Levers and Counterweights for Raising Beams
Three beam-and-pulley diagrams working out the counterweight forces
Three diagrams analyse how a beam hauled up by a cord over a pulley is balanced. In the top figure the lettered lines a r, x K, n K and p K are identified as the lever-arms of beams b K, c K, d K and e K, the weightless beam b K having reached the end of its travel. The lowest figure reduces a horizontal beam to a lever m n and counter-lever n a: since the beam's four weights act at its middle and the lever fits eight times into the counter-lever, the required counterweight is 4 × 8 = 32.
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Lever-arms of four beams b K, c K, d K, e K
Each lettered line is assigned to a beam: a r to b K, x K to c K, n K to d K and p K to e K. Beam b K carries no weight and its lever is spent, because it can descend no lower than a.
Counterweight for a horizontal beam m p
Because the beam's weight acts at its middle, its four weights are taken as concentrated at a. The thickness m n is the lever and n a the counter-lever; the lever fits eight times into the counter-lever, so every weight at the load demands eight against it.
Reckoning the counterweight: 4 × 8 = 32
With four weights at a and a lever ratio of eight, Leonardo multiplies: '4 times 8 is 32.' That 32 is the counterweight to be hung on the counter-lever to raise the beam.