Three Suspended Rods: Combining Movers and Counter-Levers
Joining powers 4 and 6 gives a motion, and a counter-lever, midway between
Leonardo studies a beam hung with three loads, labelled with the pairs 2-a-4, 4-n-6 and 6-m-10. He reasons that if 4 moves 2 and 6 moves 4, then joining the movers (10) and the moved bodies (6) yields a motion intermediate between the two; the same holds for the balance, where the counter-lever of 10 against 6 falls midway at m between the counter-levers of 2 and 4. He warns that when two powers of 8 and 1 act against resistances of 1 and 1, the rule below does not hold as promised.
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Joined movers and moved bodies give a mean motion
If 4 moves 2 and 6 moves 4 in a certain proportion of motion, then joining the movers 4 and 6 into 10, and the moved bodies 2 and 4 into 6, produces a motion intermediate to each of the first. The same reasoning is carried to the balance and the counter-lever.
The combined counter-lever lies midway at m
If 4 resists 2 in the balance and 6 resists 4 in a proportion of counter-lever as at n, then joining the powers into 10 and the resistances into 6, the counter-lever of 10 against 6 stands midway between the first two, shown at m halfway along a-n.
Where the rule breaks down
Leonardo cautions that if you set two powers against two resistances — 8 and one against one and one — the rule given below will not perform as it promises. It is a limiting case flagged before the main demonstration.