# Henri Poincaré

Henri Poincaré
Henri Poincaré's signature

Jules Henri Poincaré [ pwɛ̃kaˈʀe ] (born April 29, 1854 in Nancy , † July 17, 1912 in Paris ) was an important French mathematician , theoretical physicist , theoretical astronomer and philosopher . From 1880 until his death and afterwards he was regarded as one of the most important mathematicians, in which David Hilbert competed with him only in Germany , and also as a leading theoretical physicist and astronomer.

## Family and childhood

Poincaré was born the son of Léon Poincaré (1828-1892), a professor of medicine at the University of Nancy, and his wife Eugénie Launois (1830-1897), who came from a wealthy family in Arrancy-sur-Crusne on the border with Luxembourg came from where the family had a large estate (today Château Reny) and Poincaré often spent his holidays as a child with many relatives. The center of the household there was the grandmother Lanois, who had a reputation for being unbeatable at arithmetic and card games. His paternal grandfather, Jacques-Nicolas (1794-1865), came from Lorraine, had a pharmacy in Nancy and a large, still existing town house on the corner of today's Rue de Guise (Hôtel de Martigny), where Henri Poincaré grew up. The grandfather also dealt with botany, which he brought closer to his grandchildren. Poincaré had a sister Aline (1856–1919), with whom he was closely associated throughout his life. She later married the philosopher Émile Boutroux . The Poincaré family (with different spellings, Poincaré preferred the pronunciation after the form of the name Pontcarré) was widespread and influential in Lorraine, an older cousin of Poincaré was the future French President Raymond Poincaré , and he was the cousin of Lucien, who was also a physicist and inspector general of the secondary schools Poincaré (1862-1920). Both were sons of Poincaré's uncle Antoni Poincaré (1829-1911), who was a graduate of the Ecole Polytechnique and civil engineer (inspector of the bridges in Bar-le-Duc). The chemist Albin Haller , a cousin and close friend of Poincaré, came from the maternal family . The Poincaré family was Catholic.

Birthplace in Nancy, Hôtel de Martigny

In 1859 he fell ill with life-threatening diphtheria . He was paralyzed for some time after that and had problems speaking for a longer period of time. Poincaré was first raised privately and went to school from 1862. Poincaré was an excellent student and had a photographic memory. He only needed to have read a book once to be able to reproduce the contents exactly to the page number. In 1865 he traveled with his parents to the Vosges, to Cologne and Frankfurt, to the 1867 World's Fair in Paris, in 1869 to London and the Isle of Wight. At the age of 14, the teachers noticed his extraordinary mathematical talent; but he himself did not yet know which way to go. He had wide-ranging interests, not just in the natural sciences. As a schoolboy he had written a play about the Maid of Orleans , which he performed with his sister and cousins, he also learned the piano (with little success) and was an enthusiastic dancer.

During the Franco-Prussian War, his father ran an ambulance in Nancy, in which Henri Poincaré assisted him. During the German occupation, a senior officer was quartered in their house, which Poincaré used to improve his knowledge of German, which he also used to better inform himself politically. The war brought much suffering and bitterness to the Poincaré family, especially the Arrancy branch. For a long time, people in Lorraine feared annexation by the German Empire. Among the refugees who sought refuge in Nancy was the Alsatian Paul Appell , who attended school with Poincaré and became a close friend. During the war he prepared himself at the Lyceum in Nancy for the completion of the Bachelor of Arts, which he graduated with good grades (in a Latin essay even with very good) in August 1871. An article by Poincaré on the resurgence of states, a topic that occupied many French people at the time after the defeat in the war of 1870/71, was well received.

He narrowly passed his bachelor's degree in science and mathematics in November because he had not prepared himself sufficiently. Then he began to prepare for the entrance examination (Concours général) of the elite universities in Paris, for which he seriously began to study mathematics. He learned from the analysis textbooks of Jean Duhamel and those of geometry from Michel Chasles . In 1872 he drew attention to himself in the preparatory class when he was able to solve a difficult math problem from the entrance exam at the elite university École polytechnique that year. The result of the exams for the École normal supérieure was not good (he finished fifth, his friend Appell third), while those for the Ecole Polytechnique went very well in both written and oral terms, and he received first place. On the day of his written exam, there was cheering in Nancy because from then on it was known for sure that no annexation would take place.

Poincaré was married to Louise Poulain d'Andecy (1857-1934) since April 20, 1881, with whom he had three daughters and a son.

## Studies and time as a mining engineer

Magny-Danigon coal mine in the background

From 1873 Poincaré studied mathematics at the École polytechnique , where Charles Hermite , Edmond Laguerre , Pierre-Ossian Bonnet and Georges Henri Halphen , Amédée Mannheim in descriptive geometry , Jean Résal in mechanics , Edmond Frémy in chemistry and Alfred in physics Cornu were among his teachers. He continued to achieve good grades (except in descriptive geometry, as he was poorly drawn) and graduated in 1875 as the second best. While still a student, he published his first scientific essay on geometry in 1874. He continued his studies at the École des Mines . The director, a distant relative, insisted that he do not study math while studying. But Poincaré had the support of Bonnet and Jean-Claude Bouquet , so that in 1876 he also passed the mathematics exams at the Sorbonne . While studying at the mining school, he visited mines and metallurgical plants in Austria and Hungary in 1876 and in Sweden and Norway in 1878. In the 1878 exam he came third and from March 1879 worked initially as a mining engineer in Vesoul not far from Nancy. The activity was dangerous, and Poincaré investigated with criminal precision the causes of a mining accident in Magny-Danigon ( Ronchamp coal mines ) on September 1, 1879, in which over twenty miners died. Poincaré examined the pit while the rescue work was still going on. He concluded that a worker accidentally damaged a miner's lamp with a pick, which later caused the mine gas to explode.

## University professor for mathematics, astronomy and physics

His time as an active mining engineer was short. Formally he remained a member of the Corps des Mines, became chief engineer in 1893 and even became inspector general in 1910, which was probably only an honorary title. In December 1879 he became a lecturer in mathematics at the University of Caen (then Faculté des Sciences). In 1878 he submitted his dissertation to the Sorbonne on a topic from the theory of partial differential equations, which was based on the work of Charles Briot and Jean-Claude Bouquet. Reviewers were Laguerre, Bonnet and Darboux, and the doctorate took place in 1879. The work contained a lot of new material, which he later in a large essay on the qualitative theory of differential equations of 1881 and on the Fuchsian functions mentioned by him (to the annoyance of Felix Klein) expanded automorphic functions.

Just two years later, in 1881, Poincaré was appointed Maître de conférences for mathematical physics at the Sorbonne in Paris. In addition, he was a tutor at the Ecole Polytechnique from 1883 to 1897. In March 1885 he became professor of mechanics at the Sorbonne, which also included experimental lectures, which he was less interested in since he was not very skilled at demonstrating the experiments. In 1886, he succeeded Gabriel Lippmann as professor of mathematical physics and probability (Lippmann himself switched to experimental physics). He changed the lecture topic annually, mostly according to his current research direction, and many of the lectures were edited by his students. His students included René Baire , Émile Borel , Louis Bachelier , Mihailo Petrović , Dimitrie Pompeiu and Jules Drach . In 1896 he succeeded Félix Tisserand as professor of mathematical astronomy and celestial mechanics. He held the chair until his death in 1912.

