Victor J. Katz

What is algebra? For some, it is an abstract language of x's and y's. For mathematics majors and professional mathematicians, it is a world of axiomatically defined constructs like groups, rings, and fields. Taming the Unknown considers how these two seemingly different types of algebra evolved and how they relate. Victor Katz and Karen Parshall explore the history of algebra, from its roots in the ancient civilizations of Egypt, Mesopotamia, Greece, China, and India, through its development in the medieval Islamic world and medieval and early modern Europe, to its modern form in the early twentieth century. Defining algebra originally as a collection of techniques for determining unknowns, the authors trace the development of these techniques from geometric beginnings in ancient Egypt and Mesopotamia and classical Greece. They show how similar problems were tackled in Alexandrian Greece, in China, and in India, then look at how medieval Islamic scholars shifted to an algorithmic stage, which was further developed by medieval and early modern European mathematicians. With the introduction of a flexible and operative symbolism in the sixteenth and seventeenth centuries, algebra entered into a dynamic period characterized by the analytic geometry that could evaluate curves represented by equations in two variables, thereby solving problems in the physics of motion. This new symbolism freed mathematicians to study equations of degrees higher than two and three, ultimately leading to the present abstract era. Taming the Unknown follows algebra's remarkable growth through different epochs around the globe.

Ancient mathematics. Sherlock Holmes in Babylon / R. Creighton Buck -- Words and pictures: new light on Plimpton 322 / Eleanor Robson -- Mathematics, 600 B.C.-600 A.D. / Max Dehn -- Diophantus of Alexandria / J.D. Swift -- Hypatia of Alexandria / A.W. Richeson -- Hypatia and her mathematics / Michael A.B. Deakin -- The evolution of mathematics in ancient China / Frank Swetz -- Liu Hui and the first golden age of Chinese mathematics / Philip D. Straffin, Jr. -- Number systems of the North American Indians / W.C. Eells -- The number system of the Mayas / A.W. Richeson -- Before the conquest / Marcia Ascher -- Medieval and renaissance mathematics. The discovery of the series formula for [pi] by Leibniz, Gregory and Nilakantha / Ranjan Roy -- Ideas of calculus in Islam and India / Victor J. Katz -- Was calculus invented in India? / David Bressoud -- An early iterative method for the determinationof sin 1° / Farhad Riahi. Leonardo of Pisa and his Liber Quadratorum / R.B. McClenon -- The algorists vs. the abacists: an ancient controversy on the use of calculators / Barbara E. Reynolds -- Sidelights on the Cardan-Tartaglia controversy / Martin A. Nordgaard -- Reading Bombelli's x-purgated algebra / Abraham Arcavi and Maxim Bruckheimer -- The first work on mathematics printed in the New World / David Eugene Smith -- The seventeenth century. An application of geography to mathematics: history of the integral of the secant / V. Frederick Rickey and Philip M. Tuchinsky -- Some historical notes on the cycloid / E.A. Whitman -- Descartes and the problem-solving / Judith Grabiner -- René Descartes' curve-drawing devices: experiments in the relations between mechanical motion and symbolic language / David Dennis -- Certain mathematical achievements of James Gregory / Max Dehn and E.D. Hellinger -- The changing concept of change: the derivative from Fermat to Weierstrass / Judith V. Grabiner. The crooked made straight: Roberval and Newton on tangents / Paul R. Wolfson -- On the discovery of the logarithmic series and its development in England up to Cotes / Josef Ehrenfried Hofmann -- Isaac Newton: man, myth and mathematics / V. Frederick Rickey -- Reading the master: Newton and the birth of celestial mechanics / Bruce Pourciau -- Newton as an originator of polar coordinates / C.B. Boyer -- Newton's method for resolving affected equations / Chris Christensen -- A contribution of Leibniz to the history of complex numbers / R.B. McClenon -- Functions of a curve: Leibniz's original notion of functions / David Dennis and Jere Confrey -- The eighteenth century. Brook Taylor and the mathematical theory of linear perspectives / P.S. Jones -- Was Newton's calculus a dead end? The continental influence of Maclaurin's treatise of fluxions / Judith Grabiner -- Discussion of fluxions: from Berkeley to Woodhouse / Florian Cajori. The Bernoullis and the harmonic series / William Dunham -- Leonhard Euler 1707-1783 / J.J. Burckhardt -- The number e / J.L. Coolidge -- Euler's vision of a general partial differential calculus for a generalized kind of function / Jesper Lützen -- Euler and the fundamental theorem of algebra / William Dunham -- Euler and differentials / Anthony P. Ferzola -- Euler and quadratic reciprocity / Harold M. Edwards.