




History of Mathematics / L'histoire des mathématiques (Len Berggren, Organizer)
 TOM ARCHIBALD, Acadia University, Wolfville, Nova Scotia B0P 1X0
Hermite and Jacobi

Charles Hermite (18221901) began his research career by successfully
extending Jacobi's theory of elliptic functions. Jacobi, by then
wellestablished, responded with encouragement and interest, setting
Hermite on a path which would make him one of France's most important
pure mathematicians as well as its leading promoter of interest in
German mathematical research. In this paper we discuss Hermite's early
work, emphasizing Jacobi's influence, and examining its relation to
some of his mature work. We also discuss the position of Hermite in the
French mathematical community of his day, and consider the events
leading to his appointment to the Académie des Sciences.
 CRAIG FRASER, University of Toronto, Institute for the History and Philosophy of
Science and Technology, Victoria College, Toronto, Ontario M5S 1K7
Adolph Mayer's proof of the multiplier rule in the calculus of
variations

The multiplier rule is involved in the formulation of the most general
problem of the calculus of variations. Almost ninety years elapsed
between the first formulation by Lagrange in 1797 of this rule and
Adolph Mayer's proof in 1886. Mayer's derivation may be characterized
as the first satisfactory proof of the rule. Since Mayer there have
been several other demonstrations, by Hilbert, Turksma, Bliss and
Radon. The proof adduced in several modern textbooks is due to Bliss.
The paper contends that Mayer's original proof possesses exceptional
explanatory power and merits renewed attention. The paper also examines
the historical cirumstances associated with the long delay in the
formulation and first satisfactory proof of the rule.
 JUDITH GRABINER, Pitzer College, Claremont, California, USA
MacLaurin and Newton

Colin Maclaurin (16981746) based his major work on Newton's for
instance, Maclaurin's foundations of the fluxional calculus,
infiniteseries treatments of extrema, the EulerMaclaurin summation
formula, work on those `inverse fluxions' now called elliiptic
integrals, and the gravitational theory of the shape of the earth. He
also owed some of his early success in publication and in jobhunting
to the patronage of Newton. But more importantly, Maclaurin pursued
what I. B. Cohen has called the Newtonian style. This approach,
which involves a particular relationship between sophisticated
mathematical modeling and empirical data, was responsible not only for
Maclaurin's scientific successes but for his ability to solve problems
on other subjects, ranging from navigation to taxation to insurance
(not to mention theology), for the benefit of Scotland. All these
successes strengthened Maclaurin's as a natural philosopher, the
authority of mathematics in Scottish thought, and the prestige of
Newtonianism throughout the Scottish Enlightenment.
 HARDY GRANT, York University, North York, Ontario M3J 1P3
The parallel postulate revisited

That Aristotle adumbrated noneuclidean geometry has been repeatedly
urged by Imre Toth, and just as persistently denied by G. E. R. Lloyd,
perhaps our leading historian of Greek science. I shall try to
describe the competing positions and to set the debate amid wider
issues of axiomatics and of mathematical necessity, in Aristotle and
beyond.
 ALEXANDER JONES, Classics, University of Toronto, Toronto, Ontario M5S 2E8
Ptolemy's numerical fudgings

The Almagest of Ptolemy (c. A.D. 150) is the unique example surviving
from Classical antiquity of an advanced treatise on mathematical
astronomy relating observational data to mathematical models. In modern
times, Ptolemy's bona fides as an observer and original theoretician
has been hotly debated. The Almagest contains a great deal of logically
interconnected quantitative data, and close study of the mathematical
details reveals over and over again that Ptolemy cannot have originally
arrived at his theoretical results by the deductive routes that he
presents. Not only do the quantified models of the Almagest turn out to
be prior to the particular deductions that ostensibly lead from the
observations to the models, but also at least part of the observational
evidence cited in the Almagest appears to have been tampered with, or
even fabricated, in order to make the deductions work. In the present
paper, I examine specific instances in both Ptolemy's Almagest
and his Geography of ``sharp practice'' in deriving ostensibly
definitive results from refractory data, and I attempt to explain them
in the light of Ptolemy's professed scientific principles and the
historical background of his work.
 GREG MOORE, McMaster University, Hamilton, Ontario L8S 4KA
The early development of Cantor's continuum problem:
Hausdorff and Sierpinski

This historical paper analyzes the contributions, so fundamental but so
different in approach, of Felix Hausdorff and Waclaw Sierpinski to
solving Cantor's Continuum Problem during the period 19001930.
Hausdorff was concerned with the problem in terms of ordertypes,
classifying in particular the ordertypes of power of the continuum
and those of power alephone. Here in 1907 he formulated the key
notion of cofinality, on which later work on transfinite cardinals
depended, and discovered the Generalized Continuum Hypothesis in 1908.
He also worked toward a solution from below, showing in 1916 that the
Continuum Hypothesis (CH) holds for all Borel sets, i.e., every
uncountable Borel set has the power of the continuum.
During the 1920s Sierpinski connected CH directly to descriptive set
theory and real analysis, extending in this way the work of Luzin.
Sierpinski proved many propositions to be equivalent to CH,
e.g., the proposition that there is an analytic subset of the reals
which is not the union of less than continuum many Borel sets.
Likewise, he discovered many consequences of CH, such as the
existence of a Sierpinski set. But throughout, Sierpinski's work on
CH has a hypothetical aspect that connects it to more recent work on
models of set theory. By contrast, Hausdorff's contributions are
general results that are true, not merely hypothetical.
 RAM MURTY, Queen's University, Kingston, Ontario K7L 3N6
The work of Brahmagupta and Jayadeva

We will discuss the `bhavana' method of Brahmagupta (6th century) and
the `chakravala' method of Jayadeva (11th century) in their treatment
of the equation x^{2}  Dy^{2} = ±1 for a given squarefree number D
and integers x,y. Their combined work gives the complete algorithm
for determining all the solutions of the above equation. The first
rigorous proof of this was given by Lagrange in the 18th century.
 GLEN VAN BRUMMELEN, Bennington College, Bennington, Vermont 05201, USA
Mathematical computation in Medieval Islamic astronomy:
a survey of approximation methods

The geometric models of the Greek astronomical tradition, inherited by
medieval Islam, are not always amenable to numerical computation.
Motivated by necessity, the Muslim astronomers conceived a number of
ingenious ``solutions'' to the intractable mathematical problems that
often arose. These included (among others) functional approximations,
iteration methods, and interpolation schemes; it is in fact tempting,
but misleading, to refer to them as a prehistory of numerical
analysis. This group of techniques, considered together, are impressive
enough to rate as a sort of underground discipline: not sufficiently
independent to justify a separate existence, but cohesive enough to
indicate some focus.

