Science: The Mozart of Mathematics

A Friend of Amicable Numbers
(1737 Portrait by Johann Georg Brucker)
David Brown and Ronald S. Calinger
Washington Post Staff Writer and Professor, Catholic University
Tuesday, April 10, 2007; 11:00 AM

Washington Post staff writer David Brown and Professor Ronald S. Calinger, a historian of mathematics at Catholic University, were online Tuesday, April 10 at 11 a.m. ET to discuss the life and work of mathematician Leonhard Euler.

A transcript follows.

Known as the "Mozart of Mathematics," Euler published more than 800 papers of pure and applied mathematics before his death in 1783 (this Sunday marks the although anniversary of his birth). Within the math and physics communities, he is known for having just as much of a significant contribution as Archimedes, Isaac Newton and Carl Friedrich Gauss. Read more in: The Countless Achievements of a Math Master (Post, April 9).


David Brown: Welcome math whizzes! This is the online chat about Leonhard Euler, the mathematician whose 300th birthday is coming up this weekend. I did a story in Monday's paper on Euler. And that pretty much ran out my string of Eulerian knowledge. Luckily, however, Professor Ronald Calinger has agreed to be on this chat with me. He is a historian of mathematics at the Catholic University of America, in Washington, and a trained mathematician. It will pretty much be his chat because it has been a very long time since I took college calculus---and of course that doesn't even get me into the same room as Professor Calinger, most of you, or (needless to say) Leonhard Euler. This appears to be a very smart group of chatters today. So let's begin.


Washington, D.C.: I have a degree in mathematics and have known about Euler since before taking courses in college. I am so thrilled that the Post wrote this story and is sponsoring an on-line chat about him. Math holds incredible beauty for me, which is difficult to express to my non-mathematical friends and colleagues. Thanks for fleshing out and humanizing this great man in a way that makes it possible for most non-mathematicians to appreciate him.

Ronald S. Calinger: Euler was indeed an impressive person. Computations were a first love of his. These were also a way for him to try to attain human perfection. We still have a way to go to bring out the multidimensional Euler. But work is underway. The works of Fellmann, Thiele and my efforts in Bradley and Sandifer's TERCENTENARY VOLUME deal with his life.


Derwood, Md.: Yay, Euler! It's nice to see that the Post is honoring Euler's 300th birthday. Can you talk about how mathematicians are celebrating it?

David Brown: There are a lot of activities, some underway, some later this spring. The Mathematical Association of America (it has a great Web site) and also the Euler Archive, have activities listed. There is a tour/pilgrimage from Basel to Berlin to St. Petersburg later this spring, with activities in each place. I believe every Swiss consulate in the U.S. is doing something (or has already). The Science and Technology section of the consulate in Boston is having a poster art contest pegged to his tercentenary.


Sherdstwon, W.V.: Gentlemen... Thanks for the article concerning the life and works of Leonhard Euler. As a teacher of mathematics for 40 years, I have tried to explain to my students that mathematics is not a lot of hocus-pocus, but a "Beautiful Developed Science". Newspaper articles like your only make my job a lot easier. Since we are celebrating Euler's 300 birthday this year, I have mentioned many of the topics discussed in your article. He certainty was a prolific mathematician. Thanks again for your article and I will certainly mention it this week in my classes.

Victor H. Hughes III

Adjunct Instructor of Mathematics

Shepherd University

Shepherdstown, West Virginia

Ronald S. Calinger: Hi,

Please see Bradley and Saandifer, eds., EULER'S LIFE, WORk, AND LEGACY (2007). It has new material on his life and his prolific contributions.


Sugar Land, Texas: Hello. My name is Danny. I'm a 7th grader at First Colony Middle School. I had a question about Euler. I am trying to prepare for a math competition. Could you tell me all you know about Euler's totient function, his theorems, and how they can help in solving number theory problems, please? Thank you so much.

Ronald S. Calinger: Hi Danny,

Euler was quite fascinated with a myriad of ways to solve problems. In number theory he particularly worked with a correspondent named Goldbach. His research largely involves something called Diophantine equations. He proves conjectures from Fermat. The best source on this is Andre Weil, NUMBER THEORY FROM HAMMURAPI TO GAUSS. But a better source to begin with is William Dunham, EULER THE MASTER OF US ALL.

David Brown: Danny---

Thanks for this. (I hope you're not skipping class to listen.) Good luck in your competition; my money's on you and the First Colonials!



Madrid, Spain: Was Euler a nice person or of a difficult character?

