Henri Poincaré, photograph from the frontispiece of the 1913 edition of "Last Thoughts"
April 29, 1854
|Died||July 17, 1912|
|Field||Mathematician and physicist|
|Institutions||Corps des Mines|
Bureau des Longitudes
|Alma mater||Lycée Nancy|
École des Mines
|Academic advisor||Charles Hermite|
|Notable students||Louis Bachelier|
|Known for||Poincaré conjecture|
|Notable prizes||Matteucci Medal (1905)|
Jules Henri Poincaré (April 29, 1854 – July 17, 1912), generally known as Henri Poincaré, was one of France's greatest mathematicians and theoretical physicists, and a philosopher of science. He is often described as a polymath and as 'The Last Universalist' in mathematics, because he excelled in all fields of the discipline as it existed during his lifetime. He is known for his early formulation of the theory of relativity and for formulating the Poincaré conjecture, one of the most famous problems in mathematics. He also laid the groundwork for chaos theory in the process of attempting to solve the important problem of the motion of three or more bodies acting under mutual gravitation. In addition, he is considered one of the founders of the field of topology.
Early life and education
Poincaré was born on April 29, 1854, into an influential family in the Cité Ducale neighborhood of Nancy, France. His father, Leon Poincaré (1828-1892), was a professor of medicine at the University of Nancy (Sagaret, 1911). His younger sister, Aline, married the spiritual philosopher Emile Boutroux. Another notable member of Jules' family was his cousin Raymond Poincaré, who became the President of France (from 1913 to 1920) and a fellow member of the Académie Française.
During his childhood, he was seriously ill for a time with diphtheria and lost his voice for the better part of a year. He received special instructions from his mother, Eugénie Launois (1830-1897) and excelled in written composition.
In 1862, Henri entered the Lycée in Nancy, now renamed the Lycée Henri Poincaré in his honor. He spent eleven years at the Lycée and during this time proved to be one of the top students in every topic he studied. 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" However, poor eyesight and a tendency toward absentmindedness may have contributed to these difficulties. He graduated from the Lycée in 1871 with a Bachelor's degree in letters and sciences.
During the Franco-Prussian War of 1870, he served alongside his father in the Ambulance Corps, tending to the wounded and learning German.
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 and went on to the École des Mines, continuing to study mathematics in addition to the mining engineering syllabus, and received the degree of ordinary engineer in March of 1879.
As a graduate of the École des Mines, Poincaré 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. Because he rushed into the mine after the accident, he was at first listed among the dead in the accident. In spite of his other activities, Poincare stayed loyal to his mining career, which in later life led to important government appointments.
While undertaking his professional responsibilities, Poincaré prepared for his doctorate in mathematics under the supervision of Hermite. His doctoral thesis was in the field of differential equations. Poincaré devised a new way of studying the properties of these expressions. He not only faced the question of determining the solution of such equations, but also was the first person to study their general geometric properties. He realized that they could be used to model the behavior of multiple bodies in free motion within the Solar System. Poincaré was awarded his doctorate from the University of Paris in 1879.
Start of career in mathematics
Soon after graduation, he was offered a post as junior lecturer in mathematics at Caen University, but he never fully abandoned his mining career to mathematics. 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 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 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.
Gravitation, chaos, and the three-body problem
In 1887, Poincaré won the Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies. Although he did not solve this problem, the insights he offered were striking and original enough for him to merit the prize.
The problem of finding the general solution to the motion of more than two orbiting bodies in the Solar System had eluded mathematicians since Isaac Newton's time. This was known originally as the three-body problem and later, the n-body problem, where n is any number of more than two orbiting bodies. The n-body solution was considered very important and challenging at the close of the nineteenth century. Indeed in 1887, in honor of his 60th birthday, King Oscar II, advised by Gösta Mittag-Leffler, established a prize for anyone who could find the solution to the problem. The announcement was quite specific:
|“||Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series converges uniformly.||”|
In case the problem could not be solved, any other important contribution to classical mechanics would then considered worthy of the prize. Based on this stipulation, Poincaré's contribution was found to merit the prize. 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."
The first version of Poincaré's paper contained a serious error. When he realized this, he used his own money to purchase copies of the work that contained the error to take them out of circulation. The version finally printed contained many important ideas that led to the theory of chaos. The problem as stated originally was finally solved by Karl Sundman for n = 3 in 1912, and it was generalized to the case of n > 3 bodies by Qiudong Wang in the 1990s.
Time and the theory of relativity
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 (Galison 2003). This post led him to consider the question of establishing international time zones and the synchronization of time between bodies in relative motion.
