Subtracting one cube from another, and summing two cubes
Removing cube a from cube b c via a cylinder, then combining unequal cubes into one greater cube
Two propositions (numbered 8a and 9a) on cube arithmetic done geometrically. The first subtracts a given cube a from a cube b c so that the remainder stays cubic: b c is turned into a cylinder, the part equal to a is cut off, and the rest is returned to a cube. The second reduces two cubes of any sizes to a single cube by joining their sides at a right angle, drawing the hypotenuse, forming a cylinder, and building up the greater cube that contains both. The lower right shows a long extruded solid with a curved end.
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Removing a cube from a cube so the remainder stays cubic (a, n m, b c)
The given cube a is to be taken from cube b c, leaving a cube. Leonardo turns cube b c into the cylinder n m of the thickness of a, cuts off the part m equal to a, and returns the remainder n to the cube b. He is left with b and m, two cubes together equal to cube b c.
Combining two cubes of any size into one greater cube
The sides of the smaller and greater cubes are joined at a right angle and the hypotenuse drawn out; a cylinder made on the smaller side is fixed in length by the third proposition. The smaller cube is then added to the cylinder's length and built up, by the first of the second, into a greater cube that holds within itself both the middle and the smaller cube.
Extruded solid with a curved end as construction device
The lower right shows a long extruded solid whose end runs into a curved (semicircular) profile. It is the graphic aid for fixing the cylinder's length in the cube-summing construction.