*The Chinese Roots of Linear Algebra*, by Roger Hart, 2011. 286 pp. Illustrations, bibliographies, appendices, index. $65 hardcover, ISBN 13:978-0-8018-9755-9. The Johns Hopkins University Press, Baltimore. www.press.jhu.edu

David Eugene Smith, the noted historian of mathematics, at the beginning of the 20th century, urged a need for research on the historical origins and development of Chinese mathematics. The confused political scene in China at this time, the intervention of the Second World War and the prevailing lack of scholarly interest resulted in Smith's request being ignored. With the appearance of Joseph Needham’s comprehensive and monumental *Science and* *Civilization in China* (7 vols., published from 1954 to 2008), a new interest in Chinese mathematical accomplishments was resurrected. Since the initial appearance of Needham’s series in 1954, several surveys on the development of Chinese mathematics have been published. Most of these works are translations and commentaries on Classical Chinese mathematical texts. Their authors indicate an advanced level of Chinese mathematical achievement before the time of the European Renaissance. Such noted accomplishments include: an accurate numerical estimate for the value of π, exact root extraction methods, computational techniques for solving systems of linear equations, and a sophisticated proficiency in applying the “Pythagorean theorem.” However, due to the authors’ priorities and/or limited abilities, especially in the Chinese language, the details of these accomplishments remained elusive. A major research task remains for historians of Chinese mathematics to ascertain, clarify, and document these achievements. Such an effort now appears in Roger Hart's *The Chinese Roots of Linear Algebra*.

Hart, a scholar in the history of science fluent in the Chinese language, is on the faculty of the University of Texas, Austin. In a reading of the Chinese mathematical classic, *Nine Chapters* *on the Mathematical Art* (*ca.* 100), Hart was particularly struck by the computational procedures he found in this book's eighth chapter, entitled *Fangcheng.* This word can roughly be translated as “rectangle arrays” or “matrices,” and this chapter considers eighteen problems dealing with systems of linear equations and their solution techniques. Chinese computation was performed on a computing board or surface with a set of wooden or ivory rods. Rod configurations represented numbers. *Fangcheng* solution directions required the setting up and manipulation of rod numeral rectangular arrays, or augmented matrices. When the described manipulations are carried out and analyzed, they approximate what would be known centuries later as *Gaussian elimination. *Further, some problems are solved by a method of “cross-multiplication” that approximates a modern use of determinants. Investigating *Fangcheng* and its required computing rod manipulations in depth, Roger Hart has ascertained that Chinese computers of the first century used and understood matrix solution techniques for systems of linear equations. His research is well documented and explained, and this is an excellent and much needed research study. A particularly noteworthy conclusion of Hart’s investigation is that the visualization of rod configurations on a computing surface stimulated Chinese development of highly efficient computational algorithms. Scholars interested in the history of mathematics, and particularly the history of Chinese mathematics, will find this book a wonderful resource. I do feel that the title of this work is a bit ambitious and perhaps the book more appropriately could have been named *Matrix Computation in Ancient China*. Despite this personal bias, *The Chinese Roots of Linear Algebra *is a very useful and thought provoking book. I highly recommended it for personal reading and library acquisition.

See also the MAA Review by Charles Ashbacher.