The figure of the Earth was measured

In 1735 the Académie des Sciences sent one expedition to Lapland and another to Peru, each to measure the length of one degree of latitude, in order to settle whether the Earth bulges at the equator or at the poles. The Peru party was away for ten years.

They were arguing about one part in a hundred.

The measurement, and what it was worth #

A degree of latitude is longer where the surface is flatter, because latitude is defined by the direction of the normal to the surface — the geodetic latitude of geodetic against geocentric latitude and a flatter surface turns its normal more slowly. So on a body flattened at the poles, the polar degree is the long one.

Integrating the meridian radius of curvature over one degree gives the length:

$$L(\varphi) = \int_{\varphi - 0.5°}^{\varphi + 0.5°} \frac{a(1-e^2)}{(1 - e^2\sin^2\varphi')^{3/2}} \,\mathrm{d}\varphi'$$

On WGS84 that runs from 110,574 metres at the equator to 111,694 at the pole. The whole difference is 1,119 metres in 111,000 — one per cent.

And they were not comparing the two extremes. Lapland is at about 66° and Peru at about 1.5°, so the available separation was around 1,000 metres — out of arcs each measured by triangulating a chain more than a degree long across mountains or tundra, tying it to a baseline measured with wooden rods, and determining the latitudes of the ends astronomically.

The inversion, and what it costs #

Two measured degree lengths at two latitudes determine both $a$ and $f$, and the inversion is straightforward: Newton’s method on the pair of equations, with both partial derivatives taken numerically because the analytic ones are a page of algebra that would need its own check.

Given exact inputs it works perfectly. Feed in the WGS84 degree lengths at 1.5° and 66.33° and it returns $a = 6{,}378{,}137.000$ metres and $1/f = 298.2572$ — the radius to the millimetre and the flattening to six decimal places.

Then perturb one arc and watch:

The amplification is constant at 118: the relative error in the flattening is 118 times the relative error in the arc. Ten metres of error in a 110-kilometre measurement is ninety parts per million, and it produces a one per cent error in the flattening.

That is the conditioning of the problem, and it is the entire explanation of the historical dispute.

Why the amplification is 118 #

There is no mystery in the number and it is worth deriving, because the same reasoning applies to any two-point determination of a small difference.

The equatorial radius is carried by the sum of the two arcs, which is about 222 kilometres. The flattening is carried by their difference, which is about 937 metres. So an absolute error $\delta$ in one arc is a relative error of $\delta/111{,}000$ in the sum and $\delta/937$ in the difference — a ratio of 118.

The amplification is the ratio of the signal to the difference being extracted from it, and nothing about the algorithm changes it. A better inversion, a better integrator, more decimal places: none of them help, because the information is not in the measurements.

Which points at the two things that do help, and both were used historically:

• separate the arcs further, which increases the difference and therefore the signal — hence Lapland and Peru rather than two European sites;
• measure more arcs, so that errors partially cancel — which is what the nineteenth-century figures of Airy, Bessel and Clarke did, each fitting several arcs at once.

What a better-separated pair buys #

The amplification is not a constant of nature. It depends on how far apart the two arcs are, and the dependence is the reason the expeditions went where they went.

Comparing the two figures is why the generator takes the two latitudes as parameters. Moving from a 65° separation to a 20° one shrinks the difference between the arcs from 937 metres to 377, and the amplification rises from 118 to 295. Push it further, to the 6° of latitude a survey of France covers, and the difference falls to 117 metres and the amplification reaches 950.

At an amplification of 950, an arc measured to a hundred metres gives a flattening wrong by 85 per cent, which cannot determine the sign of anything. The expeditions were not a demonstration of superior technique. They were the recognition that the experiment had to be redesigned rather than repeated.

The prolate answer was not stupid #

The last row of the table is the one that connects the arithmetic to the history.

At a kilometre of error the inversion returns a negative flattening: an Earth longer through the poles than across the equator, like a lemon. That is precisely what the Cassini family at the Paris Observatory concluded from French arc measurements between 1683 and 1718, and defended for a generation against Newton’s theoretical prediction of an oblate Earth.

Given the conditioning, they were not being foolish. Their arcs spanned France — about 6° of latitude, so a difference of 117 metres between the two degree lengths, at an amplification of 950 — measured with rods and a chain of triangles over thirty-five years. A systematic error of a couple of hundred metres across the whole chain is entirely ordinary for that work, and it is more than enough to invert the sign. They were reporting what their measurements said.

