Letter to Ser Giuliano and notes on pyramid geometry
A personal letter about Lesandra's illness, with geometric studies of a pyramid inscribed in a double cube
The upper part of the sheet carries a personal letter, headed 'Lesandra in Florence', to a Ser Giuliano, reporting that his wife Lesandra is gravely ill and has lost her wits, and commending the writer to Giuliano's brother Leonardo. In the space left free below the letter, Leonardo has added a set of geometric notes on the relationship between a pyramid and a cube built on the same base, especially a pyramid passing through two cubes stacked one on the other (a 'double cube'). The propositions compare the volumes and heights of pyramid and cube and show how a pyramid may be made equal to a cube by tripling its height. Small diagrams of pyramids and cubes accompany the reasoning.
On this page
Letter to Ser Giuliano about the sick Lesandra
The letter tells Ser Giuliano that his wife Lesandra has a grave illness and is near death from grief, has lost her wits and seems to have grown big. The writer repeatedly commends himself to Giuliano and asks to be remembered to Giuliano's brother Leonardo, insisting that Florence is as beautiful as Rome.
Pyramid and cube on one base within a double cube
Labelled a. If a pyramid and a cube share the same base and the pyramid's axis is double the cube's, the excess of the pyramid's height over the cube equals the thickness of the cube left outside the pyramid. A pyramid passing through two stacked cubes occupies as much of the upper cube as it leaves of the lower.
The section at mid-height is a quarter of the base
The cut made at the mid-height of the pyramid is worth a quarter of its base. This is proved because the sides of the two cuts are in double proportion, so the smaller pyramid is a quarter of the larger. A related note adds that the upper half of the pyramid's height is worth a quarter of its remainder.
Making a pyramid equal to a cube
Four figures of cubes and pyramids on an equal base carry the labels a sixth, a third, two thirds and three thirds. To give a pyramid equal to a cube, triple the height of the greatest pyramid that can be drawn from the cube on the same base, and that pyramid is worth the cube.