He hid a number that would not fit. It was not his mistake
The metre came out of a 1,075-kilometre line measured tower by tower between Dunkirk and Barcelona, through the years of war and the Terror. At the southern end Pierre Méchain read the latitude from two places less than two kilometres apart and got two different answers.
His colleague worked out why only after reading his notebooks, years after he died.
Paris, 1791
One unit for everybody
Before the metre, France measured in king's feet, ells, rods and fathoms, and every town kept its own version of each. A merchant crossing from one market to the next changed units the way he changed coins. The Assembly asked for a measure nobody could own. A king had his own foot; the new measure would come out of the Earth.
The commission chose a quarter of a meridian, the run from the equator to the pole, and decided the metre would be a fraction of it. That left the hard part. Nobody could walk to the pole. But a piece of the meridian could be measured very precisely, and the rest worked out from it.
The piece they chose crossed France end to end, from Dunkirk to Barcelona, because it straddled the 45th parallel and had the sea at both ends.
What fraction of a quarter meridian was the metre meant to be?
One ten-millionth. The quarter meridian was to be exactly ten million metres, which is also where the kilometre comes from: ten thousand of them from the equator to the pole.
Dunkirk and Barcelona, 1792
Two men, two directions
Jean-Baptiste Delambre set off south from Dunkirk. Pierre Méchain set off north from Barcelona. They were to meet near Rodez, in the middle.
Nobody measured the line by pacing it. They measured angles. From one tower you sighted two others and recorded the angle between them, then moved to the next tower. The route was covered in triangles sharing their sides, and a single distance measured on the ground, somewhere along the way, set the size of all the rest.
The instrument was Borda's repeating circle. Its idea was simple and powerful: instead of reading the angle once, you measure it dozens of times, letting the readings accumulate around the dial, and divide at the end. Small errors of reading fall on both sides and cancel.
An observer measures the same angle many times over. What happens to the error?
It falls, though not as fast as you might think: as the square root of the number of readings. Halving the error takes four times as many measurements. And there is one thing the circle cannot mend: if the instrument has a fault that pushes every reading the same way, repeating does not touch it.
Melun, 1798
The only distance measured on the ground
Triangles give the chain its shape but not its size. For size you need one straight line, measured on the ground, with rules laid end to end.
They picked a level road near Melun and laid platinum and copper rules along it, one after another, for days. Every rule expands in the heat, so every laying meant a temperature reading too. A second line, near Perpignan at the far end of the chain, served as the check: its length could be calculated through the triangles all the way from Melun, and then compared with what the rules said.
This is the only part of the story in which a distance was measured directly on the ground.
The two ground-measured lines, at Melun and near Perpignan, are more than six hundred kilometres apart. What was the second one for?
As a check. The second line's length could be calculated through the triangles starting from the first. If the calculated and the measured length agreed, the whole chain between them was sound.
Orléans, 23 December 1793
Struck off the commission, in the middle of the survey
Two men climbing towers and belfries with brass instruments, in a country at war, passed easily for spies. Delambre was stopped, his passports demanded, the seals on his cases broken.
The serious blow came from Paris. A letter from the president of the temporary commission on weights and measures, written on 9 nivôse, told him that a decree of 3 nivôse had struck off six members, himself among them, and that he was to stop work. In our calendar the decree is dated 23 December 1793.
Delambre asked leave to finish at least the stations at Châtillon and Pithiviers and to carry the triangles as far as two belfries that could be found again at any time. Meanwhile, he writes, he hurried to do what he had proposed without waiting for an answer.
What did Delambre do once he learnt he had been struck off?
He carried on. He says so himself: he hurried to do what he had proposed, so that three months of work would not be lost. The survey stopped all the same, for more than a year.
Barcelona, 1792 and 1793
The number that would not fit
Méchain set up in the castle on Montjuïc, above the harbour, and read the latitude from the stars. Then the war between France and Spain kept him in the city, and to use the time he measured the latitude again, from the Fontana de Oro, a building down in the town.
The distance between the two places was known from the triangles, so the second value could be worked back to the first place and set beside the one measured there. Delambre gives them as 900 toises apart.
They did not agree. The two answers for the same point stood 3.4 arcseconds apart. On the ground that much latitude is about 105 metres.
You are holding two latitudes for the same end of the meridian, and they disagree. What do you do?
Méchain mentioned the Barcelona observations to the commission only in passing, as measurements less to be trusted. About the difference he said nothing. Delambre later wrote that Méchain “made a mystery of it to the Commission, and to me as well” (“dont Méchain a fait mystère à la Commission, ainsi qu'à moi”), and that it was why he wanted to go back to Spain.
