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Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [http://www.bartleby.com/61/wavs/3/P0400300.wav Poincaré pronunciation example at Bartlby.com ), generally known as Henri Poincaré, was one of France's greatest mathematicians and theoretical physicists, and a philosopher of science. Poincaré is often described as the last "universalist" (after Gauss) capable of understanding and contributing in virtually all parts of mathematics.

As a mathematician and physicist, he made many original fundamental contributions to pure and applied mathematics, mathematical physics, and celestial mechanics. He was responsible for formulating the Poincaré conjecture, one of the most famous problems in mathematics. In his research on the three-body problem, Poincaré became the first person to discover a chaotic deterministic system which laid the foundations of modern Chaos theory.

Poincaré introduced the modern principle of relativity and was the first to present the Lorentz transformations in their modern symmetrical form. The Poincaré group was named after him. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Lorentz in 1905. Thus he obtained perfect invariance of all of Maxwell's equations, the final step in the discovery of the theory of special relativity.

Life

Poincaré was born on April 29, 1854 in Cité Ducale neighborhood, Nancy, France into an influential family (Belliver, 1956). His father Leon Poincaré (1828-1892) was a professor of medicine at the University of Nancy (Sagaret, 1911). His adored younger sister Aline married the spiritual philosopher Emile Boutroux. Another notable member of Jules' family was his cousin Raymond Poincaré, who would become the President of France, 1913 to 1920, and a fellow member of the Académie française.

Education

During his childhood he was seriously ill for a time with diphtheria and received special instruction from his gifted mother, Eugénie Launois (1830-1897). He excelled in written composition.

In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée Henri Poincaré in his honour, along with the University of Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. His mathematics teacher described him as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France. (His poorest subjects were music and physical education, where he was described as "average at best" (O'Connor et al., 2002). However, poor eyesight and a tendency towards absentmindedness may explain these difficulties (Carl, 1968). He graduated from the Lycée in 1871 with a bachelors degree in letters and sciences.

During the Franco-Prussian War of 1870 he served alongside his father in the Ambulance Corps.

Poincaré entered the École Polytechnique in 1873. There he studied mathematics as a student of Charles Hermite, continuing to excel and publishing his first paper (Démonstration nouvelle des propriétés de l'indicatrice d'une surface) in 1874. He graduated in 1875 or 1876. He went on to study at the École des Mines, continuing to study mathematics in addition to the mining engineering syllabus and received the degree of ordinary engineer in March 1879.

As a graduate of the École des Mines he joined the Corps des Mines as an inspector for the Vesoul region in northeast France. He was on the scene of a mining disaster at Magny in August 1879 in which 18 miners died. He carried out the official investigation into the accident in a characteristically thorough and humane way.

At the same time, Poincaré was preparing for his doctorate in sciences in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field of differential equations. Poincaré devised a new way of studying the properties of these functions. He not only faced the question of determining the integral of such equations, but also was the first person to study their general geometric properties. He realised that they could be used to model the behaviour of multiple bodies in free motion within the solar system. Poincaré graduated from the University of Paris in 1879.

Career

Soon after, he was offered a post as junior lecturer in mathematics at Caen University. He never fully abandoned his mining career to mathematics however. He worked at the Ministry of Public Services as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of the Corps de Mines in 1893 and inspector general in 1910.

Beginning in 1881 and for the rest of his career, he taught at the University of Paris, (the Sorbonne). He was initially appointed as the maître de conférences d'analyse (associate professor of analysis) (Sageret, 1911). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy.

Also in that same year, Poincaré married Miss Poulain d'Andecy. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893).

In 1887, at the young age of 32, Poincaré was elected to the French Academy of Sciences. He became its president in 1906, and was elected to the Académie française in 1909.

In 1887 he won Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies. (See Three Body Problem section below)

In 1893 Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronization of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalization of circular measure, and hence time and longitude. (see Galison 2003) It was this post which led him to consider the question of establishing international time zones and the synchronization of time between bodies in relative motion. (See Relativity section below)

In 1899, and again more successfully in 1904, he intervened in the trials of Alfred Dreyfus. He attacked the spurious scientific claims of some of the evidence brought against Dreyfus, who was a Jewish officer in the French army charged with treason by anti-Semitic colleagues.

In 1912 Poincaré underwent surgery for a prostate problem and subsequently died from an embolism on July 17, 1912, aged 58. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris.

