The Measure of All Things

by Ken Alder

  There was only one snag. The Spanish general refused to let him leave. “But alas, where am I? In irons! And yet I speak like a man free to indulge his passionate zeal for the success of this superb mission. No matter; at least I have tried to make my slavery useful, if not to the mission itself, then at least to astronomy.”

  Méchain was too obsessive an astronomer to remain idle. They had barred him from Mont-Jouy, but they had not forbidden astronomical work at his Barcelona hotel. So in December he reorganized his observatory on the terrace of the Fontana de Oro. This time he would take advantage of the winter solstice to measure the angle of the earth’s rotation relative to its orbit around the sun. For this, he would also need an exact determination of the latitude of his hotel terrace; last winter’s data, gathered at Mont-Jouy, would not serve his purpose. Mont-Jouy, although readily visible to the south of the city, was over a mile distant. Equipped with the world’s most exact astronomical instrument, he intended to make this measurement with greater precision than any investigator in the past four thousand years. As an added bonus, his observations would also offer a double-check on his latitude results for Mont-Jouy.

  Méchain’s motives for undertaking these observations—the results of which were to haunt him for the rest of his life—appear to have been mixed, as motives often are. Certainly he wished to prove to his Paris colleagues and his Spanish hosts that he remained the same meticulous astronomer as before, and that the accident of the previous April had not diminished his abilities. This would silence any talk of replacing him on the meridian expedition. It would also demonstrate his diligence at a time when individuals who refused to serve the public good risked execution.

  There was also something about his earlier results that nagged at Méchain. To calculate the latitude at Mont-Jouy, he had measured the heights of six different stars: Polaris, Kochab, Thuban, Mizar, Elnath, and Pollux. More was always better. Thoroughness was always rewarded. For his final analysis, he had used the results of the first four of these stars, those for which he had gathered the most data. Of these, the results for the first three converged to a remarkable degree, giving average latitudes of 41°21‘44.91” (Polaris), 41°21'45.19” (Thuban), and 41°21'45.19” (Kochab). The total spread in these values came to an infinitesimal 0.3 seconds of a degree of arc. This suggested that Méchain had determined the global location of the castle tower of Mont-Jouy to within thirty feet. It was a stunning display of astronomical virtuosity, the kind of precise result that had won him the leadership of the southern expedition.

  Results based on the fourth star, Mizar, however, diverged from this pattern, and indicated a latitude of 41°21'41.00”, which differed from the others by four seconds of arc, or some four hundred feet. This anomaly irritated Méchain. Why did the readings for this one star differ tenfold from the rest? It was a natural question for a natural philosopher to ask. Yet even then, he might have let sleeping data lie. Only a decade before, a discrepancy of four seconds would have been a remarkable achievement. It came to little more than 0.01 percent of the six-hundred-mile arc from Dunkerque to Mont-Jouy. Besides, he had already summarized these astronomical results for his Spanish hosts and sent a précis to Borda in Paris.

  On the other hand, Méchain had a hypothesis that might explain this discrepancy. Ah, that other hand! Why is it always with that “other hand” that scientists open Pandora’s box? They do not open it to make their lives more difficult. More often they are simply seeking to reassure themselves, to confirm what they think they already know to a finer degree of certainty. But, for better or worse, they do not always know what they think they know. Sometimes they even have the good fortune to be mistaken. And then, as Enrico Fermi once said, they may make a discovery.

  The problem with the Mizar data, Méchain hypothesized, was refraction. The corrections for the bending of light had been worked out by astronomers in London and Paris. Perhaps their corrections did not apply to southerly towns like Barcelona, where those stars crossed the meridian closer to the horizon and the higher temperatures distorted their sighting through the atmosphere. Of all the stars he had measured, Mizar crossed the meridian closest to the horizon. The corrections were small in the first place, of course, and any adjustment would necessarily be smaller still. But the meridian expedition was operating with an unheard-of degree of precision. The repeating circle promised precision limited only by the patience of the observer. And Méchain refused to believe the fault lay with the stars.

