Jules Henri Poincaré (29 April 1854 – 17 July
1912) ( ) was a French
mathematician
and theoretical
physicist, and a
philosopher of science. Poincaré is
often described as a
polymath, and in
mathematics as
The Last Universalist, since he excelled in
all fields of the discipline as it existed during his
lifetime.
As a mathematician and physicist, he made many original fundamental
contributions to
pure and
applied mathematics,
mathematical physics, and
celestial mechanics. He was responsible
for formulating the
Poincaré
conjecture, one of the most famous problems in mathematics. In
his research on the
threebody
problem, Poincaré became the first person to discover a chaotic
deterministic system which laid the foundations of modern
chaos theory. He is considered to be one of the
founders of the field of
topology.
Poincaré introduced the modern
principle of relativity and was the
first to present the
Lorentz
transformations in their modern symmetrical form. Poincaré
discovered the remaining relativistic velocity transformations and
recorded them in a letter to Lorentz in 1905. Thus he obtained
perfect invariance of all of
Maxwell's equations, an important step
in the formulation of the theory of
special relativity.
The
Poincaré group used in
physics and mathematics was named after him.
Life
Poincaré
was born on 29 April 1854 in Cité Ducale neighborhood, Nancy, MeurtheetMoselle into an influential family (Belliver, 1956).
His father Leon Poincaré (1828–1892) was a professor of medicine at
the
University of Nancy
(Sagaret, 1911). His adored younger sister Aline married the
spiritual philosopher
Emile Boutroux.
Another notable member of Jules' family was his cousin,
Raymond Poincaré, who would become the
President of France, 1913 to 1920, and a fellow member of the
Académie
française.
Education
During his childhood he was seriously ill for a time with
diphtheria and received special instruction from
his mother, Eugénie Launois (1830–1897).
In 1862 Henri entered the Lycée in Nancy (now renamed the Lycée
Henri Poincaré in his honour, along with the University of Nancy).
He spent eleven years at the Lycée and during this time he proved
to be one of the top students in every topic he studied. He
excelled in written composition. His mathematics teacher described
him as a "monster of mathematics" and he won first prizes in the
concours général, a
competition between the top pupils from all the Lycées across
France. His poorest subjects were music and physical education,
where he was described as "average at best" (O'Connor et al.,
2002). However, poor eyesight and a tendency towards
absentmindedness may explain these difficulties (Carl, 1968). He
graduated from the Lycée in 1871 with a Bachelor's degree in
letters and sciences.
During the
FrancoPrussian War
of 1870 he served alongside his father in the Ambulance
Corps.
Poincaré
entered the École Polytechnique in 1873. There he studied mathematics as a
student of
Charles Hermite,
continuing to excel and publishing his first paper
(
Démonstration nouvelle des propriétés de l'indicatrice d'une
surface) in 1874. He graduated in 1875 or 1876. He went on to
study at the
École des Mines,
continuing to study mathematics in addition to the mining
engineering syllabus and received the degree of ordinary engineer
in March 1879.
As a graduate of the École des Mines he joined the
Corps des Mines as an inspector for the
Vesoul region in northeast France. He was on
the scene of a mining disaster at
Magny in
August 1879 in which 18 miners died. He carried out the official
investigation into the accident in a characteristically thorough
and humane way.
At the same time, Poincaré was preparing for his doctorate in
sciences in mathematics under the supervision of
Charles Hermite. His doctoral thesis was in
the field of
differential
equations. It was named
Sur les propriétés des fonctions
définies par les équations différences. Poincaré devised a new
way of studying the properties of these equations. He not only
faced the question of determining the integral of such equations,
but also was the first person to study their general geometric
properties. He realised that they could be used to model the
behaviour of multiple bodies in free motion within the
solar system. Poincaré graduated from the
University of Paris in 1879.
The young Henri Poincaré
Career
Soon
after, he was offered a post as junior lecturer in mathematics at
Caen
University, but he
never fully abandoned his mining career to mathematics. He
worked at the Ministry of Public Services as an engineer in charge
of northern railway development from 1881 to 1885. He eventually
became chief engineer of the Corps de Mines in 1893 and inspector
general in 1910.
Beginning
in 1881 and for the rest of his career, he taught at the University of
Paris (the Sorbonne). He
was initially appointed as the
maître de conférences
d'analyse (associate professor of analysis) (Sageret, 1911).
