Poincaré entered the mine almost immediately after the explosion, in the midst of the rescue operation, despite the risk of a secondary detonation. As the assigned mining engineer, it was his job to sort out what had caused the disaster. Searching for that culpable first spark, he turned first to the lamps. Designed by Humphrey Davy in 1815, these “safety lamps” surrounded an illuminating flame with a closely woven wire mesh that passed light and let air in but kept the flame from escaping. A punctured lamp in a methane-filled mine is a disaster in waiting. Numbers 414 and 417 had belonged to Victor Félix and Emile Doucey; they were never recovered. Lamp 18 was totally destroyed by a cave-in—its mesh and glass no longer anywhere to be found, the rods twisted and broken, and the top entirely detached from the base. His attention, Poincaré wrote in his investigation report, was particularly drawn to lamp 476: it had no glass and two ruptures. The first tear, long and wide, seemed to have come from an interior pressure. The second, by contrast, was rectangular and clearly came from the outside; in fact the puncture was a dead ringer, according to all the workmen, for a strike from a standard-issue miner’s pickax. That particular lamp had been signed out to Auguste Pautot, a thirty-three-year-old miner, but it was not found next to his body. Instead, Poincaré observed, lamp 476 still hung from a timber support a few inches from the ground and in the immediate proximity of workman Emile Perroz’s corpse. Poincaré and the rescuers found Perroz’s own lamp intact, elsewhere.
Throughout his report Poincaré mixed a factual with a more personal tone, one not yet hardened by years of accident inquiries. He recommended the chief miner for compensation in virtue of his bravery, and concluded the medical section with the mournful addendum that he hoped the miners had died instantly, sparing them a long agony. Poincaré’s conclusion listed not only the victims but the nine women and thirty-five young children they left behind: “Even the generous efforts of the company will perhaps be insufficient to relieve so much misery.”15
Exploring the causes of the accident, Poincaré’s language turned analytic, advancing hypotheses and counterhypotheses, facing them one by one against the evidence. He acknowledged, for example, the commonplace that miners upstream of an explosion in the air supply are usually burned, while those downstream are asphyxiated. In the Magny disaster, all the deceased suffered burns, so it was logical to think that the explosion had occurred down the airstream from the last man, Doucey. Cave-ins seemed to reinforce this deduction and to suggest that the gas explosion occurred in the half-moon (see figure 2.2). Against this seemingly plausible notion stood another hypothesis that “equally well accounts for the facts.” In particular, Poincaré considered whether the explosion might have occurred near the timber peg immediately adjacent to the point where lamp 476 stood:
So here we are in the presence of two hypotheses up until this time equally plausible: explosion in the half-moon, explosion at the top of the lift. Without the Doucey lamp one cannot prove directly that it had not caused the initial ignition of the gas. But diverse considerations nonetheless support the idea that the primary accident took place in the work space of Perroz.16
Since Perroz was a coal loader and therefore had no pick, Poincaré reasoned that it must have been Pautot who had accidentally punctured the lamp with his axe, and then inadvertently exchanged lamps with Perroz. Sometime after that switch, the punctured lamp 476 had lit the atmospheric methane, initiating the conflagration, and setting off a secondary explosion at the point where the incompletely burnt gas encountered the principal flow of air.
