An Ancient Chinese Mathematical Guidebook - Essay Example

Name Course Institution Date History of the Nine Chapters The Nine Chapters is an ancient Chinese mathematical guidebook comprising of 246 issues. The issues range from aimed at providing procedures to be applied to resolve routine life problems such as issues of surveying, taxation, trade and engineering (O'Connor and Robertson, 2003). It is comprised of nine chapters, hence its name. The Nine Chapters have contributed significantly to the advancement of Chinese mathematics; similar to the contribution made by Euclid’s Elements in western mathematics that originated from the establishments formed by primeval Greeks. The Nine Chapters are the earliest Chinese mathematical handbook written about 200 AD towards the end of the Han Dynasty. The Nine Chapters were composed by various generations of scholars from the 10th to the 2nd century BCE with its modern form being drawn from the 1st century CE. Basically the Nine Chapters is a book of problems that tutors could provide students with solutions and general guide on how to solve the problems. Entries in this handbook normally take a procedure of an account of a problem, followed by an account of the resolution, and a description of the process that lead to the solution. Overview of the Nine Chapters The Nine Chapters are divided into three crucial parts; the first three chapters are concerned with engineering and surveying, or city planning and field planning. In these chapters, students were taught how to understand the area of triangle, a circle, trapezoid, and a rectangle. It also taught how to calculate the area of a portion of a circle. Other issues include problems in subtraction, addition, multiplication and division fractions. Students also learnt how to understand percentages and on how to determine certain measures of rice (O'Connor and Robertson, 2003). The following three chapters 4-6 entailed how to figure out taxes and for the authorities to understand budgets. There was a challenge in finding the sides of figures when one side and the area are known, and how to calculate square and cube roots. This section exhibited how to figure out the volumes of pyramids, cylinders among other shapes. The last three chapters; chapters 7-9 were about how to figure out the solution to less precise problems. These chapters give out how to solve issues concerning more than one equation. To understand the components of the Nine Chapters, it is essential to look at individual chapters in detail. Chapter 1: land surveying The chapter comprises of 38 issues relating to field surveying. First, the chapter focuses on problems relating to areas, then principles of division, addition, multiplication and subtraction of portions. Also the Euclidean system approach for calculating the Greatest Common Divisor (GCD) of two figures is provided in this chapter (Rik and Keimpe, 2007). The forms of figures for which the area is determined comprise circles, rectangles, triangles and trapeziums. In issue 32, a precise estimate is provided for pi. Chapter 2: Millet and Rice The chapter discusses a total of 46 issues which mainly concern the exchange of products, mostly the exchange rates of 20 various grains, seeds and cereals. The calculations entail an evaluation of fractions and percentages and bring about the principle of three for resolving proportion issues (Rik and Keimpe, 2007). Chapter 3: Distribution by proportion In this chapter, 20 issues that also concern proportion, most of which involve various sums provided to or owed by authorities of varying positions. Inverse, direct and compound proportions are considered. Particularly, calculation and geometry are applied in certain issues. Chapter 4: Short width The chapter has 24 issues and derives its title from the leading 11 issues which examine what the dimension of a pitch will become if the breadth is stretched, but its area constantly maintained. The leading 11 problems concern component fractions. On the other hand, issues 12 to 18 concern the withdrawal of square root and the rest of the issues concern the removal of cube roots. Concepts infinitesimals and parameters are also discussed in chapter 4 (O'Connor and Robertson, 2003). Liu Hui comment of 263 AD came to be part and parcel of the script efforts to calculate the volume of a sphere, and provided an estimated method which he proves to be faulty. Chapter 5: Civil engineering The chapter contains 28 problems regarding building of ditches, canals and dykes among other constructions. Volumes of various solid bodies including pyramids, prisms, wedges, cylinders, tetrahedrons and truncated cones are calculated (O'Connor and Robertson, 2003). In his commentary, Liu Hui discussed an approach of enervation he had discovered to calculate the right method for the volume of pyramids. Chapter 6: Fair distribution of products The chapter also contains 28 issues which focus on quantity and ratios. These issues vary and regard issues such as travelling, sharing, and taxation among other proportions. Chapter 7: Excess and deficit This is a 20 problem chapter which involves the “rule of double false position”. Basically, rectilinear equalities are resolved by making two assumptions at the answer followed by calculation of the right answer from both faults. Chapter 8: Calculation by square tables In this chapter, 18 issues that are deduced to resolving processes of instantaneous equalities are provided. Nevertheless, the approach provided is fundamentally for resolving the method by use of the improved matrix of coefficients. The issues entail six equations in six indefinite, and the only different concept with the contemporary approach is that, the coefficients are positioned