The Lapland and Peru expeditions were the correct experimental response: not better instruments, but a longer baseline in latitude, which increases the signal without requiring any improvement in precision. Maupertuis’s Lapland arc came back in 1737 and settled the sign; Bouguer and La Condamine’s Peruvian arc, published in 1749, fixed the magnitude.

That is a general and portable lesson about ill-conditioned problems. When an estimate is dominated by conditioning rather than precision, change the geometry rather than the instrument.

What was computed, and how #

Three things, and each has a check attached.

The degree length, by quadrature of the meridian radius over a one-degree interval, at 2,000 sample points. That agrees with the collection’s existing meridian-arc machinery, built for the transverse Mercator series, which is itself checked two ways — a series and an independent quadrature — after the series was found to be wrong by 11 millimetres at 45° because its leading coefficient had been applied to every term rather than to the first.

The inversion, by Newton on the two-by-two system. Its check is that it must recover the ellipsoid it was given: exact inputs must return $a$ to a millimetre and $1/f$ to six decimals, and the assertion requires exactly that.

The sensitivity, by perturbing one input and re-inverting. Its check is the last row: the inversion must return a negative flattening somewhere in the tested range. That is the refusal — an amplification that had quietly stopped amplifying would still pass a check that only asked for large errors, and would not pass one that asks for the answer to change sign.

That figure is the conditioning argument applied to the historical record. Airy gives $1/f = 299.32$, Bessel 299.15, Clarke 294.98, Hayford 297 exactly. The spread is about 1.5 per cent, and 1.5 per cent of the flattening corresponds to about 0.013 per cent of an arc — fourteen metres in 110 kilometres. Fourteen metres is a very respectable disagreement between two nineteenth-century triangulation chains, and it produces the entire spread of the table.

The corollary is that these figures are not evidence of a debate about the Earth’s shape. They are evidence of an experiment whose amplification made every careful attempt land somewhere different, and a datum is fitted to a region shows why each of them was nevertheless the right choice for the ground it was fitted to.

Where the model stops #

This inverts two arcs, and the historical determinations inverted more. Airy, Bessel and Clarke each fitted several arcs by least squares, which reduces the amplification by averaging but does not change its character — the flattening is still carried by differences between long, nearly equal quantities.

Real arc measurements have correlated errors. The model here perturbs one arc independently. In practice a baseline error, a scale error in the measuring rods, or a systematic error in the astronomical latitudes affects both arcs, and a common error largely cancels in the difference. That makes the historical situation somewhat better than the table suggests for some error sources and no better at all for others — an error in one expedition’s latitude determination is exactly the independent case modelled here.

The deflection of the vertical is not modelled. The latitudes at the ends of an arc were determined astronomically, so they are astronomic latitudes and differ from geodetic ones by the deflection — several arcseconds, and much more near mountains. The Peruvian arc ran through the Andes. That is a systematic error of exactly the kind that does not cancel, and it is the plumb line is not the normal.

Nothing here uses gravity. The other route to the flattening runs through Clairaut’s theorem and a pendulum, and it is both cheaper and better conditioned — the flattening is not a free parameter sets out the comparison. The geodetic route got the attention because it was heroic.

What the metre was defined from #

One consequence of this measurement deserves its own paragraph, because it inverted the relationship between the two quantities for a century and a half.

In 1791 the French Academy defined the metre as one ten-millionth of the distance from the pole to the equator along the meridian through Paris. That made the unit of length a statement about the Earth’s figure — so every measurement of a degree of latitude thereafter was, in a sense, a measurement of the definition of the units it was expressed in.

The definition was retired in 1960 and the modern metre is defined by the speed of light. The quarter-meridian, on modern values, is 10,001,966 metres — so the eighteenth-century determination was out by about two parts in ten thousand, which is a very good measurement of an extremely difficult quantity and is far better than the flattening they got from the same campaigns. That asymmetry is the conditioning again: the sum of the arcs was well determined and the difference was not.

The generalisation #

Estimating a small difference between large measurements amplifies error by the ratio of the two, and that ratio is a property of the problem rather than of the method. Computing it before designing the experiment is the difference between an experiment that can succeed and one that cannot.

The arithmetic is trivial: signal over difference. Here it is 111,000 over 937, giving 118. That single number tells anybody, before setting out, that arcs measured to a hundred metres will give a flattening good to ten per cent — which is enough to settle the sign and not enough to build a map on.

