


History of Mathematics
Org: Tom Archibald (SFU) [PDF]
 TOM ARCHIBALD, Simon Fraser University
MittagLeffler's Theorem: Genesis and Development of a
Mathematical Fact
[PDF] 
Gösta MittagLeffler (18461927) remains wellknown, in part for
the theorem that bears his name. Roughly this states that one can
define a function, meromorphic on a subset of the complex plane, such
that the principal parts of the function's Laurent series at a given
collection of poles are specified in advance. MittagLeffler's
original work on the theorem was undertaken exactly in the pattern of
his mentor Weierstrass, whose famous 1876 paper on singlevalued
analytic functions constitutes a kind of template for MittagLeffler's
research and presentation. Between 1876, when the first version of
the theorem was published in Swedish, and 1884, when a long paper on
the subject appeared in MittagLeffler's new journal Acta
Mathematica, MittagLeffler progressively worked on generalizing
the theorem to apply to larger collections of poles and essential
singularities. It was in this context that he came to view the work
of Cantor on infinite point sets as important, a position which was
shared by very few researchers of the time. Indeed, the development
of MittagLeffler's research allows us to see how the currents of
opinion and mathematical fashion shaped the focus of MittagLeffler's
labours, and to some degree the nature of his results. Based on
published and unpublished correspondence with Hermite, Weierstrass,
Cantor, Poincaré and others, this paper will look at some
highlights of the development and reception of MittagLeffler's
theorem.
This paper describes joint work with Laura E. Turner
(laurat@sfu.ca, Mathematics, Simon Fraser University).
 MARCUS BARNES, Simon Fraser University, Department of Mathematics,
8888 University Drive, Burnaby, BC, V5A 1S6
Some Aspects of the Mathematical Work of J. C. Fields
[PDF] 
Most mathematicians know of J. C. Fields (18631932) as the person
for whom the Fields medal is named. Few mathematicians, however, know
anything about Fields' actual mathematical work. After a biographical
sketch and an outline of Fields' educational background, we will give
a synopsis of Fields' mathematical work. If time allows, we will
compare Fields' approach to algebraic function theory to that of other
approaches that were followed during his lifetime.
 CRAIG FRASER, University of Toronto
Analysis and the Emergence of Analytic Mechanics in the
Eighteenth Century
[PDF] 
By the middle of the eighteenth century the term analysis had largely
lost its original meaning of "solution backwards". As is well
known, during the early modern period analysis came to denote algebra
and the use more generally of symbolic methods in the solution of
problems. When one introduces a variable and derives an equation, one
is assuming logically at the outset that the thing that is sought is
at hand, even if one does know its value. Hence all methods in which
the existence of the thing sought is first assumed as an unknown
variable and its value is derived by means of some mathematical
process is analytic. Analysis came to encompass algebraic symbolic
methods that yielded equations and was contrasted with geometric modes
of solution. It was seen not simply as a method but as a subject area
in its own right employing processes that were linguistic or symbolic
in character. Above all, analysis avoided geometric modes of
representation. Synthesis denoted a geometrical conception of the
mathematical object in which this object has a whole was taken as
given and in which its known properties were used in the course of the
investigation.
The elevation of analysis within mathematics was paralleled by the
promotion of what was called the method of analysis in other areas of
inquiry. In the writings of Isaac Barrow (16301677), Etienne
Condillac (17141780) and Thomas Reid (17101796) analysis,
understood as a very general process of investigation, was put forward
as the way to truth in all branches of inquiry. The paper explores
the meaning of analysis in eighteenthcentury exact science for the
case of analytical mechanics and considers mathematical analysis in
reference to the larger intellectual context of Enlightenment
thought.
 ALEXANDER JONES, Classics, University of Toronto, 97 St. George Street,
Toronto, ON, M5S 2E8
Some Properties of Arithmetical Functions in Ancient
Astronomy
[PDF] 
Babylonian mathematical astronomy and its GrecoRoman continuation
employed arithmetical functions to model aspects of phenomena. The
most characteristic type was the linear zigzag function, according to
which a quantity alternately increases and decreases by constant
differences between a fixed minimum and maximum value. In modern
discussions, zigzag functions are typically described as a sequence of
equallyspaced discrete values of a continuous "ideal" function, in
which the independent variable is time (not necessarily measured in
units of constant duration). For the ancient astronomers, however,
the tabulated values of a zigzag function were generated
algorithmically, each value from its immediate predecessor.
All zigzag functions used in ancient astronomy had parameters that can
be expressed by terminating sexagesimal fractions, and the sequence of
generated values repeats exactly after an integral number of steps.
