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Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
, theoretical physicist,
engineer Engineers, as practitioners of engineering, are professionals who Invention, invent, design, analyze, build and test machines, complex systems, structures, gadgets and materials to fulfill functional objectives and requirements while considerin ...
, and philosopher of science. He is often described as a
polymath A polymath ( el, πολυμαθής, , "having learned much"; la, homo universalis, "universal human") is an individual whose knowledge spans a substantial number of subjects, known to draw on complex bodies of knowledge to solve specific pro ...
, and in mathematics as "The Last Universalist", since he excelled in all fields of the discipline as it existed during his lifetime. As a mathematician and
physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...
, he made many original fundamental contributions to pure and
applied mathematics Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathemati ...
,
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
, and
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
. In his research on the three-body problem, Poincaré became the first person to discover a chaotic
deterministic system In mathematics, computer science and physics, a deterministic system is a system in which no randomness is involved in the development of future states of the system. A deterministic model will thus always produce the same output from a given ...
which laid the foundations of modern
chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
. He is also considered to be one of the founders of the field of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. Poincaré made clear the importance of paying attention to the invariance of laws of physics under different transformations, and was the first to present the
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
in their modern symmetrical form. Poincaré discovered the remaining relativistic velocity transformations and recorded them in a letter to Hendrik Lorentz in 1905. Thus he obtained perfect invariance of all of
Maxwell's equations Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits ...
, an important step in the formulation of the theory of
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
. In 1905, Poincaré first proposed gravitational waves (''ondes gravifiques'') emanating from a body and propagating at the speed of light as being required by the Lorentz transformations. The Poincaré group used in physics and mathematics was named after him. Early in the 20th century he formulated the
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
that became over time one of the famous unsolved problems in mathematics until it was solved in 2002–2003 by Grigori Perelman.


Life

Poincaré was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, Meurthe-et-Moselle, into an influential French family. His father Léon Poincaré (1828–1892) was a professor of medicine at the University of Nancy. His younger sister Aline married the spiritual philosopher
Émile Boutroux Étienne Émile Marie Boutroux (; 28 July 1845 – 22 November 1921) was an eminent 19th-century French philosopher of science and religion, and a historian of philosophy. He was a firm opponent of materialism in science. He was a spiritual phil ...
. Another notable member of Henri's family was his cousin,
Raymond Poincaré Raymond Nicolas Landry Poincaré (, ; 20 August 1860 – 15 October 1934) was a French statesman who served as President of France from 1913 to 1920, and three times as Prime Minister of France. Trained in law, Poincaré was elected deputy in ...
, a fellow member of the
Académie française An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosop ...
, who was President of France from 1913 to 1920.The Internet Encyclopedia of Philosophy
Jules Henri Poincaré article by Mauro Murzi – Retrieved November 2006.


Education

During his childhood he was seriously ill for a time with
diphtheria Diphtheria is an infection caused by the bacterium '' Corynebacterium diphtheriae''. Most infections are asymptomatic or have a mild clinical course, but in some outbreaks more than 10% of those diagnosed with the disease may die. Signs and s ...
and received special instruction from his mother, Eugénie Launois (1830–1897). In 1862, Henri entered the Lycée in Nancy (now renamed the in his honour, along with
Henri Poincaré University The Henri Poincaré University, or Nancy 1, nicknamed UHP, was a public research university located in Nancy, France. UHP was merged into University of Lorraine in 2012, and was previously a member of the Nancy-Université federation, belongi ...
, also in Nancy). He spent eleven years at the Lycée and during this time he proved to be one of the top students in every topic he studied. He excelled in written composition. His mathematics teacher described him as a "monster of mathematics" and he won first prizes in the concours général, a competition between the top pupils from all the Lycées across France. His poorest subjects were music and physical education, where he was described as "average at best". However, poor eyesight and a tendency towards absentmindedness may explain these difficulties. He graduated from the Lycée in 1871 with a
baccalauréat The ''baccalauréat'' (; ), often known in France colloquially as the ''bac'', is a French national academic qualification that students can obtain at the completion of their secondary education (at the end of the ''lycée'') by meeting certain ...
in both letters and sciences. During the Franco-Prussian War of 1870, he served alongside his father in the
Ambulance Corps Emergency medical services (EMS), also known as ambulance services or paramedic services, are emergency services that provide urgent pre-hospital treatment and stabilisation for serious illness and injuries and transport to definitive care. ...
. Poincaré entered the École Polytechnique as the top qualifier in 1873 and graduated in 1875. There he studied mathematics as a student of
Charles Hermite Charles Hermite () FRS FRSE MIAS (24 December 1822 – 14 January 1901) was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra. ...
, continuing to excel and publishing his first paper (''Démonstration nouvelle des propriétés de l'indicatrice d'une surface'') in 1874. From November 1875 to June 1878 he studied at the
École des Mines École may refer to: * an elementary school in the French educational stages normally followed by secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing in région Île-de-France * École, S ...
, while continuing the study of mathematics in addition to the
mining engineering Mining in the engineering discipline is the extraction of minerals from underneath, open pit, above or on the ground. Mining engineering is associated with many other disciplines, such as mineral processing, exploration, excavation, geology, a ...
syllabus, and received the degree of ordinary mining engineer in March 1879. As a graduate of the École des Mines, he joined the
Corps des Mines The ''Corps des mines'' is the foremost technical Grand Corps of the French State (grands corps de l'Etat). It is composed of the state industrial engineers. The Corps is attached to the French Ministry of Economy and Finance. Its purpose is to e ...
as an inspector for the
Vesoul Vesoul () is a commune in the Haute-Saône department in the region of Bourgogne-Franche-Comté located in eastern France. It is the most populated municipality of the department with inhabitants in 2014. The same year, the Communauté d'aggl ...
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 Science Doctor of Science ( la, links=no, Scientiae Doctor), usually abbreviated Sc.D., D.Sc., S.D., or D.S., is an academic research degree awarded in a number of countries throughout the world. In some countries, "Doctor of Science" is the degree used f ...
in mathematics under the supervision of Charles Hermite. His doctoral thesis was in the field of
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
. It was named ''Sur les propriétés des fonctions définies par les équations aux différences partielles''. Poincaré devised a new way of studying the properties of these equations. He not only faced the question of determining the integral of such equations, but also was the first person to study their general geometric properties. He realised that they could be used to model the behaviour of multiple bodies in free motion within the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
. Poincaré graduated from the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
in 1879.


