Henri Poincaré was a mathematical genius who made the greatest advances in celestial mechanics since the time of Isaac Newton.
His work on the n-body problem, whose ultimate aim was to determine if the solar system was stable, led to Chaos theory – Poincaré gave the first mathematical description of a dynamic system behaving chaotically.
His work in special relativity yielded the modern form of the Lorentz transformations.
Poincaré showed how information can be obtained from otherwise insoluble differential equations. He founded the field of algebraic topology; and, early in his career, he discovered automorphic functions.
The Poincaré conjecture was one of mathematics’ great unsolved problems until Grigori Perelman provided a solution in 2002.
Poincaré was a public intellectual who authored a number of books, such as Science and Methods, explaining the work of mathematicians and physicists to the public.
Beginnings
Jules-Henri Poincaré was born into a wealthy family on April 29, 1854 in the city of Nancy, France.
Henri’s mother was Eugénie Launois, a notably intelligent woman from a wealthy family. Henri’s father was Léon Poincaré, a physician and professor of medicine at the University of Nancy who lectured in anatomy and physiology; his particular interest was neurology.
The Poincarés had two children: Henri and his younger sister Aline.
Like most French families the Poincarés were Roman Catholic. However, by age 18, Henri described himself as a free thinker and not religious. He declared himself opposed to clerical dogmatism and state interference; he supported equal political rights for everyone, including the rights to carry out research and tell the truth.
The Poincaré children lived in a large home with their parents, grandparents and servants. Henri started talking when he was just nine months old.
Henri Poincare and his sister Aline with their dog Tom.
Although the family was well-off and the children were happy, wealth and happiness could not ward off infectious diseases: in an age with no antibiotics, these were often deadly.
At age five, Henri contracted diphtheria. He lost the ability to walk for two months and could not talk for almost seven: diphtheria covers the back of the throat with thick mucus that blocks the airway and causes difficulty breathing. He invented a private sign language to communicate with Aline and his parents.
Henri was noticeably shy and he preferred inventing and playing games with Aline to competing in rougher games. This may have been a consequence of his diphtheria – the infection can damage the heart, the kidneys and nerves permanently.
In later life Henri continued to be shy, but opened up when discussing intellectual ideas.
School
At age eight, Henri began school for the first time.
His high school was the Lycée Nancy, now renamed the Lycée Henri Poincaré. A teacher there told Henri’s mother that her 14-year-old son was destined to become one of the world’s great mathematicians.
Henri did not know what he wanted to do. He was gifted in philosophy, physics, acting, writing literature, engineering and the arts; and he found them all interesting.
Invasion
While Henri was at the Lycée, the Prussian Army invaded France. French casualties began pouring into Nancy, a city in the line of the Prussian advance. Many civilians fled from the city, but the Poincaré family remained. Henri’s father became a medic-ambulance driver; the ambulance was, of course, horse-drawn. 16-year-old Henri helped treat the wounded – a gruesome task for a boy who, until then, had led a sheltered life.
Nancy fell to the Prussians in August 1870.
The war ended in January 1871. Later that year France agreed to surrender territory to the new German Empire. To the relief of the Poincaré family, Nancy was returned to France.
Exam Flop
In his Lycée graduation exams in 1871, Henri’s overall mark was ‘good,’ a far cry from expectations. Even in mathematics, his scores were mediocre.
“I must admit I am absolutely incapable of adding numbers without making a mistake.”
Redemption
Henri Poincaré did not lose heart. He started to prepare for the concours, a difficult set of university entrance examinations. He studied advanced mathematics books.
In the summer of 1872, the École Polytechnic in Paris, a prestigious university, set a public mathematics problem. Poincaré was the first to solve it.
Henri Poincaré in uniform at the École Polytechnic.
Now he had to put on a uniform – the school was a military one. At first other people at the Polytechnic were skeptical about Poincaré’s attitude. Although he turned up at lectures, he barely took any notes. He also did not seem interested in socializing with other students unless they asked him about mathematics; then he would talk with them in a friendly way.
Poincaré took courses in analysis, astronomy, chemistry, geometry, mechanics, physics, and history & literature.
Better than Laguerre
One day, a fellow student asked Poincaré about a proof in analysis that Professor Edmond Laguerre, a renowned mathematician, had written for them on the blackboard.
Poincaré had taken no notes – his poor eyesight prevented him seeing the blackboard well enough. Instead, he constructed his own proof. When the student showed Poincaré’s proof to Laguerre, the professor sent for Poincaré and congratulated him, telling him his proof was simpler than his own and he would use Poincaré’s proof in future.
Poincaré graduated in 1875. He immediately enrolled at the École des Mines in Paris, a highly regarded engineering school, from which he graduated in 1878 as a mining engineer.