Paris, 1799 and 1810
What the silence cost
In 1799 the commission gathered the results and fixed the metre. The platinum bar went into the archives, and the value chosen was 443.295936 lignes of the toise of Peru.
Delambre states plainly, in the third volume printed in 1810, what the commission had not had in front of it: “La commission, qui n'avoit pas connoissance des observations de Barcelone, et qui n'a pris que les nombres de Bouguer, en a conclu l'aplatissement 1/334, et le mètre 443ˡ295936.” It had not seen the Barcelona observations, and it drew the flattening of the Earth from Bouguer's numbers, measured in Peru.
Then Delambre does the subtraction that matters. Reckoned from Montjuïc alone the metre comes to 443.3076 lignes. Reckoned with the Barcelona observations as well it comes to 443.328. The difference, he writes, is 0.0204 lignes.
That is what the measurement Méchain kept to himself was worth. How much is 0.0204 lignes?
Forty-six thousandths of a millimetre. Having read everything, Delambre writes: “Après le plus mûr examen je ne trouvais rien ou presque rien à changer aux moyennes qu'il avait adoptées” — nothing, or almost nothing, to change in Méchain's means, so all the commission's calculations stood.
Castellón de la Plana, 20 September 1804
Going back
Méchain asked, with an insistence that surprised everyone, to be the one to carry the meridian further south, past Barcelona. He said the plan was his and that nobody could dispute it. They could not resist him.
He took the triangles as far as Tortosa, reconnoitred the stations to Cullera, and found a site for a six-thousand-toise line on the shore of the Albufera. Six or seven triangles remained.
An epidemic fever stopped him, on top of years of exhaustion. He died on 20 September 1804. Delambre writes that in his delirium he asked for his manuscripts over and over, and that they were laid on his bed to calm him. After his death they were sent back to Paris.
What did Delambre find when he finally read Méchain's notebooks?
“Après le plus mûr examen je ne trouvais rien ou presque rien à changer aux moyennes qu'il avait adoptées,” Delambre writes. All the commission's calculations stood. Méchain had, though, left out part of the Paris observations and the field observations that departed too far from the means.
Paris, 1810 and 1912
What Delambre put in print
What Delambre made of the Barcelona difference can be read in print, as early as 1810, and it leaves no room for doubt: “Cette différence est une preuve évidente des irrégularités de la terre.” This difference is clear evidence of the irregularities of the Earth.
The explanation involving the instrument was not his. He records it as other people's view, “on a pensé”, and then argues against it: such a fault would be possible in one circle, but much less likely in two, and at Barcelona both had been used.
Delambre knew perfectly well what can shift a latitude read from the stars. Elsewhere in the work he discusses the view of the Jesuits Maire and Boscovich that at Perpignan the nearness of the Pyrenees could have pulled the instrument's plumb line to the south. A plumb line does not point at the centre of the Earth; the mountains and the heavy rock nearby pull it sideways. Montjuïc is a hill beside the sea.
And there is one more thing. The latitude Delambre calculates for Perpignan, a point far up the same chain, “s'accorde beaucoup mieux avec les observations de Barcelone qu'avec celles de Montjouy”. It fits the hidden measurement far better than the published one.
The judgement of the man stayed in the manuscript Delambre left at his death in 1822, printed only in 1912. There he writes that Méchain could have appreciably reduced “the irregularities which he found it shorter to hide from our knowledge entirely”.
Three things to try
The repeating circle: which errors shrink and which do not
Move the number of readings. The dots are individual readings and the thick line is their mean. Then switch on a fault in the instrument, which pushes every reading the same way, and watch what happens to the mean.
Three printed values for the same metre
All three are in the books of the time. Pick one and see how far it stands from the others, in lignes of the toise of Peru and in millimetres.
Where your town falls on their arc
The measured arc runs from Dunkirk to Montjuïc, nine degrees and forty minutes of latitude. Romania lies in the same band.
