Ed Sandifer
Western Connecticut State University
Danbury, CT 06810

sandifer@wcsu.ctstateu.edu
http://vax.wcsu.edu/~Sandifer/homepage.html

Introduction

The role of problems in promoting and guiding progress in mathematics is well known. Over the centuries, the ways have changed by which problems are set before the mathematical community. I admit to a gross simplification of a considerably more complex situation as I try to condense the last five centuries of this evolution into just a few sentences.

In the 16th Century, the times of Tartaglia and Cardano, there were the problem contests, by which people won and kept their academic positions. The next century brought named problems posed by individuals, Debeaune's problem of inverse tangents, the Brachystichrone Problem, the Basel Problem, Fermat's Last Theorem. The 18th and 19th Centuries saw the spread of academic societies and their prize problems. The best known is probably the Paris Prize, won so many times by Euler. It seems, though, that prizes were offered by almost all the important academic societies, including St. Petersburg, Leipzig, Madrid, Copenhagen, among others. Prize problems were sometimes rather general. One of the Paris Prize problems that Euler won was to explain the nature of fire [E2]. As time went on, academy problems became more specific. In 1862, the Danish Academy offered a gold bar for explaining the breeding behavior of the hagfish [G]. The New York Times reports (01/02/2001) that the prize is still unclaimed.

A thorough account of the history of academy prize problems would probably make dull reading, but it would be nice to have all the information about them consolidated into a single reference.

In the 20th Century, it can be said, we got to pose our own problems. That, essentially, is how a grant proposal works. The applicant poses a problem and offers to work on it, and the granting agency sends money, whether the problem gets solved or not.

Another form of presenting problems, lists of problems, appears only occasionally in this time period. Hilbert's 23 Problems are probably the most famous. The seven Clay Mathematical Institute problems, each backed with a million dollar prize, may prove to be significant, though it can be argued that those problems had already been posed and named, and that CMI has only anthologized the problems, rather than posing them.

Erdös, of course, published more lists of problems than you can count. Most of his problems were in number theory and combinatorial mathematics, and he attached monetary prizes up to $1000 to many of these problems.

It should come as no surprise to anyone in this audience, particularly those of you who have read my abstract, that long before Clay or Erdös or Hilbert, Leonhard Euler also posed lists of problems. You will not have discerned from the abstract that Euler posed not just one, but at least four lists of problems!

Euler read his first list before the Klassensitzung in Berlin, essentially the weekly seminar of the Mathematics Department, on September 6, 1742. [W] It is a short list, just seven problems, of things he thought were important and that he was working on, and isn't a challenge to other mathematicians to work on these problems.

Euler's list of 1742

To put a context to these problems, in 1742, Euler had been in Berlin for just two years. His solution to the Basel Problem, about the sum of the reciprocals of the squares [E1], had been published in 1740 (after a six year publication delay) and Euler's fame was spreading rapidly. In June of 1742, he had received the letter from Goldbach posing the Goldbach Conjecture about even numbers always being the sum of two primes, still unsolved and not one of the Clay Math problems. A couple of years ago, a publishing company offered a million dollars for a solution to the Goldbach Conjecture. [D]

Most of these problems became Euler's published papers in the next couple of years, and he wasn't really challenging the rest of the mathematical community to join him in working on the problems.

The two "Quaestiones Physicae" lists

Euler's next two lists were two drafts of "Quaestiones Physicae". The first draft of 17 problems was included in a letter from Euler in Berlin to Schumacher in Petersburg dated August 13/24, 1751. [W] In that letter, Euler specifically recommends that the Petersburg Academy use the questions as prize problems.

Euler sent a second draft of "Quaestiones Physicae" to the Petersburg Academy on November 18/29, 1755, this time 18 questions. These questions were read before the Petersburg Academy almost two years later, on July 7, 1757. Among the questions in the second draft were:

Here, my translations are rather casual, while my other translations are more literal.

These problems were clearly intended to motivate and guide scientific research for the next several years, the same motives claimed by Hilbert and by the Clay Mathematical Institute. Still, they are "Questions of Physics", and we are interested in mathematics.

