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June/July 2001

**The Lost Squares of Dr. Franklin**

by Paul C. Pasles

paul.pasles@villanova.edu

While President Garfield found a novel proof of the Pythagorean theorem, and Napoleon had his own impressive result on triangles, we expect little from today's political figures in the way of mathematical achievements. Ben Franklin created impressive magic squares (http://www.pasles.org/Franklin.html) of orders 8 and 16. Since Franklin had little formal schooling, it might be assumed that he stumbled upon his discovery. However, it turns out that a few more examples survive, and these show a more varied set of tricks. We unearth heretofore undiscovered work of Franklin, and the construction of the famed "magic circle" ( http://www.pasles.org/circle.html) is finally brought to light.

**Numerical Differentiation as an Example for Inverse Problems**

by Martin Hanke-Bourgeois and Otmar Scherzer

hanke@math.uni-mainz.de and otmar.scherzer@uni-bayreuth.de

We present an error analysis for the numerical differentiation of noisy data via smoothing cubic splines. Our treatment is elementary enough to be included in a course on numerical problems that arise in the solution of much more complex ill-posed inverse problems.

**Differential Forms, the Early Days; or the Story of Deahna's Theorem and of Volterra's Theorem**

by Hans Samelson

samelson@math.stanford.edu

This is a short informal history of differential forms from the time when they barely existed to the time of de Rham's paper (1931) when the "modern" phase began. Many well known mathematicians and some unknown ones make their appearance. We begin with a brief description of what differential forms are and do. The story then starts with the question "which one-forms are differentials of functions?", touches on Pfaff's problem, goes through the development of the theory of "complete integrability" of systems of one-forms (Theorem of Frobenius), and moves on to the Poincaré Lemma, the basis for the modern uses of differential forms. On the way there are some surprises, the biggest one being that the two main results just mentioned are in fact due to two other people.

**How Many Subspaces Force Linearity?**

by J.H. Meyer and C.J. Maxson

MeyerJH@wis.nw.uovs.ac.za and cjmaxson@math.tamu.edu

The concept of a linear transformation on vector space and, more generally, the concept of a module homomorphism, are basic in mathematics. If *V* is a vector space over a field *K*, a function *f* on *V* that commutes with scalar multiplication for all and is a *homogeneous* fuction. How much local linearity must be imposed on the vector space* V* to force a homogeneous function to be a linear transformation? A measure of this local linearity is the *forcing linearity number* of *V,* which belongs to the set . These numbers have been obtained for vector spaces and several classes of ring modules. At present, the only known forcing linearity numbers are and positive integers of the form *p*^{n} + 2, where *p *is prime. Is there a module with forcing linearity number 3?

**Some Properties Related to Mercator Projection**

by Wim Pijls

pijls@few.eur.nl

Mercator projection is the most widely used mapping method in cartography. We first give a short introduction to the Mercator projection along with an introduction to the stereograhical projection, another well known mapping method.

We then discuss two mathematical issues that are closely related to Mercator projection. The first issue deals with complex numbers. A geographical map can be viewed as (a part of) the plane of complex numbers. Both Mercator and stereographical projection are conformal mappings. Consequently, the relation between a Mercator and a stereographical map can be expressed by a complex conformal function. It turns out that this function is the exponential function exp (*z*).

The second issue concerns the tractrix, the curve described by an object dragged by a string held by a person who moves along a straight line. We establish a close link between Mercator projection and the tractrix

**Necessary and Sufficient Conditions for Differentiating Under the Integral Sign**

by Erik Talvila

etalvila@math.ualberta.ca

If we have an integral that depends on a parameter, when it is valid to differentiate under the integral sign? For the Henstock integral we can easily formulate necessary and sufficient conditions for differentiating under an integral. The Henstock integral is defined in terms of Riemann sums in a manner very similar to the Riemann integral. However, it includes the Riemann, improper Riemann, and Lebesgue integrals as special cases. A property it has that these other integrals do not share is that every derivative is Henstock integrable. This leads to an improved version of the Fundamental Theorem of Calculus that is instrumental in proving our result about differentiating under the integral sign.

**NOTES**

**Pythagoras' Theorem for Areas**

by Jean-P Quadrat, Jean B. Lasserre, and Jean-B Hiriart-Urruty

jbhu@cict.fr

**An Alternative to the Shooting Method for a Certain Class of Boundary Problems**

by David Sanchez

dsanchez@math.tamu.edu

**Probability and Combinatorics**

by Wen-Jin Woan

wwoan@fac.howard.edu

**From Ford to Faá**

by Harley Flanders

harley@umich.edu

**A Vector Proof of a Theorem of Bang**

by Mowaffaq Hajja

mhajja@sharjah.ac.ae

**PROBLEMS AND SOLUTIONS**

**REVIEWS**

**The Heart of Mathematics: An Invitation to Effective Thinking.**

By Edward B. Burger and Michael Starbird

Reviewed by Marion Cohen

mcohen@mcs.drexel.edu

**The Integral: An Easy Approach after Kurzweil and Henstock**

By Lee Peng and Rudolf Vyborny

Reviewed by J. Alan Alewine and Eric Schechter

aalewine@math.vanderbilt.edu and schectex@math.vanderbilt.edu