An adjustment hides the sphere only in a strip of equal triangles

A strip of triangles carries the sphere’s error to the next base found that a chain of triangles computed on a substituted sphere does not accumulate an error that depends on the triangles. It accumulates the sphere’s own scale, read at the two ends. On the conformal sphere, a strip of thirty-kilometre triangles running north from 45° hands each side on 7.9 parts per million longer than the last, and after forty triangles the side it computes is 315 parts per million too long — the change in the sphere’s meridian stretch between the two ends, to within a part in three hundred. A check base three triangles along would already see it.

That was the unadjusted strip, and nobody published one. A chain between two measured bases was adjusted to close on the second, and the adjustment is exactly the operation that takes a misclosure found at a check base and spreads it back through the chain. The strip essay ended by asking what the spreading leaves. It guessed that spread evenly, the drift becomes a scale varying linearly between the bases, very nearly the stretch itself, and that only the stretch’s curvature would survive.

For a strip of equal triangles that guess is right, and what survives is smaller than it expected. For any other chain it is wrong, and what survives is larger than the stretch’s curvature by two orders of magnitude and still invisible at every single side.

What an adjustment spreads, and by what #

An adjustment of a triangulation chain is a least-squares correction of its observed angles, subject to conditions: each triangle’s angles must sum to 180° plus its excess, and the side computed at the far end must equal the base measured there. The second is the one that matters here. It says that the chain’s computed ratio of far side to near side, a product of one sine ratio per triangle, must come out at the ratio of the two measured bases, and the misclosure is the logarithm of how far it misses.

Least squares distributes that misclosure in a definite way. The side condition is linear in the angle corrections, with coefficient $\cot B$ on the angle opposite each computed side and $-\cot C$ on the angle opposite each base, and the corrections that satisfy it with the least sum of squares are proportional to those coefficients. Triangle $i$’s side ratio therefore moves by an amount proportional to $\cot^2 B_i + \cot^2 C_i$: each triangle takes a share of the misclosure in proportion to how uncertain its own side ratio is. That is the rule the two ways to spread a misclosure describes for a traverse, and it is right. A well-shaped triangle, whose ratio the angles fix firmly, is corrected little; a thin one is corrected a great deal.

The sphere’s error was not put in that way. An angle is a difference, and the difference doubles the error traced it into single angles; the strip essay found that each triangle adds the change in the sphere’s meridian stretch across it, so a triangle that carries the chain further north adds more. For the conformal sphere that holds triangle by triangle whatever the triangle’s shape, because the conformal sphere stretches every direction alike: the error in a computed side is exactly the difference in $\ln m$ between the latitude of the computed side and the latitude of the base. Checked on four irregular triangles of different shapes, sizes and orientations at 30°, 45° and 60°, it agrees with the full computation against the ellipsoid to 0.05 parts per million.

So an adjustment takes the error out by one rule and the sphere put it in by another. Whatever is left is the difference between the two, added up along the chain.

In a strip of equal triangles the two rules agree #

In the strip of the first figure every triangle is the same equilateral triangle, so every triangle’s share of the misclosure is the same, one fortieth of 315 parts per million. Every triangle’s advance north is also the same, fifteen kilometres, and the sphere’s error in each is the change in its stretch over fifteen kilometres — nearly the same at every latitude the strip crosses, because over five degrees the stretch is nearly a straight line.

What the adjustment leaves is the difference between the stretch and a straight line drawn between its values at the two bases: the stretch’s bow. At its largest, near the middle, it is 0.63 parts per million. The adjusted strip’s own noise — what one-second angles leave once the adjustment has spread their misclosure too — has a standard deviation of 12.5 parts per million at the middle, falling to nothing at each base, where the measured length holds it. The bow is twenty times smaller than one standard deviation and sixty times smaller than three.

The strip essay’s guess about the adjustment was that it would leave “the second-order part of the stretch, a few parts per million over five degrees”. It leaves less, and the reason is the latitude.

The conformal sphere’s stretch changes fastest at 45° — that is where the strip drifts most, and where the curvature of the Earth is not one number found the ellipsoid’s curvature changing fastest too — so at 45° its rate of change is at a maximum and its curvature, the rate at which the rate changes, passes through zero. A bow is the curvature’s doing. With no curvature at the middle, the bow of a strip centred near 45° comes from the next term, and it grows as the cube of the span rather than the square: 0.08 parts per million over 2.7 degrees, 0.66 over 5.4, 5.5 over 10.8, eight times for every doubling. From 20° or 60° the stretch has curvature and the bow grows as the square, four times for every doubling, and starts larger — 5.2 parts per million over 5.4 degrees from 20°, 4.3 from 60°.

