Measuring: lengths, weights, volumes

A marble plaque fixed to a stone wall, bearing a horizontal ruler between two stops and the inscription “mètre”. — A public standard metre from the eighteenth century, rue de Vaugirard in Paris: anyone could come and check that their ruler really measured a metre. Airair, CC BY-SA 3.0, source

How long is a metre?

You can show it with your hands, roughly. But “roughly” is not enough when it comes to selling cloth, building a bridge or paying a tax. So people had to agree. Agreeing on a length turned out to be far more complicated than one might imagine.

Whose foot?

The first measures come from the body. The foot, the cubit — from the elbow to the fingertips — the span — the distance between the thumb and the little finger, with the hand open — the fathom — both arms outstretched.

It is practical: you always carry your instruments with you. But the flaw is obvious. Whose foot? An adult’s foot and a child’s foot are not the same length. Two people measuring the same field do not get the same result.

As long as you measure for yourself, it does not matter. As soon as you trade, it does.

Making a reference object

The first solution was a material one: make an object that serves as a model, and compare all the others to it. This is called a standard.

In Egypt, rulers of stone or wood were cut to give the official cubit. Some survive. They measure about 52 centimetres. On them the cubit is divided into seven palms, and each palm into four fingers: the hand serves as the scale.¹

An aside, because it is instructive. The most famous of these rulers, kept in Turin, has imprecise subdivisions. The museum therefore thinks it was a prestige object, given to a high official, and not a tool for building work. The most photographed standard may not be a standard that was ever used.²

The most impressive case comes from the Indus Valley, more than four thousand years ago. At more than forty sites spread over two thousand kilometres, small cubes of hard stone used as weights have been found. They are practically identical from one site to another.

They follow a simple rule: each weight is worth double the previous one. One, two, four, eight, sixteen, thirty-two. The smallest weighs less than a gram. The most common weighs a little under fourteen grams.³

We cannot read this civilisation’s script. So we cannot read what they wrote on their weights. But the objects themselves speak: someone, somewhere, had decided the value of the local gram, and everyone kept to it.

A little further west, in Mesopotamia, people weighed with a beam balance: a bar suspended at its middle, with a pan at each end. First the empty pans are balanced, then the object is placed on one side and known weights on the other. The principle is so convenient that it is found everywhere, from Egypt to China.⁴

The weights themselves are often carved from a black, polished stone, in the shape of a duck or a lion. Sixty small ones make a medium one, sixty medium ones make a large one: the shekel, the mina, the talent. Base sixty once again: the base of the Babylonian tablets and of our sixty minutes. The smallest weight found weighs fourteen hundredths of a gram — so people weighed tiny things, and did so deliberately.⁵

The standard that carries its own law

In China, in 221 BCE, a king had just conquered all the others and proclaimed himself emperor. His very first act was neither a war nor a temple: it was an edict unifying the weights and measures of the empire.

The most interesting part is how he went about it. The edict was not merely posted: it was engraved, or cast in metal, into the weights and measures themselves. The object carries the text that makes it compulsory. An iron weight found in the north weighs thirty-two kilograms and bears forty characters inlaid in copper.⁶

This is how we know that the measure was really applied: such objects have been found in provinces very far apart from one another. The order from the capital reached the furthest borders, and it arrived in the form of an object that could be placed on a balance.

The lord’s bushel

This is where measurement stops being only a matter of science.

For a very long time, taxes were paid in grain. The peasant brought his harvest and poured it into a wooden container, the bushel. If this container was a little larger than it should be, the peasant gave more without it showing. Should it be filled to the brim, or heaped up until it overflowed? The difference could reach several per cent.

The bushel, as it happens, belonged to the lord. Measuring, weighing, checking the instruments: these were rights he held, and they earned him money.

Before the Revolution, France had roughly eight hundred unit names. The worst was not their number, but that the same name did not mean the same thing everywhere. The king’s pound weighed 489 grams, but other pounds in use weighed from 404 to 456 grams. The pint was less than a litre in Paris and nearly a litre and a half in Saint-Denis, a few kilometres away. A setier held twelve bushels of wheat, but sixteen of salt, twenty-four of oats and thirty-two of charcoal.⁷

In 1789, when the French were asked to write down their grievances, the demand came up everywhere. In the Pas-de-Calais, it appears in several cahiers. The one from the village of Oppy calls for the establishment of a single weight and a single measure throughout the kingdom. (The cahier writes “poid”, without an s: spelling was not yet fixed.) Elsewhere, the lord was targeted directly: everyone, says one cahier, has his own different bushel.⁸

These papers have been preserved, and you can go and read them. The originals even have shelf marks — 2 B 881 to 2 B 884 — and anyone can request them from the departmental archives.⁹

The story should not, however, be oversimplified. The demand also appears in the cahiers of the nobility and the clergy, often presented as a hindrance to trade rather than as an injustice. Not everyone wanted it: one municipality feared that standardising grain measures would drive away the merchants, part of whose profit came precisely from these discrepancies.

