Estimating the earthwork: cutting the hill and moat
Volume of a columnar wedge and the cost in ducats of leveling and excavation
Folio 32v works out the volume of earth to be removed for a fortification and its cost. Treating the hill as part of a columnar (cylindrical) figure of diameter about 100 braccia, Leonardo multiplies half the diameter by half the circumference and then by the height to get 3000 square canne, then subtracts fractions (2/3, then 3/8) to isolate the wedge of 375 canne, priced at 4 lire the canna. A red-chalk section of the hill and moat tops the page, and the totals are summarized: leveling the hill costs 428 4/7 ducats and excavating the 2720-canne moat 4662 6/7 ducats, before its walls.
On this page
Volume of the columnar wedge: half diameter times half circumference times height
The hill is treated as part of a columnar figure of diameter about 100 braccia. Half the diameter (15) times half the circumference (50) gives 750, and this times the height b c (4) gives 3000 square canne. The wedge (conio) a n m is then isolated as a fraction of this whole.
Section of the hill and its moat (n c, 20, f, m 20, a 30 b)
A labelled figure at the top of the sheet lays out the cross-section of the hill to be cut and the moat, keyed with the letters and dimensions n c, 20, f, m 20 and a 30 b.
Cost summary: leveling 750 canne of hill, excavating a 2720-canne moat
The hill amounts to 750 canne, whose leveling costs 428 and 4/7 ducats. The moat is 2720 canne and its excavation 4662 and 6/7 ducats, after which the walls of the moat still remain to be built.
Fractional reductions (2/3 and 3/8) and lire-to-ducat conversion
From the 3000 canne, two thirds are removed to leave a wedge of 1000 canne, then 3/8 of that wedge leaves 375 canne. At 4 lire the canna this is 1500 lire, and the sums are converted into ducats (428 4/7; 4662 6/7).