Foundations, and the Nature and Breaking of Arches
Why uneven foundations crack corners, why an arch is a strength born of two weaknesses, and where it breaks
The folio pairs a study of building foundations with a series of propositions on arches. Leonardo warns that a foundation of even width carrying an unequal load—heavier under a corner pier at b e than under the wall e o—sinks more there and tears away from the lighter wall, cracking the corners of buildings, and that foundations should instead spread equally beyond their load at m, a, and b. He then defines the arch as a strength born of two weaknesses, each quarter-circle weak alone but holding up the other, and states that it stays in equilibrium only if its halves and their loads are equal. Two further propositions locate where an excessive central load breaks the arch (at c and e, two-thirds up) and warn that thrusting an arch sideways weakens it further.
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Unequal foundations tear away at the corners (m a b; b e o)
The heavier part of a foundation sinks deeper into the ground, leaving the lighter parts higher and detached. A foundation of even width loaded unequally—burdened by the corner pier at b e—presses down more than wall e o and pulls away from it, which is why buildings crack around their piers. Always spread foundations equally beyond the load of their walls and piers, as at m, a, and b.
What an arch is
An arch is nothing but a strength born of two weaknesses: it is composed of two quarter-circles, each very weak of itself and tending to fall, and as each opposes the other's ruin the two weaknesses are converted into a single strength.
An arch keeps equilibrium only if its halves are equal
Once composed, the arch remains in equilibrium, for each half pushes the other as much as it is pushed. If one quarter-circle weighs more than the other, its permanence is lost, since the greater weight overcomes the lesser.
Load the halves equally; where the arch breaks
After giving the two quarter-circles equal weight, they must be loaded equally on top, or the same error returns. The arch will break at that part whose middle passes below the centre.
The second breaking of the arch (a, b, c, e)
An excessive weight placed at the middle of the arch, at point a, tends to fall toward b, and the arch breaks at the points two-thirds of its height, at c and e; and g e is as many times stronger than e a as m o enters into m n.
Sideways thrust as another cause of ruin
The arch grows even weaker when pushed sideways, for when the load is not directed toward the feet of the arch, the arch lasts but little.