Unrolling the Surfaces of Cylinders and a Pyramid
Developing curved solids onto the plane, with a counterweighted beam and calculations
Leonardo studies how solids can be 'unrolled' onto the plane. A line spiralling once around a cylinder is shown to equal the circumference of one face together with the square of the line, while the surfaces (spoglie) of a square pyramid and of a four-sided prism are laid out as the flat figures a b c d and e f g h. At the left a cylinder hangs from a cord over a pulley with a counterweight, and the lower right is packed with columns of arithmetic attached to a horizontal beam-and-counterweight problem.
On this page
A line wound once around a cylinder
The line e f d that makes a single full turn about its cylinder is said to equal the circumference of one face of the cylinder with the square of the line added to it — the unrolled length of a helix combining girth and height.
Unrolled surfaces of a pyramid and a prism
Turning the sheet on its side, Leonardo lays out a square pyramid inside a cylinder and triangles inside a rectangle. The figure a b c d equals the developed surface of the square pyramid, and e f g h equals the developed surface of its four-sided cylinder (prism).
Cylinder hung with a counterweight
At the left a cylinder is suspended from a cord that passes over a pulley, an ovoid counterweight hanging above it. The drawing accompanies the note headed 'horizontal beam with a counterweight.'
Columns of calculation
The lower right of the sheet is filled with columns of figures — additions and multiplications running into the thousands — worked out to support the beam-and-counterweight reasoning.