A straight baseline moves more water inward than sea outward

The sea is measured from a tide, and the headlands decide how much measured a territorial sea from the coast as it is. The low-water line was a height, the tidal datum moved it, and the headlands, standing furthest out, decided almost everywhere how far the twelve-mile limit moved in turn. The limit listened to the heads of bays only where a bay was wide enough for its own shore to be the nearest land.

The same convention allows a second way of drawing the line the sea is measured from, in which the shore between chosen points stops mattering at all. A state whose coast is deeply indented, or fringed with islands, may join appropriate points with straight lines and measure outward from those. This essay measures what that is worth on a stated coast, and finds that the answer depends much less on the geometry than on three numbers the article does not give.

The seventh article, and the three numbers it does not give #

The rule has a precise origin. Norway’s coast is a fringe of thousands of islands, rocks and reefs, the skjærgaard, and a royal decree of 1935 drew its fisheries limit from straight lines joining the outermost of them. The United Kingdom challenged it, and in December 1951 the International Court of Justice upheld the method in the Anglo-Norwegian Fisheries case, with a condition that became the rule’s central phrase: the drawing of baselines “must not depart to any appreciable extent from the general direction of the coast.” The Geneva Convention of 1958 copied the Court’s reasoning into its fourth article, and the United Nations Convention on the Law of the Sea repeated it in 1982 as its seventh.

In its present form the article lets “the method of straight baselines joining appropriate points” be used “in localities where the coastline is deeply indented and cut into, or if there is a fringe of islands along the coast in its immediate vicinity”. The lines “must not depart to any appreciable extent from the general direction of the coast”, and the sea inside them “must be sufficiently closely linked to the land domain to be subject to the regime of internal waters”. Lines may not be drawn to or from a low-tide elevation unless a lighthouse or something like one stands on it.

Three things a surveyor would need to draw the lines are not there. No length. A segment of the seventh article may be as long as the state likes; the archipelagic baselines of the forty-seventh article are capped at a hundred nautical miles, with a few at up to a hundred and twenty-five, but no such cap applies here, and segments of more than a hundred nautical miles have been drawn and protested. No angle. “Any appreciable extent” is not a number of degrees. No scale. A coast’s general direction is a direction over some stretch of coast, and a stretch of ten kilometres and one of three hundred give different directions wherever the coast bends.

Commentators and the protests of other states have filled the gaps with figures that recur: segments no longer than twenty-four nautical miles, twice the breadth of the territorial sea and the width the tenth article allows across the mouth of a bay; and departures of the order of twenty degrees. Those are readings, not law. Each can be taken as a parameter and swept, and the sea each decides can be priced — which is what one sentence, and the ground between its readings did for a land boundary, and what the rest of this essay does for a line drawn at sea.

A segment buys sea as the cube of its length #

Start with one segment and nothing else. Two points stand some distance apart, and the territorial sea measured from them alone is the union of two discs twelve nautical miles in radius. Between the points, seaward, the edge of that union dips in a notch where the two circles meet. Join the points with a straight baseline and the limit becomes a straight line twelve miles out from it: the notch fills.

How much sea that is has a closed form. The band within twelve miles of the segment on its seaward side is a rectangle of the segment’s length L times the breadth R, with a quarter-disc at each end; the two points already held a half-disc each. Once the segment is at least 2R long the discs do not meet at all, and the gain is $LR - \pi R^2/2$. Below that it is the same expression plus half the lens the two discs share, and expanding it for a short segment gives

$$\Delta A \approx \frac{L^3}{24R}.$$

The cube is the useful part. A segment of ten kilometres adds under two square kilometres; twenty adds fifteen and a half; thirty adds fifty-five. The rate at which a kilometre of segment buys sea is $R - \tfrac12\sqrt{4R^2 - L^2}$, which is zero for a very short segment, 2.98 square kilometres a kilometre at twelve nautical miles — thirteen per cent of the full rate — and exactly the full breadth, 22.2, at twenty-four. Beyond twenty-four miles the circles no longer touch, every kilometre of line buys the same strip, and the gain is linear.

So the twenty-four-mile figure is not an arbitrary round number from the point of view of the sea. It is the length at which a segment stops filling a notch and starts manufacturing a strip. Below it, a state gains little by joining points; above it, a state gains the full breadth of the sea for every kilometre, over water no disc from any point would have reached.

Land behind the gap takes most of it back #

A fringe of islands has a mainland behind it, and the mainland has a limit of its own. If the shore is d kilometres behind the islands, its limit stands R − d beyond them, and every part of the notch deeper than that is already territorial sea. The segment can only claim what lies between the two, so its gain can never exceed L·d, whatever its length.

For islands five kilometres off a straight shore, a twenty-four-mile segment adds 126 square kilometres rather than 212. At ten kilometres off it adds 183, and at twenty the mainland is too far back to matter and the gain is the full 212. A fringe close inshore is precisely the kind of coast the seventh article was written for, and it is also the kind of coast on which straight lines add least sea at the outer edge.

A fringe of islands, sixty-eight square kilometres out and nine hundred and eighty-eight in #

No real coast is used here, because every real straight baseline system already carries a state’s choices and the question is what the choices are worth.

