"Study the masters," wrote Niels Abel, "and not the pupils." With this goal in mind, a conference was organized in June of 2002 at Gimlekollen Mediacentre, Kristiansand, Norway both to commemorate the 200th anniversary of the birth of Abel and in memory of John Fauvel, whose work on the history and pedagogy of mathematics inspired so many. The volume under review, the proceedings of that conference, covers a wide range of topics related to the history of mathematics and its use in pedagogy, from biographical material on little known mathematicians to research results of studies in mathematics education. The papers vary tremendously in both their relationship to the goal of "study the masters" and in their overall quality.
Among the best articles are two by Harold Edwards. In the first, he discusses Abel's own interpretation of the great theorem of his Paris memoir of 1826, a theorem which today is usually stated in terms of Riemann surfaces (introduced, of course, many years after Abel's death) and which introduces, at least implicitly, the notion of the genus of an algebraic curve.
In the second, Edwards answers the question "what does it mean to solve a polynomial equation?", essentially asked and partially answered by both Abel and Galois. Edwards' answer is to give what he calls the "truly fundamental theorem of algebra", that given any polynomial F(X) over the rational numbers, for example, there is a monic, irreducible polynomial G(X) over the rational numbers with the property that adjoining one root of G to the rationals gives a field in which F splits completely as a product of linear factors. In other words, adjoining one root of G enables one to calculate with all the roots of F. Edwards then explains the relationship of his result to Galois's Lemma III in his famous memoir introducing Galois theory.
In another pair of articles, David Pengelley describes his use of original sources in teaching mathematics. First, he discusses in some detail his graduate seminar on the role of history in teaching mathematics in which the students produce modules designed to use history in teaching. Secondly, he gives us a brief summary of a chapter in his forthcoming book (with Reinhard Laubenbacher) on the relationship between the search for formulas for sums of integral powers in relation to integration and Euler's summation formula in relation to infinite series. In fact, he provides insight into Euler's use of divergent series in a meaningful way.
Michel Helfgott takes "study the masters" very seriously as well, as he describes his course on the history of mathematics given at SUNY Oswego. As a major component of the course, he has students read selections and work problems from some of the most significant mathematics books ever written, ranging from Euclid's Elements through Euler's Introduction to the Analysis of the Infinite to Polya's Induction and Analogy in Mathematics. In another article, Helfgott discusses Fermat's approach to proving Snell's law of refraction as well as new technological approaches to the same problem.
Among the other historical articles are several of interest. Nils Voje Johansen describes the origins of Caspar Wessel's 1797 treatise on the geometrical interpretation of complex numbers in his earlier work on surveying. In fact, it turns out that Wessel had used complex numbers as coordinates in the plane ten years earlier. He had, however, given little explanation of this, so Johansen speculates that he may well have explained it more thoroughly in a surveying report that is not extant.
Steinar Thorvaldsen shows that some of Kepler's computational work in solving what is now called Kepler's equation is essentially a modern iterative algorithm converging to a solution. Kepler in fact was among the first scientists to use logarithms to calculate tables. These tables, the Rudolphine tables, were far superior to earlier ones and enabled European astronomers to view the 1631 transit of Mercury. Thorvaldsen also discussed the correspondence between Kepler and Schickard about the latter's mechanical calculator, noting that the calculator's accuracy would not have been sufficient for Kepler's use.
Finally, there are several articles dealing directly with the use of history in the classroom. Wann-Sheng Horng discusses a teaching experiment using Euclid's theorem that "prime numbers are more than any assigned multitude of prime numbers." Noting that Euclid's statement is somewhat different from the modern theorem that there are infinitely many prime numbers and that his proof strategy is therefore also somewhat different, the author explains the difficulties his students had with understanding the respective proof methodologies.
Fulvia Furinghetti and Annamaria Somaglia show how they used historical material on tangents to help their students understand the notion of derivative. In particular, they used both Roberval's kinematic construction of tangents and Fermat's method of "adequating" to develop this notion and sketched the results of their research findings on the effect of this teaching method on their students' understanding.
Finally, Reinhard Siegmund-Schultze describes how he used Descartes's Geometry and his Rules for the Direction of the Mind in helping students understand the relationship of algebra to geometry. In particular, he asked his students to consider Descartes's proof of the linearity of the distance functions, as described in his solution of the Pappus problem of four lines, in terms of its generality and its methodology.
The community of mathematics teachers using history in the classroom owes a great deal of thanks to Otto Bekken, Sten Kaijser, and Bengt Johansson for organizing this conference and for publishing this proceedings. We hope that a way can be found to circulate the major articles of this volume to the wider audience they deserve.
Victor J. Katz is the author of A History of Mathematics
, one of the more popular textbooks in the subject. A brief version
of that book has appeared recently. Katz teaches at the University of the District of Columbia.