The Meridian and the Revolution

Two carriages in opposite directions

In the last days of June 1792, while in Paris the monarchy was living its last summer, two berlines left the city in opposite directions. They carried identical crates, and inside the crates stood the most precise instruments ever built by human hand: graduated circles of brass and telescopes signed by the craftsman Étienne Lenoir, platinum rules, thermometers, chronometers. On the first carriage, bound north for Dunkirk, travelled Jean-Baptiste Delambre, forty-three years old, an astronomer come late to science after a youth as a tutor, a scholar able to read the heavens and Greek with the same tranquil voracity. On the second, bound south for Barcelona, travelled Pierre Méchain, forty-eight, the most scrupulous observer in France, a hunter of comets esteemed throughout Europe, a melancholy and meticulous man who had never desired anything but his own observatory and his own family [Alder 2002].

They were an improbable and perfect pair. Delambre had come to astronomy past thirty, from philology: a favourite pupil of Lalande, he had made a name calculating celestial tables that had won him the prizes of the Académie, and he brought to his observations the patience of one who has collated manuscripts — nothing disturbed him, everything interested him. Méchain was the establishment of observation: editor of the national astronomical almanac, the Connaissance des Temps, discoverer of a dozen comets, famous throughout Europe for the maniacal fineness of his determinations. One had the nerves to cross a revolution, the other the eyes to measure it to the second of arc; fate, with its customary irony, assigned the revolution to the second and the seconds of arc to the first.

Their task was written in the decrees of the Assembly, and in the passports and proclamations of protection signed by Louis XVI a few weeks before the monarchy fell — documents that would soon become, in the pockets of the two astronomers, more dangerous than useless. The task was to measure, with precision never attempted, the arc of meridian that crosses France from Dunkirk to Barcelona — a little under ten degrees of latitude, a thousandth part of the Earth’s circumference — so that from that arc the length of the quarter of the meridian might be deduced, and from the quarter of the meridian its ten-millionth part: the metre, the new measure of a new world. They reckoned on finishing in a year, perhaps two. It took seven years, a war, a regime of Terror, two careers worn to the bone, and, for one of the two, a secret carried to the grave. The previous chapter closed on the promise of a measure “for all times, for all peoples”; this one recounts what it cost to keep, and what exactly, in the end, was kept.

The choice of the meridian

To understand why two astronomers had to endure seven years of roads and bell-towers, one must return for a moment to the halls of the Académie, where the decision had been taken. The commission appointed in 1791 — Borda, Lagrange, Laplace, Monge, Condorcet: hardly a more authoritative scientific jury had ever gathered round a table — had before it three candidates for the new unit: the pendulum that beats the second, an arc of the equator, an arc of meridian. The pendulum started as favourite: it was measured in a laboratory, it cost little, and half of Enlightenment Europe had been speaking of it for a century. It was set aside with two arguments, one physical and one logical. The first we know from the previous chapter: after Richer, it was known that the pendulum changes with latitude, and to fix it “at forty-five degrees” meant privileging one parallel — a French one, as it happened — and renouncing the universality promised. The second was subtler, and it was Borda who urged it: the seconds pendulum presupposes the second, that is, a unit of time inherited from the Babylonian division of the day — sixty by sixty by twenty-four — which the same commission was planning to abolish in favour of decimal time. To found the new measure on an old and condemned unit would have been to build the house on sand: length, they said, must depend on space alone, not on time [Alder 2002].

There remained the Earth. The equator was picturesque and unreachable; the meridian, on the other hand, crossed France. The arc from Dunkirk to Barcelona had every technical virtue one could wish: the two extremes at sea level, the forty-fifth parallel nearly at the midpoint, and a chain of triangles already laid down half a century earlier by the Cassinis, which it was “merely” a matter of redoing with better instruments. It also had, it is useless to deny, less universal virtues: it gave the enterprise the grandeur of a national monument, entrusted the standard of the world to the meridian of Paris, and closed the door on the cooperation with London that Talleyrand had imagined — a pendulum is negotiable, one’s own meridian is not. The science was sincere and so was the pride: in the great metrological decisions, as we shall see again, the two ingredients do not let themselves be separated.

