Taming the Unknown

Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century

Copyright Date: 2014
Pages: 480
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    Taming the Unknown
    Book Description:

    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 Unknownconsiders 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 Unknownfollows algebra's remarkable growth through different epochs around the globe.

    eISBN: 978-1-4008-5052-5
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-x)
  3. Acknowledgments
    (pp. xi-xvi)
    Victor J. Katz and Karen Hunger Parshall
  4. 1 Prelude: What Is Algebra?
    (pp. 1-11)

    What is algebra? It is a question to which a high school student will give one answer, a college student majoring in mathematics another, and a professor who teaches graduate courses and conducts algebraic research a third. The educated “layperson,” on the other hand, might simply grimace while retorting, “Oh, I never did well in mathematics. Wasn’t algebra all of thatxandystuff that I could never figure out?” This ostensibly simple question, then, apparently has a number of possible answers. What do the “experts” say?

    On 18 April 2006, the National Mathematics Advisory Panel (NMAP) within the...

  5. 2 Egypt and Mesopotamia
    (pp. 12-32)

    The earliest civilizations to have left written mathematical records—the Egyptian and the Mesopotamian—date back thousands of years. From both, we have original documents detailing mathematical calculations and mathematical problems, mostly designed to further the administration of the countries. Both also fostered scribes of a mathematical bent who carried out mathematical ideas well beyond the immediate necessity of solving a given problem. If mathematics was thus similarly institutionalized in Egypt and Mesopotamia, it nevertheless took on dramatically different forms, being written in entirely distinct ways in the two different regions. This key difference aside, the beginnings of algebra, as...

  6. 3 The Ancient Greek World
    (pp. 33-57)

    Mathematics began to develop in Greece around 600 BCE and differed greatly from what had emerged in the more ancient civilizations of Egypt and Mesopotamia. Owing perhaps to the fact that large-scale agriculture was impossible in a topography characterized by mountains and islands, Greece developed not a central government but rather the basic political organization of thepolisor city-state with a governmental unit that, in general, controlled populations of only a few thousand. Regardless of whether these governments were democratic or monarchical, they were each ruled by law and therefore fostered a climate of argument and debate. It was...

  7. 4 Later Alexandrian Developments
    (pp. 58-80)

    Euclid and, as we saw, most probably Apollonius and Archimedes participated in the vibrant research and cultural complex that began to coalesce around the Museum and Library in Alexandria around 300 BCE.¹ Conceived by Alexander the Great (356–323 BCE) and built on a strip of land just to the west of the Nile delta with the Mediterranean on its northern and Lake Mareotis on its southern shore, the city of Alexandria had rapidly taken shape in the decades after Alexander’s death in the hands of his boyhood friend and senior general, Ptolemy I. As students of Aristotle, both Alexander...

  8. 5 Algebraic Thought in Ancient and Medieval China
    (pp. 81-104)

    Although archaeological evidence supports the fact that the Chinese engaged in numerical calculation as early as the middle of the second millennium BCE, the earliest detailed written evidence of the solution of mathematical problems in China is theSuan shu shu(orBook of Numbers and Computation), a book written on 200 bamboo strips discovered in 1984 in a tomb dated to approximately 200 BCE.¹ Like Mesopotamian texts much earlier, theSuan shu shuwas a book of problems with their solutions.

    China was already a highly centralized bureaucratic state by the time theSuan shu shuwas composed, having...

  9. 6 Algebraic Thought in Medieval India
    (pp. 105-131)

    Unlike China, India was never unified completely during the ancient or medieval period.¹ Frequently, rulers in one or another of the small kingdoms on the Indian subcontinent conquered larger areas and established empires, but these rarely endured longer than a century. For example, shortly after the death of Alexander the Great in 323 BCE, Chandragupta Maurya (ca. 340–298 BCE) unified northern India, while his grandson Ashoka (304–232 BCE) spread Mauryan rule to most of the subcontinent, leaving edicts containing the earliest written evidence of Indian numerals engraved on pillars throughout the kingdom. After Ashoka’s death, however, his sons...

  10. 7 Algebraic Thought in Medieval Islam
    (pp. 132-173)

    In the first half of the seventh century a new civilization arose on the Arabian Peninsula. Under the inspiration of the prophet Muḥammad, the new monotheistic religion of Islam quickly drew adherents and pushed outward. In the century following Muḥammad’s capture of Mecca in 630 CE, Islamic armies conquered an immense territory while propagating the new religion both among the previously polytheistic tribes of the Middle East and among the followers of other faiths.¹ Syria and then Egypt were taken from the Byzantine Empire, while Persia was conquered by 642 CE. Soon, the victorious armies reached India and central Asia...