Ronald S. Calinger: My view is that Euler was a fascinating person. He was widely read. He could tell stories and jokes. He loved music. He played chess. He loved the out of doors. He ordered trees and shrubs for the Berlin Academy. Frederick the Great was upset with him because Euler would take a pencil and paper to plays and work on problems during some scenes. He did mathematics out of a deep love for it. Work on it was a joy for him. A research circle formed around him, especially in St. Petersburg. Their relations were warm and engaging. See especially his secretary Nicholas Fuss, who married Euler's granddaughter. I also think that Euler was a sharp dresser, contrary to what is sometimes thought. He had his own tailor.

David Brown: He also appears to have had an unusual milliner--or possibly it was just a bath attendant. One of the more widely reprinted portraits of him shows him wearing what appears to be a towel on his head.



Virginia: Where can mathematicians work outside of academia, beside the NSA?

Ronald S. Calinger: I would begin with IBM and Microsoft. They are only the beginning.

David Brown: I believe Google also bought some billboard space in which it posed a difficult mathematical question, which then somehow provided a clue to another site where another question was posed, and the chain eventually led to an invitation for a job interview. (Or something like that.) So you might want to consider Google too.


Lorton, Va.: I really enjoyed the article about Euler. I recently helped my grandson, David, on a science project about Euler. We studied what we called Euler's formula, the one that relates the number of faces, edges, and verticies of a polyhedron. We underestimated how general this formula might be and thought, initially, that it only applied to Platonic Solids. We were surprised when we built some of the Archimedean Solids that it worked for them too. There are these wonderful plastic shapes made in Australia called "Geofix". We could build all of these shapes out of Geofixes. David and I really take our hats off to Euler, he certainly was a genius.



Ronald S. Calinger: Your project sounds challenging. You might want to check at some point Henri Poincare's generalization of Euler's polyhedron formula.


Washington, D.C.: As one who has always struggled with advanced mathematics, on rare occasions I have glimpsed the wonder of the science. Is there an underlying theme to Euler's work and did his work have a major impact on any of the other sciences?

David Brown: My sense is that Euler contributed to so many fields--calculus, number theory, geometry, physics, mechanics, astronomy--that in an ironic way he "disappears" behind this mass of achievement. It is difficult to label him with just one or two well-known achievements. That said, my reporting suggested that calculus and mathematical analysis is probably the area of his greatest practical contribution, as he brought that form of mathematical understanding and manipulation into science in a way that hadn't happened. This gave physicists, astronomers, etc. huge new tools for understanding and calculating. At the same time, Euler made myriad discoveries in calculus that were of both pure and practical importance.


York, Pa.: How did Euler help shape Calculus, especially what's taught in Colleges? He's not as much of a household name, nor do students have a clear understanding of his contributions to Calculus as they do for Newton, Leibniz, Riemann, Taylor, etc.

Ronald S. Calinger: Euler contributed perhaps more than the other names you gave. There was no arrangement for the basic theorems when Euler appeared. His INTRODUCTION IN ANALYSIS INFINITORUM began to provide a systematic framework to the field. He made functions central to the calculus. Previously geometric curves had that role. He was the principal inventor of the calculus of variations. He helped found differential geometry. He made hundreds of discoveries in differential and integral calculus. He even came up with the integral test for convergence of infinite series. He was phenomenal in applications of the new calculus to astronomy, mechanics, and optics, helping transform these into modern exact sciences. The reason we don't yet know Euler well may have more to do with the inaccessibility of his works, especially during the Cold War, and the sheer extent of his research. Not since Claudius Ptolemy had a scholar so dominated all of mathematics and associated sciences.


Arcata, Calif.: This is a great idea- too bad it's not broadcast in audio or video as well.

We at Humboldt State University will be having a birthday celebration with talks about Euler's life, work and times all Sunday morning of his birthday- April 15th.

One source for a quick and excellent presentation about his life and work was given by Professor Robin Wilson of Gresham College in England in 2002 available on line in Video

Best regards,

Martin Flashman

Professor of Mathematics

Humboldt State University

David Brown: Thanks for this. It is kind of thrilling to think that all kinds of people around the world will be taking a little time on Sunday and Monday to admire and honor someone who was born 300 years ago---even if most of us can't really engage his genius on its own terms.

And speaking of honoring Euler, one reader e-mailed me a photograph of his Virginia license plate: L Euler. It takes vanity to a new level of brilliance. Keep an eye out for it!


Alexandria, Va.: I'm not a mathematician, but enjoyed the article immensely even though I had not heard of Euler before. I appreciated the question posed to mathematicians regarding "beautiful" equations -- an adjective not frequently (or even infrequently) attributed to math (at least by non-math folks). What is it about Euler and his approach that he developed such beautiful equations?

