Pyramid, cylinder and cube each a third of one another
The largest pyramid in a cylinder equals a third of it; converting a cylinder to a cube
This page continues Leonardo's studies of the volumes of solids. He states that the largest pyramid contained in a cylinder is one third of that cylinder, and proves the analogous relation inside a cube whose six faces are equal, showing that pyramid q p r o a is double pyramid n m o p a because the first spans the whole height of the cube and the second only half. Further notes treat a cylinder inscribed in a cube (a b c d e f g h) and set out a procedure for turning a cylinder into a cube by chaining earlier propositions. Diagrams show pyramids within cube frames and small stepped solids at lower right.
On this page
The largest pyramid in a cylinder is a third of it
Leonardo states that the greatest pyramid that fits inside a cylinder is one third of that cylinder. He proves it in a cube whose six faces are all equal: pyramid q p r o a is double pyramid n m o p a, because the first extends through the whole height of the cube while the second reaches only half its height.
Cylinder inscribed in a cube equals the pyramid
The cylinder labelled g c a e - b d f h is declared equal to the pyramid: since the pyramid is one third of its cylinder, the named part is likewise one third of the quantity of that cylinder. The block ends with a new problem, to make a cube from a pyramid of given height.
Turning a cylinder into a cube
Marking 'a is the required cube', Leonardo chains prior results: from the third part of the cylinder, equal to the pyramid of the fourth proposition, he forms its cylinder by the fifth of the first book, and from that cylinder he makes the cube by the first of the second book.