Scott A Walter

Karl-Heinz Schlote

Martina Schneider

Henri Poincaré, theoretical physics, and relativity theory in Paris

Introduction

In studies of the emergence of relativity theory, historians have sought to characterize the reception of relativist ideas with respect to national communities of physicists and mathematicians, in an effort to reveal underlying features of these communities, such as their openness to new ideas, and their capacity for change. Stimulating this activity are the basic publication counts, that tell us that the reception of relativity theory in academic journals varied markedly from one country to another. Periodicals based in Germany accounted for roughly half of all relativist publications before 1916, while Germany-based authors published two of every five articles on relativity during the same period, and made up two-fifths of the total number of scientists (one hundred) contributing to relativity theory. France, the fifth most active country from a quantitative point of view, accounted for seven percent of relativist articles, and counted eight relativist scientists, or about a twelfth of the total. 1 Examination of the content of these publications and their context of production allows for a finer-grained understanding of the differences revealed by quantitative analyses, and gives rise to theories of reception. For example, scholars of the reception of relativity theory in Germany and England have proposed explanatory models in which the details of post-secondary training in physics are seen as decisive. 2 Historians of French physics consider the muted reception of relativity 1 in France as a consequence of a pervasive positivist outlook among French scientists, which would have favored the development of mathematics, while leaving little intellectual space for the distinct melange of experimental acumen, physical and mathematical reasoning that characterized the work of a Boltzmann, a Lorentz, or an Einstein. 3 Attention to the conceptual foundations of early relativist publications reveals a marked difference in approach on the part of two contributors in particular: Henri Poincaré and Albert Einstein. Understanding this difference in approach has occupied historians and philosophers of science for over half a century, without reaching a consensus on its significance for the history of physics. 4 Poincaré's philosophical writings, published for the most part prior to the discovery of relativity, weigh heavily in these analyses, and according to one commentator, constituted an obstacle to the reception of Einstein's theory of relativity in France until the 1920s. 5 By the same token, Poincaré's philosophical writings ought to have benefited his theory of relativity, but the above-mentioned publication counts indicate that they did not do so, either in France or elsewhere in the world.

The outlines of an alternative account of French contributions to relativity during the years from 1905 to 1912 are drawn in this paper. Poincaré's intellectual and institutional leadership in French physics at the turn of the twentieth century is reviewed, and related to the emergence of Paul Langevin as his successor. Drawing on quantitative data and previously-unexploited manuscripts from Parisian archives, the paper compares the fate in France of Poincaré's theory of relativity to that of the Einstein-Minkowski theory of relativity championed by Langevin, and links these events to Langevin's rise to leadership of French theoretical physics.

Poincaré and theoretical physics in Paris

Compared to the situation of French physics in the first decades of the nineteenth century, in 1898 the future did not appear promising to Henri Poincaré. His pessimism stemmed from a perceived mismatch between the cognitive habits of the French scientist and the turbulent state of theoretical physics brought about by the discoveries of the past decade, including the null-result of the Michelson-Morley ether-drift experiment, the discovery of x-rays, the electron, the Zeeman effect, and radioactivity. A certain boldness was called for to explain such results, and Poincaré feared that the French were not up to the task at hand, as he expressed it in an official report to the Paris faculty of science:

The French mind, avid of clarity and logic, is repugnant of excessively temerarious adventures.

A new type of physicist was called for, according to Poincaré, in order to "discern the simplicity of laws beneath the complexity of phenomena". 6 The type of physicist Poincaré had in mind, although probably not the archetype, was Jean Perrin, whose candidacy he evaluated for a lectureship in physical chemistry on the Paris faculty of science. To some extent, Poincaré may have described here his own approach to the laws of physics, although his prowess in mathematics clearly set him apart from even the most mathematically-sophisticated of his colleagues in physics.

Paris did not yet dispose of a chair in theoretical physics per se, and would not create one until 1928, when the Rockefeller Foundation volunteered to finance a new institute. 7 The first French chair nominally devoted to theoretical physics dates from 1894, when the faculty of science in Bordeaux hired Pierre Duhem. This is not to say that theoretical physics was neglected in Paris. At the Paris faculty of science, the chair of probability calculus and mathematical physics, dating from 1834, was devoted to the subject. Poincaré held this chair for a decade, from 1886 to 1896, and single-handedly brought French theoretical physics to international attention. The work in theoretical optics and fluid mechanics by his successor Joseph Boussinesq, however, found little echo outside of France. 8 For the sake of comparison, across the border, a modest institute for theoretical physics was created at the University of Berlin in 1889, at the behest of Hermann von Helmholtz, and by the turn of the century, such institutes had been created in Königsberg, Leipzig, Göttingen and Munich. 9 At the turn of the twentieth century, only two of these institutes were led by full professors: Max Planck in Berlin, and Woldemar Voigt in Göttingen. 10 There were other outstanding theorists in Germany, notably Paul Drude, Willy Wien, and Arnold Sommerfeld, but some of these theorists felt Germany had lost its preeminence in the field since the time of Gustav Kirchhoff. 11 Since the death of Heinrich Hertz in 1894, and Ludwig Boltzmann's departure from Munich the same year, the brilliance of H. A. Lorentz in Leiden and Boltzmann in Vienna had cast shadows over their counterparts in Germany and France alike.

At least one theorist in Paris was prepared to meet the challenge posed by the recent results of experimental physics: Henri Poincaré. The fact that after 1896, Poincaré no longer occupied a chair of mathematical physics did not prevent him from lecturing and theorizing in this domain, just as earlier, he cultivated subjects of pure mathematics (function theory, algebraic topology), and celestial mechanics while nominally a professor of probability calculus and mathematical physics. Beginning in the late 1880s, Poincaré helped introduce Maxwell's electromagnetic theory to French readers, and in the late 1890s, he exhibited a keen interest in Lorentz's theory of electrons.12 Interest in Lorentz's theory grew rapidly when Lorentz used it to explain the splitting of Sodium D lines in an external magnetic field, an unexpected phenomenon discovered in 1896 by Lorentz's former student, Pieter Zeeman (1865-1943). Poincaré communicated to the Paris Academy of Sciences a paper by Zeeman (1897) describing his discovery, and soon engaged with the explanation of the effect offered by Lorentz. Others in France soon took up studies of the Zeeman effect, including Alfred Cornu (1841-1902), Poincaré's former physics professor at the École polytechnique, and Alfred Liénard (1869-1958), a former student of Poincaré's, who taught mathematics and physics at the School of Mines in Saint-Étienne.

Poincaré's engagement with electrodynamics was enduring, and remarkably innovative, featuring applications of sophisticated mathematical methods (complex analysis, group theory), and the reformulation of key concepts of Maxwell's electromagnetic theory and Lorentz's electron theory, along with applications of these methods and theories. For example, in the 1890s, Poincaré was among the first to use retarded potentials in Maxwell's theory, and proposed the first electromagnetic theory of diffraction, which was soon extended by Arnold Sommerfeld. His demonstration of the recurrence theorem was recognized to have fundamental repercussions on physics, particularly for kinetic theory. He also provided a theory of multiple resonance for Hertzian oscillations, and the first exact solution of Maxwell's equations for charged particles in a strong magnetic field. 13 By the turn of the century, Poincaré's contributions to physics had won the admiration and respect of his peers across Europe and in the USA. Poincaré was one of only two Frenchmen invited to contribute to a volume in honor of G.G. Stokes in 1899, alongside Cornu. 14 The next year, Poincaré was one of the three Frenchmen on the scientific committee of the international physics congress organized in Paris by the French Society of Physics, and presided by Cornu. One of ten vicepresidents of the physics congress, Poincaré presided the international congress of mathematicians, which conveniently took place in Paris the same week in Au-gust. 15 The following year, Poincaré was elected vice-president of the French Society of Physics, and in 1902, served as its president. A few years later, the Society made Poincaré one of its ten honorary members. 16 This recognition from Poincaré's peers in physics did not mean that his authority in physics went uncontested, either at home or abroad. There were those, like the Scottish natural philosopher P. G. Tait, who found his lectures on mathematical physics to be excessively analytical, and unreliable on foundational issues. 17 Near the end of the decade, when Lorentz explained the Zeeman effect on the basis of his theory of electrons, Poincaré proposed an alternative formula, which was mathematically sound, but for Lorentz, uncompelling from a physical standpoint. 18 In France, Poincaré's views on questions of mathematics or physics were very rarely challenged in public. The case of Marcel Brillouin (1854-1948) 20 At the turn of the century, Poincaré's physical acumen was severely tested, when Gabriel Lippmann's doctoral student, Victor Crémieu (1872-1935) published a result casting doubt on Rowland's effect, whereby, in line with Maxwell's theory, convected electricity produces a certain magnetic effect. Poincaré wrote the official report on Crémieu's thesis, communicated several of his results to the Paris Academy of Sciences, and argued that if the result was confirmed, Maxwell's theory would have to be abandoned. None of Europe's leading physicists gave any credence to Crémieu's findings, which if true, would have overturned the electron theories of Lorentz and Larmor, as well as Maxwell's theory. In France, Brillouin shared Poincaré's high opinion of Crémieu's results, but Poincaré's colleague at the Sorbonne, the physicist Henri Pellat (1850-1909) remained doubtful, as did 15 C.-E. Guillaume and L. Poincaré, eds. (1900-1901). 16 Three other French physicists had attained this status by 1909: Jules Violle, Gabriel Lippmann, and Émile-Hilaire Amagat; see Bulletin de la société française de physique (1908), p. 3*. 17 See the review by P. G. Tait (1892) of Poincaré's Thermodynamique (1892), reedited with annotations (and Poincaré's replies) in Walter et al., eds, eds (2007, § 2-62). 18 Poincaré 1897a; Buchwald 1985, 226. 19 See Poincaré's correspondence with Brillouin in Walter et al., eds, 2007, § 2-12;Brillouin 1891a, 1891b. 20 Poincaré 1901;Brillouin 1901. Poincaré's former teacher at the École polytechnique, Alfred Potier (1840-1905). When Harold Pender, who was Rowland's last doctoral student, confirmed Rowland's effect in Baltimore, Poincaré saw to it that Pender and his equipment were transported to the Edmond Bouty's laboratory in Paris, in order to perform experiments side-by-side with Crémieu. Pender emerged victorious from the encounter; to the French Society of Physics Pender explained not only how the Rowland effect manifested itself, but why Crémieu's apparatus had failed to detect it. [START_REF]Sur la théorie des oscillations hertziennes[END_REF] The result of the encounter suggests that Poincaré had misjudged the situation; nonetheless, he obtained what he required as a theoretical physicist: an experimental decision between Maxwellian and non-Maxwellian electrodynamics.

