Counting rods

Abstract: The Mathematical Book in Nine Chapters, written by Qin Jiushao in 1247, is a masterpiece that is representative of Chinese mathematics at that time. Most of the previous studies have focused on its mathematical achievements, while few works have addressed the counting diagrams that Qin used as a writing system. Based on a seventeenth-century copy of Qin’s treatise (i.e., Zhao Qimei’s 1616 handwritten copy), this paper systematically analyzes the writing system, which includes both a numeral system and a linear system. It argues that Qin provided a new representation of mathematics in addition to textual procedures, detailed solutions, and operations carried out with counting rods. Moreover, this new representation was used to connect mathematical practices within and outside the text and should be understood in its textual context. Therefore, Qin’s writing system represents an intermediate phase in the textualization and symbolization of Chinese mathematics in thirteenth-century China.

Abstract: We saw in the preceding chapter and earlier that from about the eighth to the fourteenth century, the Arabs had the most advanced science in the world. Consequently, in the fields of astronomy, mathematics, optics, and physical experimentation – which led directly to modern science – Chinese science was second to that of the Arab-Islamic world. Present scholarship regarding Chinese science suggests, moreover, that China developed along lines quite independent of the West and the Arabic Middle East. The Chinese knew nothing of Aristotle, Euclid, Ptolemy, or Galen. Nevertheless, there were areas in which the Chinese did accomplish great things, though in almost no case was there continuous and progressive development. As we saw in Chapter 2, discussions can be found in the writings of Chinese mathematicians on arithmetic fractions, the statement of formulas for the computation of areas and volumes, the solution to systems of simultaneous equations, and procedures for square and cube extraction. These are to be found in The Nine Chapters on the Mathematical Procedures (from about the first century). During the Sung dynasty (ca. 960–1279) Chinese mathematics underwent another period of creative growth, especially in algebraic computation. It is said that “a general technique was found for the solution of numerical equations containing any power of a single unknown.” However, it has also been claimed that the Chinese system of representation and positional notation, as well as its techniques of computation (with counting rods), were cumbersome and not nearly as generalizable and easy to use as the Arabic-Hindu numeral system. These Arabic-Hindu numerals, located in a decimal place value system had been available in al-Khwarizmi's work since about 825. In contrast, the course of development of mathematics in China required a move from computation with counting rods to the use of the abacus (generally in about the sixteenth century) and the incorporation and use of the zero (in the thirteenth and fourteenth centuries). Only in the seventeenth century was the method of paper-and-pen calculation (and hence recorded arithmetic operations) introduced into mathematics with the arrival of the Jesuits.

Abstract: The chapter is devoted to visual representations of two counting instruments, the counting rods (suanzi 算子, chousuan 籌算) and the abacus (suanpan 算盤), found in Chinese mathematical treatises compiled prior to the early 17th century. The former instrument is generally believed to be replaced by the latter in the mid-second millennium AD. The role of the visual representations of these two instruments in didactical practices has never been given due attention by historians of mathematics education. The author provides a preliminary analysis of the images of the instruments in two mathematical textbooks, the Shendao dabian lizong suanhui 神道大編曆宗算會 compiled in 1558 by Zhou Shuxue 周述學 and Panzhu suanfa 盤珠算法 completed in 1573 by Xu Xinlu 徐心魯 and conjectures that these images were used mainly for didactical purposes.

Abstract: The early eighteenth-century Korean mathematical sources testify that there were two types of authorship, the mathematical officials in the lower class and the literati in the upper class. This paper aims to show how each authorship, from dissimilar educational background, affected and transformed the algorithms and grounds of the computation differently in spite of the usage of the same computational tool, based on the analysis of two early eighteenth-century mathematical texts, the Writings of Nine and One (Kuiljip 九一集) by a skilled mathematical official, Hong Chǒng-ha 洪正夏 (1684–?), and the Summary of Nine Numbers (Kusuryak 九數略) by a renowned member of the literati, Ch’oe Sǒk-chǒng 崔錫鼎 (1646–1715). In their texts on the computational techniques using counting rods, Hong appraised the adeptness in handling counting rods and expanded the existing algorithms based on the real practice, while Ch’oe approved the algorithms in which he could find the meaning close to that conveyed by texts and images of Confucian philosophical tradition.