Step by step, Poincaré reasoned on through the investigation, eliminating other possible sources of gas one by one: Some sources were outside the flow of air, other veins of coal too old to be degassing. Clinching the argument, he added, was the fact that any source of gas from a distant source would have risen to the top of the tunnel and not, therefore, been in contact with the low-hanging lamp 476. Any slow venting of gas would have asphyxiated Perroz who, according to the medical personnel, had died standing. No, Poincaré concluded, the gas must have entered suddenly, probably from a natural gas vent just a single step from lamp 476. When the gas hit the punctured lamp, the miners were doomed.17
Poincaré worked the investigation for months. He returned to the mine on 29 November 1879 to follow up on the mechanics of aeration, conducting tests, making measurements of flow, and determining the relative air pressure at different points in the shafts. He submitted his report at Vesoul on the first of December 1879. That same day the ministry of public instruction advised him that he would be given a junior lectureship in the faculty of sciences at Caen. But neither mathematics nor any of his other pursuits ever completely diverted Poincaré from his interest in mining. At the end of 1879 he still hoped to continue his mining engineer career simultaneously with his mathematical one. In fact, he never left the Corps des Mines: named chief engineer in 1893, Poincaré became inspector general on 16 June 1910. Just before his death in 1912, he published an article titled “Les Mines” in a book he and a few colleagues produced to join the cultural with the technological and scientific. Poincaré’s entry is preceded by a small picture of a Davy lamp without comment, but no doubt a sign for him of the still-smoldering pits of Magny thirty-five years earlier. “One spark,” Poincaré wrote later in the piece, “is enough to ignite . . . ; well, I refuse to describe the horrors that follow.”18
Figure 2.2 Magny Explosion Map. Poincaré drew this map to trace the flow of air through the mineshaft at Magny, during his days as a mining safety engineer. On the basis of his investigation, he concluded that the fatal explosion of 31 August 1879 had been caused by a miner’s inadvertent puncture of his Davy “safety” lamp rather than because of an explosion in the “half moon” shaft located at the top of the map. SOURCE: ROY AND DUGAS, “HENRI POINCARÉ” (1954), P. 13.
Chaos
Even while sketching and improving the machinery of mining—lamps, lifts, and pit ventilators—Poincaré wrestled with mathematical problems that were, at the same time, physical ones. Those problems converged on what had become the great challenge of celestial mechanics: the three-body problem. It is easy enough to state. The motion of a single body is given by Newton’s injunction that a body in motion tends to stay in motion. The motion of two bodies, attracted to one another by Newtonian gravity, could also be solved. With the simplifying assumption that the planets were only attracted by the sun (and not by each other), it was a straightforward exercise for Newton and his successors to calculate the precise trajectory of these bodies around the sun. But for a system of three or more mutually attracting objects, such as the sun, the moon, and the earth, the situation was far more difficult. Eighteen interrelated equations had to be satisfied to solve the problem. If space is measured by three axes x, y, and z, then a full description of the motion of the orbs would require the positions x, y, z at each moment in time for each of the three heavenly bodies (that makes nine equations), along with the momentum of each in each direction (another nine). By choosing the right coordinates, these eighteen equations could be reduced to twelve.
For many mathematicians of the mid-nineteenth century, the trend of their discipline was toward an ever-more-rigorous formulation—precise definitions, proofs designed to annihilate the smallest shred of doubt. Such a passion for airtight logical proofs was not what drove the curriculum at Polytechnique, and it never formed one of Poincaré’s abiding concerns. Nor was he after better methods to solve equations, though in astronomy such studies could increase the accuracy of predictions about where a planet might be found. Charts of the ephemerides (as the juxtapositions of heavenly objects were called) were very useful to a ship’s navigator. Finding such numbers that were both scientifically and practically important might have been just the sort of thing that a Polytechnician like Poincaré wanted. But continuing in the usual way was not an option: he could show that the usual approximation methods for finding the ephemerides gave dramatically wrong predictions of where the planets would be. Never h
aving been drawn to the pure mathematician’s fixation on rigor and now convinced of the futility of the applied astronomer’s traditional death grip on numerical methods, he needed a dramatically new approach.
Poincaré found that new entry into celestial mechanics though diagrams: he focused on what he called the qualitative featues of differential equations. A differential equation tells how a system of things—points, planets, or water—changes from one moment to an infinitesimally later moment. By itself this is not much use in making predictions: knowing where a planet will be an instant from now will not help a ship’s navigator. For a useful longer-range prediction, the astronomer had to add up many infinitesimal changes to calculate, for example, where Mars would be next June. To many mechanicians, solving such problems meant doing this adding up (integrating) and putting the final result in a simple, recognizable form. This was not Poincaré’s goal.