Three consequences worth carrying:

An answer of the wrong sign is not evidence of incompetence. It is what an ill-conditioned inversion does when its input error exceeds its signal, and the Cassinis’ prolate Earth is the clean historical example.

The remedy is geometric, not instrumental. Better instruments reduce $\delta$ linearly; a longer baseline in the parameter that carries the signal increases the difference, often by much more. Maupertuis went north rather than buying better rods.

A well-conditioned route may exist and look less impressive. The same judgement runs through which projection is best: the impressive answer and the useful one are different questions, and naming the purpose settles which is wanted. The pendulum route to the flattening required no expedition and gave a better-conditioned answer, and it was regarded as the lesser experiment for two centuries.

The same shape recurs in this collection wherever a quantity is extracted from a near-cancellation. Vincenty’s formulae stop converging for nearly antipodal points because the geodesic’s azimuth is determined by a difference that vanishes; the conformality tolerance sits sixty times above a noise floor and 3,850 below the smallest real failure precisely so that the quantity being measured is not a near-cancellation of two larger ones. Conditioning is the question to ask of any measurement expressed as a difference.

The measurement, and what it senses #

Why a degree is longer where the surface is flatter is one line of geometry. Latitude is the angle of the normal, and a flatter surface turns its normal more slowly, so the same one-degree turn covers more ground near the pole.

That sentence is the whole experiment, and it also states its difficulty: at the true flattening the difference between the two normals is invisible at any drawing scale, and the effect separating the two hypotheses had to be measured across a hundred and eleven kilometres of ground to be seen at all.

Who found it, and when #

Newton predicted an oblate Earth in the Principia in 1687, from the argument that a rotating fluid must bulge, and put the flattening at about 1/230. Christiaan Huygens got 1/578 from a different assumption about the mass distribution. Both were reasoning from mechanics rather than measuring.

Jean-Dominique Cassini and his son Jacques measured French arcs between 1683 and 1718 and concluded the opposite — a prolate Earth — which set off fifty years of argument between Newtonian and Cartesian physics conducted largely through geodesy.

The Académie’s resolution was the two expeditions. Pierre Louis Maupertuis, Clairaut and Anders Celsius went to the Torne valley in 1736 and were back in 1737 with a degree length of about 111.9 kilometres. Pierre Bouguer, Charles Marie de La Condamine and Louis Godin went to the Viceroyalty of Peru in 1735 and were there for a decade, returning a degree of about 110.6 kilometres. The difference settled it: the polar degree is longer, so the Earth is oblate, and Newton was right.

The Peruvian expedition produced two other things worth noting. Bouguer observed that a plumb line near Chimborazo was deflected towards the mountain by far less than its bulk implied, which is the first evidence for isostasy. And the arc’s length was one of the measurements used to define the metre in 1791 — as one ten-millionth of the distance from the pole to the equator, which made the unit of length a statement about the shape of the Earth. That definition survived until 1960, and the modern metre is still within 0.02 per cent of it.

What defining a unit from a shape costs #

The metre’s original definition is worth one more paragraph, because it contains a difficulty the eighteenth century could not have avoided and the twentieth had to resolve.

A unit defined as a fraction of a meridian quadrant is a unit whose length depends on the shape of the Earth. So the quadrant is exactly ten million metres by construction, and the question how long is the quadrant stops being answerable: every subsequent measurement of it is a measurement of how well the original survey was done, expressed in a unit derived from that same survey.

The circularity was resolved by abandoning the definition rather than by improving it. The metre became a platinum-iridium bar in 1889 — an artefact, with no claim about the planet — and then a wavelength of krypton in 1960 and a fraction of a light-second in 1983. Each step moved the unit further from anything about the Earth, and each made it possible to state the quadrant’s length as a result.

The modern figure is that the quadrant is about 10,001,966 metres, so the original survey was high by about two parts in ten thousand, which is a remarkable result for the instruments involved and is a statement that could not have been made while the definition stood.

That is the general shape of the difficulty: a unit tied to a measured quantity cannot be used to measure it. The same reasoning is why a reference ellipsoid’s parameters are defined rather than measured — as the ellipsoid is a definition rather than a measurement puts it — and why the departures from it are then reportable as numbers.

Where this goes next #

That is the end of the datums field: the four commitments in a coordinate, the vertical that the horizontal assumed away, and the two essays under the ellipsoid itself. What follows is practice — grids, zones, and what a surveyor holds in their hands — which is where all of this is used, and where the errors measured here turn into the tolerances a job is written to.