in columns and not in rows. The matrix is also minimized to triangular system, with the use of elementary column operations as in today’s Gaussian elimination method. The answer is explained for the initial problem. Negative numbers are applied in the matrix, and the chapter contains procedures for their calculation. Chapter 9: right angled triangle This is the last chapter with 24 issues concerning right angled triangles. Pythagoras theorem or (Gougo rule) is employed to solve the first 13 issues (O'Connor and Robertson, 2003). Pythagorean triples are studied using two problems in this chapter, while the remaining problems use the concept of similar triangles. History The whole designation of the Nine Chapters on Mathematical Art is presented on two bronze typical measures dated 179 CE although there are claims that, the book existed before in various titles. According to intellectuals, Chinese mathematics and ancient Mediterranean world mathematics had evolved more or less unconventionally up to the moment when the Nine Chapters reached it ultimate stage. The approach of Chapter Seven came into play in Europe in the 13th century while that of Chapter Eight utilized Gaussian elimination long before Fredrick Gauss (1777-1855). Aspects of primeval Western mathematics are not reflected in antique China. The Nine Chapters were part of the Nine Arithmetic works of the Zhou Dynasty. The origin of the Nine Chapters is not clear, and it is anonymous. Liu Hui who wrote several and useful commentaries on this work however acknowledged ancient mathematicians; Zhang Cang and Geng Shouchang for their initial contribution and comments on the handbook, despite Han Dynasty archives not showing any authors and commenters (Straffin, 1998). The establishment of Chinese mathematics is based on the amazing work of the Nine Chapters on Mathematical Art. This work was the first Chinese accords dedicated to mathematics. Its inspiration on development of mathematics can only be compared to the Euclid’s Elements of the Greek mathematics. Just like Euclid’s is coined as the foundation of the Western branch of mathematics, so is Chapter Nine, which is seen as the cornerstone of the Chinese mathematics. Besides, the Nine Chapters largely influenced the growth of primeval mathematics in Japan and Korea. The Nine Chapters give arithmetical principals which focus on practical applications, presented in question and answer format. However, it is important to note that the arithmetical principles were presented in words, unlike today when they are expressed in algebraic notations. Had Chinese mathematicians had independent endurance of tradition, some of the remarkable anticipations of contemporary approaches might have significantly influenced the development of mathematics. Chinese culture was critically interfered by abrupt breaks (Merzbach, 2011). For example in 213 BCE, the Chinese emperor authorized the burning and destruction of books, an international practice in times of political unrest. Nevertheless, some works may have been spared either through oral transmission or through the existence of other copies, and learning thus continued with emphasis put on mathematical problems of trade and the calendar (Merzbach, 2011). China seems to have had contact with both India and the West, but scholars argue over the extent and nature of borrowing. For instance, the temptation to see Greek or Babylonian influence in China is faced with the idea that, the Chinese never adopted sexagesimal fractions. For a long time, Chinese numbers remained primarily decimal with notations rather remarkably variant from those in other regions (Merzbach, 2011). As a routine in the ancient China, scholars commented on conventional work. Thus, various comments were made on the Nine Chapters, for instance commentary by Liu Hui. Liu Hui was a brilliant mathematician, whose comments are considered the most influential on the Nine Chapters (Rik and Keimpe, 2007). Liu, unlike other scholars provided proof of the rules in the Nine Chapters and elaborated on how the manual went about in getting the answers. He is also known to contribute to Sea Island mathematical Manual, which was initially anticipated as an extension of the Nine Chapters, but it independently became a work. In addition, father and son, Zu Chongzi and Zu Geng are among other notable mathematicians who made noteworthy comments on Liu Hui’s comments and pointed out inaccuracies in the Nine Chapters (Straffin, 1998). They contributed to the development of the Da Ming calendar, which was the best calendar at the time, and they were also able to calculate pi to the accuracy of 7 decimal points (Rik and Keimpe, 2007). Conclusion The Nine Chapters of Mathematical Art is one of the most dynamic mathematical manual in the antiquity of Chinese mathematics. It was written during the end of Han Dynasty with several scholars contributing to its existence. The manual comprises of 9 chapters which discuss and seek solution to life problems. Various commentaries were made on the Nine Chapters but the most notable were from Liu Hui. He provides proof of the existence of the Nine Chapters and the method used to reach the answers. In the mathematical world, the Nine Chapters are likened to Euclid’s in Greek mathematics. Its contribution is fundamental and applicable to date. Works Cited Merzbach, Uta. A history of mathematics. Hoboken, N.J: Wiley, 2011. O'Connor, J. and Robertson, E. Nine Chapters on the Mathematical, 2003 Art available Rik Brandenburg, and Keimpe Nevenzeely The Nine Chapters: History of Chinese Mathematics, 2007 available Straffin, Philip D. "Liu Hui and the First Golden Age of Chinese Mathematics," Mathematics Magazine 71, 3, (1998): 163–181. Read More