In addition to his professorship at the Sorbonne, from 1904 he was professor of general astronomy at the Ecole Polytechnique. He was a member from 1893 and from 1899 head of the Bureau des Longitudes in Paris.

He had particularly close contact with Gösta Mittag-Leffler , with whom he exchanged letters from 1881 to 1911. Mittag-Leffler was editor of Acta Mathematica (in which Poincaré published a lot), had good relationships with both German and French mathematicians and also mediated the exchange of knowledge between the two countries. They first met in 1882 when Mittag-Leffler was on his honeymoon in Paris, and the wives got along well too. Poincaré visited Mittag-Leffler several times in Sweden (so in 1905). Mittag-Leffler supported Poincaré in receiving the prize publication from 1889 (see below) and later tried to secure Poincaré the Nobel Prize; At first, however, they were dismissive of theorists.

Poincaré mainly worked alone and had relatively few research students, of whom he also made high demands. He could become absent-minded in the middle of a conversation or at a party, pondering math problems, and a conversation with him could be erratic. Even as a student, he often only spoke briefly, made little notes in the lectures and, as in studying literature, preferred to reconstruct the results himself. Most of the time he thought about a problem in his head before writing it down (often having multiple problems preoccupying him) and didn't like the tedious proofreading of an essay. If, in his opinion, he came to a conceptual solution, he often did not bother to work out details, but rather impatiently moved on to the next problem. He was not resentful in conflicts and generally well-meaning, but was able to consistently represent his point of view, as the correspondence with Felix Klein shows, in which he wanted to enforce his view of the naming of mathematical objects. He was a patriot on political issues, but did not subscribe to any political party, took an independent stand and campaigned for tolerance and against prejudice, for example in a speech three weeks before his death to the French Society for Moral Education in which he spoke out against hatred started between social groups. He could be ironic, but not on scientific matters.

His publishing activity includes more than 30 books and many scientific papers. He also published popular science articles that were collected in several volumes. With Darboux and Appell, Poincaré was a member of a scientific commission of the Academie des Sciences, which assessed the evidence in the Dreyfuss Affair from a scientific point of view (in particular the handwriting analysis by Alphonse Bertillon , which Poincaré described as unscientific).

## death

Family tomb of the Poincarés in Paris

Poincaré died after prostate surgery, which initially appeared successful; however, a week later he died of an embolism . His family grave can be found in the Cimetière Montparnasse .

## plant

Poincaré's work is characterized by diversity and high originality; his extraordinary mathematical talent was characterized by a high level of intuition . In the mathematical field he developed the theory of automorphic functions, the qualitative theory of differential equations and is considered the founder of algebraic topology. Other areas of his work in pure mathematics were algebraic geometry and number theory. The Applied Mathematics benefited from Poincaré's ideas. In the field of physics, his contributions range from optics to electricity, from quantum to potential theory, from thermodynamics to special relativity , which he co-founded. In the field of epistemology (philosophy) Poincaré u. a. with his work Science and Hypothesis made significant contributions to the understanding of the relative validity of theories. In his book Poincaré presents various geometric systems, all of which are logically coherent but contradict one another. Which of these apply is not decided by mathematics, but by the natural sciences.

### mathematics

#### topology

Poincaré is considered to be the founder of algebraic topology . He introduced the concept of the fundamental group and further developed the concept of homology contained in Enrico Betti's work (although his methodology was primarily of a combinatorial nature and the algebraic perspective was poorly developed). He gave a definition of the manifold (but only embedded in a Euclidean space) and formulated the Poincaré duality for it . For an n-dimensional compact , oriented manifold, this means that the i-th homology group is isomorphic to the (ni) -th cohomology . Just as he did not rigorously formulate most of his topological terms and results, he did not rigorously prove them either.

His algebraic-topological work also includes the Poincaré conjecture , which was only proven in 2002 by Grigori Perelman for three dimensions (in the higher-dimensional cases it had already been proven). His work on differential forms is also important . Poincaré was the first to recognize that they can be used to define the De Rham cohomology , which under certain circumstances is isomorphic to the singular, but he was unable to prove this. His oeuvre also contains approaches to Morse theory and symplectic geometry .

In total, his topological work comprises 13 specialist articles , the most important of which is the Analysis Situs published in 1895 and its complements.

#### n -body problem

On the occasion of his 60th birthday, the Swedish King Oskar II , on the advice of the mathematician Magnus Gösta Mittag-Leffler , announced a prize that consisted of four individual questions. The first question was about the n -body problem . It was hoped that answering the question would provide insights into the stability of the solar system . This problem was considered so difficult that other significant results in celestial mechanics were accepted. The award committee consisted of Gösta Mittag-Leffler, the editor of Acta Matematica, where the competition was published, Charles Hermite and Karl Weierstrass . The second problem involved a detailed analysis of Fuchs's theory of differential equations , the third required studies of nonlinear first-order differential equations considered by Charles Auguste Briot and Jean-Claude Bouquet , and the last, finally, the study of such algebraic relationships of the Fuchsian function, the had the same automorphic group.

Although Poincaré had already made significant contributions to the theory of Fuchs' differential equations, he decided to investigate the first question. The n-body problem was posed as follows (the formulation came from Weierstrass):

“For a given system of n mutually attracted particles that follow Newton's laws of motion, assuming that there is no collision of two, a general solution is to be found in the form of a power series in the time and space coordinates that is common to all values ​​of the Time and space coordinates converge uniformly. "

The hope that the problem could be solved was further justified by the fact that shortly before his death Peter Gustav Lejeune Dirichlet had informed a mathematician friend that he had found a new method of integrating the differential equations of mechanics, with which he had also proven the stability of the planetary system . If this proves to be too difficult, another contribution to the mechanics could also be awarded. In 1885 Leopold Kronecker , who was not a friend of Weierstrass, criticized the awarding of the prize: The question about Fuchs' differential equations was, as he himself showed, insoluble; and the mathematician who was not mentioned in the prize question and whom Dirichlet allegedly confided in this would have been himself, and the quotation was wrong. Kronecker temporarily threatened Mittag-Leffler to go public, but then did not pursue this any further.

Contributions had to be received before June 1, 1888. The award winner's contribution should be published in the Acta . In the end, twelve contributions were received, five on the first problem and one on the third; the remaining six were devoted to other questions of celestial mechanics. Poincaré's contribution, which was unusually long at 158 ​​pages, did not quite meet the prescribed formalities, but was nevertheless accepted.

##### Complications in awarding the prizes

The award committee quickly realized that only three of the twelve entries were worthy of the award. That of Poincaré, that of Paul Appell , like Poincaré a former student of Hermite, as well as a contribution from Heidelberg. In his contribution Poincaré had concentrated on the investigation of the restricted three-body problem , in which there are two bodies of great mass and one body so small that it does not affect the movements of the other two. He simplified the problem even further to motion in a plane and, in parts of his investigation, to the fact that the two large masses move on a circular path around each other. In this form the problem was reduced to the treatment of a system of four ordinary differential equations of the first order in two variables with periodic coefficients. The methods developed by Poincaré could, however, be used far beyond that.

