Making a Cube from a Tall Pyramid, and Its Converse
Problems 9 and 10: pyramid to cube, then cube to a taller pyramid
Two paired problems of solid geometry. First, from a pyramid taller than three sides of its square base, make a cube: build the solid on the pyramid's base (rising a third of its height), then convert that cylinder into a cube. The second is the converse and, Leonardo notes, the harder investigation: from a given cube, make a pyramid whose height exceeds the three sides of that cube. He begins the pyramid at the set height, draws its sides down to the cube's base, raises the solid to a third of the pyramid's height, and clothes the pyramid in its whole cylinder to compare it with the cube. A pyramid inscribed in a wire-frame cube is drawn at upper right.
On this page
From a tall pyramid to a cube
A pyramid taller than three sides of its square base is turned into a cube. The solid on its base is built first, taller than the target cube because it always takes a third of the pyramid's height; that solid, or cylinder, is then made into the cube by the first of the second of this book.
The converse: cube to a taller pyramid
The harder converse takes a given cube and makes a pyramid exceeding its three sides in height. Begun at the set height, the pyramid's sides are drawn down to the cube's base; the solid raised to a third of the pyramid's height is a cylinder larger than the cube, so the pyramid is clothed in its whole cylinder and the same is done with a like pyramid to compare them.