Around the unit of length the commission then designed the whole edifice, and it is the edifice, more than the metre, that is the invention we use every day: coherent, decimal units, derived one from another without spurious factors. From the metre the litre, its cubic decimetre, and the are for fields; from the litre the gramme, the weight of a cubic centimetre of water; multiples and submultiples named with the Greek and Latin prefixes — kilo, hecto, deca, deci, centi, milli — christened by the law of 18 Germinal Year III, 7 April 1795. The names, too, had their small odyssey: the unit of mass was to be called the grave and to be what is now the kilogram, but the word sounded unwelcome — too close to the German titles of nobility, said some; too learned, said others — and the law fell back on the gramme, demoting the practical unit to a multiple. A scar of this remains visible to this day in the International System: the only base unit that carries a prefix in its name, the kilogram — a linguistic fossil of the Revolution, and a logical thorn that will prick again in the seventh chapter. The same rationalizing fury fell upon the circle, divided into four hundred decimal degrees (the geodesists of the expedition worked in these), upon the calendar, remade in equal months and decades, and upon the day itself: ten hours of a hundred minutes of a hundred seconds. The comparison among the fates of these reforms is a small lesson in the anthropology of measure. The decimalization of money won everywhere and almost at once: the Russia of Peter the Great had opened the way with the rouble of a hundred kopeks, the United States had taken it with the dollar in 1792, France followed with the franc of a hundred centimes — because money is already pure arithmetic, and he who counts prefers to count in comfort. The revolutionary calendar, with its poetic months and its ten-day weeks, held for twelve years of proclamations and fell on the first of January 1806, brought down less by nostalgia than by the arithmetic of rest: one Sunday every ten days instead of every seven is a reform that makes itself hated all on its own. Here reason met its limit, and it is an instructive limit. Decimal time, compulsory on paper from 1793, was suspended after a few months: one could recast a bushel, but not the bell-towers, the clocks, the habits, and the prayers of a whole people.

Measures, the first chapter taught, are embodied in the life they must serve; time is more so than all of them, and indeed it was the one conquest that decimalization never managed to take. Meanwhile, so as not to halt the country, the law of August 1793 had launched a provisional metre, calculated on the old data of the Cassinis: 443.44 lines of the toise du Pérou, the historic length-standard of the Académie. The definitive one would have to await the two astronomers — and would prove, a detail to which we shall return, a breath shorter. And so that the people might learn the new measure, the Republic made a gesture the reader of the first chapter will recognize at once: it walled the standard into the public way. Sixteen marble standard-metres, with their dutiful graduated groove, were affixed between 1796 and 1797 in the busiest spots of Paris, so that every citizen might lay his wooden metre against them and become the judge of his own span; two survive still, one in the rue de Vaugirard and one under the arcades of the place Vendôme, places of pilgrimage for sentimental metrologists. The most modern measure in the world made its debut with the most ancient gesture in the world: the revolution of units, like all successful revolutions, knew when not to innovate.

The art of triangles

How does one measure a tenth of the Earth’s circumference without unrolling a cord from Dunkirk to Barcelona? The answer has an elegance that deserves a moment’s attention, because it is the same with which all the continents of the planet would be measured, down to the age of satellites. One does not measure distances: one measures angles. One chooses points high and visible to one another — bell-towers, towers, summits — and thinks of them as the vertices of a chain of imaginary triangles covering the country. From each vertex one measures the angles under which the others are seen; and since a triangle is wholly determined by two angles and one side, it suffices to know the length of one single side of one single triangle for the whole chain to resolve itself, side after side, by trigonometry: in the triangle with vertices $A$, $B$, $C$, the sides stand to one another as the sines of the opposite angles, $a/\sin\alpha = b/\sin\beta$. Once that single side — the “base” — is measured on the ground, all the rest is geometry and patience [Alder 2002].

