Einstein's Clocks and Poincare's Maps

by Peter Galison

  At a time when the division of labor had penetrated science, Poincaré wondered aloud what the Ecole had done to hold together those disparate specializations. True, he allowed, the competitive exams selected gifted students with diverse aptitudes; true too that the traditional “nobility” of the school served as an inspiration (“Those who fight for their country, like those who combat for truth. . . .”) But something more must be at work to produce a worldview that pervaded such scattered pursuits. Perhaps it was a rubbing of shoulders—certainly chemists, physicists, and mineralogists all benefitted from the high mathematical culture of their formative schooling. Even among the most abstract of Polytechnique mathematicians, moreover, “we see the constant concern with applications”—not just among the obvious figures who had made applications their metier, but also among the most scholarly of the school’s products, including one of the most powerful mathematicians in the history of Polytechnique, Augustin Louis Cauchy. Cauchy, despite his well-known opposition to a socially engaged mathematics, drew strongly on mechanics. Poincaré then invoked his own teacher, Alfred Cornu, who liked to repeat that mechanics was the cement that held together the various parts of the Polytechnician’s soul. “There it is,” he concluded, “that factory stamp that I was looking for; our physicists, our mathematicians are all a bit mechanicians.”2

  If the Polytechnicians—Poincaré foremost among them—emerged from their training with the “factory stamp” of mechanics imprinted on their outlook, this already differentiated them from scientists trained in the universities. But for Poincaré that was not all. University professors fretted about the unity of science; Polytechnicians had another concern, that of joining thought with action. It was action, Poincaré insisted, that inoculated Polytechnique’s graduates against the melancholy that assailed many university researchers over the purported failure of science. That bond between the abstract and the concrete was the most essential feature of the “factory.” Science changed; so too did the world. Poincaré was fully prepared to allow the education to evolve: “The Ecole must transform itself bit by bit like all things human, but it must not touch that which makes its soul, the alliance of theory with practice must not be broken; it must not be mutilated, for without it there would only remain a hollow name.”3

  When Poincaré had come to Polytechnique back in 1873, the problem of balancing pure knowledge against useful applications was gripping France as never before. Just two years earlier, Germany had pounded France into a humiliating defeat, leaving Germany with new lands, united, euphoric, building scientific institutes and monuments to hail the victory.4 France, now minus Alsace-Lorraine, struggled desperately to understand the roots of its catastrophic defeat. The technical infrastructure of the State came into question: book after book lamented the sorry state of the country’s railroads and, more generally, its inadequate preparation for a new age of fighting. But more than any particular piece of technology, critics dissected the institutions of technical learning, which appeared to need reconstruction, and fast. None of these institutions had greater weight than Ecole Polytechnique, the Grande Ecole Poincaré had just entered.

  Polytechnique, founded in 1794, had no parallel in the United States, Britain, or Germany. Built to train an elite corps of engineers able to mold the military into a force of modernity by using an enlightened science, Polytechnique was to be a scientific institution where mathematics, physics, and chemistry would prosper with mathematics above all. Its students chosen by a competitive examination (the famous concours), Polytechnique produced a fraternity rigorously schooled in a mathematics-based engineering. After their graduation, generations of these students joined the highest levels of the State’s administrative structure, overseeing at first the new nation and, later, in the nineteenth century, an empire. Part Oxbridge and part Sandhurst? Half MIT, half West Point? Such comparisons fail. The Polytechnician was far more scientific than his British equivalent schooled in the Classics, more mathematical than a rising American engineer, and less inclined than an elite German physics student toward precision laboratory work. Polytechnique was, and remains, a unique institution, possessing by the 1870s a mythological status in France.

  For the first years after its founding, the revolutionary engineer and mathematician Gaspard Monge used geometry to successfully bind the conflicting demands of practical engineering and advanced mathematics. Projective geometry, he believed, trained the mind, revealed scientific truth, and sculpted stone, wood, and fortifications. Over the course of the early decades of the nineteenth century, however, Monge’s program began to falter. Mathematics, significantly propelled by the conservative Cauchy, self-consciously began to shed its ambition of joining the world with the mind. Science ascended, applications retreated—a trend encouraged by the construction of new, more specialized engineering schools.5

  When Paris fell to the Prussians, the French found no shortage of institutions to blame. Pasteur, joined by a chorus around France, lent his prestigious voice to those who saw the German triumph as a failure of French science. At Polytechnique the charges echoed loudly—it was a major center of scientific engineering and the students were already in uniform. Alfred Cornu, a former student of the school who had returned as a rising star among the physicists, articulated the school’s position after the disaster, one delicately balancing science and worldly engagement. He became, so to speak, an ideal type of the new Third Republic Polytechnician, gliding easily from the pure to the applied. As Poincaré later remarked with admiration, Cornu’s work traversed a wide field of optics, bringing new instruments and techniques not only to physicists but also to astronomers, meteorologists, and even clockmakers. Having designed and built at Nice an extraordinarily precise astronomical clock with a massive pendulum, Cornu traveled there every year to perfect his clock. In addition, Cornu developed a detailed mathematical theory of the coordination of clocks by the exchange of electric signals.6 Neither his joining of the pure to the applied nor his specific interest in electric time coordination was lost on Poincaré.

