The Measure of All Things

by Ken Alder

  There is, however, one other possibility. What if nothing and no one was to blame? Indeed, what if there was no meaningful discrepancy at all? That is: what if the error lay neither in nature nor in Méchain’s manner of observation, but in the way he understood error? Twenty-five years after Méchain’s death, a young astronomer named Jean-Nicolas Nicollet showed how this might be the case.

  Nicollet reanalyzed Méchain’s data in a series of steps. First, he threw out Méchain’s data for the inferior transit of the star Mizar, whose passage near the horizon was indeed overly distorted by refraction. Second, he recalculated Méchain’s other stellar heights using accurate tables of stellar declination, tables that had been completed at the very end of Méchain’s life and that circumvented the iterative method by which both Méchain and Delambre had assessed their latitudes. Both these changes were relatively minor, however, compared with the way Nicollet reconceptualized Méchain’s treatment of the data.

  Méchain and his contemporaries did not make a principled distinction between precision (the internal consistency of results) and accuracy (the degree to which those results approached the “right answer”). The two are not the same: precise results may appear “reliable” in the sense that they give very nearly the same answer when measured again; yet they may lack “validity” in that they deviate consistently from the “right answer.” Of course, in practice, distinguishing between the two can be extremely difficult because the “right answer” is unknown.

  Repetition using the Borda circle was designed to improve precision by reducing those errors that stemmed from the imperfect senses of the observer or the imperfect construction of the instrument’s gauge—the sort of errors we would today characterize as falling into a random distribution. The Borda circle, however, was still subject to errors caused by the basic setup of the instrument as a whole; the sort of errors we would characterize today as those constant (or systematic) errors which made results inaccurate, whatever their level of precision. Constant errors generally go undetected, of course, as long as they stay constant. And in an intuitive way Méchain and Delambre, like all Ancien Régime astronomers, understood this. That is why they were so vigilant about maintaining a consistent setup for their apparatus from one series of observations to the next. What they failed to appreciate was that the same repetition that enhanced precision might reduce accuracy. For instance, constant manipulation of the circle might wear down the instrument’s central axis and, over time, cause the circle to tilt ever so slightly from the perpendicular. It was this unanticipated drift in the constant error, Nicollet suggested, that was the source of Méchain’s discrepancy. This had kept his results for each winter location more or less internally consistent (i.e., precise), while making the results from the two successive winter locations discordant (i.e., inaccurate). Without a concept of error to help him identify the source of this contradiction, Méchain was in torment.

  Oddly enough, Nicollet noted, it was Méchain’s own obsessiveness which made it possible to confirm the cause of the discrepancy—and to correct for it. The trick was to compensate for any change in the instrument’s verticality by balancing the data for stars which passed north of the zenith (the highest point of the midnight sky) against those which passed south of it. Because Méchain had measured so many extra stars, such an operation was possible.

  To calculate the latitude, Méchain had first calculated the average latitude implied by each star he measured, and then averaged all the averages, giving equal weight to each. Nothing could be simpler—or more naïve. Nicollet, by contrast, first analyzed the data for the stars Méchain had measured that passed north of the zenith (his many observations of Polaris and Kochab, plus those of Thuban and Capricornus), taking the average of the average latitude implied by each. Then Nicollet separately did the same thing for the stars Méchain had measured to the south of the zenith (his sparser observations for Pollux and Elnath). Clustered in this manner, the results seemed to lack precision: at Mont-Jouy the average latitude implied by the north-going stars differed from the average latitude implied by the south-going stars by 1.5 seconds. At the Fontana de Oro they differed by a dismaying 4.2 seconds. But when the northern average and the southern average were themselves combined at each location, they suggested a remarkable accuracy: the combined latitude for the Fontana de Oro agreed with the combined latitude from Mont-Jouy to within a stunning 0.25 seconds—twelve times as accurate as Méchain’s 3.2-second discrepancy! In sum, Nicollet proved that there was no discrepancy and that Méchain’s reported value for Mont-Jouy was within 0.4 seconds (or forty feet) of the answer indicated by his data, when properly analyzed.

