A loaded cord and its counterweight: correcting the 32 for 4
The sagitta of a sagging cord fixes the counterweight; a proportion from the 'Theory' gives 64 pounds
Leonardo analyses a cord a m S that hangs 4 pounds at its middle and sags into an arc whose rise (sagitta) is a thirty-second part of its chord. Invoking the eighth proposition of his 'Theory', that the counter-lever stands to the lever as the weights they carry, he computes the counterweight needed at a. The upper note first admits an error, having set 32 against the whole 4 without halving the 4 at its supports; the fuller working then corrects the figure, giving 64 pounds at a S against the 4 at m.
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Correcting the counterweight: 32 set against the 4
Leonardo admits a slight error below: he had set 32 against the whole 4 that is attached, forgetting that the 4 divides in half at its supports. The correction is to reckon 16 against 2 of m rather than against 4, so that for m o against m a, 4 times 8 gives 32, and 32 weights are placed at a S against the 4 of m.
The counter-lever proportion yields a 64-pound counterweight
By the eighth proposition of his 'Theory', the counter-lever stands to its lever as the attached weights stand to one another, so the weights balance. Because o m is the middle of the pyramid a n S it is half of a n; and since a n enters 8 times into a m, m o enters 16 times, so the 4 at m multiplied by 16 makes 64, and 64 pounds must stand at a S against the 4 of m.