The Fourteen "Quaestiones Mathematicae"

Euler's fourth list of problems, the fourteen "Quaestiones Mathematicae" are on the back side of the same sheet of paper that contain the second draft of the "Quaestiones Physicae" and accompanied his letter of November, 1755. They, too, were read before the Petersburg Academy two years later, and were intended as potential problems for the Academy's prize.

Quaestiones Mathematicae

I have not yet been able to determine how many of these problems were actually used as prize problems, though some of them, numbers 6 and 8 for example, were certainly used, and others, number 12, for example, were probably pre-empted by Euler's own work before they could be used as prize problems.

As we take a closer look at the problems, we ask, like Clara Pell, "Where's the math!" There is no number theory, no differential equations, no infinite series.

The answer lies in the meaning of the word "Mathematics." To Euler, mathematics included what we now call mathematical physics and mathematical modeling. Topics we might call "pure mathematics" now, he might call geometry, arithmetic or analysis.

Now that we've accepted the idea that "Quaestiones Mathematicae" means applied mathematics, not pure mathematics, let's look at some of the questions themselves.

The problem here doesn't seem very interesting, but the comment, "Even if this machine is used most frequently, still its theory is desired," is very illuminating. It tells us that, to Euler, applied mathematics was more than just a design tool. It was a key to understanding. Even after a machine has worked well for thousands of years, it is important to know why it works. I think that Euler would agree with R. W. Hamming, who wrote 200 years later [H2] that "The purpose of computing is insight, not numbers."

Euler worked on problems of fluids for much of his career. This question, along with questions 6, 7 and 8, all reflect that continuing interest. Fluid flow still presents major mathematical problems, and one of the Clay Mathematical Institute problems, the question about the Navier-Stokes equations [F], is a modern version of the same question. The second word in Charles Fefferman's explanation of the problem is "Euler."

Euler had published Tentamen novae theoriae musicae in 1739, about 15 years earlier. These questions show Euler's continued interest in subjects of acoustics. Moreover, the great scientific controversies of the times were based on the different world views of the Newtonians and the so-called Wolffians, the disciples of Descartes and Leibniz. The main venues of their disputes were the nature of light and the action of gravity. I believe that Euler felt that if they could establish a similarity between the nature of light and the nature of sound, then they might be able to make some progress on the fundamental questions of the nature of light [H1]. For example, does light require a medium for its propagation?

Euler also worked a good deal on this problem [S1] and made good progress. However, he did not discover the principle of superposition, and that left the way open for Fourier, in the next century, to win a prize with his solution to the problem.

This is the question of the meniscus, which we now understand to be caused by surface tension. In Euler's time, questions of the meniscus were near the center of the Wolfian-Newtownian controversies. Like gravity, the meniscus seemed to be a mysterious force capable of action at a distance. Thus, this apparently innocuous little question lies at the heart of one of the day's greatest issues.

Euler's suspicions on the direction from which a solution for his follow-up question suggests that the phenomenon is related to the surface of the fluid.

I would caution against misreading Euler's remarks here. When he wrote of the "motion of bodies which are floating on those surfaces", far sighted though he was, he was not speaking of Brownian motion. Instead, he was asking for an explanation of the phenomenon that small objects floating near a concave meniscus, will seem to float "up hill" to the edge of the container. A dual phenomenon occurs near a convex meniscus.

Euler is asking that the results of his then-recent 1749 book Scientia navalis be extended. Indeed, they were, and he published them again in 1773 as Théorie complette de la construction et de la manoeuvre des vaisseaux.

It is not clear to me what Euler expected from this problem. Does he want a differential equation that explains the decrease in air pressure with altitude? That seems too simple. Is it a chemistry question? Is it a meteorology question? I don't know, but I would be happy to pay close attention to anybody who might have some ideas.

Both of these questions have the same practical objective, the measurement of time and, from that, the measurement of longitude for navigation. [S2] With these two questions, Euler was promoting both of the leading programs of the for the determination of longitude. The clock would keep a standard time, like Greenwich Mean Time, and the traveler would use astronomical observations to determine latitude and local time. The difference between the two times would determine longitude, and the navigator would know where he was.

