Doubling the cube: a mental, not instrumental, proof
Nested long squares in continued proportion and the Delian problem
Leonardo pursues the doubling of the cube through nested long squares. In the left column he inscribes a smaller long square beneath a larger so that the two are in the same proportion as their perfect squares, and hence as the cubes of which they are the diagonals. The right column states the key rule: double the square generated in a cube's diagonal cut and you obtain the diagonal cut of the double cube. A famous marginal note contrasts his method with the proof Plato gave to those of Delos, which uses compass and ruler and is only shown by experience, whereas his own is 'entirely mental and consequently geometric.'
On this page
Nested long squares in continued proportion
Producing the line d n to touch the angle of the long square, Leonardo fits another long square in the space beneath, touching the two squares, so that the two smaller long squares stand in the same proportion as their perfect squares, and hence as the cubes of which they are the diagonals.
Doubling the diagonal-cut square doubles the cube
Double the square generated in the diagonal cut of the given cube and you will have the diagonal cut of the cube double the given one; double one of the two square surfaces generated in that diagonal cut.
A mental, not instrumental, solution to doubling the cube
Leonardo notes that the other proof Plato gave to those of Delos is not geometric, since it proceeds with compasses and ruler and is shown only by experience, whereas his own method is entirely mental and consequently geometric. He tests whether the marginal double squares match squares S r in the angle of proportions.