March 2007 Contents
A New Method of Trisection
David Alan Brooks
One of the most interesting episodes in the saga of angle trisection was Archimedes' discovery that he could trisect an angle if he could make two marks on his straightedge. This note gives a method of trisection without marks; of course, one does have to "cheat" a little.
An Iterative Angle Trisection
Donald L. Muench
The problem of angle trisection continues to fascinate people even though it has long been known that it can't be done with straightedge and compass alone. However, for practical purposes, a good iterative procedure can get you as close as you want. In this note, we present such a procedure. Using only straightedge and compass, our procedure produces a sequence of angles that converge rapidly to a trisector.
"Shutting up like a telescope": Lewis Carroll's "Curious Condensation Method for Evaluating Determinants"
Adrian Rice and Eve Torrence
Charles Dodgson (Lewis Carroll) discovered a "curious" method for computing determinants. It is an iterative process that uses determinants of 2 × 2 submatrices of a matrix to obtain a smaller matrix. When the process ends, the result is the determinant of the original matrix. This article discusses both the algorithm and what may have led Dodgson to it.
Which Way is Jerusalem? Navigating on a Spheroid
Given two points on a spheroidal planet, what is the direction from the first to the second? The answer depends, of course, on what path you take. This paper compares two paths which suggest themselves, namely, the loxodrome, which is the path in which the direction stays constant, and the geodesic, which is the shortest path. The geodesic does surprising things when the eccentricity of the rotated ellipse is large.
The Origins of Finite Mathematics: The Social Science Connection
Arguably the first significant innovation in the undergraduate mathematics curriculum of the second half of the twentieth century was the finite mathematics course. The origins of this course lie in the excitement that arose, in the period around World War II, about applying mathematics to the social sciences. In this article we tell some of that story, a tale that shifts back and forth between intellectual and organizational factors and that resulted in the appearance in 1957 of the book that created the finite mathematics course, Introduction to Finite Mathematics, by John Kemeny, Laurie Snell, and Gerald L. Thompson.
Sums of Consecutive Integers
Wai Yan Pong
We begin by answering the question, "Which natural numbers are sums of consecutive integers?" We then go on to explore the set of lengths (numbers of summands) in the decompositions of an integer as such sums.
Integrals of Fitted Polynomials and an Application to Simpson's Rule
Allen D. Rogers
This article explores phenomena related to fitting polynomials with data sets with equally spaced x-values.
Fallacies, Flaws, and Flimflam
Ed Barbeau, editor
Ricardo Alfaro and Steven Althoen, editors
Doublecakes: An Archimedean Ratio Extended
Vera L. X. Figueiredo, Margarida P. Mello, and Sandra A. Santos
Archimedes proved that both the volume and surface area of a sphere are two thirds of those of a cylinder circumscribing the sphere. We show that this two-thirds property extends to other objects and their circumscribing prisms.
Pythagorean Triples with Square and Triangular Sides
Fermat proved that there are no Pythagorean triples in which the "legs" are both squares, while Sierpinski show that there are infinitely many in which the legs are consecutive triangular numbers. This capsule explores Pythagorean triples in which one leg is a square and the other a triangular number.
Bernstein's Examples on Independent Events
In 1946, Bernstein gave two examples showing that pairwise independence of a set of events does not imply joint independence. In this note, we show that his examples are the smallest ones possible.
An Improper Application of Green's Theorem
Robert L. Robertson
In this capsule, Green's theorem is used to give a calculus-level proof of the integral result
Partial Fractions by Substitution
David A. Rose
In this note, the author gives a quick method for finding the partial fraction decomposition of a rational function when the denominator is a power of just one linear or irreducible quadratic polynomial.