Patience, naturally, was the point. The angles were measured with the technological jewel of the expedition, the repeating circle conceived in Germany and perfected by Borda: two telescopes mounted on a graduated circle, with a mechanism that allows the same angle to be summed many times running along the graduation, so that the inevitable reading errors, spread over scores of repetitions, dissolve into the average. It is an idea we shall meet again, for it contains in germ the whole nineteenth-century theory of errors: unable to eliminate the mistake, one multiplies it in order to divide it. The bases, then, were measured in the most brutal and most delicate way at once: by laying on the ground, one after another, thousands of times, four bimetallic rules two fathoms long, platinum below and copper above, in which the differing expansion of the two metals served as a built-in thermometer, correcting of itself the effect of heat and cold. The base of Melun, near Paris, and that of Perpignan, near the Pyrenees — about ten kilometres each — cost weeks of that Carthusian labour stretched out along the high roads. And here geometry offered the enterprise its moment of glory, which no revolutionary rhetoric could have invented better: of the two bases only one would have sufficed, because the second could be calculated by propagating the triangles along the whole chain; to measure it nonetheless meant subjecting seven years of angles to an examination beyond appeal. When the calculated value of Perpignan, made to travel hundreds of kilometres of trigonometry, was compared with the one measured rule by rule on the ground, the discrepancy proved to be of the order of a few decimetres over eleven kilometres: a few parts in a hundred thousand. It was the proof, internal and merciless, that the chain held — and it remains one of the finest examples of what distinguishes a scientific measure from an estimate: the capacity to predict its own check [Alder 2002]. Finally, for the chain of triangles to become an arc of meridian, at the extremes one had to raise one’s eyes: the latitude of Dunkirk and that of Barcelona had to be read in the stars, pursuing night after night the passage of the heavenly bodies across the meridian with the circle mounted vertically. Geometry on the ground, astronomy at the extremes: the geodesy of the age stood wholly in this pincer.

There was then all the kitchen of the craft, which the official reports compress into a line and which devoured the months. The vertices had to be made visible at twenty, thirty, sometimes fifty kilometres: pyramids of wood and canvas hoisted on the summits, reflectors that sent back the sun, lamps for nocturnal observations when the daytime air shimmered too much. The measured angles then had to be corrected and reduced: brought back to the exact centre of the signal when the instrument could not stand there, projected onto the horizon, and purged of a fineness that tells the level of the enterprise — over triangles of that size the Earth’s surface is no longer a plane, and the angles sum something more than a hundred and eighty degrees, a “spherical excess” minute but systematic, which Legendre’s new mathematics taught how to calculate and distribute. Finally the whole chain, which wound through France touching the summits where the summits were, had to be projected by calculation onto the ideal meridian of Paris and reduced to sea level. Every link of this chain of corrections was fresh theory, often written for the purpose: the expedition was not merely using the science of its time, it was producing it as it went.

It is worth noting what, in this method, depends on trust: everything. No commissioner would ever retrace the triangles; no citizen would ever verify a register. The guarantee lay in the procedures — repetitions, cross-checks, the sum of the angles of each triangle compared with its hundred and eighty degrees — and in the reputation of two men. The old world entrusted measure to the walled standard of the market; the new entrusted it to protocols and to persons. It is a transfer of trust, not its abolition: the reader of the first chapter will recognize the theme, and the drama of Méchain will lie wholly in the point where protocol and person crack together.

The astronomer in time of Terror

Geodesy in time of peace is toil; in time of revolution it is a permanent misunderstanding. A man who posts himself on bell-towers with telescopes, makes luminous signals toward the horizon, and fills notebooks with figures resembles a spy in the most embarrassing way, and the France of 1792 saw spies everywhere, having in fact a good many. Delambre was stopped a first time a few leagues from Paris, at Épinay-sur-Seine, by volunteers marching to the front: there arose one of the most memorable scenes in the history of science, the astronomer who at dusk, surrounded by an armed and suspicious crowd, improvises a public lecture in geodesy — the meridian, the triangles, the measuring of the Earth — with his life hanging on the clarity of his teaching. He was released, stopped again, released once more: his safe-conducts, signed by an authority that changed its name every season, aged faster than his triangles [Alder 2002]. He soon learned the practical details of survival: the geodetic signals, traditionally white cloths, in a country where white was the colour of the Bourbons had to be edged with blue and red; certain bell-towers that served him as vertices had been decapitated as symbols of superstition, and had to be replaced with poles and platforms; certain others collapsed of old age beneath the observer.

