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.
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- 1.
I am grateful for the helpful comments and kind interest of Lee Jongtae, Lee Jung, Jun Yong Hoon, and Jun’s anonymous student. Special thanks go to C. and two anonymous commentators, who read drafts several times and gave precious comments and corrections, and two editors, who invited me to contribute this chapter.
For the romanization of Korean and Chinese, the McCune-Reischauer system and the Pinyin system, respectively, are used, with exception for the English articles in which authors’ names and titles have already been romanized in other ways. Korean terms coined by Korean authors in Korean texts written in traditional Chinese will appear in the McCune-Reischauer system first, and then, the Pinyin system. Korean names, their titles of posts, and titles of texts will appear only in the McCune-Reischauer system. Korean and Chinese names are cited in their original order, family name first.
This paper does not explain the circumstantial differences between Korea and China, which eventually ended up with the different usage of the counting rods, but we could offer, until now, just a rough assumption that Korea didn’t witness the rapid economic and commercial growth enough to cause the transformation of the tools from counting rods into abacus like China. This is a rich field that deserves much more detailed attention of the historians of mathematics and sciences. Also we can consider the long-lasting maintenance of the educational institution in firm support of the counting rods as one of the environmental factors that influenced their longevity for ages in Korea.
- 2.
Counting rods as material objects have been designated differently by diverse historians. For example, Li and Du (1987) called them “devices,” while Martzloff (1997) put them into the “calculating instruments” section along with abacus. Needham (1959) also put them into the “mechanical aids” section with abacus that he referred to as “instruments.” Meanwhile, what Netz (2002), as a cognitive historian, called “cognitive tools” included abacus as well as numerals. Even if there had been no sharp difference in the definitions of “tools” and “instruments,” Taub (2011) acknowledged that the appellation “instrument” is now distinguished from the term “tool,” as being used for more delicate work or for artistic or scientific purpose, and quoted an example of usage from the Oxford English Dictionary: “a workman or artisan has his tools, a draftsman, surgeon, dentist, astronomical observer, his instrument.” In this chapter, I would like to use the word “tools” in order to avoid the image of “instruments” as being too sophisticated, or advanced.
- 3.
The list of the three textbooks for the education of students was as follows: Detailed Explanation of Mathematical Methods (Xiangming suanfa 詳明算法, 1373) by An Zhizhai 安止齋 (fl. fourteenth century), Yang Hui’s Mathematical Methods (Yang Hui suanfa 楊輝算法, ca. 1270) by Yang Hui 楊輝 (fl. end of the 13th century), and Introduction to Mathematical Learning (Suanxue qimeng算學啓蒙, 1299) by Zhu Shijie 朱世傑 (fl. end of the 13th century).
- 4.
At least, two literati, Im Chun 任濬 (1608–?) and Pak Yul 朴繘 (1621–1668), were said to have written handbooks for this text.
- 5.
In the Introduction to Mathematical Learning, the problems on simple multiplication and division in the first five chapters made around 30% (79 problems out of total 259 problems) and the problems on the extraction of roots in the last chapter made up around 13% (34 problems out of 259 problems), whereas in Writings of Nine and One, the problems on simple multiplication and division in the first chapter made up around 4% (19 problems out of 473 problems; in this case the total sum of problems is calculated without the problems in the last “Miscellaneous Writing” chapter), and the problems on the extraction of roots in the last three chapters made up about 35% (166 problems out of 473 problems), the largest proportion of the entire text, as Kawahara (2010, 109–110) also pointed out.
- 6.
For the detailed explanation of this formula, see Martzloff (1997, 286).
- 7.
This kind of method for extracting the roots by iterated multiplication and addition was called “zengcheng kaifangfa 增乘開方法” and could be found in many Chinese mathematical texts from the twelfth century on, including, among others, Qin Jiushao’s 秦九韶 (ca. 1202–1261) Mathematical Treatise in Nine Chapters (Shushu jiuzhang 數書九章, 1247). For more detailed study of this method, see Martzloff (1997, 231–249), Libbrecht (1973). Also, for the general explanation of the other operations with counting rods, including the method of the celestial element, see Marzloff (1997, 217–271).
- 8.
