Written in numbers, the archipelago rule still leaves eighteen

A straight baseline moves more water inward than sea outward measured the seventh article of the law of the sea, which lets a coast fringed with islands draw straight lines between chosen points and gives no length, no angle and no scale for the choice. On a stated coast the readings a state might adopt moved the internal waters between six hundred and four thousand six hundred square kilometres, and the phrase that bound — the lines must not depart appreciably from the general direction of the coast — was the one with no number in it at all.

The forty-seventh article governs the same act for a different kind of state. An archipelagic state, one made wholly of islands, may join the outermost points of its outermost islands by straight baselines and treat everything inside as archipelagic waters. Unlike the seventh, this article writes its tolerances down. No baseline may exceed a hundred nautical miles, except that up to three per cent of the baselines enclosing an archipelago may run to 125. The main islands must be inside. And the ratio of the area of the water inside to the area of the land must lie between one to one and nine to one.

The article is younger than the seventh and was fought for. The idea that an archipelago’s islands and the sea between them form one unit was pressed by Indonesia, whose declaration of 1957 drew straight lines round its outermost islands, and by the Philippines, and it entered the 1982 Convention as its own part, with the tolerances in the forty-seventh article as the price of acceptance by states that wanted the high seas between islands kept open. The numbers are negotiated numbers, and they were meant to bind.

A rule with numbers ought to choose far more narrowly than a rule with phrases. Whether it does, which of its numbers does the choosing, and how much the systems it allows still differ, are questions only a stated archipelago can answer, because every real one comes with its own negotiating history attached.

A stated archipelago and every way to close it #

The archipelago is sixteen islands, stated as ellipses on a plane in kilometres about its own centre. Three are large and lie near the middle; together they carry seven eighths of the land. Twelve small ones make a rough ring between 230 and 290 kilometres out, and one more, P, lies to the south-east beyond a long gap. None of it is any real archipelago, and the ratios and lengths are chosen to sit where the article’s numbers can be tested against them rather than to resemble any state.

A system is a closed chain of straight segments through basepoints on the islands, taken in order of bearing round the centre, one basepoint an island, at the island’s outermost point — the point farthest from the centre, which is where a state drawing the widest lawful system would put it. A chain may skip any island. It may pass through a main island’s outermost point instead of round it. It may not leave any part of a main island outside, and no segment may span half a turn, so every system is a star-shaped polygon about the centre.

Every such chain is enumerated: 8,799 of them. Each is then tested. Its segments’ lengths are measured in nautical miles. Its area comes from the triangles each segment makes with the centre, and the land inside from a one-kilometre raster of the islands counted triangle by triangle, so the ratio of water to land is exact to the raster for every system.

The ratio passes everything and the length test passes eighteen #

The ratio test passes all 8,799 systems. The water inside lies between 4.88 and 8.04 times the land whichever islands a system uses, comfortably inside one to nine at both ends. On this archipelago the ratio is a test nothing fails.

The length test passes eighteen. It is the only number the article writes that removes anything here, and it removes 99.8 per cent of what the geometry allows. A hundred and two more systems have their longest segment between a hundred and 125 nautical miles and would pass if the exception could reach them. It cannot, for a reason that has nothing to do with this archipelago and is taken up below.

The shape of the distribution explains why. Neighbouring islands of the outer ring are between 69 and 87 nautical miles apart, so a system that visits every one of them keeps every segment under a hundred — the widest lawful system does exactly that — while a system that skips any ring island has to span two gaps in one segment, 140 miles or more, over even the exception’s ceiling. Almost every combination the geometry allows skips at least one, which is why the typical system’s longest segment is 150 to 200 miles. Skipping is not even profitable: the systems with a segment over a hundred enclose less water on average than the eighteen without, 166,593 square kilometres against 173,010, because an outer island left out takes its bulge of the ring with it. The shortest longest segment any system can have is 86.6 nautical miles, the gap from L to M, which every system spans somewhere.

So on a compact archipelago with plenty of land, the article’s ratio is idle and its length does all the work. That is the first answer to whether the numbers choose, and it is a partial one: they choose, and one of them does all of it.

The eighteen still differ by twenty-eight thousand square kilometres #

The eighteen are not interchangeable. The widest runs round the outer ring through twelve islands — G, H, I, J, K, L, M, N, O, D, E and F — and encloses 186,617 square kilometres of water. The narrowest turns inward three times, to the outermost points of the main islands C, B and A, and encloses 159,032. Between them the article allows a choice of 27,585 square kilometres, 17 per cent of the smaller system, and the other sixteen fill the range in steps set by which main islands a system turns in to and which small ones it passes.

