Solid Geometry: The Cube Divided into Pyramids
Cone in a cylinder, Euclid XII.3, and the cube resolved into six five-based bodies
This is a page of solid geometry filled with drawings of three-dimensional figures. Along the left margin stands a cone inscribed in a cylinder, followed by cubes decomposed into pyramids, right and oblique pyramids inscribed in a triangular prism, a figure drawn from Euclid XII.3, and cubes crossed by face diagonals, together with an octahedron/icosahedron and a large tetrahedron. Leonardo states that the cube resolves into six equal bodies of five faces each (four triangular, one square), lettered a b c d o and so on, that pyramids on equal bases between parallel planes are equal to one another, and that the greatest pyramid drawn from a cube is a third of the whole cube.
On this page
A cone inscribed in a cylinder
Among the seven figures ranged along the left margin, the first is a cone inscribed within a cylinder. It heads a sequence of solids in which one figure is set inside another.
The cube decomposed into six pyramids
A cube is shown broken into six pyramids, its vertices lettered a b d c and e f h g with apex o. The construction prepares the claim that the cube is the sum of six equal pyramidal bodies.
A figure drawn from Euclid XII.3
One diagram is drawn from Euclid, Book XII, proposition 3, lettered a, c K i, f d h, b g c. It bears on the division of a prism or pyramid into equal parts.
The cube resolves into six five-based bodies
The cube resolves into six equal bodies called bodies of five faces, of which four are triangular and one square, namely a b c d o, g e h f o, a b g e o, c d h f o, a c g h o, and b d e f o. These are the six pyramids meeting at the cube's centre o.
Pyramids on equal bases are equal; the greatest is a third of the cube
All pyramids raised on an equal base within parallel planes are equal to one another. The greatest pyramid that can be drawn from a cube is a third of the whole cube.