The Measure of All Things

by Ken Alder

  THE BORDA REPEATING CIRCLE

  The Borda circle is here shown in its horizontal configuration for geodetic triangulation. (From J.-B.-J. Delambre, Base du système métrique décimal, 2, plate VII; photograph by Roman Stansberry)

  THE USE OF THE BORDA REPEATING CIRCLE

  This diagram illustrates the method of reiteration, which made the repeating circle a precision instrument. In the example that begins with the top–down view in (1) the sighting targets G (for “gauche,” left) and D (for “droit,” right) are separated by 10 degrees, as indicated by the placement of scopes F and L. Both scopes are initially coupled to the graduated circle. In (2) the lower scope F has been rotated clockwise to fix on the target D, thereby moving the upper scope L clockwise the same distance. In (3) the upper scope L has been decoupled from the circle and independently rotated to focus on G, such that it must pass through twice 10 degrees relative to its point of origin. Then in (4) the scope L has been recoupled to the circle, so that when it is refocused on D it moves the lower scope F as well. Thus when the scope F is independently refocused on G (5), it adds yet another 10 degrees to the gauge, while bringing the observer back to the original situation—except that instead of having only measured the 10-degree separation once, twice the original angle has been added to the original reading. This process can then be reiterated as many times as the observer desires—see (6) and (7)—without any need to reset the gauge. The final cumulative angle is then divided by the number of iterations to give a precise reading of the angular separation. This method has the advantage of reducing the uncertainties inherent in any single angle observation and of minimizing the impact of irregularities in the manufacture of the calibrated gauge. The side view, shown in (9) and (10), demonstrates how the same process of iteration could be used to measure the vertical height of a star S with scope AB relative to the horizontal defined by the level scope MN. In this situation the process begins by rotating the entire circle 180 degrees horizontally on its axis, and then sighting the star again with the scope AB (not pictured). (From Jean-Dominique Cassini IV, Pierre-François-André Méchain, and Adrien-Marie Legendre, Exposé des opérations faites en France en 1787, pour la jonction des observatoires de Paris et de Greenwich [Paris: Institution des Sourds-Muets, 1790], plate 3; photograph from the Houghton Library, Harvard University, Cambridge, MA)

  Méchain began his measurements at the hermitage of Notre-Dame-du-Mont. He measured on the summits of Puig-se-Calm, Matagall, and Roca-Corra, where Alvarez slept in a tent on the peak until the station was surveyed from all the surrounding sites. He measured at Puig Rodos, where he and Gonzales lodged in a cowshed, making over a dozen four-hour trips up the mountain in hope of finding clear weather for their observations. He measured at Mount Matas, at the chapel above Montserrat and, in late October, over the Valvidrera ridge down into the basin of metropolitan Barcelona. On the night of November 28, 1792, residents in the city reported lights burning on Mont-Jouy, above the General’s Palace, and in other high spots around the perimeter of the city. Méchain and his colleagues were lighting parabolic mirrors to take accurate nighttime readings across the basin.

  With these angles in hand, Méchain had every reason to consider the 1793 season a success. He had closed seven stations in less than three months, traversing with meteoric speed nearly half the total distance to his final rendezvous at Rodez—and the most difficult sector of the meridian to boot. If anything, he felt he had been somewhat rushed in his labors. The Spaniards, he complained, did not like to linger at the more “arduous” stations. Had he been working at his own deliberate pace, he would surely have conducted many more observations, despite the adverse conditions. Méchain was striving for a degree of precision that had eluded all other investigators, a precision that would be the best guarantor of the meter’s universality. He was not about to let icy winds or steep mountain climbs diminish the accuracy of a measure meant for all people, for all time. But compromise was the price one paid for collaborative work.

  It was true that he had been as yet unable to measure the high mountain stations along the crests of the Pyrénées. But these frontier stations could always be approached later from the French side of the border. And once he returned to France he would have Cassini’s maps and precedent to follow, and would be able to observe to his own satisfaction. This left him only one remaining task in Spain: the latitude measurements that would fix the southern terminus of his arc in Barcelona. For this, Méchain decided to conduct his measurements at Mont-Jouy, the hilltop fortress on the southern edge of the city.