A state would draw the widest, and the widest here is also the most natural, following the ring. But the ordering of the eighteen matters for a different reason. Every one is lawful, so the article cannot be used to object to any of them; a neighbour contesting a system that turns out to the ring cannot point to a number it breaks. On this archipelago the numbered rule narrows the choice from 8,799 to eighteen and leaves among those eighteen an area more than a sixth the size of the smallest. That is not a defect in the drafting so much as the ordinary condition of a rule meant to be usable: a criterion worth using is one whose answer is not unique found the same of the criteria that choose a projection, and a rule that picked exactly one system would pick it for the state rather than let the state choose.

Two of the article’s words are not tested here and would narrow it further. The lines must not depart to any appreciable extent from the general configuration of the archipelago, which is the seventh article’s phrase for a coast, and a line has a length only at a scale is the reason no number can be read out of it without first choosing the scale the configuration is to be read at. And basepoints must be on the outermost islands and drying reefs, which a system that turns in to a main island’s outermost point arguably is not. Neither is a number, and the question here was what the numbers do.

A hundred nautical miles, measured how #

A limit written as a number moves the argument to how the number is measured, and the article does not say. A nautical mile is 1,852 metres exactly, but a segment’s length depends on the surface it is measured on, and the difference is not small against the margins here.

On the ellipsoid the Earth’s surface curves by two radii at every point, and the radius of curvature is two numbers found them 42.70 kilometres apart at the equator, where most archipelagic states lie. A segment computed as exactly a hundred nautical miles on a sphere of the Earth’s mean radius measures 99.44 on the ellipsoid if it runs north and south near the equator and 100.11 if it runs east and west. The spread between those, two thirds of a mile, is most of the 0.9 mile by which the outlier’s second segment exceeds the limit from K’s outermost point.

The basepoints themselves are coordinates, and a coordinate belongs to a datum. Datum shifts dwarf projection errors measured the difference between a local datum and a global one in hundreds of metres, and a basepoint published on one and the segment computed on another moves each end by that much. None of this reopens the eighteen, whose longest segment is 86.6 miles, but it decides the segments within a mile of the limit, and those are exactly the segments a state drawing the widest lawful system is driven towards. One sentence and the ground between its readings found a boundary treaty’s “straight line” admitting at least three curves; a baseline’s hundred miles admits at least three lengths.

Thin the land and the ratio decides instead #

The ratio passes everything here because the land is large: the three main islands carry most of it and sit inside every system. An archipelago with less land, or with its islands spread further apart, is a different test.

The two tests bind in different regions and rarely together. Spacing decides the length test. Pull the outer islands in by a fifth and 120 systems keep every segment within a hundred miles; move them out by a fifth and none does, whatever the land. Land decides the ratio. Halve every island’s area and the eighteen systems the length test allows have water between 12.4 and 14.6 times the land, every one over nine, and the archipelago cannot close at all. At seven tenths of the stated land a single system passes. At the stated land, eighteen.

The region where both matter is narrow. Pulled in by a fifth with the land halved, the length test allows 120 systems and the ratio removes 25 of them — the ones enclosing the most water, 123,097 square kilometres on average against 110,808 for those it keeps. That is the only place in the sweep where the two numbers share the choosing.

The lower bound decides nothing anywhere in the sweep. Doubling every island’s area brings the lawful-length systems down to between 2.97 and 3.20 to one and no further, because the basepoints stay on the outer islands and the water between them grows with the ring rather than with the land. For the ratio to fall below one, the land inside a system would have to exceed the water inside it, and that happens only when the outermost islands are themselves the large ones, close together, with little sea between them. That is a different geography: a few large islands with narrow channels rather than a scatter round a centre. The bound exists to exclude it, and no arrangement of a ring of small outer islands can reach it.

So the article’s three numbers answer to three different properties of an archipelago. The hundred-mile limit answers to spacing, the nine-to-one ceiling to how thinly the land is spread, and the one-to-one floor to whether the land is concentrated at the edge. An archipelago is tested by at most one of them for most shapes, and which one depends on the shape before any line is drawn.