Eventually, he held the chairs of Physical and Experimental
Mechanics, Mathematical Physics and Theory of Probability, and
Celestial Mechanics and Astronomy.
Also in that same year, Poincaré married Miss Poulain d'Andecy.
Together they had four children: Jeanne (born 1887), Yvonne (born
1889), Henriette (born 1891), and Léon (born 1893).
In 1887, at the young age of 32, Poincaré was elected to the
French Academy of
Sciences. He became its president in 1906, and was elected to
the
Académie française
in 1909.
In 1887 he won
Oscar II, King of
Sweden's mathematical competition for a resolution of the
threebody problem concerning the
free motion of multiple orbiting bodies. (See
#The threebody problem section
below)
In 1893 Poincaré joined the French
Bureau des Longitudes, which engaged
him in the synchronisation of time around the world. In 1897
Poincaré backed an unsuccessful proposal for the decimalisation of
circular measure, and hence time and
longitude (see Galison 2003). It was this post
which led him to consider the question of establishing
international time zones and the synchronisation of time between
bodies in relative motion. (See
#Work on relativity section below)
In the
year 1899, and again more successfully in 1904, he intervened in
the trials of Alfred
Dreyfus. He attacked the spurious scientific claims
of some of the evidence brought against Dreyfus, who was a Jewish
officer in the French army charged with treason by antiSemitic
colleagues.
In 1912 Poincaré underwent surgery for a
prostate problem and subsequently died from an
embolism 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.
A former
French Minister of Education, Claude
Allègre, has recently (2004) proposed that Poincaré be reburied
in the Panthéon in Paris, which is reserved for French citizens
only of the highest honour.
Students
Poincaré had two notable doctoral students at the University of
Paris,
Louis Bachelier (1900) and
Dimitrie Pompeiu (1905).
Work
Summary
Poincaré made many contributions to different fields of pure and
applied mathematics such as:
celestial mechanics,
fluid mechanics,
optics,
electricity,
telegraphy,
capillarity,
elasticity,
thermodynamics,
potential theory,
quantum theory,
theory of relativity and
physical cosmology.
He was also a populariser of mathematics and physics and wrote
several books for the lay public.
Among the specific topics he contributed to are the following:
The threebody problem
The problem of finding the general solution to the motion of more
than two orbiting bodies in the solar system had eluded
mathematicians since
Newton's time.
This was known originally as the threebody problem and later the
nbody problem, where
n is any number of more than two orbiting bodies. The
nbody solution was considered very important and
challenging at the close of the nineteenth century. Indeed in 1887,
in honour of his 60th birthday,
Oscar
II, King of Sweden, advised by
Gösta MittagLeffler, established
a prize for anyone who could find the solution to the problem. The
announcement was quite specific:
In case the problem could not be solved, any other important
contribution to classical mechanics would then be considered to be
prizeworthy. The prize was finally awarded to Poincaré, even though
he did not solve the original problem.One of the judges, the
distinguished
Karl Weierstrass,
said,
"This work cannot indeed be considered as furnishing the
complete solution of the question proposed, but that it is
nevertheless of such importance that its publication will
inaugurate a new era in the history of celestial
mechanics."(The first version of his contribution even
contained a serious error; for details see the article by Diacu).
The version finally printed contained many important ideas which
lead to the
theory of chaos. The
problem as stated originally was finally solved by
Karl F. Sundman for
n = 3 in 1912
and was generalised to the case of
n > 3
bodies by
Qiudong Wang in the
1990s.
Work on relativity
Local time
Poincaré's work at the Bureau des Longitudes on establishing
international time zones led him to consider how clocks at rest on
the Earth, which would be moving at different speeds relative to
absolute space (or the "
luminiferous
aether"), could be synchronised.
At the same time
Dutch theorist
Hendrik Lorentz was developing
Maxwell's theory into a theory of the motion of charged particles
("electrons" or "ions"), and their interaction with
radiation. He had introduced in 1895 an auxiliary quantity
(without physical interpretation) called "local time" t^\prime =
tvx^\prime/c^2, where x^\prime = x  vt and introduced the
hypothesis of
length contraction
to explain the failure of optical and electrical experiments to
detect motion relative to the aether (see
MichelsonMorley
experiment).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.
