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American Mathematical Monthly - May 1997

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Was Newton's Calculus a Dead End? The Continental Influence of Maclaurin's 'Treatise of Fluxions' Judith V. Grabiner


Here's the standard picture of the history of the calculus. Newton and Leibniz independently invented it. But then the action was all on the Continent. The Leibnizian tradition flourished there, with the Bernoullis, Euler, d'Alembert, Lagrange, Laplace, Cauchy, and Weierstrass--and then on to modern analysis. Meanwhile, the Newtonian tradition in Britain came to a halt. The place of Colin Maclaurin (1698-1746), if he has one at all, is as a reactionary advocate of Newtonian fluxions and ancient geometry--or as the man who set x = 0 in the Taylor series.

Yet Maclaurin, and through him the Newtonian calculus, turns out to have played an important role in Continental analysis. Maclaurin was, after all, a key figure in the vigorous eighteenth-century Scottish intellectual tradition--think of David Hume and Adam Smith--with its influential links to Continental thought.

This paper explains some of Maclaurin's contributions to calculus and applied mathematics, and demonstrates their Continental influence. The reader will see that Maclaurin's geometric way of thinking was a help, not a hindrance. Finally, we respond to the question, "Why haven't we heard about all of this before?"

Yet Another Definition of Chaos
Pat Touhey

This paper introduces a new, simple definition of chaos, equivalent to the definition given by Devaney. Namely, a self map on a metric space X is chaotic on X if every pair of non-empty open subsets of X shares a periodic orbit. This definition is then used to give an intuitive characterization of chaos: A self map on a metric space X is chaotic on X if and only if it mixes together, via periodic cycles, any finite number of non-empty open subsets in infinitely many ways.

Hereditary Classes of Operators and Matrices
Leiba Rodman and Scott A. McCullough


A two-by-two Jordan cell with real eigenvalue is the usual first example given of a matrix that is not normal and so is not unitarily similar (or even similar) to a diagonal matrix. But matrices that are unitarily similar to an orthogonal sum of such Jordan cells--such matrices are called 3-selfadjoint--are characterized by a simple identity in the matrix and its adjoint. On separable Hilbert space there is a remarkable connection between the 3-selfadjoint operators and Sturm-Liouville problems. This is just one example in the theory of hereditary classes of operators developed by Jim Agler. The paper is devoted to an accessible exposition of the general framework of hereditary classes, including the main ideas, results, several particular cases, examples, open problems, and applications. In particular, some hereditary classes in spaces with an indefinite scalar product are introduced; since this is essentially uncharted territory, there are many open problems here, and a few of them are explicitly stated.

Major Centers of Triangles
Clark Kimberling


Recently, it was said about triangle geometry that "this subject has more miracles per square inch than any other area of mathematics." We present problems that make sense only if triangle centers are functions, not mere points. Among the unexpectedly simple solutions to these problems are major centers, which are defined algebraically in terms of the vertices of the reference triangle ABC. Among the well known major centers are the four ancient Greek centers (centoid, incenter, circumcenter, orthocenter). For numerous examples of triangle centers, visit

On Lambert's Proof of the Irrationality of Pi
Miklós Laczkovich


The irrationality of pi was proved by J. H. Lambert in 1761 as follows. Lambert first gives the continued fraction expansion of tan x and then shows, by an argument of infinite descent, that if x is rational and nonzero then tan x is irrational. Since tan (pi/4) = 1, this implies that pi is irrational.

Since a rigorous version of Lambert's proof is rather tedious, the irrationality of pi is usually proved by (versions) of a different argument due to Hermite.

In this paper we combine an idea of Popken with Gauss' functional equation in order to obtain a very simple proof of the irrationality of tan x (and also of f(x) for a wide class of other functions) whenever x is rational and nonzero. The irrationality of pi follows. Then, we give a simple and self-contained proof of Lambert's continued fraction expansion for tan x.


A Simple Congruence modulo p
by Winfried Kohnen


A Geometrical Method for Finding an Explicit Formula for the Greatest Common Divisor
by Marcelo Polezzi

The Evolution of...
On the Historical Development of Infinitesimal Mathematics
by Detlef Laugwitz

Problems and Solutions


Mathematical Circles (Russian Experience). By Dmitri Fomin, Sergey Genkin, and Ilia Itenberg
by Andre Toom

Vita Mathematica: Historical Research and Integration with Teaching. Edited by Ronald Calinger
by Hardy Grant

Telegraphic Reviews

The Authors

Editor's Endnotes