This website presents historical evidence to confute the long-held view that linear algebra developed independently in the West. This website will communicate to a broader audience technical findings from my research monographs:

- The earliest extant records of linear algebra are from China, dating from about 150 BCE;
- The main methods taught today in modern linear algebra can be found in mathematical texts in early imperial China; and
- These practices circulated to Europe by the thirteenth century.

These findings suggest that we reconsider the assumption that mathematical and scientific practices of this period remained confined within the boundaries of what we now anachronistically call “civilizations.”

Today, linear algebra, along with calculus, is one of the most important core courses in modern mathematics at the university level.

The main problem of linear algebra is the solution of systems of *n* linear equations in *n* unknowns. There are two main solutions:

- Determinants, through what is known as Cramer's rule, offer an elegant solution for simple systems of linear equations;
- Gaussian elimination is the more general solution, and arguably the most important of all matrix algorithms.

The earliest extant records of these two approaches to solving linear equations can be found in extant mathematical treatises from imperial China.

Traditional historical accounts attributed the origins of these techniques to the work of two preeminent European mathematicians, namely, the work of Gottfried Wilhelm Leibniz (1646–1716) on determinants, and the work of Carl Friedrich Gauss (1777–1855) on elimination. More recent research into the history of mathematics, however, has increasingly been revising the traditional accounts. It is now well-known that the earliest record of the algorithm we call Gaussian elimination is found in the *Nine Chapters on the Mathematical Arts* (*Jiuzhang suanshu* 九章算術, c. 1st century CE). Gaussian elimination in fact had many European contributors prior to Gauss; recent research has demonstrated conclusively that credit for the discovery of elimination cannot be attributed to Gauss himself. By the early twentieth century it was widely known that the Japanese mathematician Seki Takakazu 關 孝和 (c. 1642–1708) was working on determinants prior to Leibniz.

Relatively little has been published on the history of linear algebra before 1700, however, because the sources are in Chinese.

My own research on linear algebra in imperial China demonstrates that the elimination algorithms found in Chinese mathematical treatises are considerably more sophisticated than has been previously recognized, and that Chinese mathematical treatises also record determinantal-style calculations and solutions.

This website explains the history of linear algebra recorded in early Chinese treatises, leading to the following conclusions:

- The essentials of the methods used today in “Western” linear algebra—augmented matrices, elimination, and determinantal-style calculations—were known by the first century CE in imperial China.
- Simple two-dimensional patterns were used to calculate the solutions on the counting board, and in particular, determinantal-style solutions to a special class of distinctive problems.
- These practices were non-scholarly—they did not require literacy and were not transmitted by texts.
- Some of these practices spread across the Eurasian continent and are recorded in texts from as early as the thirteenth century in Italy. In other words, these practices were not confined by the boundaries we anachronistically term “civilizations.”

This website is divided into several sections.

- The first uses extant written records to reconstruct
*fangcheng*as a mathematical practice on the counting board, summarizing research from my recent book on*fangcheng*, in order to demonstrate how adepts of this practice could quickly solve complex mathematical problems with*n*conditions in*n*unknowns with little more than facility with counting rods and the rote application of simple patterns on the counting board. - The second inquires into the provenance of extant written records of
*fangcheng*practices. Written records of*fangcheng*practices are preserved in treatises on the mathematical arts, which were compiled by aspiring literati and presented to the imperial court as essential to ordering the empire. These literati understood only the rudiments of these practices, yet they derided methods used by adepts as arcane. - The third presents a summary of determinantal-style calculations and solutions to
*fangcheng*problems, solutions seemingly so arcane that they were only rarely recorded by the literati who compiled mathematical treatises. - The fourth presents evidence that these arcane determinantal-style calculations and solutions—which are so distinctive that they can serve as “fingerprints”—circulated across the Eurasian continent, to be recorded in the works of Leonardo Pisano (c. 1170–c. 1250), more commonly known today by the name Fibonacci.

**Published Sources**: This website is based on work published in my first two research monographs:

Roger Hart, *The Chinese Roots of Linear Algebra* (Johns Hopkins University Press, 2010). Project Muse: doi:10.1353/book.487. ISBN: 9780801897559. Amazon ASIN B07DFMM95S (Kindle) and 0801897556 (hardcover).

———, *Imagined Civilizations: China, the West, and Their First Encounter* (Johns Hopkins University Press, 2013). Project Muse: doi:10.1353/book.23819. ISBN: 9781421406060. Amazon ASIN: B00DTTWGUQ (Kindle) and
1421406063 (Hardcover).

Some of this research can be freely accessed through this website, published in the following article, available through the following link:

———, “Tracing Practices Purloined by the ‘Three Pillars,’” *Korean Journal for the History of Science* 34, no. 2 (2012): 287–358.

**Further information**: For complete details, including comprehensive bibliographies, footnotes, and technical information not provided in this website, please consult my research monographs (above).

**Citations**: In research to be published, please cite my published works, which provide complete information, rather than citing this website, which will be constantly under revision. Suggestions for proper citation are available through the following links for *Chinese Roots of Linear Algebra* and *Imagined Civilizations*, providing Chicago, MLA, APA, and Endnote formats.

© 2020 Roger Hart