Although the committee was well aware of the quality of Poincaré's contribution, it had considerable difficulty understanding all the details. Hermite expressed this frankly in a letter to Mittag-Leffler:

“One must admit that in this work, as in his other investigations, Poincaré shows the way and provides ideas, but that he leaves it to others to fill in the gaps and thus to complete the work. Picard often asked him for explanations and explanations for his work in Comptes rendu without getting any answer, except, "That is evident ..." So he appears like a prophet for whom the truth is obvious, but only for him . "

Poincaré first looked at formal solutions in terms of trigonometric series and claimed that they were divergent . Then he used his geometric theory of differential equations, which he had developed in the Journal de Mathématique in the years 1881–1886 , and claimed to be able to prove the stability of the restricted three-body problem with it. This was followed by the introduction of integral invariants, with which he believed he had found a general theory of periodic solutions. In addition, the work contained a theorem about the non-existence of certain algebraic first integrals (conserved quantities) of the three-body problem, this was a generalization of Bruns' theorem .

Since Weierstraß himself worked on the solution of the three-body problem in the form of a convergent series, he was particularly interested in Poincaré's assertion about their divergence. Since he was not convinced by Poincaré's statements on this point, a lively correspondence ensued. Mittag-Leffler in particular sought direct contact with Poincaré before the official award ceremony, which was only possible at the expense of his impartiality. Poincaré wrote a series of nine commentaries (which were later included in the final printed version). The first of these comments dealt with the divergence of general series of disorders; Poincaré argued that these series were divergent, otherwise the problem would be integrable. But that would contradict the fact that, as Poincaré showed, the first integrals of the problem are not algebraic integrals. This reasoning was wrong, however, as the later work of Karl Sundman and Qiudong Wang showed.

Nevertheless, the committee realized that Poincaré's work was to be honored with the prize. Weierstrass took on the task of writing a report on Poincaré's work. Due to Weierstraß's poor health, however, its production was delayed. It was never published.

The king announced the award of the prize to Poincaré on his birthday, January 21, 1889. The appeal, whose prize book dealt with Abelian functions, received an honorable mention. The French press celebrated this as a victory for French science, and Poincaré and Appell became Knights of the Legion of Honor.

##### The priority question

After the award was announced, a priority dispute arose with the astronomer Hugo Gyldén , who had also carried out investigations into the restricted three-body problem with the help of series of disturbances. Gylden asserted (without ever proving it) not only that these series converged, but also that from this convergence the stability of the constrained three-body problem should follow. Mittag-Leffler, who defended Poincaré (and thus also the decision of the award committee), in turn asked Poincaré for arguments. The dispute dragged on and only subsided after Poincaré's final version was published.

##### The bug in the first version: homoclinic points

The publication of Poincaré's contribution, which eventually won the prize, was delayed until November 1890. When it was published, it differed significantly from the original work.

In his simplification of the three-body problem, Poincaré considered a system of differential equations with equilibrium solutions and periodic solutions, whereby some solutions, for example, approached a periodic solution asymptotically (they formed the stable manifold), and others that were initially close to the periodic solution were rejected (they formed the unstable manifold). He called stable and unstable manifolds asymptotic surfaces . On these he used convergent series solutions to trace the solutions for times on the stable and unstable manifolds, and he first suspected that stable and unstable manifolds combine to form a manifold that is now called homoclinic , a word that Poincaré first used in his later lectures on celestial mechanics. In this case it would have a connected invariant manifold and thus an integration invariant for the solution of the system, which would have ensured the stability of this model of the three-body problem. ${\ displaystyle t \ to \ pm \ infty}$

In July 1889, Lars Phragmén , later co-editor of Acta , asked Poincaré to explain some unclear points. In his reply to Phragmén, Poincaré discovered a major flaw in his work, which he immediately reported to Mittag-Leffler. Poincaré had overlooked that the intersection of homoclinic manifolds or orbits that occurs in the event of disturbances can also be transversal . The dynamics became very complicated ( homoclinic network , English: homoclinic tangle), and the stability of the system was no longer guaranteed. In fact, this was the first example of chaos in a dynamic system. Poincaré was so shocked by his mistake that he allowed Mittag-Leffler to withdraw the award. Mittag-Leffler was still convinced of the quality of Poincaré's work, but was also very concerned about the reputation of the price and that of the Acta, as well as his own reputation.

##### The final version of Poincaré's work

The final version of his treatise appeared in number 13 of the Acta in December 1890. In this version there is no longer any discussion of stability. The emphasis is rather on the results of the periodic, the asymptotic and the doubly asymptotic solutions, and also on the results of the nonexistence of the first integrals and the divergence of the Lindstedt series. Probably the most interesting change concerns the asymptotic surfaces. Poincaré shows that they cannot be closed, but that they intersect infinitely often in a complicated way. This was the foretaste of the chaotic behavior of the solutions.

The full scope of the work was only understood by a few (including the young Hermann Minkowski ). The criticism of astronomers like Hugo Gyldén and Anders Lindstedt mainly related to astronomical series developments.

##### The aftermath

Two years later Poincaré published his monumental work Les méthodes nouvelles de la mécanique céleste. This work is largely an elaboration of his price script. In the last chapter of the third part he considers doubly asymptotic solutions. As explained above, he considered non-periodic solutions, which however asymptotically approached a periodic solution (asymptotic surfaces, from today's perspective homoclinic orbits to a hyperbolic fixed point with stable and unstable manifolds). The correspondingly complicated behavior of the paths arises when the stable and unstable curves intersect transversely at a homoclinic point. The behavior of the river in the vicinity of such homoclinic orbits was examined by George David Birkhoff in 1937 and explained by Stephen Smale in 1965 by comparison with his horseshoe illustrations .

The question of stability was partially answered by the KAM theorem . Here it was proven that the tori of integrable systems (like the two-body problem) are stable to almost all perturbations. This almost means that disturbances with commensurable frequencies lead to instabilities, as Poincaré described them, while in the case of disturbances with incommensurable frequencies, invariant tori exist in the phase space. The existence of such stable tori also seems to have been foreseen by Poincaré.

The question of the existence of solutions that can be represented by convergent power series (the original question of price) was proved for the case n = 3 by Karl Sundman in 1912 and for n> 3 by Qiudong Wang in 1990. However, the ranks converged so slowly that they were practically useless. Wang described his solution, which formally solved the price problem, as tricky but surprisingly simple and thus the original formulation of the price as the actual mathematical error, and called the awarding of the prize to Poincaré more than justified due to the other mathematical content.

#### Other contributions to mathematics

In mathematics, he has also made important contributions to the qualitative theory of differential equations , the theory of analytical functions in several complex variables, the theory of automorphic forms, hyperbolic and algebraic geometry and number theory (which, however, is only a selection of his contributions).

