The Measure of All Things

by Ken Alder

  Two millennia would pass before anyone came up with a better one. Jean Fresnel, doctor to Henry II of France, took a comparatively crude approach in the sixteenth century. He measured the distance from Paris to Amiens by counting the number of times his carriage wheel turned along the route (a mechanical ticker inside his carriage helped him keep track of the rotations). Because he knew Amiens was located one degree of latitude due north of the capital, and the road between the two ran straight, he simply multiplied the number of rotations by the circumference of his wheel and again by the full 360 degrees of the globe. Considering his methods, he was not far wrong either. The modern technique for using triangles to measure earthly distances, however, was introduced in 1617 by Willebrord Snell, “the Dutch Eratosthenes,” on the frozen fields outside Leyden, and his method persisted for the next 340 years.

  One of the earliest official acts of the French Academy of Sciences had been to remeasure Fresnel’s itinerary from Paris to Amiens by triangulation, and this measure of the globe’s regularity had inspired Gabriel Mouton, a chaplain-astronomer in Lyon, to propose that the earth serve as the basis for all human measurements. In 1670 he suggested that the fundamental unit of length, which he dubbed the mille, be set at the length of one degree of the earth’s arc, with all subunits determined by decimal division, such that the virgula (about the length of the king’s pied or foot) would equal one ten-thousandth of the mille. The marvel of nature’s regularity suggested that human activities be aligned according to the same metric.

  But all these learned folk, from Greek astronomers through European scholars, had assumed that the earth was a perfect sphere until Isaac Newton—without ever setting sail from Cambridge—announced that our round planet was slightly flattened at the poles. His hypothesis of an oblate earth began as a theoretical prediction. Calculating the effect of rotation upon a homogenous liquid sphere (our earth) in which all particles attract one another all the time, Newton estimated that centripetal forces would produce an earthly eccentricity of 1/230. In other words he suggested that the earth’s radius at the pole was 1/230 shorter than its radius at the equator. Newton then corroborated this prediction by several choice pieces of evidence. He reanalyzed the French Academy’s meridian survey to show that the earth had flattened slightly as the triangulation proceeded north. He noted that the pendulum clock carried by another French savant to the Caribbean in 1672 had beat more slowly as it approached the equator, suggesting that gravity weakened slightly as the earth bulged, because the point was farther from the earth’s center. Finally, he pointed out, astronomers had noticed that Jupiter was flattened at the poles. On earth, then, as it is in heaven. And to cap it all off, Newton made one final startling deduction. The bulge around the earth’s middle, he surmised, explained a phenomenon that had baffled astronomers for two thousand years. The pull of the sun and moon on the earth’s bulgy middle was responsible for the precession of the equinoxes, the slow but steady 26,000-year swivel of the earth’s axis of rotation. Newton had banished the earth’s spherical perfection along with the planets’ circular orbits. We do not live on a perfectly round orange, but on a flattened tomato. Nature’s perfection lay not in childish geometry, but in nature’s forces deeply hidden—and by Newton revealed.

  The century-long debate that followed proved to be the golden age of geodesy; that is to say, an age of bitter controversy and world-shaking reversals. A survey of the meridian of France, undertaken by Cassini I in 1700, seemed to confirm Newton’s hypothesis, until his son, Cassini II, reviewed the data and dared to suggest the opposite, that Newton had in fact erred and the globe, if anything, was elongated at the poles: prolate rather than oblate, a long lemon rather than a flat tomato. The question was not just academic; it affected mapping projects on land and sea. The difficulty was that a 1 percent error in the determination of the latitudes was enough to flip the earth from oblate to prolate, from tomato to lemon—or the other way round.

