Pyramids in cylinders and cubes, with dies and a cutting template
Solid-geometry proofs on the greatest pyramid, beside notes on a copper-and-steel die and a filed template
The leaf is filled with solid-geometry demonstrations: the greatest pyramid that can be set in a cylinder is shown to equal one third of it, and related propositions treat squares and their subduple diameters, cubes, and the transformation of a solid into a pyramid. The figures include triangles inscribed in rectangles and a square subdivided into thirty-six little squares. Mixed in at the top are craft notes: a die whose female is of copper and whose male is of tempered steel, and a method for making a cutting template from thin papers glued with fish glue onto a steel plate, then cut and corrected with a file. Sketches of screwed plates and upright mechanisms accompany the text.
On this page
The greatest pyramid in a cylinder equals one third of it
First: the greatest pyramid that can be made within the cylinder equals a third of that cylinder's quantity. It follows that a third of the cylinder's height equals its greatest pyramid, and that the greatest pyramid on the base of a cube is three times taller than the cube. Leonardo works this through the labelled figures a b c, d e b e and f g b c.
The square and its subduple diameter
The side of the square is stated to be the diameter of a square subduple to it, and the semidiameter of the square to be the side of a square likewise subduple to the first. The proposition governs the proportions between the squared figures drawn nearby.
Transforming a solid into a pyramid
Using the letters b, c n, d a and h f e g, Leonardo asks what a like solid a d e f will do when set on the same base and continually widened, whereas the equiangular solid a n c d narrows into the pyramid a b d. He answers that it will diminish by two thirds of its thickness as it widens by the same obliquity.
A die: copper female, tempered-steel male
Beneath the drawing of laminae and screwed plates a note directs that the female part be of copper and the male part of tempered steel. It labels the pressing die sketched at the top of the sheet.
Making and correcting a steel cutting template
Very thin absorbent papers are to be glued with fish glue to the thickness of half a knife-back and dried between hot boards in a press; the pattern is best glued and marked out on a steel plate, which will not shift with heat or damp, then cut carefully with a file. The file is corrected and kept for marking out and re-correcting others once they are tempered and ground.