Search Loci: Convergence:
"When I use a word,"
Humpty Dumpty said,
in a rather scornful
tone, "it means just
what I choose it to
mean - neither more
Through the Looking Glass.
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Euler: The Master of Us All
Euler, The Master of Us All, William Dunham, 1999, xxviii + 185 pp., $31.95 MAA members $25.95, ISBN 0-88385-328-0, The Mathematical Association of America, P.O. Box 91112, Washington,D.C. 20090-1112. 1-800-331-1622 or www.maa.org.
Leonhard Euler (1707-1783) was a man of faith: by evening he led the family Bible study, and by day he manipulated infinite series and assigned limits also by faith, if not by sight. Lacking clear definitions and useful theorems for the concepts of function, limit, and convergence, unencumbered by logical rigor, and despite progressive blindness, Euler did not hesitate to invent extraordinarily creative ways to manipulate equations and discover new truths in all fields of mathematics. Later generations have marveled at Euler’s insight and creativity, even as they have established rigorous verifications for his results.
Euler’s greatest early fame came in 1735 when he solved Jakob Bernoulli’s “Basel problem” by establishing the remarkable result that the sum of the reciprocals of the squares of the positive integers converges to one-sixth the square of pi. His collected works, written in Latin, French, and German, comprise more than 70 volumes. Thus, despite Laplace’s famous advice to “Read Euler, read Euler,” many modern inquirers will choose instead to read Dunham’s superb introduction to Euler’s accomplishments in eight selected areas of mathematics (number theory, logarithms, infinite series, analytic number theory, complex variables, algebra, geometry, and combinatorics). Dunham writes for a mathematically literate reader who has mastered calculus, but not necessarily much beyond that. For each topic, Dunham sets the mathematical context, provides clear, concise, and sometimes beautiful explanations of Euler’s accomplishments, and mentions subsequent developments by other mathematicians.
Dunham includes a short biography, and repeatedly envisions how Euler must have enjoyed his unexpected twists of thought. The reader also should smile at Euler’s inventiveness, such as when he used the divergence of the harmonic series to show that there are infinitely many primes.
For a captivating introduction to the work of Leonhard Euler and the mathematics of the eighteenth century, read Dunham, read Dunham. For a quick warm up, read about Euler at The MacTutor History of Mathematics Archives, www-gap.dcs.st-and.ac.uk/~history , and for interesting additional details see the monthly column by Ed Sandifer, How Euler Did It, www.maa.org.
Clifford H. Wagner, Associate Professor of Mathematics and Computer Science, The Pennsylvania State University at Harrisburg