# The College Mathematics Journal

### September 2001 Contents

An Interview with Dame Mary Cartwright, D. B. E., F. R. S.
James Tattersall and Shawnee McMurran

Mary Cartwright (1900-1998) was a British mathematician who made important contributions to the theory of functions and differential equations. G. H. Hardy was her thesis adviser, she collaborated with J. E. Littlewood, and was Mistress of Girton College, Cambridge. Here is an interview, giving some details of her life and times, and some information that does not ordinarily get into print, e.g., "I learned that if Lefschetz stopped asking questions for five minutes he was asleep."

Arctangent Sums
Louis Bragg

You probably didnŐt know that the sum of the arctangents of (sinh 1)/(cosh n) as n goes from 1 to infinity is 3p/4, take away the arctangent of e, and you are excused if you don't care, but here is a clever method for generating such identities.

Models for Growth
Elizabeth Appelbaum

Exponential and logistic growth appear in almost all textbooks, but the growth of some tumors follows the Gompertz model, of which hardly anyone is aware. Here is what it is. Infinitely many other models can be constructed.

Magic Squares, Finite Planes, and Points of Inflection on Elliptic Curves
Ezra Brown

The author, a two-time winner of the Pólya Prize for mathematical exposition, here shows that every elliptic curve has nine points of inflection that can be arranged in a natural way to form a three-by-three magic square.

The Sun, the Moon, and Convexity
Noah Samuel Brannen

We go around the sun and the moon goes around us. Look down from above at the path of the moon about the sun. It has little loops when the moon is going backwards relative to our path around the sun, right? Wrong! The moonŐs path is everywhere convex. On the other hand, the orbit of Io, a moon of Jupiter, does have loops. The author shows how to tell one sort of moon from the other.

Another Look at Factoring Polynomials
Scott J. Beslin and Douglas J. Baney

We are used to factoring polynomials into products of other polynomials, but not so used to factoring them into compositions of other polynomials. Cubics cannot be so factored (in a non-trivial way) but quartics sometimes can. The authors tell us when.

Rational Approximations to Power Expansions
Maria Cecilia K. Aguilera-Navarro, Valdir C. Aguilera-Navarro, Ricardo C. Ferreira, and Neuza Teramon

An introduction to Padé approximations, which are often better than other kinds. The authors give an example of one with four terms, all of degree three or less, whose error is 1/500,000th that of a Taylor expansion with fifty terms.

Fallacies, Flaws, and Flimflam
Ed Barbeau, editor

A system of differential equations that is both stable and unstable, l'Hôpital's Rule used to show that 1 = -1, and other items of interest.

Classroom Capsules
Tom Farmer, editor

Interactive Teaching Aids for Multivariable Calculus
David E. Bailey and Gerald Kobylski

Directional derivatives you can feel and other physical aids for teaching calculus in more than two dimensions. And, unfortunately, in fewer than four.

Integration from First Principles

Here is how to integrate powers of x, including the power one-half, directly from the definition of the integral.

Heron's Formula via Proofs Without Words
Roger Nelsen

A derivation of Heron's formula for the area of a triangle using only elementary algebra, and not very many words.

Joseph B. Dence and Thomas P. Dence

Sum the squares of the sides of a quadrilateral and subtract the squares of its diagonals. The difference is zero if the quadrilateral is a parallelogram and positive otherwise, the more positive the further the quadrilateral departs from parallelogramhood. The authors show what the difference is.

The Volume of a Tetrahedron
Cho Jinsok

The author shows how to get the volume of a tetrahedron given the lengths of three coincident edges and the angles between them.

Problems and Solutions
Irl Bivens and Ben Klein, editors

### Media Highlights

Warren Page, editor

### Book Review

Review by Patricia Clark Kenschaft of The Education of a Mathematician by Philip J. Davis.