The stated coast runs for two hundred and twenty kilometres as a long gentle wave, ten kilometres in amplitude and a hundred and eighty in period, so that its direction depends on how much of it is looked at. Six islands stand between five and twelve kilometres offshore, from six to fourteen kilometres long. Two small rocks stand much further out, thirty and twenty-six kilometres. Every island and rock is permanently above water, so none is excluded as a low-tide elevation.

A reading of the article is three numbers: a longest segment, a largest departure from the general direction, and the stretch of coast the general direction is read over. For each reading, every chain of straight segments from mainland to mainland that keeps to the three numbers and never crosses land is a permitted system, and the one taken is the one enclosing the most water — the most a state could claim under that reading. The sea that system yields is then measured on a grid of hundred-metre squares: land, the open sea a flood from offshore reaches past land and lines, the internal waters it does not reach, and the territorial sea within twelve nautical miles of the lines and of every stretch of shore the open sea still touches.

Under the reading of twenty-four miles and twenty degrees, with the general direction taken over the whole coast, the most generous system is the one in the first figure: six segments, the longest 44.4 kilometres, running from the mainland along the seaward faces of four of the inshore islands and back. It turns 988 square kilometres of water into internal waters, twelve per cent of the 8,122 square kilometres of territorial sea the coast has measured from its shore. At the outer edge it adds 68, under one per cent.

The two numbers are different kinds of thing. The 68 is new: sea that was beyond the limit and is now inside it. The 988 was already territorial sea, the water between the islands and the mainland, and what changes is its regime. Internal waters are treated as the land is, and a foreign ship has no general right to pass through them; the eighth article softens this for waters enclosed by straight baselines that had not been internal before, where the right of innocent passage survives. On this coast, and on any fringe close inshore, the seventh article’s main work is to change the status of water the state already had, not to extend how far its sea reaches.

The test of direction is the one that binds #

The rocks are what the readings are really arguing about. A chain that reaches a rock thirty kilometres out and comes back has to leave the coast’s direction at a steep angle twice, and whether it may do so is the whole question of whether the rocks are basepoints.

With the length held at twenty-four miles, widening the angle from five degrees to forty moves the internal waters from 618 to 1,479 square kilometres and the sea added from 29 to 86. Each step lets a segment follow the wave a little more closely or jump a little further between islands, and each is worth tens or hundreds of square kilometres of internal water and a few of sea.

Then at forty-five degrees the picture changes in one step. A chain can now run out to the nearer rock and back, the internal waters jump by 785 square kilometres and the sea added more than triples. At fifty degrees it reaches both rocks: 2,960 square kilometres enclosed and 463 added. Past that point widening the angle changes little, because nothing further out remains to reach.

That is the same shape as the drying bank at the end of the essay on the tide: a continuous parameter, a smooth response while the same features are in play, and a jump the moment a new basepoint becomes admissible. There the switch was a crest clearing low water by a centimetre. Here it is an angle crossing a threshold nobody wrote down.

A length cap with nothing beside it holds nothing back #

The length cap looks like the stricter of the two readings, because it is a number where the angle is a phrase. Measured, it is the weaker.

With the test of direction in place, lifting the length cap from twenty-four miles to anything at all takes the sea added from 68 to 121 square kilometres and the internal waters from 988 to 1,066. The most generous chain within twenty degrees never uses a segment longer than seventy kilometres, however long a segment is allowed to be. The angle is holding the system in, and the length cap has almost nothing left to do.

Take the angle away and the length cap alone holds almost nothing in. At twenty-four miles a chain may zigzag: out from the mainland to a rock in one segment under forty-four kilometres, straight back in the next, and out again. That system adds 448 square kilometres and encloses 2,939. Lift the length cap as well and three segments do it directly, the longest 118 kilometres: 1,834 square kilometres of sea added, and 4,655 of water enclosed, an area more than half the size of the whole territorial sea measured from the shore.

The second map is what the Court’s condition exists to prevent. Its lines have nothing to do with the coast: they are a polygon round two rocks that happen to stand offshore. And it is also what a pure length cap, the one reading that looks like a number, would permit if the lines were short enough and numerous enough. On this coast the test of direction is the one that binds, and the length cap only matters once it has.

The general direction has a scale that nobody wrote down #

Every departure above was measured against one general direction, the trend of the whole coast. The article does not say that. A coast has a direction at every scale, and on this one the direction over a short stretch follows the wave, turning through nearly twenty degrees each way, while the direction over the whole coast is almost level.

The direction used for each segment is the least-squares line through the mainland shore over a stretch of W kilometres centred on the segment. Each value of W is a defensible reading of “the general direction of the coast”, and none is written anywhere.

Read over ten kilometres, the direction follows the wave, and so may the lines. The most generous system then has eight segments, and although several of them depart from the local direction by seventeen to nineteen degrees, every one stays inside twenty. It encloses 1,473 square kilometres. Read over the whole coast, the direction is the overall trend, the system has six segments, and none of them departs from that trend by more than thirteen degrees; it encloses 988.

