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American Mathematical Monthly -April 2007

April 2007

Euler's 'Mistake': The Radical Product Rule in Historical Perspective
By: Alberto A. Martinez Writers commonly note that Leonhard Euler committed embarrassing errors in his Algebra of 1770, concerning the multiplication of the square roots of negative numbers. I review the early and influential objections against Euler's formulation, analyze his expressions, and conclude that he committed no mistake on the matter, because his approach is internally consistent, even though it was not widely adopted.

The Flaw in Euler's Proof of His Polyhedral Formula
By: Christopher Francese and David Richeson, In 1750 Leonhard Euler noticed that a polyhedron with F faces, E edges, and V vertices satisfies F-E + V = 2, and a year later he discovered a proof. This beloved theorem, now called Euler's polyhedral formula, is arguably the most important contribution to the theory of polyhedra since the work of the ancient Greeks. Unfortunately, Euler's proof of this theorem is flawed. In this paper, we present Euler's proof, complete with numerous excerpts from the original paper. We show that the shortcoming of Euler's proof stems from his failure to define polyhedron (or "solid enclosed by plane faces,"as he calls such an object) and to address issues of convexity. We conclude the paper by showing how Euler's proof can be salvaged.

The Gamma Function: An Eclectic Tour
By: Gopala Krishna Srinivasan This article is a historical guided tour through a period of 280 years since the birth of the gamma function. The gamma function has inspired mathematicians of the nineteenth and early twentieth centuries and continues to do so with its vast store of beautiful identities. Among several roads, the most popular is the one discovered in 1922 by Bohr and Mollerup and traveled by E. Artin, characterizing the gamma function in terms of the functional relation and deriving therefrom its principal properties. Sixty years later, R. Remmert took a road less traveled and found dwelling among its untrodden ways a function-theoretic characterization given by H. Wielandt in 1939.

We present in detail two other approaches to the study of the gamma function and derive some of the less known results such as the reciprocity formula of Schobloch and the interpolation formula of M. A. Stern. The first is the additive approach reminiscent of Eisenstein's development of trigonometric and elliptic functions. The second is due to Gauss. Gauss obtained all the well-known properties of the gamma functions as byproducts of his researches on the hypergeometric series. We present the Gaussian approach and, in particular, discuss his remarkable proof of Euler's reflection formula. In addition, we sketch in parallel the Eulerian approach wherein the properties are derived through integrals defining the beta and gamma functions.

Semidiscrete Geometric Flows of Polygons
By: Bennett Chow and David Glickenstein, Curve shortening flow is a way of continuously untangling complicated curves into curves that eventually resemble planar circles shrinking to a point. We explore a linear system of differential equations on the vertices of a polygon that exhibits similar behavior: solutions to the equation deform complicated polygonal curves, possibly with self-intersections, into regular polygons (up to a linear transformation of the entire polygon) that shrink to a point.

When Soap Bubbles Collide
By: Colin Adams, Frank Morgan, and John M. Sullivan,, In soap bubble froths, bubbles meet in threes along curves and in fours at points. Is it possible to decompose space into sets that meet only in threes? We show that the answer is no, for locally finite decompositions into sets of bounded diameter. More generally, there are no such decompositions of n-space into sets meeting at most n at a time. It follows that in a flat n-dimensional torus, for example, tilings cannot occur as minimizing m-bubble clusters when m


A Combinatorial Proof of Vandermonde's Determinant
By: Arthur T. Benjamin and Gregory P. Dresden,

Evaluation of Some Improper Integrals Involving Hyperbolic Functions
By: Michael A. Allen

When is 0.999... Equal to 1?
By: Robin Pemantle and Carsten Schneider,

Ghys's Theorem and Semi-Osculating Conics of Planar Curves
By: Brendan Foreman

An Absolutely Continuous Function in (P)/W¹¹(P)
By: Gilbert Helmberg

Problems and Solutions


Geometric Function Theory: Explorations in Complex Analysis
By: Steven G. Krantz
Reviewed by: John P. D'Angelo