One sentence, and the ground between its readings

The Treaty of Mesilla, signed in 1853, describes part of what is now the boundary between the United States and Mexico as running in a straight line from a point on the Rio Grande to a point on the Colorado. The words are clear, both endpoints are monumented, and there is no dispute about which two points are meant.

There is still no curve. On a sphere, “straight” is not a word with a single referent, and the treaty’s draughtsmen did not need to know that because the two candidates they might have had in mind differ by less than the ground they were surveying could resolve.

They differ by more than that. Between the two named points there are at least three curves that answer honestly to “a straight line”, and the widest pair of them run 7.53 kilometres apart.

Three thousand nine hundred square kilometres is not a rounding error. It is about the area of Rhode Island, and it is the whole of what a sentence four words long leaves undecided.

What the four readings are #

This site has already measured each of these curves for its own sake, in the field about routes. What is new here is putting them between the same two points and asking which one a sentence names.

The geodesic is the shortest path on the body, which on a sphere is an arc of a great circle. The shortest route is not straight is the paths field’s first rung and it is about this curve; on almost every map it draws as a bow.

The rhumb line holds one compass bearing the whole way. Why Mercator exists is the essay about the projection built so that this curve is straight on the page, and it is the reading a nineteenth-century surveyor with a compass and no astronomical control would most naturally have run.

The straight line on the sheet is whatever curve comes back when a ruler is laid on a particular map and the result is transferred to the ground. The line drawn straight on the page is a route prices it as a route; here it is a reading of a description, and it depends on which sheet. The figures in this essay use a Lambert conformal conic fitted to the segment, because that is what a boundary commission’s working sheet very often was, and because a cylindrical sheet’s straight line is the rhumb line exactly — which would make it a duplicate reading rather than a fourth.

The parallel of latitude exists only when the two monuments share a latitude, and then it is a fourth curve. It is also, on a sphere, the same curve as the rhumb line between those two points, exactly — a constant bearing of due west is a parallel — which is a coincidence of that case and not a general fact.

That last row is worth noticing. On this segment the geodesic and the conic sheet’s straight line are almost the same curve — sixteen metres apart over seven hundred and eighty kilometres — because a conformal conic fitted to a segment is very nearly the gnomonic construction that draws great circles straight. The ambiguity is essentially binary here: geodesic or rhumb, and nothing in the treaty chooses.

Why this is a practice question and not a paths one #

The four curves above all belong to the paths field, which owns them and has priced each one for its own sake. Nothing in this essay improves on any of those measurements, and the reason it is not a rung of that anchor is a difference in what is being measured.

The paths field asks which curve a traveller should follow, and every question it poses has an answer once the curve is chosen; its essays choose it, and the choice is made on purpose — shortest, or steerable, or quickest in a flow. The unit of measurement is the kilometre and the thing being priced is a route.

Here the curve is not chosen. The whole quantity is the SPREAD between readings of one description, the unit is the square kilometre, and the thing being priced is territory. That is this field’s own subject arriving on a boundary: a coordinate is the end of a chain of conventions and what a grid is made of is the same observation about a national grid — the conventions are real, they are decided by somebody, and they are not in the number.

The separating test is what a figure has to show. A route figure draws one curve on a map. A figure here draws several and shades what lies between, because a picture of this subject with one line in it is a picture of a decision somebody has already made.

The parallel, where the spread is a hundred times larger #

The Oregon Treaty of 1846 sets the boundary at the forty-ninth parallel of north latitude from the Lake of the Woods to the Strait of Georgia. That is a much more specific sentence than “a straight line”, and it produces a much larger spread.

The reason the spread is so much larger is not that the segment is longer, though it is. It is that a parallel is very far from being a geodesic, and a great circle between two points at the same latitude bows towards the pole. Its highest latitude satisfies tan φ_max = tan φ / cos(Δλ/2), which at 49 degrees over 28.17 degrees of longitude gives 49.8637 — and 0.8637 degrees is 96.04 kilometres.

Two exact zeros in a table of measured quantities are worth more than they look. A separation computed by comparing points at equal fractions of two curves’ lengths does not produce exact zeros, because two curves of the same shape sampled differently are not opposite each other point for point. Getting nil requires measuring each point of one curve to the nearest point of the curve rather than to the nearest sample, and the first version of the machinery here did not — it reported 225 metres on a segment where the answer is zero.

