Reducing a square board to a cylinder of given length
Finding the cylinder's thickness from a board (a b c d, f g a e); note on making a board from a cylinder
A proposition (numbered 6a) on reducing a square board to the form of a cylinder of a given length and finding the cylinder's thickness. The board's square face a b c d and the cylinder's square face are treated as mere surfaces; if the required cylinder is longer than the one derived from the board it is extended, and if shorter it is cut down, so the thickness for either case is obtained. A closing line proposes the converse, making a board from a cylinder along its length to a given thickness. The upper right shows a board carrying an angled block beneath a semicircle.
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A square board reduced to a cylinder of set length (a b c d, f g a e)
The square face a b c d of the board and the square face of the cylinder are used as simple surfaces because board and cylinder share the same length. If the required cylinder is longer, the cylinder f g a e is extended by the converse of the fourth of the second; if shorter, it is cut down by the fourth, giving the thickness in each case.
Determining the thickness of the resulting cylinder
The problem fixes the cylinder's length and asks for the thickness that conserves the board's volume. Working the surfaces against one another yields that thickness for both a longer and a shorter required cylinder.
Board with angled block and semicircle
The upper right shows a board carrying an angled solid block set beneath a semicircular arc. It is the kind of surface figure — the faces named a b c d and f g a e in the text — used to fix the cylinder's thickness.