Poincaré's work at the Bureau des Longitudes on establishing international time zones, led him to consider how to synchronize clocks at rest on the Earth—clocks that would be moving at different speeds relative to absolute space (or the "luminiferous aether"). 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. What Lorentz realized is that, to make his equations applicable to a translation of uniform velocity, he had to introduce a different time variable for each reference frame. He called this "local time," given by the following equation:
where the primed variables refer to a reference frame in uniform motion relative to that of the unprimed variables. Lorentz was using it to explain the "failure" of the Michelson-Morley experiment—an experiment that failed to detect motion relative to the aether, the hypothetical medium that was thought to be the carrier of electromagnetic waves.
In The Measure of Time (Poincaré 1898), Poincaré 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. In 1900, Poincaré discussed Lorentz's concept 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.
Thereafter, Poincaré was a constant interpreter (and sometimes friendly critic) of Lorentz's theory. As a philosopher, Poincaré was interested in the "deeper meaning" of the theory. Thus he interpreted Lorentz's theory in terms of the principle of relativity, and in so doing, he came up with many insights that are now associated with the theory of special relativity.
Relationship between mass and energy
In his 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 .
Poincaré considered the recoil of the emitter to be an unresolved feature of Maxwell-Lorentz theory, which he discussed again in "Science and Hypothesis" (1902) and "The Value of Science" (1905). 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 , 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 , and the corresponding mass-energy conservation law, E = mc², that 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, Poincaré explained a mathematical property of the transformations that 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 from what mathematicians call 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 June 5, 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 form2:
He then wrote that in order for the Lorentz transformations to form a group and satisfy the principle of relativity, the arbitrary function must be unity for all (Lorentz had set by a different argument). Poincaré's discovery of the velocity transformations, allowed him to obtain perfect invariance, the final step in the discovery of his theory of 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 is invariant, and he introduced the 4-vector notation for which Hermann Minkowski became known.
Einstein and Poincaré
Albert Einstein's first paper on relativity in 1905 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 in 1906. Although Einstein 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 relativity, but Einstein acknowledged Poincaré's somewhat belatedly in the text of a lecture in 1921 titled Geometrie und Erfahrung. Later, Einstein referred to Poincaré as one of the pioneers of relativity, saying that "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 …"
Einstein formulated a general theory of relativity, which gave an expanded explanation of systems accelerating with respect to one another. The theory relating reference frames in uniform motion with respect to one another then became known as the special theory of relativity.
Father of relativity: Lorentz, Poincaré or Einstein?
Poincaré's work in the development of Special Relativity is well recognized. Most historians, however, stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work. 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.
Poincaré consistently credited Lorentz's achievements, ranking his own contributions as minor. Thus, he wrote (Poincaré 1905):
|“||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 [Lorentz 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. [emphases 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 [the 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 [to detect] motion either in relation to absolute space or even in relation to the aether. [emphasis 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.
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. Some of these arguments involved probability, and Poincare noted that they were improperly applied to the evidence.
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.
In 2004, Claude Allegre, the French Minister of Education, proposed that Poincaré be reburied in the Pantheon in Paris, which is reserved for French citizens deserving of the highest honor.
Poincaré made many contributions to different fields of physics and 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:
- algebraic topology
- the theory of analytic functions of several complex variables
- the theory of abelian functions
- algebraic geometry
- Poincaré was responsible for formulating one of the most famous problems in mathematics. Known as the Poincaré conjecture, it is a problem in topology.
- Poincaré recurrence theorem
- Hyperbolic geometry
- number theory
- the three-body problem
- the theory of diophantine equations
- the theory of electromagnetism
- the special theory of relativity
- In an 1894 paper, he introduced the concept of the fundamental group.
- In the field of differential equations Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the Poincaré sphere and the Poincaré map.
- Poincaré on "everybody's belief" in the Normal Law of Errors,
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.
Poincaré's mental organization interested not only Poincaré himself but also 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.
- 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 visualize 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 consciously worked on another problem.
In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré was the type that started from basic principle each time.
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.)
While a brilliant researcher, Poincaré's abilities to critique work in his field were limited, evidenced in what are considered Poincaré's decision-failures. For example, Poincaré was resistant to contributions from mathematicians like Georg Cantor and underestimated mathematical work that was highly relevant to fields such as economics and finance. In 1900 Poincaré misvalued Louis Bachelier's thesis "The Theory of Speculation," saying: "M. Bachelier has evidenced an original and precise mind [but] the subject is somewhat remote from those our other candidates are in the habit of treating." However, Bachelier's work explained what was then the French government's pricing options on French Bonds and anticipated many of the pricing theories in financial markets used even today.
Poincaré had the opposite philosophical views of Bertrand Russell and Gottlob Frege, who believed that mathematics was 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.
Selected quotations of Poincaré are given below.
"Normal Law of Errors"
- Everybody firmly believes in it because the mathematicians imagine it is a fact of observation, and observers that it is a theory of mathematics.
Quotes from La Science et l'Hypothèse (Of Science and Hypotheses) (1901)
- To doubt everything or to believe everything are two equally convenient solutions; both dispense with the necessity of reflection.