  The outcrop of Mont-Jouy, with its castle fortress perched at the sea-cliff’s edge, is visible from everywhere in downtown Barcelona. The six-hundred-foot ascent from the Mediterranean is unassailable. Even on the city side, the slope is a stiff climb. Today, if you prefer, you can ride up in one of the bright red gondolas that shuttled spectators to the 1992 Olympic Games. Or you can drive up the switchback road through the terraced parkland, a legacy of the World’s Fair of 1929 and still the route of one of Spain’s most famous bicycle races and Grand Prix competitions. On quieter days, however, when the amusement rides are abandoned and stray cats sleep amid the disheveled palms and pines, you can easily imagine the hillside two hundred years ago, when eighteenth-century tourists hunted for fossils in the limestone, or unearthed heavy fragments of Hebrew gravestones, remnants of the pre-exile necropolis that gives the site its Catalan name: Montjuïch, or “mountain of the Jews.” In French it is known as “Mont-Jouy,” evoking a hillside of play and pleasure. In Latin it is “Mons Jovis,” the miniature Olympus where a temple to Jove once stood. In sum, it is a place of layered histories.

  For centuries the Mont-Jouy site had been occupied by a lighthouse. Then in the seventeenth century, as France and Spain fought for control of Catalonia and the city periodically rebelled against both kingdoms, the hill had been fortified. In 1714 the Spanish monarch began building a fortress along modern lines, which English visitors considered so “perfect in its kind” as to make Barcelona “untenable by an enemy.” Massive walls were arrayed in a pentagon so defenders could train maximum firepower on assailants. The fort could accommodate 3,000 men and 120 cannon at one time. It enclosed a large parade ground and a series of well-appointed barracks for officers and men. Between the Mont-Jouy castle to the south and the massive new Ciutadella fortress to the north, the town was protected from attack and dissuaded from rebellion. From the heights of impregnable Mont-Jouy, where Méchain and his team were lodged for the winter, the town’s rising clamor against the French was a distant din. It was a fortress poised between the city and the sky.

  BARCELONA HARBOR, LOOKING TOWARD MONT-JOUY

  (From Alexandre Laborde, Voyage pittoresque et historique de l’Espagne [Paris: Didot, 1806–22], plate I, photograph courtesy of Special Collections, Golda Meir Library, University of Wisconsin-Milwaukee)

  FROM THE HEIGHTS OF MONT-JOUY, OVERLOOKING BARCELONA

  (From Alexandre Laborde, Voyage pittoresque et historique de l’Espagne [Paris: Didot, 1806–22], plate IV, photograph courtesy of Special Collections, Golda Meir Library, University of Wisconsin-Milwaukee)

  The view from the Mont-Jouy battlements takes in a vast sweep of the Costa Dorada. Far to the south, where the oceanic blue fades to a pale horizon, you can see, if the day is clear and you have the right kind of equipment, a yellowish break in the hazy line between sea and sky. That is Mallorca, the largest of the Balearic Islands, and if you can see it, then you are looking around the curvature of the earth, relying on the refracted light that bends through the low damp air of the Mediterranean atmosphere.

  That long view was itself a temptation. In the flush of ambition, Méchain dreamed of surpassing the Academy’s mission and extending the meridian as far south as Mallorca. The means, moreover, were to hand. Commander Gonzales had offered to sail his Corzo across the one-hundred-mile straits to light flares on the island peaks. Méchain wrote to France and secured permission from his colleagu
es to try the extension. In December he put that plan into action. While Gonzales set off with his crew of sailors to plant the special reflecting mirrors atop Mallorca’s five-thousand-foot Sillas Torellas peak, and Tranchot and Planez scouted locations down the mainland coast, Méchain took up his position on the battlements of Mont-Jouy. On the evening of December 16 he spotted through his telescope a faint light trembling on the horizon. Unfortunately, as he informed his colleagues back in Paris, the resolution was insufficient, and this geodetic triangle, the largest ever attempted, would only be feasible if his repeating circle were refitted with more powerful scopes and he were given larger parabolic mirrors. In the meantime, he assured them, his first priority was to complete the mission he had been sent on.