His work on the qualitative theory of ordinary differential equation of 1881/82 was a scientific milestone; in it he introduced, among other things, boundary cycles (and proved the Poincaré-Bendixson theorem ), classified fixed points and introduced the Poincaré mapping . When examining dynamic or discrete systems for fixed points and stability, the Poincaré mapping proves to be very useful and helpful. In complex dynamics , the Poincaré functional equation, which is closely related to Schröder's functional equation and introduced in 1890, is significant, the solutions of which are also called Poincaré functions.

In 1901, in number theory, he investigated the construction of rational points on elliptic curves using the tangent-secant method (which goes back to Isaac Newton ). In doing so, he provided essential impulses for this field of number theory research (arithmetic geometry), which is still very topical today; in particular, he also recognized the group structure of the points created in this way.

In function theory, he proved - in intense competition and in correspondence (from 1881) with Felix Klein , who was already older at the time , but who had to give up the competition due to a mental breakdown due to overwork - a uniformization theorem for Riemann surfaces using the theory of the automorphic functions (which he constructed with his Poincaré series). This generalizes the Riemann mapping theorem to Riemannian surfaces of the higher sex. The work of Poincaré and Klein from the 1880s did not satisfy David Hilbert, for example, who included the problem of uniformity in 1900 as the 22nd problem in his list of mathematical problems - Poincaré himself and Paul Koebe then gave more satisfactory solutions in 1907 . Poincaré's contribution to the theory of automorphic functions at the end of the 19th century was considered by many contemporaries to be his most important contribution to mathematics. He mainly published on this from 1881 to 1884, initially inspired by an essay by Lazarus Fuchs and an exchange of letters with Fuchs (who was then teaching in Heidelberg). In the 1860s, Fuchs investigated solutions to a differential equation of the 2nd order in complexes with only regular singularities (see Fuchs 'differential equation , an example of this is the hypergeometric differential equation ), and Poincaré expressed them using what he called Fuchs' (automorphic) functions. He later recalled a crucial flash of inspiration that came to him in Caen during a geological excursion while boarding a horse-drawn bus: the transformations he used to define Fuchsian functions were the same as those in non-Euclidean (hyperbolic) geometry (Poincarésches Half-plane model of hyperbolic geometry). Incidentally, the name Fuchs' function for automorphic functions did not prevail in the end, even if they are sometimes still called that. Klein had rightly pointed out to Poincaré that these had already been dealt with by Hermann Amandus Schwarz and Klein himself, while Fuchs had not published anything about them, but did not get through to Poincaré, who wanted to show his gratitude for Fuchs by naming them. The naming of another functional family after Klein was rather irritating for him and a testament to a lack of literary knowledge (Poincaré's designation of the Fuchs group has remained valid to this day). The tone remained polite and respectful (even if the correspondence only lasted until 1882), and Poincaré contributed an overview of his results for Felix Klein's Mathematical Annals, followed by a presentation of Klein's views.

### Physics and astronomy

#### theory of relativity

Poincaré turned increasingly to mathematical physics towards the end of the 19th century. In the context of the electrodynamics of moving bodies, he anticipated the special theory of relativity (1900–1905) in many points. Poincaré recognized the difficulties of classical physics, the elimination of which later led to the special theory of relativity. But unlike Albert Einstein , the more pragmatic Poincaré did not want to overturn the old mechanics, but rather rebuild it.

• From 1895 Poincaré was of the opinion that it was impossible to ever discover the aether or to prove an absolute movement. In 1900 he used the terms "principle of relative motion" and in 1904 the expression " principle of relativity " and defined this in such a way that the laws of physical processes should be the same for a stationary observer as for one in uniform translation, so that we have or cannot have any means of distinguishing whether or not we are engaged in such a movement. In 1905 he spoke of the “ postulate of the complete impossibility of determining an absolute movement”; In 1906 he introduced the term “postulate of relativity”. Despite these terminology, Poincaré stuck to the fact that an ether was necessary as a light medium, which however, due to the principle of relativity, was not recognizable.
• From 1898 onwards, Poincaré considered the concept of “absolute time” and “absolute simultaneity” to be meaningless. Building on this, he explained the mathematical auxiliary variable of "local time" introduced by Hendrik Antoon Lorentz in 1900 as the result of a signal exchange with light, the constant and absolute speed of which he postulated on this occasion. In his opinion this led to the fact that in a moving system clocks assumed to be synchronous are no longer synchronous from the point of view of a system resting in the ether, which practically leads to the relativity of simultaneity . However, since he clung to the etheric thought, it was more convenient in his opinion to designate the time of the clocks resting in the ether as the "true" time.
• In 1905 he simplified the spelling of the transformation equations introduced by Joseph Larmor and Lorentz, which he called the Lorentz transformation . By taking the principle of relativity as a basis, Poincaré recognized its group property, from which the perfect mathematical equality of the reference systems results, and coined the name Lorentz group . He was also able to fully demonstrate the Lorentz covariance of the Maxwell-Lorentz equations.
• In 1905/6 he was the first scientist to consider the possibility that the Lorentz transformation represented "a rotation in a four-dimensional space", whereby he expanded the three spatial dimensions by the time coordinate to four to form the space-time continuum and thereby introduced four- vectors. However, he refrained from doing this again because three “were more comfortable”.${\ displaystyle ct {\ sqrt {-1}}}$
• As early as 1900 he recognized that due to actio and reactio the electromagnetic energy behaves like a "fictitious" fluid with the mass , whereby the movement of the center of gravity remains uniform. However, Poincaré did not arrive at the complete equivalence of Einstein's mass and energy , since he did not recognize that a body loses or gains mass when it emits or absorbs energy. An application of an early form of the Lorentz transformation therefore led Poincaré to a radiation paradox: a change in the reference system leads to the fact that the conservation of momentum is not fulfilled, which not only makes a perpetual motion machine possible, but also violates the principle of relativity. However, if one assumes with Einstein that bodies can lose or gain mass, the paradox disappears. In 1904, however, Poincaré again distanced himself from the idea that electromagnetic radiation could be associated with ground.${\ displaystyle m = E / c ^ {2}}$
• Originally (1904) Poincaré was not sure about an instantaneous effect of gravitation . Later (1905/1906), however, he came to the conviction that a Lorentz-invariant law of gravitation with a maximum speed of propagation of gravitation at the speed of light is possible.
##### Poincaré and Einstein

Einstein knew some of Poincaré's relevant work; whether he read it before 1905 is unclear. In any case, he had knowledge of “science and hypothesis” - and thus of the main features of Poincaré's ideas on the absoluteness and relativity of time. Because the German edition contained excerpts from “La mesure du temps” (The Measure of Time, 1898). In his scientific writings Einstein refers to Poincaré in connection with the mass-energy equivalence (1906) and particularly appreciated some of Poincaré's observations on non-Euclidean geometry (1921). However, he did not appreciate his contributions to the Lorentz transformation, the synchronization of clocks or the principle of relativity. Only in 1953 and 1955 did he mention Poincaré in connection with his contributions to the theory of relativity:

"1953: Hopefully it will be ensured that the merits of HA Lorentz and H. Poincaré will also be properly recognized on this occasion."
"1955: There is no doubt that the special theory of relativity, if we look back at its development, ripe in 1905 was to be discovered. Lorentz had already recognized that the transformation, which was later named after him, was essential for the analysis of Maxwell's equations, and Poincaré deepened this knowledge. As far as I am concerned, I only knew Lorentz 'important work from 1895' La theorie électromagnétique de Maxwell 'and' Attempt at a theory of the electrical and optical phenomena of moving bodies ', but not Lorentz' later works, nor the subsequent study of Poincaré. In this sense, my work from 1905 was independent. "

Conversely, Poincaré also ignored Einstein's contributions to the special theory of relativity until his death (1912) and only honored the work of Lorentz. The two only met once, at the first Solvay Congress in Brussels in 1911. This led to differences between the two of them in their views on quantum theory, which Einstein alluded to in a letter to Heinrich Zangger:

"Poinkare [sic] was simply generally negative, but with all his acuteness showed little understanding for the situation."