Principle of relativity and Lorentz transformations
He discussed the "principle of relative motion" in two papers in
1900and named it the
principle
of relativity in 1904, according to which no mechanical or
electromagnetic experiment can discriminate between a state of
uniform motion and a state of rest.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 chargeoccupied space, and
also questioned the time dilation factor given by Lorentz.In a
second letter to Lorentz, Poincaré gave his own reason why
Lorentz's time dilation factor was indeed correct after all: it was
necessary to make the Lorentz transformation form a group and gave
what is now known as the relativistic velocityaddition
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:
{{cquote1=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{1\varepsilon^2}.}}
and showed that the arbitrary function \ell\left(\varepsilon\right)
must be unity for all \varepsilon (Lorentz had set \ell = 1 by a
different argument) to make the transformations form a group. In an
enlarged version of the paper that appeared in 1906 Poincaré
pointed out that the combination x^2+ y^2+ z^2 c^2t^2 is
invariant. He noted that a Lorentz transformation
is merely a rotation in fourdimensional space about the origin by
introducing ct\sqrt{1} as a fourth imaginary coordinate, and he
used an early form of
fourvectors.
Poincaré’s attempt at a fourdimensional reformulation of the new
mechanics was rejected by himself in 1907, because in his opinion
the translation of physics into the language of fourdimensional
geometry would entail too much effort for limited profit. So it was
Hermann Minkowski who worked out
the consequences of this notion in 1907.
Massenergy relation
Like
others before, Poincaré (1900) discovered a relation between
mass and electromagnetic energy. While studying the conflict
between the
action/reaction principle
and
Lorentz ether theory, he
tried to determine whether the
center
of gravity still moves with a uniform velocity when
electromagnetic fields are included. He noticed that the
action/reaction principle does not hold for matter alone, but that
the electromagnetic field has its own momentum. Poincaré concluded
that the electromagnetic field energy of an electromagnetic wave
behaves like a fictitious
fluid ("fluide
fictif") with a mass density of
E/
c^{2}.
If the
center of mass frame is
defined by both the mass of matter
and the mass of the
fictitious fluid, and if the fictitious fluid is indestructible —
it's neither created or destroyed — then the motion of the center
of mass frame remains uniform. But electromagnetic energy can be
converted into other forms of energy. So Poincaré assumed that
there exists a nonelectric 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 when changing
frames: if a Hertzian oscillator radiates in a certain direction,
it will suffer a
recoil from the inertia of
the fictitious fluid. Poincaré performed a
Lorentz boost (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, 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.
Poincaré himself came back to this topic in his St. Louis lecture
(1904). This time (and later also in 1908) he rejected the
possibility that energy carries mass and also the possibility, that
motions in the ether can compensate the above mentioned
problems:
He also discussed two other unexplained effects: (1)
nonconservation of mass implied by Lorentz's variable mass \gamma
m, Abraham's theory of variable mass and
Kaufmann's experiments on the
mass of fast moving electrons and (2) the nonconservation of
energy in the radium experiments of
Madame
Curie.
It was
Albert Einstein's concept of
mass–energy
equivalence (1905) that a body losing energy as radiation or
heat was losing mass of amount
m =
E/
c^{2} that
resolved Poincare'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.
Poincaré and Einstein
Einstein's first paper on relativity was published three months
after Poincaré's short paper, but before Poincaré's longer version.
It relied on the principle of relativity to derive the Lorentz
transformations and used a similar clock synchronisation procedure
(
Einstein synchronisation)
that Poincaré (1900) had described, but was remarkable in that it
contained no references at all. Poincaré never acknowledged
Einstein's work on
Special
Relativity. Einstein acknowledged Poincaré in the text of a
lecture in 1921 called
Geometrie und
Erfahrung in connection with
nonEuclidean geometry, but not in
connection with special relativity. A few years before his death
Einstein commented on Poincaré as being one of the pioneers of
relativity, saying "Lorentz had already recognised that the
transformation named after him is essential for the analysis of
Maxwell's equations, and Poincaré deepened this insight still
further ...."
Assessments
Poincaré's work in the development of special relativity is well
recognised, 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 etherconcept in his papers and argued that
clocks in the ether show the "true" time, and moving clocks show
the local time. So Poincaré tried to bring the relativity principle
in accordance with classical physics, 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.