The French Minister of Education, Claude Allegre, has recently (2004) proposed that Poincaré be reburied in the Pantheon in Paris, which is reserved for French citizens only of the highest honor. *

Work

Poincaré made many contributions to different fields of applied mathematics such as: celestial mechanics, fluid mechanics, optics, electricity, telegraphy, capillarity, elasticity, thermodynamics, potential theory, quantum theory, theory of relativity and physical cosmology.

He was also a popularizer of mathematics and physics and wrote several books for the lay public.

Among the specific topics he contributed to are the following:

The three-body problem

In 1887, in honor of his 60th birthday, Oscar II, King of Sweden sponsored a mathematical competition with a cash prize for a resolution of the question of how stable is the solar system, a variation of the three-body problem. Poincaré pointed out that the problem was not correctly posed, and proved that a complete solution to it could not be found. His work was so impressive that in 1888 the jury recognized its value by awarding him the prize. He found that the evolution of such a system is often chaotic in the sense that a small perturbation in the initial state, such as a slight change in one body's initial position, might lead to a radically different later state. If the slight change is not detectable by our measuring instruments, then we will not be able to predict which final state will occur. One of the judges, the distinguished Karl Weierstrass, said, "This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."

Weierstrass did not know how accurate he was. In Poincaré's paper, he described new mathematical ideas such as homoclinic points. The memoir was about to be published in Acta Mathematica when an error was found by the editor. This error in fact led to further discoveries by Poincaré, which are now considered to be the beginning of Chaos theory. The memoir was published later in 1890.
His research into orbits about Lagrange points and low‐energy transfers was not utilised for more than a century afterwards. See Interplanetary Transport Network.

Work on relativity

Poincaré's work on establishing international time zones, led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "luminiferous aether") could be synchronized. At the same time Dutch theorist Hendrik Lorentz was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. He had introduced the concept of local time

t^\prime = t-vx^\prime/c^2,\; \mathrm{where}\; x^\prime = x - vt
and was using it to explain the failure of optical and electrical experiments to detect motion relative to the aether (see Michelson-Morley experiment). Poincaré (1900) discussed Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronized by exchanging light signals assumed to travel with the same speed in both directions in a moving frame*. In "The Measure of Time" (Poincaré 1898), he discussed the difficulty of establishing simultaneity at a distance and concluded it can be established by convention. He also discussed the "postulate of the speed of light", and formulated the Principle of relativity, according to which no mechanical or electromagnetic experiment can discriminate between a state of uniform motion and a state of rest.

Thereafter, Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. Poincaré as a philosopher, was interested in the ”deeper meaning”. Thus he interpreted Lorentz's theory in terms the Principle of relativity and in so doing he came up with many insights that are now associated with Special relativity.

In the paper of 1900 Poincaré discussed the recoil of a physical object when it emits a burst of radiation in one direction, as predicted by Maxwell-Lorentz electrodynamics. He remarked that the stream of radiation appeared to act like a "fictitious fluid" with a mass per unit volume of e/c2, where e is the energy density; in other words, the equivalent mass of the radiation is m = E/c^2. Poincaré considered the recoil of the emiiter to be an unresolved feature of Maxwell-Lorentz theory, which he discussed again in "Science and Hypothesis" (1902) and "The Value of Science" (1904). In the latter he said the recoil "is contrary to the principle of Newton since our projectile here has no mass, it is not matter, it is energy", and discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass \gamma m, Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Madame Curie. It was Einstein's insight that a body losing energy as radiation or heat was losing mass of amount m = E/c^2, and the corresponding mass-energy conservation law, which resolved these problems.

In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance." In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz. In a second letter to Lorentz[http://web.archive.org/web/20050224225216/http://www.univ-nancy2.fr/poincare/chp/text/lorentz4.html, Poincaré explained the group property of the transformations, which Lorentz had not noticed, and gave his own reason why Lorentz's time dilation factor was indeed correct: Lorentz’s factor was necessary to make the Lorentz transformation form a group. In the letter, he also gave Lorentz what is now known as the relativistic velocity-addition law, which is necessary to demonstrate invariance. Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that short paper he wrote

The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form:
x^\prime = k\ell\left(x + \epsilon t\right),t^\prime = k\ell\left(t + \epsilon x\right),y^\prime = \ell y, z^\prime = \ell z, k = 1/\sqrt{1-\epsilon^2}.
He then wrote that in order for the Lorentz transformations to form a group and satisfy the principle of relativity, the arbitrary function \ell\left(\epsilon\right) must be unity for all \epsilon (Lorentz had set \ell = 1 by a different argument). Poincaré's discovery of the velocity transformations, allowed him to obtain perfect invariance, the final step in the discovery of the Theory of Special Relativity.

