Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century
Victor J. Katz, Karen Hunger Parshall
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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.
sine and cosine, asserted in a paper in the Philosophical Transactions of the Royal Society in 17071 that an equation of the form where n is odd, had the solution To see that this is not an unreasonable assertion, take n = 5 and consider the expansion of (m − n)5. With a little manipulation,2 this can be written as or, letting x = m − n, as Now, for n = 5, de Moivre’s equation (11.1) is 5x + 20x3 + 16x5 = a or, equivalently, . Comparing this3 with equation (11.2) then yields three equations
essentially translating the discussion into purely set-theoretic terms. In so doing, he replaced Kummer’s “entirely legitimate” but philosophically problematic notion of ideal prime factors by equivalent and explicitly definable sets.44 This represented a major shift in algebraic thinking that would increasingly come to hold sway and that would come to characterize the so-called modern algebra in the early twentieth century.45 By 1882, Dedekind and his collaborator, Heinrich Weber, had also
theory of, 317–328, 427; of transformations, 327–328 Hahn, Hans, 432 Hamilton, William Rowan, 10, 358–359, 360, 366, 402–403, 405; work of on biquaternions, 408; work of on quaternions, 406–407, 415; work of on vectors, 408–410 Han dynasty, 81 Hankel, Hermann, 413 Harriot, Thomas, 8, 248–249, 258, 259, 261, 264, 268, 276; and the structure of equations, 249–252; work of on quartic equations, 251–252 Heaviside, Oliver, 410 Hensel, Kurt, 433, 435–436 Hermite, Charles, 344, 357–358, 394;
where he had first introduced problems involving quadratic expressions, Diophantus approached the matter of cubics gently, gradually building up to more and more complicated manipulations in terms of combinations of squares and cubes. In IV.1, for example, he asked his readers simply “to find two cubic numbers the sum of which is a square number.” The approach was straightforward, especially in light of the techniques he had presented earlier in the text. He took one of the cubes to be x3 and the
Alexandrian context that follows draws from Pollard and Reid, 2006. 2 Heath, 1964, p. 129. The quotations that follow in this paragraph are also from this page. 3 Heath, 1964 is an edition of this sort, while Sesiano, 1982 provides the first English translation and critical edition of the Arabic manuscript of the original Books IV through VII. In what follows, we will use this ordering and notation for the books: I, II, and III for the first three books of Heath’s translation; IV, V, VI, and