How many triangles it takes

How big a triangle it takes prices the angle-excess route to Gaussian curvature and finds that no survey ever built could have reached one per cent. It also records, in the same breath, that its own table is wrong in an important way: every number in it is stated per triangle, and a triangulation observes hundreds.

That is the shortfall this rung pays. Averaging $n$ independent estimates divides the noise by $\sqrt{n}$, which for the Struve arc’s chain is a factor of twelve — real, and not obviously enough.

What makes it more than an arithmetic correction is that $n$ is not free. A survey has an extent, and a fixed extent holds fewer large triangles than small ones, so raising the side lowers the count. The two effects pull opposite ways and the result is a different exponent.

Three regimes, three exponents #

The estimator is $\hat K = E/A$, with $E$ the angle excess and $A$ the area. Three measured angles of standard deviation $\sigma$ give an excess of standard deviation $\sigma\sqrt3$, so

$$\varepsilon = \frac{\sigma\sqrt3}{K\,A\,\sqrt n}, \qquad A = \frac{\sqrt3}{4}s^2.$$

One triangle. $n = 1$, so $\varepsilon \propto s^{-2}$. Doubling the side quarters the error. That is rung eleven’s regime and it is the reason geodetic triangles are as large as sightlines allow.

A chain of fixed length. Consecutive triangles share a side and alternate which way they point, so each advances about half a side along the chain and $n \approx 2L/s$. Then $\sqrt n \propto s^{-1/2}$ and $\varepsilon \propto s^{-3/2}$. Doubling the side improves the answer by 2.83 rather than 4, because half the triangles have gone.

A network covering a fixed area. $n \propto s^{-2}$, so $\sqrt n \propto s^{-1}$ and $\varepsilon \propto s^{-1}$. Doubling the side buys a factor of two.

The fitted exponents are −2.0000, −1.5026 and −0.9991. They are exact rather than empirical, and the middle one is the one that applies to every measurement the eighteenth and nineteenth centuries actually made, because every great arc was a chain.

The great arcs, re-priced #

Two things in that table are worth stopping on.

The two ends of the table are worth reading as history as well as arithmetic. Snellius in 1615, with a thirty-second instrument and eight triangles, could resolve the curvature of the Earth to within a factor of eight — which is to say not at all, and which is exactly right for a survey whose purpose was to measure a degree rather than a curvature. By 1787 the Anglo-French survey’s two-second instruments bring it to 22 per cent, which is enough to say the surface is curved and nothing about how much. The step from Snellius to the Great Trigonometrical Survey is a factor of two hundred, and almost all of it is the instrument: the sides grew by half and the counts by a factor of twelve, and neither of those is a factor of two hundred.

The correction is large and it does not change the verdict. The best historic chain reaches 3.22 per cent where a single one of its triangles reaches 32. That is a factor of ten and it leaves the answer on the same side of one per cent — which is the honest shape of a paid shortfall: the deferred calculation mattered, it was worth doing, and it moved a margin rather than a conclusion.

Gauss and Struve arrive at the same number by opposite routes. Gauss’s Hanover survey is seven triangles of 85 kilometres and reaches 4.13 per cent. The Struve arc is 141 triangles of 40 kilometres and reaches 4.15. A chain nine times longer, built of triangles half the size, is worth exactly as much.

Why the middle exponent is the one that matters #

It is worth saying why the chain regime, rather than either of the others, is the one every historical number here belongs in.

A single triangle is not a survey. Nobody builds one triangle: a triangle needs a measured base to give it a scale, and a base is measured over a few kilometres and carried to the triangle by a chain of smaller triangles, so the smallest real object in geodesy is already a network. The $s^{-2}$ regime describes an estimator rather than an undertaking.

A network of fixed area is a national triangulation — the kind that fixes a country’s coordinates — and it is the regime a modern control network sits in. There the count rises as fast as the area falls, the two nearly cancel, and the exponent is one: doubling the side buys a factor of two and nothing more.

An arc is a chain, and every measurement of the figure of the Earth before the twentieth century was an arc, because the quantity being measured was how the length of a degree changes with latitude and that needs length in one direction. So the middle exponent is not a special case chosen for interest. It is the one the entire historical record lives in, and it is the one nobody has written down.