Neugebauer demonstrated in the 1940s that such sequences have certain
properties that are not obvious from consideration of the ideal
function; for example, the mean rate of increase of the running totals
of a zigzag function may not be equal to the mean of the ideal maximum
and minimum. The present paper will consider whether evidence exists
that ancient astronomers were aware of these properties and developed
mathematical methods for handling them.
 DEBORAH KENT, Simon Fraser University, Department of Mathematics,
8888 University Drive, Burnaby, BC, V5A 1S6
Analytic Mechanics, Astronomy, and Linear
Associative Algebra?: Context and Motivation for B. Peirce's
Anomalous Paper
[PDF] 
Throughout the nineteenth century, the mathematical work of Harvard
professor Benjamin Peirce primarily involved analysis and astronomy.
Nonetheless, Peirce's most wellknown work today is Linear
Associative Algebra, which appeared in the American Journal
of Mathematics in 1881. This paper contains results foundational
to the structure theory of algebras. Considered in context, it comes
into focus as the culmination of a lifetime of thought and
mathematical work, rather than a departure from earlier research
interests.
 TOKE LINDEGAARD KNUDSEN, Brown University
Jñ¯anar¯aja and mathematical astronomy in early
16th century India
[PDF] 
One of the important astronomers and mathematicians in Indian history
was Jñ¯anar¯aja (fl. 1503) from what is now the state of
Maharashtra. His two surviving works, often cited by later Indian
writers, are a treatise on astronomy and a treatise on mathematics,
none of which has been published.
Based on my research on Jñ¯anar¯aja's astronomical work, the
talk will focus on the mathematical astronomy of Jñ¯anar¯aja
and its place in the history of astronomy and mathematics in India.
 SHAWNEE MCMURRAN, US Military Academy, Department of Mathematical Sciences,
West Point, NY 10996
The Impact of Ballistics on Mathematics
[PDF] 
In 1742 Benjamin Robins published New Principles of Gunnery,
the first book to deal extensively with external ballistics.
Subsequently, Frederick the Great asked Euler for a translation of the
best manuscript on gunnery. Euler chose Robins' book and, being true
to form, tripled the length of the work with annotations. The
annotated text was translated back into English, which, two and a half
centuries later, brings us to the theme of this lecture.
This work was done in collaboration with V. Frederick Rickey and Maj. Patrick Sullivan (United States Military Academy).
 DUNCAN MELVILLE, St. Lawrence University
Reflections on Sargonic Arithmetic
[PDF] 
The mathematical corpus from the Sargonic period of Mesopotamia (ca. 23502200 BC) is modest and limited, comprising some dozen problem
texts involving sides and areas of rectangular fields. However, this
period is situated between the earlier phases of metrological
computation and the development of the abstract sexagesimal system in
the subsequent Ur III period, and is thus of great interest to
historians of mathematics attempting to trace the development of
abstraction in arithmetic. Recently, two contrasting theories of
Sargonic mathematics have been proposed. We review these theories and
analyze the remaining difficulties in interpreting these texts.
 V. FREDERICK RICKEY, United States Military Academy
Some History of the Calculus of the Trigonometric Functions
[PDF] 
Can you evaluate the integral of the sine using Riemann sums? Do you
think Archimedes could? Is it intuitively clear to you that the
derivative of the sine is the cosine? If not, why not? What did
Newton and Leibniz know about sines and cosines? When did sines
become the sine function? Who is the most important individual in the
history of trigonometry? Answers will be provided.
 DIRK SCHLIMM, McGill University, Dept. of Philosophy
On the importance of asking the right research questions:
Could Jordan have proved the JordanHölder Theorem?
[PDF] 
In 1870 Jordan proved that the composition factors of two composition
series of a group are the same. Almost 20 years later Hölder (1889)
was able to extend this result by showing that the factor groups,
which are quotient groups corresponding to the composition factors,
are isomorphic. This result, nowadays called the JordanHölder
Theorem, is one of the fundamental theorems in the theory of groups.
The fact that Jordan, who was working in the framework of substitution
groups, was able to prove only a part of the JordanHölder Theorem
is often used to emphasize the importance and even the necessity of
the abstract conception of groups, which was employed by Hölder.
However, as a littleknown paper from 1873 reveals, Jordan had all the
necessary ingredients to prove the JordanHölder Theorem at his
disposal (namely, composition series, quotient groups, and
isomorphisms), despite the fact that he was considering only
substitution groups and that he did not have an abstract conception of
groups. Thus, I argue that the answer to the question posed in the
title is "Yes".