  Cornu’s course was filled with stylized demonstrations of particular laboratory phenomena. Although he did not train his students to do the handwork themselves, experiments were for Cornu a vital part of science—its alpha if not its omega. Nor was Cornu alone: he and his colleagues made sure that a young scientist emerging from Polytechnique in the 1870s did so with a deep respect for experiments (though no great familiarity with the manipulation of devices, complex measurements, or data analysis). Instead, students learned to see the great mathematical structures that embraced data as the endpoint of scientific work. They learned to put little faith in the literal truth of particular assumptions about atomism or electric fields. It is telling that when Polytechique filled a position in chemistry, the faculty aimed above all to maintain the balance between those who believed in atoms and those who did not.7

  This was the world that Poincaré entered in November 1873. Competitive, alert, engaged, Poincaré watched his fellow students like a hawk, writing home about his rivals’ grades, occasionally commenting on student hazings or musing over the political machinations of the Jesuits. Mechanics seized his attention, and in this all-important subject his grades rose rapidly toward nineteen or twenty (out of a possible twenty). With pride, he wrote home that he had found a simpler demonstration than the one his professor had presented in class. He wrote, too, of his progress in both technical and freehand drawing. When, to the surprise of the workers, he and a group of Polytechnique students visited a local crystal factory, Poincaré, told of his fascination with the workers’ dexterity and the technology of their Siemens ovens.8 Even as a student, Poincaré recognized the “factory stamp” of Polytechnique. But in the account reported to his mother, the teaching held none of its later glow:

  It is as if we are in an immense machine whose movement we must follow on pain of being passed by; we have to do what 20 generations of X [Polytechnique] did before us and what 2n + 1 generations of conscripts will do after us.
Here you use only two faculties of your intelligence: memory and elocution; understanding a course is something anyone can do with some work and that is why everyone can get me to cram if they want. . . . I am therefore condemned to this choice: give up personal work or stay in my place; as this alternative lasts only two years, the choice I have made is not in doubt; because the advantage I will take from my position is incommensurable; but it has to be guarded and [switching to English:] this is the question.9

  On Cornu’s death, Poincaré spoke, quite personally, of how much Cornu would be mourned, as a friend, an advisor, and as a master teacher—and Poincaré’s career followed in many respects a similar path. Both had been star students at Polytechnique, both had gone on to the Ecole des Mines, both accepted the call back to Polytechnique to teach, both served the State in engineering-administrative capacities. The lessons stamped on them were deep. Poincaré, Cornu, and their contemporary Polytechnicians maintained a lifelong commitment to the link between abstract and concrete knowledge. It was a testimony to their training that both served on the board of the Bureau of Longitude and together along with their fellow Polytechnicians, drove a myriad of technologically progressive projects, from the electrotechnical journal they helped run to the scientific commissions on which they served.

  Of course, the relation between pure science and engaged technology was not frozen once and for all at Polytechnique—or anywhere else. It was much more like a vast and slowly undulating sea, sometimes with olympian science looming high above a trough of technology; at other times with a triumphalist military and industrial technology cresting over the claims of pure knowledge. For a brief moment after the great defeat of 1871, Cornu and his allies flattened this sea into a smoother surface of roughly equivalent claims, at least around Polytechnique. Inside this “immense machine” where technology and mathematical physics circulated with equal force, Poincaré became Poincaré.

  Coal

  Before turning to clocks it is worth exploring two moments early in Poincaré’s career, for they tell us a great deal about who Poincaré was, how he thought, and where he stood in the turbulent flux of technologies and sciences. As a shorthand, we may think of these two episodes as moments of coal and chaos, for during the first decade of his work—from the late 1870s to 1890 or so—Poincaré was grappling not only with a new, wildly unstable mechanics of the Solar System but also with the grubby, dangerous world of mining in late-nineteenth-century France.

  Graduating in 1875, Poincaré followed tradition by moving with the other two valedictorians of his class to the Ecole des Mines. There, the three—Poincaré, Bonnefoy, and Petitdidier—began their studies in October.10 Poincaré’s mentor, mathematician Ossian Bonnet, tried to get Poincaré’s course load reduced because of his mathematical work, but the School of Mines would hear nothing of it. So Poincaré finished his mathematical thesis while learning the ins and outs of ventilator shafts. Geological fieldwork took him, for example, to Austria and Hungary in 1877.

