The Mis-measure Of All Things

TRACK STAR MICHAEL JOHNSON MAY NOT BE AWARE OF IT, BUT when he runs the 400-meter dash, he is covering exactly one hundredthousandth of the circumference of the planet. Well, not exactly. More than 200 years ago the men who created the metric system, some of the greatest scientific minds of Revolutionary France, called for a new unit of measurement, the meter, that would equal precisely one ten-millionth of the distance from the North Pole to the equator.

The creators of the metric system included the likes of Antoine-Laurent Lavoisier, the founder of modern chemistry; Pierre-Simon Laplace, the age’s pre-eminent physicist; and the Marquis de Condorcet, the creator of mathematical social science. These men were appalled by the diversity of measures they saw all around them. In those days measures varied not only from nation to nation but within nations as well, often from one town to the next, even from one parish to the next. For measures to enable cross-border communication and facilitate free trade, they would have to be defined with precision and universally accepted. Only a unit derived from nature, reasoned the Revolutionary scientists, could be determined with sufficient precision and appeal equally to people all over the world. And what natural unit could have wider appeal than the size of the world itself? The challenge was to measure the earth precisely.

In June 1792 the Revolutionary French government launched an expedition to determine the distance along the Paris meridian that ran from Dunkirk to Barcelona, so as to extrapolate to the rest of the globe. A cosmopolitan and erudite astronomer, Jean-Baptiste-Joseph Delambre, was sent north to measure the longer portion that ran from Dunkirk to Rodez; a cautious and scrupulous astronomer, Pierre-François-André Méchain, was sent south to measure the shorter, uncharted, mountainous portion from Rodez to Barcelona. For the next seven years the two men labored to extract from the curvature of the planet a single value that might be enshrined in a platinum meter bar and thereby serve as the universal standard “for all people, for all time.”

Today their vision has largely come to pass. More than 90 percent of the world’s population lives in nations that require use of the metric system. The system has become the language of international science and a lubricant of the global economy, as well as the norm for international sports. Only three countries lie outside the metric domain: Burma, Liberia, and the United States. And even in the United States, metric measurements have become routine in many, though not all, industries and professions—if not in our daily lives. How has this come to pass? How did the meter come to rule the world but not the nation that leads the global economy?

Some blame American technophobia and ignorance of science. But the metric system has met with passionate resistance whenever and wherever it has been proposed—even in its country of origin. Although created during the French Revolution, it was rescinded by Napoleon and only readopted in France 50 years later. The conflict there can teach us about the likelihood of its adoption in America. It also has something to say about the role of standards in our scientific and commercial dealings and the degree to which such standards are the products of human choice and social conflict.

THE METER, IT TURNS OUT, IS IN ERROR. TODAY WE define it as the distance traveled by light in a vacuum in 1/299,792,458 seconds (with time, the fundamental unit, determined by an atomic clock). This current definition, like all the intervening redefinitions proposed during the past two centuries, was carefully designed to preserve the original value found by Delambre and Méchain in their meridian expedition of the 1790s. After all, any change would lead to economic and scientific chaos.

Yet today we know that the actual distance from the pole to the equator is approximately 10,002,000 meters, meaning that the meter is short by about 0.2 millimeters, roughly the width of two pieces of paper. This may not seem like much, but it is far from the precision the two French astronomers hoped for. Had Michael Johnson really been timed over a hundredthousandth of the circumference of the planet, instead of 400 meters, he would not have set the world record in 1999.

Do such errors matter? So long as some kind of fixed standard is declared—a standard enshrined in a platinum meter bar, for instance—what difference does it make whether that bar in fact equals its original definition? So long as everyone runs the same race, what does it matter what units are used to measure the distance? Standards, by this view, are social conventions subject only to the consent of those who use them. Yet getting the precise answer mattered greatly to the men sent out on that meridian expedition. Indeed, their very failure helped establish a new understanding of error, one that continues to shape the practice of scientists to this day.