First scientific achievements

After receiving his degree, Poincaré began teaching as junior
lecturer Lecturer is an academic rank within many universities, though the meaning of the term varies somewhat from country to country. It generally denotes an academic expert who is hired to teach on a full- or part-time basis. They may also conduct re ...
in mathematics at the University of Caen in Normandy (in December 1879). At the same time he published his first major article concerning the treatment of a class of automorphic functions. There, in
Caen Caen (, ; nrf, Kaem) is a commune in northwestern France. It is the prefecture of the department of Calvados. The city proper has 105,512 inhabitants (), while its functional urban area has 470,000,Isidore Geoffroy Saint-Hilaire and great-granddaughter of
Étienne Geoffroy Saint-Hilaire Étienne Geoffroy Saint-Hilaire (15 April 177219 June 1844) was a French naturalist who established the principle of "unity of composition". He was a colleague of Jean-Baptiste Lamarck and expanded and defended Lamarck's evolutionary theories ...
and on 20 April 1881, they married. Together they had four children: Jeanne (born 1887), Yvonne (born 1889), Henriette (born 1891), and Léon (born 1893). Poincaré immediately established himself among the greatest mathematicians of Europe, attracting the attention of many prominent mathematicians. In 1881 Poincaré was invited to take a teaching position at the Faculty of Sciences of the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
; he accepted the invitation. During the years 1883 to 1897, he taught
mathematical analysis Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions. These theories are usually studied ...
in the École Polytechnique. In 1881–1882, Poincaré created a new branch of mathematics:
qualitative theory of differential equations In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relativel ...
. He showed how it is possible to derive the most important information about the behavior of a family of solutions without having to solve the equation (since this may not always be possible). He successfully used this approach to problems in
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
and
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
.


Career

He never fully abandoned his career in the mining administration to mathematics. He worked at the Ministry of Public Services as an engineer in charge of northern railway development from 1881 to 1885. He eventually became chief engineer of the
Corps des Mines The ''Corps des mines'' is the foremost technical Grand Corps of the French State (grands corps de l'Etat). It is composed of the state industrial engineers. The Corps is attached to the French Ministry of Economy and Finance. Its purpose is to e ...
in 1893 and inspector general in 1910. Beginning in 1881 and for the rest of his career, he taught at the
University of Paris , image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and a ...
(the Sorbonne). He was initially appointed as the ''maître de conférences d'analyse'' (associate professor of analysis). Eventually, he held the chairs of Physical and Experimental Mechanics, Mathematical Physics and Theory of Probability, and Celestial Mechanics and Astronomy. In 1887, at the young age of 32, Poincaré was elected to the
French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at ...
. He became its president in 1906, and was elected to the
Académie française An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosop ...
on 5 March 1908. In 1887, he won Oscar II, King of Sweden's mathematical competition for a resolution of the three-body problem concerning the free motion of multiple orbiting bodies. (See
three-body problem In physics and classical mechanics, the three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's ...
section below.) In 1893, Poincaré joined the French
Bureau des Longitudes Bureau ( ) may refer to: Agencies and organizations * Government agency *Public administration * News bureau, an office for gathering or distributing news, generally for a given geographical location * Bureau (European Parliament), the administ ...
, which engaged him in the synchronisation of time around the world. In 1897 Poincaré backed an unsuccessful proposal for the decimalisation of circular measure, and hence time and
longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...
. It was this post which led him to consider the question of establishing international time zones and the synchronisation of time between bodies in relative motion. (See work on relativity section below.) In 1904, he intervened in the
trials In law, a trial is a coming together of parties to a dispute, to present information (in the form of evidence) in a tribunal, a formal setting with the authority to adjudicate claims or disputes. One form of tribunal is a court. The tribun ...
of
Alfred Dreyfus Alfred Dreyfus ( , also , ; 9 October 1859 – 12 July 1935) was a French artillery officer of Jewish ancestry whose trial and conviction in 1894 on charges of treason became one of the most polarizing political dramas in modern French history. ...
, attacking the spurious scientific claims regarding evidence brought against Dreyfus. Poincaré was the President of the Société Astronomique de France (SAF), the French astronomical society, from 1901 to 1903.


Students

Poincaré had two notable doctoral students at the University of Paris,
Louis Bachelier Louis Jean-Baptiste Alphonse Bachelier (; 11 March 1870 – 28 April 1946) was a French mathematician at the turn of the 20th century. He is credited with being the first person to model the stochastic process now called Brownian motion, as part ...
(1900) and
Dimitrie Pompeiu Dimitrie D. Pompeiu (; – 8 October 1954) was a Romanian mathematician, professor at the University of Bucharest, titular member of the Romanian Academy, and President of the Chamber of Deputies. Biography He was born in 1873 in Broscăuți, ...
(1905).


Death

In 1912, Poincaré underwent surgery for a
prostate The prostate is both an accessory gland of the male reproductive system and a muscle-driven mechanical switch between urination and ejaculation. It is found only in some mammals. It differs between species anatomically, chemically, and phys ...
problem and subsequently died from an
embolism An embolism is the lodging of an embolus, a blockage-causing piece of material, inside a blood vessel. The embolus may be a blood clot (thrombus), a fat globule (fat embolism), a bubble of air or other gas ( gas embolism), amniotic fluid (am ...
on 17 July 1912, in Paris. He was 58 years of age. He is buried in the Poincaré family vault in the Cemetery of Montparnasse, Paris, in section 16 close to the gate Rue Émile-Richard. A former French Minister of Education,
Claude Allègre Claude Allègre (; born 31 March 1937) is a French politician and scientist. Scientific work The main scientific area of Claude Allègre was geochemistry. Allègre co-authored an ''Introduction to geochemistry'' in 1974. Since the 1980s, he ...
, proposed in 2004 that Poincaré be reburied in the
Panthéon The Panthéon (, from the Classical Greek word , , ' empleto all the gods') is a monument in the 5th arrondissement of Paris, France. It stands in the Latin Quarter, atop the , in the centre of the , which was named after it. The edifice was b ...
in Paris, which is reserved for French citizens of the highest honour.


Work


Summary

Poincaré made many contributions to different fields of pure and applied mathematics such as:
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
,
fluid mechanics Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them. It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
,
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultrav ...
,
electricity Electricity is the set of physical phenomena associated with the presence and motion of matter that has a property of electric charge. Electricity is related to magnetism, both being part of the phenomenon of electromagnetism, as describe ...
,
telegraphy Telegraphy is the long-distance transmission of messages where the sender uses symbolic codes, known to the recipient, rather than a physical exchange of an object bearing the message. Thus flag semaphore is a method of telegraphy, whereas ...
, capillarity, elasticity,
thermodynamics Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws ...
,
potential theory In mathematics and mathematical physics, potential theory is the study of harmonic functions. The term "potential theory" was coined in 19th-century physics when it was realized that two fundamental forces of nature known at the time, namely gra ...
, quantum theory,
theory of relativity The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
and
physical cosmology Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of f ...
. He was also a populariser of mathematics and physics and wrote several books for the lay public. Among the specific topics he contributed to are the following: *
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
(a field that Poincaré virtually invented) * the theory of analytic functions of several complex variables * the theory of abelian functions *
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
*the
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
, proven in 2003 by Grigori Perelman. *
Poincaré recurrence theorem In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (fo ...
*
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
*
number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Ma ...
*the three-body problem * the theory of diophantine equations *
electromagnetism In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
* the special theory of relativity *the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
*In the field of
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
Poincaré has given many results that are critical for the qualitative theory of differential equations, for example the
Poincaré sphere Poincaré sphere may refer to: * Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light ** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system * Poincaré homology s ...
and the Poincaré map. *Poincaré on "everybody's belief" in the ''Normal Law of Errors'' (see
normal distribution In statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is : f(x) = \frac e^ The parameter \mu ...
for an account of that "law") *Published an influential paper providing a novel mathematical argument in support of
quantum mechanics Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
.