The geodesic bows towards the pole, and the treaty gained by it #

Which side the geodesic bows to is not a detail, because on a boundary the side decides who gains.

A great circle between two points at the same northern latitude passes north of the parallel joining them, everywhere between them. The reason is that a parallel is not a shortest path: to stay at constant latitude, a traveller heading west must continuously steer left, towards the pole they are circling — so the shortest path, which does not steer, drifts poleward. The great-circle vertex is the paths essay about the highest latitude a great circle reaches and gives it in closed form from the departure bearing alone.

So a boundary described as the forty-ninth parallel and realised as a geodesic between its endpoints would have put 130,972 square kilometres on the southern side of the line that the words put on the northern. Which is to say the choice of reading is worth a piece of ground the size of England, and its sign is fixed by the hemisphere.

That was never at issue on this boundary, because the line was run as a chain of short segments rather than as one long one, and the spread is quadratic in segment length. It is very much at issue on any boundary described as a parallel and realised in one go, and the two conventions differ by an amount that has to be computed rather than assumed small.

The segment that has no ambiguity at all #

That failure was found by the case this whole file is checked against, and it is the third real segment in the set.

The Anglo-Russian Convention of 1825 fixes part of the Alaska boundary at the meridian line of the 141st degree of west longitude. Along a meridian, the geodesic, the rhumb line and a straight line on a conic sheet whose axis is that meridian are all the same curve — a meridian is a great circle, and a constant bearing of due north stays on it.

A measurement with no case that must return zero is a measurement nobody can check. The meridian panel is that case, and it earned its place twice: once when the separation was measured to samples rather than to curves, and once when the conic sheet was drawn with its axis at Greenwich rather than at the segment — which put a straight line on the page 2.24 kilometres from the meridian and was entirely an artefact of the figure’s own defaults.

The list of readings is not complete, and finding that out cost one assertion #

Three or four readings is not obviously a complete list, and there is an argument that it is.

A curve between two named points is a reading of “the line between them” only if it can be constructed from the sentence alone. That rules out most curves at once: a river’s course, a watershed and a road are all perfectly good boundary descriptions and none of them is a reading of this sentence, because each needs the ground. What is left is the curves a rule plus two endpoints determine, and the sheet reading is a family rather than a single answer — one member per projection.

The argument for completeness is that the family is bounded by two of its own members. A gnomonic page draws every great circle straight, so its straight line is the geodesic; a Mercator page draws every rhumb straight, so its straight line is the rhumb. Every other page should then sit somewhere between the two, and the widest pair in the tables above would be a bound rather than a largest observed value.

So the argument is wrong, and it is wrong in the direction that matters. The two named readings do not bracket the sheet readings; they are two members of the family that happen to be nameable, and a page whose distortion is severe enough in the right direction puts its straight line outside them.

The mechanism is visible in which pages do it. The six are the cylindrical and pseudocylindrical projections with the strongest north–south compression — Gall–Peters, Lambert’s cylindrical, the plate carrée, Miller, Eckert IV and Robinson — and on a segment running west-north-west a page that squashes latitudes relative to longitudes bends the straight line further from the great circle than even the rhumb does. On the forty-ninth parallel, where the two endpoints share a latitude, not one page goes outside: every projection that draws parallels as straight horizontal lines reaches exactly 1.0000, because the parallel is the straight line on those pages.

This collection’s own gate on the file now asserts the finding rather than the argument. It requires that the gnomonic and Mercator ends be exact, that some page leave the envelope — a run reporting the envelope tight is a run reading the pages wrongly — and that the overshoot stay under six per cent. The wrong assertion would have passed on the forty-ninth parallel and failed on the Gadsden line, which is exactly why the file carries three segments rather than one.

And the readings differ in length as well as in place #

There is a second currency the spread is paid in, and it is the one a fence is costed against.

The two quantities do not rank the readings the same way and it is worth seeing why. The conic sheet’s straight line is a kilometre from the geodesic in place and a metre from it in length, because the two curves cross and the excess of one over the other cancels. The rhumb line is twelve kilometres longer and ninety-six from the parallel in place — except that the rhumb line and the parallel are the same curve, so on this segment the length difference and the position difference are measuring the same choice from two sides.

On the Gadsden line the numbers separate cleanly: the rhumb runs 193 metres further than the geodesic over 782 kilometres, a difference of 0.025 per cent, while running 7.53 kilometres away from it in place. Three hundred times more error in position than in length, on the same pair of curves.

