The Strength of Beams and Supports
Shorter supports are stronger; bundling nine or a hundred beams multiplies their resistance
Studies on the strength of structural supports (sostentaculi). Leonardo states that among supports of equal material and thickness the shorter is the stronger, and shows that removing nine tenths of a beam's height makes the remainder resist ten times as much, so that binding nine equal beams together equals the strength of a ninth-part of one. A further rule holds that a hundred supports bound tightly together each come to resist a hundred. Diagrams of a balance weighing 100 against 1000 and of loaded horizontal beams labelled with their capacities illustrate the ratios.
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Shorter supports are stronger
Under the heading De sostentaculi, Leonardo states the principle: among supports of equal material and thickness, the one of greater strength is the one whose length is shorter.
Cut the height by nine tenths, resist ten times more
If a crosswise support of equal thickness and material resists 100, then removing nine tenths of its height and supporting the remainder at its ends makes it resist a thousand. Equivalently, one finds the same force in nine bound beams as in a ninth-part of one of them. The figures are marked 100 and 1000.
Three beams and the ninth-part rule
For the three lettered beams, a b supports 27 and is made of 9 beams; therefore c d, its ninth part, supports 3; and e f, the ninth part of the length of c d, supports 27 because it is nine times shorter. The loads are noted f e -27, d c -3, b a -27.
A hundred bound supports each resist a hundred
Because body B is proportional to A, it is of equal resistance (A -9; B -1/9). And if a hundred upright supports each resisting one are joined with tight binding, each comes to resist a hundred, since the combined bundle is not only a hundredfold in mass but a hundred times lower in figure than a single support.