Ground Truth · the standard re-derived from the real Earth
Short by a Hair
The metre was supposed to be exactly one ten-millionth of the distance from the North Pole to the equator. Measure that distance against the real Earth and the metre comes up about 0.2 mm short of its own founding definition — a gap narrower than a sewing pin, baked into every ruler ever made.
In 1791 the French Academy of Sciences gave the new unit a poetic anchor: a metre would be one ten-millionth of the meridian quarter — the arc from the pole, through Paris, down to the equator. Delambre and Méchain spent seven years (1792–1798) surveying the Dunkirk-to-Barcelona stretch of that arc to pin it down. But the Earth is not a perfect sphere, and the figure they extrapolated to was slightly wrong. So the metre we fixed forever in 1799 is not quite the metre the definition asked for.
Pull the slider below, or press use the real Earth, to drop in the true meridian quarter and watch the legal metre fail to close the gap. Every number is integrated live from the defining constants — nothing here is a canned animation.
Meridian quarter (assumed)
10,001,965.7 m
the real (WGS84) Earth
Metre this would define
1.0001966 m
= quarter ÷ 10,000,000
Legal metre is short by
0.197 mm
vs the 1799 metre (= 1.000000 m)
Slide to exactly 10,000,000 m and the gap closes — that is the metre the definition wanted. The real Earth sits ~1,966 m past it.
The stage above draws the two metres end to end at a magnification you can read — the legal metre (the bar locked into the National Archives in 1799, our 1.000000 m) against the ideal metre the assumed Earth would demand. At true scale the gap is invisible; the ruler magnifies it (by the factor printed on the stage) so the eye can referee what the readout already knows.
Whose fault is the hair?
Ken Alder's beloved history, The Measure of All Things, hangs the shortfall on human drama: the astronomer Méchain, tormented, hiding a fatal latitude error in his notebooks. The drama is real — but the documented error budget says his mistake is the smallest sliver. Most of the 0.197 mm is physics no one in the 1790s could have modelled.
Where the 0.197 mm comes from
A scholarly attribution estimate (geodetic analyses, via Alder 2002) — not an exact ledger. The bars are drawn to the published split below.
And then, in 1960, the metre let go of the Earth entirely — first pinned to a wavelength of krypton-86, then in 1983 to a fixed number: light travels exactly 299,792,458 metres per second, so a metre is simply the distance light covers in 1 ⁄ 299,792,458 of a second. The standard that began as a piece of the planet is now a story about light — and the 0.2 mm hair it was born with is frozen into it forever.
The check — every number recomputed in front of you
The meridian quarter is integrated live in your browser (composite Simpson, 4,000 steps) from the WGS84 defining constants — a = 6,378,137 m, 1/f = 298.257223563 — not read from a table. The shortfall falls straight out of it:
The dated facts, cross-checked offline (green = lands on the published / legislated value):
| fact | value | source |
|---|
Run it yourself: node research/short-by-a-hair/verify-short-by-a-hair.mjs — the same integral, two independent methods, plus the ligne and speed-of-light checks (22/22).
What is proven, what is model-dependent, and what is interpretation
Proven (recomputed here). Integrate the meridian arc of the WGS84 ellipsoid and the quarter comes to 10,001,965.7 m; dividing by ten million gives 1.0001966 m, so a metre defined that way would be 0.1966 mm longer — i.e. the legal metre is ~0.197 mm short. Two independent methods (Simpson integration and the third-flattening series) agree, and land on Wikipedia's published figure, which itself states "0.197 millimetres shorter." The 1799 metre was fixed at 443.296 lignes; the 1795 provisional value was 443.44 lignes — a gap of 0.144 lignes ≈ 0.32 mm. And c = 299,792,458 m/s exactly, by the 1983 SI definition.
Model-dependent. "0.197 mm" depends on which Earth you measure against. We use the modern WGS84 mean ellipsoid; a different ellipsoid or a geoid-based figure shifts the last digits. The original definition aimed at the quarter-meridian through Paris (the Dunkirk–Barcelona arc, extrapolated), not the WGS84 global mean — a real subtlety: the historical target and the modern yardstick are not exactly the same curve, so treat 0.197 mm as the comparison against today's best Earth, the cleanest public reference point, not as a recovered 18th-century quantity.
Attribution estimate, not a ledger. The 95% / 3% / <2% split is a scholarly estimate from later geodetic analyses (reported via Alder), not an exact accounting. We show it as the published apportionment; the headline that survives is the ordering — gravity deflection dominates, Méchain's measurement error is the smallest term.
Interpretation. Alder's reading that Méchain deliberately concealed a latitude error is a historian's interpretation of the notebooks, not a proven motive. We present it as interpretation. What is not in dispute is that whatever Méchain did or hid, it accounts for under 2% of the shortfall — the drama is true but incomplete.
The earlier corrections. The crisp digit-swap trap: the provisional/definitive gap is 0.144 lignes (≈0.32 mm), not 0.114 (which would be ≈0.26 mm). We use the legislated 443.44 → 443.296 figures and the Paris ligne (pied du roi / 144 ≈ 2.2558 mm).