Cubes double one another: the diagonal cut
Squaring the oblong of a cube's diagonal and the triangle of proportions
This page treats cubes that are double one another. Leonardo observes that cutting a cube along its diagonal yields an oblong (longer than wide) because the diagonal a c exceeds the side a b, so it must be squared 'with the rule of the fifth.' He then builds a triangle of proportions, placing the long square and the perfect square in contact on line a b and continuing the contact line c d until the two lines intersect at right angles. A right-margin note compares the diagonals of the squarings of a larger and a smaller cube, showing a b equal to a c. Numerous diagrams show cubes halved vertically and diagonally.
On this page
Why the diagonal cut yields an oblong, not a square
Because the diagonal of the square is longer than its side (a c longer than a b), the quadrilateral made from the cube cut by the diagonal is not a perfect square but longer than wide, and must be squared with the rule of the fifth.
The triangle of proportions
Each of these squares must be right-angled. Leonardo puts the long square and the perfect square in contact on line a b and continues the contact line c d so the two lines intersect within 4 right angles; such squares are r S.
Building a cube half the given one
He cuts the cube through the middle equidistant from each side into a n m o, cuts that square in two by diagonal a o (giving n), makes the long square perfect with the fifth (giving m), then carries it to the angle of proportions where square v sits in contact with square S double to it, touching line c d.
Equal diagonals of the two squarings
The square p q a m is the squaring of the diagonal cut of the larger cube, and a n o c that of the smaller. The diagonal of half the larger square (a r b m) equals the diagonal of the whole smaller square (a n o c); the equal diagonals are a b and a c.