On the night of 4 August 1789, privileges were abolished — and with them the seigneurial right to measure, to weigh and to check the instruments.

Measuring the Earth to define the metre

On 26 March 1791, a decision was taken in France: a unit would be created that depended on no one. Not on a king’s foot, not on an object kept in some city. The text says so explicitly: a unit containing nothing arbitrary, and nothing particular to any nation on the globe.

The idea adopted was to derive it from the Earth itself. The metre would be the ten-millionth part of the distance between the North Pole and the equator.¹⁰

That distance still had to be measured. Two astronomers, Delambre and Méchain, set out in 1792 to measure, triangle by triangle, the arc of meridian between Dunkirk and Barcelona. The mission was supposed to last a year. It lasted six. They worked in the middle of a war, and were arrested, suspected, held up.¹¹

In 1795, the metric system was established by law. In 1799, the definitive length was fixed and cast in a platinum bar.

Then comes the part of the story that is rarely told: this measurement is wrong.

In Barcelona, Méchain obtained two series of results. Each was carefully made. They did not agree.

Today, we would know what to do with such a discrepancy: there is a science of measurement errors. In his day, it did not yet exist — it would be born in the following century. Méchain therefore believed he had made a mistake. He did not dare say so, and it would torment him for the rest of his life.

He did worse than keep quiet. His papers, found much later, show that he altered his figures to bring them closer to those of his colleague. The remarkable agreement between the two astronomers, so reassuring at the time, was partly fabricated.¹²

The irony of history is cruel. The data he set aside because they troubled him are closer to today’s accepted value than those he kept.¹³

He suppressed the good ones.

After his death, Delambre went through his notebooks and understood. He wrote in the margin that he would not speak of it to the public, and put the papers under seal.

What this story eventually produced

The 1799 metre is too short by about two tenths of a millimetre — the thickness of two pages of a book.¹⁴

Méchain’s error, incidentally, has little to do with it: the real problem is that the Earth does not have the regular shape it was assumed to have. Their measurements were so fine that they no longer fitted any neat curve. They were measuring very accurately an object they pictured wrongly.

A cruel detail: their result, obtained over seven years, turned out to be less accurate than a provisional value adopted in 1793 on the basis of an earlier, and yet less precise, survey.¹⁵

Yet it was from this failure that a crucial idea emerged, and it is used on every page of this site.

No measurement is free of error. A researcher must therefore not only reduce their error: they must estimate it and state it. A published result is not a true value, it is a value accompanied by its uncertainty — one acknowledges the error in order to control it.¹⁶

This seems obvious to us. It was not. Méchain died without ever having been granted that right.

Units that can no longer be distorted

A platinum bar is better than a king’s foot. But it is still an object, and an object has a life of its own.

The cylinder that defined the kilogram, kept under three glass bell jars near Paris, did not stay equal to its copies. Over a century, a gap of about fifty micrograms opened up between it and its official copies — the weight of an eyelash. Had the copies grown heavier, or had it grown lighter? It is impossible to decide. Since all the masses in the world were referred to it, if it lost a little, the entire universe grew heavier by the same amount, simply by definition.¹⁷

So, in 2018, an international conference decided to take the final step: units would no longer rest on any object. They would be defined from constants of nature — numbers that never change, wherever one is in the universe. The change came into force in 2019.¹⁸

To do this, these constants had to be measured with extreme precision. In the United States, the physicist Darine Haddad and her team spent three years measuring one of them on a balance of a special kind, in which the weight of an object is offset by an electric current. It is this number that defines the kilogram today.

No one can distort the kilogram any more. It is the culmination of an idea two centuries old, and of a problem several thousand years old.

That said, the body has not disappeared. Screens are still measured in inches, and aircraft give their altitude in feet. The nautical mile, by contrast, owes nothing to the body: it is worth about one minute of arc of meridian, and it was fixed at exactly 1,852 metres by an international conference in 1929.¹⁹

To finish, a detail that should be reassuring about the Mesopotamians. Their shekel, so well defined, weighed 7.8 grams here and 9.4 grams there, and some fifteen different values are known for it. Exactly like the pint of Paris and that of Saint-Denis, three thousand years later. The same name not meaning the same thing from place to place: this is the permanent problem of measurement, and it took until 2019 to truly escape it.

Four ways to fix a measure

What it rests on	Example	Weak point
The human body	foot, cubit, span	varies from person to person
A grain, a seed	the Mesopotamian shekel, about 180 grains of barley	grains are not all alike
A reference object	Indus weights, Egyptian cubit, Qin weights, platinum metre	the object can be deformed or lost
A constant of nature	current units since 2019	none, but very difficult to measure

Note 1

Surviving cubit rods: […] From these examples, it has been calculated that the cubit corresponded to 52 centimetres. Divisions of the cubit: One cubit = 7 palms (Egyptian Ssp) One cubit = 28 fingers (Egyptian Dba)

University College London, “Measuring length in Ancient Egypt”, Digital Egypt for Universities, 2002