Three-body problem

The problem of finding the general solution to the motion of more than two orbiting bodies in the
Solar System The Solar System Capitalization of the name varies. The International Astronomical Union, the authoritative body regarding astronomical nomenclature, specifies capitalizing the names of all individual astronomical objects but uses mixed "Solar ...
had eluded mathematicians since Newton's time. This was known originally as the three-body problem and later the ''n''-body problem, where ''n'' is any number of more than two orbiting bodies. The ''n''-body solution was considered very important and challenging at the close of the 19th century. Indeed, in 1887, in honour of his 60th birthday, Oscar II, King of Sweden, advised by
Gösta Mittag-Leffler Magnus Gustaf "Gösta" Mittag-Leffler (16 March 1846 – 7 July 1927) was a Swedish mathematician. His mathematical contributions are connected chiefly with the theory of functions, which today is called complex analysis. Biography Mittag-Leffl ...
, established a prize for anyone who could find the solution to the problem. The announcement was quite specific:
Given a system of arbitrarily many mass points that attract each according to Newton's law, under the assumption that no two points ever collide, try to find a representation of the coordinates of each point as a series in a variable that is some known function of time and for all of whose values the series
converges uniformly In the mathematical field of analysis, uniform convergence is a mode of convergence of functions stronger than pointwise convergence. A sequence of functions (f_n) converges uniformly to a limiting function f on a set E if, given any arbitrarily s ...
.
In case the problem could not be solved, any other important contribution to classical mechanics would then be considered to be prizeworthy. The prize was finally awarded to Poincaré, even though he did not solve the original problem. One of the judges, the distinguished
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
, said, ''"This work cannot indeed be considered as furnishing the complete solution of the question proposed, but that it is nevertheless of such importance that its publication will inaugurate a new era in the history of celestial mechanics."'' (The first version of his contribution even contained a serious error; for details see the article by Diacu and the book by Barrow-Green). The version finally printed contained many important ideas which led to the theory of chaos. The problem as stated originally was finally solved by Karl F. Sundman for ''n'' = 3 in 1912 and was generalised to the case of ''n'' > 3 bodies by Qiudong Wang in the 1990s. The series solutions have very slow convergence. It would take millions of terms to determine the motion of the particles for even very short intervals of time, so they are unusable in numerical work.


Work on relativity


Local time

Poincaré's work at the Bureau des Longitudes on establishing international time zones led him to consider how clocks at rest on the Earth, which would be moving at different speeds relative to absolute space (or the "
luminiferous aether Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
"), could be synchronised. At the same time Dutch theorist Hendrik Lorentz was developing Maxwell's theory into a theory of the motion of charged particles ("electrons" or "ions"), and their interaction with radiation. In 1895 Lorentz had introduced an auxiliary quantity (without physical interpretation) called "local time" t^\prime = t-v x/c^2 \, and introduced the hypothesis of
length contraction Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as Lorentz contraction or Lorentz–FitzGera ...
to explain the failure of optical and electrical experiments to detect motion relative to the aether (see
Michelson–Morley experiment The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 188 ...
). 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 and in so doing he came up with many insights that are now associated with special relativity. In The Measure of Time (1898), Poincaré said, " A little reflection is sufficient to understand that all these affirmations have by themselves no meaning. They can have one only as the result of a convention." He also argued that scientists have to set the constancy of the speed of light as a postulate to give physical theories the simplest form. Based on these assumptions he discussed in 1900 Lorentz's "wonderful invention" of local time and remarked that it arose when moving clocks are synchronised by exchanging light signals assumed to travel with the same speed in both directions in a moving frame.. See also th
English translation
/ref>


Principle of relativity and Lorentz transformations

In 1881 Poincaré described
hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P ...
in terms of the
hyperboloid model In geometry, the hyperboloid model, also known as the Minkowski model after Hermann Minkowski, is a model of ''n''-dimensional hyperbolic geometry in which points are represented by points on the forward sheet ''S''+ of a two-sheeted hyperbo ...
, formulating transformations leaving invariant the
Lorentz interval In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
x^2+y^2-z^2=-1, which makes them mathematically equivalent to the Lorentz transformations in 2+1 dimensions. In addition, Poincaré's other models of hyperbolic geometry (
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk ...
,
Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincar� ...
) as well as the Beltrami–Klein model can be related to the relativistic velocity space (see
Gyrovector space A gyrovector space is a mathematical concept proposed by Abraham A. Ungar for studying hyperbolic geometry in analogy to the way vector spaces are used in Euclidean geometry.Abraham A. Ungar (2005), "Analytic Hyperbolic Geometry: Mathematical Fo ...
). In 1892 Poincaré developed a
mathematical theory A mathematical theory is a mathematical model of a branch of mathematics that is based on a set of axioms An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reason ...
of
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
including polarization. His vision of the action of polarizers and retarders, acting on a sphere representing polarized states, is called the
Poincaré sphere Poincaré sphere may refer to: * Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light ** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system * Poincaré homology s ...
. It was shown that the Poincaré sphere possesses an underlying Lorentzian symmetry, by which it can be used as a geometrical representation of Lorentz transformations and velocity additions. He discussed the "principle of relative motion" in two papers in 1900 and named it the
principle of relativity In physics, the principle of relativity is the requirement that the equations describing the laws of physics have the same form in all admissible frames of reference. For example, in the framework of special relativity the Maxwell equations ha ...
in 1904, according to which no physical experiment can discriminate between a state of uniform motion and a state of rest. available i
online chapter from 1913 book
/ref> In 1905 Poincaré wrote to Lorentz about Lorentz's paper of 1904, which Poincaré described as a "paper of supreme importance." In this letter he pointed out an error Lorentz had made when he had applied his transformation to one of Maxwell's equations, that for charge-occupied space, and also questioned the time dilation factor given by Lorentz. In a second letter to Lorentz, Poincaré gave his own reason why Lorentz's time dilation factor was indeed correct after all—it was necessary to make the Lorentz transformation form a group—and he gave what is now known as the relativistic velocity-addition law. Poincaré later delivered a paper at the meeting of the Academy of Sciences in Paris on 5 June 1905 in which these issues were addressed. In the published version of that he wrote:
(PDF) Membres de l'Académie des sciences depuis sa création : Henri Poincare. Sur la dynamique de l' electron. Note de H. Poincaré. C.R. T.140 (1905) 1504–1508.
The essential point, established by Lorentz, is that the equations of the electromagnetic field are not altered by a certain transformation (which I will call by the name of Lorentz) of the form: ::x^\prime = k\ell\left(x + \varepsilon t\right)\!,\;t^\prime = k\ell\left(t + \varepsilon x\right)\!,\;y^\prime = \ell y,\;z^\prime = \ell z,\;k = 1/\sqrt.
and showed that the arbitrary function \ell\left(\varepsilon\right) must be unity for all \varepsilon (Lorentz had set \ell = 1 by a different argument) to make the transformations form a group. In an enlarged version of the paper that appeared in 1906 Poincaré pointed out that the combination x^2+ y^2+ z^2- c^2t^2 is Invariant (mathematics), invariant. He noted that a Lorentz transformation is merely a rotation in four-dimensional space about the origin by introducing ct\sqrt as a fourth imaginary coordinate, and he used an early form of
four-vector In special relativity, a four-vector (or 4-vector) is an object with four components, which transform in a specific way under Lorentz transformations. Specifically, a four-vector is an element of a four-dimensional vector space considered as a ...
s. (Wikisource translation) Poincaré expressed a lack of interest in a four-dimensional reformulation of his new mechanics in 1907, because in his opinion the translation of physics into the language of four-dimensional geometry would entail too much effort for limited profit. So it was
Hermann Minkowski Hermann Minkowski (; ; 22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen. He created and developed the geometry of numbers and used geometrical methods to solve problems in number t ...
who worked out the consequences of this notion in 1907.


