Rule of Transverse Supports: the Resistance of Beams
A support of double diameter is four times thicker and bears four times the weight
The heading gives a rule for transverse supports held immovable at their fixings, illustrated by a horizontal beam diagram at the top with weights hanging from it, labelled a, n, b and d, c and the numbers 24 and 6. The text argues that the support c d, being of double the diameter of the support a b, is four times thicker and therefore bears four times the weight at an equal distance from its fixing. It concludes that the midpoint n b of the upper support has the same bearing strength as the end c of the thicker support, since both keep the same proportion of thickness to length. Further faint construction figures and reversed show-through text appear at the left, not transcribed here.
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Rule of transverse supports fixed at their ends
Leonardo states a rule for transverse supports that are held immovable at their fixings. The horizontal beam of the diagram, loaded with hanging weights, is the case to which the rule is applied.
Resistance scales with the square of the diameter
The support c d, being of double the diameter in thickness of the support a b above, comes by the fifth proposition to be four times thicker, and so at an equal distance from its immovable fixing it sustains four times the weight. The midpoint n b of the upper support therefore matches the end c of the thicker support in bearing strength, because each keeps the same proportion of thickness to its length. Labels a, b, n and d, c and the numbers 24 and 6 mark the beams.