Cubing a cylinder by lending and restoring
Cube roots and one-quarter (subquadruple) proportions to square the cylinder
Two subdivided box-like solids are drawn at the right: a quadrilateral cylinder divided into twenty parts (a b) and a cube divided into parts. The text shows how, 'by lending and restoring', a cube can be made from the quadrilateral cylinder a b, first ensuring the face is an aliquot part of the side. Leonardo argues that the resulting greater cube d f is four times the cylinder, so its subquadruple (one-quarter) cube root gives the sought cube, by a rule he admits is not yet geometric.
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From the quadrilateral cylinder a b to a cube
A cylinder divided into twenty parts (a b) and a cube divided into parts are drawn at the right. Leonardo proposes to make a cube from the quadrilateral cylinder a b by 'lending and restoring', first making the face of the cylinder an aliquot part of one of its sides so the construction can proceed.
Cube roots and the subquadruple rule
Taking c d as the side, double the face c, Leonardo reasons that the greater cube d f is 4 times the cylinder; so a cube one-quarter of d f cubes the cylinder c d. He takes the cube root subquadruple to the root of the greater cube d f, obtained by a given rule which he notes 'is not yet geometric', adding that the effect cannot be achieved any other way.