  To resolve the question, the Academy of Sciences launched an expedition to Peru to determine whether the earth bulged at the equator. It also dispatched rival savants to Lapland to measure the earth’s curvature as it approached the pole. These stirring voyages cast science in a heroic light, bringing Newtonian physics to public attention and entertaining the salons of Paris with the splenetic quarrels of the academicians. In 1740 the French Crown also sponsored Cassini III’s survey of the meridian from Dunkerque to Perpignan to help settle the controversy—and recast the map of France. On his return, Cassini abjured his father’s prolate earth and acknowledged that we live on Newton’s flattened globe—although no two savants could quite agree on just how flattened it was.

  As one chieftain of eighteenth-century science confessed, this controversy had seemed to dash the “flattering dream” of a universal measure based on the perfection of nature. But natural philosophers do not give up so easily on nature’s perfection. Several savants—Laplace in particular—remained convinced that the earth, flattened though it was, might still serve as the basis for a perfect meter. As Condorcet explained to the National Assembly, these arguments had convinced him to switch his allegiance from a simple pendulum standard to a geodetic mission. And he pleaded with the Assembly likewise to embrace this revised plan. The meridian project was based on the soundest science, he noted, of such universal principles that in future years no one would even be able to say which nation had performed the task. And he went on to urge them, in somewhat contradictory terms, not to wait for “the concourse of other nations” before settling on a standard. As the representatives of a great and enlightened nation, one whose vision reached out to all people and all times, it was incumbent on the French to reject the easy path, and instead “approach perfection.” On March 26, 1791, despite some grumbling about the likely cost and delays, the National Assembly adopted this meridian standard.

  This decision had lasting repercussions. In the short run, it ended any chance of international cooperation. To savants outside France the meridian project smacked of self-interest. Those savants who favored the pendulum standard refused to concede its inferiority. Geodesers, they pointed out, also relied on many other units like time and angles to measure the globe, so no unit could ever be truly fundamental. The leaders of London’s Royal Society accused their French colleagues of seeking to “divert the attention of the European public from the true amount of their proposal, which in fact is that their measurement of 9 or 10 degrees of a meridian in France shall be adopted as the Universal standard.” Jefferson likewise withdrew his support from the metric system when he learned that the French would survey their own meridian. As he pointed out: “If other nations adopt this unit, they must take the word of the French mathematicians for it’s [sic] length. . . . So there is an end to it.”

  For the French savants themselves, however, the expedition paid handsome dividends. Thanks to the extravagance of the meridian project, the budget for the creation of the metric system was revised upwards to 300,000 livres, roughly three times the annual operating costs of the entire Academy under the Ancien Régime. Government funds also flowed into the coffers of instrument-makers like Lenoir, who had been hit hard by the disruption of the luxury trade at the advent of the Revolution. Almost every one of the Academy’s savants who worked in the physical sciences found employment on the metric system project. All, that is, except for Lalande, who refused to participate in a project he considered pointless—though he wished his former students well.

  Even within France, some criticism was heard. The literary critic Louis-Sébastien Mercier thought the meridian expedition smelled of charlatanism. The savants, he said, had “preserved their pensions and salaries . . . under the pretext of measuring the arc of the meridian.” Other commentators were more scathing. Jean-Paul Marat, bilious enemy of the Academy (“those cowardly lackeys of despotism”) deemed the 300,000-livre budget “a little gâteau they will share out among confederates.” There were even some savants who (privately) ascribed the change in plans t
o ulterior motives. Delambre himself later speculated that Borda had pushed the meridian project to enhance the reputation of his repeating circles. Others wondered whether Laplace and the other physicists had primarily promoted the project as a way to pin down the exact shape of the earth, rather than as an attempt to ascertain the length of the meter.

