Solid geometry: pyramid in a cube and squaring a pyramid base
Two numbered propositions on the height and base of pyramids inscribed in solids
This leaf carries two numbered geometrical propositions on pyramids and their solids. The first asks whether a pyramid built on one face of a cube can reach an arbitrary height, and concludes it cannot: such a pyramid's height must be exactly three times the cube's, no more and no less. The second explains how to convert a pyramid with a many-sided base into a square pyramid of the same height, by resolving the base into triangles, squaring them, and combining the squares into one. Two pen diagrams accompany the text: a pyramid inscribed in a cube-like frame at upper right and an octagon crossed by its diagonals below.
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A pyramid on a cube's face must be three times the cube's height
The problem is to raise a pyramid whose base is one face of a cube and which extends to a given height. Leonardo answers that this is impossible: the pyramid's height must contain the height of its cube exactly three times, neither more nor less. He cites the third proposition of the third book as proof.
Reducing a many-sided pyramid to a square pyramid
To turn a pyramid of many sides into a square-based pyramid, resolve its base into as many triangles as it has sides, square those triangles, and join all the squares into a single square using the penultimate proposition of the geometric Elements. Upon that square build the pyramid at the same height as the first. Leonardo insists it can be of neither lesser nor greater height than the original.