Thursday, June 27, 2013

Erudite Euler

Leonhard Euler (1707-1783)
Up this week is Euler, the great mathematician. It is impossible for me to do him justice in this post, because of the shear volume of his work and my lack of knowledge about mathematics. If you would like to read more about him, please look at some of the articles in the references. These ones are particularly good.

I mostly know of Euler from two things-Euler's method, which was an approximation method I used in calculus, and Euler's formula, which could actually mean many things, but in this case I refer to his formula that relates exponential and trigonometric functions, eix = cos(x) + isin(x), and is perhaps seen more commonly in less scientific circles in the particular form where x = π, where it simplifies to e = 1. These are both definitely important, and I used the latter in about every other homework that I did this year, but Euler considered many more applied problems than this small sample size would suggest.

Leonhard Euler was the son of a minister and originally intended to study theology. Fields of study were different in the 18th century, though, and at the conclusion of his master's degree he gave a lecture comparing the natural philosophy of Newton and Descartes. At the University of Basel he studied, among other things, mathematics under the tutelage of Johann Bernoulli, who encouraged him to study mathematics more pointedly and was of much help in later years as well. As a young man, Euler competed for the prize question of the Paris Academy of Sciences, a competition open to the greatest scientific minds in Europe, and came in second. Not bad. In later years, he came in first twelve times. He applied to a position in physics at the University of Basel and failed, but then was invited to the Academy of Sciences in St. Petersburg, where he was devoted mainly to mathematics. He stayed in St. Petersburg from 1727-1741. He was then invited to help found the Academy of Sciences in Berlin, Prussia. He did not get along well with Frederick II, though, and when rebuffed from the position of president of the Academy, returned to St. Petersburg in 1766 at the invitation of Catherine II, and stayed there until his death in 1783.

In setting down to consider the accomplishments of Euler, I found it interesting to note who had gone before. Nicholas Fuss, one of Euler's students, in his eulogy on Euler, said:
At the time when Mr. Euler entered into mathematics, nothing could be more discouraging. A mediocre talent simply could not expect to make a name for it and it was best to choose another career or to distinguish one brilliantly. The memory of the recently deceased great men that had been part of the past century and the beginning of ours was still particularly fresh in our minds. Hardly had Newton and Leibniz altered the face of geometry when they died and we had not yet forgotten the important services that the discoveries of Huyghens, Bernoulli, Moivre, Tschirnhausen, Taylor, Fermat and so many other mathematicians had provided to all the branches of mathematics.
Euler clearly chose the second option: "to distinguish one brilliantly". Rather than consider that mathematics had been exhausted, as one might think, or even that certain areas had, he pursued many different avenues and pushed mathematics in many new and old directions.

He tackled the field of mechanics in two volumes, introducing to it integral and differential calculus. He was also very interested in sound, and had written a thesis on it when applying for the position in Basel. But he returned to the subject in St. Petersburg, and extended his writings to include the emotions that sounds can evoke. There he also developed the Γ function (which gives factorials for positive integers, but can be applied to non- and negative integers as well), and the constant γ, called Euler's constant. He also developed the concept of the fuction, and the notation still used today of f(x). While writing on complex and novel mathematical ideas, Euler also wrote works on more basic subjects, like textbooks on arithematic for use in Russian schools, and Théorie complete de la construction et de la manœuvre des vaisseaux, a text for sailors on navigation. Other problems that he dealt with included optimal profiles for the teeth on gears, why disks (think pennies being spun) seem to spin faster as they fall down, the critical load for a rod to buckle, and the number of vertices, edges, and faces for polyhedra. In investigating these, he also often returned to a topic for many years after he first looked at it.

Something that undoubtedly helped his work was his prodigious memory. He was reported to be able to recite the entirety of Virgil's Aeneid (which, having read, I can assure you is no mean feat). It helps, of course, that he could read and write Latin. He was also very good at doing calculations in his head and remembering the results afterwards. This was vital to his work, especially in later years, since he lost the sight in one eye in 1735 and suffered from cataracts in the other, eventually losing his sight almost completely. That did not, however, stop his productivity, and he had his sons and others take dictation, or copy large letters from a slate.