Geodesy is the science of measuring the size and shape of the earth. In the late eighteenth century the practice of geodesy was based on triangulation, a technique already 100 years old and still used up until the advent of the satellite. The method involved selecting a series of elevated observation sites close to the line of the meridian—church towers, mountaintops, or specially built platforms— so that the geodeser might survey all the angles of a chain of imaginary triangles, each with sides roughly 40 miles long. A single side of one of the triangles was then measured along the ground with rulers, so that the lengths of all the other sides might be calculated. For instance, from the dome of the Panthéon, on a hill in central Paris, Delambre measured the angular distance separating a church belfry 40 miles north and the pavilion of a chateau 40 miles east. He then took analogous angular measurements at each of those places. Next he made an adjacent triangle with its corners at the Panthéon, the chateau, and a medieval fortress tower to the south, and continued that way down along the meridian. Finally, the latitudes of the northern and southern endpoints would be determined, by astronomical observations so that the surveyed portion of the meridian arc could be extrapolated to the circumference of the earth. The principles involved are as simple as basic geometry, but getting precise results means coping with many complicating factors, including the bending of light and the curvature of the earth. The modern science of geodesy had been born with Newton’s shocking theory that the earth was not a perfect sphere but was flattened at the poles. Throughout the eighteenth century scientists across Europe bitterly debated the extent of this eccentricity. One of the secondary motives behind the Delambre and Méchain expedition was to settle this question to a new degree of exactitude. To that end the two astronomers were supplied with an innovative new instrument, the Borda repeating circle, that promised to reduce error to the vanishing point by allowing the same angle to be measured repeatedly.

Delambre and Méchain were well aware of the importance of their expedition to the future of science and commerce. As Lavoisier informed Méchain (with typical French understatement), “You must not forget that you are carrying out the most important mission that any man has ever been charged with, that you are working for all nations of the world, and that you are the representative of the Academy of Sciences and all the savants of the universe.”

At times, the responsibility became a burden almost too heavy to bear. Although the two men came from remarkably similar backgrounds, their temperaments were quite different, and that difference would decide their fates. Both were skillful astronomers in their mid-forties. Both had been born to humble families in the French provinces and had risen by dint of long labor to the summit of French science, the Paris Academy of Sciences. Delambre was the eldest son of cloth sellers. As an infant he had been afflicted by smallpox and nearly blinded, losing his eyelashes. Only gradually did his sight recover enough for him to win a scholarship to study classics in Paris and take a position as a tutor. He did not begin his study of science until he was in his thirties. Yet in one short decade he became one of France’s leading astronomers. He was a tolerant bachelor, as curious about his fellow human beings as he was about the stars.

Méchain, five years his senior, was a man of narrower vision and more concentrated passion. A keen observer of the stars, he spent two decades in the cartography department of the French navy, preparing maps of a Mediterranean coastline he had never visited, while avidly stargazing in his off hours. After discovering many comets and other celestial novelties, he was rewarded with membership in the academy and a pleasant home on the grounds of the Paris Observatory, where he lived with his wife and three children.

The year 1792 was an inauspicious time to set out to measure the world. Not long after the expedition got under way, the French monarchy collapsed and the Republic was declared. A few months later, Revolutionary France went to war with its neighbors, and the government took a radical turn, abolishing venerable institutions including the Academy of Sciences. Several of the country’s greatest scientists went to the guillotine.

Out in the French countryside, Delambre and Méchain struggled against many obstacles. They were harassed as counterrevolutionaries, threatened as foreigners, arrested as spies, and hampered by the world’s first bout of hyperinflation.They also had to contend with the mistrust with which peasants always greeted surveyors, whom they viewed as the emissaries of landowners bent on removing them from their fields and livelihoods. Through it all, Delambre maintained the calm and cheerful detachment that saw him through every crisis.

