Balances, counterweights and the compound lever
Levers and double pulleys; the simple rod shown as the root of the compound (5x5=25)
Two balance diagrams explore how counterweights and pulleys equalise a loaded rod. A weight of 36 pounds set at the rim of a disc above b is reckoned as if it hung at a; where the counter-lever a b goes six times into the lever, 36 pounds at a hold six. Leonardo then applies the ninth proposition on simple versus compound rods — the simple rod is the 'root' of the compound weight — computing 5 times 5 equals 25 of force in e m, and 25 plus 5 equals 30 for the whole.
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36 pounds reckoned as if at a
The 36 pounds are treated as if they were at a, even though they hang at the extremity of the disc above b.
Balance with counterweight
A balance beam carrying a counterweight, lettered c b a and marked with the values 6 and 36.
Double-pulley balance: the counter-lever entering six times
With a double pulley (o r n; f t m e d), if the counter-lever a b goes six times into the lever b 6, then 36 pounds at a hold 6, with the whole rod in balance.
The simple rod as root of the compound
By the ninth proposition, the simple rod is the root of the compound rod's weight: for a lever of 5 weights in 5 spaces, 5 times 5 makes 25 of force in e m, which balances the rod's 5 weights; 25 in m e and 5 in m f make 30.