Turning cubes into boards by irrational proportion
Deleted geometrical notes on transforming a cube into a board via cylinders and surd proportions.
A working sheet of solid-geometry notes, most of them struck through. Leonardo repeatedly poses the problem of transforming a given cube into a board (tavola) whose length, width and thickness stand in an irrational or 'surd' proportion to another board, using an intermediate cylinder to pass between board and cube. He lists paired transformation problems, cube to board and board to cube, asking how much a solid shrinks, widens, lowers or shortens under each, and sketches cube and board figures. The reasoning is geometric throughout, framed as ratios between solids.
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Turning a cube into a board of irrational proportion
The opening problems, all struck through, ask that from a given cube a board be made whose length, width and thickness stand in an irrational or 'surd' proportion to those of another given board. Leonardo restates the demand three times in slightly varied wording.
Board to cylinder to cube construction
The given proportion of sides is put in the form of a board; by one proposition the board is made into its cylinder, and by another the cylinder into its cube. With two cubes of different sizes in hand, the greater is reduced to a cylinder as thick as the lesser cube to reveal the difference of the proportion, 4) 2.
Proportions carried from line to solid at equal dimensions
Making proportions from line to line is like making them from cylinder to cylinder of equal thickness, from surface to surface of equal width, from board to board of equal length and thickness, and from solid to solid of equal width and thickness.
A catalogue of cube-and-board transformation problems
A struck-through list pairs the transformations: from a board make a cube, how much does it shrink; from a cube a square board, how much does it widen; and further boards of given width, thickness or length, asking how much each lowers, narrows or shortens. It then reverses the direction, making cubes from boards of given dimensions.
Surd proportions between solids
One deleted note calls it the difficult case: from the given cube to make the board in a given surd proportion to a surd proportion of a board of given width, length and thickness. A related note directs putting the given proportion in the form of a board, making its natural cube, then diminishing or increasing that cube to match the given one.