Mathematics in Ancient China
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EVARISTE GALOIS the Long Road to Galois CHAPTER 1 Babylon
A radical life EVARISTE GALOIS The long road to Galois CHAPTER 1 Babylon How many miles to Babylon? Three score miles and ten. Can I get there by candle-light? Yes, and back again. If your heels are nimble and light, You may get there by candle-light.[ Babylon was the capital of Babylonia, an ancient kingdom occupying the area of modern Iraq. Babylonian algebra B.M. Tablet 13901-front (From: The Babylonian Quadratic Equation, by A.E. Berryman, Math. Gazette, 40 (1956), 185-192) B.M. Tablet 13901-back Time Passes . Centuries and then millennia pass. In those years empires rose and fell. The Greeks invented mathematics as we know it, and in Alexandria produced the first scientific revolution. The Roman empire forgot almost all that the Greeks had done in math and science. Germanic tribes put an end to the Roman empire, Arabic tribes invaded Europe, and the Ottoman empire began forming in the east. Time passes . Wars, and wars, and wars. Christians against Arabs. Christians against Turks. Christians against Christians. The Roman empire crumbled, the Holy Roman Germanic empire appeared. Nations as we know them today started to form. And we approach the year 1500, and the Renaissance; but I want to mention two events preceding it. Al-Khwarismi (~790-850) Abu Ja'far Muhammad ibn Musa Al-Khwarizmi was born during the reign of the most famous of all Caliphs of the Arabic empire with capital in Baghdad: Harun al Rashid; the one mentioned in the 1001 Nights. He wrote a book that was to become very influential Hisab al-jabr w'al-muqabala in which he studies quadratic (and linear) equations. -
Hong Jeongha's Tianyuanshu and Zhengcheng Kaifangfa
Journal for History of Mathematics http://dx.doi.org/10.14477/jhm.2014.27.3.155 Vol. 27 No. 3 (June 2014), 155–164 Hong JeongHa’s Tianyuanshu and Zhengcheng Kaifangfa 洪正夏의 天元術과 增乘開方法 Hong Sung Sa 홍성사 Hong Young Hee 홍영희 Kim Young Wook* 김영욱 Tianyuanshu and Zengcheng Kaifangfa introduced in the Song–Yuan dynasties and their contribution to the theory of equations are one of the most important achieve- ments in the history of Chinese mathematics. Furthermore, they became the most fundamental subject in the history of East Asian mathematics as well. The opera- tions, or the mathematical structure of polynomials have been overlooked by tra- ditional mathematics books. Investigation of GuIlJib (九一集) of Joseon mathemati- cian Hong JeongHa reveals thatQ Hong’s approach to polynomials is highly struc- n tural. For the expansion of k=1(x + ak), Hong invented a new method which we name Hong JeongHa’s synthetic expansion. Using this, he reveals that the pro- cesses in Zhengcheng Kaifangfa is not synthetic division but synthetic expansion. Keywords: Hong JeongHa, GuIlJib, Hong JeongHa’s synthetic expansion, Tianyuan- shu, Structure of polynomials, Binomial coefficients, Zhengcheng Kaifangfa, Shisuo Kaifangfa; 洪正夏, 九一集, 洪正夏의 組立展開, 天元術, 多項式의 構造, 二項係數, 增乘開 方法, 釋鎖開方法. MSC: 01A13, 01A25, 01A45, 01A50, 12–03, 12E05, 12E12 1 Introduction The theory of equations in Eastern mathematics has as long a history as that in the West and divides into two parts, namely constructing equations and solving them. For the former, Tianyuanshu (天元術) was introduced in the early period of the Song dynasty (960–1279) and then extended up to Siyuanshu (四元術) to repre- sent polynomials of four indeterminates by Zhu Shijie (朱世傑) in his Siyuan Yujian (四元玉鑑, 1303). -
On Mathematical Symbols in China* by Fang Li and Yong Zhang†