Mass–energy relation

Like others before, Poincaré (1900) discovered a relation between
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
and
electromagnetic energy In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions o ...
. While studying the conflict between the action/reaction principle and
Lorentz ether theory What is now often called Lorentz ether theory (LET) has its roots in Hendrik Lorentz's "theory of electrons", which was the final point in the development of the classical aether theories at the end of the 19th and at the beginning of the 20th cent ...
, he tried to determine whether the
center of gravity In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force ma ...
still moves with a uniform velocity when electromagnetic fields are included. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. Poincaré concluded that the electromagnetic field energy of an electromagnetic wave behaves like a fictitious
fluid In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
(''fluide fictif'') with a mass density of ''E''/''c''2. If the
center of mass frame In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system is ...
is defined by both the mass of matter ''and'' the mass of the fictitious fluid, and if the fictitious fluid is indestructible— it's neither created or destroyed—then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions. However, Poincaré's resolution led to a
paradox A paradox is a logically self-contradictory statement or a statement that runs contrary to one's expectation. It is a statement that, despite apparently valid reasoning from true premises, leads to a seemingly self-contradictory or a logically u ...
when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a
recoil Recoil (often called knockback, kickback or simply kick) is the rearward thrust generated when a gun is being discharged. In technical terms, the recoil is a result of conservation of momentum, as according to Newton's third law the force r ...
from the inertia of the fictitious fluid. Poincaré performed a
Lorentz boost In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation ...
(to order ''v''/''c'') to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow
perpetual motion Perpetual motion is the motion of bodies that continues forever in an unperturbed system. A perpetual motion machine is a hypothetical machine that can do work infinitely without an external energy source. This kind of machine is impossible, a ...
, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold. Therefore, he argued that also in this case there has to be another compensating mechanism in the
ether In organic chemistry, ethers are a class of compounds that contain an ether group—an oxygen atom connected to two alkyl or aryl groups. They have the general formula , where R and R′ represent the alkyl or aryl groups. Ethers can again ...
. Poincaré himself came back to this topic in his St. Louis lecture (1904). He rejected the possibility that energy carries mass and criticized his own solution to compensate the above-mentioned problems: In the above quote he refers to the Hertz assumption of total aether entrainment that was falsified by the Fizeau_experiment but that experiment does indeed show that that light is partially "carried along" with a substance. Finally in 1908 he revisits the problem and ends with abandoning the principle of reaction altogether in favor of supporting a solution based in the inertia of aether itself. He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass \gamma m, Abraham's theory of variable mass and
Kaufmann Kaufmann is a surname with many variants such as Kauffmann, Kaufman, and Kauffman. In German, the name means '' merchant''. It is the cognate of the English ''Chapman'' (which had a similar meaning in the Middle Ages, though it disappeared from ...
's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of
Marie Curie Marie Salomea Skłodowska–Curie ( , , ; born Maria Salomea Skłodowska, ; 7 November 1867 – 4 July 1934) was a Polish and naturalized-French physicist and chemist who conducted pioneering research on radioactivity. She was the fir ...
. It was
Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theor ...
's concept of
mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
(1905) that a body losing energy as radiation or heat was losing mass of amount ''m'' = ''E''/''c''2 that resolvedDarrigol 2005, Secondary sources on relativity Poincaré's paradox, without using any compensating mechanism within the ether. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. However, concerning Poincaré's solution of the Center of Gravity problem, Einstein noted that Poincaré's formulation and his own from 1906 were mathematically equivalent.


Gravitational waves

In 1905 Poincaré first proposed
gravitational waves Gravitational waves are waves of the intensity of gravity generated by the accelerated masses of an orbital binary system that propagate as waves outward from their source at the speed of light. They were first proposed by Oliver Heaviside i ...
(''ondes gravifiques'') emanating from a body and propagating at the speed of light. He wrote:


Poincaré and Einstein

Einstein's first paper on relativity was published three months after Poincaré's short paper, but before Poincaré's longer version. Einstein relied on the principle of relativity to derive the Lorentz transformations and used a similar clock synchronisation procedure ( Einstein synchronisation) to the one that Poincaré (1900) had described, but Einstein's paper was remarkable in that it contained no references at all. Poincaré never acknowledged Einstein's work on
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
. However, Einstein expressed sympathy with Poincaré's outlook obliquely in a letter to
Hans Vaihinger Hans Vaihinger (; September 25, 1852 – December 18, 1933) was a German philosopher, best known as a Kant scholar and for his ''Die Philosophie des Als Ob'' ('' The Philosophy of 'As if), published in 1911 although its statement of basi ...
on 3 May 1919, when Einstein considered Vaihinger's general outlook to be close to his own and Poincaré's to be close to Vaihinger's. In public, Einstein acknowledged Poincaré posthumously in the text of a lecture in 1921 titled "''Geometrie und Erfahrung'' (Geometry and Experience)" in connection with
non-Euclidean geometry In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean g ...
, but not in connection with special relativity. A few years before his death, Einstein commented on Poincaré as being one of the pioneers of relativity, saying "Lorentz had already recognized that the transformation named after him is essential for the analysis of Maxwell's equations, and Poincaré deepened this insight still further ...."


Assessments on Poincaré and relativity

Poincaré's work in the development of special relativity is well recognised, though most historians stress that despite many similarities with Einstein's work, the two had very different research agendas and interpretations of the work. Poincaré developed a similar physical interpretation of local time and noticed the connection to signal velocity, but contrary to Einstein he continued to use the ether-concept in his papers and argued that clocks at rest in the ether show the "true" time, and moving clocks show the local time. So Poincaré tried to keep the relativity principle in accordance with classical concepts, while Einstein developed a mathematically equivalent kinematics based on the new physical concepts of the relativity of space and time. While this is the view of most historians, a minority go much further, such as E. T. Whittaker, who held that Poincaré and Lorentz were the true discoverers of relativity.


Algebra and number theory

Poincaré introduced
group theory In abstract algebra, group theory studies the algebraic structures known as groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces, can all be seen ...
to physics, and was the first to study the group of
Lorentz transformations In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
. He also made major contributions to the theory of discrete groups and their representations.


Topology

The subject is clearly defined by
Felix Klein Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and grou ...
in his "Erlangen Program" (1872): the geometry invariants of arbitrary continuous transformation, a kind of geometry. The term "topology" was introduced, as suggested by Johann Benedict Listing, instead of previously used "Analysis situs". Some important concepts were introduced by Enrico Betti and
Bernhard Riemann Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...
. But the foundation of this science, for a space of any dimension, was created by Poincaré. His first article on this topic appeared in 1894. His research in
geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...
led to the abstract topological definition of
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deform ...
and homology. He also first introduced the basic concepts and invariants of combinatorial topology, such as Betti numbers and the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, o ...
. Poincaré proved a formula relating the number of edges, vertices and faces of ''n''-dimensional
polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all o ...
(the Euler–Poincaré theorem) and gave the first precise formulation of the intuitive notion of dimension.


