Ancient Chinese Mathematics (a very short summary) • Documented civilization in China from c.3000 BCE. Un- til c.200 CE, ‘China’ refers to roughly to the area shown in the map: north of the Yangtze and around the Yellow rivers. • Earliest known method of enumeration dates from the Shang dynasty (c.1600–1046 BCE) commensurate with the earliest known oracle bone script for Chinese char- acters. Most information on the Shang dynsaty comes from commentaries by later scholars, though many orig- inal oracle bones have been excavated, particularly from Anyang, its capital. • During the Zhou dynasty (c.1046–256 BCE) and especially towards the end in the Warring States period (c.475–221 BCE) a number of mathematical texts were written. Most have been lost but much content can be ascertained from later commentaries. • The warring states period is an excellent example of the idea that wars lead to progress. Rapid change created pressure for new systems of thought and technology. Feudal lords employed itinerant philosophers1) The use of iron in China expanded enormously, leading to major changes in warfare2 and commerce, both of which created an increased need for mathematics. Indeed the astronomy, the calendar and trade were the dominant drivers of Chinese mathematics for many centuries. • The warring states period ended in 221 BCE with the victory of victory of the Qin Emperor Shi Huang Di: famous for commanding that books be burned, rebuilding the great walls and for being buried with the Terracotta Army in Xi’an. China was subsequently ruled by a succession of dynasties until the abolition of the monarchy in 1912. While this simple description might suggest a long calm in which Chinese culture and technology could develop in comfort, in reality the empire experienced its fair share of rebellions, schisms and reprisals. The empire of the Qin dynasty was tiny in comparison to the extent of modern China, and its expansion did not happen smoothly or without reversals. • East Asia (modern day China, Korea, Japan, etc.) is geographically isolated from the rest of the world and, in particular, from other areas of developing civilization. To the north is Siberia, to the west the Gobi desert, southwest are the Himalaya and to the south is jungle. During the Han dynasty (c.200 BCE–220 CE) a network of trading routes known as the silk road connected China, India and, through Persia, Europe. Indeed part of the purpose of the Great Wall was the protection of these trade routes. Knowledge moved more slowly than goods, and there is very little evidence of mathematical and philosophical ideas making the journey until many 1Confucius was one such and served for a period as an advisor to Lu, a vassal state of the Zhou. He died at the beginning of the warring states period, around 479 BCE, but his followers turned his teachings into a major philosophical school of thought Confucianism. Its primary emphasis was stability and unity as a counter to turmoil. The other major philosophical system dating from this time is Taoism which is more comfortable with change and adaptation. 2Sun Tzu’s military classic The Art of War dates from this time. centuries later. For instance, there is no evidence of the Chinese using sexagesimal notation to aid their calculations, which meant that essentially none of Babylonian and Greek work on astronomy made it to China. Similarly, there are many mathematical ideas which saw no ana- logue in the west until many centuries after Chinese mathematicians had invented them. It seems reasonable to conclude that Chinese and Mediterranean mathematics developed essen- tially independently. Important Mathematical Texts The oldest suspected3 mathematical text is the Zhou Bi Suan Jing (The Mathematical Classic of the Zhou Gnomon4 and the Circular Paths of Heaven). It was probably compiled some time in the period 500–200 BCE. The text contains, arguably, the earliest statement of Pythagoras’ Theorem as well as simple rules for computing fractions and conducting arithmetic. The book was largely concerned with astronomical calculations and presented in the form of a dialogue between the 11th century BCE Duke of Zhou5 and Shang Gao (one of his ministers, and a skilled mathematician). As with other early texts, there is no rigorous notion of proof or axiomatics, merely strong assertions or ‘ob- vious’ appeals to pictures. For instance, the ‘proof’ of Pythagoras’ Theorem is purely pictorial: taking the sides of the triangle to be a, b, c in increasing order of length, the picture essentially claims that 1 c2 = 4 · ab + (b − a)2 2 giving the familiar result when multiplied out. Of course the Chinese do not attribute this result to Pythagoras! Instead it is known as