Making a cube from a quadrilateral prism
The 'angle of proportionality' and the diagonal cut of a cube
Leonardo continues his solid-geometry study of turning a quadrilateral prism back into a cube. He divides a prism by its diagonal, transmutes half of it into a square with 'the fifth,' then sets that square into the 'angle of proportionality' where the diagonal cut of a cube meets its squaring. A closing problem asks how to make one cube out of two given cubes of different size by lengthening the smaller cube to the height of the larger and swinging a compass arc a n. Diagrams show a tall prism, a rectangle-and-square pair, the proportion angle with squares d, f, and stacked cubes.
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Cubing the quadrilateral prism
To make a cube of the prism b, Leonardo divides it by the diagonal and puts the half at a, then with the fifth transmutes it into the square m.
The angle of proportionality
The square m is placed in the angle of proportionality, on whose sides is first set the diagonal cut of a cube upon its squaring (a c), where m goes at site d. He then makes the diagonal cut b of the cube he seeks, a rectangle whose minor sides are the cube's edges and whose major sides give the cube's lateral diagonal.
One cube from two cubes of different size
To make a single cube from two given cubes of different size, Leonardo lengthens the smaller cube to the height of the larger along line a c, placing it at c o b d; he carries the larger cube's height a c down to a b, sets the compass foot at b and draws the curve a n where it cuts the prism.
