Poincaré, Jules Henri

Poincaré, Jules Henri (1854-1912), French
physicist and one of the foremost mathematicians of the 19th century.

Poincaré was a cousin of the French statesman and author Raymond Poincaré. He was born in Nancy and educated at the École Polytechnique and the École Supérieur des Mines in
Paris. He taught at the University of Caen from 1879
to 1881 and was lecturer at the University of Paris from then until 1885, when
he became professor there of physical mechanics, mathematical physics (1886),
and celestial mechanics (1896).

Poincaré made important original contributions to differential equations,
topology, probability, and the theory of functions. He is particularly noted
for his development of the so-called Fuchsian
functions and his contribution to analytical mechanics. His studies included
research into the electromagnetic theory of light and into electricity, fluid
mechanics, heat transfer, and thermodynamics. He also anticipated chaos theory.
Among Poincaré's more than 30 books are *Science
and Hypothesis* (1903; trans. 1905), *The Value of Science* (1905;
trans. 1907), *Science and Method* (1908; trans. 1914), and *The
Foundations of Science* (1902-8; trans. 1913). In 1887 Poincaré
became a member of the French Academy of Sciences and served at its president
in 1906. He also was elected to membership in the French Academy in 1908.

**Microsoft ® Encarta ® Reference Library 2003.**
© 1993-2002 Microsoft Corporation. All rights reserved.

**Poincaré****, Henri**

**b. April**** 29, 1854, Nancy,
Fr.**

d. July 17, 1912, Paris

in full JULES-**HENRI** **POINCARÉ**, French mathematician, theoretical astronomer, and philosopher of
science who influenced cosmogony, relativity, and topology and was a gifted
interpreter of science to a wide public.

**Poincaré** was from a family
distinguished by its contributions to government and administration. His first
cousin was Raymond **Poincaré**, president of the
French Republic during World War I. **Poincaré**
was ambidextrous and was nearsighted; during his childhood he had poor muscular
coordination and was seriously ill for a time with diphtheria. He received
special instruction from his gifted mother and excelled in written composition
while still in elementary school. Becoming deeply interested in mathematics
during adolescence, he attended in 1872-75 the École Polytéchnique in Paris, where he easily won top honours in
mathematics, but was undistinguished in physical exercise and in art. **Poincaré** had an unusually retentive memory for
everything he read; moreover, he could visualize what he heard, a useful
faculty because he could not clearly see at a distance the mathematical symbols
that were on the blackboard. Throughout his life he was able to perform complex
mathematical calculations in his head and could quickly write a paper without
extensive revisions. He received, in 1879, a doctorate from the École Nationale Supérieure des Mines with a thesis on differential
equations.

Following a brief
appointment in mathematical analysis at the University of Caen,
in 1881 **Poincaré** joined the University of
Paris, where, during the rest of his life, he lectured and wrote
prolifically--almost 500 papers--on mechanics and experimental physics, in all
branches of pure and applied mathematics, and in theoretical astronomy.
Changing his lectures every year, he would review optics, electricity, the
equilibrium of fluid masses, the mathematics of electricity, astronomy,
thermodynamics, light, and probability. Many of these lectures appeared in
print shortly after they were delivered at the university. In his extensive
writings on probability, **Poincaré** anticipated
the concept of ergodicity that is basic to
statistical mechanics.

Applying his
mastery of analysis to the question of the solvability of algebraic equations, **Poincaré** developed before age 30 the idea of the automorphic function--one that is invariant under a
group of transformations that are characterized algebraically by ratios of
linear terms. He showed how these functions can be used to integrate linear
differential equations with rational algebraic coefficients and how to
express the coordinates of any point on an algebraic curve as uniform functions
of a single algebraic variable (or parameter). Some of these automorphic functions he called Fuchsian,
after the German mathematician Immanuel Lazarus Fuchs, who was one of the
founders of the theory of differential equations; he found that they were
associated with transformations arising in non-Euclidean geometry. In
recognition of his fundamental contributions to mathematics, he was elected, in
1887, to membership in the Académie des Sciences in
Paris.

