The line a commission can actually run

The rung below this one leaves a boundary description with four readings 96 kilometres apart and no rule for choosing between them. The rule the subject actually uses is not geometric: where a described line and a monumented line differ, the monuments govern.

That resolves the dispute and replaces it with a measurement. If the boundary is the chain of monuments, then the question is how far the chain departs from the words — and unlike the choice between readings, that question has a closed form.

A commission cannot run a parallel. It can occupy a monument, determine its latitude astronomically, sight a straight line to the next monument, and clear the intervening ground. A straight line between two points of the same latitude is a geodesic, and a geodesic between two points at the same northern latitude passes north of the parallel joining them, everywhere between them.

The closed form, and the factor it was missing #

The geodesic’s highest latitude between two points at latitude φ separated by Δλ satisfies tan φ_max = tan φ / cos(Δλ/2). Expanding for small Δλ gives a rise of R·sin φ·cos φ·(Δλ/2)²/2, and substituting Δλ = s/(R cos φ) for a chord of ground length s collapses the whole thing to

offset = s² · tan φ / 8R

Quadratic in the monument spacing, with the latitude entering only through a tangent, and with the radius in the denominator. At 49 degrees a chord of 20 kilometres rises 9.03 metres above the parallel at its midpoint.

That last sentence is the one worth carrying out of the figure. A ladder built to check an exponent will not catch a wrong constant, because the constant divides out of every ratio the exponent is fitted from. The error survived until the closed form and the sampled geodesic were required to agree row for row rather than merely to share a slope, and it would have survived a great deal longer if the check had been a plot rather than a comparison.

Why the geodesic goes poleward, and why that decides who gains #

The direction of the bow is not a detail. On a boundary it decides which side the disputed ground falls on, and the direction is fixed by the hemisphere rather than by anything a commission chooses.

A traveller holding constant latitude while heading west is not going straight. To stay on a parallel they must steer continuously towards the pole they are circling, because the parallel curves away from a straight path — the great-circle vertex puts it the other way round and gives the highest latitude a great circle reaches in closed form from its departure bearing. A geodesic does not steer, so it drifts poleward, and the drift is the offset.

In the northern hemisphere that puts every scallop north of the described line. On the forty-ninth parallel that is ground the treaty’s words assign to Canada and the marks assign to the United States, at every one of the twelve hundred and eighty-four scallops. In the southern hemisphere the sign flips and the gain goes the other way, for the same reason and by the same formula, since tan φ is negative below the equator.

That systematic sign is what makes the total area worth computing at all. If the departures alternated, they would cancel in the aggregate and only the largest single one would matter. They do not alternate, so the areas add.

Where on the Earth it costs most #

The tangent is the whole of the latitude dependence, and a tangent is a violent function near ninety degrees.

That is the opposite of where such boundaries were mostly drawn. The parallel boundaries in the world are at 49 north between Canada and the United States, at 60 north between the Northwest Territories and the provinces, at 22 north across the Sahara, at 26 north across Namibia — and the northern ones are the long ones. The description is doing its worst work exactly where it is most used.

The reason is not carelessness about geometry. A parallel is attractive as a boundary precisely where the ground is unsurveyed and featureless, which correlates with high latitude and with desert, and in both cases the alternative was not a better description but no description.

The spacing at which the geometry becomes visible #

Everything above is a computable departure from a described line. The question that decides whether any of it ever mattered is whether the people running the line could see it.

The instrument that fixes a monument’s latitude is an astronomical observation, and its error is quoted in arcseconds. An arcsecond of latitude is 30.89 metres on the ground. So the comparison is between the scallop and that.

The actual monument spacing on the forty-ninth parallel boundary is of the order of a mile in cleared country, and at 1.6 kilometres the offset is fifty-eight millimetres. That is five hundred times below an arcsecond and about ten thousand times below the accuracy anybody was working to. The whole scallop over the two-thousand-kilometre boundary encloses 0.077 square kilometres — not quite eight hectares, spread over one thousand two hundred and eighty-four separate lens-shaped slivers each a few centimetres wide.