  Nicollet was a typical French astronomer of the early nineteenth century: a student of Laplace’s, well versed in error theory. In 1828, when he reviewed Méchain’s results, he was forty-two years old and working part time at the Paris Observatory. Unfortunately, his statistical skills did not stand him in good stead outside astronomy, and he lost his fortune a few years later playing the stock market. He emigrated to America, where his astronomical and mathematical skills earned him the leadership of the first geodetic survey of the upper Missouri and Mississippi valleys. A generation after Lewis and Clark passed through the territory of the Louisiana Purchase, Nicollet compiled the first accurate maps of the Upper Midwest.

  The irony was that the very stars Méchain cursed himself for measuring had vindicated his exactitude. By contrast, Delambre had measured only Polaris and Kochab, both of which passed north of the zenith, so his results could not be corrected retrospectively. Between them, Méchain and Delambre had identified most of the sources of error that had produced the discrepancy: the refraction correction, the verticality of the circle, the setup of the instrument. What they lacked was a way to disentangle these errors. Méchain had not so much erred as he had misunderstood what error meant. Yet by his very misunderstanding he had inadvertently contributed to our own understanding of error, forever altering what it means to practice science.

  What is error? And who decides when it is too great to bear?

  Modern science accepts error as its lot. It does not demand truth from its practitioners, only honesty. It assumes that the truth will emerge eventually from a collective effort—so long as everyone is honest. Certainly scientists care passionately about getting the right answer. But when theory and experiment align too tightly, suspicion is warranted. Thus, the statistician R. A. Fisher concluded that Gregor Johann Mendel’s pea-breeding data could hardly have come as close to the 1:3 genetic ratio as he claimed. The same holds for Robert Millikan, who won a Nobel Prize for an electron experiment in which he suppressed anomalous data. In Mendel and Millikan’s day—from the nineteenth through the early twentieth century—such fudges were common, though frowned on. Today they continue, despite official censure. In Méchain’s day they were not only common, they were considered a savant’s prerogative. It was error that was seen as a moral failing.

  Méchain took his science personally. His observations were his to publish or suppress, keep or destroy. He felt no obligation to parade his accomplishments before an anonymous public. Rather, he sought to impress his fellow savants with his ability to approach perfection. Recording his observations in pencil and on loose scraps of paper only facilitated the task of refining the data. Even after he had handed over his scribal report to the Commission, he considered the raw expedition data to be his private property, part of the accumulated experience he carried with him in his trunk wherever he traveled. The data were his legacy and his tombstone; they were all he had. The truth belongs to everyone, but error is ours alone.

  Delambre’s scrupulousness was cut of a different cloth. For him, investigators answered to both their colleagues and their sponsors. The Revolutionary government had sponsored their lavish meridian mission, and it deserved a full accounting. That was why Delambre recorded his results like a public official: in ink, in a bound notebook with numbered pages, in order of observation, and with each page signe
d and dated. He believed that just as the republic operated with transparency—with roll-call votes, public laws, and open trials—so science should open itself to scrutiny. Delambre considered his data public property. All he asked was that he get credit for his labor. Thus, he was forthright about the discrepancies he discovered and the approximations he used. He never pretended that his results were definitive. What mattered was that he had tried to be as exacting as he could, and that the results were sufficiently precise to solve the problem at hand.

  Once, on the eve of presenting some astronomical tables to the Academy, Delambre discovered a trivial error in his calculations and spent the next three weeks working day and night to eliminate an error that was “to all intents and purposes imperceptible.” It was the most tedious task in a career of monumental labor, but when it was done, he could hand over the tables with a clear conscience. Then, when a fellow savant detected yet another error—as one colleague soon did—Delambre could freely and publicly acknowledge his mistake and correct it yet again.