- Science is built up with facts, as a house is with stones. But a collection of facts is no more a science than a heap of stones is a house.
- Sociology is the science with the greatest number of methods and the least results.
Quotes from Valeur de la Science (The Value of Science) (1904)
- It is not nature which imposes [time and space] upon us, it is we who impose them upon nature because we find them convenient.
- If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous.
- Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No, beyond doubt, a reality completely independent of the mind which conceives it, sees or feels it, is an impossibility.
- The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the past centuries.
- Oscar II, King of Sweden's mathematical competition (1887)
- American Philosophical Society (1899)
- Gold Medal of the Royal Astronomical Society of London (1900)
- Matteucci Medal (1905)
- French Academy of Sciences (1906)
- Académie Française (1909)
- Bruce Medal (1911)
Named after him
- The Poincaré group used in physics and mathematics was named after him.
- Poincaré Prize (Mathematical Physics International Prize)
- Annales Henri Poincaré (Scientific Journal)
- Poincaré Seminar (nicknamed "Bourbaphy")
- Poincaré crater (on the Moon)
- Asteroid 2021 Poincaré
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., 1892-1899; English trans., 1967)
- Lessons of Celestial Mechanics. (1905-1910).
In popular writings he helped establish the fundamental popular definitions and perceptions of science by these writings:
- Science and Hypothesis, 1901. (complete text online)
- The Value of Science, 1905. (For more, see the French version.)
- Science and Method, 1908.
- Dernières pensées (Eng., "Last Thoughts"); Edition Ernest Flammarion, Paris, 1913.
- Celestial mechanics
- Fluid mechanics
- Quantum mechanics
- Special relativity
- ↑ Mauro Murzi, Jules Henri Poincaré. The Internet Encyclopedia of Philosophy. Retrieved November 12, 2007.
- ↑ J. John O'Connor et al., 2002. "Jules Henri Poincaré". University of St. Andrews, Scotland. Retrieved November 13, 2007.
- ↑ Michael N. Macrossan, 1986-01-01, A Note on Relativity Before Einstein. British Journal for the Philosophy of Science 37 : 232-234. The University of Queensland, Australia.
- ↑ 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 in Einstein's meaning. In response J. Riseman and I. G. Young (1953) defended Einstein's derivation and physical insight, and Ives (1953) replied.
- ↑ Poincaré à Lorentz. Poincaré's letter to Lorentz. (In French). Retrieved November 12, 2007.
- ↑ Poincaré à Lorentz. Poincaré's letter to Lorentz. (In French). Retrieved November 12, 2007.
- ↑ Sur la Dynamique de l'Électron. Retrieved November 12, 2007.
- ↑ Sur la Dynamique de l'Électron. Retrieved November 12, 2007.
- ↑ Olivier Darrigol, The Mystery of the Einstein-Poincaré Connection. Isis 95(4) (2004): 614-428.
- ↑ Peter Louis Galison. 2003. Einstein's Clocks, Poincaré's Maps: Empires of Time. (New York, NY: W.W. Norton)
- ↑ Kragh 1999
- ↑ E.T. Whittaker. 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. (London, UK: Nelson, 1953)
- ↑ Lorentz, Poincaré et Einstein. lexpress.fr. (In French.) Retrieved November 12, 2007.
- ↑ J. John O'Connor et al., 2002, "Jules Henri Poincaré". University of St. Andrews, Scotland. Retrieved November 13, 2007.
- ↑ André Belliver. 1956. Henri Poincaré ou la vocation souveraine. (Paris, FR: Gallimard)
- ↑ Bernstein, 1996, 199-200
- ↑ Henri Poincaré. Wikiquote. Retrieved November 12, 2007.
- ↑ On the "Normal Law of Errors" in the Introduction to his book Thermodynamique (1892); as told by J.H. Gaddum in Nature 156 (1945): 463-466.
ReferencesISBN links support NWE through referral fees
This article incorporates material from Jules Henri Poincaré on PlanetMath, which is licensed under the GFDL.
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- Poincaré, H. 1905. Sur la dynamique de l'electron. Comptes Rendues. 140:1504-8. Retrieved November 13, 2007.
- Poincaré, H. 1906. Sur la dynamique de l'electron. Rendiconti del Circolo matematico di Palermo. 21:129-176. Retrieved November 13, 2007.
- Poincaré, H. 1913. Mathematics and Science: Last Essays. New York, NY: 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. London, UK: Nelson.
All links retrieved December 15, 2017.
- Works by Henri Poincaré. Project Gutenberg.
- Henri Poincaré at the Mathematics Genealogy Project.
- John J. O'Connor and Edmund F. Robertson. Henri Poincaré at the MacTutor archive.
- Jules Henri Poincaré.
- Jules Henri Poincaré (1854—1912) - Internet Encyclopedia of Philosophy
- Bruce Medal page.
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