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Neither in the subjective nor in the objective world can we find a criterion for the reality of the number concept, because the first contains no such concept, and the second contains nothing that is free from the concept. How then can we arrive at a criterion? Not by evidence, for the dice of evidence are loaded. Not by logic, for logic has no existence independent of mathematics: it is only one phase of this multiplied necessity that we call mathematics.

How then shall mathematical concepts be judged? They shall not be judged. Mathematics is the supreme arbiter. From its decisions there is no appeal. We cannot change the rules of the game, we cannot ascertain whether the game is fair. We can only study the player at his game; not, however, with the detached attitude of a bystander, for we are watching our own minds at play.

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# Euler Tercentenary Volumes

The MAA Tercentenary Euler Celebration: 2 vols: The Genius of EulerThe Early Mathematics of Leonhard Euler.

Volume I:  The Early Mathematics of Leonhard Euler, C. Edward Sandifer (ed): 2007, 391 pp. hard cover, $49.95, (member price$39.95) ISBN 10: 0-88385-559-3;

Volume II: The Genius of Euler: Reflections on his Life and Work, William Dunham, (ed); 2007, 307 pp. hard cover, $47.95 (member price$38.50) ISBN 10: 0-88385-559-5;

Mathematical Association of America, Washington, D.C. 1-800-331-1622 or www.maa.org

The year 2007 marks three hundred years since the birth of one of the greatest mathematicians of all time, Leonhard Euler (15 April 1707 – 18 September 1783).  His work was both prolific and enormously broad in scope.  He wrote textbooks in algebra, analytic geometry, calculus, and differential equations as well as authoring nearly five hundred research articles on topics ranging from complex variables, topology, algebra, probability, number theory, to mechanics and optics.  Another 300 of his papers were published posthumously. Many of the terms and symbols Euler introduced have become standard in our mathematical lexicon and practice, for example: “ e”, “Π” , “i”  and the functional notation  f(x). His notions of function, logarithms and trigonometric and other transcendental functions have remained with us.  Leonhard  Euler accomplished much of this work while he was blind.  Certainly, he was an unusual man worthy of tribute.

Ed Sandifer has surveyed Euler’s early work and purposefully chosen a collection of papers published between the years 1725 to 1741.  The published collection, The Early Mathematics of Leonhard Euler, contains descriptions of forty-nine of Euler’s articles spanning the richness and breadth of the man’s genius.  From his first paper, “Construction of isochronal curves in any kind of resistance,” written as he was leaving Johann Bernoulli’s tutelage to his statement “On the utility of higher mathematics” offered as he was moving on from the St. Petersburg Academy in 1741, the reader is introduced to a sweeping landscape of mathematics touching on such topics as continued fractions, the Chinese Remainder Theorem, Gamma functions, the calculus of variations, the Ricatti equation and many more.  The articles are presented in chronological order but clustered into segments upon which the editor has added illuminating commentary on the world events of the time and Euler’s life.  This commentary sets an excellent perspective for further appreciating Euler’s accomplishments.

Bill Dunham has chosen thirty published articles and items (poems, illustrations, etc.) to survey Euler’s life and work.  The resulting volume, The Genius of Euler, is a very readable and enjoyable book.  An eclectic group of authors ranging from the well known:  Rouse Ball, George Polya, Carl Boyer, Morris Kline, etc to lesser known contributors, a high school teacher, Charlie Marion, to graduate students, Dominic Klyne and Lee Stemkoski, share their findings and impression of this great mathematician.  A glossary (pp 289-301) supports the text together with varied bibliographies.  Photos and diagrams help make this book visually appealing and even more informative.

Good books are not just written or edited, they are crafted.  The exposition is carefully chosen around instructional goals; a story is told.  Supporting materials complement and enhance the presentation.  The whole is greater than the sum of its parts.  These are two good books, masterly crafted by their editors.  They comprise a worthy tribute to the mathematical genius of Leonhard Euler.  The books are informative, attractive and quite readable.  We owe the editors, Edward Sandifer and William Dunham, an acknowledgement of thanks for their efforts on behalf of the MAA on this project.  Every college and university library, as well as any individual truly interested in the history of mathematics, should have copies of these two books

Frank Swetz, Professor Emeritus, The Pennsylvania State University