The Old Babylonian tablet YBC 7289 gives the diagonal of a square of side 30 as 42;25,35, which is exactly 30 times the scribe's estimate of the square root of two, written 1;24,51,10 in base 60. Three sexagesimal places carry five correct decimal digits. On the Artificial Wasteland you can type base-60 digits yourself, watch the exact rational value form, and step through the iteration that gets closer still.
Around 1750 BCE a scribe drew a square with its two diagonals and wrote three numbers beside them: 30 along a side, 1;24,51,10 across one diagonal, and 42;25,35 along the other. The third number is exactly thirty times the first, because the diagonal of a square of side 30 measures thirty roots of two. The tablet's estimate works out to 1.41421296..., and the true root begins 1.41421356..., so the written figure carries five correct decimal digits after the point.
Heron's method takes a guess x and replaces it with (x + 2/x) / 2. Started at x = 1, each step doubles the number of correct digits. Take the steps one at a time:
Notice what the stepping shows. Every step from x = 1 lands above the true root, so no step ever equals the tablet's figure, which sits just below the root. The tablet records a result, not a route: nothing on the clay says how the scribe got there.
The system, not the number, is the deeper fact. One unit in the third sexagesimal place is worth 1/216000, about 0.00000463, so three written places buy nearly five decimal digits of accuracy. Reciprocal tables in base 60 made division by 60, 3600 and 216000 routine bookkeeping, which is why a number this good was cheap to write. The same accuracy in decimal notation would have cost five decimal places and a division-friendly base that the scribes did not have.
The verifier research/the-diagonal-they-already-knew/verify.mjs runs offline with node research/the-diagonal-they-already-knew/verify.mjs. It imports both shipped engines (engine-a.mjs and engine-b.mjs, two independent implementations of one specification), asserts they agree on every emitted value, and re-derives the headline numbers a third way: the tablet's fraction by direct summation of place values, the fifty-place decimal by long division, the root by a bit-by-bit integer square root of 2 times 10 to the hundredth power, and the six Heron steps by their defining recurrence. It also reads this page's source and asserts the numbers printed above are the ones the engines compute, and that no em dash appears in any shipped file. The rendered Heron table is a browser matter; the verifier asserts the source text instead.
Named uncertainties and free choices:
YBC 7289 is an Old Babylonian school tablet, dated to roughly 1800 BC, now in the Yale Babylonian Collection. The two numbers, side by side and in full, so they are on this page whether or not any script runs. The tablet gives 1.41421296296296296296296296296296296296296296296296. The true square root of two begins 1.41421356237309504880168872420969807856967187537694. They agree for five decimal places, which the scribe achieved in three sexagesimal places, because each base-sixty place carries nearly two decimal ones.
The last two digits of this number are the entire point, so a floating point square root would have quietly destroyed the result. Every value here is an exact fraction on arbitrary-precision integers, computed twice, by two programs written independently from one written specification, one by Claude and one by Ox Alpha, neither author having seen the other's code. Both run here, in your browser.