Then politics reached the numbers too. At the end of 1793 the Committee of Public Safety purged the commission of measures: Delambre was removed from his post together with Borda, Laplace, Coulomb, and Brisson, with a justification Delambre himself preserved and which deserves to be remembered — those purged did not offer sufficient guarantees of “republican virtues and hatred of kings” [Alder 2002]. The measuring of the Earth, suspended for want of hatred: the Enlightenment had not foreseen this item in its budgets. Lavoisier, who had been the treasurer and organizational engine of the metric system and who with Haüy had begun the precision weighings for the unit of mass, was guillotined in May 1794, not for the measures but for the tax farms he had kept alive as a financier.

After Thermidor the political pendulum swung back: the law of Germinal Year III set the metric system on its feet again and recalled the astronomers to their triangles. Delambre set off again without a word of resentment — resentment subtracts time from observations — and ground out, station after station, his half of France. In Paris he had turned into a geodetic vertex the most solemn monument available: the dome of the Panthéon, where the fatherland was burying its great men, surmounted for the occasion by a provisional observatory — beneath him Voltaire and Rousseau, before him the horizon of bell-towers to be measured; it is hard to compose a better allegory of the century. From there the chain descended through the Sologne and the Berry to the meeting: in 1797 the two halves of the arc, his descending and Méchain’s ascending, joined near Rodez, beneath the red-sandstone bell-tower that dominates the uplands of the Rouergue. The geometric junction was made. The human one was another matter.

Méchain’s secret

The southern half of the arc was shorter, and it fell to Méchain because it was the harder: the Pyrenees, and the new extension as far as Barcelona, where the Cassinis had never reached. The first months were the best of his scientific life. With the engineer-geographer Tranchot opening the way summit by summit, and in concert with the Spanish commissioners — the war had not yet come — he pursued his signals up a mountain that did not forgive: stations reached on muleback, wooden pyramids rebuilt after every storm, nights of waiting for the air to grow calm enough not to make the lamps dance on the horizon. He closed the triangles of Catalonia in a single season, and in the winter of 1792–93 he installed himself on the hill of Montjuïc, in the fortress that looks over Barcelona and the sea, to measure its latitude. It was, by the canons of the time, a masterpiece: nights upon nights of stellar transits, thousands of readings at the repeating circle, a latitude determined with a fineness no one had ever reached. Méchain was this: the man who did not stop until the last figure ceased to tremble [Alder 2002].

Then 1793 overturned upon him, in order, a war and a machine. The war was that between the Convention and Spain, which suddenly made of him an enemy in hostile territory: no more fortress, no more triangles, a French astronomer confined to Barcelona with his instruments under seal. The machine was a hydraulic pump that a friend insisted on showing him: a gear seized him and flung him against a wall, shattering his ribs and collarbone and leaving him for days between life and death. He healed crooked and in pain, with his right hand half useless; and it was during his convalescence, a luxury prisoner in a city at war, that he committed the one act of his life he could never forgive himself: he redid the observations. From the roof of the inn that lodged him, out of scruple, out of boredom, out of that incapacity to leave his own numbers in peace which was at once his greatness and his sickness, he remeasured the latitude of Barcelona. And he found another value. The discrepancy with Montjuïc was about three seconds of arc — a hundred metres or so on the ground: nothing for a traveller, an abyss for a man who had promised the world the exact ten-millionth part of the quarter of the meridian [Alder 2002].