In an enlarged version of the paper that did not appear until 1906, he published his group property proof, incorporating the velocity addition law that he had previously written to Lorentz. The paper contains many other deductions from, and applications of, the transformations. For example, Poincaré (1906) pointed out that the combination x^2+ y^2+ z^2- c^2t^2 is invariant, and he introduced the 4-vector notation that Hermann Minkowski became known for.

Einstein's first paper on relativity derived the Lorentz transformation and presented them in the same form as had Poincaré. It was published three months after Poincaré's short paper, but before Poincaré's longer version appeared. Although Einstein (1905) relied on the Principle of relativity and used the same clock synchronization procedure that Poincaré (1900) had described, his paper was remarkable in that it had no references at all.

Poincaré never acknowledged Einstein's work on Special Relativity, but Einstein acknowledged Poincaré's in the text of a lecture in 1921 called Geometrie und Erfahrung. Later Einstein commented on Poincaré as being one of the pioneers of relativity:

"Lorentz had already recognized that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ..."

Poincaré's work in the development of Special Relativity is well recognized (e.g. Darrigol 2004), though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work (see Galison 2003 and Kragh 1999). A minority go much further, such as the historian of science Sir Edmund Whittaker who held that Poincaré and Lorentz were the true discoverers of Relativity (Whittaker 1953). Poincaré consistently credited Lorentz's achievements, ranking his own contributions as minor. Thus, he wrote:

"Lorentz has tried to modify his hypothesis so as to make it in accord with the hypothesis of complete impossibility of measuring absolute motion. He has succeeded in doing so in his article 1904. The importance of the problem has made me take up the question again; the results that I have obtained agreement on all important points with those of Lorentz; I have been led only to modify or complete them on some points of detail. The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation" (Poincaré 1905) added.
In an address in 1909 on "The New Mechanics", Poincaré discussed the demolition of Newton's mechanics brought about by Max Abraham and Lorentz, without mentioning Einstein. In one of his last essays entitled "The Quantum Theory" (1913), when referring to the Solvay Conference, Poincaré again described special relativity as the "mechanics of Lorentz":
"... at every moment twenty physicists from different countries could be heard talking of the new mechanics which they contrasted with the old mechanics. Now what was the old mechanics? Was it that of Newton, the one which still reigned uncontested at the close of the nineteenth century? No, it was the mechanics of Lorentz, the one dealing with the principle of relativity; the one which, hardly five years ago, seemed to be the height of boldness ... the mechanics of Lorentz endures ... no body in motion will ever be able to exceed the speed of light ... the mass of a body is not constant ... no experiment will ever be able detect motion either in relation to absolute space or even in relation to the aether." added

On the other hand, in a memoir written as a tribute to Poincaré after his death, Lorentz readily admitted the mistake he had made and credited Poincaré's achievements:

"For certain of the physical magnitudes which enter in the formulae I have not indicated the transformation which suits best. This has been done by Poincaré, and later by Einstein and Minkowski. My formulae were encumbered by certain terms which should have been made to disappear. I have not established the principle of relativity as rigorously and universally true. Poincaré, on the other hand, has obtained a perfect invariance of the electro-magnetic equations, and he has formulated 'the postulate of relativity', terms which he was the first to employ." [emphasis added

In summary, Poincaré regarded the mechanics as developed by Lorentz in order to obey the principle of relativity as the essence of the theory, while Lorentz stressed that perfect invariance was first obtained by Poincaré. The modern view is inclined to say that the group property and the invariance are the essential points.

Character

Poincaré's work habits have been compared to a bee flying from flower to flower. Poincaré was interested in the way his mind worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in Paris. He linked his way of thinking to how he made several discoveries.

The mathematician Darboux claimed he was un intuitif (intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation. He did not care about being rigorous and disliked logic. He believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.

Toulouse' characterization

His mental organization was not only interesting to him but also to Toulouse, a psychologist of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled Henri Poincaré (1910). In it, he discussed Poincaré's regular schedule:

  • He worked during the same times each day in short periods of time. He undertook mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m.. He would read articles in journals later in the evening.

  • He had an exceptional memory and could recall the page and line of any item in a text he had read. He was also able to remember verbatim things heard by ear. He retained these abilities all his life.

  • His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper.

  • He was ambidextrous and nearsighted.