Character
Photographic portrait of H.
Poincaré's work habits have been compared to a bee flying from
flower to flower. Poincaré was interested in the way his mind
worked; he studied his habits and gave a talk about his
observations in 1908 at the Institute of General Psychology in
Paris. He linked his way of thinking to how he made several
discoveries.
The mathematician Darboux claimed he was
un intuitif
(intuitive), arguing that this is demonstrated by the fact that he
worked so often by visual representation. He did not care about
being rigorous and disliked logic. He believed that logic was not a
way to invent but a way to structure ideas and that logic limits
ideas.
Toulouse' characterisation
Poincaré's mental organisation was not only interesting to Poincaré
himself but also to Toulouse, a psychologist of the Psychology
Laboratory of the School of Higher Studies in Paris. Toulouse wrote
a book entitled
Henri Poincaré (1910). In it, he discussed
Poincaré's regular schedule:
 He worked during the same times each day in short periods of
time. He undertook mathematical research for four hours a day,
between 10 a.m. and noon then again from 5 p.m. to 7 p.m.. He would
read articles in journals later in the evening.
 His normal work habit was to solve a problem completely in his
head, then commit the completed problem to paper.
 He was ambidextrous and nearsighted.
 His ability to visualise what he heard proved particularly
useful when he attended lectures, since his eyesight was so poor
that he could not see properly what 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:
 "Habitué à négliger les détails et à ne regarder que les
cimes, il passait de l'une à l'autre avec une promptitude
surprenante et les faits qu'il découvrait se groupant d'euxmêmes
autour de leur centre étaient instantanément et automatiquement
classés dans sa mémoire."("Accustomed to neglecting details
and to looking only at mountain tops, he went from one peak to
another with surprising rapidity, and the facts he discovered,
clustering around their center, were instantly and automatically
pigeonholed in his memory.") Belliver (1956)
Attitude towards Cantor
Poincare was not attracted to the work of
Georg Cantor.
View on economics
Poincaré saw mathematical work in
economics and finance as peripheral. In 1900
Poincaré commented on
Louis
Bachelier's thesis "The Theory of Speculation", saying: "M.
Bachelier has evidenced an original and precise mind [but] the
subject is somewhat remote from those our other candidates are in
the habit of treating." (Bernstein, 1996, pp. 199–200) Bachelier's
work explained what was then the French government's pricing
options on French Bonds and anticipated many of the pricing
theories in financial markets used today.
Honours
Awards
Named after him
Philosophy
Poincaré had philosophical views opposite to those of
Bertrand Russell and
Gottlob Frege, who believed that mathematics
was a branch of
logic. Poincaré strongly
disagreed, claiming that
intuition was the life of mathematics.
Poincaré gives an interesting point of view in his book
Science
and Hypothesis:
 For a superficial observer, scientific truth is beyond the
possibility of doubt; the logic of science is infallible, and if
the scientists are sometimes mistaken, this is only from their
mistaking its rule.
Poincaré believed that
arithmetic is a
synthetic science. He
argued that
Peano's axioms cannot be
proven noncircularly with the principle of induction (Murzi,
1998), therefore concluding that arithmetic is
a priori
synthetic and not
analytic. Poincaré then went
on to say that mathematics cannot be deduced from logic since it is
not analytic. His views were similar to those of
Immanuel Kant (Kolak, 2001, Folina 1992). He
strongly opposed Cantorian
set theory,
objecting to its use of
impredicative 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
nonEuclidean 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.
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 nonEuclidean measured by rigid
rulers, or as a Euclidean space where the rulers are expanded or
shrunk by a variable heat distribution. However, Poincaré thought
that we were so accustomed to
Euclidean geometry that we would prefer
to change the physical laws to save Euclidean geometry rather than
shift to a nonEuclidean 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 as the source for
the idea that
creativity and
invention consist of two mental stages, first
random combinations of possible solutions to a problem, followed by
a critical evaluation.
Although he most often spoke of a deterministic universe, 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's
twostage model of
free will.
See also
References
Footnotes and primary sources
 [1] Poincaré pronunciation example at
Bartleby.com
 The Internet Encyclopedia of Philosophy Jules
Henri Poincaré article by Mauro Murzi — accessed November
2006.
