General rules of blow and force: loaded beams and a lifting screw
Statics of inclined beams, a weight-lifting screw, and Pythagorean proportion applied to force
This mechanics sheet sets out 'general rules of the blow and of force', proposing that the proportions Pythagoras used in music can also govern force. Several figures of beams in various inclinations are analysed for how much weight they bear, with the principle that a less-inclined support carries more and that load can be found from a perpendicular 'thread' dropped between beams. A weight-lifting screw is studied for how an off-centre load bears unevenly on the threads of the screw and its nut, and a small column of arithmetic (320, 4, 1280, 256, 4000, 16, 64) accompanies the reasoning. The labelled figures use letters such as e d c b a, n m, and b a n m.
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General rules of blow and force; Pythagorean proportion
Leonardo states a general rule of the blow and a general rule of force, and proposes that in both one may use the same proportions that Pythagoras used in his music. The note frames the whole sheet as an attempt to bring force under proportional law.
Less-inclined supports bear more weight
Among supports of equal quality but different obliquity, the one that is less inclined bears more weight upon itself. The nearer the perpendiculars dropped from a beam's ends come together as it is made oblique, the more weight it bears on its middle. The figures carry the labels e d c b a, n m, t, S and numerical series such as 40 320 and 80 1280.
Finding a beam's load from a plumbed thread
If beam b a bears 100 pounds, look at the perpendicular thread b m: if the thread touches twice as much wood at n m as at b a, then multiplying the two threads (one times two makes two) shows that n m will bear 200. Of beams of uniform quality and different obliquity, the one touched by the greater sum of a plumbed thread is the stronger and more resistant.
Weight-lifting screw with an off-centre load
If the centre of the weight is attached outside the centre of the screw that moves it, the threads of the screw and of its nut are loaded from below on the far side and from above on the near side. The more the weight hangs to one side, the more of its weight it gives to the nut, in proportion as it is farther from the screw's centre; with c the distance of weight n from centre a and b the opposite contact, the excess load follows how many times a b enters c a.
Column of numerical working
A short arithmetical column accompanies the mechanical figures: 320, 4, 1280, 256, 1000/4000, 16, 64. The numbers echo the doubling series (40 320, 80 1280) attached to the beam figures.