Ground Truth · 2026-08-11

The Line Drawn With a Ruler

In 1977 a court of arbitration fixed 170 nautical miles of the seabed boundary between Britain and France by laying a straight edge on a Mercator chart. Britain's hydrographers replied that a straight line on that chart is not straight on the Earth, and that the faithful line lay about four nautical miles further south. Both figures are recomputed here from the award's own coordinates. The larger finding is underneath them: the award defines its boundary as the bisector of two lines whose direction it never finishes specifying.

On 30 June 1977 five judges settled where the continental shelf of the United Kingdom stops and that of France begins. Most of the boundary runs up the English Channel, and most of it was uncontroversial. The last stretch was not. West of the Isles of Scilly the two coasts stop facing each other and start running side by side, and the Scillies, sitting twenty-one nautical miles out into the Atlantic, drag the dividing line a long way south if they are allowed to count in full.

The tribunal decided they should count half. It borrowed a construction from state practice: draw the line equidistant between the two coasts without the islands, draw it again with the islands, and put the boundary between the two. Then it did something a mathematician would not have done, and a navigator would have found obvious. It drew the answer as a single straight line on an Admiralty chart.

The method of giving half effect consists in delimiting the line equidistant between the two coasts, first, without the use of the offshore island as a base-point and, secondly, with its use as a base-point; a boundary giving half-effect to the island is then the line drawn mid-way between those two equidistance lines. Decision of 30 June 1977, paragraph 251

Everything on this page comes from that award and its 1978 sequel, printed in volume XVIII of the United Nations Reports of International Arbitral Awards. The three rocks that fix the Atlantic segment, the turning points, the two bearings and the final bisector were read off page images rather than machine-extracted text, because the volume's own text layer confuses zero with O and one with I. Nothing here is taken on anyone's word: the numbers are recomputed, and the recomputation disagrees with the award in places that turn out to be the interesting ones.

What the award says the boundary is

The Technical Report annexed to the decision gives the construction in full. Two equidistance lines, each defined by exactly two rocks:

Then the arithmetic. The angle between the two directions is halved, and the half is added to the southern one. That arithmetic is exact, to the arcsecond, and it is the only part of the construction that is.

The trouble is in the two numbers the arithmetic starts from. Each of them is called the direction westwards of the course followed by one of the equidistance lines. But an equidistance line between two points on a round Earth is not straight. Its bearing changes continuously along its own length. Asking for its direction is like asking for the direction of a river.

The instrument: choose where to measure, and watch the boundary move

Below is the South-Western Approaches. The two equidistance lines are computed live in your browser from the award's three rocks, not traced from a picture. Drag the slider to choose a point along line (a); the instrument takes the true bearing of each line at the corresponding meridian, halves the angle between them exactly as the Technical Report does, and draws the boundary that follows. The award's own line stays fixed in amber for comparison.

Two things are worth doing with it. First, slide the reference point from one end to the other and watch the readout: the direction of a single named line runs through nearly three degrees, and the boundary that follows from it sweeps a fan several nautical miles wide at the far end.

Second, switch the projection. Do not trust your eye for this one: over 170 nautical miles the bend is about three per cent of the line's length, which is real and nearly invisible. So the instrument lays a straight edge between the same two endpoints on whichever chart is showing, and measures how far each candidate line departs from it. On Mercator the award's boundary is the straight one and the great circle sags away to the south. Switch to the gnomonic chart, on which great circles are straight lines by construction, and the two measurements trade places. Neither line moved. Only the paper changed, and the paper is what decided which line was called straight.

Britain's two numbers, recomputed

In September 1977 the British Embassy in Paris handed the French Ministry of Foreign Affairs a note saying the tribunal's expert had drawn the boundary with the technique of constant bearings on a Mercator chart, and that this was not the line the tribunal had described in words. The note carried two quantities, and both are checkable.

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Both reproduce. The separation between a rhumb line and a geodesic leaving Point M on the same bearing, measured where the award puts its terminal Point N, comes out at just under five nautical miles with the geodesic to the south, against a note that said approximately four to the south; and continuing to a two hundred mile run gives just over six, against a note that said about six. The hydrographers were right, and they were right by a margin that their own hedging words covered.

The four lines the award's own words allow

The award describes its boundary three times, and the three descriptions are three different constructions. Paragraph 251 calls it the line drawn mid-way between the two equidistance lines. Paragraph 254 and the Technical Report call it the line which bisects the area formed by them. The Technical Report then computes it as the bisector of the angle at Point M. Mid-way is itself two readings, because a curve can be halfway between two curves by latitude or by distance. Here is where each lands, at the meridian of the award's own terminal point:

They are spread across six nautical miles: more than the discrepancy Britain went back to court about. The angle bisector is not the area bisector, and it does not become one on a charitable reading, because the award never says where the region between the two lines is supposed to stop. Cut it at the meridian of Point N and the area bisector runs a degree and a half off the award's line; cut it at a constant distance from M instead and it runs a degree off. And the vertex is wrong in a smaller way as well: the award bisects the angle at Point M, but the two lines it is bisecting actually cross somewhere else.

