Lowering a pyramid while widening its base at equal volume
Transforming a solid via its cylinder, cube and square while keeping the same quantity of material
This page continues Leonardo's studies on transforming one solid into another of equal volume. It treats a pyramid lowered to a given height, whose base must grow to compensate: the pyramid is 'clothed' in its equivalent cylinder, the cylinder is shortened, and it is then divided into three equal parts to recover the largest pyramid it can hold. Small diagrams show a pyramid inscribed in a cylinder, a cube, and successive squarings labelled with letters. A marginal note and a further worked figure (a, b, c, d, e) extend the method.
On this page
Lowering the pyramid o m and enlarging its base
If a pyramid is lowered to a given height, its base must grow to keep the same volume. Leonardo clothes the pyramid m o in its whole equivalent cylinder, shortens that cylinder to the given length o n, then divides the thickened cylinder into three equal parts to draw out the largest pyramid that fits inside it.
Reducing the cylinder to a square and drawing it out (b, c, d, e)
Let a be the cylinder that clothes the pyramid. He reduces it to its square so that b becomes c; should the square c fall short of the required height on side p, he turns side p toward side d, squares it again to obtain e, and finally draws e out in length.