What the numbers depend on that the sentence does not mention #

Every figure above is computed on a sphere of radius 6,371.0088 kilometres, and both of those choices are conventions the treaties do not make.

The radius does not matter much. Every separation here is a length on the ground and would scale with the radius; the site’s mean radius is 559 parts per million small for a distance along a meridian, which on a 96-kilometre separation is 54 metres and on a 7.5-kilometre one is four. Neither changes a conclusion.

The shape matters more. Geodesics on the ellipsoid shows that the shortest path on a flattened body is not a plane curve, has no closed form, and can be longer or shorter than the spherical answer depending on which way it runs. The 96-kilometre bow on the forty-ninth parallel would change in the third digit on WGS84 — the ellipsoidal geodesic between two points at the same latitude also bows poleward, and by a very slightly different amount, because the meridional radius of curvature exceeds the spherical one at that latitude.

What does not change is the ordering or the magnitude, and that is the claim this rung makes. The point is not that the geodesic bows by exactly 96.04 kilometres; it is that a sentence naming two monuments leaves a hundred and thirty thousand square kilometres undecided, and no refinement of the body’s shape closes that gap by more than a fraction of a per cent. The tolerance decides the model is the essay about choosing how much geometry a job needs, and by its rule a sphere is more than enough here: the quantity being measured is three orders of magnitude larger than the difference between the models.

Three sentences, three sizes of ambiguity #

Putting the three segments side by side gives the practical rule this rung is for, and it is not the rule a reader would guess.

A boundary along a meridian is unambiguous. Every reading of it is the same curve, exactly, and it stays exact for any length of segment. That is a property no other description has, and it is why so many colonial-era boundaries in Africa and North America are meridians — not because a meridian is easy to survey, which it is not, but because it is the one description that cannot be argued about once the longitude is agreed.

A boundary described as “a straight line” between two points is ambiguous by an amount quadratic in its length. On the Gadsden segment’s 782 kilometres it is 7.53. The two readings that matter are the geodesic and the rhumb, and no map sheet enters — the sheet’s straight line follows the geodesic to sixteen metres.

A boundary along a parallel is ambiguous by an amount an order of magnitude larger, because a parallel is not a shortest path and the departure is not a small correction to one. Ninety-six kilometres over two thousand, and the sign is fixed by the hemisphere.

And the sheet reading adds a fourth possibility to all three: a straight line on a page can fall outside both named readings, so “straight on whichever sheet was in front of the draughtsman” is not merely a third answer but one that can be more extreme than either of the two anybody would think to argue about.

That ordering runs opposite to how specific the three sentences sound. “The forty-ninth parallel of north latitude” is the most precise description of the three and leaves the most ground undecided; “the meridian line of the 141st degree” is a description of the same form and leaves none. The difference is entirely that one of the two coordinate curves is a geodesic and the other is not.

Why the ambiguity survived #

None of this was invisible to the people who drew these boundaries, and it is worth being careful about what they did and did not know.

A nineteenth-century boundary commission knew perfectly well that a parallel is not a great circle. What it did not have was a reason to write the distinction into a treaty, because the instrument that fixes a boundary on the ground is a monument, and a monument’s position is decided by an astronomical observation with an error of its own. An arcsecond of latitude is thirty-one metres, and a commission whose fixes were good to a few arcseconds could argue about a hundred metres and not about ninety-six kilometres — as long as the monuments were close enough together that the choice of curve between them did not matter.

That is the resolution, and it is the subject of the rung above this one. The treaty’s line is a description; the boundary on the ground is a chain of monuments; and where the two differ, the practice of the subject is that the monuments govern. The description sets out to name a curve, fails, and is superseded by an object that has no ambiguity in it at all.

That principle has a name in the law of boundaries — monuments control over courses and distances — and it is older than any of the geometry above. It exists because a monument is a thing a person can stand next to and a description is a thing a person can read differently, and the whole history of boundary disputes is a history of that preference being asserted. A published coordinate is a result makes the same distinction one level in: a coordinate is the output of a process and the mark in the ground is the thing itself.

What that leaves is a quantity worth measuring rather than a dispute worth having: how far the monumented line departs from the described one, which is a computable geometric question with a closed form and a square law. The forty-ninth parallel’s monuments do not sit on the parallel, they sit poleward of it, and by an amount that depends on nothing but how far apart they are and what latitude they are at.