The Strength of Supports and the Scaling of Loads
Why a column enlarged in proportion cannot bear proportionally more weight
Across this sheet Leonardo sets out propositions on the strength of upright supports: when the lower of two stacked bodies is already at the extreme of its resistance, the pair cannot be enlarged proportionally without failing. He argues that a body which just sustains itself at a given length can never sustain proportionally more once scaled up, because resistance does not grow as fast as load — a support ten times as thick (10 x 10 = 100 in cross-section) carries only a thousand where the unit carried a hundred. A companion note observes that twelve equal props bound tightly together each resist twelvefold. The Ambrosiana catalogue also associates the sheet with brief notes on lenses (eyeglasses) and on the elements that are not part of the present transcription.
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A scaled-up support cannot bear proportionally more
Leonardo states that a body which just sustains its own weight at a given length will never sustain more of itself once enlarged in the same proportion. The reason he gives is that resistance does not scale with bulk: if the thickness of one unit resists a hundred, a hundred such units do not go on to resist a hundred thousand.
Twelve props bound together each resist twelvefold
A note observes that if twelve supports of equal length are joined in a tight bundle, each resists far more than it did alone. Where each prop by itself withstood a load of one, the bound and compressed group makes each withstand twelve.
Worked scaling example: thickness 1 bears 100, tenfold bears 1000
A worked example sets a support's thickness at one and the greatest weight it can bear at a hundred. Multiplying the dimensions tenfold, Leonardo reckons 10 x 10 = 100 for the cross-section and 10 x 100 = 1000 for the load placed on top, so the enlarged prop bears a thousand where the unit bore a hundred.
Two stacked bodies at the limit cannot grow proportionally
The opening propositions declare it impossible to enlarge two stacked bodies proportionally, in shape and in action, when the lower one already stands at the extreme of its resistance. Three parallel statements restate the rule in terms of action, resistance and power.