The equivalent triangle #

That coincidence is the exponent stated the other way round, and it is the useful form.

A chain of $n$ triangles of side $s$ is worth one triangle of side $s\,n^{1/4}$, because $\varepsilon \propto s^{-2}n^{-1/2}$ and the two exponents are in the ratio four to one. The fourth root is what makes counting cheap and unrewarding. Multiplying the count by sixteen is worth doubling the side; the Struve arc’s 141 triangles are worth multiplying its side by 3.44.

The fourth root also settles a question about survey design that the anchor has not been able to ask. Given a fixed amount of observing — a fixed number of measured angles, which is three per triangle — is it better to spend it on many small triangles or few large ones? The answer is unambiguous and it is few large ones: the count enters as the fourth root of itself and the side enters directly, so trading four triangles for one of twice the side is a straight gain of $2/4^{1/4} = 1.41$. Every doubling of the side is worth four triangles and costs one, and the only thing that stops the argument is that a sightline has a length beyond which no tower is tall enough. The historical practice of building the largest triangle the terrain allows is exactly right, and this is the first calculation here that says so with a number.

That is why the great arcs cluster so tightly in the last column of the table. Their sides differ by a factor of two and their counts by a factor of thirty-five, and after the fourth root the whole spread collapses to between 55 and 157 kilometres of equivalent side.

What it would have taken #

Since the count buys so little, the question rung eleven asked comes back sharpened: what combination of instrument and survey would have reached one per cent?

A one-second instrument does not get there at any length that could be built. Ten thousand kilometres of chain — longer than the Struve arc by a factor of three and a half, longer than any survey ever attempted — returns 2.2 per cent. Three tenths of a second reaches one per cent at ten thousand kilometres, and a tenth of a second reaches it at a thousand.

So the answer is not “a longer survey”. It is an instrument the nineteenth century did not have, and the same chain that Struve built would clear one per cent comfortably today: with a modern 0.1-second instrument its 141 triangles return 0.415 per cent.

The obstruction was the instrument, and the count does not substitute for it. That is rung eleven’s conclusion with the shortfall paid, and the payment strengthens it: the √n a network buys was the one remaining hope that observation could have made up the difference, and it is worth a factor of twelve against a shortfall of a factor of thirty.

What a modern network reaches #

The other end of the same calculation is worth having, because it says how much of this is history.

Satellite geodesy does not measure angles between mountain tops, so the estimator priced here is not how anybody determines the shape of the Earth now. But the question the estimator answers — can a flatlander confined to the surface measure its curvature — is not a historical one, and the answer has changed.

A modern one-second theodolite is routine and a tenth-second instrument is available. On the Struve arc’s own geometry — 141 triangles of forty kilometres — a tenth-second instrument returns 0.415 per cent, and a hundredth-second one returns 0.04. The measurement rung three describes as possible in principle is now possible in practice, on a chain that was actually built, by an instrument a survey department owns.

So the impossibility this anchor is about was never the geometry, and it is no longer the instrument either. What remains is that nobody has a reason to do it: the shape of the Earth is known to nine figures from orbits, and confirming it from inside would be an exercise. That is a change in the character of the obstruction rather than its removal, and it is worth being precise about which kind of obstruction a result is describing.

What was computed, and how #

Everything is on WGS84, with $K = 1/(MN)$ from the ellipsoid’s own radii of curvature and no sphere anywhere. The noise term is the exact standard deviation of a sum of three independent angles — the curvature of the Earth is not one number is where the varying $K$ that the second term is about comes from; the variation term — that a large triangle returns the area mean of $K$ rather than its value at a point — is carried through from rung eleven and is negligible at every size, as that rung established.

The triangle counts are derived rather than quoted. Each survey is described in the literature by its length and its typical side, and quoted triangle counts vary between sources in ways that would put a third uncertain number into a table already about uncertainty. The chain relation $n \approx 2L/s$ comes from the geometry of a chain and is drawn above; applying it to each survey’s own two numbers gives counts of 4 to 141 which are consistent with what the surveys are described as having.

