Squaring a pyramid's base and proportional division of solids
Continuation: recovering a cube's volume in a square base, and cutting solids in equal proportion
A single continuous demonstration, carried over from the previous page, finishes a construction on the equivalence of a pyramid and a cube. Leonardo shows that after removing part of a solid, the remainder i K g h q keeps the true quantity of material of the first cube; because the pyramid's base is not square he squares it and raises it upright as u y z, the excess over the base shown at x y z o. He then states a general 'right-angle' rule: if one line or quantity grows or diminishes by a certain amount, another diminishes in the same proportion, illustrated by cylinder L m f g and cube b c f g, where line c g cut in the same ratio as L b f halves the pyramid's base c d g h q. The whole sheet is filled with text and carries no diagram.
On this page
Recovering the cube's volume in a squared base (u y z, x y z o)
After the cut, the remaining figure i K g h q keeps the true quantity of material of the first cube. Because the pyramid's base does not stay square, Leonardo squares it in the same quantity and raises it upright at the given length as u y z, with the excess of the cube's base over the pyramid's base shown at x y z o.
The right-angle rule of proportional growth and diminution (L m f g, b c f g)
The same right-angle figure serves to ask: if a line or quantity grows or diminishes by a certain amount, by how much does another diminish? In the cylindrical figure L m f g, from which the first cube b c f g (half the cylinder) was drawn, the line c g cut in the same proportion as line L b f halves the pyramid's base c d g h q, taking as much from the pyramid as from the cylinder.