Turning a cylinder into a square board, and back again
Converse transformations of solid to board and board to solid, tracking thickness (a b, b c, b e)
Two converse propositions (numbered 6a and 7a) on passing between a cylinder and a square board of equal volume. The first makes a square board of given width from a cylinder a b: because the cylinder is shorter than the board's side b e, it is lengthened, then its face is worked against the board's face to yield the required thickness. The second reverses the operation, making a cylinder from a square board to the length of a given space and asking how much it draws in. Two board-and-block diagrams with arcs sit in the right margin.
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A square board of given width made from a cylinder (a b, b c, b e)
Given cylinder a b and the required board side b c, the cylinder is first lengthened to match the board's side b e by the converse of the fourth of the second. Its face is then used together with the board's face as a surface, and with the sixth of the first the required thickness of the square board is found.
Converse: a cylinder drawn from a square board, and how far it narrows
Reversing the previous problem, a cylinder is made from the square board b e to the length of a given space, and it is asked how much it draws in. The board is reduced to its natural cylinder and that cylinder shortened to the given height, so the amount of narrowing follows.
Two board-and-block diagrams with arcs
The upper and lower right each show a board carrying a raised angled block set beneath a quarter- or semicircular arc. These are the geometric means used to lengthen or shorten the cylinder in the two converse constructions.