Making a Proportional Slab from a Cube by Three Lines
Turning a cube into a rectangular slab whose sides keep the irrational ratio of three given lines
The problem set here is to turn a cube into a rectangular slab (tavola) whose thickness, width, and length stand in the same irrational proportion as three given lines. The construction is carried through lettered figures: a cube a, three vertical lines b c d, and two cylinders e and f, invoking numbered propositions of earlier books. By making a slab from the three lines, converting it into cylinder f and then into a cube, and joining cubes and cylinders face to face into a single cylinder, Leonardo prepares the "judicial" proportional lines used on the facing page. Small diagrams of the cube, the cylinders, and a graduated bar illustrate the steps.
On this page
The problem: a slab in the irrational ratio of three lines
From a cube a rectangular slab is to be made whose thickness, width, and length lie in the same irrational proportion as that which three given lines have among themselves. The task frames the whole construction that follows.
The lettered figures: cube a, lines b c d, cylinders e and f
The page keys its steps to a cube a, three vertical lines b c d, and two cylinders e and f. A slab e is first made from the three lines, then turned into cylinder f, and from that cylinder its cube is drawn.
Joining cube and cylinder into a single cylinder
Because the new cube may be larger or smaller than the first given cube, a cylinder is made from the larger cube to the thickness of the smaller, and the pieces are joined fully face to face into one cylinder. The three judicial lines then complete the work, as described in the penultimate proposition.