Henri Poincaré

Henri Poincaré
File:Poincare jh.jpg Henri Poincaré, photograph from the frontispiece of the 1913 edition of "Last Thoughts"
Born	(1854-04-29)April 29, 1854 Nancy, France
Died	July 17, 1912(1912-07-17) (aged 58) Paris, France
Nationality	French
Alma mater	Lycée Nancy École Polytechnique École des Mines
Known for	Poincaré conjecture Three-body problem Topology Special relativity
Awards	Matteucci Medal (1905)
Scientific career
Fields	Mathematician and physicist
Institutions	Corps des Mines Caen University La Sorbonne Bureau des Longitudes
Doctoral advisor	Charles Hermite
Doctoral students	Louis Bachelier

Jules Henri Poincaré (April 29, 1854 – July 17, 1912) (IPA: [pwɛ̃kaʀe]) was one of France's greatest mathematicians and theoretical physicists, and a philosopher of science. Poincaré is often described as a polymath, and in mathematics as The Last Universalist, since he excelled in all fields of the discipline as it existed during his lifetime.

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. He is considered to be one of the founders of the field of topology.

Poincaré introduced the modern principle of relativity and was the first to present the Lorentz transformations in their modern symmetrical form. 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 formulation of the theory of special relativity.

The Poincaré group used in physics and mathematics was named after him.

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, 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).

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. He excelled in written composition. 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 Bachelor's 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 equations. 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.

The young Henri Poincaré

Career

Soon after, 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 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 #The three-body problem section below)

In 1893 Poincaré joined the French Bureau des Longitudes, which engaged him in the synchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalisation 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 synchronisation of time between bodies in relative motion. (See #Work on 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, in Paris. He was 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 Panthéon in Paris, which is reserved for French citizens only of the highest honour.

hi mom!

Work

Poincaré made many contributions to different fields of pure 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 populariser 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 (see normal distribution for an account of that "law")

The three-body problem

The problem of finding the general solution to the motion of more than two orbiting bodies in the solar system had eluded mathematicians since 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 honour of his 60th birthday, Oscar II, King of Sweden, 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 be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. 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 his contribution even contained a serious error; for details see the article by Diacu). The version finally printed contained many important ideas which lead to the theory of chaos. The problem as stated originally was finally solved by Karl F. Sundman for n = 3 in 1912 and was generalised to the case of n > 3 bodies by Qiudong Wang in the 1990s.

Work on relativity

Main article: Lorentz ether theory

Local time

Poincaré's work at the Bureau des Longitudes 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 synchronised. 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 synchronised 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 of the principle of relativity and in so doing he came up with many insights that are now associated with special relativity.

Marie Curie and Poincaré talk at the 1911 Solvay Conference.

Inertia of energy

In 1900 Henri Poincaré studied the conflict between the action/reaction principle and Lorentz ether theory. He tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. The electromagnetic field energy behaves like a fictitious fluid ("fluide fictif") with a mass density of E/c² and velocity c. If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible--it's neither created or destroyed--- then the motion of the center of mass frame remains uniform.

But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions.

But Poincare's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. In the framework of Lorentz ether theory Poincare performed a Lorentz boost to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow a perpetuum mobile, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold.

Poincaré came back to this topic in "Science and Hypothesis" (1902) and "The Value of Science" (1905). This time he rejected the possibility that energy carries mass: "... [the recoil] is contrary to the principle of Newton since our projectile here has no mass, it is not matter, it is energy". He also 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 $m=E/c^{2}$ that resolved Poincare's paradox. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. Einstein noted in 1906 that Poincaré's solution to the center of mass problem and his own were mathematically equivalent.