Ronald S. Calinger: Since antiquity scholars have seen beauty as an essential aspect of mathematics. Plato was among these. Archimedes must have been. As G. H. Hardy said, "There is no such thing as ugly mathematics." Euler had deep interests in music and art that must have heightened this sense.

David Brown: As I have said, I don't know much math, but I know enough to realize that an equation that has e, i, pi, 1, zero, and multiplication and addition is breathtaking. It's hard to believe it exists. It's proof of the beauty and order in the universe, to say the least.


Washington, D.C.: Was Euler's interest primarily in pure mathematics or did he also use math to solve specific problems. Did he know he would be contributing to space travel and the other things you mentioned in your article?

Ronald S. Calinger: Euler contributed to all known areas of pure and applied mathematics of his time. He was well aware that much of his work would find future application. This is usually a point of debate in funding scientific research to the present day. Science fiction was different in Euler's day, but he read about people visiting other planets, for example.


Tucson, Ariz.: I seem to recall that Euler was a "fast computer" and that many of his conjectures were formed in a way similar to how conjectures would be created today on a computer. Unlike a computer/human combination, he had a whole understanding of the underlying math. I've noticed more and more PhDs appealing to simulation while ignoring - and being ignorant of - heuristic tools that Euler would have used. As historians, do you feel computers are doing any damage to math and science in addition to the help that they provide?

Ronald S. Calinger: If computers are seen as helpful tools, they are great. Since antiquity mathematicians have had tools to help in their computations. Consider the abacus. Just think of the computations that astronomers had to do by hand before computers. I think there are issues that still have to be resolved with computers, for example, what constitutes a proof. Michael Mahoney has written on some of these issues. An interesting science fiction book giving a different view is THE HUMANOIDS.


Washington, D.C.: Did Euler leave any notable descendants, or are they lost to history? -- Bill B.

David Brown: Euler apparently had many descendants, including, I believe, direct descendants in Germany and Russia, as well as collateral descendants in Switzerland. I am told that two of his descendants won Nobel Prizes---Hans von Euler-Chelpin, for the Prize in Chemistry in 1929, and Ulf von Euler, in Physiology or Medicine in 1970. But this is all hearsay testimony--I haven't actually reported it out.


San Diego: The University of San Diego and the Consulate General of Switzerland in Los Angeles are among those who will be celebrating Euler's life and work with several events.

There will be a week long scavenger hunt on the USD campus with prizes for the math-informed winners. There will be a lecture, reception, and birthday cake-cutting.

Re the lecture the announcement says: In honor of Euler's tercentennial, this talk will explore Euler's life and world, touch on some of his mathematics and seek to explain why the great Laplace exclaimed, "Read Euler! Read Euler! He is the teacher of us all."

David Brown: More Euler activities from the West Coast. A mathematical scavenger hunt---very cool. There's a script to a great thriller out there somewhere.


York, Pa.: Follow-up about Euler's contributions to Calculus ...

Didn't know that Euler was the one who developed the integral test for series. Great to know.

Do you think your book, or other factors, will increase Euler's visibility to the mathematical community & to text writers enough that his role in shaping Calculus will become more widely known and clearly understood?

Ronald S. Calinger: I think the Bradley and Sandifer, eds., EULER: LIFE, WORK AND LEGACY improves our knowledge of Euler. My biography will attempt to present a fuller understanding of Euler's vast contributions across the mathematical sciences within the context of his time and the status of mathematics in the Enlightenment. Please note that the Birkaeuser OPERA OMNIA for Euler continues to publish new volumes.


Gainesville, Va.: Hello, just wondering if you have some idea of how he would go about developing his mathematical ideas. Some of the series of number go into the millions. Would he think away about the series doing trial and error hour after hour day after day, or do you think he had flashes of insight telling him where the series would go? Or both?

Ronald S. Calinger: Euler persistently and constantly developed his mathematical ideas. That does not mean that he did not do other things as well. His general approach was to develop the computations in a field until he could go no further. He then laid that work aside. Either he or someone else would make a breakthrough that allowed a host of new computations. In the psychology of mathematics, these solutions are often said to emerge from the subconscious. That is, the problems are laid aside in the conscious realm but the subconscious takes over and in time pushes up the solution. There seems to be a need for both, that is, intense research and a subconscious response. Not Euler but some mathematicians become quite distressed when a problem resists solution.

David Brown: Poets do this to. The subconscious is a great teacher.


Rockville, Md.: Do you mean:

The Humanoids: A Novel (Paperback)

by Jack Williamson

That is the only one I know.

