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.

illustrate his point, Cauchy explicitly wrote out the determinant in the cases of two, three, and four variables. In the three-variable case, for example, he obtained the third-degree polynomial in s.27 In the general case, however, D is an nth-degree polynomial in s, and the problem was to solve the equation D = 0 for the eigenvalues s of the characteristic equation. By the fundamental theorem of algebra (recall chapter 10), D = 0 has n real and/or complex roots s1, s2, …, sn, each of which

ideas in broadly philosophical terms, carrying on the debate of his predecessors, Peacock, Hamilton, De Morgan, and others. He opened by defining mathematics as “the science which draws necessary conclusions,” elaborating that this definition of mathematics is wider than that which is ordinarily given, and by which its range is limited to quantitative research. The ordinary definition, like those of other sciences, is objective; whereas this is subjective. Recent investigations, of which

N simultaneously satisfying for integral values x, y, z, or (what amounts to the same thing) satisfying the congruences Master Sun gave not only the answer but also his method of solution: “If you count by threes and have the remainder 2, put 140. If you count by fives and have the remainder 3, put 63. If you count by sevens and have the remainder 2, put 30. Add these numbers and you get 233. From this subtract 210 and you get 23.” He further explained that “for each unity as remainder when

papyri are still extant. Besides the display of arithmetic techniques, the most important mathematical concept apparent in the two mathematical papyri is proportionality. As noted in the opening chapter, this idea generally underlies the earliest problems introduced in school mathematics that require the determination of an unknown quantity, so the fact that problems involving proportionality are present on the Egyptian papyri makes these perhaps the simplest and earliest algebra problems in

root equal to −4, he concluded that the sum of the remaining roots must be 4 and their product 5. Earlier, however, he had given a general formula showing how to solve this kind of problem,9 namely, if the sum of the roots is x and the product is xx − df, then the roots are, in fact, In this case, since x = 4, we must have d f = 11, so the two desired values are and . Therefore, the complex roots of the original equation are and . Harriot’s notational innovations made Viète’s ideas an even more