  Of course, as many commentators have pointed out then and since, the decision to base the meter on one ten-millionth of a quarter of the earth’s meridian was itself arbitrary. To begin with, it was not even a real distance, but a calculated distance along a portion of the surface of an imaginary sea-level geoid that would have to be extrapolated from one small segment of the arc to the whole. And metric kibitzers suggested many alternative ways of slicing up the globe. Some preferred a meter based on the circumference of the equator. Not only was the equator unique, it was apparently circular and unchanging. A meridian, by contrast, was arbitrary, elliptical, and possibly subject to change over time. Still others agreed with the choice of a meridian, but wondered why the savants had not chosen their standard to equal one hundred-millionth of the total meridian (rather than one ten-millionth of the quarter meridian) to make the meter more nearly come out to the length of the foot, a more manageable size for daily use. Occasionally—as the meridian project suffered one delay after another—politicians and ordinary citizens even had the temerity to ask whether a natural standard was necessary at all. Nature was changeable and irregular, some said. “Everything in nature is unequal,” another complained. Even the shape of the earth might change over time, as Laplace himself admitted—though surely the measure of the meridian would not take that long to complete.

  All along, Lalande, the old iconoclast, stuck to his preference for a physical standard, like the copper toise of Paris which the Academy had in its keeping. Such a standard could be defined with much greater ease and accuracy, whereas any attempt to find a standard in nature would prove ephemeral because too many factors influenced the investigation of natural phenomena. For instance, there were many factors that might influence the length of a pendulum besides the latitude at which it was measured; from the arc of its swing and the ambient temperature to the air resistance. Moreover, he noted, savants could not even be sure that the pendulum’s periodicity was the same at every spot along the same latitude, since the tug of nearby mountains or other deformities of the earth might affect its oscillation. The same factors would affect any measure based on geodesy. Given these uncertainties, Lalande predicted that the progress of science (in which he most fervently believed) would produce more accurate results twenty years hence. And what would happen then? Would the natural standards have to be periodically revised to take account of the improved results? Under the circumstances, what was the point of striving for precision now?

  Eager as he was to resume his campaign as soon as the spring weather allowed—delay might give the bureaucrats an excuse to cancel the mission, and the cost of transportation was rising—Delambre still needed official permission before he set out. The barricades of the previous year had taught him the value of a valid passport. In March, he petitioned the Paris municipal council for permission to move freely throughout the Republic. The council—now in the hands of the radical sans-culottes party and hostile to the Academy as an elitist institution—voted unanimously against his petition. “Is it possible,” he wrote back, “that in Paris, at the center of enlightenment and the arts, the executors of a law applauded across Europe find themselves stymied in their tracks?” He resubmitted his petition, countersigned this time by prominent administrators. The council voted unanimously to issue him a passport. As an extra precaution, he wrote in advance to all the towns along his route to assure them that his mission was benign.

  The Republican government had proclaimed France a nation of “one law, one weight, and one measure.” It had promised to end the shameful inequalities of Ancien Régime justice and taxation. It had promised to open careers to talent and to liberate commerce. But the Revolution had also shattered the royal authority that governed France from the center. Placing sovereignty in the hands of the people had made every town its own master. Markets were in disarray, and food prices were rising. The towns were suspicious of the countryside; the peasants mistrustful of the towns. At every step Delambre had to present his papers. When he passed through his hometown of Amiens, an old friend had to prepare a dossier of official documents for him, all signed, stamped, and dressed with fancy seals.

  At least Delambre had a strategy for this campaign season. Rather than circle Paris in futility, he and his two collaborators, Lefrançais and Bellet, would begin at the beginning. They would start at Dunkerque, the northernmost station, and work their way south. The strategy was logical, but circumstances have their own logic. The optimal season for geodetic campaigning coincided with the optimal season for military campaigning. All that spring, while Delambre had been waiting for a passport in Paris, the Prussian-Austrian army had been massing on the frontier for another drive to restore the monarchy. By the time he arrived in the north country, the plains of Flanders had become a battlefield, with the invaders again advancing toward Paris.

