Artificial Wasteland · physical

The Island Under the Horizon

A Carolinian navigator fixes his position by the bearing of an island he cannot see. Compute that bearing on the real atolls and the real stars and two things fall out: the star compass is an exact ruler of the sky at the equator and stretches as the tangent of your latitude, and the rule of thumb Gladwin recorded in 1970 about why reference islands must not be too close is a cotangent, written in prose. Then a crossing two ethnographers counted differently, settled by the islands themselves.

There are 45.8 nautical miles of open Pacific between Satawal and West Fayu, and nothing on them to look at. A navigator trained in the Carolinian school can tell you, at any point on that crossing, how much of it is behind him. He does it by keeping track of a third island. Lamotrek lies 35.9 miles off to one side of the course, far below the horizon, and it stays invisible for the whole passage. As the canoe runs north the bearing of that unseen island swings backward through the named points of a star compass, and how far it has swung is how far you have come.

The technique is called etak. This page is not an account of it: better ones exist, they were written by people who sailed with the navigators, and they are listed at the foot. What follows is an attempt to compute the geometry the technique sits inside, from the surveyed positions of the actual atolls and the catalogued declinations of the actual stars, and then to put that arithmetic against two things the ethnographers wrote down. One of them it confirms exactly. The other it does not.

One. The horizon is a ruler, and it is exact in exactly one place

A star rises in the same place every night. Not the same time, but the same place: pick a notch on the horizon and a given star will come out of the sea there, tonight, next month, and in twenty years. That is the fact a sidereal compass is built on, and it is why a ring of star names can do the work a magnetic card does.

Where on the horizon depends on two things: the star's declination, which is how far north or south of the celestial equator it sits, and your own latitude. The relation is one line of spherical trigonometry, and on the equator it collapses into something startlingly plain. Writing A for the bearing at which the star rises, measured round from north, and d for its declination:

A = 90° − d   (on the equator, exactly)

A star twenty degrees north of the celestial equator rises exactly twenty degrees north of east. The horizon, seen from the equator, is an undistorted ruler laid along the sky: every degree of declination is one degree of bearing, with no scale factor and no correction. All 32 points of the compass drawn below sit at exactly zero degrees of that prediction when the observer is on the equator, which is to say the identity is not approximate.

Move off the equator and the ruler stretches. Here is the whole compass, and a latitude you can drag. The faint marks are where each point sits on the equator; the bright ones are where it sits at the latitude you have chosen.

At Satawal, where this compass is used, nothing has gone far. The worst-displaced point on the whole ring has moved 1.79 degrees from where it would sit on the equator, and that point is the one nearest the pole, where the geometry is always weakest. Sail to Saipan, the northern limit of the documented Carolinian voyaging range, and the worst point has moved 11.28 degrees. Carry the same ring to Honolulu and 2 of its points have stopped rising altogether. Carry it to Auckland and 8 have gone, and the worst survivor has moved 12.7 degrees, which is more than a house and a half.

Why it holds still where they sailed

The rate at which a compass point slides as you change your latitude has a closed form, and it is short enough to be worth looking at:

dA/dφ = − cot(A) · tan(φ)

Two things fall straight out of it, and both are visible in the instrument above. The first is that the whole expression is zero when φ is zero. On the equator every point on the compass is stationary in latitude, to first order: not merely small, but exactly zero rate of change. The Caroline Islands sit between about 6° and 10° north, where tan(φ) is still around a tenth, so the instrument is being used in the flattest part of its own error surface. Not because anyone chose that. Because that is where the islands are.

The second is that the rate also vanishes when A is 90°, which is to say due east. A star sitting on the celestial equator rises due east from everywhere on Earth, at every latitude, with no exception and no approximation. In this compass that star is Mintaka, the westernmost of the three in Orion's Belt, which sits 0.30 degrees from the celestial equator.

David Lewis noticed the pattern from the other end. Sailing with the navigators in the late 1960s he wrote that not all the star bearings shift equally with latitude, "and certain ones not at all, namely Polaris, the Southern Cross, and the east and west bearings of Orion". The formula says why his three are his three. Polaris is not a rising point at all but the pole itself. Orion's Belt sits on the celestial equator, so cot(A) is nearly zero. The Southern Cross is used upright as a southern marker rather than by its bearing, which takes it out of the question entirely: as a rising point it is in fact the worst-behaved thing on the ring, and the instrument above will show you it going first.

Why the points bunch near east, and a third answer nobody needed

The ring above is conspicuously uneven. Its narrowest gap is 1.8 degrees and its widest is 26.6, against the 11.25 they would all be if the horizon were cut into equal houses. (It is worth saying plainly that it is not: the equal-house compass with 11.25° sectors is the Hawaiian one, designed by Nainoa Thompson in the 1970s. The Carolinian points sit where the stars are actually seen to rise, and the two get conflated constantly.)