A solution to question 11, the three body problem would allow the calculation of a standard time from almanac data about the position of the moon or of the moons of Jupiter. The almanac data required the accurate solution of a three body problem, either a Sun-Earth-Moon system or a Sun-Jupiter-moon system. The solutions given in his 1753 Theoria motus lunae (Theory of the motion of the moon) were too complicated to be useful, but by 1772, in Theoria motuum lunae, (Theory of the motions of the moon), the techniques were almost practical.

A good deal of Euler's other mathematics were motivated by such questions of navigation. For example, when he wrote a paper about using infinite series to evaluate sines or tangents or logarithms of trigonometric functions, he surely had navigational applications in mind.

This is the "spinning quarter" or "spinning hoop" problem. Why does the hoop seem to hover there instead of falling down flat? A toy demonstrating this phenomenon was marketed a couple of years ago in the "educational toy" stores under the name "Euler Disk."

It is a little surprising to see this problem on this list, prepared in 1755, read to the Academy in 1757, since Euler's well-known paper on the subject, E 257 "De frictione corporum rotantium", was mailed to Schumacher in Petersburg 23 Aug/3 Sept 1754. It almost seems that Euler posed the problem after he himself had solved the problem. Perhaps he was dissatisfied with his solution, or perhaps he had prepared the list before 1755.

In these last two problems, Euler poses practical problems in optics, questions which he himself addressed over and over again in several important papers and in his three-volume Dioptrica, published in 1769.

Conclusions

It seems that the fourteen problems that Euler sent to the Petersburg Academy in 1755 were, like Hilbert's, intended to motivate progress in mathematics, particularly applied mathematics, for the next several decades. Some of them became problems for the Academy's prize. Others remained on Euler's own research agenda and became some of his own important contributions. A form of one of Euler's problems remains a major unsolved problem to day.

Bibliography

[B] Burckhardt, J. J., E. A. Fellmann and W. Habicht, eds., Leonhard Euler: Beiträge zu leben und Werk, Birkhäuser, Basel, 1983.
[D] Doxiadis, Apostolos K., Uncle Petros and Goldbach's Conjecture, Bloomsbury, NY, 2000.
[E1] Euler, L., "De summis serierum reciprocarum", Opera Omnia, Series 1 volume 14 pp. 73-86. Commentarii academie scientarum Petropolitanae, 7, (1734/5) 1740, pp. 123-143.
[E2] Euler, L., RP Lozeran de Fiesc, Comte de Créquy, Émilie du Châtelet and Voltaire, De la nature et de la propagation de feu, cinq mémoires couronnés par l'Académie Royale des Sciences Paris 1738, Association pour la Sauvegarde et la Promotion du Patrimoine Métallurgique Haut-Marnais, France, 1994.
[F] Fefferman, Charles, "Existence & Smoothness of the Navier-Stokes Equation", http://www.claymath.org/prize_problems/navier_stokes.htm, as of January 1, 2001. posted May 1, 2000.
[G] Goldberg, Carey, "Students Pursue One of the Ocean's Slimy Mysteries", The New York Times, p. F4, January 2, 2000.
[H1] Hakfoort, Casper, Optics in the Age of Euler: Conceptions of the nature of Light, 1700-1795, Cambridge University Press, 1995.
[H2] Hamming, R. W., Numerical Methods for Scientists and Engineers, 2ed., McGraw-Hill, 1973.
[S1] Schot, Steven, "Vibrating Strings, Functions and Distributions", unpublished manuscript distributed at the Institute for the History of Mathematics and Its Uses in Teaching, summer, 1996.
[S2] Sobol, Dava, and William J. H. Andrews, The Illustrated Longitude, Walker and Company, NY, 1998.
[W] Winter, Eduard, Die Registres der Berliner Akademie der Wissenschaften 1746-1766, Dokumente für das Wirken Leonhard Eulers in Berlin Zum 250. Geburtstag, Akademie-Verlag, Berlin, 1957.