Fairbank Center for Chinese Studies Harvard University The Evolution of Mathematics in Ancient China. From the newly discovered 數Shu and 算數書 Suan shu shu Bamboo Texts to the Nine Chapters on the Art of Mathematics Joseph W. Dauben Department of History Herbert H. Lehman College and Ph.D. Program in History The Graduate Center City University of New York Fairbank Center for Chinese Studies Harvard University The Evolution of Mathematics in Ancient China. From the newly discovered 數Shu and 算數書 Suan shu shu Bamboo Texts to the Nine Chapters on the Art of Mathematics 1934: 楚帛書 Chu boshu The Chu Silk Manuscript State of Chu, Warring States period Excavated: 1934/1942 Zidanku, Changsha radiocarbon date: 305±30 BCE Purchased: 1965 Arthur M. Sackler 數術 “Numerals and Arts/Skills” If […] and the length of the lunar months becomes too long or too short, then they will not fit the proper degree and spring, summer, autumn, and winter will [not] be […] regular; the sun, moon, and planets will erratically overstep their paths. When (the months) are too long, too short, contrary, or chaotic, (the growth of) the grasses and trees have no regularity. This is [called] yao, “demonic” (influences or omens). When heaven and earth create calamities, the Heaven’s Cudgel (Tianpou) star creates (sweeping) destruction, sending (the destruction) down through all four regions (of the earth). Mountains collapse, springs gush forth geysers. This is called “contravention.” If you contravene the years (and) the months, then upon entering the seventh or eighth day of the month there will be fog, frost, and clouds of dust, and you will not be able to function according (to heaven’s plan) [Li and Cook 1999: 174] 1972–1974: 馬王堆 Mawangdui , Changsha, Hunan (186-168 BCE) Excavated: 1972 to 1974 Three Western Han dynasty tombs: Li Cang, chancellor to the prince of Changsha State and the Marquis of Dai; Li Cang’ wife, the Marquise of Dai; and their son. Tomb No. 1 Tomb No. 2 Tomb No.3 Wooden Tablet with Burial Date Length 30cm Unearthed from Tomb No.3 “On the first Wuchen day of Yisi, the second month of the twelfth year, the chamberlain in charge of funeral services, holding a memorial with funeral articles, presents a list of the delivery for inspection.” i.e. the twelfth year in the reign of Emperor Wendi of Han (168 BCE) Wooden Outer Coffin from Tomb No.1 Length: 672cm Width: 488cm Height: 280cm Coffin with Painted Design on Black Lacquer Coating Length 256cm width 118cm height 114cm Unearthed from Tomb No.1 Second Inner Coffin with Painted Design on Vermilion Lacquer Coating Length 230cm width 92cm height 89cm Innermost Coffin Decorated with Brocade Length:202cm Width:69cm Height:63cm Xin Zhui Marquise of Dai Scene of Banquet on the T-shaped Paintings on Silk Book of Changes (Zhou Yi ) Length 30cm width 21.5cm Unearthed from Tomb No.3 “Harmony of Yin and Yang” (He Yin Yang) Drawing of Daoyin, a Physical Exercise Chart Length 100cm width 50cm (Tomb No. 3) 44 Daoyin positions are illustrated, such as breathing exercises, limb stretching and gymnastics, and each position is labeled with its particular term and function. The exercise of Five Fowls was created by Hua Tuo, a famous doctor