The following year, age 25, he obtained a doctorate in mathematics from the University of Paris. His thesis, concerning partial differential equations, introduced the new mathematical concepts of:
- resonance of eigenvalues, and
- algebroid functions
Henri Poincaré’s Contributions to Science
After spending a few months working as an inspector of mines, in December 1879 Poincaré become a lecturer in mathematical analysis at the University of Caen. Interestingly, he continued taking occasional assignments as a mining inspector.
Automorphic Functions
In 1880, Poincaré became known internationally. His discovery of automorphic functions – he called them Fuchsian and Kleinian functions – marked him as a mathematician to watch.
Many years later he described how, after struggling for two weeks, he made his first major discovery in Fuchsian functions in 1880:
“Every day I spent an hour or two at my desk, trying a lot of combinations without result. One evening, I could not get to sleep because, unusually, I had drunk black coffee. Ideas flooded into my head; I could feel them colliding with one another until two of them bonded to make a stable combination.”
Another great breakthrough followed quickly. He had been preoccupied with a mining conference he was attending and not thinking about mathematics.
Stepping on to a bus:
“…just as I put my foot on the step, the idea came to me. Nothing in my former thoughts seemed to have prepared me for it: the transformations I had used to define Fuchsian functions were identical with those of non-Euclidian geometry. I had no time to verify the thought, because the conversation started again as soon as I sat down, but I felt absolute certainty at once.”
Poincaré was certain the subconscious played a tremendous part in mathematical invention. He never worked consciously for more than an hour or two at a time on any research problem. He would then give his subconscious mind plenty of time to work on it before returning to the problem again consciously.
In 1881, the University of Paris enticed Poincaré back to the French capital with a professorship. He would stay in Paris for the rest of his life.
Qualitative Theory of Differential Equations
Much of physical science is built on differential equations. The majority of differential equations cannot be solved exactly.
In 1881, Poincaré showed how highly useful information could be extracted from insoluble differential equations. He did this using methods from analysis and topology. For example, he showed how he could obtain singular points – saddle, focus, center, and node. For finite-difference equations he invented asymptotic analysis of the solutions.
A stable focus – one of the singular points of integral curves produced by Poincaré’s qualitative methods.
Poincaré’s discoveries paved the way for his crucial work in celestial mechanics, including the n-body problem discussed below.
The n-body problem was not merely theoretical. Practitioners of celestial mechanics had still to prove (or otherwise) that our solar system is stable!
Chaos Theory and the n-Body Problem
In 1887, King Oscar II of Sweden announced a prize challenge. The task was to find out how to best calculate the behavior of any number of masses obeying Newton’s law of gravitation. This n-body problem was an extension of the famous 3-body problem in celestial mechanics.
Poincaré’s entry won the prize. Despite this, one of the judges, Charles Hermite, who was Poincaré’s former doctoral advisor, wrote to another judge:
“But it must be acknowledged, in this work as in almost all his researches, Poincaré shows the way and gives the signs, but leaves much to be done to fill the gaps and complete his work.”
In fact, Poincaré’s winning entry contained a rather significant error, which none of the judges noticed. Poincaré noticed it when the judges asked him for clarification of certain sections of his work. Printing new versions of his winning entry cost Poincaré more than the value of his prize!
The major consequence of his error correction was his discovery of solutions to the 3-body problem that were stunningly sensitive to initial conditions. This was the basis of Chaos theory – tiny changes to the way a process begins have enormous consequences for how it ends.
This is often characterized by the butterfly effect: a butterfly’s wings flapping on one continent are the cause of a hurricane on another. Poincaré foreshadowed this. Regarding weather forecasting he wrote:
“Here again we find the same contrast between a minimal cause, too small to be seen by an observer, and substantial effects, which are sometimes terrible disasters.”
Poincaré could not take his discovery of chaotic equations much further.
Chaos theory only took off in the second half of the twentieth century when electronic computers allowed mathematicians to study chaotic systems for the first time. These systems are devastatingly complex, requiring significant computing power to study them. They were simply impractical to investigate in Poincaré’s time – something as simple as a solitary butterfly’s wings flapping can lead to an enormous number of possible outcomes.
So, Is the Solar System Stable?
In studying the 3-body problem, Poincaré learned that he could not find a rigorous closed-form solution – in fact such solutions are impossible for three bodies and n bodies. Hence, he could not say that the solar system is stable.
Today, with the benefit of computers, we can solve the relevant equations numerically to predict the orbits of planets with high precision. This allows astronomers to state that the solar system will be stable for the next few tens of millions of years. Beyond this, since gravitational interactions between planets produce a chaotic state, we cannot say the solar system is stable.
This means, for example, that at some point in the distant future one or more of the planets could possibly be ejected from the sun’s orbit.
“The methods employed by Poincaré were incomparably more profound and powerful than any previously used in celestial mechanics, and mark an epoch in the development of the science.”
The Curious Case of Decimal Angles and Decimal Time
The Sumerians and Babylonians divided the circle into 360 degrees. Our modern system for measuring angles – partly sexagesimal and partly decimal – came to us from these ancient cultures via the geometry of the Ancient Greeks.