More or less from the time Poincaré left the mine pits, he aimed to attack differential equations in his own way. Instead of following one water droplet down a stream, so to speak, he wanted to characterize the flow pattern of all the drops making up the surface of the water. He was after the general pattern of flow to extract the features of the system as a whole. For example, how many vortices formed—six, two, none? Such an approach would neither provide an elegant formula to yield the velocity of a particular droplet so many centimeters downstream, nor offer a new numerical scheme to approximate the position of Mars next 12 April. Instead, Poincaré sought a picture that would capture the physiognomy of the equation and the physical system it represented. Under what conditions would an asteroid or planet fly off into space? Career into the sun? Of course such studies were abstract, mathematical; but they were, at the same time, also concrete. Poincaré wanted, above all, to grasp curves and their qualitative behavior—later could come the details of formulas, numerical predictions, or the last degree of rigor.19
By staking his reasoning to the fulness of the visual-geometrical approach rather than the icy edge of algebra, Poincaré was returning, now with far greater sophistication, to the mathematical ambitions of a much older Polytechnique. Not for him the great abstract formulas of Euler, Laplace, or Lagrange. It was Lagrange, after all, who so distrusted geometry that he swore his great work on analytic mechanics would stand on algebra alone, never would it rest on geometrical constructions. Not a mechanical analogy, not a single diagram would sully its pages.
By contrast, Poincaré worked precisely in such geometrical ways, and mechanical analogies were near to hand. Already in 1881, when focusing differential equations that were linked to the three-body problem, he highlighted his qualitative, intuitive ambition:
Could one not ask whether one of the bodies will always remain in a certain region of the heavens, or if it could just as well travel further and further away forever; whether the distance between two bodies will grow or diminish in the infinite future, or if it instead remains bracketed between certain limits forever? Could one not ask a thousand questions of this kind which would all be solved once one understood how to construct qualitatively the trajectories of the three bodies?20
Here, in the joint realm of the geometrical-visualizable and the physical, were concerns Poincaré returned to time after time, just as he came back throughout his career to questions in the mathematics of differential equations and the physics of the three-body problem. As he put it in 1885 (about one of his most important pieces of mathematics), one simply “cannot read . . . parts of this memoir without being struck by the resemblance between the various questions which are treated there and the great astronomical problem of the stability of the solar system.”21 Mechanics—machines—always lay near. It was the factory stamp.
Poincaré pushed ahead with his qualitative program for understanding the behavior of differential equations, with success sufficient to catch the attention of the world’s leading mathematicians. When, in mid-1885, Nature printed an announcement for a mathematical competition honoring the 60th birthday of Oscar II, the King of Sweden, Poincaré was a leading contender to win. Gösta Mittag-Leffler, a well-known mathematician and editor of Acta Mathematica, had responsibility for gathering the committee of judges. First he recruited Charles Hermite, one of Poincaré’s Polytechnique teachers, and then Hermite’s own professor, the formidable German mathematician Karl Weierstrass, whose mathematical life had long stood for relentless logical rigor. (Mittag-Leffler was himself on friendly terms with Poincaré.) Entries were due 1 June 1888: the very first question was on the three-body problem.22
Between the announcement of the prize and the submission date, the French Academy of Sciences elected Poincaré to its ranks. This was a signal honor. It meant that, as of 31 January 1887, at the age of thirty-two, he was a fixture of the French scientific elite. Membership in the Academy meant he could be (and frequently was) called to a wide range of administrative functions, from the Bureau of Longitude to interministerial commissions ranging across legal, military, and scientific domains. Adjusting easily to this new public role, Poincaré began writing for a broader audience. Of the nearly 100 nontechnical articles and books he penned for newspapers, journals, and scientific reviews, you can count on one hand those written before his appointment to the Academy of Sciences.
Throughout the whole of his career, visual, intuitive methods served as Poincaré’s pole star. He had used non-Euclidean geometry as a way of cracking problems at the very beginning of his work. And now, using visual (topological) techniques that he had honed during a decade considering differential equations, Poincaré entered the prize competition under his Latin banner, “For the stars do not cross their prescribed limits.” In that adage, much was conveyed. Technically, Poincaré was aiming to set bounds on the motions caused by the mutual attraction of the planets, reaffirming the stability of the solar system. But beyond the mathematics, Poincaré’s motto reflected his deep faith in the basically stable nature of the world around him. Despite the excellence of several runner-up contributions, Poincaré won the competition hands down; offering not only results but also a host of new methods that placed his achievement at the pinnacle of mathematics. All was, or seemed, right with the world.
