Mathematics

Friday, October 3

12–12:30 p.m.

Coffee and refreshments in the Reinhart Center Lobby

Susan Kelly (University of Wisconsin-La Crosse) Winifred Edgerton Merrill—She Opened Doors

Winifred Edgerton Merrill was the first American woman to receive her Ph.D. in mathematics. She earned her degree from Columbia University in 1886, a time when women were not allowed to be officially admitted to this male only university. Her thesis was on multiple integrals, and this talk will present some of her results. Her work uses a nice application of Green’s Theorem, which could be presented in a Multidimensional Calculus course. Highlights of her life, obtained from published works, archives, and a personal family journal, will detail some of her contributions to the advancement of women.

Dan Curtin (Northern Kentucky University) Fermat vs Descartes: The Calculus Wars

Fermat and Descartes both came independently to some of the fundamental ideas that lead to the Calculus. Their difference in style, desire to publish, and personality led to an acerbic priority dispute. This talk focuses on some of the mathematical misunderstanding that fed the dispute. Oddly enough Descartes falls into errors that are reminiscent of those of Calculus students, and Fermat's less than clear presentations are part of the problem.

Lawrence D'Antonio (Ramapo College) The Sums of Squares Problem: a Study in Contrasts

In this talk we give an overview of the history of the sums of squares problem, namely, the problem of counting the number of representations of a positive integer as a sum of squares. The history of this problem reveals interesting contrasts between algebraic and analytic methods. Especially important is to compare the methods of Gauss, using quadratic forms, and Jacobi, using elliptic functions. Girard and Fermat considered the problem of determining which integers are the sums of two squares. Euler provided the first proof of Girard’s Theorem that primes of the form 4n+1 are the sum of two squares. Euler also worked for many years on the problem of showing that all positive integers are the sum of four squares, the proof being first supplied by Lagrange. This work can be seen as a jumping off point for Jacobi’s calculation of the number of representations of a positive integer as a sum of two or four squares. Gauss, using the method of quadratic forms, had previously computed the number of sums of three squares. We sketch the methods of Jacobi and Gauss and their influence on future work on the problem.

Charlotte Simmons (University of Central Oklahoma) Augustus De Morgan: The Man Behind the Scenes

Augustus De Morgan was a nineteenth century mathematician whose contributions to mathematics are not fully appreciated by many historians. “[C]ertainly he made no lasting original contribution to mathematics,” says Smith. This talk will explore the significant contributions made by De Morgan from “behind the scenes.” De Morgan was a close friend and correspondent of Sir William Rowan Hamilton and George Boole, two of the greatest algebraists of the nineteenth century. He significantly impacted both of their careers, and it is doubtful that they would have attained the level of success that they ultimately achieved without his help. We also explore his contributions to the field of actuarial science, both direct (via his own papers) and indirect (via his friendship and support of actuary pioneer Benjamin Gompertz). If he had done nothing else noteworthy in mathematics besides supporting the efforts of these three men, we would still owe him a great debt.

Rich Maresh (Viterbo University) Pre-Calculator Era Computation Techniques

Prior to the introduction of the electronic calculator into the school setting, numerous algorithmic techniques were employed to do calculations that are today done at the push of a button. Perhaps modern calculator techniques do not allow students the experiential understanding of what is happening in some calculations. We will explore methods used for extracting square roots and why those methods work. We will also consider some of the tricks employed in using logarithm tables to do arithmetical computations.

Wally Sizer (Minnesota State University Moorhead) Numerals in History

There have been probably close to one hundred different numeral systems over the years in different parts of the world. More are currently in use than most people imagine. There seem to be four different aproaches to constructing all of these numeral systems. I classify the approaches as repetitive, alphabetic, multiplicative, and positional. Some systems use a combination of these principles. I will indicate how these principles work and illustrate them with some less well known examples, like Aramaean and Egyptian demotic numerals.

6–6:30 p.m.

Banquet in the Fine Arts Center Hospitality Suite

Plenary Talk:  Chris Christensen (Northern Kentucky University) “The Theorem that Won the War” and other work of the World War II Polish mathematician-codebreakers

World War II marked the first recruitment of mathematicians as codebreakers. In the 1930s, the Polish Cipher Bureau recruited a team of mathematicians to attack the German Enigma cipher machine. In a remarkable application of mathematics and mathematical thought, the Poles solved the wiring of the Enigma rotors and developed methods for determining Enigma settings. One of the theorems from permutation theory that they used was later referred to as “The Theorem that Won the War.” Of course World War II was not won by a theorem; it was won by the men and women in the Allied armed forces, but the work of Allied mathematician-codebreakers saved lives and brought the war to an end much sooner than would have otherwise been the case. We will consider the work of the Polish mathematician-codebreakers.

Saturday, October 4

8–8:40 a.m.

Coffee and refreshments in the Reinhart Center Lobby

Thomas Drucker (University of Wisconsin-Whitewater) Origins and Roots: What Makes Somewhere Home

Reference works frequently provide brief geographical identification for mathematicians. In some cases these emerge easily from the stories of their lives, but in other cases they are hard to determine. One can ask about the usefulness of such identifications, but they will continue to be provided even when the fit is not ideal. This talk will look at some mathematicians of central European origin with an eye toward determining what constitutes their "home."

Steve Hinthorne (Principia College)

Numeration methods led computers (those who compute) to devise clever strategies for logistics (aka arithmetic). Egyptian multiplication ("duplacion") and division ("dimydician") used binary representations. The Greek Ptolemy preferred Sumerian numeration to Greek or Egyptian c. 150. The Lilavati of Bhaskara (1150) gave five plans for multiplication using the Hindu system. Early texts such as The Crafte of Nombrynge (c. 1300) influenced the transition from Roman to Hindu/Arabic numerals; attribution of invention of this "crafte" is mistakenly given to "a king of Inde the quych heyth Algor." Seven "spices or partes of this crafte" were presented with "algoryms" for computers – leading to four methods of subtraction, seven methods of multiplication, and at least nine methods of division, which, as described by Hylles (1600), “is esteemed one of the busiest operations of Arithmetick, and such as requireth a mynde not wandering, or settled vppon other matters.” [Smith, Vol II, p. 132]. This session takes a quick look at some interesting snapshots from the fascinating history of how we came to compute with decimal digits.

Colin McKinney (University of Iowa) Eutocius' Commentary on the Conics of Apollonius

Eutocius of Ascalon, a mathematician in the late 5th and early 6th centuries CE, wrote several commentaries on earlier Greek mathematical works. Of these, his commentary on Archimedes' On the Sphere and the Cylinder has only recently been published, by Reviel Netz in 2004. However, to date, only small parts of Eutocius' commentary on Apollonius' Conics have been translated. And as Heath says, "...Eutocius' commentary on Apollonius' Conics is extant for the first four Books, and it is probably owing to their having been commented on by Eutocius, as well as to their being more elementary than the rest, that these four Books alone survive in Greek." In this talk, I shall present my preliminary translation work from this commentary, specifically the "introduction" to Book I, and examine it from both mathematical and historical points of view.

Daniel Otero The History of the Determinant

The determinant, a central concept from linear algebra, coalesced out of the work of numerous mathematicians across the span of the 18th century: Leibniz, Maclaurin, Cramer, Bézout, Vandermonde, Laplace, Gauss, Binet, Cauchy, Jacobi, and others.  While the account of this history is not new, it is not well known, as it is not often told.  The members of the ORESME (Ohio River Early Sources in Mathematical Exposition) Reading Group devoted their September 2008 meeting to a study of this history; this paper is a record of their reexamination of the early history of the determinant.