The story's numbers, in brief
| The measured arc, Dunkirk to Montjuïc | 551,571.6 toises, about 1,075 kilometres |
|---|---|
| The arc, in latitude | 9 degrees 40 minutes 24 seconds |
| Principal stations in the chain | 96, plus 34 secondary points |
| The difference Méchain kept to himself | 3.4 arcseconds, about 105 metres on the ground |
| Distance between the two places in Barcelona | 900 toises, about 1,754 metres |
| The metre fixed by the commission, 1799 | 443.295936 lignes of the toise of Peru |
| The metre reckoned from Montjuïc alone, Delambre 1810 | 443.3076 lignes |
| The metre reckoned with Barcelona too, Delambre 1810 | 443.328 lignes |
| What the hidden observations were worth, by Delambre's own subtraction | 0.0204 lignes, or 46 thousandths of a millimetre |
| How far the metre falls short of a ten-millionth of the quarter meridian | 0.197 millimetres |
Quiz: what happened and what did not
Eight statements. Choose, then see what the sources say.
The metre was paced out over the whole length of the meridian.
Only two straight lines were measured on the ground, near Melun and near Perpignan. The rest of the route, more than a thousand kilometres, was calculated from angles taken tower to tower.
Delambre, Base du système métrique décimal, tome III, Paris, 1810
The commission fixed the metre without having seen the Barcelona observations.
Delambre says so himself, in 1810: “La commission, qui n'avoit pas connoissance des observations de Barcelone, et qui n'a pris que les nombres de Bouguer, en a conclu l'aplatissement 1/334, et le mètre 443ˡ295936.”
Delambre, Base du système métrique décimal, tome III, Paris, 1810
Delambre found that Méchain had invented numbers.
Delambre writes that he found “rien ou presque rien à changer” in Méchain's means. What he did find is different: Méchain had left some observations out, had sometimes altered observed angles so that series would agree better, and had said nothing about the Barcelona difference.
Delambre, Grandeur et figure de la Terre, the manuscript published by G. Bigourdan, Paris, 1912
Delambre writes that the two places in Barcelona are 900 toises apart.
Nine hundred toises is 1,754 metres, and Delambre stresses how close they are: so large a difference between two points that near looked to him like clear evidence of the Earth's irregularities. The latitudes in his own table put them somewhat further apart, so the figure stays his.
Delambre, Grandeur et figure de la Terre, the manuscript published by G. Bigourdan, Paris, 1912
Delambre left it open whether the Barcelona difference came from the instrument or from the Earth.
He printed the answer in 1810 without hedging: “Cette différence est une preuve évidente des irrégularités de la terre.” The explanation involving the instrument he gives as other people's view, and argues against it.
Delambre, Base du système métrique décimal, tome III, Paris, 1810
Delambre was struck off the commission measuring the metre while he was measuring it.
A decree of 3 nivôse year II, 23 December 1793, struck off six members, himself among them, and told him to stop work. He learnt of it by a letter of 9 nivôse.
Delambre, Base du système métrique décimal, tome I, Paris, 1806
The measurement Méchain hid would have changed the length of the metre a great deal.
Delambre does the subtraction himself: from Montjuïc alone the metre comes to 443.3076 lignes, and with the Barcelona observations to 443.328. The difference is 0.0204 lignes, forty-six thousandths of a millimetre.
Delambre, Base du système métrique décimal, tome III, Paris, 1810
Delambre came to think the hidden measurement fitted the calculated latitude of Perpignan better.
He writes that the latitude he obtains for Perpignan “s'accorde beaucoup mieux avec les observations de Barcelone qu'avec celles de Montjouy”.
Delambre, Grandeur et figure de la Terre, the manuscript published by G. Bigourdan, Paris, 1912
What the sources cannot say
- Why Méchain kept quiet. No explanation of his own is known. Everything here about his reasons comes from Delambre's notes, written after Méchain's death.
- Exactly how large the difference was. The work printed in 1810 gives it as 3.24 arcseconds, the manuscript published in 1912 as 3.4. This page uses 3.4 and says where it comes from.
- How far apart the two places in Barcelona are. Delambre writes 900 toises, but the latitudes in his own table put them further apart. The figure here is his.
- The second volume of the work, where the Barcelona observations are printed, could not be read for this page. Nothing here depends on it.
Sources
The chain on the map is drawn from the fifth table Delambre printed, which gives for each of the 96 principal stations its distance from the Dunkirk meridian and its observed latitude. Positions are computed from the measured distances, and the printed longitudes beside them serve as the check. Eight stations where Delambre's two columns disagree with each other are recorded as such.
- Delambre, Base du système métrique décimal, tome I, Paris, 1806. india.history.resource.17954
- Delambre, Base du système métrique décimal, tome III, Paris, 1810. india.history.resource.14827
- Delambre, Grandeur et figure de la Terre, the manuscript published by G. Bigourdan, Paris, 1912. grandeuretfigure00dela
- Coastlines and rivers: Natural Earth, 10 m, public domain.