  So he spent the winter of 1793–94 on his back on the terrace of his Barcelona hotel, taking nighttime observations. While Tranchot held the lantern and verified the spirit level, Méchain hunkered down as before, whirling the circle, then the scope; listening for the clock to beat the moment of the star’s meridian transit; then whirling the circle, then the scope, and repeating his eye-and-ear measurement. He conducted observations on Christmas Eve, on Christmas night, on New Year’s Eve, and on the first night of the new year, plus every clear night in December, January, February, and March. He took 910 stellar readings, each with ten or more repetitions, for a Herculean total of some ten thousand observations. Then, inside his hotel, during the daylight hours, he calculated his way through this mass of data, his refraction tables and logarithmic tables continually by his side. By early March he had determined the north latitude of his hotel to be 41°22'47.43” (based on Polaris), 41°22'48.38” (based on Kochab), and 41°22'44.10” (based on Mizar). Once again, the results for the first two stars (those in which he had the most confidence) agreed to within an impressive one second of arc (or one hundred feet), making the Fontana de Oro the most accurately located hotel on the face of the planet. Yet once again the Mizar data gave discordant results, differing some four hundred feet from the others.

  One final step would clarify the mystery. Méchain would now need to compare his new latitude results at the Fontana de Oro with his old results for Mont-Jouy by subtracting the distance between them. Calculating this distance, of course, was just the sort of task his expedition had been equipped to perform. He laid out a triangulation that included his hotel, Mont-Jouy, and the cathedral of Barcelona, and to make doubly sure, a second triangulation which included his hotel, Mont-Jouy, and the Lanterna that served as the port lighthouse. There was only one snag: to carry out the triangulation accurately, he would need to take angle measurements at each station, and the Mont-Jouy castle was closed to him as a Frenchman. It was so near, yet just out of reach.

  By mid-March, with Tranchot’s help, he had taken measurements from his hotel, the cathedral, and the Lanterna. In the meantime, he apparently persuaded the Mont-Jouy commander to grant him a single day at his old observatory tower at the castle. On Sunday, March 16, 1794, a slightly overcast spring day, Méchain climbed the hill of Mont-Jouy to perform a final triangulation—while several hundred of his fellow citizens languished in the prison below. Then he returned to his hotel to calculate.

  The numbers were quickly tallied. According to the triangulations, Mont-Jouy was located 59.6 seconds of arc from his hotel, or a distance of 1.1 miles south. Comparing this distance with the two latitude measurements was a matter of simple subtraction. After subtracting 59.60 seconds from the average latitude of the Fontana de Oro (41°22'47.91”), the result should equal the average of his most reliable latitude data from Mont-Jouy (41°21'45.10”). It was a moment’s work.

  MÉCHAIN’S BARCELONA TRIANGULATIONS OF 1794

  This is Méchain’s own map of the triangulations he made within the city limits of Barcelona in 1794 to verify the distance between the Fontana de Oro and the fortress of Mont-Jouy. The Fontana de Oro is located at the center of the diamond. Méchain constructed two triangles that included Mont-Jouy and his hotel: one using the north tower of the cathedral and the other using the Lanterna (lighthouse). (From the Archives de l’Observatoire de Paris)

  Imagine then his horror when the results fell short by 3.2 seconds of arc. Not 3.2 seconds to be folded into the six-hundred-mile arc from Dun
kerque to Mont-Jouy, which would have been an insignificant difference of 0.01 percent, but 3.2 seconds over the course of a 1.1-mile arc, for a stunning discrepancy of 5.4 percent. Instead of explaining away the anomaly within his Mont-Jouy results, Méchain now confronted an anomaly of horrific proportions. Having pinned down his latitude to within forty to one hundred feet on two different occasions, he had now discovered that his two average results diverged by a horrifying three hundred to four hundred feet. He must have erred in his observations or his calculations. But which? Which set of data could he believe? Most horrible of all: he had already mailed a summary of one set of these results, those for Mont-Jouy, to his colleagues in Paris. From this they would want to calculate the length of the meter, the supreme standard for all people, for all time.

  It was as if he had set out to fine-tune a Stradivarius and snapped the instrument’s neck. His integrity had plunged him into a crisis. His effort to rehabilitate his reputation had only caused him to doubt his own abilities. What had gone wrong?