  • His ability to visualise what he heard proved particularly useful when he attended lectures since his eyesight was so poor that he could not see properly what his lecturers were writing on the blackboard.

However, these abilities were somewhat balanced by his shortcomings:

  • He was physically clumsy and artistically inept.

  • He was always in a rush and disliked going back for changes or corrections.

  • He never spent a long time on a problem since he believed that the subconscious would continue working on the problem while he worked on another problem.

In addition, Toulouse stated that most mathematicians worked from principle already established while Poincaré was the type that started from basic principle each time. (O'Connor et al., 2002)

His method of thinking is well summarized as:

Habitué à négliger les détails et à ne regarder que les cimes, il passait de l'une à l'autre avec une promptitude surprenante et les faits qu'il découvrait se groupant d'eux-mêmes autour de leur centre étaient instantanément et automatiquement classés dans sa mémoire. (He neglected details and jumped from idea to idea, the facts gathered from each idea would then come together and solve the problem.) (Belliver, 1956)

Honors

Awards Named after him

Publications

Poincaré's major contribution to algebraic topology was Analysis situs (1895), which was the first real systematic look at topology.

He published two major works that placed celestial mechanics on a rigorous mathematical basis:

  • New Methods of Celestial Mechanics ISBN 1563961172 (3 vols., 1892-99; Eng. trans., 1967)
  • Lessons of Celestial Mechanics. (1905-10).

In popular writings he helped establish the fundamental popular definitions and perceptions of science by these writings:

  • Science and Hypothesis, 1901.
  • The Value of Science, 1904. (For more, see the French version.)
  • Science and Method, 1908.

  • Dernières pensées (Eng., "Last Thoughts"); Edition Ernest Flammarion, Paris, 1913.

Philosophy

Poincaré had the opposite philosophical views of Bertrand Russell and Gottlob Frege, who believed that mathematics were a branch of logic. Poincaré strongly disagreed, claiming that intuition was the life of mathematics. Poincaré gives an interesting point of view in his book Science and Hypothesis:

For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rule.

Poincaré believed that arithmetic is a synthetic science. He argued that Peano's axioms cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is a priori synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were the same as those of Kant (Kolak, 2001). However Poincaré did not share Kantian views in all branches of philosophy and mathematics. For example, in geometry, Poincaré believed that the structure of non-Euclidean space can be known analytically.

See also

Notes

H. E. Ives (1952) wrote that Einstein's derivation was a tautology due to Einstein's use of approximations, and credited Planck (1907) with the first correct derivation of E = mc^2 in Einstein's meaning. In response J. Riseman and I. G. Young (1953) defended Einstein's derivation and physical insight, and Ives (1953) replied. Lorentz (1904) had written x^\prime = k\ell x^{\prime\prime}, t^{\prime} = \ell t/k - k\ell wx^{\prime\prime}/c^2, k^2 = c^2/\left(c^2-w^2\right). Later in the paper he deduced that \ell = 1. Lorentz's x^{\prime\prime} was, in Poincaré's notation, equal to x - wt. Eliminating x^{\prime\prime} and putting \varepsilon = -w/c yields the Lorentz transformations as Poincaré wrote them.

References

  • Bell, Eric Temple, 1986. Men of Mathematics (reissue edition). Touchstone Books. ISBN 0671628186.
  • Belliver, André, 1956. Henri Poincaré ou la vocation souveraine. Paris: Gallimard.
  • Boyer, B. Carl, 1968. A History of Mathematics: Henri Poincaré, John Wiley & Sons.
  • Olivier Darrigol (2004): "The Mystery of the Einstein-Poincaré Connection". Isis: Vol.95, Issue 4; pg. 614, 14 pgs
  • Ewald, William B., ed., 1996. From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford Uni. Press. Contains among others:
  • Grattan-Guinness, Ivor, 2000. ''The Search for Mathematical Roots 1870-1940. Princeton Uni. Press.
  • Gray, Jeremy, 1986. Linear differential equations and group theory from Riemann to Poincaré, Birkhauser
  • Kolak, Daniel, 2001. Lovers of Wisdom, 2nd ed. Wadsworth.
  • Murzi, 1998. "Henri Poincaré".
  • O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland.
  • Peterson, Ivars, 1995. Newton's Clock: Chaos in the Solar System (reissue edition). W H Freeman & Co. ISBN 0716727242.
  • Poincaré, Henri. 1894. "On the nature of mathematical reasoning," 972-81.
  • ________. 1898. "On the foundations of geometry," 982-1011.
  • ________. 1900. "Intuition and Logic in mathematics," 1012-20.
  • ________. 1905-06. "Mathematics and Logic, I-III," 1021-70.
  • ________. 1910. "On transfinite numbers," 1071-74.
  • Sageret, Jules, 1911. Henri Poincaré. Paris: Mercure de France.
  • Toulouse, E.,1910. Henri Poincaré. - (Source biography in French)