Three seconds of arc, and no way to understand. To return to Montjuïc, a military fortress in time of war, was impossible; to reproduce from an inn’s roof the conditions of an observatory, equally so. Which of the two Barcelonas was the true one? Méchain did what his century, in the end, suggested to him: he held the discrepancy to be a fault. His own, his instrument’s, of the diligence that must have failed somewhere. And he hid it. He did not report it to Borda, nor to Delambre, nor to the commission: he handed over the latitude of Montjuïc, the best, the first, and kept the other in his notebooks and in his stomach. From that day the most scrupulous man in France lived like one with a crack in his foundations: he took refuge in Italy when Spain became unliveable for him, returned to France in 1795, and began to slow down. Every triangle of the south became a pretext for not finishing: to redo, to verify, to winter over, to redo again. His wife Thérèse, who had not seen him for years, had to join him in the Midi to tear him from the siege of his own scruples; Delambre, with a tact that moves one today, covered his slowness by redistributing stations and deadlines. Méchain finished his chain out of exhaustion more than out of peace made, and when the enterprise closed he long refused to hand over the original registers of the observations — the very place where the secret dwelt. There was, in that refusal, also the habit of an age: the observers of the eighteenth century handed over results, not observations — cleaned averages, not columns of raw figures with their deviations on display — because the raw datum seemed a back room the public was not to see. Méchain carried that habit to its pathological extreme, and involuntarily demonstrated its vice: where the raw data are private, the discrepancy becomes a personal secret instead of a scientific fact, and the keeper of the secret consumes himself. Later science took the opposite road — to publish the observations, the deviations, the methods — not out of virtue, but because cases like this taught it the price of reticence [Alder 2002].

Let us pause a moment, for here the story touches one of the nerves of the book. Méchain falsified nothing, in the sense that he invented no numbers: he chose, between two honest measures, the one he believed better, and kept silent about the existence of the other. Today any laboratory student knows that two series of observations made under different conditions must differ, that the deviation is declared, estimated, and published, and that the discrepancy is not a shame but a datum. But this wisdom — the idea that error is a statistical property of the measurement and not a moral defect of the measurer — did not yet exist in 1794: it would be built, in the sixth chapter, precisely by the astronomers of the following generation, meditating among other things on cases like this. Méchain lived exactly on the ridge between two ages: he had the instruments capable of seeing deviations no one had ever seen, and he did not yet have the theory that would have let him look at them without feeling guilty for them. His tragedy was not imprecision: it was precision, arrived before the word to name it.

The metre of platinum

While Méchain slowed, France was in a hurry. In 1798 the government convened in Paris what would be, with hindsight, the first international scientific conference in history: delegates of the allied and neutral countries — from Holland to Denmark, from Spain to the Italian states — invited to verify calculations and procedures and to christen the result, so that the measure “of all peoples” might not seem the measure of one alone. For months, mixed commissions redid the sums: the angles, the bases, the latitudes, the weighings of the future kilogram. The final report was drafted by the Dutchman van Swinden, and the figure that issued from it had the dry savour of sentences: the quarter of the meridian was worth 5,130,740 toises, and the metre, its ten-millionth part, 443.296 lines of the same toise du Pérou — four hundredths of a line less than the provisional metre, a difference of a tenth of a millimetre that almost no citizen would ever perceive and that cost seven years of lives [Alder 2002].