After submitting the prize essay (perhaps thinking of the role that his own qualitative, visually oriented work had played in showing the stability of the Solar System), Poincaré stepped back to survey the role of logic and intuition in mathematical science and pedagogy. Looking over older mathematical books, he said, contemporary mathematicians saw work punctuated with lapses in rigor. Many of the older concepts—point, line, surface, space—now seemed absurdly vague. The proofs of “our fathers” looked like frail structures unable to support their own weight. Poincaré acknowledged that mathematicians now knew, as their fathers did not, that there are crowds of bizarre functions that “seem to struggle to resemble as little as possible the honest functions that have some useful purpose.” These new functions might be continuous but they are constructed in such a peculiar way that one cannot even define their slope. Worse yet, Poincaré lamented, these strange functions seem to be in the majority. Simple laws seem to be nothing but particular cases. There was a time when new functions were invented to serve practical goals; now we mathematicians invent them to show off the faulty reasoning of our fathers. If we were to follow a strictly logical path, we would familiarize beginners, from their start in mathematics, with this “teratological museum” of freakish new functions.
But this teratological path was not one Poincaré counseled his readers to follow—whether students or pure mathematicians. In mathematical education, he argued, intuition should not be counted least among the faculties of mind to be cultivated. However important logic was, it was by way of intuition “that the mathematical world remains in contact with the real world; and even though pure mathematics could do without it, it is always necessary to come back to intuition to bridge the abyss [that] separates symbol from reality.” Practicians always needed intuition, and for every pure
geometer there were a hundred in the trenches. Even the pure mathematician, however, depended on intuition. Logic could provide demonstrations and criticisms, but intuition was the key to creating new theorems, new mathematics. Poincaré was blunt: a mathematician without intuition was like a writer locked in a cell with nothing but grammar. His view was therefore that instruction (he referred explicitly to Polytechnique where he was by then teaching) should emphasize the intuitive and abandon these formal, unintuitive functions that only served to haunt the mathematical legacy of our mathematical ancestors.23
This plea for mathematical intuition appeared in print as Poincaré’s prize paper was about to be published. But in July 1889, Edvard Phragmén, a twenty-six-year-old Swedish mathematician working as an editor of Acta under Mittag-Leffler noticed some problems with the prize proof. He relayed them to Mittag-Leffler, who cheerfully told Poincaré on 16 July that, with a single exception, “one could make them disappear almost immediately.”24 Poincaré soon understood that the exception was not easily repaired; this was neither a typographical error nor a simple gap easily plugged with a few lines of more careful mathematics.25 Something was deeply wrong with his work. Not only the prize, but his, the journal’s, and the judges’ reputations were at stake; Poincaré had to figure out what was wrong.
Here was the issue. As in his studies of differential equations, Poincaré had considered three bodies: a little asteroid hurtling around the orbiting system of Jupiter and the Sun. What could the orbit of the asteroid do? One particularly simple behavior would be to return every time to the same spot with the same velocity: simple periodic motion. To represent such repetitive orbits in a dramatically simpler way, Poincaré proposed a startling idea: Don’t think about the trajectory itself. Instead, Poincaré realized that he could examine the situation once each time the asteroid came around—creating a stroboscopic picture, so to speak, that has come to be known as a “Poincaré map.” Properly speaking, his map plotted the asteroid’s momentum and its position each time around, but we can capture the idea by picturing the map as a vast sheet of paper, much bigger than any planet, spread in space perpendicular to the asteroid’s wandering. Every time the asteroid comes back, picture it punching a hole F in this cosmic sheet. In a simple periodic orbit, the asteroid would punch through the sheet and then travel through that same hole over and over again for all eternity. That hole is termed a fixed point. More generally, Poincaré’s idea was to study the patterns punctured in the two-dimensional sheet by the asteroid rather than its full orbit in space.
Einstein's Clocks and Poincare's Maps Page 6