Pender and Crémieu's account of their parallel investigations of convected electricity appeared in a leading journal of French physics: the Journal de physique théorique et appliquée (JPTA), founded in 1872. During the first decade of the twentieth century, the JPTA's editorial board counted two professors of physics from the Paris faculty of science, Edmond Bouty and Gabriel Lippmann, along with a pair of senior theorists, Alfred Potier and Élie Mascart, neither of whom survived the decade. Filling out the editorial board were a trio of associate editors, former students of the École normale supérieure: Lucien Poincaré, Bernard Brunhes, and Georges Sagnac; and one non-Normalien associate editor, Marcel Lamotte, an associate professor of physics at Clermont-Ferrand, who like Brunhes had helped edit Poincaré's volumes on mathematical physics in the early 1890s.

The JPTA did not publish contributions in theoretical physics that involved sophisticated mathematical elaboration, in order to remain accessible to "isolated" physicists, which is to say, those out of range of science faculties and their libraries. 22 This approach manifested itself in the selection of articles for publication, and in the abstracts of articles published abroad. More often than not, when an article featured a mathematical argument, the JPTA abstract revealed this fact alone, with no judgment of merit or meaning. Consequently, readers were illinformed of current work in theoretical physics, beyond what might be guessed from reading the name of the author, and the title of the contribution.

There were other venues in France for publishing research in theoretical physics, including the Annales de chimie et de physique (or ACP for short), Le Radium, the Annales scientifiques de l'École normale supérieure, and the Journal de mathématiques pures et appliquées. The latter two journals attracted few papers on physics, unlike Le Radium, founded in 1904 by H. Becquerel, P. Curie, E. Rutherford, Ch.-E. Guillaume and others. Le Radium effectively competed for readers with the JPTA, providing translations of German and French contributions, and abstracts of various periodicals, until the two journals fused in 1920. The ACP, founded in 1816, attracted significant communications in the first decade of the twentieth century from Paul Langevin, Jacques Hadamard, Marcel Brillouin, and the latter's student, Jean Perrin. At the beginning of the decade, the ACP was directed by the venerable trio of Marcelin Berthelot, Élie Mascart, and Henri Moissan, none of whom were still alive in 1910. At the end of the first decade, a different trio of editors directed the ACP: the chemist Albin Haller (1849-1925) and his two colleagues on the Paris faculty of science, Lippmann and Bouty, who continued to edit the JPTA.

During this period the ACP published doctoral theses in physics, as well as extended summaries of experimental and theoretical investigations. Two examples may be mentioned here. One of these is the Swiss theorist Walter Ritz's long memoir, "Critical investigations in general electrodynamics", in which Ritz gave an overview of the work of Lorentz, Poincaré, Einstein, and others, and sketched an alternative approach to the electrodynamics of moving bodies, based on retarded potentials and a principle of superposition. [START_REF] Cuvaj | A History of Relativity: The Role of Henri Poincaré and Paul Langevin[END_REF] Another is Perrin's "Brownian motion and molecular reality", where he presented the results of experiments that confirmed Einstein's formula for Brownian motion of a particle in a fluid, work for which Perrin was awarded the Nobel Prize in physics in 1926. 24 The only other publishing outlet for research in theoretical physics in France, but one more widely cited than the ACP or any other French scientific journal, was the organ of the Paris Academy of Sciences, the Comptes rendus hebdomadaires (hereafter CRAS). This was where Poincaré published most often, averaging nine papers a year throughout his career, including a signal contribution to relativity theory on 6 June 1905. The CRAS enforced a page limit on its contributors, and Poincaré's four-page summary was no exception to the rule. The memoir summarized in the CRAS appeared in the Rendiconti del Circolo matematico di Palermo, a journal in which since 1888 Poincaré had published on the theory of differential equations, analytical mechanics, and algebraic topology. Until 1906, Poincaré published all his articles on physics (excluding notes in the CRAS) either in foreign journals, or in a Paris-based journal of electrical engineering, Éclairage électrique, on the editorial board of which he served beginning in 1899.

One consequence of this habit was that until 1906, Poincaré's latest research in theoretical and applied physics was known best to French electrical engineers, and readers of CRAS and foreign research journals. Students of physics knew Poincaré best through his lectures on mathematical physics, published in thirteen volumes (not counting translations to German, or reeditions). The effect of these volumes was described somewhat breathlessly by the mathematician (and former Poincaré student) Maurice d'Ocagne, for whom Poincaré had, in addition to being the world's premier theoretical astronomer, . . . carved for himself an unequaled position as a theoretical physicist, projecting a new light, emanating from the most unexpected sources, upon every part of mathematical physics: heat, optics, electricity, elasticity, capillarity, etc. . . . . He has covered everything, renewed everything, extended everything. [. . . ] What is more, there are many experimentalists who make no mistake in recognizing all they owe to the theoretical views introduced to science by Mr. Poincaré, and who have quite often reoriented their laboratory investigations to the great benefit of the general advance of our knowledge. (Maurice d 'Ocagne 1909, 541) What d'Ocagne's remark suggests most clearly is the source of Poincaré's preeminence in French theoretical physics, and his influence on research agendas in experimental physics. Physicists who acknowledged such an influence included, among others in France, Henri Becquerel, René Blondlot, Gustave Le Bon, Paul Langevin, Georges Sagnac, Alfred Perot, and Victor Crémieu; in Geneva, Lucien de la Rive and Édouard Sarasin; in Kristiania (now Oslo), Kristian Birkeland. 25 While Poincaré's influence on the agenda of experimentalists is apparent, what can be said of his mark on the agenda of theorists? Some of the aforementioned experimentalists also wore a theorist's cap on occasion, like Birkeland, Langevin, and Sagnac. All three of these physicists published on subjects stemming from those taken up earlier by Poincaré, notably in the domains of Hertzian waves and electron theory; all were former students of Poincaré. According to another former student of Poincaré's, Arthur Korn, there was not a single physicist anywhere whose work had not found fundamental stimulation in Poincaré's lectures. 26 Poincaré is often characterized by historians as a leading critic of theories of physics, and indeed, his lectures in mathematical physics offered a magisterial discussion of rival theories in the several branches of physics, that compared relative strengths and weaknesses. 27 His lectures on Maxwell's theory were eagerly read in Germany (in German translation), and exercised a profound influence on the first German textbooks on Maxwellian electrodynamics. 28 Some of his nontechnical analyses were reedited for a larger audience in the four anthologies of his epistemological writings on mathematics and the exact sciences edited by 1913, 25 Poincaré's interaction with experimental physicists is well-documented in his correspondence; see the introduction to Walter et al., eds, 2007. 26 Korn 1912. 27 Such a characterization is offered by Darrigol 1995Darrigol , 2000. 28 . 28 See Darrigol 1993; 2000, 354. which were widely read and appreciated by both specialists and the general reading public alike. 29 Poincaré's critical acumen in theoretical physics was appreciated by his peers, including Joseph Larmor, who contributed a preface to the English translation of the first of the anthologies: Science and Hypothesis.

On an international level, with the discovery of x-rays, the electron and radioactivity in the closing years of the nineteenth century, the physics of charged particles filled the pages of physics journals. French prowess in experimental microphysics received international recognition following work by Henri Becquerel and the Curies on radioactive matter, and René Blondlot on electrical convection, although the latter's reputation was later tarnished when what he called "N-rays" proved spurious. On the theoretical side, Poincaré and Alfred Liénard were among the first theorists to contribute to Lorentz's electron theory, and to apply it to dispersion phenomena and the Zeeman effect. 30 Outside of France, respected theorists at the turn of the twentieth century included, first and foremost, Lorentz in Leiden, Boltzmann in Vienna, Joseph Larmor and J.J. Thomson in Cambridge, Ernest Rutherford in Montreal, Paul Drude in Giessen, Max Planck in Berlin, Sommerfeld in Aachen, Wilhelm Wien in Würzburg, Woldemar Voigt, Emil Wiechert, Max Abraham, and Walther Nernst in Göttingen.

Critical analysis of physical theories was an activity at which Poincaré was skilled and accomplished, and for which he was amply rewarded. His contributions to physics, however, went well beyond writing textbooks and critiques of others' work, into the creative realm of theory construction. Among the theoretical physicists mentioned above, Sommerfeld and Abraham found significant inspiration in Poincaré's theories of physics. Sommerfeld's electromagnetic theory of diffraction of plane waves (1896) improved on Poincaré's groundbreaking paper of 1892, while Abraham borrowed on the Frenchman's conception of electromagnetic momentum to form his theory of electron dynamics. 31 Last but not least, in the summer of 1907, Hermann Minkowski took up the elements of Poincaré's four-dimensional approach to relativity theory, in what became a game-changing theory of physics: the theory of spacetime. 32 The latter three contributions were among those cited in support of an ultimately unsuccessful campaign to award Poincaré the Nobel prize in physics in 1910, in addition to work on the propagation of Hertzian waves, and the theories of vibrating plates, rotating fluid masses, and electron stability. The failure of 29 According to Lebon (1912, 84), the first of these anthologies, entitles La science et l'hypothèse (1902), sold twenty thousand copies by 1911. On the composition of Poincaré's anthologies, see Rollet (2001, chap. 4).