  11. 8 Transmission, Transplantation, and Diffusion in the Latin West
    (pp. 174-213)

    When Hypatia died her violent death in 415 CE, the rise of Islam was more than two centuries in the offing, but the Roman Empire, of which her native Alexandria was a part, had long been in decline. One manifestation of that decline, to which her work in the Diophantine tradition served as a notable counterexample, was a waning interest in learning and in the preservation of classical Greek knowledge. By 500 CE, the Roman Empire had completely ceased to exist in the West, and the centralized government, established commercial networks, and urban stability that had characterized it had been...

  12. 9 The Growth of Algebraic Thought in Sixteenth-Century Europe
    (pp. 214-246)

    In his compendiumSumma de arithmetica, geometria, proportioni, e pro-portionalita, Luca Pacioli, like his fifteenth-century contemporaries and successors, devoted most of his algebraic efforts to solutions of equations of the first and second degree but could not resist treating those higher-degree equations for which solutions were known.¹ He noted, for example, that given any one of the six types of linear and quadratic equations, multiplication by any given power of the unknown generated a solvable higher-order equation. More interestingly, he considered eight cases of fourth-degree equations, which he wrote as

    (1) “censo de censo equale a numero” orax4=e,


  13. 10 From Analytic Geometry to the Fundamental Theorem of Algebra
    (pp. 247-288)

    Although Viète’s work represented an advance on earlier algebraic work in the generality of its treatment of equations, his continued use of words and abbreviations rather than true symbols made it difficult for his successors actually to realize his aim to “solve every problem.” In particular, the problems of motion on earth, as studied by Galileo Galilei (1564–1642), and motion in the heavens, as explored by Johannes Kepler (1571–1630), required new algebraic ideas in order for their solutions to be understood and for further questions about motion to be resolved. It was one thing to solve a cubic...

  14. 11 Finding the Roots of Algebraic Equations
    (pp. 289-334)

    As we saw in chapter 9, Cardano and others had published methods for solving cubic and quartic polynomial equations by radicals in the middle of the sixteenth century. Over the next two hundred years, other mathematicians struggled to find analogous algebraic methods to solve higher-degree equations, that is, methods, beginning with the coefficients of the original polynomial, that consist of a finite number of steps and involve just the four arithmetic operations together with taking roots. Of course, certain particular equations could be solved, including those arising in circle-division problems, but, for more general equations, new methods were sought. Some...

  15. 12 Understanding Polynomial Equations in n Unknowns
    (pp. 335-380)

    As mathematicians in the sixteenth, seventeenth, eighteenth, and nineteenth centuries worked to understand the solvability of polynomial equations in one unknown of degree three and higher, they also naturally encountered systems of equations for which they sought simultaneous solutions. This should come as no surprise, since, as we have seen, such systems had arisen inmathematical problem solving since ancient times. Mesopotamian mathematicians simultaneously solved two equations in two unknowns using their method of false position. Diophantus developed a whole series of techniques for tackling more complicated systems involving squares and higher powers and treated examples that were both determinate and...

  16. 13 Understanding the Properties of ʺNumbersʺ
    (pp. 381-426)

    If understanding polynomials and polynomial equations defined two intertwining threads of what came to be recognized as the fabric of algebraic research by the end of the nineteenth century, a third thread was spun from the efforts of mathematicians to understand the properties of “numbers.” As early as the sixth century BCE, this quest had motivated the Pythagoreans to inquire into the nature of the positive integers and to define concepts such as that of theperfectnumber, that is, a positive integer like 6=1+2+3 or 28=1+2+4+7+14, which satisfies the property that it equals the sum of its proper divisors....

  17. 14 The Emergence of Modern Algebra
    (pp. 427-448)

    As the previous three chapters have documented, by the first decade of the twentieth century, the topography of algebra had changed significantly. Galois’s ideas on groups, which had developed in the specific context of finding the roots of algebraic equations, had not only entered the algebraic mainstream but had developed over the course of the last half of the nineteenth century—thanks to the work of Arthur Cayley, Camille Jordan, Ludwig Sylow, and Heinrich Weber, among many others—into an independent and freestanding theory of groups. Weber, in particular, had also abstracted out and explicitly expressed a viable set of...

  18. References
    (pp. 449-476)
  19. Index
    (pp. 477-486)