*The Greate Invention of Algebra: Thomas Harriot’s Treatise on Equations*, Jacqueline A. Stedall, 2003. 322 pp., illustrations, bibliography, $124.50 cloth. ISB 0-19-852602-4. Oxford University Press, 198 Madison Avenue , New York , NY 10016 . (800) 451-7556.www.oup.com/us/.

Thomas Harriot (c. 1560-1621) is perhaps best remembered as a member of Walter Raleigh’s expedition to Virginia in 1585. He served as navigator and surveyor. But Harriot was an able mathematical scientist versed in astronomy, optics, geometry and algebra, as well as navigation and surveying. Today, mathematics history books usually associate his name with the development of the “greater than” and “less than” symbols.

Although he was a prodigious writer and calculator, he did not publish any of his scientific or mathematical findings during his lifetime. Upon his death, he left over four thousand pages of work to be deciphered. The wide scope and general disorder of these papers have defeated most attempts to organize them for publication. In 1631, a collection of his work appeared under the title Artis analyticae praxis, however, it was not very informative and shed little light on his methods or mathematical insights. In *The* *Greate Invention of Algebra*, Jacqueline Stedall has carefully selected, ordered and translated from the Latin, one hundred forty pages of Harriot’s writings dealing with the structure and solutions of polynomial equations. This cogent collection actually follows its original author’s sequencing and forms the basis of an algebra text, which Stedall has chosen to name Treatise on Equations.

Harriot’s recognition as a mathematician did not evolve until after his Virginia expedition. By 1593, he was singled out by his associates as a “profounde Mathematician” in the company of his British colleagues Thomas Digges and John Dee. Through the efforts of a fellow mathematician, Nathaniel Torporley, Harriot learned about the work of the Frenchman, François Viète. He built upon Viète’s system of symbolism and introduced some of this own innovations into algebraic exploration. For example, he popularized the use of joined letters, e.g. *ab*, to represent a product (*a* X *b*) and similarly the use of a repeated variable to represent exponentiation, i.e. *aaa* = *a* ^{3}. After Harriot’s death, Torporley was given the task of organizing and publishing his friend’s papers. He failed at this endeavor.

Jacqueline Stedall, in her limited effort, has succeeded, and in doing so she has provided a valuable resource for serious researchers of mathematics history. Her book consists of a brief introduction into the life and work of Thomas Harriot followed by the translation of the Treatise. Facsimiles of some of Harriot’s original pages occasionally punctuate the text. The translation is straightforward – there are no explanatory comments added. The reader must draw his or her conclusions. However, this excerpt of Harriot’s work captures the spirit of mathematical grappling and exploration of the period. A reader is left with a permeating sense of inquiry. ‘If this is the result, what happens if I expand by another factor?’ Three bibliographies support the work. While The Greate Invention of Algebra is an excellent accomplishment, its usefulness will be limited by its scholarly scope and its cost.

Frank J. Swetz, Professor Emeritus, The Pennsylvania State University

See also the MAA Review by Fernando Q. Gouvêa.