Centre of gravity of a suspended triangle divided in nine
Balancing 4 pyramids against 5, from surface to solid by multiplying by 8
A triangle sketched at the upper right is divided by crossing lines into nine equal small triangles (pyramids), keyed a b m c and c n o, to locate its centre of gravity. Leonardo weighs the small pyramid m o c at one pound and, doubling, gets pyramid b n c at 8 pounds, then argues that point b is the accidental centre of gravity of the figure treated as a plate of equal thickness. He proves it by counting pyramids on each side (4 against 5), converting surface to solid by multiplying by 8 to get 32 against 40, which at the final reduction returns to the ratio 4 against 5.
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b is the centre of gravity, 4 parts against 5
Leonardo finds that point b is the accidental centre of gravity of this figure, treated as of equal thickness, as if sawn from a plank. Four parts on one side make a counterweight to five on the opposite side, and the same holds if the body were solid with faces like those drawn.
Triangle divided into nine equal pyramids
The triangle a b m c is subdivided by crossing lines into nine equal small triangles, or pyramids, keyed with letters c n o and others. The small pyramid m o c is taken as the unit and larger regions are counted as multiples of it.
From surface to solid by multiplying by eight
Doubling the unit pyramid m o c (1 pound) gives pyramid b n c at 8 pounds; from the surface, 4 pyramids give 8, so the 5 of a b c n give 10, i.e. four fifths. Converting to solid, the 4 pyramids from b to c become 32 pounds and the 5 from b to a become 40, so 32 stands against 40 in the balance b, reducing finally to 4 against 5.