Ronald S. Calinger: Yes, the book is a paperback. Years ago it was required reading for artificial intelligence courses at a few colleges.


David Brown: Nobody's asked, but just for the record: The Faulkner quote about Joyce and the illiterate Baptist preacher is from a Paris Review interview done in 1956. It is in the first collection of Paris Review interviews with writers, and is wonderfully interesting and full of other quotable observations.


Arlington, Va.: With someone as prolific as Euler, I wonder about his activities outside of mathematics. Did he have any ? I would guess gardening and fishing weren't things he took part in (unless he was working on mathematics in his head)?

Ronald S. Calinger: While Euler concentrated on mathematics, he did have other interests. He was a devout Christian. He loved to play the clavier. He would invite young composers to his house to play their works and ask for responses. He attended dinners with his wife at the home of Maupertuis, the president of the Berlin Academy. He was a talented chess player. He liked to take his children to the zoo. He seemed to love gardening and was quite well informed about plants. In his later years in Berlin, he often took walks to visit his children who were residing with their grandmother, as was customary in the 18th century.


Falls Church, Va.: Whata wonderful chat! I would cross seven bridges to participate.

Prof. Calinger, it seems to be a pattern in our time that mathematicians complete their best and most original work very early in life (in their twenties or early thirties). Was that true of Euler as well, or did he sustain his creativity throughout his life?

Ronald S. Calinger: Euler continued his original work throughout his life. I have always questioned the view that good mathematics comes mainly early.

David Brown: Gloss on this question: Euler addressed a famous problem, called the Konigsberg problem, which posed the question (I believe) whether one could cross all seven bridges in the Prussian city of Konigsberg without crossing one twice and with returning to the starting point.


Bronx, N.Y.: In your opinion, who had a more interesting personal life Euler or Leibniz, and how did their life experiences affect their mathematical thinking.


Ronald S. Calinger: I would select Euler. Leibniz had a difficult life. He opened the way for Euler by proposing the Berlin and St. Petersburg Academies of Science.Leibniz was phenomenal. He never got the position that he wanted in Paris or England. The dispute with Newton was difficult. Euler usually won his competitions with people like d'Alembert.


Chicago, Ill.: If Euler is the Mozart, then Gauss is surely the Beethoven of mathematics. Why do you think that Gauss was so reluctant to publish so many of his deep results after Euler had shown the way to be a "modern" academic mathematician by publishing his results? Didn't Gauss learn anything from the needless Newton/Liebniz squabble?

Ronald S. Calinger: Gauss was very concerned about rigorous foundations. His motto was "few, but ripe." He was the opposite of Euler, who wanted to not hold information back for too long.


Washington, D.C.: In an add on to yesterday's article, the Post notes that Euler discovered two numbers which were in the millions, whose proper factors added up to each others' sums. I forget the name for this term but is there any real importance to this exercise, and even if there isn't, how did he investigate such high numbers?

Ronald S. Calinger: These are amicable numbers, such as 220 and 284. They are rare in the number system. But work with them involves prime numbers and perfect numbers. Thus they are important in number theory. The prime number theorem is one of the two elemental theorems of number theory.

David Brown: Amicable, but very hard to get to know.


New York, N.Y.: Dear David and Ronald:

Are there Euler texts that would be suitable for a layperson interested in mathematics, but years out from having studied it in school?

Ronald S. Calinger: I would recommend beginning with Euler's LETTERS TO A GERMAN PRINCESS, just to see Euler in a general phase. You could begin with Ed Sandifer's early manuscripts of Euler and his part on the MAA Web site titled "What would Euler do?" There are otherwise a host of works.


Rumford, Maine: Professor Calinger,

The public, in general, tend to shy away when mathematics is mentioned specifically. Euler wrote mathematics incessantly, but there is so much more to this man than just his mathematics. What about his cultural significance as a man of the church?

Ronald S. Calinger: Yes, Euler did not fit the general makeup of an Enlightenment figure. He was deeply religious. He saw his pursuit of the secrets of nature as a way to know God better. He was Basel reformed in his youth, but this did not exist elsewhere, so he was a Calvinist in Berlin and St. Petersburg.


Miami, Fla.: If Euler were to come back to life on the event of his 300th birthday, which recent developments in mathematics do you think he would be most intrigued by?

Ronald S. Calinger: I would say the new applications of differential equations in physics over the last 30 years. But I think he would like to see number theory as it has developed.


Bethesda, Md.: Thanks for the interesting article and what I assume will be a great book.

