The Law of the Balance: Beam, Arms and Counterweight
Steelyard proportions, a note on converse proportion, and a week's household expenses
This sheet works out the law of the balance or steelyard: how the proportion between the arms of a beam and the point at which it is suspended fixes the counterweight needed to hold it level with the horizon, illustrated by small balance diagrams and worked numbers ('take the half of 12, which is 6...'). A note at the top sets out the 'converse' in proportionality — that if 8 is double 4 and 6 double 3, then 4 is subdouble to 8 and 3 to 6 — as one of '69 modes of arguing in due manner.' In the lower left a faint pencil account lists a week's household spending, day by day, in soldi and denari (meat, wine, bran, greens, ricotta, oranges, bread). Further faint numeric columns and untranscribed jottings are present on the sheet.
On this page
A weight that holds an unequally suspended beam level with the horizon
If an equal beam is suspended between unequal parts, the weight hung at the end of the shorter portion that keeps the beam level with the horizon stands to the whole beam as the shorter portion does to half the beam. Equivalently, that same proportion equals the ratio of the shorter portion to the amount by which it is exceeded by half the beam. Leonardo states the rule several ways, tying the counterweight to the geometry of the suspension point.
The 'converse' in proportion: double and subdouble
Under the heading '69 modes found to argue in due manner,' Leonardo pairs 8 to 4 and 6 to 3 as double proportions, then argues the converse: 4 is subdouble to 8 and 3 is subdouble to 6. He explains that it is called the 'converse' because the first and second antecedents are turned about and converted into consequents, and the consequents into antecedents, interchangeably.
Worked balance example: rods against one
A numeric worked example: take the half of 12, which is 6; subtract the shorter arm, which is one; five remains; divide by the one of the shorter arm, giving five, so five rods stand against one. A companion figure divides the greater arm by the lesser and places the result at the end of the shorter arm as the counterweight of the greater.
A week of household expenses in soldi and denari
A day-by-day pencil account of household spending: Sunday, Monday, Tuesday, Wednesday, each with recurring items — meat, wine, bran, greens, ricotta, oranges, bean-pods, bread and candles — priced in soldi (s.) and denari (d.). Running sums appear at the head of each day and a total is struck at the foot.