the gou gu where these words refer first to the shorter and then the longer of the two non-hypotenuse sides of the triangle. There are several other early works; here are two of the most important. Suanshu Shu (A Book on Arithmetic) Compiled c.300–150 BCE, covering, amongst other topics, frac- tions, the areas of rectangular fields, and the computation of fair taxes. Jiu Zhang Suan Shu (Nine Chapters on Mathematical Arts) Written c.300 BCE–200 CE. Many top- ics are covered, including square roots, working with ratios (false position and the rule of three6), simultaneous equations, areas/volumes, right-angled triangles, etc. The Nine Chapters is a hugely influential text, in no small part due to the creation of a detailed commentary and solution manual to its 246 problems written by the mathematician Liu Hui in 263 CE. 3Ancient Chinese texts are extremely difficult to date: we have no original copies and most seem to have been compiled over several hundred years. Most of what we know of these texts is in the form of commentaries by later authors. 4Gnomon: “One that knows or examines” — strictly the elevated piece of a sun/moondial which would have been used for measuring said circular paths of heaven. 5Credited with writing the I Ching, the ‘classic of changes.’ 6 bc Given equal ratios a : b = c : d, where a, b, c known, then d = a . 2 The style of both text consists of laying out methods of solution which have wide application, rather than on proving that a particular method is guaranteed to work. Indeed there is no notion of ax- iomatics on which one could construct a proof in the modern sense of the word. Liu made other contributions to mathematics, including accurate estimates of p made similarly to Archimedes. He made particu- Ao lar use of the out-in principle which essentially describes how to Bi compare areas and volumes: Co 1. Areas and volumes are invariant under translation. 2. If a figure is subdivided, the sum of the areas/volumes of Bo the parts equals that of the whole. C A For instance, Liu gave the argument shown in the picture as an i i alternative proof of the gao gu: the green square is subdivided and the in pieces Ai, Bi, Ci translated to new positions Ao, Bo, Co to assemble the required squares. Liu extended the out-in principle to analyze solids, comparing the volumes of four basic solids: • Cube (lifang) • Right triangluar prism (qiandu) • Rectangular pyramid (yangma) • Tetrahedron (bienuan) These could be assembled to calculate the volume of, say, a truncated pyramid: The Bamboo Problem This problem is taken from the Nine Chapters: indeed the picture shows the problem as depicted in Yang Hui’s famous Analysis of the Nine Chapters from 1261. A bamboo is 10 chi high. It breaks and the top touches the ground 3 chi from the base of the stem. What is the height of the break? In modern language, if a, b, c is the triangle, we know that b + c = 10 and a = 3. We want b. 1 a2 91 The solution is b = b + c − = chi: think about why. 2 b + c 100 3 Chinese Enumeration The Chinese had two parallel systems of enumeration. Both are essentially decimal. Oracle Bone Script and Modern Numerals The earliest Chinese writing is known as oracle bone script dates from around 1600 BCE. The numbers 1–10 were recorded with distinct symbols, with extra symbols for 20, 100, 1000 and 10000. These were decorated to denotes various multiples. Some examples are shown below. The system is quite complex, given all the possibilities for decoration, and therefore more advanced than other contemporary systems. Standard modern numerals are a direct descendant of this script: Observe the similarity between the expressions for the first 10 digits. The second image denotes the number 842, where second and 4th symbols represent 100’s and 10’s respectively: literally eight hundred four ten two. No zero symbol is required as a separator: one could not confuse 205 with 250. The system is still partly positional: the symbol for, say, 8 can mean 800 if placed correctly, but only if followed by the symbol for 100. Rod Numerals The second dominant form of enumeration dates from around 300 BCE and was in very wide use by 300 CE. Numbers were denoted by patterns known as Zongs and Hengs. These represented alternate powers of 10: Zongs denoted units, 100’s, 10000’s, etc., while Hengs were for 10’s, 1000’s, 100000’s, etc. Rod numerals were immensely practical. In extremis they could easily be scratched in the dirt. More commonly they were created using short bamboo sticks or counting rods, of which any merchant worth their salt would carry a bundle. They were often used in conjunction with a counting board: a grid of squares on which sticks could be placed for ease of calculation.