On Quadratures: Squaring the Sphere and the Cone
The sphere's surface as twelve squares; the cone's surface rectified
Under the heading 'On quadratures,' Leonardo squares curved surfaces. A hemisphere with parallels, a fan-shaped sector and a beehive dome accompany the text. He reasons that the triangle a b c is one eighth of a sphere's surface, its third part the triangle c a twenty-fourth, so the square d c e (double the triangle c) is one twelfth of the surface; a parallelogram holding twelve such squares equals the whole spherical surface. He then rectifies a cone whose slant equals its base diameter, obtaining a rectangle equal to two circles like its base and, doubled, the surface of a cylinder.
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The sphere's surface as twelve squares d c e
The triangle a b c equals one eighth of a sphere's surface, and its third part, the triangle c, a twenty-fourth. The square d c e, made double the triangle c by motion, therefore equals one twelfth of the surface, so a parallelogram containing that square twelve times equals the whole surface of the sphere.
The cone's lateral surface rectified into a semicircle
The full revolution of the base of a cone whose slant equals its base diameter traces a straight line which, multiplied by the cone's axis, gives a rectangle equal to two circles like the base, or a semicircle equal to the cone's lateral surface. Doubling that surface yields a cylinder, whose full revolution leaves a trace equal to a circle double the cone.