Rule for Computing How a Beam Distributes Its Load
Worked rule-of-three arithmetic for supports a, b, n and f, checked "from experience"
Folio 166r sets out a "Rule" (Regola) with worked arithmetic for finding how a compound beam and its added simple rod n s distribute their load among the supports a, b, n and f. Using rule-of-three steps and fractions, Leonardo computes that b sustains 5/6 while n takes 3 and 2/3, then cross-checks a second method drawn "from experience" by dividing the four braccia into eight parts. Numbered balance-beam diagrams and a projecting-arm sketch accompany the calculations, which he verifies with the note "e sta bene" (and it is correct).
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Load shared between supports b and n of a compound beam
Working the rule for the beam n s, Leonardo finds that support b, which formerly held 1 1/2 of the three braccia b n, is lightened by 2/3 to sustain 5/6, while support n takes the remainder, 3 and 2/3. Support f simply carries the four braccia of c b.
Rule-of-three check "from experience"
A second method divides the four braccia into eight equal parts and takes half of the projecting braccio, reasoning by proportion: if 3 gives one, one whole gives 1/3. Converting to sixths, 1 1/2 becomes 9/6, and subtracting 1/6 leaves 8/6 = 1 1/2, confirming the result.
Restated rule for a n against a p
Leonardo restates the calculation more clearly: one times 3 makes 3 for a n against a p; putting one under p and 3 under a, if 3 becomes one in a n it becomes 1/3 in p, and a third is subtracted from 1 1/2.