The exponents are fitted from the ladders rather than asserted, and required to match −2, −3/2 and −1 to within a fiftieth. That matters because the three-halves is the whole content of the rung: an exponent that came back at −1.7 would mean the chain relation is wrong, and one at −2 would mean the count is not binding.

The assertions require five things separately: that $n$ triangles be exactly $\sqrt n$ better than one; that the three fitted exponents match their closed forms; that the largest historic chain’s count be worth more than a factor of ten; that it still fall short of one per cent, since a paid shortfall that overturned its own rung would need saying loudly; and that a modern instrument on the same chain clear it, which is what locates the obstruction in the instrument.

Where the model stops #

The triangles are treated as independent and they are not. Consecutive triangles in a chain share a side and therefore share two measured angles, so their excess estimates are correlated and the true improvement is somewhat less than $\sqrt n$. The correlation is not large — each triangle has three angles and shares at most two of them with each neighbour — but it is positive, so every network figure here is an optimistic bound. Correcting it needs the survey’s actual observation schedule, which is a different kind of object from the two numbers each survey is described by.

A real adjustment does not average excesses. A coordinate is the output of a solve is the honest description of what a triangulation does with its angles, and a least-squares network estimates coordinates rather than a curvature. Extracting $K$ from an adjusted network is a different estimator with a different variance, and the one priced here — average the per-triangle excesses — is the one a flatlander could actually run.

And nobody was trying to do this. The great arcs were measuring the flattening, by comparing the length of a degree at different latitudes, which is a far more sensitive test of the same shape. That every one of them could also have estimated the curvature to a few per cent is a fact about the data they collected rather than a fact about what they were doing.

The generalisation #

The rule is that a budget constraint turns one exponent into another, and the constraint is usually unstated.

The bare estimator says $s^{-2}$: make the triangle bigger. That is true and it is not a plan, because nobody has an unlimited supply of triangles of any size. The plan needs to know what is scarce — sightlines, stations, time, ground — and each scarcity gives a different exponent. Three halves for a chain, one for an area, two for a single triangle somebody has already built.

The same shape runs through every measurement this collection prices. How many sheets an atlas needs is the same trade with tolerance against sheet count; the worst point is not on the grid is a refinement whose cost per improvement has an exponent of its own. In each case the naive rate — refine and it gets better — is right, and the rate at which it gets better per unit of the scarce thing is a different number that nobody quotes.

There is a second half to the rule, which is what to do when the constraint is not known. Fit the exponent. A ladder of measurements at different sizes has a slope, the slope names the regime, and the regime names what was scarce — so an exponent of −1.5 on somebody else’s data is a statement that they were working along a line, and one of −1 that they were covering ground. That is a diagnostic rather than an assumption, and it is the same move the deficit’s rate makes to tell a boundary extremum from an interior one.

The habit is one question: what is fixed while the thing being improved is being improved? An accuracy that improves as the square when nothing else moves may improve as the first power when the thing that actually constrains the work is held still, and the difference between those two is the difference between a plan that works and one that does not.

Who found it, and when #

The $\sqrt n$ is not a discovery and neither is the chain relation. What is missing from the literature is the join, and the reason is the one this ladder keeps meeting: nobody was estimating curvature from a triangulation, so nobody wrote down its error budget.

The surveyors knew the arithmetic perfectly well in its own setting. The whole design of a triangulation chain — its side lengths, its base measurements, its check bases every few hundred kilometres — is a series of decisions about propagating error along a chain, and Gauss’s least squares was invented for exactly this. What was being propagated was position and scale, and the excess was a nuisance to be corrected for rather than a quantity to be estimated.

So the calculation that would have told a nineteenth-century geodesist how well their own network measured the curvature of the Earth from the inside was entirely within reach and had no reason to be done. The answer, computed here, is between three and five per cent for the best of them — which is a good deal better than a single triangle suggests, and is still an instrument’s width away from being able to tell a sphere from a plane by an argument that never leaves the surface.

Where the ladder goes next #

Thirteen rungs have measured the curvature of idealised surfaces: a sphere, an ellipsoid, a triaxial body, a surface with a hole. The surface heights are actually measured against is none of those. It is an equipotential fitted to gravity, it has its own Gaussian curvature, and nothing in this collection has computed it.