 Lorentz, Poincaré et Einstein — L'Express
 Mathematics Genealogy Project North Dakota State
University, Accessed April 2008
 Reprinted as The Measure of Time in "The Value of
Science", Ch. 2.
 . See also the English translation
 . Reprinted in "Science and Hypothesis", Ch. 9–10.
 . English translation in Reprinted in "The value of science",
Ch. 7–9.
 Letter from Poincaré to Lorentz, Mai 1905
 Letter from Poincaré to Lorentz, Mai 1905
 Reprinted in Poincaré, Oeuvres, tome IX, S. 489–493.
 Partial English translation in Dynamics
of the electron.
 Walter (2007), Secondary sources on relativity
 Miller 1981, Secondary sources on relativity
 Darrigol 2005, Secondary sources on relativity
 . See also English translation.
 Darrigol 2004, Secondary sources on relativity
 Galison 2003 and Kragh 1999, Secondary sources on
relativity
 Holton (1988), 196206
 Hentschel (1990), 313
 Miller (1981), 216217
 Darrigol (2005), 1518
 Katzir (2005), 286288
 Whittaker 1953, Secondary sources on relativity
 Hadamard, Jacques. An Essay On The Psychology Of Invention
In The Mathematical Field. Princeton Univ Press (1949)
 Science and Method, Chapter 3, Mathematical Discovery,
1914, pp.58
 Dennett, Daniel C. 1978. Brainstorms: Philosophical Essays on
Mind and Psychology. The MIT Press, p.293
Poincaré's writings in English translation
Popular writings on the
philosophy
of science:
 ; This book includes the English translations of Science and
Hypothesis (1902), The Value of Science (1905), Science and Method
(1908).
On
algebraic topology:
On
celestial mechanics:
 1892–99. New Methods of Celestial Mechanics, 3 vols.
English trans., 1967. ISBN 1563961172.
 1905–10. Lessons of Celestial Mechanics.
On the
philosophy of
mathematics:

 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.
General references
 Bell, Eric Temple, 1986.
Men of Mathematics (reissue edition). Touchstone Books.
ISBN 0671628186.
 Belliver, André, 1956. Henri Poincaré ou la vocation
souveraine. Paris: Gallimard.
 Bernstein, Peter L, 1996. "Against the Gods: A Remarkable Story
of Risk". (p. 199–200). John Wiley & Sons.
 Boyer, B. Carl, 1968. A History of Mathematics: Henri
Poincaré, John Wiley & Sons.
 GrattanGuinness, Ivor,
2000. The Search for Mathematical Roots 1870–1940.
Princeton Uni. Press.
 Folina, Janet, 1992. Poincare and the Philosophy of
Mathematics. Macmillan, New York.
 Gray, Jeremy, 1986. Linear differential equations and group
theory from Riemann to Poincaré, Birkhauser
 Kolak, Daniel, 2001. Lovers of Wisdom, 2nd ed.
Wadsworth.
 Murzi, 1998. "Henri Poincaré".
 O'Connor, J. John, and Robertson, F. Edmund, 2002, "Jules Henri Poincaré". University of St.
Andrews, Scotland.
 Peterson, Ivars, 1995.
Newton's Clock: Chaos in the Solar System (reissue
edition). W H Freeman & Co. ISBN 0716727242.
 Sageret, Jules, 1911. Henri Poincaré. Paris: Mercure
de France.
 Toulouse, E.,1910. Henri Poincaré. — (Source biography
in French)
Secondary sources to work on relativity
 Nonmainstream
External links
 Online English translations of whole works by Poincaré:
 Free audio download of Poincaré's Science and
Hypothesis, from LibriVox.
 Internet
Encyclopedia of Philosophy: " Henri Poincare"—by Mauro Murzi.
 Henri Poincaré on Information Philosopher
 A timeline of Poincaré's life University of Nancy (in
French).
 Bruce Medal page
 Collins, Graham P., " Henri Poincaré, His Conjecture, Copacabana and
Higher Dimensions," Scientific American, 9 June
2004.
 BBC In Our Time, " Discussion of the Poincaré conjecture," 2
November 2006, hosted by Melvynn Bragg. See Internet Archive
 Poincare Contemplates Copernicus at
MathPages