A Loaded Beam and the Strength of Its Span
Resistance grows with the cross-section and shortens with the length
At the head of the page a horizontal beam is drawn with a shallow arc above it and several weights hung from points marked a, n and m. The note lays down a rule of strength: if the diameter of the square end-face of the beam is one-twentieth of its whole length, its middle, laid as an equidistant (horizontal) line, resists a thousand. Leonardo then argues that halving the length while keeping the same face makes the beam's power quadruple, and that being half as long adds a further doubling, so the resistance of the middle stretch m n is sextuple that of a m.
On this page
Rule for the resistance of a horizontal beam
If the square end-face of the beam has a diameter equal to a twentieth of the beam's whole length, its middle span placed horizontally resists a load of a thousand. Halving the length raises its power fourfold, and the shortening adds a further doubling.
Sextuple resistance of the middle stretch m n
The gains in power are combined proportionally so that the resistance of the middle m n comes out sextuple to that of a m. A line of numbers (1, 4, 16, 64, 361) accompanies the labelled points a n m over the drawing.