In 1893, Poincaré joined the Bureau de longitudes in Paris. In 1897, this organization established a commission to decimalize angles; Poincaré, a wholehearted supporter of decimalization, was secretary. He urged that a circle should contain 400 degrees.
A right-angle in the new decimal system.
Moreover, Poincaré favored decimal time. Each day would still have 24 hours, but each hour would contain 100 minutes, and each minute a 100 seconds. A decimal day would have 2,400 minutes instead of the more familiar 1,440 minutes.
Needless to say, there was opposition in France: everyone’s maps, charts, watches and clocks would be worthless! The rest of the world rolled its collective eyes and rejected the idea out of hand. In 1900, the commission was disbanded.
Relativity Theory
In 1905, his miracle year, Albert Einstein created the special theory of relativity. Among other things, he told a largely uncomprehending world that all observers do not measure time passing at the same rate, and he gave us that most famous, illuminating, and ominous equation:
However, Einstein was not the only scientist working in this field.
In 1898, Poincaré had hit a big nail squarely on the head:
“…we have no direct intuition about the equality of two time intervals. The simultaneity of two events or the order of their succession, as well as the equality of two time intervals, must be defined in such a way that the statements of the natural laws be as simple as possible.”
Poincaré also gave us the Lorentz transformations in their modern form.
In fact, Poincaré created much of special relativity independently of Einstein. However, only Einstein’s formulation:
- established the relativity of space and time – while Hendrik Lorentz and Poincaré believed the Lorentz transformations arose from interactions of matter and the (non-existent) aether, Einstein derived the transformations from the nature of space and time.
- offered Paul Dirac the means to unite special relativity and quantum mechanics in the Dirac equation – the extraordinary equation that naturally produced the quantum property of spin and foretold the existence of antimatter.
Topology
In 1895, in a paper entitled Analysis situs, Poincaré founded algebraic topology. Algebraic topology investigates topological space using the techniques of abstract algebra.
A Mobius strip is a simple structure. It has just one surface and one edge. Topology considers the properties of geometric objects when they are deformed by bending, crumpling, stretching and twisting.
The Poincaré conjecture – a problem in topology – stood out as one of the great unsolved problems in mathematics until Grigori Perelman provided a solution in 2002. Perelman refused the $1 million prize awarded by the Clay Mathematics Institute, stating that Richard S. Hamilton’s contribution to the solution was equally important.
Honors for Poincaré
1887: Elected to the French Academy of Sciences
1894: Elected Officer of the Legion of Honor
1900: Royal Astronomical Society Gold Medal
1901: Royal Society Sylvester Medal
1901: President of the French Astronomical Society
1902: President of the French Society of Physics
1903: Commanders of the Legion of Honor
1905: Matteucci Medal
1905: Bolyai Prize
1906: Elected President of the French Academy of Sciences
1908: Elected to the French Academy
1911: Bruce Medal
“We prove using logic; we create using intuition.”
Personal Details and The End
Music gave Poincaré enormous pleasure. His favorite composer was Richard Wagner.
For much of his life Poincaré suffered from severe indigestion, which forced him to rest for up to three hours after a meal. He disapproved of smoking.
Poincaré conformed to the archetype of a mad, work-obsessed professor : he didn’t listen to social conversations properly, was forgetful, and his mind often seemed to be elsewhere.
Ambidextrous from an early age, he found telling his right from his left challenging.
In April 1881, one day before his 27th birthday, Poincaré married Louise Poulain d’Andecy, who came from an academic family. Louise looked after nearly all the family’s affairs, which was just as well given Poincaré’s absent-mindedness. He famously once wrote a letter and sent an envelope containing blank paper rather than the letter.
The couple had four children: Jeanne, Yvonne, Henriette and Léon.
Henri Poincaré died in Paris at age 58 on July 17, 1912. A prostate operation a few days earlier seemed to have been successful and Poincaré appeared to be almost fully recovered. He died suddenly of an embolism when he was dressing in the morning.
Poincaré was much loved and admired. His funeral attracted a huge turnout from all walks of life. He was buried in the Poincaré family vault in the Montparnasse Cemetery, Paris.
“Knowing how to criticize is good; knowing how to create is better.”
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Further Reading
Henri Poincaré
Science et méthode
E. Flammarion, Paris, 1908
June Barrow-Green
Poincaré and the Three Body Problem
American Mathematical Society, 1997
Ferdinand Verhulst
Henri Poincaré: Impatient Genius
Springer, 2012
Jeremy Gray
Henri Poincaré: A Scientific Biography
Princeton University Press, 2013
C. Marchal, (Editors: B. Steves, A. J. Maciejewski)
The Restless Universe Applications of Gravitational N-Body Dynamics to Planetary Stellar and Galactic Systems
CRC Press, 7 May 2019