Toward the quadrature of the circle: sphere, cylinder and cone
Solids inscribed in a cube, and the squaring of the circle 'which I intended to demonstrate'
This page gathers Leonardo's solid-geometry constructions leading to his claimed quadrature of the circle. He states that a sphere's surface equals the lateral surface of a cylinder as tall and wide as the sphere, and inscribes a sphere and a cylinder within a cube. A cone folded from a semicircle is unrolled into a triangle, and a right-angled figure n made from half that triangle m is set equal to the circle of the cone's base S. Leonardo declares this to be 'the quadrature of the circle, which I intended to demonstrate,' achievable in several ways as shown at f.
On this page
A sphere's surface equal to an equal cylinder's surface
The surface of a sphere equals the surface of a cylinder as tall and wide as the sphere. This equivalence underlies the cube constructions that follow.
Largest sphere and largest cylinder within a cube
The largest sphere a cube can contain has a surface equal to the lateral surface of the largest cylinder that fits within the same cube. The point is illustrated by a sphere and a cylinder each drawn inside a cube.
Cone folded from a semicircle and its swept triangle
A cone whose base diameter equals its slant describes a semicircle when revolved about its fixed apex; rolling its base along a straight line yields an equilateral triangle. The triangle r o c has base r o from the periphery of circle S and axis from the axis of cone S b.
The claimed quadrature of the circle at f
The right-angled figure n, made from half the triangle m, equals the circle of the base of cone S. Leonardo calls this the quadrature of the circle he intended to demonstrate, achievable in several ways as shown at f.