The award audited against itself

The Technical Report publishes seventeen turning points, and for each one it names the two or three rocks that the point is equidistant from. That is a redundancy, and a redundancy is a test: the rocks determine the point, so every published point is a claim that can be checked without any outside information at all. All seventeen are checked below, under five different readings of the word distance.

Two figures are given for each. The spread is the difference between the largest and smallest distance from the published point to its own defining rocks; it is zero for a point that is truly equidistant, and it is well behaved. The residual is how far the published point sits from the position its definition requires, which is the more intuitive quantity but inherits the conditioning of the intersection: where two of the rocks are close together the residual is amplified, which is why point D3 has a residual of three and a half nautical miles on a spread of one and a quarter.

The result is not a smear. It is two populations.

Ten of the sixteen constructed points are equidistant to a precision the award cannot even print: their worst error is smaller than the rounding forced by quoting coordinates to the arcsecond. The expert's trigonometry was, for those points, exact. The other six are wrong by half a nautical mile to nearly two, and they are wrong because they were never meant to be right. The Technical Report says so on its own first page, in a sentence that is easy to read past:

The actual course of this boundary is determined by construction lines of equal length drawn from opposing points on the two coasts. It is, in principle, a median line, but is one adjusted in certain respects, relatively unimportant, so as to form a more convenient, simplified, boundary based on the true median line. Expert's Technical Report, paragraph (3)

So the audit recovers, and measures, a simplification the award admits to in words and never quantifies. The largest of those relatively unimportant adjustments moves a boundary turning point by about a nautical mile and a half of its own defining geometry.

Which geometry the expert actually used

The same seventeen points answer a question the 1978 court answered differently. Asked whether the two Atlantic equidistance lines were true equidistance lines, the court replied that they had been calculated on Mercator projection, and that the divergence Britain complained of related exclusively to the scale error inherent in that projection. If the expert really worked in the plane of the chart, the published points should sit closer to plane-chart solutions than to solutions on a curved Earth. They do not, and the comparison needs no points set aside to make the case:

Spherical trigonometry fits an order of magnitude better than the plane of the chart, over all sixteen points with no selection at all. Restrict to the ten the expert did not simplify and the fit tightens to under a hundredth of a nautical mile, roughly one second of latitude, while the plane-chart reading gets no better. The two published bearings tell the same story: read as the true spherical bearing of each line at the meridian of its own British base-point, they come back to within eighteen and eighty arcseconds; read as constant bearings in the plane of the chart they are out by twenty-five and ten arcminutes.

That reconstruction also recovers something the award simply omits. Line (b) is defined as equidistant from Le Crom and Runnelstone, south of Land's End, and the award never states Runnelstone's coordinates: not in the decision, not in the dispositif, not in the Technical Report, though every other base-point it uses is given to the arcsecond. Half of the construction that fixes 170 nautical miles of international boundary cannot be rebuilt from the award alone. Supplying the actual rock's position from an independent source reproduces the award's published bearing for that line to eighty arcseconds, which is how we know the rock the award meant is the rock of that name.

What the court decided, which is the part worth sitting with

Britain asked the tribunal to interpret its own decision so that the boundary followed the geometry rather than the chart. On 14 March 1978 the tribunal declined. Its headnote is unusually blunt:

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It is tempting to read this as a court failing at arithmetic, and that reading is wrong. Nothing in the 1977 numbers is careless: the arithmetic is exact, ten of the turning points are computed to the limit of their own printed precision, and the two bearings are recoverable today from three rocks and a sphere. What the tribunal refused was the premise that a maritime boundary is the sort of object that has an exact geometric answer waiting to be found. It had already said that equidistance is applied in practice in a somewhat qualified manner, that boundaries are routinely simplified, and that simplification is what its own expert had done.

And the measurements on this page support the refusal rather than undermining it. Britain's four nautical miles were real. But the award's own words leave the direction of the boundary undetermined across nearly three degrees, which is a fan nine and a half nautical miles wide at Point N before any question of projection arises; and the three verbal definitions of the boundary pick out lines six nautical miles apart. The precision Britain was asking the definition to deliver was never in the definition. You cannot correct a line to a tolerance finer than the words that define it, and the four miles sit inside that tolerance.

Which leaves the thing that is genuinely strange, and true. Somewhere west of the Scillies there is a line on the seabed that decides which of two states may drill. It is not the line equidistant from the coasts, and it is not the line half-way between two such lines, and it is not any line a geometer would derive. It is the line that was straight on one particular sheet of paper, in Mercator's projection, and it is straight nowhere else. The map did not represent the boundary. The map was the boundary.

Show the check

Artificial Wasteland · this layer · every figure recomputed from the award's own coordinates · colophon