This edition of Books IV to VII of Diophantus' Arithmetica, which are extant only in a recently discovered Arabic translation, is the outgrowth of a doctoral dissertation submitted to the Brown University Department of the History of Mathematics in May 1975. Early in 1973, my thesis adviser, Gerald Toomer, learned of the existence of this manuscript in A. Gulchln-i Macanl's just-published catalogue of the mathematical manuscripts in the Mashhad Shrine Library, and secured a photographic copy of it. In Sep tember 1973, he proposed that the study of it be the subject of my dissertation. Since limitations of time compelled us to decide on priorities, the first objective was to establish a critical text and to translate it. For this reason, the Arabic text and the English translation appear here virtually as they did in my thesis. Major changes, however, are found in the mathematical com mentary and, even more so, in the Arabic index. The discussion of Greek and Arabic interpolations is entirely new, as is the reconstruction of the history of the Arithmetica from Diophantine to Arabic times. It is with the deepest gratitude that I acknowledge my great debt to Gerald Toomer for his constant encouragement and invaluable assistance.

This volume offers an English translation of all ten extant books of Diophantus of Alexandria’s Arithmetica, along with a comprehensive conceptual, historical, and mathematical commentary. Before his work became the inspiration for the emerging field of number theory in the seventeenth century, Diophantus (ca. 3rd c. CE) was known primarily as an algebraist. This volume explains how his method of solving arithmetical problems agrees both conceptually and procedurally with the premodern algebra later practiced in Arabic, Latin, and European vernaculars, and how this algebra differs radically from the modern algebra initiated by François Viète and René Descartes. It also discusses other surviving traces of ancient Greek algebra and follows the influence of the Arithmetica in medieval Islam, Byzantium, and the European Renaissance down to the 1621 publication of Claude-Gaspard Bachet’s edition. After the English translation the book provides a problem-by-problem commentary explaining the solutions in a manner compatible with Diophantus’s mode of thought. The Arithmetica of Diophantus provides an invaluable resource for historians of mathematics, science, and technology, as well as those studying ancient Greek, medieval Islamic and Byzantine, and Renaissance history. In addition, the volume is also suitable for mathematicians and mathematics educators.

This witty introduction to number theory deals with the properties of numbers and numbers as abstract concepts. Topics include primes, divisibility, quadratic forms, and related theorems.

Geometry and the theory of numbers are as old as some of the oldest historical records of humanity. Ever since antiquity, mathematicians have discovered many beautiful interactions between the two subjects and recorded them in such classical texts as Euclid's Elements and Diophantus's Arithmetica. Nowadays, the field of mathematics that studies the interactions between number theory and algebraic geometry is known as arithmetic geometry. This book is an introduction to number theory and arithmetic geometry, and the goal of the text is to use geometry as the motivation to prove the main theorems in the book. For example, the fundamental theorem of arithmetic is a consequence of the tools we develop in order to find all the integral points on a line in the plane. Similarly, Gauss's law of quadratic reciprocity and the theory of continued fractions naturally arise when we attempt to determine the integral points on a curve in the plane given by a quadratic polynomial equation. After an introduction to the theory of diophantine equations, the rest of the book is structured in three acts that correspond to the study of the integral and rational solutions of linear, quadratic, and cubic curves, respectively. This book describes many applications including modern applications in cryptography; it also presents some recent results in arithmetic geometry. With many exercises, this book can be used as a text for a first course in number theory or for a subsequent course on arithmetic (or diophantine) geometry at the junior-senior level.

With the publication of this book I discharge a debt which our era has long owed to the memory of a great mathematician of antiquity: to pub lish the /llost books" of the Conics of Apollonius in the form which is the closest we have to the original, the Arabic version of the Banu Musil. Un til now this has been accessible only in Halley's Latin translation of 1710 (and translations into other languages entirely dependent on that). While I yield to none in my admiration for Halley's edition of the Conics, it is far from satisfying the requirements of modern scholarship. In particular, it does not contain the Arabic text. I hope that the present edition will not only remedy those deficiencies, but will also serve as a foundation for the study of the influence of the Conics in the medieval Islamic world. I acknowledge with gratitude the help of a number of institutions and people. The John Simon Guggenheim Memorial Foundation, by the award of one of its Fellowships for 1985-86, enabled me to devote an unbroken year to this project, and to consult essential material in the Bodleian Li brary, Oxford, and the Bibliotheque Nationale, Paris. Corpus Christi Col lege, Cambridge, appointed me to a Visiting Fellowship in Trinity Term, 1988, which allowed me to make good use of the rich resources of both the University Library, Cambridge, and the Bodleian Library.

This is an introduction to diophantine geometry at the advanced graduate level. The book contains a proof of the Mordell conjecture which will make it quite attractive to graduate students and professional mathematicians. In each part of the book, the reader will find numerous exercises.