Against the noise it hardly matters. The adjusted noise at the middle grows as the square root of the number of triangles, and for every start latitude and every span measured but one, three standard deviations of it exceed the bow. The one exception is a strip from 60° across twenty-two degrees, whose bow of 92 parts per million clears three standard deviations of 75 — a chain of 2,400 kilometres with no base anywhere between its ends. In a strip of equal triangles an adjustment hides the sphere completely. A re-observation of the adjusted coordinates would find differences from the modern values at the level of the old angles’ noise and nothing systematic in them.

A real chain is not made of equal triangles #

First-order chains were designed for well-shaped triangles and could not have them all. The ground decided where a station could be seen from, and a chain across real country is a sequence of triangles of different sizes and shapes: some reaching far north and some hardly at all, some with angles near 60° and some with an angle of 40°.

The figure is the beginning of such a chain, forty triangles long, stated by the only two properties of a triangle the argument uses: how far north it carries the chain, between 8 and 22 kilometres, and its angles, every one between 40° and 100°. For each triangle the left bar is the error the conformal sphere put in and the right bar the correction the adjustment takes out. In the equilateral strip the two would be equal to about a part in a hundred everywhere. Here they are unrelated. The fourteenth triangle reaches an average distance north and has two of its angles at 41° and 42°: the sphere put 8.7 parts per million into its ratio and the adjustment takes out 19.7, because its ratio is the chain’s least certain. The eighth reaches far north with a well-shaped figure: 9.7 in, 2.6 out.

Each triangle’s difference is a first-order quantity, a few parts per million. Added up along the chain, they are the adjusted residual.

A residual of first order, below the noise at every side #

The adjusted residual of the irregular chain wanders, as a sum of unrelated terms does, and reaches 10.1 parts per million at its largest, 4.6 root-mean-square. That is fifteen times the equilateral strip’s bow over the same number of triangles. It is also, at every single side, inside the noise: within one standard deviation of what the old angles leave at 38 of the 40 sides, and within three at all of them. Read side by side against a re-observation, it would look like nothing. A survey comparing old coordinates with new ones would find the differences unremarkable everywhere, and the blunder the network cannot see is the general form of that: a pattern spread thinly enough is below every local test.

But this residual is not unknown. It is fixed completely by three things the old network’s own records contain — each triangle’s position, its advance, and its angles — together with the substitution being tested. The sphere’s share in each triangle is the change in its stretch across it; the adjustment’s share is the misclosure times that triangle’s cot² sum over the chain’s. The network’s answer is decided before it is measured found an adjustment’s treatment of an error fixed by the network’s design, and this is the same fact in its most useful form. The residual is a template, computable for any archived chain and any hypothesised sphere without a single new observation.

Matched against its template, the chain shows the sphere #

A signal that is small at every point and known in shape is found by matching rather than by looking. Take the differences between the old adjusted sides and a modern re-observation, weight each triangle’s step by the inverse of its variance, and project onto the template’s steps. With no sphere in the old computation the result is pure noise, with a standard deviation that is computable; with the sphere, it is displaced by the template’s own length measured in that noise, $\sqrt{\sum_i (d_i - D v_i/V)^2 / v_i}$, where $d_i$ is the sphere’s error in triangle $i$, $v_i$ the variance of its side ratio, and $D$ and $V$ their sums over the chain.

For the irregular chain of forty triangles that length is 7.0 standard deviations. Across eight irregular chains of the same specification it runs from 5.1 to 7.0, median 5.9; at twenty triangles, 4.0; at ten, 2.9; at eighty, 8.0, growing roughly as the square root of the count, since every triangle adds an independent piece of the template. For the equilateral strip the same statistic is 0.06 at forty triangles and 0.36 at eighty. It is not that the equilateral strip’s template has not been looked for hard enough. There is almost nothing in it to find.

That reverses the practical reading of the strip essay. There, the unadjusted strip’s drift was large and simple, and an adjustment looked like the operation that would bury it. Here, the equal triangles an ideal chain would have are the one arrangement in which an adjustment does bury it, and every real chain’s irregularity is what keeps it recoverable.

The latitude matters in the obvious way. Each triangle’s share of the sphere’s error is the change of the stretch across it, so the template is largest where the stretch changes fastest — 5.9 standard deviations for chains starting at 45°, 5.3 at 30°, 4.9 at 60° — and falls towards the equator and the pole, to 0.8 from the equator, where the conformal sphere’s stretch is stationary and a chain crossing it carries almost nothing. A chain in the tropics computed on the conformal sphere is, for this purpose, a chain computed on the ellipsoid.