  Poincaré’s cultural circle was also expanding. His adored younger sister, Aline, introduced him to the philosopher Emile Boutroux, whom she would eventually marry. Boutroux and Poincaré immediately began discussing philosophy. Having studied in Heidelberg until the Franco-Prussian War, Boutroux had made his own the German philosophical commitment to join the humanities and the sciences. Productive, enthusiastic, religiously inclined, Boutroux argued (following the Kantianism he imbibed in Heidelberg) that much in the scientific domain was more of the mind than “out there.” Through Boutroux, Poincaré came to meet other philosophers, including the philosophically inclined mathematician Jules Tannery. The group did not share Boutroux’s religiosity (Tannery was an ardent, secular Republican), but all sought a middle way between science as simple observation and science as mental creation. Poincaré seemed to find the view congenial; for years he, too, argued that science needed just such a mix of induction and deduction. As Poincaré put it in a letter to Aline sometime around 1877, observation and induction ought be treated with “reserve.” He added, “you will say induction can only give us knowledge of the same nature as the observations themselves; observation cannot teach us anything about Substance . . . it can only show us the phenomenon by itself; and not even the phenomenon in itself; but only the sensations that it produces in us.” Experience, Poincaré told his sister, could never be sufficient to ground the full generality of knowledge. “Eh what do you want me to do about it; let us always take what belongs to us, and as for the rest, we must resign ourselves to admit that it will remain to us forever but a dead letter.”11

  Figure 2.1 Poincaré’s Curve of Happiness (with detail). This letter, written by Poincaré to his mother during the summer of 1879, includes both a hand-drawn map of his geological travels and a geometrical curve showing the “limits of his joy” in “normal times,” “yesterday,” “in trains,” and “right now.” SOURCE: ARCHIVES HENRI POINCARÉ, M021.

  Another of Poincaré’s fellow Polytechnicians, the philosopher-scientist Auguste Calinon, held a similarly cautious view about knowledge that “did not belong to us.” In 1885, Calinon published a treatise on the foundations of mechanics and geometry. He and Poincaré seemed to have been on good terms; when they saw each other in early August 1886, Calinon followed up with a copy of his recent Critical Study of Mechanics. It began, straightaway, with a cautionary note about absolutes in space and time:

  Many authors, in mathematics as in philosophy, accept the notion of absolute movement as a first principle (idée première). This opinion has been much contested; . . . it is worth remarking that from the point of view of rational mechanics, this question has no importance; the movement of a point, considered in isolation, is a purely metaphysical conception, because, even allowing that one could imagine such a movement, it is impossible to certify it and to determine its geometric conditions, for example, the form of its trajectory.12

  Because of the inaccessibility of the absolute, Calinon would only speak of relative movement. Similarly, he suggested that simultaneity too had to be accessible: two moving stellar objects at particular positions would be called simultaneous only if one saw them “at the same time.” For Calinon, the human registration of the events was so important that he wanted to take into account the time it took for the sensations to be registered by the brain: “The very idea of time is therefore inherent to the mode by which our brain functions and has no sense except for minds made like ours.”13 A Kantianism, no doubt, but one that was distinctly more psychological (or psychophysiological) than that which dominated the German-speaking scene. Apparently Poincaré wrote back almost immediately addressing the work point-by-point. Though that letter is lost, Calinon’s reply survives, from which it is clear that Poincaré’s very first comment addressed the notion of “at the same time.” It appears that Poincaré agreed that “it is by our sensations alone that we judge simultaneity or successiveness.”14

  Poincaré’s philosophical speculations about the limits of scientific knowledge, the restricted power of observation, and the active role that the mind must play in making science, were themes that remained with him for the rest of his life. But none of these metaphysical musings interrupted either his mathematical work (he submitted his mathematics thesis in 1878) or his work on mining. In March 1879, Poincaré received his degree of “ordinary engineer” from the Ecole des Mines, arriving on 3 April for his new position at Vesoul, where his inspections began the next day and continued intensively over the following months. On 4 June 1879, he reported that Saint-Charles had just about exhausted its pits—“veins both poor and irregular.” On 25 September at the pits of Sainte-Pauline, he focused on aeration, removal of gas, and sources of water—just the sort of engineering tasks that Ecole des Mines had emphasized. A month later, on 27 October, Poincaré arrived at the pits of Saint-Joseph to inspect the smelting works. His last mining visit took place on 29 November 1879.

  But one stop on the itinerary, at Magny, was anything but routine. In the e
vening of 31 August 1879, at 6:00 P.M., twenty-two men descended for their shiftwork into the coal pits. About 3:45 A.M. an explosion rocked the mine, instantly blowing out the miners’ lamps. Two miners in the cage were violently shaken, two were knocked into the sump (which, luckily, was covered with a board about five feet down). These four survivors staggered to the earth’s surface. Master miner Juif, who had been off duty near the pits, immediately led the men back into the mine, where they discovered a pile of clothes burning without flames like a piece of smoldering punk. Juif headed straight for them, extinguishing the material before it could ignite the wooden retaining structures, the coal, or, worst of all, another catastrophic gas explosion. Following cries, they discovered Eugène Jeanroy, a sixteen-year old, who died of his wounds the next day. Every other miner the team found during the search was already dead, some of appalling burns.