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History of Mathematics / L'histoire des mathématiques
(Len Berggren, Organizer)

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TOM ARCHIBALD, Acadia University, Wolfville, Nova Scotia  B0P 1X0
Hermite and Jacobi

Charles Hermite (1822-1901) began his research career by successfully extending Jacobi's theory of elliptic functions. Jacobi, by then well-established, 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 (1698-1746) based his major work on Newton's for instance, Maclaurin's foundations of the fluxional calculus, infinite-series treatments of extrema, the Euler-Maclaurin 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 job-hunting 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 1900-1930. Hausdorff was concerned with the problem in terms of order-types, classifying in particular the order-types of power of the continuum and those of power aleph-one. 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 x² - Dy² = ±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 pre-history 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.