Strength of beams: length, thickness and weight borne
A fan diagram and calculations show beam a b carrying 2560 lb, r S only 10
This sheet studies the load-bearing strength of wooden beams, arguing that carrying capacity rises steeply with thickness and falls with length. The large fan diagram and the number series 10–40–160–640–2560 show beam a b bearing 2560 pounds while beam r S bears only 10, and Leonardo reasons that halving a beam's length while doubling its thickness multiplies the weight it sustains. Columns of multiplication and division at lower right work these figures out, and a small inclined-beam sketch asks where to hang a weight so that both ends carry equally. Additional faint notes in the lower margins are not included in the transcription.
On this page
Beam a b bears 256 times more than beam r S
The note above the large drawing states that beam a b supports 256 times more weight than beam r S, because a b bears 2560 pounds while r S bears only 10. The fan of radiating lines expresses this proportional growth of load across a graded series of beams.
Beam i K bears 8 times the load through doubled thickness
Leonardo reasons that beam i K bears 8 times the weight of r S: were the beam only half as long it would bear 4 times more, and being compressed whole into the space of the half it is 'of doubled thickness', so if the half sustained 4, the double in thickness sustains 8. The number row runs 10 (r s a), p q, n o, l m, i k, g h, e f, c d, 40, 160, 640, 2560 (b).
Inclined beam a n: where to hang a balancing weight
The small inclined-beam sketch marks a beam a–1 2–n and states that a n carries half again as much weight at the foot as at the head. Leonardo asks where along the beam a weight could be hung so that end a carries as much as end n, and answers: hang it at the third from the head.
Worked calculations of the load figures
Columns of arithmetic at lower right multiply and divide the beam-load values, working 160 × 4, 640 × 4 and 2560 × 10, and producing intermediate figures such as 15360, 12800 and 5120. Small boxed pairs 8–4 and 2–4 record the doubling ratios discussed in the text.