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

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

## Synopsis

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.

## Excerpt

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 that *x* and *y* stuff 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 US Department of Education was established by executive order of then President George W. Bush to advise him, as well as the Secretary of Education, on means to “foster greater knowledge of and improved performance in mathematics among American students.” Among the panel’s charges was to make recommendations on “the critical skills and skill progressions for students to acquire competence in algebra and readiness for higher levels of mathematics.” Why should competence in algebra have been especially singled out?

When it issued its final report in March 2008, the panel stated that “a strong grounding in high school mathematics through Algebra II or higher correlates powerfully with access to college, graduation from college, and earning in the top quartile of income from employment.” Furthermore, it acknowledged that “although our students encounter difficulties with many aspects of mathematics, many observers of

US Dept. of Education, 2008, p. 71. The next quotation is also found here.

US Dept. of Education, 2008, p. xii. For the next two quotations, see pp. xiii and 16, respectively.