Astronomy and celestial mechanics

Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). They introduced the small parameter method, fixed points, integral invariants, variational equations, the convergence of the asymptotic expansions. Generalizing a theory of Bruns (1887), Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
ic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since
Isaac Newton Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
. These monographs include an idea of Poincaré, which later became the basis for mathematical "
chaos theory Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to hav ...
" (see, in particular, the
Poincaré recurrence theorem In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (fo ...
) and the general theory of
dynamical system In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water i ...
s. Poincaré authored important works on
astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...
for the equilibrium figures of a gravitating rotating fluid. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).


Differential equations and mathematical physics

After defending his doctoral thesis on the study of singular points of the system of
differential equations In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
, Poincaré wrote a series of memoirs under the title "On curves defined by differential equations" (1881–1882). In these articles, he built a new branch of mathematics, called "
qualitative theory of differential equations In mathematics, the qualitative theory of differential equations studies the behavior of differential equations by means other than finding their solutions. It originated from the works of Henri Poincaré and Aleksandr Lyapunov. There are relativel ...
". Poincaré showed that even if the differential equation can not be solved in terms of known functions, yet from the very form of the equation, a wealth of information about the properties and behavior of the solutions can be found. In particular, Poincaré investigated the nature of the trajectories of the integral curves in the plane, gave a classification of singular points (
saddle The saddle is a supportive structure for a rider of an animal, fastened to an animal's back by a girth. The most common type is equestrian. However, specialized saddles have been created for oxen, camels and other animals. It is not kno ...
, focus, center,
node In general, a node is a localized swelling (a " knot") or a point of intersection (a vertex). Node may refer to: In mathematics * Vertex (graph theory), a vertex in a mathematical graph * Vertex (geometry), a point where two or more curves, line ...
), introduced the concept of a
limit cycle In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity o ...
and the loop index, and showed that the number of limit cycles is always finite, except for some special cases. Poincaré also developed a general theory of integral invariants and solutions of the variational equations. For the finite-difference equations, he created a new direction – the
asymptotic In analytic geometry, an asymptote () of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the ''x'' or ''y'' coordinates tends to infinity. In projective geometry and related context ...
analysis of the solutions. He applied all these achievements to study practical problems of
mathematical physics Mathematical physics refers to the development of mathematical methods for application to problems in physics. The '' Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the developm ...
and
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
, and the methods used were the basis of its topological works. File: Phase Portrait Sadle.svg , Saddle File: Phase Portrait Stable Focus.svg , Focus File: Phase portrait center.svg , Center File: Phase Portrait Stable Node.svg , Node


Character

Poincaré's work habits have been compared to a
bee Bees are winged insects closely related to wasps and ants, known for their roles in pollination and, in the case of the best-known bee species, the western honey bee, for producing honey. Bees are a monophyletic lineage within the superfami ...
flying from flower to flower. Poincaré was interested in the way his
mind The mind is the set of faculties responsible for all mental phenomena. Often the term is also identified with the phenomena themselves. These faculties include thought, imagination, memory, will, and sensation. They are responsible for various m ...
worked; he studied his habits and gave a talk about his observations in 1908 at the Institute of General Psychology in
Paris Paris () is the capital and most populous city of France, with an estimated population of 2,165,423 residents in 2019 in an area of more than 105 km² (41 sq mi), making it the 30th most densely populated city in the world in 2020. Si ...
. He linked his way of thinking to how he made several discoveries. The mathematician Darboux claimed he was ''un intuitif'' (an intuitive), arguing that this is demonstrated by the fact that he worked so often by visual representation.
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...
wrote that Poincaré's research demonstrated marvelous clarity and Poincaré himself wrote that he believed that logic was not a way to invent but a way to structure ideas and that logic limits ideas.


Toulouse's characterisation

Poincaré's mental organisation was interesting not only to Poincaré himself but also to Édouard Toulouse, a
psychologist A psychologist is a professional who practices psychology and studies mental states, perceptual Perception () is the organization, identification, and interpretation of sensory information in order to represent and understand the pre ...
of the Psychology Laboratory of the School of Higher Studies in Paris. Toulouse wrote a book entitled ''Henri Poincaré'' (1910). In it, he discussed Poincaré's regular schedule: * He worked during the same times each day in short periods of time. He undertook mathematical research for four hours a day, between 10 a.m. and noon then again from 5 p.m. to 7 p.m.. He would read articles in journals later in the evening. * His normal work habit was to solve a problem completely in his head, then commit the completed problem to paper. * He was
ambidextrous Ambidexterity is the ability to use both the right and left hand equally well. When referring to objects, the term indicates that the object is equally suitable for right-handed and left-handed people. When referring to humans, it indicates that ...
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 the lecturer wrote on the blackboard. These abilities were offset to some extent by his shortcomings: * He was physically clumsy and artistically inept. * He was always in a rush and disliked going back for changes or corrections. * He never spent a long time on a problem since he believed that the subconscious would continue working on the problem while he consciously worked on another problem. In addition, Toulouse stated that most mathematicians worked from principles already established while Poincaré started from basic principles each time (O'Connor et al., 2002). His method of thinking is well summarised as:


