The general rule of weights on the arms of the balance
The sesquialteral rule and a brief method: the arms' proportion equals the weights' proportion
Folio 170v sets out Leonardo's 'general rule of weights upon the extreme arms of the balances.' He gives a multiplication procedure — the divisions of the shorter arm multiplied by those of the longer, and again by themselves — worked through an example in which the lesser arm a b has 2 divisions and the greater b c has 3 (2 times 3 makes 6, 2 times 2 makes 4). He names this the 'sesquialteral rule' and closes with a brief general statement: whatever proportion the balance arms have between themselves, the opposing weights will have the same. Diagrams of divided beams with hung weights and pulleys accompany the text.
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General rule of weights on the balance arms
Leonardo announces a 'general rule of weights upon the extreme arms of the balances,' and offers a 'definition of the figure a b drawn above.' The rule governs how weights on opposite arms hold one another in check.
Multiplication procedure with worked example
The divisions of the lesser arm are multiplied by those of the greater and the sum attached to the lesser arm's end; then the lesser arm's divisions are squared and attached to the greater arm's end. Example: lesser arm a b has 2 parts, greater b c has 3, so 2 times 3 is 6 and 2 times 2 is 4.
The sesquialteral rule
Written inside the figure, Leonardo labels the case the 'sesquialteral rule' — the 3-to-2 ratio between the arms and their weights that the worked example illustrates.
Brief final method: arms' proportion equals weights'
A 'general and final brief method' states the law compactly: whatever proportion the arms of the balance have between themselves, the opposing weights will have the same proportion between themselves.