So the geometry this rung is about was, on the boundary it is most famously about, entirely invisible. That is not a reason to stop measuring it. It is the reason the boundary is not in dispute, and it is a different fact from the one a reader would assume: the marked line agrees with the described line not because a commission was careful but because the scallop is quadratic and the spacing was small.

What the scallop is not #

Three things get confused with this offset and none of them is it, and separating them is most of what makes the number usable.

It is not a convergence effect. Further north on the grid is not further north is about grid north departing from true north, which is a property of a projection and reaches degrees rather than millimetres. The offset here has no projection in it: it is a comparison between two curves on the ground, and it would be the same if nobody had ever drawn a map.

It is not a misclosure. A traverse must close is about a chain of measurements failing to return to its own starting point, and the gap it measures is the accumulation of observational error. The scallop accumulates nothing. Each chord’s departure is a property of that chord alone, it returns to zero at every monument, and running the line twice as carefully changes it by not one millimetre.

It is not a flat-earth approximation. How small is flat enough prices the extent at which a plane survey stops being adequate, and the answer is in hundreds of kilometres. The offset here exists at every spacing, is exact rather than approximate, and is a difference between two curves both computed on the curved body.

What it is, is the price of the one operation a survey can perform that the description does not name. A treaty says “the parallel” and a party can sight, level and clear. Sighting produces a geodesic. The gap between what was asked for and what can be done is the whole of the quantity, and it is why this rung belongs in a field about what a coordinate is the end of rather than in a field about distortion.

What the area actually comes to #

The offset is a length and the thing at stake is ground, so the two have to be connected. The area between the chorded line and the parallel is the integral of the offset along the boundary, which for a chain of equal chords is the number of scallops times the area of one — and since the offset is quadratic in the spacing while the number of scallops is inversely proportional to it, the total area is linear in the spacing.

A linear law is a much more useful design rule than a quadratic one, because it says the total is decided by a single choice with no threshold in it. Halve the spacing, halve the disputed ground. There is no spacing below which the disagreement vanishes and none above which it becomes catastrophic; it is simply proportional, all the way down.

The same shape, one field over #

The chorded parallel is an instance of something this collection has met twice before under other names, and noticing that is worth a paragraph because it says which of the three numbers here is the general one.

A curve approximated by a chain of straight segments departs from it by an amount quadratic in the segment length, always, whatever the curve. Flying a curve in straight legs measures it for a great circle flown as constant-heading legs and finds the gap falling as the square of the number of legs. A polygon inscribed in a circle departs from it by r(1 − cos(θ/2)), which is the same expansion. The scallop here is the third instance, and the factor that distinguishes it is the curve’s own geodesic curvature.

That is the general statement: the departure of a chord from a curve is (curve’s geodesic curvature) × s² / 8, and everything specific to this rung is in evaluating the geodesic curvature of a parallel, which is tan φ / R. At the equator that curvature is zero — the equator is a geodesic — so the offset vanishes there, exactly, which the latitude figure shows as the curve running into the origin.

So the rung is not really about parallels. It is about boundaries described by a curve that is not a geodesic, and the parallel is the case that occurs in treaties. A boundary described as “the watershed” or “the line of equal distance from two coasts” has a geodesic curvature of its own, generally larger and not constant, and the same s²/8 applies to it — which is a thread the fourth rung of this anchor picks up.

What this does not explain #

The forty-ninth parallel boundary as marked does not lie exactly on the parallel, and the scallop is not why. The monuments are where nineteenth-century commissions determined the parallel to be, and the determination carries the errors of the instruments and the methods of the day — several tens of metres in places, which is three orders of magnitude larger than the fifty-eight millimetres the chord geometry contributes.

That is worth stating plainly because it inverts the usual expectation. A reader meeting the scallop for the first time assumes it is the reason a boundary wanders; it is not, and on this boundary it is the smallest of the effects in play. The wander is observational, and what a tape measures is the same distinction one level down — a measurement is the output of an instrument with its own corrections, and the geometry between measurements is usually the cleanest part of the chain.