Publications

* * * * * * * * * * *


Honours

Awards *Oscar II, King of Sweden's mathematical competition (1887) *Foreign member of the
Royal Netherlands Academy of Arts and Sciences The Royal Netherlands Academy of Arts and Sciences ( nl, Koninklijke Nederlandse Akademie van Wetenschappen, abbreviated: KNAW) is an organization dedicated to the advancement of science and literature in the Netherlands. The academy is housed ...
(1897) *
American Philosophical Society The American Philosophical Society (APS), founded in 1743 in Philadelphia, is a scholarly organization that promotes knowledge in the sciences and humanities through research, professional meetings, publications, library resources, and communit ...
1899 *
Gold Medal of the Royal Astronomical Society The Gold Medal of the Royal Astronomical Society is the highest award given by the Royal Astronomical Society (RAS). The RAS Council have "complete freedom as to the grounds on which it is awarded" and it can be awarded for any reason. Past awar ...
of London (1900) * Bolyai Prize in 1905 *
Matteucci Medal The Matteucci Medal is an Italian award for physicists, named after Carlo Matteucci from Forlì. It was established to award physicists for their fundamental contributions. Under an Italian Royal Decree dated July 10, 1870, the Italian Society ...
1905 *
French Academy of Sciences The French Academy of Sciences (French: ''Académie des sciences'') is a learned society, founded in 1666 by Louis XIV at the suggestion of Jean-Baptiste Colbert, to encourage and protect the spirit of French scientific research. It was at ...
1906 *
Académie française An academy (Attic Greek: Ἀκαδήμεια; Koine Greek Ἀκαδημία) is an institution of secondary or tertiary higher learning (and generally also research or honorary membership). The name traces back to Plato's school of philosop ...
1909 *
Bruce Medal The Catherine Wolfe Bruce Gold Medal is awarded every year by the Astronomical Society of the Pacific for outstanding lifetime contributions to astronomy. It is named after Catherine Wolfe Bruce, an American patroness of astronomy, and was fi ...
(1911) Named after him *
Institut Henri Poincaré The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrond ...
(mathematics and theoretical physics center) * Poincaré Prize (Mathematical Physics International Prize) *
Annales Henri Poincaré The ''Annales Henri Poincaré'' (''A Journal of Theoretical and Mathematical Physics'') is a peer-reviewed scientific journal which collects and publishes original research papers in the field of theoretical and mathematical physics. The emphas ...
(Scientific Journal) *Poincaré Seminar (nicknamed " Bourbaphy") *The crater Poincaré on the Moon *
Asteroid An asteroid is a minor planet of the inner Solar System. Sizes and shapes of asteroids vary significantly, ranging from 1-meter rocks to a dwarf planet almost 1000 km in diameter; they are rocky, metallic or icy bodies with no atmosphere. ...
2021 Poincaré *
List of things named after Henri Poincaré In physics and mathematics, a number of ideas are named after Henri Poincaré: * Euler–Poincaré characteristic * Hilbert–Poincaré series * Poincaré–Bendixson theorem * Poincaré–Birkhoff theorem * Poincaré–Birkhoff–Witt theorem, us ...
Henri Poincaré did not receive the
Nobel Prize in Physics ) , image = Nobel Prize.png , alt = A golden medallion with an embossed image of a bearded man facing left in profile. To the left of the man is the text "ALFR•" then "NOBEL", and on the right, the text (smaller) "NAT•" then " ...
, but he had influential advocates like
Henri Becquerel Antoine Henri Becquerel (; 15 December 1852 – 25 August 1908) was a French engineer, physicist, Nobel laureate, and the first person to discover evidence of radioactivity. For work in this field he, along with Marie Skłodowska-Curie and Pie ...
or committee member
Gösta Mittag-Leffler Magnus Gustaf "Gösta" Mittag-Leffler (16 March 1846 – 7 July 1927) was a Swedish mathematician. His mathematical contributions are connected chiefly with the theory of functions, which today is called complex analysis. Biography Mittag-Leffl ...
. The nomination archive reveals that Poincaré received a total of 51 nominations between 1904 and 1912, the year of his death. Of the 58 nominations for the 1910 Nobel Prize, 34 named Poincaré. Nominators included Nobel laureates Hendrik Lorentz and Pieter Zeeman (both of 1902),
Marie Curie Marie Salomea Skłodowska–Curie ( , , ; born Maria Salomea Skłodowska, ; 7 November 1867 – 4 July 1934) was a Polish and naturalized-French physicist and chemist who conducted pioneering research on radioactivity. She was the fir ...
(of 1903),
Albert Michelson Albert Abraham Michelson FFRS HFRSE (surname pronunciation anglicized as "Michael-son", December 19, 1852 – May 9, 1931) was a German-born American physicist of Polish/Jewish origin, known for his work on measuring the speed of light and espe ...
(of 1907),
Gabriel Lippmann Jonas Ferdinand Gabriel Lippmann (16 August 1845 – 13 July 1921) was a Franco-Luxembourgish physicist and inventor, and Nobel laureate in physics for his method of reproducing colours photographically based on the phenomenon of interference. ...
(of 1908) and
Guglielmo Marconi Guglielmo Giovanni Maria Marconi, 1st Marquis of Marconi (; 25 April 187420 July 1937) was an Italian inventor and electrical engineer, known for his creation of a practical radio wave-based wireless telegraph system. This led to Marconi ...
(of 1909). The fact that renowned
theoretical physicists The following is a partial list of notable theoretical physicists. Arranged by century of birth, then century of death, then year of birth, then year of death, then alphabetically by surname. For explanation of symbols, see Notes at end of this ar ...
like Poincaré, Boltzmann or Gibbs were not awarded the
Nobel Prize The Nobel Prizes ( ; sv, Nobelpriset ; no, Nobelprisen ) are five separate prizes that, according to Alfred Nobel's will of 1895, are awarded to "those who, during the preceding year, have conferred the greatest benefit to humankind." Alfr ...
is seen as evidence that the Nobel committee had more regard for experimentation than theory. In Poincaré's case, several of those who nominated him pointed out that the greatest problem was to name a specific discovery, invention, or technique.


Philosophy

Poincaré had
philosophical Philosophy (from , ) is the systematized study of general and fundamental questions, such as those about existence, reason, knowledge, values, mind, and language. Such questions are often posed as problems to be studied or resolved. Some ...
views opposite to those of
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, a ...
and
Gottlob Frege Friedrich Ludwig Gottlob Frege (; ; 8 November 1848 – 26 July 1925) was a German philosopher, logician, and mathematician. He was a mathematics professor at the University of Jena, and is understood by many to be the father of analytic p ...
, who believed that mathematics was a branch of
logic Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...
. Poincaré strongly disagreed, claiming that
intuition Intuition is the ability to acquire knowledge without recourse to conscious reasoning. Different fields use the word "intuition" in very different ways, including but not limited to: direct access to unconscious knowledge; unconscious cognition; ...
was the life of mathematics. Poincaré gives an interesting point of view in his 1902 book '' Science and Hypothesis'': Poincaré believed that
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
is
synthetic Synthetic things are composed of multiple parts, often with the implication that they are artificial. In particular, 'synthetic' may refer to: Science * Synthetic chemical or compound, produced by the process of chemical synthesis * Synthetic ...
. He argued that
Peano's axioms In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearl ...
cannot be proven non-circularly with the principle of induction (Murzi, 1998), therefore concluding that arithmetic is ''
a priori ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ex ...
'' synthetic and not analytic. Poincaré then went on to say that mathematics cannot be deduced from logic since it is not analytic. His views were similar to those of
Immanuel Kant Immanuel Kant (, , ; 22 April 1724 – 12 February 1804) was a German philosopher and one of the central Enlightenment thinkers. Born in Königsberg, Kant's comprehensive and systematic works in epistemology, metaphysics, ethics, and ...
(Kolak, 2001, Folina 1992). He strongly opposed Cantorian
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly concern ...
, objecting to its use of
impredicative In mathematics, logic and philosophy of mathematics, something that is impredicative is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more co ...
definitions. 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. Poincaré held that convention plays an important role in physics. His view (and some later, more extreme versions of it) came to be known as " conventionalism". Poincaré believed that Newton's first law was not empirical but is a conventional framework assumption for
mechanics Mechanics (from Ancient Greek: μηχανική, ''mēkhanikḗ'', "of machines") is the area of mathematics and physics concerned with the relationships between force, matter, and motion among physical objects. Forces applied to objec ...
(Gargani, 2012). He also believed that the geometry of physical space is conventional. He considered examples in which either the geometry of the physical fields or gradients of temperature can be changed, either describing a space as non-Euclidean measured by rigid rulers, or as a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...
where the rulers are expanded or shrunk by a variable heat distribution. However, Poincaré thought that we were so accustomed to
Euclidean geometry Euclidean geometry is a mathematical system attributed to ancient Greek mathematician Euclid, which he described in his textbook on geometry: the '' Elements''. Euclid's approach consists in assuming a small set of intuitively appealing axioms ...
that we would prefer to change the physical laws to save Euclidean geometry rather than shift to a non-Euclidean physical geometry.


