Henri Poincaré

File:Poincare jh.jpg

Henri Poincaré, photograph from the frontispiece of the 1913 edition of "Last Thoughts"

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. Poincaré (pronounced (IPA) BrE: [ˈpwæŋ kæ reɪ]; [1]) 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 theoretical 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 to 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.

The young Henri Poincaré

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 a 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. 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. [2]

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 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 still not fully resolved today.
• 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

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

Marie Curie and Poincaré talk at the 1911 Solvay Conference.

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. 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[3]. 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. He showed that according to the Maxwell-Lorentz theory the stream of radiation could be considered as 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 $m=E/c^{2}$, or $E=mc^{2}$. 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(refactored from Ives).

In 1905 Poincaré wrote to Lorentz [4], concerning Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance". In this letter he pointed out an error Lorentz (1904) had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space. In a second letter to Lorentz[5] Poincaré agreed that Lorentz had been correct about the time dilation factor. He explained the group property of the transformations, which Lorentz had not noticed, and gave his own reason why Lorentz's time dilation factor was 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(refactored from LT1904):

$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 which 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 with the speed of light normalized to 1, the combination $x^{2}+y^{2}+z^{2}-t^{2}$ 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. There was however, an implied reference in the sentence “the same laws of electrodynamics and optics will be valid for all frames of reference for which the mechanics hold good, as has already been shown to the first order of small quantities” [emphasis added]. This is almost certainly a reference to Lorentz (’’the’’ expert on electrodynamics at the time) whose previous results might have been so well known (either from the originals or the many popular commentaries by Poincaré) as to not need a citation.

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 becoming more recognized publicly, though most historians 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 (Whittaker 1953); most scholars distinguish Poincaré's and Lorentz's "relativities" as being different than Einstein's (see, i.e., Galison 2003 and Kragh 1999). 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 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" (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 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.

Work on gravity

In his 1905 paper Poincaré noted that Lorentz (1904) had "[supposed] that all forces, whatever their origin, should be affected by translation in the same manner as the electrodynamic force." He wanted to find what modifications Lorentz's hypothesis would require in the laws of gravitation. Poincaré then went on to re-examine, in the light of Lorentz's mechanics, Laplace's conjecture that the propagation of gravity is not instantaneous but "is that of light", something which Laplace had shown was contrary to observations if it were applied to the (unmodified) Newtonian force of gravity. It followed from Laplace's conjecture that "the position ... of the attracting body ... will be the position ... at the instant when the gravitational wave just leaves the body; ... the position ... of the body which is attracted ... will be the position when the wave emitted by the other body has reached [it]" (Poincaré 1905). Poincaré found using Lorentz's transformations, and Lorentz's requirement that the force must transform in the same way as an electromagnetic force, that the corrected gravity force had a component parallel to the radial vector and another parallel to the velocity of the orbiting body. Poincare expounded on this in the expanded version of his paper (Poincaré 1906).

In one of his popular science books, "Science and Method" (1908, Book III, Chapter III, "The New Mechanics and Astronomy"), Poincaré said there were many possible ways to construct a theory of gravity which satisfied the Principle of Relativity, and in which gravity propagated at the speed of light. He described one such solution, an electromagnetic theory of gravity, which he attributed to Lorentz. In this theory, an orbiting mass emitted electromagnetic radiation. He said that Lorentz's theory gave the correct direction of the precession of the perihelion of Mercury, but the value was 7" (per century) as opposed to the observed value of 43". Poincaré lectured widely on these concepts, in Paris, Lille, and Göttingen.

In a memoir dedicated to Poincaré, Langevin (1914) wrote that Poincaré had derived numerous covariant solutions for gravitation, where gravity propagates at the speed of light as Laplace had suggested, and which gave a precession of the perihelion of Mercury of 6".

Starting in 1907, Albert Einstein worked on a different approach to reconciling gravity and special relativity. This culminated in general relativity (1915), in which all forms of mass and energy influence and respond to the curvature of spacetime, and in which test particles of all types follow geodesics when not subject to any forces (gravity being identified with inertial motion, rather than being a force). David Hilbert, following discussions with Einstein, completed the final field equations almost simultaneously with Einstein. General Relativity replaced all previous theories of gravity because of the accuracy and scope of its agreement with experimental data.

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 am and noon then again from 5 pm to 7 pm. 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

• Gold Medal of the Royal Astronomical Society of London (1900)
• Bruce Medal (1911)

Named after him

• 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.
• 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 can not be a 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

• Albert Einstein
• History of special relativity
• Relativity priority disputes

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 $x^{\prime }=k\ell x^{\prime \prime },$ $t^{\prime }=\ell t/k-k\ell wx^{\prime \prime }/c^{2},$ $k=c^{2}/\left(c^{2}-w^{2}\right)$. Later in the paper he deduced that $\ell =1.$ Lorentz's $x^{\prime \prime }$ was, in Poincare's notation, equal to $x-wt.$ Eliminating $x^{\prime \prime }$ and putting $\varepsilon =-w/c$ yields the Lorentz transformations as Poincare wrote them. Thus, Poincaré was the first to rearrange Lorentz's equations into their modern form.

References

Jules Henri Poincaré at PlanetMath.

• 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.
• 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.
• 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. (1916) "Die Grundlage der allgemeinen Relativitätstheorie", Annalen der Physik, 49
• 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. O. S. A., 42, pp. 540-543.
• Ives, H. E. (1953) "Note on 'Mass-Energy Relationship'", J. O. S. A., 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 Poincare Librairie (Felix Alcan 1914) pp. 115-202.
• Lewis, G. N. (1908) Philosophical Magazine, XVI, 705
• 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. (1914) "Deux Memoires de Henri Poincaré," Acta Mathematica 38: 293, p.1921.
• 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. (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) "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. O. S. A., 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
• 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).
• Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions