Resolving the Spherical Body into Pyramids
Cubing the sphere by dividing both sphere and cube into pyramids
Continuing the cubature of the sphere, Leonardo resolves the spherical body into pyramids with triangular bases whose axes are the sphere's semidiameter. A large sphere at the top right is netted with great circles; below, a square with crossed diagonals meeting at a central point o shows a cube divided into pyramids, and marginal sketches place a pyramid inside a parallelepiped with a quartered circle and square. He argues that one sixth of the cube equals one eighth plus a third of an eighth of the sphere, so that the eight pyramids of the sphere convert into the six of the cube.
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Resolving the sphere into triangular-based pyramids
Let the spherical body be resolved into pyramids that have triangular bases, and let their axes be made from the semidiameter of its sphere. Granting two thirds of the body, each pyramid can then be cubed.
The cube divided into six pyramids around the centre o
The cube is divided into pyramids meeting at the centre o. The angles of the six inner triangles are right according to the cut through the middle of the sides a b and d c, but obtuse according to the angles a o c. The sixth and final figure shows this resolution, with the sphere as its next-to-last stage.
One sixth of the cube equals 1/8 plus 1/3 of an eighth of the sphere
By the conclusion opposite, one sixth of the cube equals 1/8 plus 1/3 of an eighth of the sphere, placed here in a b c d g (an eighth of the sphere) with b e d f added as a third of that eighth. Squared and set in place of the fourth figure, this pyramid becomes one sixth of the cube.