in the Eastern Han Dynasty and was based on Dao-yin. “Ten Questions” (十問 Shi Wen) Tomb No.3 Prescriptions for 52 Diseases (Wu Shi Er Bing Fang) Guard of Honor Length 219cm width 99cm Unearthed from Tomb No.3 A painting on silk was found on the western wall of the coffin chamber. At the top left: the tomb owner Divination by Five Stars (Wu Xing Zhan) 天文氣象雜占 Divination by Astrological and Meteorological Phenomena Excavated: 1973 Mawangdui Western Han (202 BCE – 9 CE) 29 comets (彗星 hui xing) recorded over a period of about 300 years. ca. 223 BCE 1975: 睡虎地秦 简 Shuihudi Qin jian Qin Bamboo Slips from Shuihudi Excavated: 1975 Tomb #11 at 城關睡虎地 Shuihudi, Yunmeng County, Hubei Province The tomb belonged to a Qin administrator. From Shuihudi: Bamboo strips written with documents of law; Painted lacquer spoon with wooden body in the shape of phoenix; kitchen utensil (bottom, height 13.3 cm); Flattened lacquer pot with colored painting of ox and horses: wine vessel or water vessel (in background), height 22.8 cm. 1983: 張家山漢簡 Zhangjiashan Han jian (Han Bamboo Slips from Zhangjiashan) 202–186 BCE Excavated: 1983 Tomb No. 247, Zhangjiashan Jiangling County, Hubei Province 算數書 Suan Shu Shu Wen Wu (2000) 算數書 Suan Shu Shu (A Book on Numbers and Computations) Peng Hao, ed. Beijing: Science Press, 2001. The first of the bamboo slips shown here on the right, with the title of the book clearly shown in the uppermost part of the right most slip: Suan Shu Shu. Suan Shu Shu, Peng Hao, ed. Beijing: Science Press, 2001. Title of the book: 算數書 Suan Shu Shu (A Book on Numbers and Computations) 筭 suan Suan Shu Shu, Peng Hao, ed. Beijing: Science Press, 2001. 1993: 郭店楚簡 Guodian chujian Guodian Chu Bambo Slips Excavated: 1993 Guodian, Jingmen, Hubei Warring States period (mid-4th –early 3rd-century BCE). Tu Weiming, Yenching Institute, Harvard University Guo Yi discussing the Tao Te Ching (1993) 1996: 走馬樓簡牘 Zoumalou jiandu Zoumalou Bamboo Documents Census count Wooden tablets unearthed from Ancient Well No. J22 Zoumalou, Changhsha, Hunan Wu Kingdom (222-280) of the Three Kingdoms (220-280) Excavated: 1996 2002: 里耶古城秦代 简牍 Liye gucheng Qindai jiandu Liye Ancient City Qin Dynasty Bamboo Documents Multiplication table Liye, Longshan County, Hunan “Four times eight is thirty-two; five times eight is forty…” Excavated: April, 2002 Qin Dynasty (221 BCE-206 BCE) Length 22cm width 4.5cm Unearthed at Liye city site, Longshan in 2002 2007: 嶽麓書院藏秦簡 Yuelu shuyuan cang qinjian Qin Slips Collected by the Yuelu Academy 嶽麓書院 Founded in 976 in the 9th year of the Song Dynasty during the reign of Emperor 開寶 Kaibao . Zhu Xi and . Zhang Shi . Emperor Song Taizu 宋太祖 Figure 1, p. 77 Figure 2, p. 78 Figure 3, p. 79 Figures 11 and 12, p. 83 Figure 13, p. 83 2008: 清华简 Qinghua jian Tsing-Hua Bamboo Slips Booking a Place in History: Rare ancient Chinese bamboo books dating back more than 2,000 years come home By ZAN JIFANG NO.48 NOV.27, 2008 Tsinghua Warring States Bamboo Strips Professor 李学勤 Li Xueqin 战国竹简重回故土 发布时间： 2008-10-30 13:21 来源：文摘报 近日，清华大学宣布，2100枚战国时期的竹简入藏清华，它 是由校友赵伟国从境外拍卖会买到后捐赠给清华的。 经过专家组鉴定，这批“清华简”属于战国中晚期，距 今大约2300~2400年左右，出土于古代的楚国境内。 “清华简”最长的有46厘米，最短的不到10厘米。“10 厘米就相当于我们现在的‘口袋书’了，这在 以前从未发现 过。” 