30 See Buchwald 1985. 31 Poincaré 189231 Poincaré -1893;;[START_REF] Sommerfeld | Mathematische Theorie der Diffraction[END_REF]Sommerfeld , 2004. On Abraham's and Planck's theories see [START_REF] Miller | On some other approaches to electrodynamics in[END_REF][START_REF]Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation[END_REF]. 32 Poincaré 1906;[START_REF] Minkowski | Die Grundgleichungen für die electromagnetischen Vorgänge in bewegten Körpern[END_REF][START_REF] Walter | La correspondance entre Henri Poincaré et les physiciens, chimistes et ingénieurs[END_REF]Walter , 2008. . Poincaré's Nobel campaign reflects in part the still-uncertain status of the theory of relativity in 1910, and in fact, the Nobel committee never awarded a prize in recognition of the discovery of special relativity. In context, it is curious that a Nobel prize nomination emanating from the Paris Academy of Sciences in January 1910, and including among its signatories the Academy's permanent secretary for the mathematical sciences, Gaston Darboux, should feature work "of the highest importance" by Poincaré on the principle of relativity. 33 What happened to the theory of relativity in France during the latter half of the first decade of the twentieth century? And how did Einstein's theory come to prominence in France in 1911? In the next section, I show that while Lorentz's theory was often discussed, alternative theories remained nearly invisible in France until 1911. The situation changed in 1911, as the final section will show.

The invisibility of Einstein's theory in France

In the scientific centers of Western Europe, physicists did not distinguish at first the theories of Lorentz, Poincaré, and Einstein. Of these three founders of relativity theory, Poincaré alone took care to identify the differences between his theory and that of Lorentz; Einstein's theory had not yet been published when he wrote his memoir. A year later, after Einstein's theory had been aired in the Annalen der Physik, Poincaré took care to explain to his students at the Sorbonne how his theory of relativity differed from that of Einstein, albeit without ever mentioning Einstein or his theory.

Poincaré performed a curious thought experiment for his students, in which a 33 See Darboux et al. to the Nobel Committee, ca. 1 January 1910, transcribed and annotated in Walter et al., eds, 2007, § 2-62. On the organization of the 1910 campaign, see Ph. Nabonnand's notes to the correspondence between Poincaré and G. Mittag-Leffler (Nabonnand 1999). 34 Poincaré 1905, 1906. 35 Guye and Ratnowsky (1910), originally submitted on 10 January 1910, and withdrawn by Guye, ostensibly to permit the inclusion of new data (Guye to Gaston Darboux, 30 January 1910, Archives of the Academy of Sciences, session folder, 7 February 1910). pair of inertial observers, one at rest, the other moving away in a straight line at constant speed, describe the form of a locus of light at a certain instant of time. An observer at rest with respect to the ether judges the light locus to have the form of a sphere, the radius of which increases with the speed of light. Observers in motion with respect to the ether, Poincaré explained, would conclude that the light locus at any instant of time (as determined via co-moving light-synchronized clocks) is represented by an ellipsoid of rotation, elongated in the direction of observer motion with respect to the ether. In Einstein's theory, by contrast, the light locus at any given instant of time (as determined via co-moving light-synchronized clocks) is always represented by a sphere. 36 After presenting his view of relativity to his students, Poincaré published his light-ellipsoid theory of relativity in France's leading popular-science biweekly, the Revue générale des sciences pures et appliquées. He did not mention Einstein's theory, and in the Revue générale no one else did, either, until Maurice Lémeray wrote of "Einstein's beautiful results" four years later. 37 Poincaré's silence with respect to Einstein's theory has been the subject of much historical speculation, and will not concern us here. Instead, let us ask why no one else in France saw fit to mention Einstein's theory in print before 1911. And to begin with, let us investigate why one person in particular, Paul Langevin, did not mention Einstein's theory in print before 1911. Recall that in 1905 Langevin proposed an electron theory similar in some respects to that of Alfred Heinrich Bucherer, featuring an electron model of constant volume, and velocity-dependent shape, and that Poincaré showed Langevin's theory to be incompatible with relativity. Langevin acknowledged Poincaré's judgment of his theory, but did not give it up until the experimental results presented by A.H. Bucherer in September 1908 persuaded him to do so. 38 To put it briefly, until the fall of 1908 there were several plausible alternatives available to relativity theory, some of which enjoyed, like Abraham's rigid-electron theory, better empirical support in some tests than did the theory of relativity. 39 Einstein was not unknown in French physics circles, and his name was cited in contexts other than relativity in the period from 1905 to 1910. In kinetic theory, for example, Einstein's formula of 1905 for specific heat was promoted by Jean Perrin in 1908, and referred to simply as "Einstein's formula". A look at the abstracts published by the JPTA from 1905 to 1911 reveals that the "Abraham theory" of the electrodynamics of moving bodies was mentioned twice, the "Einstein theory" 36 See my forthcoming paper in Einstein Studies. For alternative explanations of Poincaré's light-ellipsoid, see Cuvaj (1970, 74) and Darrigol (2006, 17-19). 37 Poincaré 1908;Lémeray 1912. 38 See Langevin's Notice sur les travaux scientifiques (1908, 35). 39 On the choice between alternative theories of the electrodynamics of moving bodies circa 1905, see Darrigol (2000, 391). three times, and the "Poincaré theory" or "Lorentz-Poincaré theory" four times. One notices that Poincaré's theory never stood alone in these abstracts, but was always accompanied by a reference to Lorentz's theory, which was mentioned much more often than any other, garnering a total of twenty-two independent occurrences.

Also, the paucity of detail in JPTA abstracts on relativity and electron theory, compared with that provided for other subjects, suggests a certain lack of comprehension or interest on the part of the abstract writer. A general ignorance of and disinterest in relativity theory was not unique to French physicists, as even in Germany, publication numbers remained modest in this area until 1909, when they began to climb rapidly (see Fig. 2). One difficulty for relativity theory was its poor performance in electron-deflection experiments, which led many to believe that relativity theory was empirically untenable. In a discussion of electron theory in 1906, for example, Paul Ehrenfest considered Lorentz's theory to have been definitively disproved by experiment, and Ehrenfest's opinion was duly related by Léon Bloch for readers of Le Radium. 40 In such circumstances, it is a wonder that any physicist bothered learning relativity theory before the end of 1908.

After an experimental confirmation of relativity theory was announced in September 1908, the incentive to learn the theory, and to investigate its consequences naturally increased. What is curious in the French context is that apart from Poincaré, no other physicist took up relativity, until Paul Langevin lectured on the subject at the Collège de France in 1910-1911. According to Poincaré's own report, he pursued a relativistic theory of elastic collisions, but deemed his results unworthy of publication. As he explained it to a Berlin audience in late 1910, the lack of such a theory was one reason why the new mechanics of relativity could not be considered "definitively grounded". 41 In front of French audiences, Poincaré offered a different message, designed to reassure those worried about overturning Newtonian mechanics: the "old mechanics", Poincaré announced, was still the one for "our practical life and our terrestrial technology". 42 Poincaré's measured consideration of the theory he helped create may have dissuaded a few junior French theorists from following in his tracks, but not Paul Langevin. 43 As a student of Poincaré's 1896 lectures on the elastic theory of light, Langevin had learned how a certain theorist referred to as "Somerset" extended 40 Ehrenfest 1906;Le Radium 3, 1906, p. 148. 41 Poincaré 1910, 115-116. 42 "Quoi qu'il en soit, d'ailleurs, elle restera la mécanique des vitesses très petites par rapport à la vitesse de la lumière, la mécanique donc de notre vie pratique et de notre technique terrestre." Plenary lecture, 3 August, 1909, to the meeting of the French Association for the Advancement of Science in Lille (Poincaré 1909).

43 On Langevin's relation to Poincaré and Einstein, see Paty (2002).

Poincaré's theory of polarization by diffraction. [START_REF] Langevin | Poincaré Élasticité et optique III 1896[END_REF] Ten years later, on the strength of this work, and more recent contributions to electron theory, this same theorist -better known as Arnold Sommerfeld -was named to the chair of theoretical physics in Munich, formerly held by Boltzmann. 45 Sommerfeld was in charge of the physics volume of Felix Klein's planned six-volume Encyclopedia of Mathematical Sciences with Applications, the first entries of which appeared in 1903. 46 On 16 April, 1906, Sommerfeld informed Langevin that Klein had agreed to let him co-edit the French version of the physics volume with Jean Perrin, a task that would occupy the two Frenchmen for nearly a decade. 47 Along with their editing duties, Sommerfeld and Langevin shared for several years the electromagnetic world-view, which promised a unification of all forces on an electromagnetic basis. But as mentioned above, in late 1908, theory and experiment conspired to convince Langevin of the cogency of the theory of relativity.

As a former student of Poincaré's, and an occasional dinner guest at his flat in Paris, Langevin would have been at first glance a natural candidate to take up Poincaré's theory of relativity. A similar remark may be made about Sommerfeld, who did not hear Poincaré's lectures at the Sorbonne, but who admired and emulated his approach to physics. Whatever affinity Sommerfeld and Langevin had with Poincaré and his science, they both preferred the Einstein-Minkowski theory to that of Poincaré. For Sommerfeld, it was Minkowski's spacetime theory that persuaded him of the cogency of relativity theory. 48 Langevin, too, was impressed by Minkowski's theory, and by Sommerfeld's related four-dimensional vector algebra and analysis, which he presented in his 1910-1911 lectures at the Collège de France. [START_REF] Sommerfeld ; Likewise | Minkowski admired Langevin's contributions to the kinetic theory of gases[END_REF] The elements of spacetime theory were readily available to French readers by then, since in late 1909, a pair of former students of the École normale supérieure had translated Minkowski's 1908 lecture "Space and time" for publication in the Annales scientifiques de l'École normale supérieure. 50 Like Poincaré, Langevin felt that the ether was not a wholly superfluous con-cept for modern physics. One auditor of Langevin's lectures, Léon Brillouin, recorded Langevin's remark on this subject:

The very notion of the ether loses its sense, says Einstein -this is an exaggeration. We can't discern our speed with respect to the ether, but we can discern [our] accelerations and rotations. 51 On the subject of light-waves, Langevin maintained on another occasion that a spherical light-wave in one inertial frame is actually spherical for all inertial observers. 52 The latter view signals Langevin's break with Poincaré, for whom the light locus only appeared spherical for observers in motion with respect to the ether. In fact, Langevin fully agreed with Einstein and Minkowski that the universal validity of the principle of relativity implied a new view of space and time, and he defended this view publicly, beginning in 1911.