There was, in the data, also a scientific piece of news of the first rank: comparing the arc of France with the one measured half a century earlier in Peru, the flattening of the Earth came out at about one three-hundred-and-thirty-fourth — confirmation, on new grounds, that the planet is not a sphere. It was at once a triumph and a worm: if the Earth is flattened, and flattened in a not perfectly regular way, then every meridian has its own history, and the “ten-millionth part of the quarter of the meridian” depends on which meridian, and on how one crosses it. The worm was minuted and set aside: the figure was the figure, and they could cast. The foundry, in turn, was a frontier. For the standards platinum had been chosen, a metal then at the margins of chemistry: almost unattackable by acids, refractory to melting, twice as dense as silver — the only material that seemed equal to the promised eternity, and one of the very few that no furnace in Europe yet knew how to handle with confidence. The goldsmith Janety, the recognized virtuoso of platinum, purified the metal for the kilogram; from Lenoir’s workshop issued the bar of the metre, twenty-five millimetres wide and four thick, worked and compared until it best reproduced the 443.296 lines of the sentence. As for the kilogram, its pedigree had been if possible more refined: the mass of a cubic decimetre of water, yes — but which water? The precision weighings conducted by Lefèvre-Gineau had established that the water be taken at the point of its maximum density, around four degrees, where an error of temperature does least harm: the unit of mass was born already clinging to a fine property of matter, the first hint of a dependence — mass defined by means of physics, and not the reverse — destined to explode as a problem a century and a half later. On 22 June 1799 — 4 Messidor Year VII — a procession of scientists delivered to the National Archives a platinum bar of one metre and a platinum cylinder of one kilogram: the mètre des Archives and the kilogramme des Archives, deposited beside the constitutions as what they were, founding acts. In the official speech it was said that the standard, drawn from the Earth itself, belonged “to all times, to all peoples”; in fact, from that morning, the quarter of the meridian had ceased to count. The bar counted.

The reader of the first chapter has already seen this scene, played by other actors: it is the toise of the Châtelet, it is the granite cubit, it is the arm walled in beside the door — the standard that, an instant after its consecration, detaches itself from the thing it was meant to represent and begins to be worth something by its own authority. The Revolution had promised a measure taken from nature so that no king could possess it; it obtained a measure taken once only from nature, and then kept in a state archive like the most precious of the crown jewels. It is not an accusation: it is the logical structure of every standard, and no geodesy could circumvent it. Nature delivers no units; it delivers, to whoever questions it with sufficient obstinacy, a number — and the number, to become a measure, must become an object, and the object, to endure, must become a convention. “For all times, for all peoples” did not describe the metre: it described the commitment to behave as if the metre were eternal. Which is, on reflection, what every promise does.

There is indeed, in the metre of 1799, a subtlety that distinguishes it from all the “natural” standards dreamed of before and after. The pendulum was an experiment: anyone, anywhere, in any age, could have redone it and recovered the unit. The meridian, no: it was an expedition — a historical event, dated and signed, irreproducible as all events are. Whoever remeasured the arc a century later, with better instruments, would find a different number; and indeed he did, and nothing came of it. The metre is therefore not the child of a law of nature, but of a measurement: of that measurement, with those instruments, those men, and those winters. The Revolution believed it was anchoring the unit to geography and anchored it, without knowing, to history — which, for an age convinced it was beginning time over from zero, is perhaps the most fitting outcome imaginable.

The legacy of the wrong metre

What that commitment was worth, the following decades tell, for they tested the metre in every possible way. Geodesy tested it: with ever better instruments and arcs it was understood that the Earth is still less regular than expected, that the true flattening is closer to one three-hundredth than to one three-hundred-and-thirty-fourth, and that the quarter of the meridian measures about 10,002 kilometres: the metre of the Archives is therefore shorter than its geodetic ideal by about a fifth of a millimetre. The news, when it matured in the nineteenth century, changed nothing — and this is the point that makes it precious. No one proposed lengthening the metre: the definition was let to slide, silently, from the Earth to the bar. The “wrong” metre stayed the right metre, because right, in metrology, means shared, stable, and reproducible, not faithful to a piece of planet that, moreover, will not let itself be measured twice in the same way. The deflection of the vertical that tilts the plumb lines near the mountains, the anomalous refractions, the flexures of the instruments: everything that had tormented the observers became, in time, matter for science instead of for guilt — and the standard, emancipated from its origin, went on doing its job as if nothing had happened [Alder 2002].