Lewis noticed the unevenness has a shape: "in easterly and westerly directions the points are crowded together, while further north and south they are spread widely apart". Two explanations are on record and they contradict each other. Gladwin thought the crowding "reflects the greater demands for accuracy that are placed on the navigation system as a whole by longer east-west passages". Lewis rejected that on the same page and proposed instead that the east-west stars are simply the easiest to use.

There is a third possibility neither of them raises, and it is the one that arithmetic can actually test: that the horizon does the crowding. If the projection from sky to horizon squeezed things toward east all by itself, then evenly spaced stars would produce a bunched ring and no cultural explanation would be needed at all.

It does not. At Satawal the map from declination to bearing is stretched by a factor of 1.0084, which is to say almost not at all, and the consequence is measurable: every gap between adjacent compass points reproduces the gap between those stars' declinations to within 0.88 degrees, with a median discrepancy of 0.12. The compass's spacing is the stars' spacing, passed through undistorted. Aquila's three points sit 1.8 degrees apart on the horizon because Tarazed, Altair and Alshain sit that far apart in the sky.

Which does not settle the argument, but it does clear a suspect. The crowding is inherited, not manufactured, so any explanation has to be about which stars were chosen. Gladwin's answer and Lewis's answer are both of that kind. The third one, which would have made theirs unnecessary, is ruled out. Drag the instrument to Auckland and watch the agreement fall apart, from under a degree to more than six: the identity is a fact about being near the equator, not a fact about horizons.

The compass cannot tell you which hemisphere you are in

Latitude enters the relation only through cos(φ), and a cosine does not care about sign. So the compass reads identically at 15° north and 15° south. Not nearly identically. The largest disagreement anywhere on the ring, at any latitude tested, is 0 degrees, which is zero to the last bit a double can hold.

This is a property of the instrument and not of the sky, and the difference matters. The sky is perfectly capable of telling you your hemisphere: star altitudes differ enormously between +21° and −21°, by more than forty degrees for some of these stars, and reading latitude off a star's height is exactly what Carolinian and Hawaiian navigators do. It is the ring of bearings, specifically, that is blind. Drag the instrument above from +20 to −20 and watch nothing happen.

Two. The shape Gladwin wrote down without writing it down

Now the second instrument, and the reason the reference island has to be chosen carefully.

Put the canoe on a straight track and an island off to one side of it. Let d be how far the island lies off the track and s how far along the track you are, measured from the point of closest approach. The angle θ between your bow and the island then satisfies a cotangent, and the distance you cover per degree of bearing change is its derivative:

s = − d · cot(θ)   →   ds/dθ = d · csc²(θ)

That second expression is smallest when θ is 90°, where it equals d exactly, and it grows without bound at both ends of the passage. In plain terms: the bearing of an off-track island changes fastest when the island is abeam and slowest when it is nearly ahead or nearly astern. So if you cut a voyage into equal bearing steps, the pieces of track they correspond to are long at the start, short in the middle, and long again at the end.

Thomas Gladwin, who spent 1967 on Puluwat, put it this way:

If the reference island is too close, it passes under many stars, dividing the journey into a lot of segments. Worse, the segments are of very unequal length. They start out rather long (slow) and then as the canoe passes close by, they become shorter (fast) as the reference island swings under one star after another, and then at the end they are long again, a confusing effect. A distant reference island has the opposite effect making the segments approximately equal, but so few in number that they do not divide the journey into components of a useful size. Thomas Gladwin, East Is a Big Bird (1970), p. 187, as quoted in Edwin Hutchins, Cognition in the Wild (1995), p. 76

Long, then short as you pass close by, then long again, and the effect stronger the nearer the island. That is d · csc²(θ), described in words, by a man reporting what navigators told him about which islands make good references. The function is in the paragraph. Nobody wrote it as a function because nobody needed to.

The bar under the chart is the same passage cut into its etak, drawn to scale. The pieces are visibly unequal, and they are unequal for two reasons at once, which is worth separating because only one of them is Gladwin's.

The first reason is the cotangent: the bearing swings fastest abeam. The second is that the star points are not evenly spaced, because stars are not evenly spaced. Divide each piece by the width of the house it crosses and the cotangent comes out clean: on the Satawal to West Fayu crossing the fastest swing, 1.16 km of track per degree of bearing, is in the piece that contains the moment Lamotrek passes abeam, exactly as the closed form requires. The raw lengths do not order that way at all. The shortest piece by raw length is somewhere else entirely, because a narrow house happens to sit just past the beam.

Three. Two accounts of one crossing, and what the islands say

The ethnographic record contains a crossing described twice, by two people, with two different answers. Both describe Satawal to West Fayu with Lamotrek as the reference island. Both give the same sequence of star points. They stop at different places.