Méchain did not take fortune’s blows so lightly, though his mission began smoothly enough. With the help of Spanish collaborators he managed in one short summer to measure the sector through Catalonia, surveying his way south from the crests of the Pyrenees to the city of Barcelona. Then that winter he took the crucial latitude measurements at the fortress of Mount-Joy on the outskirts of Barcelona, to define the southern anchor of the meridian arc. When war broke out that spring between monarchical Spain and radical France, he was ejected from the fortress and detained in Barcelona to prevent his geodetic information from falling into the hands of Spain’s Republican enemies. Not long after, he suffered a terrible accident. He was hurled against a wall by machinery he was being shown at a pumping station and went into a coma for three days. During his long recovery, a virtual prisoner in Spain, he decided to retake the latitude measurements at Barcelona, this time from the terrace of his hotel, only to discover, to his horror, that the results did not agree with his data of the previous year, even after correcting for the mile that separated the two sites. Then the fortunes of war changed once again, and he was obliged to leave Spain without resolving the discrepancy. He traveled first to neutral Genoa, and then, after a year on the Italian Riviera waiting out the worst excesses of the Revolution, made his way back to France.

Méchain was acutely conscious that he had been sent on a mission to bring back a measure for all people, for all time- and that he had botched the job. Worse, Delambre’s success compounded his guilt. Reading between the lines of his selfdeprecating letters, it is easy to see the jealousy with which he tracked the progress made by his northern colleague. Méchain holed up in the mountains of southern France and refused to make any progress. He lived in fear of his colleagues’ discovering the discrepancy in his data and ruining his reputation. It was a violent age. Men were executed for lesser offenses. So he made a fateful decision to conceal his error while trying every ruse to postpone his final day of reckoning.

By 1798 that day was fast approaching. The Paris academicians who had devised the metric system had called for a gathering of the world’s leading scientists, history’s first international scientific conference. They planned to meet in Paris in September 1798, review the data from the meridian expedition, and translate them into a value for the meter. Yet Méchain was unwilling to hand over his work, return to Paris, or even complete his mission. His letters became frantic, melancholic, suicidal. In desperation, his colleagues sent his wife, Barbe-Thérèse Méchain, to fetch him back from the remote mountains of southern France. Madame Méchain, a competent astronomer in her own right, persuaded him to resume his mission. And once he was done, Delambre lured him back to Paris, promising him the directorship of the illustrious Paris Observatory. But still Méchain refused to hand over his raw data. All he offered were his summary numbers.

In 1799 the international meeting adjourned, the meter was promulgated and enshrined in a platinum bar, and the government—with Napoleon now at its head—set about the still more challenging task of persuading the citizenry to adopt the metric system. Méchain, meanwhile, continued to brood. No one could understand his agony. Did he not have the most prestigious position in his disci- pline, the respect of his peers, a loving family, and a fine house on the grounds of the observatory? Yet when the chance presented itself in 1802, he insisted on extending his meridian arc south of Barcelona to the Balearic Islands, in the secret hope that he might bypass the discordant results he had obtained a decade before. By now he was 60 years old. Nothing on the new expedition went as planned, and in 1804 the unhappy man died of malaria in Valencia, his mission in shambles.

ONLY THEN DID DELAMBRE, AS MÉCHAIN’S SCIEN tific executor, come into possession of his colleague’s papers and learn the extent of his duplicity. Méchain, Delambre discovered, had recorded his data in pencil and on loose scraps of paper, so as to better fudge his observations after the fact. His goal was not to alter the final outcome so much as to make himself look good by eliminating data that did not sufficiently conform to expectation. In the case of the Barcelona data, he had suppressed the entire second season’s observations from his hotel. Delambre now faced a fateful decision of his own: whether or not to publish these data.

In the end he decided he had to. Science had an obligation to make its discoveries known. Moreover, Méchain’s data, he believed, helped demonstrate something suggested by the rest of the geodetic data: that the earth not only was flattened at the poles but was not even consistently curved between them. Rather, Delambre argued, the earth was roiled by deep underground forces that made its figure (or what scientists today call the geoid) distend, distort, and warp. It was the minute gravitational effects caused by the misshapenness of the earth that had caused the discrepancy in Méchain’s Barcelona latitude data from two nearby locations. This was something unexpected, a genuine scientific discovery. There was only one problem: The discovery invalidated the guiding premise of the entire expedition. If each meridian was irregular and of slightly different length, then the size of the world could hardly serve as a universal referent for all measures. Yet Delambre boldly insisted on publishing this inconvenient discovery. That he could do so at all can be attributed in part to a new way of treating scientific data.

