Friday, July 22, 2011

Euler: The Master of Us All

Swiss mathematician Leonard Euler (1707-1783) is a fascinating man. I discussed him once before in this blog, during an entry about the book e: The Story of a Number. Euler’s name never appears in the 4th edition of Intermediate Physics for Medicine and Biology, but his influence is there.

William Dunham describes Euler's life and work in his book Euler: The Master of Us All. In the Preface, Dunham writes
“This book is about one of the undisputed geniuses of mathematics, Leonhard Euler. His insight was breathtaking, his vision profound, his influence as significant as that of anyone in history. Euler contributed to long-established branches of mathematics like number theory, analysis, algebra, and geometry. He also ventured into the largely unexplored territory of analytic number theory, graph theory, and differential geometry. In addition, he was his century’s foremost applied mathematician, as his work in mechanics, optics, and acoustics amply demonstrates. There was hardly an aspect of the subject that escaped Euler’s penetrating gaze. As the twentieth-century mathematician Andre Weil put it, ‘All his life…he seems to have carried in his head the whole of the mathematics of his day, both pure and applied.’”
In Chapter 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss one of Euler’s best known contributions, his relationship between the exponential function, trigonometric functions, and complex numbers.
“The numbers that we have been using are called real numbers. The number i = √−1 is called an imaginary number. A combination of a real and imaginary number is called a complex number. The remarkable property of imaginary numbers that make them useful in this context is that e = cosθ + i sinθ.”
Dunham wrote about this identity:
“ ‘From these equations,’ Euler noted with evident satisfaction, ‘we understand how complex exponentials can be expressed by real sines and cosines.’ His enthusiasm has been echoed by mathematicians ever since. Few would argue that Euler’s identity is among the most beautiful formulas of all.”
Euler didn’t invent complex numbers, but he did contribute significantly to their development, including a derivation of this gem (“which seems extraordinary to me,” wrote Euler)

ii = e-π/2.

Dunham’s book gives examples of Euler’s contributions to number theory, logarithms, infinite series, analytic number theory, complex variables, algebra, geometry, and combinatorics. For instance, Dunham describes an discovery Euler made when in his 20s.
“One of his earliest triumphs was a solution of the so-called ‘Basel Problem’ that perplexed mathematicians for the better part of the previous century. The issue was to determine the exact value of the infinite series

1 + 1/4 + 1/9 + 1/16 + 1/25 + … + 1/k2 + … .

… The answer was not only a mathematical tour de force but a genuine surprise, for the series sums to π2/6. This highly non-intuitive result made the solution all the more spectacular and its solver all the more famous.”
As he grew older, Euler slowly became blind. His accomplishments despite his handicap remind me of Beethoven composing his majestic 9th symphony after going deaf. Dunham writes about Euler
“Although unable to see, he not only maintained but even increased his scientific output. In the year 1775, for instance, he wrote an average of one mathematical paper per week. Such productivity came in spite of the fact that he now had to have others read him the contents of scientific papers, and he in turn had to dictate his work to diligent scribes. During his descent into blindness, he wrote an influential textbook on algebra, a 775-page treatise on the motion of the moon, and a massive, three-volume development of integral calculus, the Institutiones calculi integralis. Never was his remarkable memory more useful than when he could see mathematics only in his mind’s eye.

That this blind and aging man forged ahead with such gusto is a remarkable lesson, a tale for the ages. Euler’s courage, determination, and utter unwillingness to be beaten serves, in the truest sense of the word, as an inspiration for mathematician and non-mathematician alike. The long history of mathematics provides no finer example of the triumph of the human spirit.”
Dunham concludes
“Euler left behind a legacy of epic proportions. So prolific was he that the journal of the St. Petersburg Academy was still publishing the backlog of his papers a full 48 years after his death. There is hardly a branch of mathematics—or for that matter of physics—in which he did not play a significant role.”

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