Squaring a rectangular board to a given width
Reducing an oblong plank through a cylinder to an equivalent square board (n m p q, o r, b)
A proposition (numbered 10a) on 'squaring' a rectangular board that is longer than it is wide: given a required width, find the thickness the equivalent square board must have. Leonardo reduces the oblong board n m p q to a cylinder as long as the board, extends it by its given thickness to the necessary length, and then squares the result so that the new board b holds the whole volume of the first. The right margin carries an extruded solid resting on a board beneath a semicircle, with a small squared panel below.
On this page
Squaring an oblong board by way of a cylinder (n m p q, o r, b)
The oblong board n m p q, longer than wide, is reduced to a cylinder of the same length by the fifth of the first. Extending it by its given thickness produces o r, which is not yet square. A final squaring by the fifth of the first yields the board b, born of the second board a, that contains the whole of the first board.
Finding the thickness a board of given width must have
The task fixes the required width and asks for the resulting thickness so that volume is conserved. By passing the oblong board through its cylinder and re-squaring, the construction returns exactly the thickness of the equivalent square board.
Marginal diagrams of board, extruded solid and semicircle
The upper right shows an extruded block set on a board beneath a semicircular arc, with a small squared panel below it. These figures accompany the passage from oblong board to cylinder to squared board set out in the text.