Balancing composite arms of a balance against each other
A rule for the equilibrium of many opposed weights, in whole numbers and fractions.
The sheet develops a 'rule' for finding the equilibrium of many weights hung on the two arms of a balance, so that each weight's share against every opposing weight can be reckoned, whether in whole numbers or in fractions. Diagrams show horizontal balance beams with weights (circled numbers) suspended at lettered points, and the spaces along each arm are compared as ratios. The large figure at right combines three simpler balances into a single one, summing their weights beneath the letters m, n, o, p, q, r.
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A rule for the equilibrium of composite balance arms
Leonardo sets out a method for placing many weights opposed to one another on the arms of a balance and knowing, for each weight by itself, what portion it holds against each opposing weight. He notes the same procedure works with fractions as with whole numbers. It is the statics of a lever generalised to many loads.
Spaces and weights reckoned as ratios (a, b, c, d, e)
Because the space bc is half the space ab, its opposing weight at a is half the weight c; equal spaces ab and bd carry equal weights, and ab being 2/3 of be makes the weight of e 2/3 of a. The figures under each point are summed and entered in the circle attached to it. This turns each balance into a proportional calculation.
Summing three balances into one
The right-hand figure is the sum of the three balances beside it, gathered into one under the letters m, n, o, p, q, r. Leonardo carries the first weights across the pole (9 + 2 + 1 = 12 under p), then the second (3 + 3 + 3 = 9 under q) and the third, so that each weight's share against its three opponents can be read. He observes that removing equal opposed amounts leaves the balance unchanged.