Between the two the answer moves in steps rather than smoothly. For every stretch up to a hundred and twenty kilometres it is the same system to within ten square kilometres. At a hundred and thirty-five and a hundred and fifty, one segment on the wave’s flank fails and the system has seven, enclosing about 1,390. From a hundred and sixty-five it is 988 for every stretch there is. The last switch sits just short of the length of the wave itself, which is the natural place for it and cannot be read off the article: a coast’s general direction is a property of the scale it is looked at, and on this coast the scale decides a third of what the lines enclose.

The sea added, meanwhile, hardly moves: between 67 and 83 square kilometres whatever the stretch. The scale of the general direction decides regime, not extent, because every system it permits stays close in.

It is the same problem a line has a length only at a scale found for a coastline’s length, arriving from the other side. A coast’s length grows without limit as the ruler shortens; a coast’s direction does not grow, but it stops meaning one thing as soon as the coast bends at more than one scale. A tolerance is a promise about the picture makes a line’s simplification a statement about a stated tolerance. The general direction of the seventh article is a simplification of the coast with the tolerance left out.

Four readings of one article #

Set side by side, the readings sort into two pairs, and the pairing is by the angle rather than by the length. Both readings with the test of direction enclose about a thousand square kilometres and add about a hundred. Both without it enclose several thousand and add several hundred or more. Within each pair, the length cap moves the result by a factor that is modest when the angle is held and large when it is not.

In every reading the water converted to internal waters exceeds the sea added at the outer edge. The ratio is fifteen for the strictest reading and two and a half for the most permissive, and the direction of that trend has a reason. A strict system stays close to the coast, where the mainland’s own limit already covers most of what a segment could add, and its effect is almost entirely on status. A permissive one reaches out, where the mainland’s limit no longer covers the notches, and its effect on the extent of the sea grows. The more a system departs from the coast, the more of its effect is new sea rather than reclassified water, which is exactly the effect the Court’s condition was written to restrain.

How the areas were checked #

The grid measures the sea in hundred-metre squares, and three controls stand behind every number above.

A straight coast of two hundred and twenty kilometres with no islands must have a territorial sea of its length times the breadth, 4,889 square kilometres, and no reading may give it internal waters. The grid reads 4,884, a difference of 0.1 per cent.

One segment between two tiny islands far out to sea must add, on its two sides together, twice the closed form. At twenty-four nautical miles the closed form gives 424.0 square kilometres and the grid 422.0; at sixty kilometres, 1,115.2 and 1,111.6. Both are within half a per cent. The limit measured from a segment is taken from the segment’s exact distance rather than from the grid squares it passes through; measured from the squares, the limit stood up to a square seaward of the line along its whole length, and on the fringe that added four square kilometres to the 68.

Halving the squares to fifty metres moves the strictest reading’s sea added from 67.7 square kilometres to 66.7 and its internal waters from 988 to 982: one and a half per cent and half a per cent. Every value quoted is at the hundred-metre grid.

One caution belongs with the choice of system rather than with the measurement. The chain taken for each reading is the one enclosing the most water, and that is not the same as the one adding the most sea at the outer edge; a state choosing lines to maximise its territorial sea’s extent could do slightly better on the second number. The internal waters are a maximum, and the sea added is what the maximum-water system yields.

What the picking is worth #

An equidistance line belongs to a surface found forty kilometres between readings of one word, and a tripoint defined three times three answers to where three boundaries meet. The seventh article adds a boundary drawn by choice, and the choice has a price that splits into two currencies. In sea added at the outer edge, the readings with a test of direction are worth under two per cent of the territorial sea; the readings without it, up to more than a fifth. In water whose regime changes, the readings range from an eighth of the territorial sea to more than half.

The single most consequential reading is the least numerical one. The angle of departure decides whether the outer rocks are basepoints, and the scale at which the coast’s direction is read decides whether two segments on the flank of a wave are legal. Neither is fixed by the article, and both matter more on this coast than the twenty-four-mile cap that is the one reading usually quoted as a number.

The other boundaries measured here were all lines on land or lines measured from a shore as it is: the line a commission can actually run, a meridian boundary moves when its datum does, a river boundary goes where the river goes, or stays where it was and the highest summits stay where they are, and the watershed does not. Every one of them had a feature on the ground that the words referred to. A straight baseline refers to no feature between its ends. It is the first boundary here that is a line drawn by choosing points, and the first where the ground between readings is not a spread around some object but a list of objects that one reading admits and another does not.

Still open: the baseline rule that is written in numbers #

The seventh article leaves its three numbers out. The forty-seventh, for archipelagic states, puts them in: no segment over a hundred nautical miles except for three per cent of them, which may reach a hundred and twenty-five; the lines must include the main islands; and the ratio of water to land inside them must lie between one to one and nine to one.

That is a rule with the tolerance written into it, which makes it the natural test of the question this measurement raises. A rule with numbers ought to pick out far fewer systems than a rule with phrases. Whether it does — how many distinct archipelagic systems satisfy the forty-seventh article on a stated archipelago, how different the sea they enclose can be, and whether the ratio test binds where the length test does not, as the angle bound here where the length did not — is a question the seventh article, which gives no number to hold a system against, cannot ask.