Jules Tannery (1848-1910) and Paul Appell (1855-1930) 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.
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 al-His¯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:
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 view-not, of course, uncontested-of mathematics as attaining a unique exactness, certainty and insight into the nature of things.
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.
I will discuss some of Fermat's major contributions to number theory, noting his intellectual debts and his legacy.
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.
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.
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 so-called Menelaus Theorem.
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 al-Kashi, 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 al-Kashi's solution in detail.