Shortly afterwards, Poincaré wrote the following recommendation to Pierre-Ernest Weiss for Einstein's engagement at the ETH Zurich , where he expressed great appreciation on the one hand, but also some reservations on the other hand:

“Einstein is one of the most original minds I have ever met; in spite of his youth he already holds a very respectable rank among the leading scholars of his time. What we have to admire most about him is the ease with which he adapts to new concepts and draws the necessary consequences from them. He does not stick to any classical principles and, in the face of a physical problem, promptly grasps all the possibilities that open up. In his mind, this immediately translates into the prediction of new phenomena that one day should be experimentally demonstrable. I am not saying that all of these predictions will pass the experimental test when that test is possible. Since he researches in all directions, one must, on the contrary, reckon with the fact that the majority of the paths he has taken will be dead ends; At the same time, however, one can hope that one of the directions he points out will be the right one, and that is sufficient. You just have to do it that way. The job of mathematical physics is to ask correct questions and only experiment can solve them. The future will show Einstein's worth more and more clearly, and the university, which succeeds in winning this young man for itself, can be sure that it will be honoring it. "

Hermann Minkowski (1907) used similar ideas as Poincaré for his space-time construction as part of his contribution to the theory of relativity. Compared to Poincaré, however, he developed this approach decisively. Whereby Minkowski mentions Poincaré's view of gravity in this context, but does not mention his thoughts on four-dimensional space. In his well-known work Raum und Zeit he does not mention Poincaré at all.

#### Chaotic paths

Astronomers associate the name Henri Poincaré with his contributions to celestial mechanics . As explained above, Poincaré discovered deterministic chaos while analyzing the stability of the solar system - a topic that is very topical today. He summarized the discussion about determinism and predictability in his book "Wissenschaft und Methode" (1912). At that time there was a mechanistic worldview in science. In his book he writes:

“If we knew exactly the laws of nature and the initial state , we could predict the state of the universe at any subsequent point in time. But even if the laws of nature had no more secrets from us, we could only approximately determine the initial conditions . If this allows us to state the following states with the same approximation, then we say that the behavior was predicted to follow regularities. But that is not always the case: it can happen that small differences in the initial conditions result in large results […, a] prediction becomes impossible and we have a random phenomenon. "

Today we know that the system of planets and smaller celestial bodies in the solar system also tends to behave chaotically in the long term, as extensive simulation calculations by Jacques Laskar , Jack Wisdom and Gerald Jay Sussman have shown. While the resulting dangers are mostly in the distant future, such behavior is a potential danger in the case of asteroids running close to the earth's orbit . These can "suddenly" drift away or just as "suddenly" become asteroids close to the earth . At the end of the 1990s, the Viennese astronomer Rudolf Dvorak calculated that the well-known minor planet Eros would plunge into the sun after 20 million years on a relatively stable orbit due to chaotic orbit disturbances .

### Engineering and Geodesy

Poincaré, who in the tradition of the “ polytechniciens ” alternated between abstract science and concrete applications, was the grand master of the French engineering scholars. He organized surveying expeditions to Peru and campaigned for the preservation of the Eiffel Tower as a radio tower. Poincaré was only unsuccessful when his “Bureau des Longitudes” tried to metrize the units of time . He participated in the World Time Conference in 1884, where it was about the definition of a zero meridian and time measurement and time synchronization. If France was still followed for a universal measure of length at the 1875 Meter Convention , the prime meridian now ran through Greenwich - a diplomatic defeat - and the "unmetric" units of 24 hours and sixty minutes or seconds remained. In 1897, Poincaré submitted another proposal to decimalize the time when adhering to the 24-hour day and to divide the circle by 400 degrees; in his opinion, taking into account the demands of expediency, conventionality and continuity and thus less radical than, for example, contributions to the discussion of his contemporary Alfred Cornu . In 1900, however, his efforts finally failed politically. Instead of a world time, an agreement was made on the (American) compromise of time zones .

At the turn of the century Poincaré (like Einstein) dealt with the problem of time not only from a physical and philosophical perspective, but also from a technical perspective. The national and international synchronization of the most important time services, which was previously based on joint observation of astronomical events, should now be done by exchanging telegraphic signals. International synchronization, realized from around 1950 through the worldwide distribution of UTC radio signals, owes an important initial spark to Poincaré. A time coordination system was initiated immediately by Poincaré using a system installed as the center at the Eiffel Tower in Paris . The global positioning system is organized according to the same logic today.

### Epistemology and the basics of mathematics

Poincaré, who from around 1900 dealt intensively with philosophy (he gave a lecture on intuition and logic at the 1900 International Congress of Mathematicians in Paris), starts from Immanuel Kant's epistemology and defends his postulate of synthetic a priori judgments . In contrast to Kant, however, he did not see the Euclidean geometry of space as a basis (the choice of geometry is instead made from experience) and instead of time as a further basis, he argued for intuition based on unlimited repetition (principle of complete induction ) from number theory as the basis of mathematics. This goes beyond a purely formal logical system (which would ultimately correspond to a tautology ). Instead of Euclidean space as the basis, he sees the concepts of continuity and the group in geometry and topology as the basis.

He rejected the concept of an actual infinite and was therefore counted among their forerunners by the intuitionists , even if he never questioned the principle of the excluded third party . He is considered a representative of constructivism in mathematics. Even before his more intensive occupation with philosophy of mathematics in the last decade of his life, he became interested in the work of Georg Cantor at an early age and suggested their translation into French (he also used his results in his treatise on Kleinsche groups of 1884). The book on the axiomatization of geometry by David Hilbert, which appeared in 1899, he reviewed positively in 1902. Like Russell, he traced the fundamental crisis of mathematics that opened up with Russell's paradox back to self-reference (Russell introduced his type theory to solve it ) and distinguished between predictive and impredicative statements (such as those in the paradox). However, he did not exclude all impredicative statements, but differentiated according to the context. According to Poincaré, they only cause problems when they are used to construct an object. According to him, there were two types of allowed non-constructive definition contexts: existence per a priori definition (whereby an already existing object is selected) and existence based on mathematical intuition - the intuitive concept of the continuum. For example, the impredicative concept of the smallest upper bound was allowed because it was definable by sets of upper bounds in the same way that real numbers were constructed. As Hermann Weyl later recognized, according to Poincaré, the restriction to merely predictive statements would be too restrictive and too cumbersome for the construction of mathematics.

