The Also-ran and the King

Combing through Greg Ross wonderful Futility Closet I found a nice post he called "Also-Ran".

I'll give you his article and then fill in the missing details...

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Arthur Conan Doyle tells us little about James Moriarty, the criminal mastermind in the Sherlock Holmes stories. But he does mention one intriguing accomplishment in *The Valley of Fear*:

Is he not the celebrated author of

The Dynamics of an Asteroid, a book which ascends to such rarefied heights of pure mathematics that it is said that there was no man in the scientific press capable of criticizing it?

Mathematicians Alain Goriely and Simon P. Norton have both pointed out that in 1887 King Oscar II of Sweden offered a bounty for the solution to the *n*-body problem in celestial mechanics. Doyle’s story was set in 1888, so it’s possible that Moriarty had intended his book as his entry in this contest.

If he did, he was disappointed — the prize went to Henri Poincaré.

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Ok, I thought, but wait.... I didn't think the n-body solution was actually solved, so I did the obvious thing, I queried Quora. Quora says, "There are no solutions to the n-body question.. " quickly followed by such things as, "If you stand on one leg and squint your eyes" ... Ok, they don't say that bit exactly, but lots of talk about assumptions that might make it "sort of"accurate if you stand on one leg... but I covered that.

So I started to research the actual contest described in his post. Turns out the idea did not suddenly appear to the king of Sweden one day. His 60th birthday they were celebrating, just to plant this thing in calendar tile, was January 21, 1889. And it wasn't exactly the King's idea from the start. The usually quite accurate Wikipedia tells it this way, " "Oscar was also particularly interested in mathematics. In 1889 he set up a contest, on the occasion of his 60th birthday, for "an important discovery in the realm of higher mathematical analysis". In truth the contest was to be decided and awarded on the Kings Birthday on, yep, January 21, 1889. That doesn't leave much time to solve, write up and deliver your solution to an unsolved problem." In fact the idea was thought up years before with not a whisper at first to the Good King Oscar II.

The whole contest was the brainchild of Magnus Gustaf "Gösta" Mittag-Leffler, who had founded in 1882, the most important mathematical periodical ever, Acta Mathematica, and would be its editor for 40 years. One of Mittag-Leffler’s inventive ideas in promoting Acta Mathematica was to turn to King Oscar II of Sweden and Norway, both for financial support of the project and as the first enlisted subscriber. In 1884 another grand idea from Mittag-Leffler had matured. Again involving King Oscar, he now wanted to arrange an international prize competition in mathematics honouring the 60th birthday of the king. It seems that Mittag-Leffler first revealed his plan to Kovalevskaya. We find a short reference in a letter to her from 4 May 1884.

The special features of this competition was the international and ambitious appeal, and the connection not to an academy or institution, but to the journal Acta Mathematica, where the winning entry finally was to be published. The prize consisted of a gold medal and 2,500 Swedish kronor. (Note the prize amount, it becomes important later.) The memoirs should be submitted before 1 June 1888 (nearly three years after the original announcement and six months before the King's birthday), with anonymity maintained through a motto on an enclosed sealed envelope containing the name of the author. (Even in the beginning the three judges, "Mittag-Leffler himself, acting as administrative and coordinative liaison with his mentors and friends Karl Weierstrass in Berlin and Charles Hermite in Paris. They were not only the two dominant mathematicians of the older generation, but there was also a special sympathy between them. This would be a prize awarded not for past contributions, but for a solution to an unsolved problem specified by the committee. In order to attract the best mathematicians from different branches of mathematical analysis they agreed on four questions." (So there were choices and the n-body problem was just one of them. )* Institut Mittag-Leffler

* Institut Mittag-Leffler |

The committee knew very well that Poincaré had the capacity to attack any of the four questions. In correspondence with Mittag-Leffler he made clear his intention to grapple with Question 1, the n-body problem. In May 1888, after hard work and many doubts, he submitted his memoir Sur le problème des trois corps et les équations de la dynamique. As for the anonymity, well …, accompanying the memoir were two letter notes, one to the prize jury and one to Mittag-Leffler.

A list of all the twelve manuscripts received by June 1888 was published in Volume 11 of Acta with their identifying epigraphs. It turned out that five of the authors had attempted the prestigious n-body problem, one had tried Question 3 (???), while six treated a subject of their own, which remained a secondary option. Only four of the authors have been identified: in addition to Poincaré also Paul Appell, Guy de Longchamps and Jean Escary.

Finally after almost 300 handwritten pages (including the appendices) his (Poincaré ) new concept of integral invariants and the subsequent geometrical arguments led to a claim of stability for this system.