To put Langevin's defense of Einstein-Minkowski theory into historical perspective, let us examine some publication numbers. In 1911, publication of articles on relativity theory in periodicals worldwide hit a peak at one hundred and seventeen titles, after a sustained increase in scientific interest beginning in 1909 (see Fig. 1). This increase is reflected on a modest scale, and with a delay of a year or two, in the United Kingdom and in France. Figure 2 shows the evolution of publication numbers from 1905 to 1916 for the top five nations in article productivity. French numbers rose slightly in 1911, and peaked at thirteen articles in 1913. 53 Bare publication numbers tell us nothing of the causes of their annual fluctuation, a fact which leads us back to the JPTA abstracts. In 1911, "Einstein theory" is mentioned in nine abstracts, six of which mention no other theory. Next comes "Lorentz theory", with five mentions, followed by one mention each for Poincaré and Minkowski. The novelty in 1911 French physics, according to this source, was Einstein's theory of relativity. A closer look at the JPTA abstracts, however, suggests that these citation figures be treated with prudence. In 1911, the 51 "La notion même d'éther perd son sens, dit Einstein-c'est exagéré. On ne peut saisir notre vitesse p[ar] rapp[ort] à l'éther, mais on peut saisir les accélérations et rotations." Léon Brillouin, Notebook "Cours de Relativité au Collège de France 1910-1911", Léon Brillouin Papers, Box 7, folder 8, American Institute of Physics, Niels Bohr Library. Langevin made the same pointwithout mentioning Einstein by name -in a lecture delivered on 10 April 1911 to the Fourth International Congress of Philosophy in Bologna, where Poincaré was present; see Langevin (1911, 233). 52 Langevin 1912, 335. 53 These publication numbers do not take into account an author's nationality or workplace. Data correlating the production of articles on relativity to nationality of the writer is presented in Walter (1996), which is also the source of the data in the figures presented here, augmented by fifty titles gleaned from the author's subsequent research. The publication database is freely available from the author's homepage. JPTA recruited a new abstract writer, a nautical engineer from Antibes, Maurice Lémeray (b. 1860), and assigned him articles on relativity published in German or English. A science teacher turned warship designer, Lémeray was himself a prolific writer on relativity, having published more articles in 1911 and 1912 than any other Frenchman. His writings show no marked allegiance to either Einstein or Poincaré, but agree in general with Einstein's theory. Indeed, Lémeray was the first to cite Einstein's publications on relativity in the Comptes rendus, in a note communicated to Academy of Sciences by Poincaré, whose name Lémeray was careful to cite. 54 In summary, the increased number of citations of Einstein's theory in the 1911 JPTA abstracts has more to do with staff changes at the JPTA than with any bound in recognition of Einstein's contributions to relativity among French physicists.

The details of Lémeray's rise to prominence in France throw light on the reception of Einstein's theory. Archival documents reveal that Lémeray sought the Paris Academy's approval for his work on relativity as early as September 1910, when he submitted a manuscript to Gaston Darboux, one of the Academy's permanent secretaries. 55 Judged unfit for publication, the four-page note entitled "On the Lorentz transformation" purported to demonstrate Lorentz's formulas for local time, length contraction, and transverse and longitudinal mass from Einstein's twin postulates of relativity and universal lightspeed invariance, and dimensional analysis. Lémeray insisted that his results were free of "any hypothesis on the mechanism of phenomena or on any electrical theory", and he cited only one paper: Einstein's first French-language publication on relativity in the Archives de Genève. His purported demonstration of time dilation from the longitudinal Doppler effect for lightwaves, however, involved circular reasoning, and probably rendered his manuscript unpublishable. What this episode suggests is that the invisibility of Einstein's theory in France until 1911 was due in part to the paucity of physicists prepared to meet the cognitive challenge of Einstein's theory, combined with the existence of a rigorous manuscript review process. Similar instances of manuscript rejection in this area of physics took place elsewhere, of course, Germany included. 56

Epilogue

With assistance from Perrin, Langevin, and Lémeray, Einstein's star was ascending over France by 1911. In November, 1911, Poincaré recommended him for a 54 Lémeray 1911. 55 Session folder, 3 October 1910, Archives of the Academy of Sciences, Paris. 56 See, for example, Lewis Pyenson's review (1985, chap. 8) of Max Planck's rejection of papers submitted to the Annalen der Physik.

chair in theoretical physics at the ETH in Zurich, commenting that "the future will show more and more what Mr. Einstein's value is", and in January 1912, Einstein was named to this chair, and elected a non-resident council member of the French Society of Physics. 57 In May, 1912, Poincaré admitted that the new mechanics of relativity could serve as a basis for a redefinition of time and space, thereby recognizing the philosophical significance of Einstein-Minkowski theory. 58 This was a giant step for Poincaré, but it came too late to make any difference for physics in France. By 1912, the leading French theorists, including Langevin, and the mathematicians Émile Borel and Élie Cartan, had already adopted Einstein-Minkowski theory. 59 The engagement of Borel, Cartan, and other French mathematicians with the theory of relativity followed an example set in Germany by Minkowski, Gustav Herglotz, and Felix Klein. To some extent, the contributions of French mathematicians compensated the feeble participation of French theoretical physicists -Poincaré excepted -in the construction and diffusion of relativity. Once again, Langevin appears to have been instrumental in attracting the attention of French mathematicians to the study of Einstein-Minkowski theory. His role in introducing Einstein's theory to French scientists was later described by Jacques Hadamard as follows:

It is well known that, under the powerful leadership of Mr. Langevin, the young French physicists rallied to the new movement of ideas created by Mr. Einstein's discoveries. But cooperation with this movement was no less important to mathematicians, whose doctrines the new theory brought into play to a higher degree than any other previous physical conception. This is just what geometers like Mr. Borel understood from the beginning. (Hadamard 1922, i) What Hadamard's remark suggests is that for us to understand the reception of relativity in France, we need to go beyond the small circle of theoretical physicists, and examine how mathematicians came to engage with the theory.

In this essay, Poincaré's influence on theoretical physicists in France has been discussed, but not his interaction with mathematicians. Nonetheless, even in the restricted domain of theoretical physics in France, the interactions between mathematics and physics appear decisive for the reception of relativity theory. The systematic appeal to sophisticated and powerful mathematics in the construction and 57 Poincaré to Pierre Weiss, ca. November, 1911, transcribed in Walter et al., eds, 2007, § 2-59-3; Procès-verbaux de la Société française de physique 1912, p. 9. 58 Poincaré 1912;Walter 2009. 59 See Borel's 1913 lectures on Minkowski spacetime at the Sorbonne (Borel 1914), and Cartan's lecture on the "new kinematics" of relativity before the French Society of Mathematics (Cartan 1912). elaboration of physical theory was a legacy Poincaré bestowed on all his physics students. In this sense, Poincaré may be said to have smoothed the path in France for both Paul Langevin and the Einstein-Minkowski theory of relativity, at the expense of his own approach to relativity.

Figure 1 :

 1 Figure 1: Global publication of articles on relativity in periodicals, 1905-1916. N = 662. Source: Walter (1996).

Figure 2 :

 2 Figure 2: Publication of articles on relativity in periodicals, 1905-1916: Germany, United Kingdom, USA, Italy, France. N = 566. Source: Walter (1996).

  is instructive from this perspective. With doctoral degrees in mathematics and physics, Brillouin was named associate professor at the École normale supérieure in 1887. In the early 1890s, Brillouin dutifully pointed out what he thought was an error in the first edition of Poincaré's lectures on Maxwell's theory, concerning Hertzian waves. Poincaré's gentle private lesson led Brillouin to retract his criticism. 19 In 1900, Brillouin replaced Joseph Bertrand as professor of general and mathematical physics at the Collège de France, and when a new edition of Poincaré's Électricité et optique appeared in 1901, Brillouin had only high praise for it.

  On 5 June 1905, Poincaré's precis of relativity theory appeared in the Comptes rendus of the Academy, announcing a longer work published in the Palermo Rendiconti. 34 Afterwards, no notes were published by anyone on this subject in the Comptes rendus until 7 February 1910, when results of cathode-ray deflection experiments by Charles-Eugène Guye and Simon Ratnowsky in Geneva appeared, tending to confirm Lorentz's predictions of velocity-dependent mass. 35 Contrary to Darboux's description, the publication record suggests that the theory of relativity was of little importance to French science, at least until February 1910.

See Paty (1987, 115),Biezunski (1987, 184), and Pestre (1992, 117). On French isolationism in electrodynamics (and its exceptions), seeDarrigol 2000, 352.

A balanced overview of the "mystery of the Einstein-Poincaré connection" is provided byDarrigol (2004).

See Borella 2002.

Faculté des sciences de Paris, pièces annexes auxprocès-verbaux, 1883-1903,

78-79, French National Archives. A transcription is available from the Henri Poincaré Papers website. 7 Siegmund-Schultze 2001. 8 Darrigol 2005, 239. 9 Jungnickel & McCormmach 1986, vol. 2, 254. 10 Schlote 2004, 86. 11Jungnickel & McCormmach 1986, vol. 2, 159.

See Darrigol 2000.