Part of the gap is bookkeeping, and it is worth clearing before the rest. The two accounts are not counting the same objects. Gladwin's other example, the one below, describes a reference island moving "from the setting Little Bear to the rising Cassiopeia, so dividing the voyage into four segments": five named points with four gaps between them, so for Gladwin a segment is an interval. Thomas's sequence instead starts at a point and names six more, so his six are transitions. Those two conventions differ by one on the same sweep before anybody disagrees about anything.

Take the two measurable things next. Finney, reporting Thomas, puts Lamotrek "thirty-five miles west of the course line"; the atolls' surveyed positions put it at 35.9, which is as close to exact as a sentence in a book chapter gets. The leg itself is 45.8 nautical miles. Gladwin's "a little over 40 miles" is close; the "fifty-five-mile crossing" in Finney's account of Thomas is not.

Now the disagreement. Gladwin's seventh etak requires Lamotrek to swing all the way to the setting point of the Southern Cross. It does not get there. On arrival at West Fayu the island bears 213.2°, and the setting Southern Cross is at 205.9°, leaving 7.3 degrees of sky that the passage never covers. Setting Shaula, where Thomas stops, is passed comfortably.

That would be a cheap result if it hung on a coordinate. These are low atolls a few kilometres across and gazetteers disagree about where their centres are, so the calculation was run again with all three islands displaced independently to every point within a disc, half a million combinations at each radius:

At two kilometres, which is larger than any disagreement between the two gazetteers consulted here (the worst was 1.29 km, once one false hit was thrown out), the answer is not close: the sweep ends more than four degrees short. At five kilometres it still holds, by a quarter of a degree. Only at ten kilometres, a displacement larger than any of these atolls, can the sweep be made to reach the Cross. So the geometry supports the shorter count, and it does so robustly, but the honest version of the claim is narrow: it settles where the bearing ends, not what a navigator called it.

Which is the useful shape of the result. The method is not being audited and it does not need auditing; it worked, for centuries, across an ocean. What two published paragraphs disagree about is a detail of transcription, seventy years and two languages downstream of the navigator who was actually counting, and the islands themselves still hold the answer.

What is contested, and what this page is not entitled to say

The passage above is arithmetic and the arithmetic is checkable. The interpretation of etak is not settled, and it would be dishonest to let a page full of computed numbers imply otherwise.

The famous framing is that the canoe stands still and the islands move. Lewis records it plainly, and he also records the navigators being wry about it: Ulutak insisted Lewis could never know his position at sea without grasping that the islands move, then said, laughing, "of course we know that islands stay in the same place". Gladwin wrote that he "would certainly not suggest that they believe the islands actually move". Hutchins argues that the moving frame is a genuine computational choice rather than a figure of speech, because updating the islands against the canoe and the stars is cheaper than the reverse. He also argues, against both Gladwin and Sarfert before him, that the etak segments are not units of distance and that reading them as units is a Western assumption smuggled in with the diagrams.

And Vicente Diaz, a Pohnpeian scholar who trained under the Polowatese navigators Soste and Manny Sikau, objects to the whole reflex of reassuring the reader that the islands are not really moving:

Incidentally, western observers who have encountered this sensation in their studies of Carolinian navigation have been quick to point out that, of course, the islands are not actually moving, but that this is a cognitive operation. But I want to assert emphatically that the islands are moving, tectonically, as well as culturally and historically. Vicente M. Diaz, "Hypo-modernity: Traditional Carolinian Navigation as Critique and Aesthetic", lecture, SUNY Binghamton, October 2001, pp. 11–12

That is a live disagreement between serious people and this page does not adjudicate it. It is worth noticing, though, what the plan-view diagrams above are doing. Hutchins and Finney both point out that drawing etak from overhead, as Gladwin, Lewis and Sarfert all did and as instrument two does, is already a translation: it takes a scheme built entirely from the navigator's own horizon and redraws it from a vantage no one in the canoe has. The chart is the Western reading, made legible. Keeping that in view is part of reading it honestly.

Three smaller things that popular accounts routinely get wrong, all of them checked here:

• The reference island is not visible. On all three passages above it stays at least 66 km away, against a geometric sighting range of about 20 km for an eye 1.5 m above the water looking for palms 20 m tall. Hutchins and Finney both state it outright; the arithmetic agrees.
• Etak segments are not equal, and were never meant to be. That is the whole content of the Gladwin paragraph quoted above.
• "Etak" probably does not mean "refuge". That gloss descends from Sarfert's 1911 Notinsel, "emergency island". Gladwin rejected it: reference islands are often uninhabited, hard to approach, or bare reefs, and Riesenberg later found that some are not islands at all but agreed phantoms. No dictionary entry for the word was found to settle the etymology either way, so it is left open here rather than repeated.