Stedall (Oxford U.) traces the transition from the traditional algebra of equation-solving in the 16th and 17th centuries to the emergence of modern, or abstract, algebra in the middle 19th century. The two poles of the change are Girolamo Cardano's 1545 Ars magna and Joseph-Louis Lagrange's "Reflections on the algebraic solutions of equations" in 1771. The intervening developments pivot on Isaac Newton in the early 18th century. Among the stages are from Viète to Descartes, roots as sums of radicals, elimination theory, numerical solution, and the outsiders. A final chapter looks at dissemination and new directions after Lagrange. Distributed in North America by the American Mathematical Society. Annotation ©2011 Book News, Inc., Portland, OR (booknews.com)