Hong’s expansion started with the similar type of problems in the Introduction to Mathematical Learning, but while the Introduction dealt only with objects of no more than three dimensions that could be easily imagined in a real space, Hong expanded this version of problems up to ten dimensions.
- 9.
This is the expression with the coinage of the terms that Hong used mostly in his text. For other various expressions used, see Martzloff (1997, 229).
- 10.
Kawahara provided a few reasons why he praised Hong’s texts as “a big jump of the celestial element method”: the expansion of the total numbers and the increasing percentage of the numbers of the problems on the extraction of roots, and the increasing portion of the problems using the celestial element method in those chapters. Kawahara (1998, 60–65).
- 11.
This anecdote was fully addressed in Horng’s paper (2002b).
- 12.
For example, problems seem like the variations on the combinations of basic astronomical situations found in “A Year with Three Hundred Days” in the Book of Documents:
(Problem 2) [Let us suppose that] now there is a circle-shaped lake, whose circumference is 365 1/4 chi 尺. If it is only known that the large ant and the small one [started] together [to] walk, and that the small one goes 1 chi a day and the big one goes 13 7/19 chi a day, then how many days will pass before they meet [again]?
(Problem 3) [Let us suppose that] now there is heaven, whose circumference is 365 1/4 du 度, surrounding the earth and rotating clockwise. The sun and the moon [started to] go together with the heaven; the sun is 1 du slower than the heaven, and the moon is 13 7/19 du [slower than the heaven]. How many days will pass before the sun and the moon meet [again]? (Hong 1985, 651–653).
This kind of problems had been circulated in a wide range of areas for centuries. For more detailed discussion on these “pursuit problems,” see Bréard (2002).
- 13.
He Guozhu had served as a staff member of the Office of Mathematics for Kangxi 康熙 (r. 1662–1722) in the Qing 淸 dynasty. For his general activities and missions, including his visit to Korea, see Jami (2012, 263–273, 277–279).
- 14.
- 15.
It was from the late eighteenth century on that it had been reported that literati had learned from, and sometimes co-worked with the chungin officials: at first, from astronomical officials, and then, in the nineteenth century, even from mathematical officials. By contrast, there were several anecdotes reporting that a member of literati, such as Hwang Yun-sǒk 黃胤錫 (1729–1791), hesitated to see any chungin officials to borrow their mathematical texts so that he could not have a chance to read the texts for years, or that another member of the literati, such as Hong Kil-ju 洪吉周 (1786–1841), learned mathematical methods from his mother when young, and soon was self-taught by mathematical texts when he grew up (Sun 2006; Koo 2012).
- 16.
- 17.
Take the other example, a procedure called “Ch’ongsŭng 總乘.” The term was coined by Ch’oe and its meaning seemed the “total multiplication,” because during the procedure, “the sun,” “the moon,” and “the planets” were multiplied all together. This procedure belonged to taiyang, whose sign was ⚌, because it underwent multiplication below, and another multiplication above. For the other categories, see Kawahara (1996, 2010).
- 18.
Ch’oe’s interpretation of the multiplication and division as a pair was not at all his unique thought. For more detailed discussion, see Chemla (2010).
- 19.
- 20.
Here I translate “yi 義” as “meaning,” but the term has already lots of other translation, such as “signification,” “justice,” and the like. In the field of history of ancient Chinese mathematics, as Chemla (2012, 481) pointed out, “the algorithms , together with the situations in relation to which they were introduced, provided means for determining the ‘meaning’ of an operation or a sequence of operations. This appears to a key act for proving the correctness of algorithms , and it is noteworthy that a term (yi ‘meaning’) seems to have been specialized to designate it in ancient China.” If we accept this view, we might say that Ch’oe’s usage of the term “meaning” expanded its “correctness” not only from the perspective of mathematics, but also from the perspective of philosophical behaviour.
- 21.
For the contents of Shuli jingyun and the written calculation in China, see Jami (2012).
- 22.
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Oh, Y.S. (2018). Same Rods, Same Calculation? Contextualizing Computations in Early Eighteenth-Century Korea. In: Volkov, A., Freiman, V. (eds) Computations and Computing Devices in Mathematics Education Before the Advent of Electronic Calculators. Mathematics Education in the Digital Era, vol 11. Springer, Cham. https://doi.org/10.1007/978-3-319-73396-8_9
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