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Mathematics Magazine - December 2003


Transits of Venus and the Astronomical Unit
Donald A. Teets
In the December 1771 edition of the Philosophical Transactions of the Royal Society of London, Thomas Hornsby wrote that “the mean distance from the Earth to the Sun [is] 93,726,900 English miles.” Amazingly, this distance differs from a modern radar-based value by a mere eight-tenths of a percent. Eighteenth century astronomers computed this distance, known as the astronomical unit, from the geometric configuration of the Earth, Venus, and the Sun during a rare astronomical event known as a transit of Venus. The method was suggested by Edmond Halley, and carried out by the English telescope maker James Short, among others. This article presents a glimpse into the very rich history surrounding observations of the transits of Venus, as well as a mathematical description of how the transits were used to determine the length of the astronomical unit.

PROOF WITHOUT WORDS: Equal Areas in a Partition of a Parallelogram
Philippe R. Richard
Equal Areas in a Partition of a Parallelogram presents two little known results in elementary geometry. The proofs via transformations are interesting as they use the development of dynamic mental images, which are controlled by the knowledge of the triangle’s area formula. Presented in the form of a comic strip, the acceptance of the proofs requires comparison of the initial and final states of each mapping.

Fits and Covers
John E. Wetzel
Precisely when does an equilateral triangle fit in a rectangle, or a square in a triangle? How large must a square be to accommodate every triangle of perimeter two, or an equilateral triangle to contain a copy of each arc of unit length? Questions of when one shape fits in another lie at the core of the investigation of shapes, and yet they seem to be little studied. We survey what is known about such “fits” and “covers” questions for elementary shapes, summarizing a few results, asking many questions, and reporting many conjectures.

The Operator (xd/dx)n and Its Applications to Series
Peter M. Knopf
Studying Taylor series in calculus, one learns how to obtain the Maclaurin series for the sine and cosine functions. But, what about the other trigonometric functions? The operator (xd/dx)n is a useful tool to deal with the remaining trig functions. We first see this operator applied to obtain the exact sum of two types of familiar infinite series from calculus. With very little work, the operator gives the Maclaurin series for the tangent function. After showing in a few lines a result due to Laplace, we find the Maclaurin series for x cot x and x csc x. A couple of extra tricks are required to produce the secant's Maclaurin series. We even derive a formula for computing Bernoulli numbers. Throughout the article, we touch on the rich history of work on these problems going back to Euler.

Symmetric Polynomials in the Work of Newton and Lagrange
Greg St. George
This article is intended to introduce the power of symmetric polynomials to undergraduates using a familiar setting. Paralleling an investigation by Newton into cubic equations, we work out the resultant of two quadratic equations. This leads to an appearance of the Fundamental Theorem of Symmetric Polynomials. To further illustrate these ideas, we present an application due to Lagrange, who used ideas of symmetry to elucidate the connection between the solutions of the cubic and quartic equations.


Deconstructing Bases: Fair, Fitting, and Fast Bases
Thomas Q. Sibley
A reinterpretation of the idea of a base using factorials instead of powers leads to finite decimal representations for all fractions. Further generalizations provide other curious results, such as a base for which π equals 3.1111...

Another Look at the Euler-Gergonne-Soddy Triangle
Raymond A. Beauregard and E. R. Suryanarayan
Every nonisosceles triangle has an associated triangle (the EGS) triangle determined as the intersection of the Euler line, Oldknow’s Gergonne line, and the Soddy Line. Using trilinear coordinates, Oldknow (Amer. Math. Monthly, 103 (1996), 319-329) showed that the EGS triangle is always a right triangle. We prove this result using Cartesian coordinates; this facilitates numerical considerations. Thus we show that there are an infinite number of nonsimilar Pythagorean triangles having EGS triangle with integer-length sides. We describe how this problem reduces to a classic question in number theory that was solved by Aubry in 1910.

Historical Mathematical Blunders: The Case of Barbaro’s Cannonballs
George M. Hollenback
A fifteenth-century diarist records the circumferences and weights of two large granite cannonballs, and modern historians don’t realize that his data yield wildly divergent densities for the cannonballs. A little metrological research and mathematical analysis resolve the problem.

An LDU Factorization in Elementary Number Theory
Warren P. Johnson
One of the changes in the introductory linear algebra course in the last 25 years is that LU and LDU factorization are now often included. It is interesting to try this with the symmetric matrix whose {ij}th entry is the greatest common divisor of i and j: L is very simple, U is evidently the transpose of L, and D contains the first several values of Euler's phi function. A determinant evaluation of the outstanding 19th century mathematician H.J.S. Smith is an immediate corollary.

A Laplace Transform Technique for Evaluating Infinite Series
James P. Lesko and Wendy D. Smith
In this article we demonstrate how the Laplace transform can be used as a tool to find closed form expressions for certain infinite series. Sometimes the summand of a series can be realized as a Laplace transform integral. If this is the case, and if the order of summation and integration can be interchanged (with justification of course!), then the series can be written as an integral; if this integral can be evaluated, then we have a closed form expression for the series! The purpose of this article is to explain this technique, and to illustrate with several examples.