Lorentz transformation

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 gave his own reason why Lorentz's time dilation factor was indeed correct after all: it was necessary to make the Lorentz transformation form a group and gave what is now known as the relativistic velocity-addition law. 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(refactored from LT1904):

$x^{\prime }=k\ell \left(x+\varepsilon t\right),~$$t^{\prime }=k\ell \left(t+\varepsilon x\right),~$$y^{\prime }=\ell y,~$ $z^{\prime }=\ell z,~$$k=1/{\sqrt {1-\varepsilon ^{2}}}.$

and showed that the arbitrary function $\ell \left(\varepsilon \right)$ must be unity for all $\varepsilon$ (Lorentz had set $\ell =1$ by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination $x^{2}+y^{2}+z^{2}-c^{2}t^{2}$ is invariant, and he introduced the 4-vector notation for which Hermann Minkowski became known.

Einstein's first paper on relativity was published three months after Poincaré's short paper, but before Poincaré's longer version. It relied on the principle of relativity to derive the Lorentz transformations and used the same clock synchronisation procedure that Poincaré (1900) had described, but was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on Special Relativity. Einstein acknowledged Poincaré in the text of a lecture in 1921 called Geometrie und Erfahrung in connection with the Non-Euclidean geometry, but not in connection with special relativity. A few years before his death Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognised that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ..."

Assessments

Poincaré's work in the development of special relativity is well recognised (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 postulate 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." (Poincaré 1905) [emphasis 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 after Poincaré's 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' characterisation

Poincaré's mental organisation was not only interesting to Poincaré himself 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.

• 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 consciously worked on another problem.

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

His method of thinking is well summarised 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)

Shortcomings

Although a brilliant researcher, Poincaré was resistant to contributions from mathematicians like Georg Cantor and saw mathematical work in economics and finance as peripheral. In 1900 Poincaré commented on 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." (Bernstein, 1996, p.199-200) 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.

Honours

Awards

• Oscar II, King of Sweden's mathematical competition (1887)
• American Philosophical Society 1899
• Gold Medal of the Royal Astronomical Society of London (1900)
• Bolyai prize in 1905
• Matteucci Medal 1905
• French Academy of Sciences 1906
• Académie Française 1909
• Bruce Medal (1911)

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

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. (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.

Philosophy

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.

See also

• Poincaré–Bendixson theorem
• Poincaré–Birkhoff–Witt theorem
• Poincaré half-plane model
• Poincaré symmetry
• Poincaré–Hopf theorem
• Poincaré metric
• Poincaré duality
• Poincaré group
• Poincaré map
• Institut Henri Poincaré, Paris
• History of special relativity
• Relativity priority dispute
• Poincaré Conjecture

Notes

Template:Ent 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. Template:EntLorentz (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

Jules Henri Poincaré at PlanetMath.

General 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.
• Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199-200). John Wiley & Sons.
• 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

Galison, P. (2003) Einstein's Clocks, Poincare's Maps.

• 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) "A 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. (1898) "La mesure du Temps", reprinted in "La valeur de la science", Ernest Flammarion, Paris.
• Poincaré, H. (1900) "La Theorie de Lorentz et la Principe de Reaction", Archives Neerlandaises, V, 252-78.
• Poincaré, H. (1905) "La valeur de la Science"
• Poincaré, H. (1904) "L'état actuel et l'avenir de la physique mathématique", Bulletin des sciences mathématiques 28(1904), 302-324 (Congress of Arts and Science, St. Louis, September 24, 1904)
• Poincaré, H. (1905) "Sur la dynamique de l'electron", Comptes Rendues, 140, 1504-8.
• Poincaré, H. (1906) "Sur 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.

External links

• Works by Henri Poincaré at Project Gutenberg
• Henri Poincaré at the Mathematics Genealogy Project
• O'Connor, John J.; Robertson, Edmund F., "Henri Poincaré", MacTutor History of Mathematics Archive, University of St Andrews
• "The Sciences and the Humanities" by Heri Poincaré (complete text online - in English)
• "Science and Hypothesis" by Heri Poincaré (complete text online - in English)
• A review of Poincaré's life and mathematical achievements - from the University of Tennessee at Martin, USA.
• A timeline of Poincaré's life University of Nancy (in French).
• Bruce Medal page
• Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions
• BBC In Our Time -- discussion of the Poincaré conjecture, November 2, 2006, hosted by Melvynn Bragg See Internet Archive

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