Free will

Poincaré's famous lectures before the Société de Psychologie in Paris (published as '' Science and Hypothesis'', '' The Value of Science'', and ''Science and Method'') were cited by
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...
as the source for the idea that
creativity Creativity is a phenomenon whereby something new and valuable is formed. The created item may be intangible (such as an idea, a scientific theory, a musical composition, or a joke) or a physical object (such as an invention, a printed lit ...
and
invention An invention is a unique or novel device, method, composition, idea or process. An invention may be an improvement upon a machine, product, or process for increasing efficiency or lowering cost. It may also be an entirely new concept. If an ...
consist of two mental stages, first random combinations of possible solutions to a problem, followed by a
critical Critical or Critically may refer to: *Critical, or critical but stable, medical states **Critical, or intensive care medicine * Critical juncture, a discontinuous change studied in the social sciences. *Critical Software, a company specializing ...
evaluation Evaluation is a systematic determination and assessment of a subject's merit, worth and significance, using criteria governed by a set of standards. It can assist an organization, program, design, project or any other intervention or initiative to ...
. Although he most often spoke of a
deterministic Determinism is a philosophical view, where all events are determined completely by previously existing causes. Deterministic theories throughout the history of philosophy have developed from diverse and sometimes overlapping motives and cons ...
universe The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univers ...
, Poincaré said that the subconscious generation of new possibilities involves chance.
It is certain that the combinations which present themselves to the mind in a kind of sudden illumination after a somewhat prolonged period of unconscious work are generally useful and fruitful combinations... all the combinations are formed as a result of the automatic action of the subliminal ego, but those only which are interesting find their way into the field of consciousness... A few only are harmonious, and consequently at once useful and beautiful, and they will be capable of affecting the geometrician's special sensibility I have been speaking of; which, once aroused, will direct our attention upon them, and will thus give them the opportunity of becoming conscious... In the subliminal ego, on the contrary, there reigns what I would call liberty, if one could give this name to the mere absence of discipline and to disorder born of chance.
Poincaré's two stages—random combinations followed by selection—became the basis for
Daniel Dennett Daniel Clement Dennett III (born March 28, 1942) is an American philosopher, writer, and cognitive scientist whose research centers on the philosophy of mind, philosophy of science, and philosophy of biology, particularly as those fields relat ...
's two-stage model of
free will Free will is the capacity of agents to choose between different possible courses of action unimpeded. Free will is closely linked to the concepts of moral responsibility, praise, culpability, sin, and other judgements which apply only to ac ...
.


Bibliography


Poincaré's writings in English translation

Popular writings on the
philosophy of science Philosophy of science is a branch of philosophy concerned with the foundations, methods, and implications of science. The central questions of this study concern what qualifies as science, the reliability of scientific theories, and the ultim ...
: *; reprinted in 1921; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908). * 1905. "," The Walter Scott Publishing Co. * 1906. "," Athenæum * 1913. "The New Mechanics," The Monist, Vol. XXIII. * 1913. "The Relativity of Space," The Monist, Vol. XXIII. * 1913. * 1956. ''Chance.'' In James R. Newman, ed., The World of Mathematics (4 Vols). * 1958. ''The Value of Science,'' New York: Dover. On
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
: * 1895. . The first systematic study of
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
. On
celestial mechanics Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, ...
: * 1890. * 1892–99. ''New Methods of Celestial Mechanics'', 3 vols. English trans., 1967. . * 1905. "The Capture Hypothesis of J. J. See," The Monist, Vol. XV. * 1905–10. ''Lessons of Celestial Mechanics''. On the
philosophy of mathematics The philosophy of mathematics is the branch of philosophy that studies the assumptions, foundations, and implications of mathematics. It aims to understand the nature and methods of mathematics, and find out the place of mathematics in people' ...
: * Ewald, William B., ed., 1996. ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics'', 2 vols. Oxford Univ. Press. Contains the following works by Poincaré: ** 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. * 1905. "The Principles of Mathematical Physics," The Monist, Vol. XV. * 1910. "The Future of Mathematics," The Monist, Vol. XX. * 1910. "Mathematical Creation," The Monist, Vol. XX. Other: * 1904. ''Maxwell's Theory and Wireless Telegraphy,'' New York, McGraw Publishing Company. * 1905. "The New Logics," The Monist, Vol. XV. * 1905. "The Latest Efforts of the Logisticians," The Monist, Vol. XV. Exhaustive bibliography of English translations: * 1892–2017. .


See also


Concepts

* Poincaré–Andronov–Hopf bifurcation *
Poincaré complex In mathematics, and especially topology, a Poincaré complex (named after the mathematician Henri Poincaré) is an abstraction of the singular chain complex of a closed, orientable manifold. The singular homology and cohomology groups of a closed, ...
– an abstraction of the singular chain complex of a closed, orientable manifold *
Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if ''M'' is an ''n''-dimensional oriented closed manifold (compact ...
*
Poincaré disk model In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which all points are inside the unit disk, and straight lines are either circular arcs contained within the disk ...
* Poincaré expansion *
Poincaré gauge In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes a mathematical procedure for coping with redundant degrees of freedom in field variables. By definition, a gauge theory represents each physically distinct con ...
* Poincaré group *
Poincaré half-plane model In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H = \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry. Equivalently the Poincar� ...
*
Poincaré homology sphere Poincaré is a French surname. Notable people with the surname include: * Henri Poincaré (1854–1912), French physicist, mathematician and philosopher of science * Henriette Poincaré (1858-1943), wife of Prime Minister Raymond Poincaré * Luci ...
* Poincaré inequality * Poincaré lemma * Poincaré map *
Poincaré residue In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions. Given a hypersurfa ...
*
Poincaré series (modular form) In number theory, a Poincaré series is a mathematical series generalizing the classical theta series that is associated to any discrete group of symmetries of a complex domain, possibly of several complex variables. In particular, they general ...
* Poincaré space * Poincaré metric * Poincaré plot * Poincaré polynomial * Poincaré series *
Poincaré sphere Poincaré sphere may refer to: * Poincaré sphere (optics), a graphical tool for visualizing different types of polarized light ** Bloch sphere, a related tool for representing states of a two-level quantum mechanical system * Poincaré homology s ...
* Poincaré–Einstein synchronisation * Poincaré–Lelong equation *
Poincaré–Lindstedt method In perturbation theory, the Poincaré–Lindstedt method or Lindstedt–Poincaré method is a technique for uniformly approximating periodic solutions to ordinary differential equations, when regular perturbation approaches fail. The method remove ...
* Poincaré–Lindstedt perturbation theory * Poincaré–Steklov operator * Euler–Poincaré characteristic * Neumann–Poincaré operator * Reflecting Function


Theorems

Here is a list of theorems proved by Poincaré: * Poincaré's recurrence theorem: certain systems will, after a sufficiently long but finite time, return to a state very close to the initial state. * Poincaré–Bendixson theorem: a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. *
Poincaré–Hopf theorem In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincar� ...
: a generalization of the hairy-ball theorem, which states that there is no smooth vector field on a sphere having no sources or sinks. * Poincaré–Lefschetz duality theorem: a version of Poincaré duality in geometric topology, applying to a manifold with boundary * Poincaré separation theorem: gives the upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. * Poincaré–Birkhoff theorem: every area-preserving, orientation-preserving homeomorphism of an annulus that rotates the two boundaries in opposite directions has at least two fixed points. * Poincaré–Birkhoff–Witt theorem: an explicit description of the universal enveloping algebra of a Lie algebra. * Poincaré–Bjerknes circulation theorem: theorem about a conservation of quantity for the rotating frame. *
Poincaré conjecture In the mathematical field of geometric topology, the Poincaré conjecture (, , ) is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. Originally conjectured ...
(now a theorem): Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. * Poincaré–Miranda theorem: a generalization of the
intermediate value theorem In mathematical analysis, the intermediate value theorem states that if f is a continuous function whose domain contains the interval , then it takes on any given value between f(a) and f(b) at some point within the interval. This has two impor ...
to ''n'' dimensions.


