On the Cubature of the Sphere
Reducing the sphere to eight pyramids in order to build an equal cube
Under the heading 'On the cubature of the sphere,' Leonardo works toward building a cube equal in volume to a sphere. He shows a series of pyramids at the right (a curved-sided wedge that is one eighth of the sphere, its rectilinear equivalent, a quadrilateral-based pyramid, and a square-based double pyramid) and, below, a sphere divided by great circles into eight triangular pyramids labelled with the letters a, b, c, d, e, f, g. To convert the sphere's eight pyramids into a cube's six, he divides 8 by 6 to obtain one and a third.
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One eighth of the sphere as a curved-sided pyramid (a)
The motion of one side of the triangle with curved sides and surface yields a triangle with straight sides. This straight triangle is a, the base of one eighth of the sphere, to which is added the pyramid that penetrates the sphere as far as its centre. Successive figures (b, c, d, e) recast this curved wedge into an equal square-based pyramid.
The sphere divided into eight triangular pyramids
The sphere is divided principally into 8 triangles with 8 pyramids: four in the upper half from the circle b e f d upward (a b c d, a b e c, a e f c, a f d c) and four below it (b d g c, b e g c, e f g c, f d g c). A sphere whose diameter equals the side of the square is claimed to compose an equal cube.
Cubing the sphere: dividing 8 by 6 to get one and a third
To cube the sphere with the help of the square pyramid e, equal to pyramid a, the eight pyramids that compose the sphere must be recast as the six that compose the cube. To do this he divides 8 by 6, which yields one and a third, then follows the construction drawn opposite in the cube.