The two contribute in different ways as well. Observational error is random between monuments, so it does not accumulate along the line and its effect on the total disputed area is small and unsigned. The scallop is systematic, always on the same side, and its contribution to the area has a sign. Over a long enough boundary a tiny systematic term outweighs a large random one — which is the two ways to spread a misclosure arriving as a fact about territory rather than about a traverse.

At the actual spacing it does not, and by a wide margin. The point at which it would is the crossing in the figures above: a commission setting monuments 37 kilometres apart at 49 degrees would have had a systematic term as large as its own random one, and beyond that the systematic term wins.

The spacing a commission would choose, if it chose on this #

Turning the two laws round gives a design rule, which is the practical form of the rung.

The offset is s² tan φ / 8R, so a stated tolerance e is met at a spacing of s = √(8Re / tan φ). At 49 degrees a one-metre tolerance permits 6.66 kilometres between monuments, a ten-metre tolerance permits 21.1, and an arcsecond permits 37.0. At seventy degrees each of those shrinks by the square root of the ratio of tangents, which is a factor of 1.55.

That is a rule with a square root in it, and a square root is a forgiving function. Halving the permitted offset costs only a factor of 1.41 in monument count, so a commission tightening its geometric tolerance by a factor of a hundred pays a factor of ten in monuments — which is real money and is not the exponential penalty a reader might fear.

Nobody chose spacing this way, and the reason is in the numbers above: at every spacing anybody has actually used, the scallop is orders of magnitude below the observational error, so the binding constraint was always how far apart two monuments could be and still be intervisible, or cleared, or maintained. The tolerance decides the model is the essay about which corrections a stated tolerance permits dropping, and by its own rule this is a term a boundary commission may drop — provided it can say what its spacing is, which is what the design rule is for.

Two monuments, and the one thing the chord does not decide #

There is a step in the argument above that has been left implicit and is worth making, because it is where a reader’s objection would land.

The chord between two monuments is a geodesic only if the commission ran it as one. What a commission actually does is sight a line, which means pointing an instrument at a target and clearing towards it — and a line of sight is a straight line in three dimensions, not a curve on the ellipsoid. The two differ by the refraction of the atmosphere and by the fact that a straight line between two points on a curved surface passes below it.

Neither affects the offset measured here. The line of sight and the geodesic between the same two monuments have the same horizontal projection to the precision of anything in this rung: refraction bends a sight line vertically rather than sideways, and the chord through the rock lies below the surface geodesic without departing from it in azimuth. So the ground cleared between two monuments follows the geodesic, and the fifty-eight millimetres stands.

What the argument does assume is that the monuments themselves are on the parallel. They are on whatever latitude the commission determined, and the difference between the two is observational — which is the term this rung has just shown to be three orders of magnitude larger. The scallop is exact and small; the monuments are approximate and less small; and the boundary is the monuments.

And the words themselves are not fixed #

Everything in this rung compares a chain of monuments against the parallel of 49 degrees north latitude. That parallel is a curve on a model of the Earth, and the model is not in the treaty.

A latitude of 49 degrees means one place on one datum and a different place on another. The commissions of the 1850s and 1870s worked in astronomical latitude, which is defined by the direction of gravity and differs from the geodetic latitude of any ellipsoid by the local deflection of the vertical; the modern description is in NAD83, which is a different surface again. Those differences are not fifty-eight millimetres.

There is a second thing the words do not fix, and it is the one this rung has quietly assumed away. “The forty-ninth parallel” names a curve of constant latitude, and there are three quantities called latitude: astronomical, geodetic and geocentric. The commissions measured the first, because a transit instrument and a star are what they had; the treaty’s readers assume the second, because that is what a coordinate on a modern map means; and the two differ by the deflection of the vertical, which in mountainous country reaches tens of arcseconds.

Thirty arcseconds of latitude is nine hundred metres. That is fifteen thousand times the scallop, it varies along the boundary with the mass distribution underneath it, and no amount of care with the geometry touches it. A coordinate is the output of a solve is the essay about how much of a published position is decided by the model rather than by the measurement, and this is that observation applied to a treaty.

Which is the rung above, and it is a larger number than anything here.