Seventeen problems on reshaping cubes, cylinders and slabs
Volume-preserving conversions asking how much a solid thickens or narrows when reshaped
A numbered list of seventeen solid-geometry problems on transforming one body into another of equal quantity: a faceted cylinder into a cube, a cube into a cylinder of given length or thickness, and square or rectangular slabs stretched or shortened to a given dimension. Each problem asks for the unknown that results from preserving volume, for instance the height of the new cube, the diameter of the new cylinder, or how much a slab thickens or narrows. The page is written entirely in Leonardo's mirror script with the problems numbered 1 to 17 down the margin.
On this page
Converting a faceted cylinder into a cube and back
The opening problems ask to turn a given faceted cylinder into a cube and find that cube's height, then to reverse the operation and make a cylinder from a cube to a given length or a given thickness. Each transformation keeps the quantity of the solid fixed while a single dimension is sought.
Combining and subtracting cubes
Later problems ask to make a single cube from two unequal cubes and find its height, to cut a cube of a given size out of another so that the remainder is still a cube, and to compound two equal cubes into one. These are classic problems of equal-volume solid construction.
How much a stretched or shortened slab narrows or thickens
The slab problems extend or shorten a square or rectangular tavola to a given length while holding width or thickness fixed, and ask by how much the remaining dimension diminishes or grows. The answers depend on keeping the total quantity of material constant, so the change in one measure is set in proportion to the change imposed on another.