The exception is paid for in segments #

The exception is the article’s most specific number and it does something the drafting may not have intended. Three per cent of the baselines, rounded down, may run to 125 nautical miles. A system of thirty-three segments has 0.99 of a segment’s allowance and rounds to none. A system can use the exception at all only if it has thirty-four segments or more, and two long segments need sixty-seven.

No system here has more than sixteen. The 102 systems whose longest segment lies between a hundred and 125 miles are unlawful not because their long segment is too long but because they have too few short ones.

The outlier P shows what that means in practice. It lies beyond a gap of 122 nautical miles from J and 101 from K, measured from the islands’ outermost points. A system through P encloses up to 199,257 square kilometres of water — 12,640 more than any fully lawful system — and needs two long segments, so sixty-seven segments in all. Moving the basepoints on J and K round to face P shortens the second gap to 95 miles, under the limit, and leaves one long segment at 114. Now thirty-four segments will do. The system through P has fifteen, counting the two short segments along J’s and K’s own coasts that the moved basepoints need.

The other nineteen can be had for nothing. A basepoint may be any point on an island’s low-water line, so a state can place a run of basepoints a few kilometres apart along the seaward side of any island and join them, and each short segment counts toward the total exactly as a long one does. Nineteen short segments along a coast enclose no extra water and move no line anybody would notice, and they buy the exception that buys 12,640 square kilometres.

Archipelagic states that have declared their baselines generally have many segments, and the reasons are mostly geographical — a large archipelago has many outer islands. The arithmetic is the same whatever the reason. A rule that grants an allowance as a share of a count rewards a larger count, and the cheapest segments to count are the shortest.

How the areas were checked #

Two ways to the same area. A system’s area from its triangles about the centre must equal the shoelace area of the same polygon. For the first system enumerated, the middle one and the largest, they agree to a part in a million.

Two ways to the same land. The land inside a system counted triangle by triangle must equal a direct count of the raster cells inside the polygon by ray casting. They agree within the cells along the polygon’s edges, which the two methods assign to different sides.

The exception’s arithmetic. One long segment must need 34 in all, two 67 and three 100. Three per cent of 34 is 1.02 and of 33 is 0.99; of 67, 2.01; of 100, exactly 3.

Every variant counted on a comparable raster. When the archipelago is spread outward the raster is coarsened in proportion, so every variant is counted on about the same number of cells and no cell of the two-way sweep is decided by resolution.

Where the stated archipelago stops #

One basepoint an island. A real state places basepoints anywhere on an island’s low-water line and uses several on one island. With one each, a system is a choice of islands, and 8,799 is a count of those choices rather than of every lawful polygon, which is uncountable. The outlier’s price shows that moving a basepoint round an island can bring a segment under the limit, so the eighteen are a lower bound on the choice: other basepoints on the same islands would widen it.

Islands as ellipses on a plane. The islands are smooth, the plane is flat and the distances are straight lines. Over 580 kilometres the difference between a plane distance and a geodesic on the Earth is under a part in a thousand, a tenth of a nautical mile on the longest segment here — enough to decide a segment that misses the limit by less than that, as the second gap to P nearly does.

No drying reefs, no atolls. The article counts the water of atolls and the land of steep-sided plateaus in special ways, and lets drying reefs carry basepoints. A reef between two islands is a basepoint in the middle of a gap, and would change the length test’s answer exactly where it binds.

The unnumbered tests are untested. The general configuration and the outermost-islands requirement are words, and either could remove some of the eighteen. What they would remove depends on how they were read, which is what the seventh article’s measurement found about its own words.

Still open: whether the choice can be priced from the outside #

The eighteen lawful systems differ by 27,585 square kilometres, and a neighbour whose own waters abut the archipelago can object to none of them on the article’s numbers. What it can do is measure. Every system moves the line from which the archipelago’s territorial sea and exclusive economic zone are then measured, and where an archipelagic system faces a neighbour’s coast, the choice among the eighteen moves the median line between them — a line which, as an equidistance line belongs to a surface found, is itself only defined once the surface it is measured on is named.

The sea is measured from the baseline outward, so the same arithmetic that measured a territorial sea from a tide applies from the other side: a segment pushed outward moves the outer limit outward by less than it moves the baseline, and the difference depends on the curvature of what lies behind. Whether the 27,585 square kilometres inside become a comparable amount of sea outside, how much of it lies on the side facing a neighbour, and whether the system a state would choose for the most water inside is also the one that moves the median line furthest, are questions the water inside the lines cannot answer on its own.