[START_REF] Poincaré | Sur le problème des trois corps et les équations de la dynamique[END_REF]Poincaré , 1891bPoincaré , 1891aPoincaré , 1892Poincaré -1893Poincaré , 1897b. .

See Cambridge Philosophical Society, ed. (1900).

[START_REF] Pender | Recherches contradictoires sur l'effet magnétique de la convection électrique[END_REF]. Historical accounts include Indorato & Masotto 1989, and Walter et al., eds. 2007, § 2-17.

This was the policy announced by the JPTA's founder, J.-C. Almeida, in the first issue of the review.

[START_REF] Ritz | Recherches critiques sur l'électrodynamique générale[END_REF][START_REF] Martínez | Einstein, and the emission hypothesis[END_REF]

[START_REF] Perrin | Mouvement brownien et réalité moléculaire[END_REF][START_REF] Nye | Molecular Reality: A Perspective on the Scientific Work of Jean Perrin[END_REF]


Chapter 2

The historical origins of spacetime

The idea of spacetime investigated in this chapter, with a view toward understanding its immediate sources and development, is the one formulated and proposed by Hermann Minkowski in 1908. Until recently, the principle source used to form historical narratives of Minkowski's discovery of spacetime has been Minkowski's own discovery account, outlined in the lecture he delivered in Cologne, entitled "Space and time" [START_REF] Minkowski | Raum und Zeit[END_REF]. Minkowski's lecture is usually considered as a bona fide first-person narrative of lived events. According to this received view, spacetime was a natural outgrowth of Felix Klein's successful project to promote the study of geometries via their characteristic groups of transformations. Or as Minkowski expressed the same basic thought himself, the theory of relativity discovered by the physicists in 1905 could just as well have been proposed by some late-nineteenth-century mathematician, by simply reflecting upon the groups of transformations that left invariant the form of the equation of a propagating light wave.

Minkowski's publications and research notes provide a contrasting picture of the discovery of spacetime, in which group theory plays no direct part. In order to relate the steps of Minkowski's discovery, I begin with an account of Poincaré's theory of gravitation, where Minkowski found some of the germs of spacetime. Poincaré's geometric interpretation of the Lorentz transformation is examined, along with his reasons for not pursuing a four-dimensional vector calculus. In the second section, Minkowski's discovery and presentation of the notion of a worldline in spacetime is presented. In the third and final section, Poincaré's and Minkowski's diagrammatic interpretations of the Lorentz transformation are compared.

Poincaré's theory of gravitation

In the month of May, 1905, Henri Poincaré (1854-1912) wrote to his Dutch colleague H. A. Lorentz (1853Lorentz ( -1928) ) to apologize for missing the latter's lecture in Paris, and also to communicate his latest discovery, which was related to Lorentz's recent paper [START_REF] Antoon | Electromagnetic phenomena in a system moving with any velocity less than that of light[END_REF] on electromagnetic phenomena in frames moving with sublight velocity [3, §38.3]. In [START_REF] Antoon | Electromagnetic phenomena in a system moving with any velocity less than that of light[END_REF], Lorentz had shown that the form of the fundamental equations of his theory of electrons is invariant with respect to the coordinate transformations:

x = γ x, y = y, z = z, t = γ t -β v c 2 x, (2.1)
where

γ = 1/ 1 -v 2 /c 2 , = f (v), = 1 for v = 0, c = vacuum speed of light.
The latter transformation was understood to compose with a transformation later known as a "Galilei" transformation: x = x -vt , t = t . (Both here and elsewhere in this chapter, original notation is modified for ease of reading.) The essence of Poincaré's discovery in May 1905, communicated in subsequent letters to Lorentz, was that the coordinate transformations employed by Lorentz form a group, provided that the factor is set to unity. Poincaré performed the composition of the two transformations to obtain a single transformation, which he called the "Lorentz" transformation:

x = γ(x -vt), y = y, z = z, t = γ(t -vx/c 2 ).

(2.2)

In his letters to Lorentz, Poincaré noted that while he had concocted an electron model that was both stable and relativistic, in the new theory he was unable to preserve the "unity of time", i.e., a definition of duration valid in both the ether and in moving frames. The details of Poincaré's theory [START_REF] Poincaré | Sur la dynamique de l'électron[END_REF] were published in January, 1906, by which time Einstein had published his own theory of relativity [START_REF] Einstein | Zur Elektrodynamik bewegter Körper[END_REF], which employed the "unified" form of the Lorentz transformation (2.2), and vigorously embraced the relativity of space and time with respect to inertial frames of motion. The final section of Poincaré's memoir is devoted to a topic he had neglected to broach with Lorentz, and that Einstein had neglected altogether: gravitation.

If the principle of relativity was to be universally valid, Poincaré reasoned, then Newton's law of gravitation would have to be modified. An adept of the group-theoretical understanding of geometry since his discovery of what he called "Fuchsian" functions in 1880 [START_REF] Gray | Henri Poincaré: Trois suppléments sur la découverte des fonctions fuchsiennes[END_REF], Poincaré realized that a Lorentz transformation may be construed as a rotation about the origin of coordinates in a four-dimensional vector space with three real axes and one imaginary axis, preserving the sum of squares:

x 2 + y 2 + z 2 -t 2 = x 2 + y 2 + z 2 -t 2 , (2.3)
where Poincaré set c = 1. Employing the substitution u = t √ -1, and drawing on a method promoted by Sophus Lie and Georg Scheffers in the early 1890s [START_REF] Lie | Vorlesungen über continuierliche Gruppen mit geometrischen und anderen Anwendungen[END_REF], Poincaré identified a series of quantities that are invariant with respect to the Lorentz Group. These quantities were meant to be the fundamental building blocks of a Lorentz-covariant family of laws of gravitational attraction. Neglecting a possible dependence on acceleration, and assuming that the propagation velocity of gravitation is the same as that of light in empty space, Poincaré identified a pair of laws, one vaguely Newtonian, the other vaguely Maxwellian, that he expressed in the form of what would later be called four-vectors. In the course of his work on Lorentz-covariant gravitation, Poincaré defined several quadruples formally equivalent to four-vectors, including definitions of radius, velocity, force, and force density. The signs of Poincaré's invariants suggest that when he formed them, he did not consider them to be scalar products of four-vectors. This state of affairs led at least one contemporary observer to conclude -in the wake of Minkowski's contributions -that Poincaré had simply miscalculated one of his Lorentz invariants [START_REF] Walter | Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905-1910[END_REF]203,238]. Poincaré's four-dimensional vector space attracted little attention at first, except from the vectorist Roberto Marcolongo (1862Marcolongo ( -1945)), Professor of Mathematical Physics in Messina. Redefining Poincaré's temporal coordinate as u = -t √ -1, Marcolongo introduced four-vector definitions of current and potential, which enabled him to express the Lorentz-covariance of the equations of electrodynamics in matricial form [START_REF] Marcolongo | Sugli integrali delle equazioni dell' elettrodinamica[END_REF]. Largely ignored at the time, Marcolongo's paper nonetheless broke new ground in applying Poincaré's four-dimensional approach to the laws of electrodynamics. Marcolongo was one of many ardent vectorists active in the first decade of the twentieth century, when vector methods effectively sidelined the rival quaternionic approaches [10,259]. More and more theorists recognized the advantages of vector analysis, and also of a unified vector notation for mathematical physics. The pages of the leading journal of theoretical physics, the Annalen der Physik, edited by Paul Drude until his suicide in 1906, then by Max Planck and Willy Wien, bear witness to this evolution. Even in the pages of the Annalen der Physik, however, notation was far from standardized, leading several theorists to deplore the field's babel of symbolic expressions. Among the theorists who regretted the multiplication of systems of notation was Poincaré, who employed ordinary vectors in his own teaching and publications on electrodynamics, while ignoring the notational innovations of Lorentz and others. In particular, Poincaré saw no future for a fourdimensional vector calculus. Expressing physical laws by means of such a calculus, he wrote in 1907, would entail "much trouble for little profit" [START_REF] Poincaré | The Value of Science: Essential Writings of Henri Poincaré[END_REF]438]. This was not a dogmatic view, and in fact, some years later he acknowledged the value of a four-dimensional approach in theoretical physics [START_REF] Walter | Hypothesis and convention in Poincaré's defense of Galilei spacetime[END_REF]210]. He was already convinced that there was a place for (3 + n)-dimensional geometries at the university. As Poincaré observed in the paper Gaston Darboux read in his stead at the International Congress of Mathematicians in Rome, in April, 1908, university students were no longer taken aback by geometries with "more than three dimensions" [START_REF] Poincaré | L'avenir des mathématiques[END_REF]938]. Relativity theory, however, was another matter for Poincaré. Recentlyrediscovered manuscript notes by Henri Vergne of Poincaré's lectures on relativity theory in 1906-1907 reveal that Poincaré introduced his students to the Lorentz Group, and taught them how to form Lorentz-invariant quantities with real coordinates. He also taught his students that the sum of squares (2.3) is invariant with respect to the transformations of the Lorentz Group. Curiously, Poincaré did not teach his students that a Lorentz transformation corresponded to a rotation about the origin in a four-dimensional vector space with one imaginary coordinate. He also neglected to show students the handful of four-vectors he had defined in the summer of 1905. Apparently for Poincaré, knowledge of the Lorentz Group and the formation of Lorentzinvariant quantities was all that was needed for the physics of relativity. In other words, Poincaré acted as if one could do without an interpretation of the Lorentz transformation in four-dimensional geometry. If four-dimensional geometry was superfluous to interpretation of the Lorentz transformation, the same was not true for plane geometry. Evidence of this view is found in Vergne's notes, which feature a curious figure that I'll call a light ellipse, redrawn here as Figure 2.1. Poincaré's light ellipse is given to be the meridional section of an ellipsoid of rotation representing the locus of a spherical light pulse at an instant of time. It works as follows: an observer at rest with respect to the ether measures the radius of a spherical light pulse at an instant of absolute time t (as determined by clocks at rest with respect to the ether). The observer measures the light pulse radius with measuring rods in uniform motion of velocity v. These flying rods are Lorentz-contracted, while the light wave is assumed to propagate spherically in the ether. Consequently, for Poincaré, the form of a spherical light pulse measured in this fashion is that of an ellipsoid of rotation, elongated in the direction of motion of the flying rods. (A derivation of the equation of Poincaré's light ellipse is provided along these lines in [START_REF] Darrigol | Poincaré and light[END_REF].)

