Transforming Cubes, Slabs, Pyramids and Cylinders
Stereometric problems: converting a cube to a pyramid, and a square slab through cylinder and cube to a pyramid of given height.
These leaves pose problems in the mensuration and transformation of solids. Leonardo asks how to turn a cube into a square-based pyramid of a given height, and by how much the pyramid's base must be narrower than the cube's. He sets out a chain of conversions, reducing a square slab first to its 'natural' cylinder, then to a cube, then extending it into a pyramid, citing earlier propositions of his 'first' and 'second' books. The facing leaf is crowded with drawings of pyramids, cubes, cylinders and slabs illustrating these transformations.
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Turning a cube into a square-based pyramid of given height
From a cube make a square-based pyramid rising to a given height that exceeds the three sides of the cube joined in one line. The question is set of the geometric transformation between the two solids.
How much narrower the pyramid's base than the cube's
Having raised the pyramid to the given height, it is asked by how much the base of the pyramid becomes narrower than the base of the cube, a quantitative measure of the reduction accompanying the increase in height.
A pyramid from a square slab to a given height
From a square slab make a pyramid extending to a given height, asking how much the slab is narrowed. The method makes the slab's cube first, then extends it to the pyramid's height, using the penultimate proposition to reach its true quantity.
Reducing a square slab to cylinder, then to cube
To make a cube from a square slab, Leonardo reduces the slab to its natural cylinder by the fifth of the first book, then by the first of the second turns the cylinder into its cube, so the whole quantity of the slab is contained in that cube, and reverses the chain to form pyramids.