A simulated re-observation #

The arithmetic above is a statement about a distribution, and it can be run. Four thousand times, the chain’s forty triangles are given angle errors of one second of arc, the adjustment spreads their misclosure as it spreads any other, and the result is matched against the conformal sphere’s template — once for a chain computed on the ellipsoid and once for the same chain computed on the sphere.

On the ellipsoid the statistic has a mean of −0.02 and a standard deviation of 0.99: a unit normal, as it should be, above three standard deviations in 0.1 per cent of trials. On the sphere its mean is 6.94 with the same spread, above three in 99.9 per cent. A re-observation of forty triangles, with the old chain’s own geometry as the only other input, tells the two computations apart on essentially every occasion.

What the archive would have to hold #

The test needs nothing that an archived first-order chain does not already record. The station positions give each triangle’s advance and latitude. The observed angles give each triangle’s shape and so its share of any adjustment. The adjustment’s own misclosure at the second base is usually printed. What the test adds is a modern re-observation of the same sides, or of enough of them, and a hypothesis about which sphere the computers used.

It also says which chains are worth testing. A chain along a parallel carries nothing, as the strip essay found, and neither does a chain near the equator. A chain of nearly equal triangles carries almost nothing once adjusted, however long. A chain running north through the middle latitudes, with triangles as unequal as ground makes them, carries a template several standard deviations deep — and that is a description of most of the great European meridian arcs.

Four radii of the Earth is the reminder that the handbooks offered several spheres. For the conformal sphere the template is exact, triangle by triangle, for the reason a bearing on a sphere is decided by its latitude, not its radius gave: a sphere that keeps angles can be wrong only in how it spaces its parallels. For a sphere whose stretch differs along the meridian and the parallel, like the geodetic-latitude or the geocentric, a triangle’s error also depends on its orientation — the zigzag the strip essay found on a strip turned from the meridian — and in an irregular chain that adds an orientation term to every triangle’s share. It makes those spheres’ templates larger, not smaller, but it means the template has to be computed triangle by triangle against the ellipsoid rather than read off the stretch.

What was checked #

The law the template rests on must hold triangle by triangle. On four irregular triangles — sides of 18 to 35 kilometres, at 30°, 45° and 60°, turned every way — the conformal sphere’s error in the computed side must equal the change in its meridian stretch between the base’s latitude and the side’s, to a tenth of a part per million. The largest disagreement is 0.05.

The equal strip computed from that law must match the full computation. The largest adjusted error of the forty-triangle equilateral strip is 0.66 parts per million from the law and 0.63 from computing every triangle against the ellipsoid.

An adjustment must close. Every chain’s residual must end at the second base at zero, and does to the precision of the arithmetic.

The finding must be there to fail. Over forty triangles the equilateral strip’s template must stand below half a standard deviation and the irregular chains’ median above three; they stand at 0.06 and 5.9. And the simulated statistic must be a unit normal without the sphere and centred on the template’s length with it: mean −0.02 and standard deviation 0.99 against 0 and 1; mean 6.94 against 6.96.

Where the chain is a model #

A chain stated by two numbers a triangle. Each triangle here is its advance north and its two base-adjacent angles, drawn at random within stated ranges. A real chain has braced figures, diagonals and more than one route between bases, and its adjustment has more conditions than one side condition; the shares would be computed from the real network’s normal equations rather than from each triangle’s cot² sum. The mechanism survives that: any adjustment spreads a misclosure by the observations’ weights, and a sphere does not put it in by them.

Only the base condition. The angle conditions of each triangle are taken as already closed by equal shares, as the strip essay closed them, and each triangle is solved by Legendre’s theorem as the correction is the smaller of the two corrections set out. A simultaneous adjustment of both moves each triangle’s share slightly, and changes the template’s length by less than its spread between chains.

One-second angles throughout. A chain observed with three-second angles has a template three times shallower, and the forty-triangle median falls from 5.9 to about 2.

The conformal sphere only. The other substitutions add an orientation term that has not been computed here.

Still open: whether a surviving arc shows it #

The template is a prediction that can be made for any archived chain today, and the prediction has a sign and a shape that no baseline error and no refraction model produces: it is zero at every base, it follows the chain’s own triangle shapes, and it changes with latitude as one named sphere’s stretch does. Whether the computers of a particular national arc used a sphere at all, and which, is a matter of record in some cases and of inference in others.

What would settle it is a first-order arc whose original angles, stations and adjustment survive and whose sides have been re-observed by satellite: the Struve arc, the French meridian of the 1870s, the Indian Great Arc. Whether any of them carries a template of the conformal sphere, or of another, at the depth the simulation says forty triangles would show — and whether the differences between old and new already attributed to base standards and refraction are partly this — are questions a stated chain cannot answer.