Arc and Bow Fields in Mathematical Texts

Source abstract

Abstract Arc-related problems in transmitted mathematical texts are often derived from the Jiuzhang suanshu ’s 九章算術 (1st c. ad ) “arc field procedure” ( hutian shu 弧田術 ), discussing the linear relationship between chord, arc, and diameter, or that between chord, arc, and area, without considering arc length as a parameter. Influenced by astronomy, arc length became a computational factor in Chinese mathematics only after the introduction of trigonometric functions through Islamic and European sources. However, the “arc field” and “bow field” ( gongtian 弓田 ) problems in the bamboo manuscript Suanshu 算術 from Shuihudi 睡虎地 tomb 77 reveal that as early as the Western Han, there existed methods to calculate arc-shaped areas from arc length. Though imprecise, these change our previous understanding of the matter. This article reviews the arc-related algorithms in mathematical texts and discusses their origins and development.