战国竹简重回故土 发布时间： 2008-10-30 13:21 来源：文摘报 近日，清华大学宣布，2100枚战国时期的竹简入藏清华，它 是由校友赵伟国从境外拍卖会买到后捐赠给清华的。 经过专家组鉴定，这批“清华简”属于战国中晚期，距 今大约2300~2400年左右，出土于古代的楚国境内。 “清华简”最长的有46厘米，最短的不到10厘米。“10 厘米就相当于我们现在的‘口袋书’了，这在 以前从未发现 过。” Conservation of the 2388 Warring States Bamboo Strips from Chu at the Tsinghua University Center for Unearthed Documents Research and Protection The strips include invaluable texts such as “Bao Xun”-- the last words of King Wen of the Zhou Dynasty to his son, and some other important contents, including a multiplication table. Tsinghua University Center for Unearthed Documents Research and Protection Peking University Qin Bamboo Slips June 14, 2012 Sackler Museum of Art and Archaeology, Peking University Guo Han Feng Shuchun Wei Lisheng 北大秦简中的数学文献 韩巍（北京大学中国古代史研究中心） 北大秦简中的数学文献 Mathematical Documents among the Qin Bamboo Strips at Peking University 韩巍（北京大学中国古代史研究中心） Han Wei (Peking University Research Center for Ancient Chinese History) 日書 Ri shu Comparable illustration in black ink only found in the “Day Book” from Shuihudi tomb 11. slips 150r1-154r1 [KALINOWSKI 2008] 湖南大學 嶽麓書院 嶽麓書院藏秦簡《數》書釋文·簡注 計算田地產量或租稅的算題 嶽麓書院藏秦簡整理小组（执筆：肖 燦 ） 1. 說 明 2. 计算土地面积的算題 《数》书經整理得到： 一，有編號的簡共二百三十余枚。 谷物换算 殘片二十余片，無編號。 3. 二，有完整算题75例。（现存题设条件和问题或答案能依据简 文列出算法式的算题就视为完整算题）。 衰分算題 又有单独成題的“术”17例。 4. 又有记录有谷物体积重量比率、兑换比率的簡32枚， 未計入算題數目。 贏不足算題 湖南大學 又有記錄量制的簡3枚，未計入算題數目。 嶽麓書院 5. 凡 例 6. 少廣算题 一 釋文中異體字、假借字一般隨文注出，外加（ ）號。 二 簡文原有錯字，一般在釋文中隨注正字，外加 計算體積的算題 〈 〉號。原有脫字或衍文，釋文不加更動， 7. 竹簡殘斷，不能復原。 在注釋中說明。 三 簡文原有殘缺字，可據殘筆或文例補足的， 8. 勾股算题 外加〔〕 號。不能辨識的殘缺字，用 □ 表示， 每字一格。其他符號： 營軍之術（此算題暫時未歸類） ……字跡模糊，字數不能確定。 9. 10. 量制 湖南大學 嶽麓書院 嶽麓書院藏秦簡《數》書釋文·簡注 計算田地產量或租稅的算題 嶽麓書院藏秦簡整理小组（执筆：肖 燦 ） 1. 說 明 2. 计算土地面积的算題 《数》书經整理得到： 一，有編號的簡共二百三十余枚。 谷物换算 殘片二十余片，無編號。 3. 二，有完整算题75例。（现存题设条件和问题或答案能依据简 文列出算法式的算题就视为完整算题）。 衰分算題 又有单独成題的“术”17例。 4. 又有记录有谷物体积重量比率、兑换比率的簡32枚， 未計入算題數目。 贏不足算題 湖南大學 又有記錄量制的簡3枚，未計入算題數目。 嶽麓書院 5. 凡 例 6. 少廣算题 一 釋文中異體字、假借字一般隨文注出，外加（ ）號。 二 簡文原有錯字，一般在釋文中隨注正字，外加 計算體積的算題 〈 〉號。原有脫字或衍文，釋文不加更動， 7. 竹簡殘斷，不能復原。 在注釋中說明。 三 簡文原有殘缺字，可據殘筆或文例補足的， 8. 勾股算题 外加〔〕 號。不能辨識的殘缺字，用 □ 表示， 每字一格。其他符號： 營軍之術（此算題暫時未歸類） ……字跡模糊，字數不能確定。 9. 10. 量制 湖南大學 嶽麓書院 嶽麓書院藏秦簡《數》書釋文·簡注 計算田地產量或租稅的算題 嶽麓書院藏秦簡整理小组（执筆：肖 燦 ） 1. 說 明 2. 计算土地面积的算題 《数》书經整理得到： 一，有編號的簡共二百三十余枚。 谷物换算 殘片二十余片，無編號。 3. 二，有完整算题75例。（现存题设条件和问题或答案能依据简 文列出算法式的算题就视为完整算题）。 衰分算題 又有单独成題的“术”17例。 4. 又有记录有谷物体积重量比率、兑换比率的簡32枚， 未計入算題數目。 贏不足算題 湖南大學 又有記錄量制的簡3枚，未計入算題數目。 嶽麓書院 5. 凡 例 6. 少廣算题 一 釋文中異體字、假借字一般隨文注出，外加（ ）號。 二 簡文原有錯字，一般在釋文中隨注正字，外加 計算體積的算題 〈 〉號。原有脫字或衍文，釋文不加更動， 7. 竹簡殘斷，不能復原。 在注釋中說明。 三 簡文原有殘缺字，可據殘筆或文例補足的， 8. Gou-Gu Problems 外加〔〕 號。不能辨識的殘缺字，用 □ 表示， 每字一格。其他符號： 營軍之術（此算題暫時未歸類） ……字跡模糊，字數不能確定。 9. 10. 量制 □有圓材薶（埋）地，不智（知）小大，斲之，入材一寸而 得平一尺，問材周大幾可（何）。即曰，半平得五寸，令相 乘也，以深 [0304] 一寸为法，如法得一寸，有（又）以深益之，即材徑也。 [0457] □有圓材薶（埋）地，不智（知）小大，斲之，入材一寸而 得平一尺，問材周大幾可（何）。即曰，半平得五寸，令相 乘也，以深 [0304] 一寸为法，如法得一寸，有（又）以深益之，即材徑也。 [0457] Suppose there is a circular [piece of] wood buried in the ground, whose size is unknown, but cutting to a depth of 1 cun gives a chord of 1 chi, it is asked how great is the circumference of the [circular piece of] wood? [The method] says: half the chord is 5 cun, multiply it by itself and using the depth of 1 cun as the divisor, dividing gives the result in cun, again adding the depth [of the cut] gives the diameter of the wood.