The light ellipse originally concerned ether-fixed observers measuring a locus of light with clocks at absolute rest, and rods in motion. Notably, in his first discussion of the light ellipse, Poincaré neglected to consider the point of view of observers in motion with respect to the ether. In particular, Poincaré's graphical model of light propagation does not display relativity of simultaneity for inertial observers, since it represents a single frame of motion. Nonetheless, Poincaré's light ellipse was applicable to the case of observers in uniform motion, as he showed himself in 1909. In this case, the radius vector of the light ellipse represents the light-pulse radius at an instant of "apparent" time t , as determined by comoving, light-synchronized clocks, and comoving rods corrected for Lorentz-contraction. Such an interpretation implies that clock rates depend on frame velocity, as Einstein recognized in 1905 in consequence of his kinematic assumptions about ideal rods and clocks [START_REF] Einstein | Zur Elektrodynamik bewegter Körper[END_REF]904], and which Poincaré acknowledged in a lecture in Göttingen on 28 April, 1909, as an effect epistemically akin to Lorentz-contraction, induced by clock motion with respect to the ether [START_REF] Poincaré | Sechs Vorträge über ausgewählte Gegenstände aus der reinen Mathematik und mathematischen Physik[END_REF]55].

Beginning in August 1909, Poincaré repurposed his light ellipse diagram to account for the apparent dilation of periods of ideal clocks in motion with respect to the ether [START_REF] Poincaré | La mécanique nouvelle[END_REF]174]. This sequence of events raises the question of what led Poincaré to embrace the notion of time deformation in moving frames, and to repurpose his light ellipse? He didn't say, but there is a plausible explanation at hand, that I will return to later, as it rests on events in the history of relativity from 1907 to 1908 to be discussed in the next section.

Minkowski's path to spacetime

From the summer of 1905 to the fall of 1908, the theory of relativity was reputed to be inconsistent with the observed deflection of β-rays by electric and magnetic fields. In view of experimental results published by Walter Kaufmann (1871-1947), Lorentz wrote in despair to Poincaré on 8 March 1906 in hopes that the Frenchman would find a way to save his theory. As far as Lorentz was concerned, he was "at the end of [his] Latin" [START_REF] Miller | Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation[END_REF]334]. Apparently, Poincaré saw no way around Kaufmann's results, either. But by the end of 1908, the outlook for relativity theory had changed for the better, thanks in part to new experiments performed by A. H. Bucherer (1863-1927), which tended to confirm the predictions of relativity theory. The outlook for the latter theory was also enhanced by the contributions of a mathematician in Göttingen, Hermann Minkowski (1864[START_REF] Minkowski | Raum und Zeit[END_REF]).

Minkowski's path to theoretical physics was a meandering one, that began in earnest during his student days in Berlin, where he heard lectures by Hermann Helmholtz, Gustav Kirchhoff, Carl Runge, and Woldemar Voigt.

There followed a dissertation in Königsberg on quadratic forms, and Habilitation in Bonn on a related topic in 1887 [START_REF] Schwermer | Räumliche Anschauung und Minima positiv definiter quadratischer Formen[END_REF]. While in Bonn, Minkowski frequented Heinrich Hertz's laboratory beginning in December, 1890, when it was teeming with young physicists eager to master techniques for the study of electromagnetic wave phenomena. and 1899, there emerged a professional niche for individuals trained in both mathematical physics and experimental physics, that very few mathematicians chose to enter. This "institutional revolution" in German physics [START_REF] Cahan | The institutional revolution in German physics, 1865-1914[END_REF] gave rise to a new breed of physicist: the "theoretical" physicist [START_REF] Jungnickel | Intellectual Mastery of Nature: Theoretical Physics from Ohm to Einstein[END_REF].

In the summer of 1905, Minkowski and Hilbert co-directed a third seminar in mathematical physics, convinced that only higher mathematics could solve the problems then facing physicists, and with Poincaré's fourteen volumes of Sorbonne lectures on mathematical physics serving as an example. This time they delved into a branch of physics new to both of them: electron theory. Their seminar was an occasion for them to acquaint themselves, their colleagues Emil Wiechert and Gustav Herglotz, and students including Max Laue and Max Born, with recent research in electron theory. From all accounts, the seminar succeeded in familiarizing its participants with the state of the art in electron theory, although the syllabus did not feature the most recent contributions from Lorentz and Poincaré [START_REF] Pyenson | Physics in the shadow of mathematics: the Göttingen electron-theory seminar of 1905[END_REF]. In particular, according to Born's distant recollections of the seminar, Minkowski "occasionally hinted" of his engagement with the Lorentz transformation, and he conveyed an "inkling" of the results he would publish in 1908 [START_REF] Walter | Hermann Minkowski's approach to physics[END_REF].

The immediate consequence of the electron-theory seminar for Minkowski was a new interest in a related, and quite-puzzling topic in theoretical physics: blackbody radiation. Minkowski gave two lectures on heat radiation in 1906, and offered a lecture course in this subject during the summer semester of 1907. According to Minkowski's class notes, he referred to Max Planck's contribution to the foundations of relativistic thermodynamics [START_REF] Planck | Zur Dynamik bewegter Systeme[END_REF], which praised Einstein's formulation of a general approach to the principle of relativity for ponderable systems. In fact, Minkowski had little time to assimilate Planck's findings (communicated on 13 June 1907) and communicate them to his students. This may explain why his lecture notes cover only nonrelativistic approaches to heat radiation. By the fall of 1907, Minkowski had come to realize some important consequences of relativity theory not only for thermodynamics, but for all of physics. On 9 October, he wrote to Einstein, requesting an offprint of his first paper on relativity, which was the one cited by Planck [START_REF] Klein | The Collected Papers of Albert Einstein[END_REF]Doc. 62].

Less than a month later, on 5 November 1907, Minkowski delivered a lecture to the Göttingen Mathematical Society, the subject of which was described succinctly as "On the principle of relativity in electrodynamics: a new form of the equations of electrodynamics" [START_REF] Walter | Hermann Minkowski's approach to physics[END_REF].

The lecture before the mathematical society was the occasion for Minkowski to unveil a new research program: to reformulate the laws of physics in fourdimensional terms, based on the Lorentz-invariance of the quadratic form

x 2 + y 2 + z 2 -c 2 t 2
, where x, y, z, are rectangular space coordinates, fixed "in ether," t is time, and c is the vacuum speed of light [START_REF] Minkowski | Das Relativitätsprinzip[END_REF]374]. Progress toward the achievement of such a reformulation had been realized by Poincaré's relativistic reformulation of the law of gravitation in terms of Lorentz-invariant quantities expressed in the form of four-vectors, as mentioned above.

Poincaré's formal contribution was duly acknowledged by Minkowski, who intended to go beyond what the Frenchman had accomplished in 1905. He also intended to go beyond what Poincaré had considered to be desirable, with respect to the application of geometric reasoning in the physical sciences. Poincaré, we recall, had famously predicted that Euclidean geometry would forever remain the most convenient one for physics [START_REF] Poincaré | The Value of Science: Essential Writings of Henri Poincaré[END_REF][START_REF] Einstein | Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips[END_REF]. Poincaré's prediction stemmed in part from his doctrine of physical space, according to which the question of the geometry of phenomenal space cannot be decided on empirical grounds. In fact, few of Poincaré's contemporaries in the physical and mathematical sciences agreed with his doctrine [START_REF] Walter | La vérité en géométrie : sur le rejet mathématique de la doctrine conventionnaliste[END_REF].

Euclidean geometry was to be discarded in favor of a certain four-dimensional manifold, and not just any manifold, but a non-Euclidean manifold. The reason for this was metaphysical, in that for Minkowski, the phenomenal world was not Euclidean, but non-Euclidean and four-dimensional:

The world in space and time is, in a certain sense, a four-dimensional non-Euclidean manifold. [START_REF] Minkowski | Das Relativitätsprinzip[END_REF]372] Explaining this enigmatic proposition would take up the rest of Minkowski's lecture.

To begin with, Minkowski discussed neither space, time, manifolds, or non-Euclidean geometry, but vectors. Borrowing Poincaré's definitions of radius and force density, and adding (like Marcolongo before him) expressions for four-current density, , and four-potential, ψ, Minkowski expressed Maxwell's vacuum equations in the compact form:

ψ j = -j (j = 1, 2, 3, 4), (2.4)
where is the Dalembertian operator. According to Minkowski, no one had realized before that the equations of electrodynamics could be written so succinctly, "not even Poincaré" (cf. [START_REF] Galison | Minkowski's spacetime: from visual thinking to the absolute world[END_REF]). Apparently, Minkowski had not noticed Marcolongo's paper, mentioned above.