  The rest of December was spent in erecting an astronomical observatory alongside the fortress tower of Mont-Jouy, overlooking the sea. The observatory was a wooden cabin, fifteen by twelve feet, with a retractable roof that could be raised by pulleys, and windows that opened out onto an unimpeded view south to the sea and north to the mountains. In this cabin, Méchain positioned his repeating circle in such a way as to capture the transit of stars that crossed the celestial meridian. This was nighttime labor, the work of a few experts. The mountain team was disbanded. While Gonzales counted off seconds from a pendulum clock and Tranchot held a lantern to verify the level of the instrument, Méchain measured the stars.

  The latitude measurements to anchor the end points of the survey were the most delicate operation of the entire mission; the slightest stumble here would distort the final result. Ordinarily, the determination of latitude posed no great challenge to eighteenth-century astronomers. Unlike the longitude problem, which had stumped the world’s navigators for centuries, mariners had long known how to calculate their distance north or south of the equator by measuring the height of the sun, Polaris (the pole star), or some other celestial object. Latitude was routine. But science thrives on transforming old routines into new problems. Méchain hoped to determine the latitude of Mont-Jouy with a degree of precision hitherto unmatched in the history of astronomy. His goal was to pinpoint its position on the globe to within one second of a degree, a distance of about one hundred feet, an accuracy comparable to today’s commercial global positioning systems. This degree of precision transformed a routine measurement into an awesome challenge.

  In this quest for celestial precision, Méchain’s sole armament was the same Borda repeating circle. But Méchain felt he was up to the challenge. “As Monsieur de Borda says, it depends only upon the patience of the observer to ensure that errors are in the end totally eliminated.” So far, Méchain’s forbearance and patient industry had surpassed all expectations. He wrote to the inventor himself to explain that he had just completed some preliminary calculations of his geodetic results and the sum of angles in his survey triangles never deviated more than 3.5 seconds from 180 degrees, a stunningly small 0.0005 percent discrepancy. Now he was ready to put the instrument to its astronomical test.

  For Méchain, the pursuit of precision was a moral quest as much as a scientific one. Its consummation proved that the investigator possessed the patience, skill, and rectitude to reveal nature’s predictability and lawfulness. But the pursuit of precision, like all moral quests, is a hazardous affair. Zooming in on nature’s fine structure can produce unexpected explosions. If the results do not converge, who will take the blame? Nature or the investigator?

  For this celestial operation, the plane of the repeating circle was flipped into a vertical position so that it stood erect like a stop sign aligned in the direction of the meridian. One of the scopes, the one that came equipped with a calibrated air-bubble level, was then directed toward the horizon and its position carefully monitored, while the other scope was angled toward the expected height of the transiting star. As the star approached the line of the meridian, the savant tracked its movement across the wire grid in the lens, listening out for the beat of the pendulum clock to mark the exact time. Prowess at this eye-and-ear method was one of the most basic and demanding astronomical skills, but the repeating circle demanded something more. Although the angle between the two scopes, when read off the graduated ring, gave a preliminary assessment of the stellar height, accurate results from the repeating circle once again depended on reiteration. To accomplish this, the savant rotated the entire instrument 180 degrees on its vertical axis, so that the stop sign faced the opposite direction, aligned once again along the meridian. Then he loosened the screws and swung the working scope back around the graduated ring until he could sight the star in the wire grid once more. In doing so, he had doubled the angle traced out along the graduated ring. This action, repeated, gave a value of four times the stellar height, and so on. The whirl of dextrous activity, with the astronomer alternately spinning the plane of the instrument one way and the scope the other, can be likened to a master mathematician solving a Rubik’s cube in the hopes that the right sequence of high-speed moves will unlock the secret combination of the heavens.