Méchain did not live long enough to be absolved. He obtained, in 1803, what he had been asking for years: to return to Spain to prolong the arc as far as the Balearics, pursuing the secret hope of settling his accounts with Barcelona and with himself. He died there of yellow fever, at Castellón de la Plana, in September 1804, far from his wife and from the triangles of home. The extension he had not been able to complete was carried to its end by two men of the new generation, Biot and the very young Arago, and it cost the latter an epic that is worth a line: surprised by the Spanish war of 1808 on his mountain stations, taken for a spy, imprisoned in Mallorca, escaped to Algiers, recaptured by corsairs, shipwrecked and hostage, Arago recrossed the Mediterranean defending above all else the registers of the observations, and delivered them to Paris in 1809 as one delivers a relic. The measuring of the Earth still demanded, of those who served it, a pilgrim’s devotion; and its notebooks had become objects for which men risked their lives. It was Delambre — by now perpetual secretary of the Académie, survivor of everything with the same composure with which he had crossed the checkpoints — who found the false bottom, sorting his colleague’s papers to write the official history of the enterprise: the second observations, the recopied pages, the discrepancy kept silent for ten years. He could have buried it all, and no one would have known; instead he published the whole affair in the Base du système métrique, with a delicacy that is itself an epistemological document. Méchain, he wrote in substance, had committed not fraud but an excess of conscience: he had demanded of his own numbers an agreement that the nature of the instruments and the places could not grant, and had mistaken for personal failure what was the normal behaviour of measures at the limit of the possible. The absolution came from a new world: in those pages, among the first of the scientific literature, the deviation between observations ceases to be a secret to be kept and becomes information to be published. Today the arc of Delambre and Méchain has been reanalysed with modern methods: their measures were excellent, and the discrepancies that tormented Méchain reflected real systematic effects — refraction, flexures, local attractions — that no diligence in the world could have eliminated, only declared. The theory capable of saying so with numbers, and of turning a man’s torment into a probability distribution, was being born in those very years in the hands of Gauss and Laplace: we shall meet it in the sixth chapter, and the reader will do well to arrive there remembering the roof of the inn at Barcelona.

The metre, meanwhile, was beginning its career. It was a slow career full of ironies: Napoleonic France, for all its proclamations, reintroduced in 1812 the “customary measures” — the foot and the pound of old, redefined on a metric basis — because the markets would have none of it, and only a law of 1837 made the metric system compulsory from 1840. The same Napoleon who shelved it at home had meanwhile sown it across half of Europe in the baggage of his armies and his codes: and indeed the first to adopt the metre by law, in 1816, were the Netherlands just out of French occupation — the measure of universal liberty, naturalized by way of conquest. The history of units never goes where its prophets send it, but it goes there with their instruments. The measure born for the people had to be imposed on the people: Kula, whom we read in the first chapter, would have remarked that no measure is ever merely an instrument, and that the peasants were being asked to surrender, along with the bushel, a whole order of relations they knew how to handle. But the long history went in another direction, and the next chapters will tell it: the metre became the lingua franca of science, then of continental commerce, and at last — with the treaty of 1875 and the international prototype of 1889 — of a planetary community of states. And when the platinum bar too showed its limits, the definition returned twice to nature, but to a nature very different from the meridian: in 1960 the metre was redefined as a number of wavelengths of an orange line of krypton-86, and in 1983 as the distance light travels in vacuum in 1/299,792,458 of a second, fixing once and for all the speed of light [BIPM 2019] . The parabola deserves to be viewed from afar: the Revolution had sought the unit in the body of the planet, the twentieth century found it in the behaviour of light — a standard that does not deform, does not age, and dwells in no archive, because it is identical in every laboratory in the universe. Condorcet’s dream came true to the letter, two centuries later and by another road: but note — it is the thread of the book — that the speed of light too became the metre by decree of an assembly, voted by delegates in a congress hall. Nature furnishes the constants; the measure, now and always, is made by men gathered together.

There remained, after space, the twin magnitude: time. While France triangulated its bell-towers, England had already spent a century pursuing on the ocean a problem apparently wholly practical — knowing what time it is aboard a ship — and had solved it with a machine destined to change the measure of the world at least as much as the metre. It is the story of a prize, a carpenter, and a clock: the next chapter.