Motions of Water and the Doubling of the Cube
Four kinds of liquid motion; squaring and duplicating cubes
Leonardo distinguishes four kinds of liquid motion: natural motions through air, faster and finer at the end than at the start; semi-natural motions along a straight, even river bed, of equal velocity; accidental motions through air, which slow at every height; and semi-accidental motions along a canal bed, longer than the simple accidental because the water leans on the bed and sheds part of its weight. The page then turns to doubling the cube (cross-referencing page 59), with box-shaped figures (a b, c, d, n o, S) showing a cube divided by its diameter, squared imperfectly, halved and reassembled through a 'judicial angle' a b c d. A final triangle figure (b e d, r f, a c, n m, d) claims to prove one square double another where the ancient method could not.
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Four kinds of motion of liquids
Natural motions of liquids through air are faster and finer at the end than the beginning; semi-natural motions along a straight, even river bed are of equal velocity; accidental motions through air slow at every degree of height; and semi-accidental motions along a canal bed run longer than the simple accidental, because the water leans on the bed and always discharges part of its weight.
Doubling the cube by squaring and transformation
Divide the cube a b by its diameter; because the division is not a perfect squaring, square it by the eighth proposition, take the half at c and square it as d, then return d to the imperfect squaring as at S. Beneath S add as much again to make a cube half the one above; the taking-away and perfecting is done with the judicial angle a b c d, where r is the imperfect squared half-cube, f the perfection, m the smaller perfect square, n the imperfect.
The judicial triangle proving one square double another
The figure of the judicial triangle (labelled b e d, r f, a c, n m, d) is offered as a proof: the other, ancient method cannot prove one square to be double the other, as this one can.