Indexing the archive…
Your Universe of Digital Possibilities
Greenhill, 1881: a column buckles under its own weight at a height proportional to its diameter to the two-thirds. Grow a tree from a seedling to a giant and choose how its trunk thickens — as a photocopy, as McMahon’s elastic similarity, or for constant stress. The silver line beside it is the height at which it would bow and fall; real trees, McMahon found, stop at about a quarter of it.
A slender column carries its load straight until Pcr, then bows sideways all at once. Stiffness E·I grows as d⁴, so doubling the length quarters the load a column of the same girth can bear.
A column standing under its own weight w = ρgπd²/4 per metre buckles when w·h³/EI reaches 7.837. Tall and thin fails first: a trunk twice as thick may stand only 22/3 ≈ 1.6 times as tall.
Grow so that every size sits the same fraction below its buckling height and the trunk must thicken as h3/2 — between the geometric twin (d ∝ h) that buckles and the stress rule (d ∝ h²) that never needs to.
With Δ = 2 the cross-section is conserved at every fork, so the branches of a tree squeezed together would make one trunk. Murray’s Δ = 3, the rule for vessels pumping fluid, thins the crown faster.
The Lemuel grammar: a quantity as a power of body size. On log–log axes every allometric law is a straight line whose slope is the exponent b — fitted, on every instrument of the edition, from the sizes you visit.
The seventh size is The Bone’s argument stood upright: a femur fails by crushing, a trunk by buckling, and both lose to weight because stiffness and strength grow slower than mass. Euler found the critical load of a column in 1744; Greenhill let the column carry itself in 1881; McMahon saw in 1973 that real trees choose elastic similarity — the shape that keeps the margin constant. Leonardo’s branching is the tree’s answer to wind, as Eloy showed in 2011, and the same area-keeping split drives The Heart’s vessels.