In celestial
mechanics, **Poincaré** made substantial
contributions to the theory of orbits,
particularly the classical three-body problem (for example, the system
involving the Sun, Moon, and Earth). This was part of the "problem of *n*
bodies" (planets, stars, etc.), set for a prize by King Oscar II of
Sweden: given the present masses, velocities, motions, and mutual distances of
"*n *bodies," how long will they remain stable in their present
spatial relationships, or will their orbits change at some future date? In his
solution, **Poincaré** developed powerful new
mathematical techniques, including the theories of asymptotic expansions and
integral invariants, and made fundamental discoveries on the behaviour of the
integral curves of differential equations near singularities. He was awarded
the prize in 1889, even though his solution to the problem was only partially
correct; in the same year he was also made a knight of the French Legion of
Honour.

**Poincaré** summarized his
new mathematical methods in astronomy in *Les Méthodes
nouvelles de la mécanique céleste,* 3 vol. (1892, 1893, 1899;
"The New Methods of Celestial Mechanics"). Another result of this
work, his *Analysis situs *("Positional
Analysis") in 1895, was an early systematic treatment of topology,
which deals with properties of a system that endure when metric distortion occurs--that
is, topology deals with the qualitative characteristics of spatial
configurations that do not vary during cumulative transformations. He also
contributed to the theory of numbers by demonstrating how the conception of
binary quadratic
forms, which was developed by the German mathematician Carl Friedrich
Gauss, could be cast in geometric form. In 1904 he lectured at the St. Louis
Exposition.

In mathematical
analysis, **Poincaré** made important
contributions to the theory of equilibrium of rotating fluid masses. In
particular, he described the conditions of stability of the pear-shaped figures
that played so prominent a part in the researches of later cosmogony,
with reference to the evolution of celestial bodies. He attempted an
application of these ideas to the stability of Saturn's rings and to the origin
of binary stars. In 1906, in a paper on the dynamics of the electron, he
obtained, independently of Albert Einstein, many of the results of the special
theory of relativity. The principal difference was that Einstein developed the
theory from elementary considerations concerning light signaling,
whereas **Poincaré**'s treatment was based on the
full theory of electromagnetism and was restricted to phenomena associated with
the concept of a universal ether that functioned as
the means of transmitting light.

After **Poincaré** achieved prominence as a mathematician, he
turned his superb literary gifts to the challenge of describing for the general
public the meaning and importance of science and mathematics. Always deeply
interested in the philosophy of science, he wrote *La Science et l'hypothèse *(1903; *Science
and Hypothesis*), *La Valeur de la science *(1905;
*The Value of Science*), and *Science et méthode
*(1908; *Science and Method*), all of which reached a wide public of nonprofessionals. His works were translated into English,
German, Hungarian, Japanese, Spanish, and Swedish. He emphasized the
subconscious, while probing the psychology of mathematical discovery and
invention. He was a forerunner of the modern intuitionist school in that he
believed that some mathematical induction is a priori and independent of logic.
In his view, sudden illumination, following long subconscious work, is a
prelude to mathematical creation. But his greatest contribution to philosophy
was to emphasize the role played in scientific method by convention--*i.e.,*
by the arbitrary choice of concepts.

**Poincaré**'s prestige and influence increased in French science, and in 1906
he was elected president of the Académie des Sciences
and in 1908 to membership in the Académie Française, the highest honour accorded a French writer.

**Poincaré**** conjecture**

This conjecture,
formulated by the French mathematician **Henri** **Poincaré**,
is a famous problem of 20th-century mathematics. It asserts that a simply
connected closed three-dimensional manifold is a three-dimensional sphere. The
term simply connected means that any closed path can be contracted to a point.
William Thurston formulated a program for the classification of three-manifolds
that included the **Poincaré** conjecture in the
more general setting of three-manifolds with finite fundamental groups. In
another direction, a higher dimensional analog of the
**Poincaré** conjecture states that any closed *n*-manifold
which is homotopy equivalent to the *n*-sphere
must be the *n*-sphere. When *n* is 3 this
is equivalent to the original formulation above. The higher dimensional
conjecture was proved by Stephen Smale (1961) when *n*
is at least 5, and by Michael Freedman (1982) when *n* is 4. However, the
original three-dimensional case has defied all attempts to solve it and remains
a *cause célèbre* in mathematics.