Monetary prizes for solving complex scientific and technical problems are not awarded as often as they used to be. Today, tenders where a committee of experts offers prizes for the best creations sent to its address by a certain date are a common way of finding solutions in the fields of architecture and other artistic/technical disciplines, but not so much in science. Every once in a while an organization or an individual still announces that a prize will be given to anyone who can solve some seemingly insoluble problem, but these awards are not really significant in terms of maintaining or increasing the financing of scientific work. In the past, however, this used to be quite different.

A little more than a hundred years ago, the Swedish mathematician Gösta Mittag-Leffler persuaded King Oscar II of Sweden to organize a mathematics competition in honor of his sixtieth birthday. As the king was a student of mathematics himself, he took to the idea quickly, especially because Mittag-Leffler had succeeded in convincing two highly distinguished European mathematicians to take part in the jury that would set the tasks and evaluate the solutions. The invitation was accepted by Karl Weierstrass of Berlin and Charles Hermite of Paris.

Is the solar system stable?

Four tasks were given, all relating to problems that were also principal subjects of mathematical research at the time. Even though the king’s birthday was not until 1889, preparations for the award already started in 1884, so that scholars from across Europe would have enough time to examine the problems thoroughly and come up with their solutions. In the middle of 1885 Nature magazine printed an announcement inviting scientists to participate in a mathematical competition in honor of King Oscar II’s sixtieth birthday. The deadline for submitting the solutions to the Swedish court and the president of the jury Gösta Mittag-Leffler was June 1, 1888. Naturally, a precondition of entry was that all submissions be made anonymously, to guarantee unbiased evaluation of the work. That is why all candidates were required to enclose with their solutions a sealed envelope containing their names.

The first of the four questions which were given in formal mathematical language pertained to the problem of the movement of three bodies. In other words, the jury was interested in the question of the stability of our solar system. Today, more than a hundred years after the event took place, what happened a couple of months after the announcement of the winner is more interesting than the initial question and its answers. When the winning solution was already being prepared for publication the editor of the magazine noticed that the jury had rewarded a work which was not completely flawless. One of the mistakes actually turned out to be of key importance.

The winner was no surprise

The winner of the competition was none other than the eminent French mathematician Henri Poincaré who, despite his young age at the time, already enjoyed a very high reputation. In the period between the announcement of the competition and the deadline for submissions he was even elected a member of the French Academy of Sciences, a great honor for a man of only thirty-two.

The competition was not about winning the money. The awarded sum came nowhere close to that of today’s Nobel Prize. The winner received 2500 Swedish kronor which was approximately one third of a professor’s yearly salary. The award improved the young mathematician’s scientific career significantly and enabled him to secure a good position at a renowned university, but it certainly did not make him rich. As we shall see shortly, the prizewinning Poincaré actually lost more than he gained from the award.

With his 158-page-long solution to the first problem Poincaré enclosed a short accompanying letter which he signed, even though all of the submitted essays were supposed to be anonymous. During the evaluation process the jury thus already knew who had written the solution which they later voted unanimously voted as the best. The winner was announced during the celebration of the king’s birthday on January 21, 1889. A part of the prize was also the publication of the solution in the prominent mathematical journal Acta Mathematica.

It was at this moment that the problem occurred. In July 1889 when the editor of the journal was preparing Poincaré’s article for publication he noticed a few minor mistakes in the manuscript. He notified Mittag-Leffler who, in turn, wrote a letter to Poincaré on July 16 to inform him that all of the mistakes but one could be fixed immediately. The one mistake was neither a typing error nor a minor mathematical error. It turned out to be of a much greater importance. As it soon turned out, Poincaré overlooked something essential in his proof which was only discovered after the prize had already been awarded and the article was almost in print.

How to avoid a scandal?

What to do? If it were to become known that a flawed solution had been awarded the prize, the reputation of both the king and all the highly esteemed mathematicians who took part in the evaluation process would be badly tarnished. The disgrace would be especially devastating to Poincaré, who had just begun to shine as the new star on the European scientific scene.

Poincaré went back to work in order to correct the mistake, but the more he examined the problem the more it seemed it was not merely a minor error that could be fixed easily. At first, he sent extensive comments to the text which were meant to clarify the issue, but it became increasingly obvious that these explanations alone would not be sufficient to deal with the difficulty.

So what kind of an error was it? In his original solution Poincaré used a completely new method that made his work much easier and brought about a minor revolution in the solving of similar mathematical problems. Instead of calculating the entire orbits of individual bodies or planets, Poincaré chose to focus only on specific moments when a planet or asteroid intersects with a chosen plane. In other words, this method would be best compared to taking an image of the solar system at the exact moment when an observed body circles the sun. In the case of Earth, this would happen once a year.

Poincaré only wanted to determine if a planet or an asteroid, after circling the sun, returns to the same spot, or to find out how the yearly positions change with time. In slightly more technical language: if a chosen plane is always intersected at the same point, the orbit of the motion is obviously stable, but if the plane is crossed at a slightly different point each time, this change has to be described and examined to determine whether these deviations occur within a stable system.

The chaotic mechanics of the stars

Poincaré’s discovery, which he had also argued for in his original essay, was that the solar system was stable, at least in a simple model of the sun with one bigger planet and one small asteroid orbiting in the same plane. His first finding was that, in this case, the orbits of motion were stable. When he examined the problem again, it became evident that he had forgotten to take into account an entire range of solutions which were not stable, but would lead to chaos.

It turned out that Poincaré had forgotten to take into consideration a geometrical configuration which leads to completely different solutions than those he had described in his original essay. Today, we would describe these new solutions, which he overlooked at first, as chaotic. Even though they are precisely determined by clear equations and the paths of the bodies in question are such that it is possible to predict their future motion, this kind of prediction depends on the knowledge of the initial conditions. If there is only a slight difference in the position of the bodies at the beginning, they are bound to move in a completely different way. On the basis of this finding Poincaré could only conclude that not all forms of movement in a simplified system of three bodies were stable, which also meant that the solar system or any other similar system is not necessarily or absolutely stable.

In the couple of months Poincaré spent striving to correct the mistake he had made in his original article, he set the foundations for a new branch of mathematics which later developed into chaos theory. The hundred and fifty-eight pages of the original essay expanded to two hundred and seventy. The problem, however, was that the original article had already been printed in Sweden. Fortunately for all those involved it was not yet distributed to the subscribers. To avoid a scandal Poincaré paid 3585 kronor for a corrected reprint of the issue, much more than the sum of the prize money he had received. The organizers also did their best to ensure that the first prints of the journal were destroyed, but a couple of the original issues survived. In 1985, during a research visit to Sweden, the American mathematician Richard McGehee was looking through some archives and came upon a couple of copies of the thirteenth issue of Acta Mathematica in an unlabeled box containing Mittag-Leffler’s letters. On looking through the contents McGehee discovered that Poincaré’s article was different to the one he was familiar with from other copies of the same journal.

Even though Poincaré won the prize for the first, flawed version of the article, the edited article, with which he made a giant leap towards the science of chaos, is much more important from today’s point of view. It might seem self-contradictory, but the award was definitely given to the right person, although for the wrong solution and the wrong essay.