  When Méchain refused to let his Spanish hosts take a crack at solving this puzzle themselves, they were understandably irritated. Planez complained that Méchain had relegated him to a “mere spectator,” insinuating that the Spanish king would not appreciate having his officers treated so high-handedly. Yet Méchain refused to relinquish the eyepiece. The task of measurement, he insisted, was his alone. Only he had the requisite skill and experience. He alone was answerable to the Academy for the results. This was the savant’s responsibility and his prerogative. It was the way Cassini had treated him on their 1788 expedition from Paris to Greenwich. In the end, Méchain had Planez dismissed and replaced with a more conciliatory assistant, the military engineer Captain Agustín Bueno.

  Over the course of the next three months, Méchain took 1,050 sightings of six different stars, each comprised of ten repeated observations. It was a Herculean effort. At night the deep celestial cold caused buckets of water to ice over. All through the Christmas holidays and into the new year, Méchain, Tranchot, and Gonzales worked under the black Mediterranean nights high above the city, pinpointing the latitude of the fortress.

  Occasionally he made other sightings. At 6:15 on the evening of January 10, 1793, he reported the discovery of a new comet. Comet-hunting was Méchain’s favorite activity and the basis of his scientific reputation, as all his colleagues knew. So lest they think he had been diverted from his official duties, he also reported that the comet was a bit to the west of Mizar, one of the stars he was using for his latitude measurements. “It’s not my fault,” he wrote. “I wasn’t looking for it.” This innocent discovery was written up in the Diario, Barcelona’s daily newspaper, where the editors felt compelled to add that a comet was a natural celestial body and its appearance did not, as the common people believed, foretell the coming of “war, pestilence, or the death of kings.” That very week the National Convention began its deliberations in Paris on the fate of Louis XVI. Ten days later, on January 21, 1793, the French king would be executed. War soon followed.

  On the evening of February 20, in plain sight of Mont-Jouy—but beyond the reach of its guns—a French privateer sacked a Spanish convoy on its way into Barcelona harbor with a treasure of American gold. Huge crowds gathered along the sea wall to watch helplessly as the French pirates absconded with 80,000 duros due to the town’s merchants. Mobs clamored for vengeance, and stoned to death a citizen of Genoa whom they mistook for a Frenchman. The French consul feared that the populace would seize French goods in retribution. “The Catalans,” he wrote, “are reckless, bold, and vindictive; money is their god.” Méchain was distressed to learn that the frigate Corzo, under the command of his friend and scientific collaborator, José Gonzales, had been ordered to give chase to the pirates. Three days later, Gonzales returned empty-handed. The citizens of Barcelona were enraged. Indeed, some historians have called this incident one of the provocations that led to war between the former allies.

  The time had come t
o wrap up the Catalan mission. On the day of Gonzales’ return, Méchain wrote for the first time to his northern collaborator, offering Delambre words of encouragement and sympathy. “Monsieur and dear colleague,” he wrote, “I heard of your misadventures, and felt for you; I learned of your successes, and they gave me much joy.” He suggested that they swap notes and advice. For instance, how did Delambre set up his signals? How had he positioned his repeating circle? For his part, Méchain described his procedures and, with elaborate modesty, assured his collaborator that his own southern triangles would never match the precision of Delambre’s northern ones. His own ineptitude, he hastened to add, was only partially to blame. Measurements in the mountains presented all sorts of difficulties such as one did not encounter in the lowlands. Up there, heavy clouds obscured stations; trails were treacherous; the cold hampered the instrument. Still, he had been ably seconded, and once his latitude data were complete he intended to return to France “sometime next month.”

  Delambre answered from Paris with best wishes for Méchain’s “health, courage, and patience.” On a more practical note he also enclosed updated passports—without the seal of the former king.

  But before Méchain could leave Barcelona, the Spanish authorities insisted that he report to them on his geodetic triangles and the latitude of Mont-Jouy. As equal partners in the expedition, the Spaniards had every right to this information. The data would enable them to draw the first accurate map of Catalonia, as well as locate their fortresses with pinpoint accuracy. Cajoling these data out of Méchain tried Gonzales’ patience, but at his insistence Méchain spent the month of March preparing a summary report for his Spanish collaborators. He also mailed a précis of his results to Borda in Paris.