


History of Mathematics / Histoire des mathématiques Org: Len Berggren (Simon Fraser)
 TOM ARCHIBALD, Acadia University, Wolfville, Nova Scotia B4P 2R6
Jules Tannery, Paul Appell, and the mathematical research
community in France, 18701914

Jules Tannery (18481910) and Paul Appell (18551930) occupied key
positions in the development of mathematics in late nineteenth and
early twentieth century France. Tannery's early research on linear
differential equations stands at the origin of much research in this
area in France, undertaken by such authors as G. Floquet, E. Picard,
and E. Goursat. His textbook writings, particularly the "Eléments
de la théorie des fonctions elliptiques", jointly written with
J. Molk, were fundamental to the French university curriculum for many
years. Most important, though, was Tannery's work as the assistant
director in charge of science at the Ecole normale supérieure, a
post he took up in 1884. In this capacity, he exerted a strong
influence on the careers of many normaliens. Appell, a student
at the ENS shortly after Tannery, became a professor at the Faculty of
Sciences in Paris and subsequently its Dean, from 1903 to 1920. Like
Tannery, he was a highly influential thesis director and textbook
author. As an example we may mention his "Thëorie des fonction
algébriques et de leurs intégrales", written jointly with Goursat
and later revised by Fatou.
In this talk, I shall survey aspects of the careers of these two
important figures and discuss their influence.
 LAWRENCE D'ANTONIO, Ramapo College, Mahwah, New Jersey 07430, USA
The solution of the BehaEddin problem: elliptic curves to the
rescue!

Fermat's Last Theorem was not the only Diophantine problem whose
solution required the use of elliptic curves. The Islamic mathematician
Beh¯a Edd¯in al'Amuli, in his early 17th century work, the
Khul¯as¯at alHis¯ab, (Essence of Reckoning),
discussed seven unsolved problems in algebra. One of the problems was,
in fact, Fermat's Last Theorem in the cubic case. Another problem, the
seventh, was to find rational solutions for the pair of equations:
x^{2}+x+2=y^{2}, x^{2}x2=z^{2}. 

In this paper we trace the history of the problem, first considering the
origins of this problem and its relationship to the questions about congruent
numbers. Next, the first systematic attempt at a solution, using the chord and
tangent method, was done by Lucas in 1877. Finally, Horst Zimmer found the
complete solution to the problem in 1983. Zimmer transforms rational solutions
of the Beh¯a Edd¯in problem into rational points on a certain elliptic
curve, and then gives a complete description of the group of rational points
on that curve.
 HARDY GRANT, York University, Toronto, Ontario M3J 1P3
Greek mathematics in cultural context

I shall attempt an overview (necessarily sketchy!) of the "career" of
Greek mathematics in its cultural setting. Features of the background
include Eleatic philosophy, the development of the "liberal arts"
tradition, the rivalry of professions, the rise of rhetoric as a
technique of demonstration, and Aristotle's formalization of logic. A
unifying thread is the emergence of the familiar viewnot, of course,
uncontestedof mathematics as attaining a unique exactness, certainty
and insight into the nature of things.
 ALEXANDER JONES, Classics, University of Toronto, 97 St. George Street, Toronto,
Ontario
Ptolemy's mathematical tones

The Alexandrian scientist Claudius Ptolemy is best known for his
celestial mechanics and his mapping of the known world. Among his less
familiar books is the "Harmonics", an attempt to apply mathematical
models to the esthetics of musical tones and tunings. We will describe
how Ptolemy reasoned that certain kinds of ratios between whole numbers
could account for the practices of the musicians of his time.
 ISRAEL KLEINER, York Univeristy, Department of Mathematics and Statistics, Toronto,
Ontario M3J 1P3
Fermat: The founder of modern number theory

I will discuss some of Fermat's major contributions to number theory,
noting his intellectual debts and his legacy.
 GREGORY MOORE, McMaster University, Hamilton, Ontario
Attacking the infinite, defending the infinite: Cantor's
strategy and tactics

Georg Cantor, the creator/discoverer of set theory, was well aware of
his predecessors' attitudes toward the actual infinite. Beginning in
1883, he undertook a spirited defense of the actual infinite against
(as Zermelo later put it when editing Cantor's collected works)
"philosophical and theological objections". We examine several of
the alleged "proofs" that there is no actual infinite (including a
"proof" by Cauchy), and Cantor's replies to them.
 RAM MURTY, Queen's University, Kingston, Ontario K7L 3N6
History of the Euclidean algorithm

We will begin with Euclid and then to Aryabhata and finally to Gauss to
see how the modern formulation of the Euclidean algorithm evolved over
the centuries. We will also study the abstraction of the notion and
address Samuel's question of the classification of rings with Euclidean
algorithm.
 NATHAN SIDOLI, University of Toronto
Menelaus' Theorem in Ptolemy & Theon

It is generally assumed that Ptolemy took the theorem upon which he
built his spherical trigonometry from Menelaus' Spherics. Neither
Ptolemy, nor his commentator Theon, however, makes any mention of
Menelaus in this regard. We have lost most of the Greek original of
Menelaus' text, in particular we have lost the section relevant to
Ptolemy's spherical astronomy. Our text of Menelaus' Spherics is
preserved in the Arabic tradition in a number of different translations
and editions. In this paper, I examine all of the relevant versions of
the Menelaus' theorem and show that the line of transmission cannot
have been as straightforward as has previously been thought. In
particular, I show that the Arabic tradition contains material that was
not available to Ptolemy and Theon as well as material that is most
likely based on their work. This technical discussion gives rise to a
more speculative inquiry into the source of Ptolemy's spherical
trigonometry based on the role of the Menelaus Theorem in the Spherics
and the Almagest. Finally, I reconsider the possibility that Hipparchus
treatise on simultaneous risings may have used metrical methods based
on the socalled Menelaus Theorem.
 GLEN VAN BRUMMELEN, Bennington College
Taking latitude with Ptolemy: AlKashi's final solution to the
determination of the positions of the planets

Although the model to determine planetary longitudes in Ptolemy's
Almagest produced elegant and satisfactory longitude computations, his
model for latitudes was, seemingly, too complicated to allow for easy
handling mathematically. As a result Ptolemy was forced into making
several approximations, leading to an unsatisfactory mathematical
theory of latitudes. While several innovations were proposed to deal
with the computation of latitudes in medieval Islam, hardly any of them
dealt with the core mathematical issues. Jamshid alKashi, perhaps the
greatest computational astronomer in the Ptolemaic tradition, achieved
a complete solution to the problem in his Khaqani Zij in the early 15th
century. We shall survey various Muslim contributions and describe
alKashi's solution in detail.