#### Conventionalism

He wrote several philosophical treatises on the philosophy of science , establishing a form of conventionalism . He rejected the separation into the two extremes of idealism and empiricism , and in his philosophy he succeeded in amalgamating questions from the humanities and the natural sciences.

Shaped by the paradigm of progress and optimism of the 19th century, Poincaré assumed a mathematical understanding of nature in connection with experiment. Science does not research the ultimate truth, but relationships between real objects, and these can be expressed mathematically at the deepest level (for example in geometry or in physics in the form of differential equations ). According to him, the usefulness of science is an indication that these relations are not chosen arbitrarily, but are given in the outside world, in experiments. The most important relationships survived changes in theory in the course of the history of science and express fundamental relationships of reality. There is also a Darwinian component in Poincaré: the correspondence between cognitive structures and reality is also a consequence of the evolutionary adaptation, which provides advantages for those who best represent the outside world.

#### Science and hypothesis

The work is divided into four parts.

“Number and Size” deals first with the possibility of mathematics. Is mathematics just a tautological undertaking, a system of analytical judgments that all trace back to identity? No, the mathematician also deduces the general from the particular. Poincaré introduces the complete induction , the "recurrent conclusion". "Mathematicians don't study objects, they study relationships between objects ..." .

The mathematician constructs a "mathematical continuum" through logical inference. He creates a system that is only limited by contradictions. The starting point of the construction are symbols that are created through intuition. Thus the mathematical continuum is in contrast to the physical continuum, which is derived from sensory experience. Poincaré's philosophy thus differs from the position of Bertrand Russell ( logicism ) and from David Hilbert's formalism, which Poincaré also criticizes.

“The room” deals with geometry (which he does not want to be treated together with mathematics). Geometry arises from the experience of solid objects in nature, but it is not an empirical science - it idealizes these bodies and thus simplifies nature. Poincaré presents various systems of axioms of geometry, which they call "languages". The human mind adapts itself to a certain extent to the observed nature; we choose the geometrical system that is most “comfortable”: “our geometry is not true, it is advantageous” .

“Die Kraft” is initially devoted to mechanics and poses the basic question of whether its basic principles can be changed - Poincaré contrasts the empiricism of British tradition with the continental deductive method. Poincaré demands the separation of hypotheses and mere conventions: space, time, simultaneity and Euclidean geometry are not absolute, they are pure conventions - convenient languages ​​of description. Mechanics is therefore anthropomorphic. He presents short conceptual stories about mass, acceleration, force and movement, connects them and leads us briefly in a circle, before disentangling the circle of ideas with the meaning of convention. However, through the introduction of (practical) agreements, through generalization, objectivity is lost. Where this goes too far, Poincaré starts with a criticism of nominalism. As with mechanics, it also uses astronomy and thermodynamics to clarify its position.

The last part "Nature" begins with his epistemology. Poincaré's source of knowledge is initially only experiment and generalization. He recognizes that this is not free of worldview and "... one must therefore never reject an examination ..." . Generalization assumes a simplicity of nature, but this simplicity can only be apparent. The hypothesis is to be subject to “verification as often as possible” , as Karl R. Popper ( Critical Rationalism ) will later formulate - given different reasons . Poincaré distinguishes three types of hypotheses: natural ones, which arise directly from perception, indifferent ones, which create useful conditions without influencing the result, and real generalizations . Poincaré justifies the role of mathematics in physics, starting from the (assumed) homogeneity of nature, from the decomposition of the phenomena into a large number of smaller phenomena (according to time, space or partial movement), the superposition of which can be described with a mathematical method. As in the other sections, Poincaré clarifies his position from the history of science, here with a theoretical history on light, electricity and magnetism, up to the "satisfactory" Lorentzian theory. A chapter on probability theory has also been incorporated , and how this - a method increasingly used in physics at the time - is possible (philosophically). According to Poincaré, it is used wherever ignorance is involved: in the case of ignorance of the initial state and knowledge of the natural law for describing the state of a system, for the formation of theories themselves and in error theory. In any case, the basis is the belief in the continuity of the phenomena. The work closes with presenting the current positions on the existence of matter, the theory of electrons and ethers at that time . Detailed comments present what has been presented to the interested reader in a deeper mathematical representation.

## Awards and honors

In 1928 the Henri Poincaré Institute was founded in his honor . The Henri Poincaré Prize for mathematical physics , which has been awarded every three years since 1997, and the Poincaré crater on the moon are also named after the mathematician .

In honor of his life's work, the asteroid (2021) Poincaré was named after him.

## Major works

Wikisource: Henri Poincaré  - Sources and full texts
• Oeuvre. 11 vols. Gauthier-Villars, Paris 1916–1954, published by the Académie des sciences , new edition by Edition Gabay:
• Volume 1: Analysis: Equations différentielles (Ed. Paul Appell , Jules Drach ), 1928, 1952, Archive.org
• Volume 2: Analysis: Fonctions fuchsienne (Eds. Niels Erik Nørlund , Ernest Lebon under the overall direction of Gaston Darboux , who contributed a biography), 1916, 1952, Archive.org
• Volume 3: Analysis: Equations différentielles. Théorie des fonctions (Ed. Jules Drach), 1934, 1965, Archive.org
• Volume 4: Analysis: Théorie des fonctions (Ed. Georges Valiron ), 1950, Archive.org
• Volume 5: Arithmétique et Algèbre (Ed. Albert Châtelet ), 1950, Archive.org
• Volume 6: Géométrie: Analysis situs (Eds. René Garnier , Jean Leray ), 1953, Archive.org
• Volume 7: Mécanique céleste et astronomie: Masses fluides en rotation. Principes de Mécanique analytique. Problème des trois corps (Ed. Jacques Lévy), 1952, Archive.org
• Volume 8: Mécanique céleste et astronomie: Mécanique céleste. Astronomy (Ed. Pierre Sémirot), 1952, Archive.org
• Volume 9: Physique mathématique (Ed. Gérard Petiau, foreword Louis de Broglie ), 1954, Archive.org
• Volume 10: Physique mathématique (Ed. Gérard Petiau, preface by Gaston Julia ), 1954, Archive.org
• Volume 11: Mémoires diverse, Hommages à Henri Poincaré, Livre du Centenaire de la naissance de Henri Poincaré, 1854–1954 (Ed. Gérard Petiau under the direction of Gaston Julia), 1956
• Les méthodes nouvelles de la mécanique céleste , 3 volumes, Gauthier-Villars, Paris, 1892–1899, volume 1 (Solutions périodiques. Non-existence des intégrales uniformes. Solutions asymptotique), Volume 2 (Méthodes de Newcomb, Gylden, Lindstedt et Bohlin ), Volume 3 (Invariants integraux. Solutions périodiques du deuxième genre. Solutions doublement asymptotiques).
• Science and hypothesis . (Original La science et l'hypothèse , Paris 1902), Berlin 1928, Xenomos Verlag, Berlin 2003. ISBN 3-936532-24-9 (repr.) Archive.org, French , Archive.org, German edition by Teubner
• The value of science . (Original La valeur de la science , Paris 1905), Leipzig 1921. Xenomos Verlag, Berlin 2003. ISBN 3-936532-23-0 (repr.), Archive.org, French , Archive.org, German edition by Teubner
• Science and method . (Original Science et méthode , Paris 1908), Berlin 1914, Xenomos Verlag, Berlin 2003. ISBN 3-936532-31-1 (repr.) Archive.org
• Final thoughts . (Original Dernières pensées , Paris, Flammarion 1913, 2nd edition 1926), Leipzig 1913, Xenomos Verlag, Berlin 2003. ISBN 3-936532-27-3 (repr.), Archive.org
• Des fondements de la géométrie , Paris, Chiron 1921, archives
• The Three-Body Problem and the Equations of Dynamics: Poincaré's Foundational Work on Dynamical Systems Theory (translator Bruce D. Popp), Springer 2017

