Distributing 12 pounds onto the three cords of a beam
Twelve pounds (5 at a, 7 at e) resolved onto cords b, c, d by proportion
A graduated horizontal beam a-b-c-d-e is suspended by three cords, its spaces marked with fractions such as 2 6/7 and 6 6/21, and weight-boxes of 5 and 7 pounds hang at the ends. Leonardo works out how the 12 pounds hung at the ends discharge onto the three cords b, c, d: he divides the spaces into 12 equal parts, draws out four proportions among the segments t x, r t and S t, sums them to 21, and converts each into a fraction of twelfths (12/21) that gives each cord its due share. He fixes t as the centre of equality between the 5 and the 7.
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Reducing four proportions to a common measure of 21
From the segments Leonardo draws four proportions: 5 of t x against 7 of r t makes 12; 5 of t x against 4 of S t makes 9; and 9 plus 12 makes 21. He then divides the 21 parts by the 12 pounds so that each part is 12/21, where 21 is the number of parts and 12 the quality of the part, and multiplies each cord's share by 12 and divides by 21.
Twelve pounds discharged onto three cords
The rod a-e carries 12 pounds, 5 in a and 7 in e, which discharge onto the three cords b, c, d. Leonardo assigns each proportion its due, consuming the 21 numbers, reduces by a third, and gives each cord its rate, noting that t is the centre of equality of 5 against 7.