References to work on relativity

  • Einstein, A. (1905) "Zur Elektrodynamik Bewegter Körper", Annalen der Physik, 17, 891. English translation
  • Einstein, A. (1915) "Erklärung der Periheldrehung des Merkur aus der allgemainen Relativitätstheorie", Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, 799-801
  • Einstein, A. (1916) "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, 49
  • Giannetto, Enrico (1998) "The Rise of Special Relativity: Henri Poincaré's Works Before Einstein", ''Atti del XVIII congresso di storia della fisica e dell'astronomia
  • Galison, Peter Louis (2003) Einstein's Clocks, Poincaré's Maps: Empires of Time. New York: W.W. Norton. ISBN 0393020010
  • Hasenöhrl, F. (1907) Wien Sitz. CXVI 2a, p.1391
  • Ives, H. E. (1952) "Derivation of the Mass-Energy Relationship", J. Optical Society America, 42, pp. 540-543.
  • Ives, H. E. (1953) "Note on 'Mass-Energy Relationship'", J. Optical Society America, 43, 619.
  • Keswani, G. H. (1965-6) "Origin and Concept of Relativity, Parts I, II, III", Brit. J. Phil. Sci., v15-17.
  • Kragh, Helge. (1999) Quantum Generations: A History of Physics in the Twentieth Century. Princeton, N.J. : Princeton University Press.
  • Langevin, P. (1905) "Sur l'origine des radiations et l'inertie électromagnétique", Journal de Physique Théorique et Appliquée, 4, pp.165-183.
  • Langevin, P. (1914) "Le Physicien" in Henri Poincaré Librairie (Felix Alcan 1914) pp. 115-202.
  • Lewis, G. N. (1908) Philosophical Magazine, XVI, 705
  • Logunov, A, (2005) Book "Henri Poincaré and Relativity Theory", Nauka, Moscow, ISBN 5-02-033964-4
  • Lorentz, H. A. (1899) "Simplified Theory of Electrical and Optical Phenomena in Moving Systems", Proc. Acad. Science Amsterdam, I, 427-43.
  • Lorentz, H. A. (1904) "Electromagnetic Phenomena in a System Moving with Any Velocity Less Than That of Light", Proc. Acad. Science Amsterdam, IV, 669-78.
  • Lorentz, H. A. (1911) Amsterdam Versl. XX, 87
  • Lorentz, H.A. (1921) 1914 manuscripts "Deux memoires de Henri Poincaré," Acta Mathematica 38, p.293.
  • Macrossan, M. N. (1986) Note on Relativity Before Einstein", Brit. J. Phil. Sci., 37, pp.232-34.
  • Planck, M. (1907) Berlin Sitz., 542
  • Planck, M. (1908) Verh. d. Deutsch. Phys. Ges. X, p218, and Phys. ZS, IX, 828
  • Poincaré, H. (1897) "The Relativity of Space", article in English translation
  • Poincaré, H. (1900) "La Theorie de Lorentz et la Principe de Reaction", Archives Neerlandaises, V, 252-78.
  • Poincaré, H. (1904) "La valeur de la Science"
  • Poincaré, H. (1905) la dynamique de l'electron", Comptes Rendues, 140, 1504-8.
  • Poincaré, H. (1906) la dynamique de l'electron", Rendiconti del Circolo matematico di Palermo, t.21, 129-176.
  • Poincaré, H. (1913) Mathematics and Science: Last Essays, Dover 1963 (translated from Dernières Pensées posthumously published by Ernest Flammarion, 1913)
  • Riseman, J. and I. G. Young (1953) "Mass-Energy Relationship", J. Optical Society America, 43, 618.
  • Whittaker, E. T (1953) A History of the Theories of Aether and Electricity: Vol 2 The Modern Theories 1900-1926. Chapter II: The Relativity Theory of Poincaré and Lorentz, Nelson, London.

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1854 births | 1912 deaths | 19th century mathematicians | 20th century mathematicians | French mathematicians | French physicists | Algebraic geometers | Topologists | Geometers | Mathematical analysts | Alumni of the École Polytechnique

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