Transforming a stretched board into a cylinder to find its thinning
Volume-preserving geometry of solids: how far a plank thins or narrows when drawn out to length
Two linked propositions (numbered 11a and 12a) on transforming a square board into its equivalent cylinder and back again in order to measure how a solid changes when reshaped. The first asks how much thinner a square board becomes when stretched to a given length while keeping its width; the second asks how much a board narrows in width when stretched to length while keeping its thickness. Leonardo works each by casting the board into its 'natural cylinder' and then re-squaring it, citing propositions of his first and second books. The sheet is drawn with rectangular boxes, extruded trough-like solids and a semicircle used as geometric aids.
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A square board cast into a cylinder, then re-squared, to measure its thinning
A square board is stretched lengthwise to a given distance without losing its original width, and the question is how much thinner it grows. Leonardo reduces the board to its natural cylinder, extends that cylinder to the given distance, then brings the extended cylinder's face back to the face of the first board. The chain of the sixth and fifth propositions of his second book yields the answer.
How much width is lost when a board is drawn out to length (p, n)
Here the square board is stretched in length to a given distance while its thickness is preserved, and it is asked how far it shrinks from its first width. Using the sixth of the first, the square board p is made to the length of board n without diminishing thickness, which reveals exactly how much the width decreases.
Boxes, extruded troughs and a semicircle as construction aids
The sheet carries drawn rectangular boxes at top right, two long extruded trough-like solids showing a board opened out, and a semicircle with an inscribed line at lower right. These serve as the geometric devices for the volume-preserving passages between board and cylinder described in the text.