Balance-arm and the proportion of combined weights
An adversary on the centres of gravity of balance arms reckoned together with their attached loads
A red-chalk diagram of a balance beam labelled b, e, a, g, f, c heads a mechanical argument about equilibrium. Leonardo reports an 'adversary' who holds that the weight of each balance-arm must be reckoned together with the load hung on it, so that the opposed centres of the combined gravities stand at distances from the pole proportioned to those combined weights. He grants the point and proves it with a worked case: each arm weighs 4 pounds, a load of 4 is hung at the middle of arm a b (making 8, centred at e), and 2 is set at the far end of the opposite arm, giving distances in the double ratio.
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Centre of gravity of a balance-arm joined to its load
Leonardo grants the adversary's claim that arm-weight and hung weight must be combined. Taking each arm as 4 pounds (a b and a c), he hangs 4 at the middle of a b so the combined weight is 8, whose centre of gravity lies at the arm's midpoint e. Against the other arm he sets 2 at its far end, so the distances of the attached weights come out double, and he builds the 'judging balance' e f g.
Distances proportioned to the combined weights
The rule at issue is proportional: the opposed centres of the joined gravities have their distances from the pole of the balance in the same ratio as the joined weights. The numerals 4, 2, 8, 6 written beside the beam track this reckoning, the 8 (arm plus load) balancing at double distance against the smaller weight.