  In mid-May of 1793 Delambre hurried to Dunkerque before the French defenses collapsed. There he was assisted in the bell tower by Monsieur Garcia, whose family had been tower masters for three hundred years. Sometime during that long interval, the 162-foot tower had been separated from the main body of the church by a road that still serves as the main thoroughfare through the city center. From the top of the red-brick belfry—a climb of 264 steps, “and we counted”—the team had a view of several nations: France, the Low Countries, and across the Channel, England. Cassini and Méchain had used the belfry in their 1788 survey to link Paris and Greenwich. Nearer to view, Delambre could see the dunes along the beach, the port made idle by war with Britain, and the long low coast that swept into the gray mists. Inland, he could see the French armies maneuvering along the border.

  From Dunkerque, Delambre made steady progress south through Picardy, his home region. It was an ideal landscape for triangulation, and summer was the ideal season for geodesy. The corrugated countryside was laced with low ridges, and each town boasted an elegant church steeple. Excellent stations were abundant, though each presented its particular challenge. In Watten, a small town a dozen miles inland on the Aa River, the church tower was not high enough to be seen from afar, so he capped it with a crown of white planks. At Cassel, to the immediate east, the summer heat in the steeple was suffocating. At Mesnil, he had to wait four days to erect his signal until the local carpenters had finished celebrating their village festival in the cabarets. At Fiefs, he had to wait for permission to punch holes in the church steeple so that he might have a clear view in all directions. At Bayonvilles, he had to chop down several trees to open up a line of sight. By mid-July he had closed ten triangles, accomplishing more in one month than in all the previous year. By his own account, this was his happiest portion of the meridian, and the most well favored. Behind him, the battle was turning against the French. The British had laid siege to Dunkerque and the Hanoverians were closing in on Lille. But by then Delambre was approaching his hometown of Amiens.

  Lefrançais never made it there. In mid-July he had to rush back to Paris. His wife (Lalande’s daughter) was approaching her due date. On July 27, she gave birth to a girl, Uranie, though the baptism was postponed until Delambre could stand in as her godfather. Delambre wrote to congratulate the young mother. “I admire you for having resumed your astronomical labors so soon; in giving us a Uranie you have accomplished enough and could have rested yourself a bit longer.” He had six more stations to complete before he could return to Paris for the baptism of “our new muse.” As for Lefrançais, grandfather Lalande wrote that he would return to the mission as soon as he was elected to the Academy in his own right, probably at the meeting set for August 7.

  Lefrançais would never return to the mission. On August 8 the Academy was abolished, and Lalande pu
t his nephew back to work on his all-important celestial chart.

  Delambre was setting up a signal in the cathedral spire of Amiens when he learned of the Academy’s demise. “I don’t know if I still have the right to call you my colleague,” Lavoisier wrote, “though I send you this letter as a fellow believer in the progress of science.” The good news was that the savants had managed to preserve the metric reform, and the meridian survey along with it. “The suppression of the Academy ought not in any way disrupt your labor, nor diminish your indefatigable activity.” The bad news was that there was no money to pay Lefrançais, and the continuation of the survey had been ransomed with a dangerous concession: the establishment of a “provisional meter.”

  FOLDING METER STICK

  This iron meter stick was made to match the specifications of the 1793 provisional meter. It reads: “Meter stick equal to one ten-millionth part of a quarter of the earth’s meridian, Borda, 1793.” (From the Musée des Arts et Métiers-CNAM, Paris; photograph by CNAM)

  The suppression of the Academy did not come as a complete shock to Delambre. For years the academicians had been attacked as self-appointed elitists who disparaged popular inventors and thinkers. In the past months, radical politicians had called for the dissolution of all royal institutions. Some legislators had tried to exempt the Academy of Sciences because of the transcendent truthfulness of science and the useful services it provided the nation—especially with regard to the reform of weights and measures—but to no avail. In the end, some academicians even came to agree that the Academy was undemocratic and applauded its fall. When Cassini IV tried a procedural motion to delay the final closing, some of the Republican savants echoed the phrase that the drunken militiaman of Lagny had hurled at Delambre, shouting: “There is no more Academy!”