Elasticity of bent wood and interlocked timber roof frames
Why a bent bow springs back; reciprocal beam networks spanning 45 braccia
The main columns develop a theory of why a bent bow or arch returns to shape: every 'elemented' body is porous, and when wood is bent one part of its air condenses while the other rarefies, the condensed pushing and the rarefied pulling until it recovers its first nature. The rest of the sheet, drawn with the page turned, works out self-supporting interlocked beam structures (incatenatura) in hexagons and squares, counting the number of 2-braccia beams in each pattern and reckoning how a span of 45 braccia can be roofed without central props using 84 short cantilevers and 164 tie-ropes, covered over with woolen cloths. Columns of arithmetic accompany the beam counts.
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Why a bent bow springs back: condensation and rarefaction
Leonardo reasons that every elemented body is porous, so bent wood has one part where the air condenses and one where it rarefies; the condensed part pushes and the rarefied pulls, driving the bow back to its first nature. If the bow is left bent for a long time, the rarefied side slowly condenses and the condensed side rarefies, and the wood keeps its acquired curvature.
Roofing a 45-braccia span without central props
This covering is to span 45 braccia without props in the middle, using 84 little beams or cantilevers, of which 24 support the other 60. The cantilevers are 10 braccia long, four in a row with three-braccia gaps, and the arch of the whole vault reduces the 50-braccia total to a 45-braccia covered space, over which interwoven woolen cloths are stretched.
Hexagonal interlaced beam frame a b c d e f
For the interlacing of six hexagons lettered a b c d e f counter-clockwise, multiply the sides: six times six makes thirty-six, and that is the number of 2-braccia beams in the assembly. The central hexagon is not counted because its sides are formed of those same 36 beams, and the whole circuit will be 50 braccia across.
Counting beams in the square interlace
For the square chaining, multiplying the 18 squares by 4 (four beams each) gives 72, but in truth there are far fewer because outer and shared squares hold beams in common. Leonardo subtracts the double-counted timbers to reach the true total, working through the same reckoning for another square linkage a b c d f.