Boolean algebra to effectively tackle categorical
syllogistic.
