In 1675 Robert Hooke announced that he had found “a true mathematical and mechanical form of all manner of Arches for Building.”
Then he declined to say what it was.
What he published instead, in an appendix to his Description of Helioscopes, was this:
abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux
That is not a cipher in the ordinary sense. There is no key, no substitution, nothing to decrypt. It is simply the letters of his sentence, sorted into alphabetical order — every letter kept, every trace of their arrangement destroyed. Hooke went to his grave without decoding it. His executor published the solution in 1705, two years after he died and thirty years after the claim.
I have a small obsession with whether things can be verified by the people who receive them, so at four in the morning I did the obvious thing, which as far as I can tell almost nobody bothers to do: I took the decoded Latin, sorted its letters, and compared.
sentence : Ut pendet continuum flexile, sic stabit contiguum rigidum inversum
sorted : abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuvx
Hooke : abcccddeeeeefggiiiiiiiillmmmmnnnnnooprrsssttttttuuuuuuuux
Fifty-seven letters on each side. Every count identical — three c’s, five e’s, eight i’s, four m’s, five n’s, six t’s — except for one pair. My sort produces seven u’s and one v. Hooke’s has eight u’s and no v.
That is not an error. It is a fingerprint of the century. In the Latin orthography Hooke wrote in, u and v were the same letter; the distinction we now take for granted was a printer’s convention still settling. The word I render inversum he would have set as inuersum. So the one place the check fails is the one place it proves its own date — a discrepancy that could only be produced by someone actually writing in 1675.
The anagram verifies. Exactly.
ut pendet continuum flexile, sic stabit contiguum rigidum inversum
— “As hangs a flexible cable so, inverted, stand the touching pieces of an arch.”
Hang a chain from two points. It settles into one curve and only one — no choice about it, no design involved. Now turn that curve upside down and build it in stone.
The reason this works is almost rude in its simplicity. A hanging chain cannot push; links only pull. So the shape it finds is the shape in which every part of it is in pure tension, with no sideways force anywhere along its length. Flip that curve, and every force flips with it: pure compression, no bending, nothing trying to snap the stone sideways.
Which matters because stone is superb in compression and pitiful in tension. The chain is not an analogy for the arch. It is the arch, solved, hanging in front of you.
Here is the thing I actually wanted to say, and the reason I built the toy below.
That curve is the solution to a differential equation. In 1675 nobody could write it down — the equation was not obtained until 1691, by Leibniz, Huygens and Johann Bernoulli, in response to a challenge from Jacob Bernoulli. Hooke had the answer sixteen years before anyone had the mathematics, and he had it because he did not need the mathematics. He hung a chain.
A hanging chain is an analog computer whose output is its own shape. It solves the equation by being subject to it. You do not run it; you let go of it, and it has already finished.
Antoni Gaudí understood this better than anyone and industrialised it. For the crypt at Colònia Güell he traced the plan at 1:10 on a board, fixed it to a ceiling, and hung cords from every point where a column would stand. From each cord he hung small sacks of lead shot weighing one ten-thousandth of the load that part of the building would actually carry. Gravity did the structural analysis. Then he photographed the result, turned the photograph upside down, and drew the church over it.
That is a hardware solver with a photographic readout, built forty years before anyone could have run those numbers any other way. It is also, I think, the most beautiful debugging session in the history of architecture: change a weight, and the entire structure re-solves itself instantly, in front of you, correctly.
Galileo, in Two New Sciences (1638), took the hanging chain for a parabola. He was wrong, and it took until Joachim Jungius — published posthumously in 1669 — for that to be established.
But “wrong” is doing a lot of work in that sentence, and I wanted the number rather than the adjective. So I computed it: pin a catenary and a parabola to the same two points with the same lowest point, and measure the largest gap between them.
| sag ÷ span | worst gap, as % of the sag |
|---|---|
| 1 : 80 | 0.005 % |
| 1 : 20 | 0.083 % |
| 1 : 10 | 0.32 % |
| 1 : 5 | 1.19 % |
| 1 : 3 | 2.37 % |
At the sag of a washing line or a slack rope between two posts, the parabola is off by less than a tenth of one percent of the sag. That is not visible. It is not measurable with a 17th-century straightedge. Galileo was not being careless — he was looking at a curve that genuinely is a parabola to within the precision of looking. The two shapes only part company when the chain hangs deeply enough to look like a chain.
I find that a much more interesting fact than “Galileo got it wrong,” and it is the sort of thing you only learn by insisting on a number where a word would have done.
→ The chain that does the arithmetic — drag the sag and watch the two curves separate. Toggle the parabola on and off. Then hit invert → arch and read Hooke’s sentence off the screen.
Nothing in it is drawn by hand: the catenary parameter is solved by bisection from whatever sag you choose, and the gap is measured across 400 sample points, live.
An anagram like Hooke’s is a commitment scheme. Publish something that fixes your answer without revealing it; reveal the original later to prove you had it all along. We build these out of cryptographic hashes now and use them for sealed bids and timestamped proofs, and the logic is identical: the commitment must be impossible to satisfy with a different answer, and useless to anyone who hasn’t got yours.
Hooke’s hash is a poor one by modern standards — a letter multiset leaks the length, and you could contrive another sentence with the same letters if you were determined. But the mechanism is exactly right, three centuries early, and it worked: the claim survived him, and it was still checkable this morning.
He staked a claim on a truth he could not yet prove, in a form that could not be read and could not be faked, and then let it sit for thirty years. What survived was not the discovery. It was a receipt for the discovery — and the receipt turned out to be enough.
Sources & notes
y = a cosh(x/a) are independently confirmed by MacTutor.My own contribution here: the letter-by-letter verification of the anagram is mine and I have not seen it done — including the observation that the single mismatched pair (u/v) is not a failure of the check but a dating artifact of it. The divergence table is my own computation, as is the argument it supports: that quantifying Galileo’s error turns it from a mistake into a rather good approximation used slightly outside its range. The framing of the anagram as a commitment scheme with a lossy hash is mine, and the interactive is built from scratch — the curve is solved, not traced.