For one with such remarkable skills, he also seems to have been quite humble and well liked, and passed up opportunities to quibble over who had discovered things first. He married twice and had 13 children, five of whom survived to adulthood. He had a fit of apoplexy while playing with one of his grandsons and drinking a cup of tea, and died a few hours later. Fuss spoke glowingly about him: how he dropped calculations for ordinary conversation, explained concepts at the level of the listener, did not hold
grudges, fought injustice where he saw it, and many other praiseworthy qualities. It has been exciting to see how a man could be at the top of his field, clearly pursuing topics that piqued his interest, and yet still  be praised as a humane, relateable, Christian man.

  • Marquis de Condorcet, "Eulogy to Mr. Euler", History of the Royal Academy of Sciences, 1783, Paris 1786, p. 37-68. A nice summary of Euler's life and work, free of equations and with some nice anecdotes.
  • Nicolas Fuss, "Eulogy of Leonhard Euler", read at the Imperial Academy of Sciences of Saint Petersburg, October 23, 1783. Fuss was a student of Euler and a grandson-in-law. His eulogy is longer than Condorcet's, but more personal. If you are interested in what Euler was like as a person, skip to the end. Fuss paints a wonderful picture of a caring, Christian family man.
  • Walter Gautschi, "Leonhard Euler: His Life, the Man, and His Works", SIAM Review 50, no. 1 (2008), 3-33. DOI: 10.1137/070702710. A relatively short summary of his life, providing both an outline of life events and some of his mathematical accomplishments. For a quick summary of some of his math, this is a good place to start, as this one actually includes some diagrams and equations.

Thursday, June 20, 2013

The Elusive Wulff

Wulff construction
A Wulff plot (the surface energies are given in red)
Drawing by Michael Schmid
and used under the GNU Free Documentation License 
I first ran into the name Wulff last year in my thermodynamics class. He gave his name to a method for constructing the shape that a single crystal will take based on the surface energies of different crystallographic directions. It is a clever construction, and that particular homework problem was probably my favorite of the whole year. So I decided to see what I could find out about Wulff, and I have found him very difficult to track down, so I put this post on the shelf. Then in winter quarter I ran into the name again in an x-ray diffraction class where we used Wulff nets, and I decided to track him down again. It proved no easier, but I did get farther! It does seem strange, though, for a man who's name is so often used, that there is so little information on him. He doesn't even have an English Wikipedia page! The first reason for confusion about Wulff is that he was Ukrainian, so there are different transliterations of his name, but worse than that, he went by two names! Georg Wulff was the name he used in German-language publications, and thus is the name that we are more familiar with, but his name in Russian (transliterated, of course) was Yuri Viktorovich. I did, at last, find a nice article on him in the Complete Dictionary of Scientific Biography, and some information in one of my x-ray diffraction textbooks.

Image of a projection from Wulff's 1902 paper
Georg Wulff was born in the Ukraine in 1863 and studied at Warsaw University. In 1907, or 1908, or 1911, he became a professor (or teacher) of crystallography at Moscow University. Nobody seems to be able to agree. What is important, I think, is that between defending his dissertation and his death, he taught in various capacities at universities in Russia and the U.S.S.R.

He published two important papers in 1901 and 1902 regarding crystal structures and stereographic projections. They are both in German, so I don't know exactly what they are getting at. Hammond says that Wulff proposed the Wulff net in 1909, but there seems to be an image of part of a Wulff net in his 1902 paper. His 1901 paper introduced the principles of the Wulff construction, which is a graphical method for determining the faces of a crystal that are expressed based on the surface energy of the different crystallographic directions. This idea built on Josiah Gibbs' proposal that materials want to minimize total surface energy. Wulff himself did not prove mathematically why his construction worked, but it was proved by Conyers Herring (1914-2009) in the 1950s.
Crystal diagram from Wulff's 1901 paper

Wulff was also in communication with William Henry Bragg and his son, William Lawrence Bragg, English crystallographers. Wulff derived an equation for x-ray diffraction in 1913 that was equivalent to the one proposed the year before by the Braggs, and so some people at the time called what is now known as Braggs' Law the Bragg-Wulff Law. Wulff appears to have lost out on the name because he published second, and, more importantly, he did not follow up with as many advancements on the topic as the Braggs. After that, he appears not to have taken on any new areas of study and faded into obscurity, though not without leaving his name for students of thermodynamics and crystallography to stumble upon.