ESHS/HSS Annual Meeting

From Zhu Shijie to Seki Takakazu: transforming the calculating-rod notation into the side- writing notation

• In Event: Diagrammatic Computations and Their Symbolic Aspects (I)

English Abstract

My presentation will trace the transformation from the calculating-rod notation (chou suan)
employed in Song-Yuan China, to the side-writing notation (Bōshohō) in Edo Japan.
In ancient China, mathematical operations were expressed in written Chinese, and from ca. 10th
century CE onwards, the calculating-rod symbols – numeral signs that represent the different
positions of the calculating rods under a decimal system – were introduced into mathematical
treatises. Thereafter, both the Chinese script and calculating-rod symbols constitute a single
mathematical language in Chinese mathematical treatises. These numeral signs are notated
according to their physical positions in the calculating board, configured in multiple rows and
columns. In ca. 12th century, a novel algebraic procedure named the tianyuan shu (Procedure of
Celestial Sources) was invented in Northern China, and subsequently reshaped the way this
mathematical language is expressed in the following two centuries.
When a Korean edition of Zhu Shijie’s (ca.13th-14th centuries CE) Introduction to Mathematics
(1299) was brought to Edo Japan from China, the tianyuan shu was then introduced to Japanese
mathematicians and eventually became the foundation of the Bōshohō (side-writing notation). The
Bōshohō was devised by the Japanese mathematician Seki Takakazu (1642?-1708), who, after
having absorbed the language employed in the tianyuan shu. came up with a writing system that
made possible the expression of multiple unknowns, without the need of physically manipulating
calculating rods; thus, adopting a pure hand-written version of the calculating-rod notation for
expressing symbolic algebra. This talk will examine how that transformation took place.

Author

• Stephen HUNG, SPHERE-CNRS, Université Paris Cité