## lectures

• Leçons sur la théorie mathématique de la lumière professées pendant le premier semestre 1887–1888, Carré 1889, archive
• Électricité et optique, la lumière et les théories électrodynamiques, leçons professées en 1888, 1890 et 1899, Carré et Naud, 1901
• Thermodynamique: leçons professées pendant le premier semestre 1888–1889, edited by J. Blondin, Paris Gauthier-Villars 1908, reprinted by Jacques Gabay 1995, archive
• Capillarité: Leçons professées pendant le deuxième semestre 1888–1889, Paris, Carré 1895, Archives
• Leçons sur la théorie de l'élasticité, Carré 1892, Archives
• Théorie mathématique de la lumière II: nouvelles études sur la diffraction. Théorie de la dispersion de Helmholtz: Leçons professées pendant le premier semestre 1891–1892, Carré 1892, archive
• Théorie des tourbillons, leçons professées pendant le deuxième semestre 1891–1892, Carré et Naud 1893, archive
• Les oscillations électriques, leçons professées pendant le premier trimestre 1892–1893, Carré et Naud 1900, archive
• Théorie analytique de la propagation de la chaleur, leçons professées pendant le premier semestre 1893–1894, Carré 1895, archive
• Calcul des probabilités, leçons professées pendant le deuxième semestre 1893–1894, Carré et Naud 1896, Gauthier-Villars 1912 archive
• Théorie du potentiel newtonien, leçons professées pendant le premier semestre 1894–1895, Carré et Naud 1899, archive
• Figures d'équilibre d'une mass fluide; leçons professées à la Sorbonne en 1900, Paris: Gauthier-Villars 1902, Archives
• Leçons sur les hypothèses cosmogoniques, Paris: Hermann 1911, Archives
• Cours d'astronomie générale: École polytechnique 1906–1907, École polytechnique (Paris), 1907
• Leçons de mécanique céleste , Gauthier-Villars 1905, 3 volumes ( Volume 1 , Volume 2 , Volume 3 )
• Six lectures from pure mathematics and mathematical physics , Teubner 1910 (held at the invitation of the Wolfskehl Commission in Göttingen, April 22-28 , 1909), Archive.org , Gutenberg project

## literature

• Paul Appeal : Henri Poincaré , Paris 1925
• Felix Browder (Ed.): The mathematical heritage of Henri Poincaré , 2 volumes, American Mathematical Society 1983 (Symposium Indiana University 1980)
• Gaston Darboux : Eloge historique d'Henri Poincaré , Mémoires de l'Académie des sciences, Volume 52, 1914, pp. 81-148.
• Éric Charpentier, Étienne Ghys , Annick Lesne (Eds.): The scientific legacy of Henri Poincaré , American Mathematical Society 2010 (French original 2006)
• Jean Dieudonné : Poincaré, Henri . In: Charles Coulston Gillispie (Ed.): Dictionary of Scientific Biography . tape 11 : A. Pitcairn - B. Rush . Charles Scribner's Sons, New York 1975, p. 51-61 .
• Bernard Duplantier, Henri Rivasseau (eds.), Henri Poincaré 1912–2012, Poincaré Seminar 2012, Birkhäuser 2015 (therein Olivier Darrigol, Poincaré's light, Alain Chenciner Poincaré and the three-body-problem, Mazliak Poincaré's Odds, Francois Beguin, Henri Poincaré and the uniformization of Riemann surfaces)
• Jean-Marc Ginoux, Christian Gerini: Henri Poincare: A Biography Through The Daily Papers , World Scientific 2013
• June Barrow-Green : Poincaré and the Three Body Problem , American Mathematical Society 1997, ISBN 0-8218-0367-0
• June Barrow-Green: Poincaré and the discovery of chaos , Icon Books 2005
• June Barrow-Green: Oscar II's prize competition and the error in Poincaré's memoir on the three body problem , Arch. Hist. Exact Sci., Vol. 48, 1994, pp. 107-131.
• Florin Diacu, Philip Holmes Celestial Encounters. The origins of chaos and stability , Princeton University Press 1996
• Giedymin, J .: Science and Convention: Essays on Henri Poincaré's Philosophy of Science and the Conventionalist Tradition . Pergamon Press, Oxford 1982, ISBN 0-08-025790-9 .
• Jeremy Gray : Henri Poincaré. A Scientific Biography , Princeton University Press, Princeton, New Jersey, USA 2012. Review by John Stillwell, Notices AMS, April 2014, pdf
• Jeremy Gray: Linear differential equations and group theory from Riemann to Poincaré , Birkhäuser 1986
• Langevin, P .: L'œuvre d'Henri Poincaré: le physicien . In: Revue de métaphysique et de morale . 21, 1913, p. 703.
• Jean Mawhin : Henri Poincaré. A life in the service of Science , Notices AMS, October 2005, pdf
• Poincaré: La correspondance d'Henri Poincaré avec des mathématiciens de A à H, Cahiers du séminaire d'histoire des mathématiques, Paris, Volume 7, 1986, pp. 59-219. numdam
• Poincaré, La correspondance d'Henri Poincaré avec des mathématiciens de J à Z, Cahiers du séminaire d'histoire des mathématiques, Volume 10, 1989, pp. 83-229. numdam
• Ferdinand Verhulst Henri Poincaré-impatient genius , Springer Verlag, New York City a. a. 2012.
• Zahar, E .: Poincare's Philosophy: From Conventionalism to Phenomenology . Open Court Pub Co, Chicago 2001, ISBN 0-8126-9435-X .

Especially for Poincaré and the theory of relativity:

Non-mainstream to Poincaré and relativity

• Keswani, GH ,: Origin and Concept of Relativity, Part I . In: Brit. J. Phil. Sci. . 15, No. 60, 1965, pp. 286-306. doi : 10.1093 / bjps / XV.60.286 .
• Keswani, GH: Origin and Concept of Relativity, Part II . In: Brit. J. Phil. Sci. . 16, No. 61, 1965, pp. 19-32. doi : 10.1093 / bjps / XVI.61.19 .
• Keswani, GH: Origin and Concept of Relativity, Part III . In: Brit. J. Phil. Sci. . 16, No. 64, pp. 273-294. doi : 10.1093 / bjps / XVI.64.273 .
• Leveugle, J .: La Relativité et Einstein, Planck, Hilbert - Histoire véridique de la Théorie de la Relativitén . L'Harmattan, Pars 2004.
• Logunov, AA: Henri Poincaré and relativity theory . Nauka, Moscow 2004, ISBN 5-02-033964-4 .
• Edmund Taylor Whittaker : The Relativity Theory of Poincaré and Lorentz . In: A History of the Theories of Aether and Electricity: The Modern Theories 1900-1926 . Nelson, London 1953, pp. 27-77.

