Dividing a wedge into three equal pyramids
A solid-geometry proof that a square-based pyramid holds two thirds of the wedge
Folio 54 verso carries a solid-geometry proof drawn beside a rectangular block (parallelepiped) with diagonals, its corners lettered b, f, d, e, a, c. Reasoning from the axiom that things equal to a third are equal to one another, Leonardo shows a wedge (conio) can be resolved into three pyramids of equal height and volume. He concludes that the square-based pyramid removed from the wedge contains two thirds of the whole. The demonstration is relevant to the codex's concern with the volumes and moulds of cast forms.
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Resolving a wedge into three equal pyramids
Base a b is shared by two pyramids a b c d e and a b c e of equal height, since the square base d b c e is halved by the diameter b c. Likewise base a b d is common to pyramids a b d f and a b d c of equal height, because side a f equals side d c. The wedge is thus split into three equal pyramids, and the square-based pyramid taken from it holds 2/3 of the whole.