Cubes and slabs turned into pyramids and cylinders
Volume-transformation problems among cube, slab, cylinder and pyramid
A dense sheet of solid-geometry transformations: from a cube to make a square-based pyramid of given height, from a square slab a pyramid or a cube, and from a cube back to a slab of given width. Each is solved by reducing the figure to its natural cylinder and cube and invoking numbered propositions, such as the fifth of the first and the second and eighth of the second. Both leaves swarm with perspective sketches of tall pyramids rising from cubes, cones, half-cylinders and vaulted blocks.
On this page
From a cube to a square-based pyramid
The problem makes a square-based pyramid from a cube at a height exceeding the three sides of the cube laid end to end in one line. It asks how much narrower the pyramid's base becomes than the cube's base.
From a square slab to a pyramid
A square slab is to be made into a pyramid extending to a given height, asking how much the base shrinks. The method first turns the slab into its cube, then extends it to the pyramid's height, arriving at the true quantity with the help of the penultimate proposition.
Reducing a slab to its cylinder and cube
The slab is reduced to its natural cylinder by the fifth of the first, then to a cube by the first of the second, capturing its whole quantity. Reversing the chain with the second and eighth of the second returns a cube to a slab of given width, and further steps narrow a pyramid's base.
Perspective studies of pyramids on cubes
Tall spike-like pyramids rise from cubes, cones and half-cylindrical vaults across both leaves. The repeated sketches visualise the volume-preserving transformations described in the text.