  Under normal circumstances, Méchain would simply have climbed back up Mont-Jouy and taken more stellar observations at the castle. But these were not normal circumstances. His one-day pass to the fortress had been a begrudging exception, not to be repeated for an enemy of the Spanish Crown. And day by day, as the Revolutionary armies advanced deeper into Catalonia, the political tensions worsened. The Republic promised the people of Catalonia a “sister” republic of their own; the Spanish Crown declared a religious war against atheism. Some residents of Barcelona supported the Revolution; others seethed against French godlessness. It was no time to be a Frenchman in Barcelona.

  Moreover, nothing now seemed to prevent the team’s departure. Ricardos, who had opposed it, was dead. Tranchot and Esteveny were eager to return to France where their duty lay, as well as their colleagues, friends, and families. Méchain, however, faced a terrible dilemma. He had told no one of his error, not even Tranchot. He was free to go, but did he dare to leave his mistake behind? Once he left Spain, how would he ever return to Mont-Jouy? Yet how could he justify staying on in a foreign country—an enemy nation—now that his work there was done? He dared not risk giving the impression that he had decided to emigrate. Even a rumor to that effect might lead the Paris authorities to cut off his salary, imprison his family, and bar his return to France forever.

  So, on the advice of his Spanish friends, he secured a passport for neutral Italy—ostensibly because it would not arouse suspicion—bypassing any need to inform General La Unión of his departure. In late May, after two years in Catalonia, Méchain booked passage on a Venetian vessel bound for Genoa, the Italian city nearest the French border. For someone who anticipated the worst, however, Méchain certainly attracted his share of calamities. His pessimism offered him no protection. On May 25, three days after he had loaded his precious repeating circles onto the vessel in Barcelona harbor, a bolt of lightning struck its mast, bursting the wooden boxes that carried the repeating circles and charring one instrument’s stand. The circles themselves appeared undamaged, but it was a fitting final salute from Catalonia. They sailed on June 4.

  Nothing of these events was known in Paris. There, everyone assumed Méchain had been placed in detention by the Spanish generals (perhaps for having smuggled out the fortress plans). He himself had written home of his “unjust detention” in Barcelona. He had complained that he was being held “in irons.” The words were metaphorical, even melodramatic—Méchain had been comfortably lodged all this time at the Fontana de Oro—but his colleagues sent word to General Dugommier that the French astronomer was being held against his will. In mid-June, two weeks after Méchain sailed for Italy, Dugommier wrote an indignant letter to his opposite commander, the devout young General La Unión, demanding that the Frenchman be freed. “In the name of the French Republic, which protects the savants of all nations, and which knows how to avenge any outrages committed against its own, I seize this occasion to demand that in the name of the arts whose free exercise should be respected at all times and by all nations, you release the citizen Méchain and his two colleagues, charged with the measure of the arc of the meridian, and detained in Barcelona by the orders of your predecessor or by you.” This was not the only lesson the Republican general thought he would teach the barbarous monarchist. “Savants must not be considered soldiers,” Dugommier wrote, “nor treated as such. The peaceable arts have nothing to do with war. And unless you wish to perpetuate an extraordinary violation of those conventions which govern even the most uncivilized people, you cannot refuse to return him and his two collaborators to liberty and their homeland.” Méchain’s mission, he insisted, “must be respected around the globe.”

  General La Unión knew no more of Méchain’s whereabouts than did the French. But he knew when his honor had been slurred. Never would he have impeded the advancement of human learning, nor dishonored his good name by holding an innocent civilian against his will. “If Méchain were to declare that he had been imprisoned by orders of either the Spanish government or myself, I would pass for an impostor in the eyes of the universe,” he wrote back. And then he added his own veiled accusation against the godless French. Like the rest of his countrymen, he announced, he appreciated “not only Méchain’s knowledge, but his moral virtues as well.” Yet just in case those virtues had gone unrecognized, he privately reminded Barcelona’s governors to treat Méchain honorably and provide him with financial assistance. Of course by then Méchain had been far from Spain for months before.