Dr. Calinger, I wonder if you could make make your case for why Euler should be considered the equal to Newton and Gauss. His life overlapped the end of Newton's and the beginning of Gauss's, but was he really as great as either of the other two?

Related question: Newton and Gauss both achieved considerable fame in their own lifetimes. How did Euler's contemporaries view him?

Ronald S. Calinger: In his lifetime, Euler was considered the greatest mathematician in Europe. I think that Newton and Guass made seminal discoveries that are sharp advances over the past. Euler did likewise in my estimation, but his work cannot be reduced to a simpler form yet. His work is both synthetic and original.


Richmond, Va.: Euler is a phenomenon, and his contributions form the basis of most things we now take for granted in the world of computing. Let's recognize the guy that gave us "e" and thereby paved the way for the actual use of natural logarithms. This is the only way mankind has been able to step up the evolution of intelligence - the natural logarithmic perspective of time series. This is the only way that physicists and mathematicians and singularitarians are able to fathom vast expanses of time and space and make sense of it. Euler is responsible for our intelligence today!

David Brown: It is interesting how some huge achievements are not associated with their inventors or discoverers. I would bet that not 1 in 100 Americans knows who invented television, although we spend an average of four hours a day in front of it!


Danbury, Connecticut: I am following this discussion with great delight, and since I am here with my books, and since I can type faster than Ron does, I can fill in a couple of gaps.

First, the commentator from Virginia is looking for jobs outside academia. Pixar has become one of the high-profile employers of mathematicians, too.

Second, the question from Washington, DC on an underlying theme for Euler's work: I learned from one of my teachers, Ron Calinger, that a question like this will have different answers in different times. What they thought of Euler's theme in 1783, when he died is different from what they thought 100 or 200 years later. I think it is that correct analysis, using the tools of calculus and algebra, based on correct principles of mechanics, can correctly predict the phenomena of the world, and he demonstrated it for us.

Ed Sandifer

David Brown: That from a distinguished mathematician and Euler scholar in his own right.


Danbury, Ct.: A participant in Washington asked if Euler had any notable descendents. According to a genealogy of the Euler family that seems to be authoritative, shown to me by Ruediger Thiele, indeed there are the two 20th century Nobel prize winners (a fact corroborated on the Nobel Prize web site) and also the very first German pilot's license, number D-1, was issued to another descendent named Euler.

Ed Sandifer

David Brown: Quite a family.


Sunabeda, Orissa state , India: Hello , I want to know whether two mathematicians have given us the same equation after studying a particular topic any where, any time? If so can you cite one or two examples. Do human mind catch hold of a knowledge which is universal or eternal? This seems to be true in case of science; what about maths?


Teacher in maths(India)

Ronald S. Calinger: The notion that there is one person who is first seems to be a part of western culture. In mathematics you can see several people working on the solution of the cubic equation by radicals at about the same time. Tartaglia and Cardano enter a debate. Cardano is later but he claims others preceded Tartaglia. Clearly the search for the solution of cubics was in the arithmetics of the 16th century. The invention of non-Euclidean geometries seems similar with Gauss and Lobachevsky.To add a little time, there is Riemann.

You are probing a question that still fascinates mathematicians. Do they discover or invent mathematics? It is safe to say that often the foundations for the discovery are in the literature or being pursued in some research project. If you accept the discovery thesis, there would be eternal truths that we occasionally grasp. The more we study mathematics, the more we see some preceding work that is vital to making a breakthrough, such as Leibniz and Newton made with differential calculus.


Oud-Turnhout, Belgium: Why is it that Euler is so in the spotlight these days? Anything to do with "Quantum Mechanics" and the renewed interest in complex numbers??

Ronald S. Calinger: I think the chief reason is the 300th anniversary of the birth of Euler. In addition, the publication of his OPERA OMNIA makes his primary sources more accessible. Mathematicians see that his contributions are vast, sometimes unexpected. For example, a divergent series of his, which diverges slowly, is important in describing the lift on an airplane wing. This is important on two levels. It showed that divergent series should not be discarded out of hand and that there was another application in aeronautics. Another reason for this interest may be the end of the Cold War. It is now easier for American and western European scholars to work with colleagues in Russia. It seems that the branch of the Russian Academy of Sciences in St. Petersburg is making impressive efforts to present the work of Euler.

David Brown: And now I'm afraid it's time for us to Turnhout too. Alas, there are still a lot of very interesting unanswered questions. But there is a huge amount of Euler material just a web search away. The Euler Archive is a good place to start. Thank you for tuning in. I think I can speak for all of us that it was a privilege to have Professor Calinger here to do the heavy lifting.


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