In the year after Méchain’s death, the French mathematician Adrien-Marie Legendre had published a new method for handling data, the method of least squares. It found the optimum fit of a curve to data by minimizing the sum of the squares of the divergences of the data points from the curve. Legendre had been led to this method, in part, while working his way painstakingly through the data of the meridian expedition. During the next few years, Laplace and Gauss coupled Legendre’s method to probability theory and thus laid the foundation for error theory, on which would be built the modern science of statistics. It was this method that Delambre invoked in publishing Méchain’s data. Méchain had not so much erred as he had not known how to interpret error. In his efforts to preserve an aura of perfection, he had inadvertently pointed the way toward a scheme for handling the inevitable shortcomings of any attempt to measure the world.

As for Méchain’s falsified data and his battle against melancholic madness, Delambre decided to conceal that story from the public. He saw no need to sully Méchain’s name, so he placed all the surveyor’s letters under seal in the archives of the Paris Observatory, alongside the doctored data, where they remained unread for 200 years.

It was on discovering and reading those letters that I began to see that Delambre and Méchain had been sent out on a quixotic mission to begin with. Established on a false premise about the shape of the world, the expedition had brought back a new understanding of the earth. Launched with unrealistic expectations of perfection, it had led to a new way to treat error. Sent out to bring back a natural and non-arbitrary value for the meter, it had shown that any measure, even if ultimately totally arbitrary, could serve as a standard, as long as it didn’t vary and was accepted by all.

IN THE DECADE AFTER THE INTERNATIONAL CONFER ence, the French government tried to persuade its citizens to adopt the metric system, but without success. Ordinary French people, it turned out, preferred to stick with their old idiosyncratic units, the livre (a pound), the aune (a yard), and so on, the products of centuries of conflict and negotiation. These measures had been woven into the fabric of daily life, and they often served to define the value of labor and the worth of objects. The creators of the metric system may have wanted to facilitate open markets for goods and the free flow of information, but they failed to understand the profound mistrust with which their fellow citizens would view such liberalization. By 1812, on the eve of his disastrous invasion of Russia, Napoleon conceded defeat on the home front and returned France to the units of the Old Regime. Not until the middle of the nineteenth century, after public education, railway transportation, and urbanization had begun to make France into a modern economy, did the French government again try to coax its people to adopt the metric system. It would take until the end of World War I for the nation to be fully converted.

Given that history in the meter’s birthplace, it is hardly surprising that Americans have been leery of switching to the measure. Thomas Jefferson did his best to make the United States the second nation to adopt it. By the end of his career, however, he could write with resignation to John Quincy Adams: “On the subject of weights and measures, you will have, at its threshold, to encounter the question.… Shall we mould our citizens to the law, or the law to our citizens?”

The American preference has clearly been the latter, at least in the realm of commerce. Not only is the American government hesitant to take a commanding role in the economic life of the nation, it has had little need to meddle with the nation’s weights and measures. Compared with most nations, the United States, thanks to its short history and colonial origins, has always had relatively consistent measures. (By contrast its money was a hodgepodge at the time of the Revolution, so for that a metriclike decimal system was created.) And so long as everyone uses the same measures, what does it matter what those measures are? Only in recent decades, as the rest of the world has converted to the metric system, have certain American industries, particularly those that rely on worldwide manufacturing, begun to climb onto the bandwagon.

Paradoxically, the introduction of the system into many domains of American manufacturing and scientific life has brought an unprecedented degree of nonuniformity in our measures. A dramatic sign of this was the disastrous failure of the Mars Climate Orbiter, which went astray in 1999 because one set of engineers was using Anglo-American units and another, metric ones. For the first time America now has a strong incentive to switch to a new set of units. No doubt the transformation must take many decades, but the United States may yet go metric in the end—not because the rest of the world uses the metric system but because America already does.