Pythagorean triples are studied using two problems in this chapter, while the remaining problems use the concept of similar triangles. History The whole designation of the Nine Chapters on Mathematical Art is presented on two bronze typical measures dated 179 CE although there are claims that, the book existed before in various titles. According to intellectuals, Chinese mathematics and ancient Mediterranean world mathematics had evolved more or less unconventionally up to the moment when the Nine Chapters reached it ultimate stage.

The approach of Chapter Seven came into play in Europe in the 13th century while that of Chapter Eight utilized Gaussian elimination long before Fredrick Gauss (1777-1855). Aspects of primeval Western mathematics are not reflected in antique China. The Nine Chapters were part of the Nine Arithmetic works of the Zhou Dynasty. The origin of the Nine Chapters is not clear, and it is anonymous. Liu Hui who wrote several and useful commentaries on this work however acknowledged ancient mathematicians; Zhang Cang and Geng Shouchang for their initial contribution and comments on the handbook, despite Han Dynasty archives not showing any authors and commenters (Straffin, 1998).

The establishment of Chinese mathematics is based on the amazing work of the Nine Chapters on Mathematical Art. This work was the first Chinese accords dedicated to mathematics. Its inspiration on development of mathematics can only be compared to the Euclid’s Elements of the Greek mathematics. Just like Euclid’s is coined as the foundation of the Western branch of mathematics, so is Chapter Nine, which is seen as the cornerstone of the Chinese mathematics. Besides, the Nine Chapters largely influenced the growth of primeval mathematics in Japan and Korea.

The Nine Chapters give arithmetical principals which focus on practical applications, presented in question and answer format. However, it is important to note that the arithmetical principles were presented in words, unlike today when they are expressed in algebraic notations. Had Chinese mathematicians had independent endurance of tradition, some of the remarkable anticipations of contemporary approaches might have significantly influenced the development of mathematics. Chinese culture was critically interfered by abrupt breaks (Merzbach, 2011).

For example in 213 BCE, the Chinese emperor authorized the burning and destruction of books, an international practice in times of political unrest. Nevertheless, some works may have been spared either through oral transmission or through the existence of other copies, and learning thus continued with emphasis put on mathematical problems of trade and the calendar (Merzbach, 2011). China seems to have had contact with both India and the West, but scholars argue over the extent and nature of borrowing.

For instance, the temptation to see Greek or Babylonian influence in China is faced with the idea that, the Chinese never adopted sexagesimal fractions. For a long time, Chinese numbers remained primarily decimal with notations rather remarkably variant from those in other regions (Merzbach, 2011). As a routine in the ancient China, scholars commented on conventional work. Thus, various comments were made on the Nine Chapters, for instance commentary by Liu Hui. Liu Hui was a brilliant mathematician, whose comments are considered the most influential on the Nine Chapters (Rik and Keimpe, 2007).

Liu, unlike other scholars provided proof of the rules in the Nine Chapters and elaborated on how the manual went about in getting the answers. He is also known to contribute to Sea Island mathematical Manual, which was initially anticipated as an extension of the Nine Chapters, but it independently became a work. In addition, father and son, Zu Chongzi and Zu Geng are among other notable mathematicians who made noteworthy comments on Liu Hui’s comments and pointed out inaccuracies in the Nine Chapters (Straffin, 1998).

They contributed to the development of the Da Ming calendar, which was the best calendar at the time, and they were also able to calculate pi to the accuracy of 7 decimal points (Rik and Keimpe, 2007).

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