Commons : Henri Poincaré  - Collection of Images, Videos and Audio Files
Secondary literature
Works

## Individual evidence

1. Before that he taught at the medical school. Nancy got a university after the Franco-Prussian War , when the French University of Strasbourg moved there after the annexation of Alsace
2. ^ Verhulst, Poincaré, p. 6. After Gaston Darboux .
3. He got along less well with Mannheim and his assistant Jules de la Gournerie. In 1886 he suggested Mannheim when he was elected to the Academie des Sciences
4. His main competitor and first to graduate was Marcel Bonnefoy (1854–1881), who died early in a mining accident (as did another student among the top three of the year, Jules Petitdidier). Poincaré and Bonnefoy were on friendly terms.
5. McTutor, Poincaré - Inspector of Mines
6. Maurice Roy, René Dugas: Henri Poincaré, Ingénieur des Mines, Annales des Mines, Volume 193, 1954, pp. 8-23. Website Annales des Mines zu Poincaré
7. ^ Verhulst, Poincaré, p. 27
8. Poincaré, Sur les propriétés des fonctions définies par les equations aux différences partielles, dissertation 1879, archive
9. Contained in his book Valeur de la Science
10. ^ Poincaré, Future of Mathematics, pdf
11. ^ Verhulst, Poincaré, p. 56f
12. ^ Verhulst, Poincaré, p. 67
13. Poincaré, Analysis situs, Journal de l'École Polytechnique, Series 2, Volume 1, 1895, pp. 1–123, and the first to fifth complements: Rendiconti del Circolo Matematico di Palermo, Volume 13, 1899, p. 285 -343 (Complement 1), Proceedings of the London Mathematical Society, Volume 32, 1900, pp. 277-308 (Complement 2), Bulletin de la Société mathématique de France, Volume 30, 1902, pp. 49-70 (Complement 3 ), Journal de mathématiques pures et appliquées, series 5, volume 8, 1902, pp 169-214 (complement 4), Rendiconti del Circolo Matematico di Palermo, volume 18, 1904, pp 45-110 (complement 5). English translation of all articles in John Stillwell (Ed.): Papers on Topology: Analysis Situs and Its Five Supplements, 2009, pdf
14. ^ Verhulst, Poincaré, p. 70
15. ^ Verhulst, Poincaré, p. 71
16. ^ Sundman, Recherches sur leproblemème des trois corps, Acta Mathematica, Volume 36, 1912, pp. 105–1979
17. ^ QD Wong, Global solution of the n-body problem, Celestial Mechanics, Volume 50, 1991, pp. 73-88
18. ^ Verhulst, Poincaré, p. 72
19. ^ Verhulst, Poincaré, p. 75
20. ^ Poincaré, Sur leproblemème des trois corps et les equations de la dynamique, Acta Mathematica, Volume 13, 1890, pp. 1-270.
21. ^ Verhulst, Poincaré, p. 74.
22. ^ Verhulst, Poincaré, p. 73
23. ^ Mémoire sur les courbes définies par une équation différentielle, Journal de Mathématiques, 3rd series, Volume 7, 1881, pp. 375-422, Volume 8, 1882, pp. 251-296
24. ^ Poincaré, Henri, Sur une class nouvelle de transcendantes uniformes . Journal de mathématiques pures et appliquées (4), Volume 6, pp. 313-366, 1890
25. Sur les proprietes des courbes algebriques planes, J. Liouville, Series 5, Volume 7, 1901, pp. 161-233
26. ^ The correspondence between Klein and Poincaré is published in Acta Mathematica, Volume 39, 1924, pp. 94-132 and in Volume 3 of Klein's Collected Treatises, SUB Göttingen
27. ^ Poincaré, Sur l'uniformisation des fonctions analytiques, Acta Mathematica, Volume 31, 1907, pp. 1–63
28. With a large essay in the first volume of Acta Mathematica: Poincaré, Theory des groupes fuchsiennes, Acta Mathematica, Volume 1, 1882, pp. 1–62, archive
29. ↑ Much of the work in English translation in John Stillwell (Ed.), Henri Poincaré, Papers on Fuchsian Functions, Springer 1985
30. ^ Poincaré, L'invention mathématique, in: Science et Méthode 1908
31. Poincaré, Sur les fonctions uniformes qui se reproduisent par des substitutions linéaires, Mathematische Annalen, Volume 19, 1882, pp. 553-564, SUB Göttingen
32. Klein, About unique functions with linear transformations in itself, Math. Annalen, Volume 19, 1882, pp. 565-568, SUB Göttingen
33. a b c Poincaré, Henri: The present state and the future of mathematical physics . In: The Value of Science (Chapters 7-9) . BG Teubner, Leipzig 1904/6, pp. 129–159.
34. Poincaré, Henri: The measure of time . In: The Value of Science (Chapter 2) . BG Teubner, Leipzig 1898/1906, pp. 26–43.
35. ^ A b Poincaré, Henri: La théorie de Lorentz et le principe de réaction . In: Archives néerlandaises des sciences exactes et naturelles . 5, 1900, pp. 252-278. . See also German translation .
36. ^ A b Poincaré, Henri: Sur la dynamique de l'électron . In: Comptes rendus hebdomadaires des séances de l'Académie des sciences . 140, 1905, pp. 1504-1508. See also German translation .
37. ^ A b Poincaré, Henri: Sur la dynamique de l'électron . In: Rendiconti del Circolo matematico di Palermo . 21, 1906, pp. 129-176. See also German translation .
38. Darrigol 2004, Galison, 2003
39. Pais 1982, chap. 8th
40. ^ Poincaré, Henri: The new mechanics . BG Teubner, Leipzig 1910/11.
41. Darrigol 2004, p. 624
42. Galison 2003, p. 314
43. Minkowski, H .: The basic equations for the electromagnetic processes in moving bodies . In: Göttinger Nachrichten . 1908, p. 53-111 .
44. Walter 2007
45. ^ Dvorak, The long term evolution of Atens and Apollos , in: J. Svoren u. a .: Evolution and source regions of asteroids and comets: proceedings of the 173rd colloquium of the International Astronomical Union, held in Tatranska Lomnica, Slovak Republic, August 24-28, 1998
46. Janet Frolina, Poincaré's philosophy of mathematics, Internet Encyclopedia of Philosophy
47. Illustration based on Frolina, Poincaré's philosophy of mathematics, Internet Encyclopedia of Philosophy, loc.cit.
48. Janet Frolina, Poincaré's philosophy of mathematics, loc. cit.
49. ^ Page on Poincaré, Academie des Sciences
50. ^ Entry on Poincare; Jules Henri (1854–1912) in the Archives of the Royal Society , London
51. ^ Fellows Directory. Biographical Index: Former RSE Fellows 1783–2002. (PDF file) Royal Society of Edinburgh, accessed March 30, 2020 .