  As to whether perfection existed “out there,” curled up in nature’s womb awaiting delivery, that was a theological question, and Delambre was a pagan. He had been raised in a devout home, he was schooled by the Jesuits, and he had considered taking holy orders. But he was neither a believer nor an atheist in the manner of his maître Lalande. He was a skeptical Stoic, for whom perfect knowledge lay beyond man’s grasp. Why then should anyone expect him to produce a perfect meter?

  This was something he had known all along. He had known it when he stood before the fiery volunteers of Saint-Denis to explain his absurd mission. Even then he had secretly agreed with the volunteers—in part, anyway. His mission was absurd. Why set out to measure the world while the old order was going up in flames, while millions of soldiers were rushing to die in battle? Why measure the world to create a unit of length when a standard “meter” could be created by legal fiat or simple agreement? It was absurd to travel so far to find what lay so near. Yet someone had to do it. Someone had to dedicate himself to the task of reconstructing the world. Otherwise there would be nothing left standing after the soldiers had finished their slaughter and the vandals had leveled the towers to the heights of the “humble cottages” of the sans-culottes. Someone had to construct a new order, a horizontal grid to enable people to keep track of where they stood, and what they made, and how much they bought and sold.

  Delambre knew that the International Commission’s boast of perfection was a sham. The Commission had claimed to know the length of the meter to six significant digits, or within 0.0001 percent. Delambre now acknowledged that this was “a precision to which we ought not to presume.” Now that the sanctified platinum bar was safely stored in the National Archives—smug and untouchable in its triple-locked box—he thought it only honest to admit as much. Thus in 1810, in the third and final volume of the Base, he described a range of plausible values for the earth’s eccentricity and a corresponding range of plausible values for one ten-millionth of the quarter meridian. He suggested that the best value for the earth’s eccentricity was probably nearer to 1/309 than the Commission’s 1/334. He took into account both of Méchain’s values for the latitude of Barcelona, thereby shifting the length of the arc by another 0.01 percent. He concluded that a better length for the meter would be 443.325 lignes (rather than the official 443.296 lignes).

  It was a trivial adjustment, less than the thickness of a piece of paper. But it was an act of remarkable integrity. Only one decade after the meter had been declared “definitive,” its chief creator, writing in the official account of its creation, acknowledged that scientific progress had undermined its validity. Today we recognize this as a step in the right direction—not because Delambre’s new value closed about a third of the shortfall of the definitive meter (after all, his new value was still less accurate than the provisional meter of 1793), but because it was a calculated homage to the transience of human knowledge.

  And Delambre did not stop there. He suggested rounding off the length of the official meter to 443.3 lignes. This may seem another trivial adjustment. But by lopping off two decimal places, Delambre implied that the meter was only accurate to within 0.01 percent. And he noted that the revision had yet another reason to recommend it: the value of 443.3 was easy to remember because it was composed of two 4s followed by two 3s. As for those who still insisted on working with six significant figures, he suggested that they consider the meter to be 443.322 lignes long, because that value too was easy to remember: two 4s, two 3s, two 2s. Nothing illustrates more clearly Delambre’s acceptance of the arbitrariness of conventional standards. As he noted in a private letter to a foreign savant: “Say what you like of the degree of precision we achieved, all I can assure you is that I conveyed every detail of our mission with the greatest possible sincerity and without the least reticence.”

  In the end, Delambre took longer to write the history of the meter than to measure France, and in a sense he traveled further in the process. He began writing in 1799 in the wake of the disturbing results of the International Commission. The 1806 volume began as an adventure story; the 1807 volume plunged him into a tale of scandal and discovery; and the 1810 volume concluded with a demonstration of the open-endedness of knowledge. Delambre had come to accept, as Méchain could not, the evanescence of earthly knowledge. So to the extent that he conspired in his colleague’s cover-up he did so for the opposite reason: because he realized that getting the perfect answer did not matter. Which is to say that Delambre understood that Méchain had agonized—and died—for nothing. We live on a fallen planet, and there is no way back to Eden. Delambre had decided to live on the surface of the earth, buckled, bent, and warped though it was.

  In coming to terms with the imperfection of earthly knowledge, Delambre had a powerful new intellectual tool at his disposal, one which he and Méchain had inadvertently inspired, but which he alone had lived to take advantage of. For the past century savants had sought to fit imperfect data to a perfect planetary curve. Geodesers had agreed that the earth was an oblate ellipsoid, but they had been unable to agree on its degree of eccentricity, which now seemed, moreover, to vary from place to place. Or were the data faulty? Assume the data then. Assume that the data had been gathered by fallible (but exacting) investigators using fallible (but ingenious) instruments on a (possibly) lumpy irregular earth, and then ask yourself: what was the best curve through the data, and how much did the data deviate from that curve? That was the question Adrien-Marie Legendre asked.

  Legendre’s answer, the method of least squares, has since become the workhorse of modern statistical analysis. It was also among the most important breakthroughs in modern science—not because it produced new knowledge of nature, but because it produced new knowledge of error.

  Legendre cloaked his personal life in obscurity and invested his clarity in his mathematics. A contemporary of Laplace and Delambre, he was elected to the Academy at the age of thirty for his work on number theory and analysis. In 1788 he showed the geodesers on the Paris–Greenwich expedition how to correct for the curvature of their triangles. Appointed with Cassini and Méchain to the Revolutionary meridian project, he withdrew in favor of Delambre. During the Terror he went briefly into hiding, only to emerge with a bride half his age. Later he codirected the Agency for Weights and Measures and was one of the savants to calculate the length of the meter for the International Commission. He was as baffled by the outcome—an unexpected eccentricity of 1/150—as the rest of them. Five years later—one year after the death of Méchain—progress slipped in sideways yet again.

  For centuries savants had felt entitled to use their intuition and experience to publish their single “best” observation as the measure of a phenomenon. During the course of the eighteenth century they had increasingly come to believe that the arithmetic mean of their measurements offered the most “balanced” view of their results. Yet many savants continued to feel, like Méchain, that any measurement that strayed to
o far from the mean ought to count for less than those near to it, and hence could be suppressed without apology. And even the most rigorous savants were flummoxed when they confronted multivariable phenomena for which they had diverse observations—such as the curvature of a nonspherical earth based on observations at various latitudes, or the elliptical orbit of a planet, especially if that orbit were perturbed. Some mathematicians had tried to find rules to compensate for results that deviated too much. The Jesuit geodeser Boscovich had proposed one method, and several other astronomers had tried their hand. Laplace had introduced a cumbersome method of minimizing the maximum deviation. But these methods remained awkward and unjustified.

  Legendre suggested a practical solution. He suggested that the best curve would be the one that minimized the square of the value of the departure of each data point from the curve. This was a general rule, and it was a feasible calculation. It was a practical dictum, and it prompted a radical reconceptualization. Legendre’s least-squares method played off the intuition that the best result should strike a balance among divergent data, much as the center of gravity defines the balance point of an object. As he noted, the least-squares method also justified choosing the arithmetic mean in the simplest cases.

  In 1805, just as Delambre was completing the first volume of the Base, Legendre tried out his method on what was now the world’s most famous data set, the one he had puzzled over ever since Delambre and Méchain had handed it to the International Commission. Legendre assumed that the earth’s meridian traced out an ellipse; he then used the least-squares rule to find the eccentricity that would minimize the square of each latitude’s deviation from that curve as it arced—in a kind of high-wire balancing act—along the data Delambre and Méchain had gathered at Dunkerque, Paris, Evaux, Carcassonne, and Barcelona. And when he did, he found that the deviations of the various latitudes from that optimal high-wire curve remained sufficiently large to be ascribed to the figure of the earth and not to the data. And in Volume 3 of the Base, Delambre echoed his analysis: it was the earth that was warped—not the data.