The next mathematical object that Minkowski introduced was a real step forward, and soon acknowledged as such by physicists. This is what Minkowski called a "Traktor": a six-component object later called a "six-vector", and more recently, an antisymmetric rank-2 tensor. Minkowski defined the Traktor's six components via his four-vector potential, using a two-index notation:

ψ jk = ∂ψ k /∂x j -∂ψ j /∂x k ,
= σ j -j , (2.5)
where σ is the four-current density for matter. Up to this point in his lecture, Minkowski had presented a new and valuable mathematical object, the antisymmetric rank-2 tensor. He had yet to reveal the sense in which the world is a "four-dimensional non-Euclidean manifold". His argument proceeded as follows. The tip of a four-dimensional velocity vector w 1 , w 2 , w 3 , w 4 , Minkowski explained, is always a point on the surface

w 2 1 + w 2 2 + w 2 3 + w 2 4 = -1, (2.6)
or if you prefer, on

t 2 -x 2 -y 2 -z 2 = 1, (2.7)
and represents both the four-dimensional vector from the origin to this point, and null velocity, or rest, being a genuine vector of this sort. Non-Euclidean geometry, of which I spoke earlier in an imprecise fashion, now unfolds for these velocity vectors. [START_REF] Minkowski | Das Relativitätsprinzip[END_REF]373] Figure 2.2: A reconstruction of Minkowski's 5 Nov. 1907 presentation of relativistic velocity space, with a pair of temporal axes, one spatial axis, a unit hyperbola and its asymptotes.

These two surfaces, a pseudo-hypersphere of unit imaginary radius (2.6), and its real counterpart, the two-sheeted unit hyperboloid (2.7), give rise to well-known models of hyperbolic space, popularized by Helmholtz in the late nineteenth century [31, Vol. 2]. The upper sheet (t > 0) of the unit hyperboloid (2.7) models hyperbolic geometry; for details, see [START_REF] Reynolds | Hyperbolic geometry on a hyperboloid[END_REF]. The conjugate diameters of the hyperboloid (2.7) give rise to a geometric image of the Lorentz transformation. Any point on (2.7) can be considered to be at rest, i.e., it may be taken to lie on a t-diameter, as shown in Figure 2.2. This change of axes corresponds to an orthogonal transformation of the time and space coordinates, which is a Lorentz transformation (letting c = 1). In other words, the three-dimensional hyperboloid (2.7) embedded in four-dimensional pseudo-Euclidean space affords an interpretation of the Lorentz transformation. Although Minkowski did not spell out his geometric interpretation, he probably recognized that a displacement on the hypersurface (2.7) corresponds to a rotation ψ about the origin, such that frame velocity v is described by a hyperbolic function, v = tanh ψ. However, he did not yet realize that his hypersurfaces represent the set of events occurring at coordinate time t = 1 of all inertial observers, the worldlines of whom pass through the origin of coordinates (with a common origin of time). According to (2.7), this time is imaginary, a fact which may have obscured the latter interpretation. How do we know that Minkowski was still unaware of worldlines in spacetime? Inspection of Minkowski's definition of four-velocity vectors reveals an error, which is both trivial and interesting: trivial from a mathematical standpoint, and interesting for what it says about his knowledge of the structure of spacetime, and the progress he had realized toward his goal of replacing the Euclidean geometry of phenomenal space with the geometry of a fourdimensional non-Euclidean manifold.

When faced with the question of how to define a four-velocity vector, Minkowski had the option of adopting the definition given by Poincaré in 1905. Instead, he rederived his own version, by following a simple rule. Minkowski defined a four-vector potential, four-current density, and four-force density, all by simply generalizing ordinary three-component vectors to their fourcomponent counterparts. When he came to define four-velocity, he took over the components of the ordinary velocity vector w for the spatial part of fourvelocity, and added an imaginary fourth component, i √ 1 -w 2 . This resulted in four components of four-velocity, w 1 , w 2 , w 3 , w 4 :

w x , w y , w z , i √ 1 -w 2 . (2.8)
Since the components of Minkowski's quadruplet do not transform like the coordinates of his vector space x 1 , x 2 , x 3 , x 4 , they lack what he knew to be a four-vector property.

Minkowski's error in defining four-velocity indicates that he did not yet grasp the notion of four-velocity as a four-vector tangent to the worldline of a particle [START_REF] Walter | Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905-1910[END_REF]. If we grant ourselves the latter notion, then we can let the square of the differential parameter dτ of a given worldline be dτ 2 = -(dx 2 1 + dx 2 2 + dx 2 3 + dx 2 4 ), such that the 4-velocity w µ may be defined as the first derivative with respect to τ , w µ = dx µ /dτ (µ = 1, 2, 3, 4). In addition to a valid four-velocity vector, Minkowski was missing a four-force vector, and a notion of proper time. In light of these significant lacunae in his knowledge of the basic mathematical objects of four-dimensional physics, Minkowski's triumphant description of his four-dimensional formalism as "virtually the greatest triumph ever shown by the application of mathematics" [START_REF] Minkowski | Das Relativitätsprinzip[END_REF]373] is all the more remarkable, and bears witness to the depth of Minkowski's conviction that he was on the right track. Sometime after Minkowski spoke to the Göttingen Mathematical Society, he repaired his definition of four-velocity, and perhaps in connection with this, he came up with the constitutive elements of his theory of spacetime.

In particular, he formulated the idea of proper time as the parameter of a hyperline in spacetime, the light-hypercone structure of spacetime, and the spacetime equations of motion of a material particle. He expressed his new theory in a sixty-page memoir [START_REF] Minkowski | Die Grundgleichungen für die electromagnetischen Vorgänge in bewegten Körpern[END_REF] published in the Göttinger Nachrichten on 5 April, 1908. His memoir, entitled "The basic equations for electromagnetic processes in moving bodies" made for challenging reading. It was packed with new notation, terminology, and calculation rules, it made scant reference to the scientific literature, and offered no figures or diagrams. Minkowski defined a single differential operator, named lor in honor of Lorentz, which streamlined his expressions, while rendering them all the more unfamiliar to physicists used to the three-dimensional operators of ordinary vector analysis. Along the same lines, Minkowski rewrote velocity, denoted q, in terms of the tangent of an imaginary angle iψ, q = -i tan iψ, (2.9)

where q < 1. From his earlier geometric interpretation of hyperbolic velocity space, Minkowski kept the idea that every rotation of a t-diameter corresponds to a Lorentz transformation, which he now expressed in terms of iψ:

x 1 = x 1 , x 3 = x 3 cos iψ + x 4 sin iψ, x 2 = x 2 , x 4 = -x 3 sin iψ + x 4 cos iψ.
(2.10)

Minkowski was undoubtedly aware of the connection between the composition of Lorentz transformations and velocity composition, but he did not mention it. In fact, Minkowski neither mentioned Einstein's law of velocity addition, nor expressed it mathematically. While Minkowski made no appeal in "The basic equations" to the hyperbolic geometry of velocity vectors, he retained the hypersurface (2.7) on which it was based, and provided a new interpretation of its physical significance. This interpretation represents an important clue to understanding how Minkowski discovered the worldline structure of spacetime. The appendix to "The basic equations" rehearses the argument according to which one may choose any point on (2.7) such that the line from this point to the origin forms a new time axis, and corresponds to a Lorentz transformation. He further defined a "spacetime line" to be the totality of spacetime points corresponding to any particular point of matter for all time t. With respect to the new concept of a spacetime line, Minkowski noted that its direction is determined at every spacetime point. Here Minkowski introduced the notion of "proper time" (Eigenzeit), τ , expressing the increase of coordinate time dt for a point of matter with respect to dτ :

dτ = dt 2 -dx 2 -dy 2 -dz 2 = dt √ 1 -w 2 = dx 4 w 4 , (2.11)
where w 2 is the square of ordinary velocity, dx 4 = idt, and w 4 = i/ √ 1 -w 2 , which silently corrects the flawed definition of this fourth component of fourvelocity (2.8) delivered by Minkowski in his November 5 lecture. Although Minkowski did not connect four-velocity to Einstein's law of velocity addition, others did this for him, beginning with Sommerfeld, who expressed parallel velocity addition as the sum of tangents of an imaginary angle [START_REF] Sommerfeld | Über die Zusammensetzung der Geschwindigkeiten in der Relativtheorie[END_REF]. Minkowski's former student Philipp Frank reexpressed both velocity and the Lorentz transformation as hyperbolic functions of a real angle [START_REF] Philipp | Die Stellung des Relativitätsprinzips im System der Mechanik und der Elektrodynamik[END_REF]. The Serbian mathematician Vladimir Varičak found relativity theory to be ripe for application of hyperbolic geometry , and recapitulated several relativistic formulae in terms of hyperbolic functions of a real angle [START_REF] Varičak | Anwendung der Lobatschefskijschen Geometrie in der Relativtheorie[END_REF]. A small group of mathematicians and physicists pursued this "non-Euclidean style" of Minkowskian relativity, including Varičak, Alfred Robb, Émile Borel, Gilbert Newton Lewis, and Edwin Bidwell Wilson [START_REF] Walter | The non-Euclidean style of Minkowskian relativity[END_REF]. The definition of four-velocity was formally linked by Minkowski to the hyperbolic space of velocity vectors in "The basic equations", and thereby to the lightcone structure of spacetime. Some time before Minkowski came to study the Lorentz transformation in earnest, both Einstein and Poincaré understood light waves in empty space to be the only physical objects immune to Lorentz contraction. Minkowski noticed that when light rays are considered as worldlines, they divide spacetime into three regions, corresponding to the spacetime region inside a future-directed (t > 0) hypercone ("Nachkegel "), the region inside a past-directed (t < 0) hypercone ("Vorkegel "), and the region outside any such hypercone pair. The propagation in space and time of a spherical light wave is described by a hypercone, or what Minkowski called a lightcone ("Lichtkegel "). One immediate consequence for Minkowski of the lightcone structure of spacetime concerned the relativity of simultaneity. In a section of "The basic equations" entitled "The concept of time", Minkowski [33, § 6] showed that Einstein's relativity of simultaneity is not absolute. While the relativity of simultaneity is indeed valid for two or three simultaneous "events" (Ereignisse), the simultaneity of four events is absolute, so long as the four spacetime points do not lie on the same spatial plane. Minkowski's demonstration relied on the Einstein simultaneity convention, and employed both light signals and spacetime geometry. His result showed the advantage of employing his spacetime geometry in physics, and later writers -including Poincaré -appear to have agreed with him, by attributing to spacetime geometry the discovery of the existence of a class of events for a given observer that can be the cause of no other events for the same observer [START_REF] Walter | Hypothesis and convention in Poincaré's defense of Galilei spacetime[END_REF]210]. Another signature result of Minkowski's spacetime geometry was the geometric derivation of a Lorentz-covariant law of gravitation. Like Poincaré, Minkowski proposed two four-vector laws of gravitation, exploiting analogies to Newtonian gravitation and Maxwellian electrodynamics, respectively. Minkowski presented only the Newtonian version of the law of gravitation in "The basic equations", relating the states of two massive particles in arbritrary motion, and obtaining an expression for the spacelike component of the four-force of gravitation. Although his derivation involved a new spacetime geometry, Minkowski did not illustrate graphically his new law, a decision which led some physicists to describe his theory as unintelligible. According to Minkowski, however, his achievement was a formal one, inasmuch as Poincaré had formulated his theory of gravitation by proceeding in what he described as a "completely different way" [START_REF] Walter | Breaking in the 4-vectors: the four-dimensional movement in gravitation, 1905-1910[END_REF]225]. Few were impressed at first by Minkowski's innovations in spacetime geometry and four-dimensional vector calculus. Shortly after "The basic equations" appeared in print, two of Minkowski's former students, Einstein and Laub, discovered what they believed to be an infelicity in Minkowski's definition of ponderomotive force density [START_REF] Stachel | Einstein and Laub on the electrodynamics of moving media[END_REF]. These two young physicists were more impressed by Minkowski's electrodynamics of moving media than by the novel four-dimensional formalism in which it was couched, which seemed far too laborious. Ostensibly as a service to the community, Einstein and Laub reexpressed Minkowski's theory in terms of ordinary vector analysis [START_REF]The Collected Papers of Albert Einstein[END_REF]Doc. 51]. Minkowski's reaction to the latter work is unknown, but it must have come to him as a disappointment. According to Max Born, Minkowski always aspired . . . to find the form for the presentation of his thoughts that corresponded best to the subject matter. [START_REF] Born | Die Physik der bewegten Materie und die Relativitätstheorie[END_REF] The form Minkowski gave to his theory of moving media in "The basic equations" had been judged unwieldy by a founder of relativity theory, and in the circumstances, decisive action was called for if his formalism was not to be ignored. In September 1908, during the annual meeting of the German Association of Scientists and Physicians in Cologne, Minkowski took action, by affirming the reality of the four-dimensional "world", and its necessity for physics [START_REF] Walter | Minkowski's modern world[END_REF]. The next section focuses on the use to which Minkowski put spacetime diagrams in his Cologne lecture, and how these diagrams relate to Poincaré's light ellipse.