  That fall, the siege of Figuères reached its climax. General Dugommier died in battle there on November 17, killed by an exploding shell as he surveyed his impending victory. “Dugommier is dead on the field of honor,” the proclamation read. “He demands vengeance and not tears.” Three days later, General La Unión followed him into the grave, killed by two musket shots during a bloody French assault. The French drove the Spaniards from the crest where Captain Bueno had his tower, forced the surrender of Figuères, and pushed east toward the coast. Their successes, however, quickly got the better of them. Their supply lines unraveled. Desertions mounted. The two nations entered into formal negotiations to end the war, and in July 1795 they signed the Treaty of Bâle, returning the border to the same ambiguous position it had occupied before the war. But by then, Méchain had no prospect of returning to Mont-Jouy.

  CHAPTER FIVE

  A Calculating People

  There are certain ideas of uniformity which sometimes seize great minds (as they did Charlemagne’s), but which invariably strike the petty. They find in them a kind of perfection which they recognize because it is impossible not to discover it; the same weights and measures in commerce, the same laws in the state, the same religion in all parts. But is uniformity always appropriate without exception?

  —CHARLES DE SECONDAT DE MONTESQUIEU, The Spirit of Laws, 1750

  [This] chapter has earned Montesquieu the indulgence of all people of prejudice. . . . Ideas of uniformity, of regularity, please all minds, and especially just minds. . . . Uniformity of measures can only displease those lawyers who fear to see the number of lawsuits diminished, and those traders who fear a loss of profit from anything which renders commercial transactions easy and simple. . . . A good law ought to be good for all men, as a true proposition [in geometry] is true for all men.

  —M.-J.-A.-N. DE CONDORCET, Observations on “The Spirit of Laws,” 1793

  Delambre had been stopped in his tracks. Méchain had been trapped behind enemy lines. Like a suspension bridge abandoned after its end supports had been raised, the meridian survey had been called off in mid-execution, leaving a span half the length of France unbuilt between them. Not that the leaders of the Revolutionary government cared. They considered the meridian arc a monument to futility. Now that they had the provisional meter in hand, they could leave the ruins of the meridian survey unfinished, a folly of scientific presumption. For them, the challenge was not to push precision to an ever narrower closure, but t
o bring the advantages of the metric system to the common people. This meant putting meter sticks in the hands of 25 million French men and women.

  Yet when the date for the obligatory use of the metric system arrived on July 1, 1794, the Revolutionary government had produced fewer than one thousand meter sticks and not a single French citizen was using the new system. Even the petty officials who answered to the dictatorial Committee of Public Safety were still filling out their reports in the old measures, making it impossible for the central government to monitor grain supplies. Prieur de la Côte-d’Or and the other members of the Committee pleaded with their subordinates to conduct the nation’s business in the new metric system. They denounced the feudal diversity of measures as a barbarous remnant of the Ancien Régime. They expressed their frustration: Why had the people who had pleaded so passionately for metric reform in the Cahiers de doléances become suddenly so reluctant to accept the metric system?

  This paradox would not have been so surprising had the politicians and savants put aside their willful disregard for the meaning of measurement in the Ancien Régime—and considered the enormity of the change they were demanding. Understanding that change will clarify the meaning of measurement, both for them—and for us. A modern system of measurement allows objects to be described in abstracted, commensurable units that relate to an absolute standard. This is true of the new metric system the French were seeking to establish, as it is of the nonmetric measures still in use in America today. In either system a measurement stays fixed, no matter where the object is measured, or which measurement instrument is used. A meter is a meter; as a foot is a foot, a pound is a pound, and a kilogram is a kilogram. The dimensions of any other object can be described by reference to these units. The ultimate guarantor of these standards is a national or international agency with precise standards and a staff of inspectors. These inspectors are rarely seen any more because they have built their supervisory role into the measuring instruments we use every day: rulers, scales, graduated cylinders, clocks, or gauges. Only in cases of extreme controversy are the inspectors obliged actually to check the calibration. Until that time, we trust the instruments. This form of measurement is adapted to our modern economy, in which buyers and sellers remote from one another in time and space conduct impersonal exchanges, quite certain that their measures are commensurable.