“實如法而一”: Division and Equality in Early Chinese Mathematical Texts

Abstract This article is about an ancient Chinese phrase, shi ru fa er yi 實如法而一, a fixed multi-word expression that conceptually and prescriptively encodes the operation of division in mathematical texts at least since Han dynasty. The formulaic character of this fixed expression was most explicit in the Nine Chapters of Mathematical Procedures ( Jiu zhang suan shu 九章算術) where it was the first choice for instructing abstract division (without indication of a unit of measurement). We investigate this formulaic and apparently linguistically obscure phrase in its historico-philosophical and linguistic dimension with a special focus upon the concept yi 一 (one, unit, unity, oneness, …). By analyzing other expressions of division in ancient mathematical manuscript sources and Han to Tang printed sources both from a philological and a linguistic point of view, we show that the phrase is intimately related to material practice and a conceptualization of division as successive substraction and counting of units. A variety of abreviated and similar phrases thereby serve as key indicators of the centrality of the unit ‘一’ in division; they also point to a more general conception of numbers, i.e. that these are constituted of a multitude of units.