Spacetime diagrams

One way for Minkowski to persuade physicists of the value of his spacetime approach to understanding physical interactions was to appeal to their visual intuition [START_REF] Galison | Minkowski's spacetime: from visual thinking to the absolute world[END_REF]. From the standpoint of visual aids, the contrast between Minkowski's two publications on spacetime is remarkable: where "The basic equations" is bereft of diagrams and illustrations, Minkowski's Cologne lecture makes effective use of diagrams in two and three dimensions. For instance, Minkowski employed a two-dimensional spacetime diagrams to illustrate FitzGerald-Lorentz contraction of an electron, and the lightcone structure of spacetime (see Figure 2.3). Minkowski's lecture in Cologne, entitled "Space and time", offered two diagrammatic readings of the Lorentz transformation, one of his own creation, the other he attributed to Lorentz and Einstein. One of these readings was supposed to represent the kinematics of the theory of relativity of Lorentz and Einstein. In fact, Minkowski's reading captured Lorentzian kinematics, but distorted those of Einstein, and prompted corrective action from Philipp Frank, Guido Castelnuovo, and Max Born [START_REF] Walter | Minkowski, mathematicians, and the mathematical theory of relativity[END_REF]. The idea stressed by Minkowski was that in the (Galilean) kinematics employed in Lorentz's electron theory, time being absolute, the temporal axis on a space-time diagram may be rotated freely about the coordinate origin in the upper half-plane (t > 0), as shown in Figure 2.4. The location of a point P may be described with respect to frames S and S , corresponding to axes (x, t) and (x , t ), respectively, according to the transformation: x = x -vt, t = t. In contradistinction to the latter view, the theory proposed by Minkowski required a certain symmetry between the spatial and temporal axes. This constraint on symmetry sufficed for a geometric derivation of the Lorentz transformation. Minkowski described his spacetime diagram (Figure 2.5) as an illustration of the Lorentz transformation, and provided an idea of a demonstration in "Space and time". A demonstration was later supplied by Sommerfeld, in an editorial note to his friend's lecture [START_REF] Blumenthal | Das Relativitätsprinzip; Eine Sammlung von Abhandlungen[END_REF], which appeared in an anthology of papers on the theory of relativity edited by Otto Blumenthal

6) and (2.7).

There is also a relation between the light ellipse and the spacetime map, although this may not have been apparent to either Poincaré or Minkowski. Their published appreciations of each other's contributions to relativity field the barest of acknowledgments, suggesting no substantial intellectual indebtedness on either side. The diagrams employed in the field of relativity by Poincaré and Minkowski differ in several respects, but one difference in particular stands out. On the one hand, the light ellipse represents spatial relations in a plane defined as a Both before and after 1909, Poincaré found that a spherical light pulse in the ether would be described as a prolate ellipsoid in inertial frames. Meanwhile, for Einstein and others who admitted the spatio-temporal relativity of inertial frames, the form of a spherical light pulse remained spherical in all inertial frames. In Poincaré's scheme of things, only for ether-fixed observers measuring wavefronts with clocks and rods at rest is the light pulse a sphere; in all other inertial frames the light pulse is necessarily shaped like a prolate ellipsoid. Comparison of Poincaré's pre-1909 and post-1909 readings of the light ellipse shows the ellipse dimensions to be unchanged. What differs in the Minkowskian representations of these two readings is the angle of the spacelike plane containing the light ellipse with respect to the lightcone. The complementary representation is obtained in either case by rotating the light ellipse through an angle ψ = tanh -1 v about the line parallel to the y-axis passing through point B.

We are now in a position to answer the question raised above, concerning the reasons for Poincaré's embrace of time deformation in 1909. From the standpoint of experiment, there was no pressing need to recognize time dilation in 1909, although in 1907 Einstein figured it would be seen as a transverse Doppler effect in the spectrum of canal rays [START_REF] Einstein | Über die Möglichkeit einer neuen Prüfung des Relativitätsprinzips[END_REF]. On the theoretical side, Minkowski's spacetime theory was instrumental in convincing leading ethertheorists like Sommerfeld and Max Abraham of the advantages of Einstein's theory. Taken in historical context, Poincaré's poignant acknowledgment in Göttingen of time deformation (and subsequent repurposing of his light ellipse) reflects the growing appreciation among scientists, circa 1909, of the Einstein-Minkowski theory of relativity [START_REF] Walter | Poincaré on clocks in motion[END_REF].

Figure 2 . 1 :

 21 Figure 2.1: Poincaré's light ellipse, after manuscript notes by Henri Vergne, 1906-1907 (Henri Poincaré Archives). Labels H and A are added for clarity.

Figure 2 . 3 :

 23 Figure 2.3: The lightcone structure of spacetime [1].

Figure 2 . 4 :

 24 Figure 2.4: A reconstruction of Minkowski's depiction of the kinematics of Lorentz and Einstein.

Figure 2 . 5 :

 25 Figure 2.5: Minkowski's spacetime diagram, after [1].

Figure 2 . 6 :

 26 Figure 2.6: Spacetime model of Poincaré's light ellipse (1906) in a spacelike plane (t =const.).

Figure 2 . 7 :

 27 Figure 2.7: Spacetime model of Poincaré's light ellipse (1909) in a spacelike plane (t =const.).

  Minkowski left Bonn for a position at the University of Königsberg, where he taught mathematics until 1896, and then moved to Switzerland, where he joined his former teacher Adolf Hurwitz (1859-1919) on the faculty of Zürich Polytechnic. In Zürich he taught courses in mathematics and mathematical physics to undergraduates including Walther Ritz (1878-1909), Marcel Grossmann (1878-1936), and AlbertEinstein (1879Einstein ( -1955)). In 1902, Minkowski accepted the offer to take up a new chair in mathematics created for him in Göttingen at the request of his good friend, David Hilbert (1862-1943)[START_REF] Rowe | Jewish mathematics' at Göttingen in the era of Felix Klein[END_REF] 436]. Minkowski's arrival in Göttingen comforted the university's premier position in mathematical research in Germany. His mathematical credentials were well-established following the publication, in 1896, of the seminal Geometry of Numbers[START_REF] Minkowski | Geometrie der Zahlen[END_REF]. During his first two years in Göttingen, Minkowski continued to publish in number theory, and to teach pure mathematics. With Hilbert, who had taken an interest in questions of mathematical physics in the 1890s, Minkowski co-directed a pair of seminars on stability and mechanics[START_REF] Corry | From Grundlagen der Geometrie to Grundlagen der Physik[END_REF]. It was quite unusual at the time for Continental mathematicians to pursue research in theoretical physics. Arguably, Poincaré was the exception that proved the rule, in that no other scientist displayed comparable mastery of research in both mathematics and theoretical physics. In Germany, apart from Carl Neumann, mathematicians left physics to the physicists. With the construction in Germany of twelve new physical institutes between 1870

  noting the antisymmetry relation ψ kj = -ψ jk , and zeros along the diagonal ψ jj = 0, such that the components ψ 14 , ψ 24 , ψ 34 , ψ 23 , ψ 31 , ψ 12 match the field quantities -iE

	p:
	∂p 1j ∂x 1	+	∂p 2j ∂x 2	+	∂p 3j ∂x 3	+	∂p 4j ∂x 4

x , -iE y , -iE z , B x , B y , B z . To express the source equations, Minkowski introduced a "Polarisationstraktor ",