Reducing Any Solid to a Cube
Transforming a parallelepiped via Euclid, and making one cube from two cubes
On this text-and-diagram page Leonardo shows how to reduce any parallelepiped (a 'parallel body') to cubic form. Taking a body of height 2, width 6 and length 8, he takes its face a e b d and, citing the last proposition of Euclid's second book, produces a solid with a square front labelled f g e a, then the cube q r g o. A further problem combines two given cubes of unknown size into a single cube by turning one into a cylinder (extended from b c to b d) and striking its diameters. The margins carry several three-dimensional diagrams of boxes, arcs and cylinders with letter labels.
On this page
A body of height 2, width 6, length 8 to be made square-fronted
Leonardo proposes to take a body of height 2, width 6 and length 8 and remake it as a body of square front and length 8. This sets up the reduction of a rectangular solid to one of equal length but square cross-section.
Square front f g e a by Euclid's Book II
He takes the face a e b d of the proposed body and, following the last proposition of the second book of Euclid, forms the body with square front f g e a.
From the square-fronted figure to the cube q r g o
By the preceding operation the square-fronted figure is turned into the cube q r g o. Leonardo then sets a further problem: from two given cubes of unknown quantity, to make a single cube.
Making one cube from two by way of a cylinder
Reduce the cube a into a cylinder b c of the thickness of the smaller cube and join it to the larger. Then add the smaller cube at c d, extending the cylinder from b c to b d, and finally strike the diameters of the cylinder and make a cube of the whole.