Other

* French epistemology * History of special relativity *
List of things named after Henri Poincaré In physics and mathematics, a number of ideas are named after Henri Poincaré: * Euler–Poincaré characteristic * Hilbert–Poincaré series * Poincaré–Bendixson theorem * Poincaré–Birkhoff theorem * Poincaré–Birkhoff–Witt theorem, us ...
*
Institut Henri Poincaré The Henri Poincaré Institute (or IHP for ''Institut Henri Poincaré'') is a mathematics research institute part of Sorbonne University, in association with the Centre national de la recherche scientifique (CNRS). It is located in the 5th arrond ...
, Paris *
Brouwer fixed-point theorem Brouwer's fixed-point theorem is a fixed-point theorem in topology, named after L. E. J. (Bertus) Brouwer. It states that for any continuous function f mapping a compact convex set to itself there is a point x_0 such that f(x_0)=x_0. The simples ...
*
Relativity priority dispute Albert Einstein presented the theories of special relativity and general relativity in publications that either contained no formal references to previous literature, or referred only to a small number of his predecessors for fundamental result ...
*
Epistemic structural realism In the philosophy of science, structuralism (also known as scientific structuralism or as the structuralistic theory-concept) asserts that all aspects of reality are best understood in terms of empirical scientific constructs of entities and their ...
"Structural Realism"
entry by James Ladyman in the ''
Stanford Encyclopedia of Philosophy The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by Stanford University. E ...
''


References


Footnotes


Sources

* Bell, Eric Temple, 1986. ''Men of Mathematics'' (reissue edition). Touchstone Books. . * Belliver, André, 1956. ''Henri Poincaré ou la vocation souveraine''. Paris: Gallimard. * Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story of Risk". (p. 199–200). John Wiley & Sons. * Boyer, B. Carl, 1968. ''A History of Mathematics: Henri Poincaré'', John Wiley & Sons. * Grattan-Guinness, Ivor, 2000. ''The Search for Mathematical Roots 1870–1940.'' Princeton Uni. Press. * . Internet version published in Journal of the ACMS 2004. * Folina, Janet, 1992. ''Poincaré and the Philosophy of Mathematics.'' Macmillan, New York. * Gray, Jeremy, 1986. ''Linear differential equations and group theory from Riemann to Poincaré'', Birkhauser * Gray, Jeremy, 2013. ''Henri Poincaré: A scientific biography''. Princeton University Press * * Kolak, Daniel, 2001. ''Lovers of Wisdom'', 2nd ed. Wadsworth. * Gargani, Julien, 2012. ''Poincaré, le hasard et l'étude des systèmes complexes'', L'Harmattan. * 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. . * Sageret, Jules, 1911. ''Henri Poincaré''. Paris: Mercure de France. * Toulouse, E.,1910. ''Henri Poincaré''.—(Source biography in French) at University of Michigan Historic Math Collection. * * Verhulst, Ferdinand, 2012 ''Henri Poincaré. Impatient Genius''. N.Y.: Springer. * ''Henri Poincaré, l'œuvre scientifique, l'œuvre philosophique'', by Vito Volterra, Jacques Hadamard, Paul Langevin and Pierre Boutroux, Felix Alcan, 1914. ** ''Henri Poincaré, l'œuvre mathématique'', by
Vito Volterra Vito Volterra (, ; 3 May 1860 – 11 October 1940) was an Italian mathematician and physicist, known for his contributions to mathematical biology and integral equations, being one of the founders of functional analysis. Biography Born in An ...
. ** ''Henri Poincaré, le problème des trois corps'', by
Jacques Hadamard Jacques Salomon Hadamard (; 8 December 1865 – 17 October 1963) was a French mathematician who made major contributions in number theory, complex analysis, differential geometry and partial differential equations. Biography The son of a teac ...
. ** ''Henri Poincaré, le physicien'', by
Paul Langevin Paul Langevin (; ; 23 January 1872 – 19 December 1946) was a French physicist who developed Langevin dynamics and the Langevin equation. He was one of the founders of the ''Comité de vigilance des intellectuels antifascistes'', an ant ...
. ** ''Henri Poincaré, l'œuvre philosophique'', by Pierre Boutroux. *


Further reading


Secondary sources to work on relativity

* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *


Non-mainstream sources

* *


External links

* * *
Henri Poincaré's Bibliography
*
Internet Encyclopedia of Philosophy The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original p ...
:
Henri Poincaré
"—by Mauro Murzi. *
Internet Encyclopedia of Philosophy The ''Internet Encyclopedia of Philosophy'' (''IEP'') is a scholarly online encyclopedia, dealing with philosophy, philosophical topics, and philosophers. The IEP combines open access publication with peer reviewed publication of original p ...
:
Poincaré’s Philosophy of Mathematics
—by Janet Folina. *
Henri Poincaré on Information Philosopher
*
A timeline of Poincaré's life
University of Nantes (in French).
Henri Poincaré Papers
University of Nantes (in French).

*Collins, Graham P.,
Henri Poincaré, His Conjecture, Copacabana and Higher Dimensions
" ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it ...
'', 9 June 2004. *BBC in Our Time,
Discussion of the Poincaré conjecture
" 2 November 2006, hosted by Melvynn Bragg.
Poincare Contemplates Copernicus
at MathPages
High Anxieties – The Mathematics of Chaos
(2008) BBC documentary directed by David Malone looking at the influence of Poincaré's discoveries on 20th Century mathematics. {{DEFAULTSORT:Poincare, Henri 1854 births 1912 deaths 19th-century essayists 19th-century French male writers 19th-century French mathematicians 19th-century French non-fiction writers 19th-century French philosophers 20th-century essayists 20th-century French male writers 20th-century French mathematicians 20th-century French non-fiction writers 20th-century French philosophers Algebraic geometers Burials at Montparnasse Cemetery Chaos theorists Continental philosophers Corps des mines Corresponding members of the Saint Petersburg Academy of Sciences Deaths from embolism Determinists Dynamical systems theorists École Polytechnique alumni Fluid dynamicists Foreign associates of the National Academy of Sciences Foreign Members of the Royal Society French male essayists French male non-fiction writers French male writers French military personnel of the Franco-Prussian War French mining engineers French geometers Lecturers Mathematical analysts Members of the Académie Française Members of the Royal Netherlands Academy of Arts and Sciences Mines ParisTech alumni Officers of the French Academy of Sciences Scientists from Nancy, France Philosophers of logic Philosophers of mathematics Philosophers of psychology Philosophers of science Philosophy academics Philosophy writers Recipients of the Bruce Medal Recipients of the Gold Medal of the Royal Astronomical Society French relativity theorists Thermodynamicists Topologists University of Paris faculty Recipients of the Matteucci Medal