Mittag-Leffler made his final presentation to the king on 20 January 1889, the day before the monarch’s 60th birthday, only the brief summary from Weierstrass was enclosed in the general report. For the winning entry Henri Poincaré received the sum of 2,500 kronor together with a gold medal. Paul Appell was also rewarded with a gold medal in addition to the honourable mention. For various reasons the prize ceremony didn’t take place on the king’s birthday, but the announcement was made public that day. Instead Poincaré received his prize from the hands of the Swedish ambassador in Paris later in March. Mittag-Leffler also informed the French Academy of Sciences of the news.

One of Mittag-Leffler’s inventive ideas in promoting Acta Mathematica was to turn to King Oscar II of Sweden and Norway, both for financial support of the project and as the first enlisted subscriber. In 1884 another grand idea from Mittag-Leffler had matured. Again involving King Oscar, he now wanted to arrange an international prize competition in mathematics honouring the 60th birthday of the king. It seems that Mittag-Leffler first revealed his plan to Kovalevskaya. We find a short reference in a letter to her from 4 May 1884.

The prize consisted of a gold medal and 2,500 Swedish kronor. (As a comparison, Mittag-Leffler’s annual salary as professor was 7,000 kronor.) The memoirs should be submitted before 1 June 1888 (almost three years after the announcement), with anonymity maintained through a motto on an enclosed sealed envelope containing the name of the author.

A list of all the twelve manuscripts received by June 1888 was published in Volume 11 of Acta with their identifying epigraphs. It turned out that five of the authors had attempted the prestigious Question 1, the n-body problem; one had tried Question 3, while six treated a subject of their own, which remained a secondary option. Only four of the authors have been identified: in addition to Poincaré also Paul Appell, Guy de Longchamps and Jean Escary.

Finally after almost 300 handwritten pages (including the appendices) his (Poincaré ) new concept of integral invariants and the subsequent geometrical arguments led to a claim of stability for this system.

Although no entry had actually solved any of the questions, Mittag-Leffler and his jury were soon of the preliminary opinion that Poincaré was in a class of his own, that Appell should be awarded a second honorary prize, and that no other entries needed much further examination.

Mittag-Leffler made his final presentation to the king on 20 January 1889, the day before the monarch’s 60th birthday, only the brief summary from Weierstrass was enclosed in the general report. For the winning entry Henri Poincaré received the sum of 2,500 kronor together with a gold medal. Paul Appell was also rewarded with a gold medal in addition to the honourable mention. For various reasons the prize ceremony didn’t take place on the king’s birthday, but the announcement was made public that day. Instead Poincaré received his prize from the hands of the Swedish ambassador in Paris later in March. Mittag-Leffler also informed the French Academy of Sciences of the news.

Amid suspicions of a flaw in the work by assistant and gifted former student Edvard Phragmén, who became an active editor in Stockholm while Mittag-Leffler traveled Europe, the printing was held up.

Finally, in July 1889, Mittag-Leffler decided that it was time to take action and print Poincaré’s dissertation, with all its added appendices. This went on until mid November, when the next volume of Acta was due to appear. Phragmén went on with the editorial work, and from the summer he was the only one who still raised objections to conclusions in the memoir that he didn’t understand, first to Mittag-Leffler and Weierstrass, and then directly in contact with Poincaré. The queries forced Poincaré more and more to confront his arguments in detail.On the last day of November 1889, an ominous telegram reached Mittag-Leffler. Poincaré briefly told him to stop the presses. He had found an error. An explanation was expected by letter the next day. After a sleepless night Mittag-Leffler could then read that the error was graver than Poincaré had first thought. “It is not true that the asymptotic surfaces are closed”, he wrote.

Poincaré was asked to pay for the first printing, which he accepted. The expenses amounted to over 3,500 kronor, i.e. 1,000 more than the prize money he had received!

After intense work in December, and over Christmas and New Year, Poincaré was ready to submit a substantially revised memoir on 5 January 1890. He had altered some of the implicit assumptions which had turned out to be precipitate. Instead of stability for the restricted three-body problem, he had come to the inevitable conclusion that chaotic motion could occur, as we would now call the phenomenon.

The printing resumed in late April 1890, but Poincaré’s final memoir of 290 pages only appeared in December 1890, in Volume 13 of Acta, together with Appell’s contribution and Hermite’s report.

Henri Poincaré and Gösta Mittag-Leffler

A remarkable epilogue to King Oscar’s prize competition occurred when the Finnish mathematician and astronomer Karl Sundman actually found a complete solution to Question 1 in the general case of three bodies. In articles between 1907 and 1912 he gave a proof of the convergence of an infinite series solution to the three-body problem for almost all initial values, using well-known results. Although the methods used are relatively simple, the very slow convergence renders the series solutions unusable for practical purposes, and they provide no qualitative insight into the motion of the bodies. Even though Sundman’s achievement was praised and received attention in the decade to follow, it soon faded into oblivion. In 1991 Qiudong Wang managed to generalize Sundman’s solution to the general n-body case.

Extensive parts of this post have been clipped and/or paraphrased from a much longer article at Mittag-Leffler Institute *PB

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