On Mathematical Symbols in China* by Fang Li and Yong Zhang† Abstract. When studying the history of mathemati- cal symbols, one finds that the development of mathematical symbols in China is a significant piece of Chinese history; however, between the beginning of mathematics and modern day mathematics in China, there exists a long blank period. Let us focus on the development of Chinese mathematical symbols, and find out the significance of their origin, evolution, rise and fall within Chinese mathematics. The Origin of the Mathematical Symbols Figure 1. The symbols for numerals are the earliest math- ematical symbols. Ancient civilizations in Babylonia, derived from the word “white” (ⱑ) which symbolizes Egypt and Rome, which each had distinct writing sys- the human’s head in hieroglyphics. Similarly, the Chi- tems, independently developed mathematical sym- nese word “ten thousand” (ϛ) in Oracle Bone Script bols. Naturally, China did not fall behind. In the 16th was derived from the symbol for scorpion, possibly century BC, symbols for numerals, called Shang Or- because it is a creature found throughout rocks “in acle Numerals because they were used in the Oracle the thousands”. In Oracle Script, the multiples of ten, Bone Script, appeared in China as a set of thirteen hundred, thousand and ten thousand can be denoted numbers, seen below. (The original can be found in in two ways: one is called co-digital or co-text, which reference [2].) combines two single figures; another one is called Oracle Bone Script is a branch of hieroglyphics. analysis-word or a sub-word, which uses two sepa- In Figure 1, it is obvious that 1 to 4 are the hiero- rate symbols to represent a single meaning (see [5]). -
PHD Thesis in Islamic Studies.Pdf
INTERNATIONAL ISLAMIC UNIVERSITY ISLAMABAD FACULTY OF USULUDIN A Comparative Study of the Practice of Integrating Chinese Traditions Bewteen Hui Hui Muslim Scholars and Jesuits Missionaries during 1600A.D.- 1730A.D. Supervised By: Prof Dr Anis Ahamd Submitted By: Wu Juan (AISHA) Reg. No.98-FU/PHD/S08 In partial gulfillment of the award of Ph.D degree in Comparative Religion Department of Comparative Religion Ramadan 1440A.H./ June,2017 2 COMMITTEE OF EXAMINERS 1. External Examiner: ...................................... .......................... Name Signature 2. Internal Examiner: ...................................... .......................... Name Signature 3. Supervisor: ...................................... .......................... Name Signature The Viva-voce of this thesis took place on ........... Day.............................. Month.............Year A. H. which is the same as ........... Day .............................. Month .............Year A. D. Contexts AKNOWLEDGEMENT………………………………….............................. 5 3 PREFACE……………………………………………..................................7 INTRODUCTION………………………………….....................................8 Chapter 1: Jesuits Missionaries in China …………...........................15 Chapter2: Main Books of Jesuits in Han Chinese……...................... 20 Chapter3: Islam and Hui Hui Muslims in China……......................... 24 Chapter4: Hui Hui Scholars’ Han Kitab Literature…......................... 34 PART ONE:EVIDENCING RELIGIOUS FAITH………........................ 50 Chapter1: The -
The Assault on the Quintic Niels Henrik Abel
The Assault on the Quintic Niels Henrik Abel • Niels Henrick Abel was born in 1802 to a Luther Minister and the daughter of a shipping merchant. • He came from a long line of pastors; she was a beautiful woman with a passion for the finer things. Niels Henrick Abel • Poverty and alcoholism haunted the family; at the same time, famine haunted Norway. • Abel attended Cathedral school is Oslo and along with many other students was beaten by the mathematics teacher. Abel • Being with his friends and going to the theater became Abel’s salvation. • Bernt Michael Holmboe replaced the sadistic math teacher and recognized Abel’s talent. They remained friends throughout Abel’s life. Abel • During these high school years he thought he’s solved the general quintic. He submitted the proof and was asked to better explain it. In trying to do so, he found his mistake. Abel • His father died in 1820, making even worse the poverty that was to remain with him all his life. Somehow, in spite of this poverty, he entered the university in 1821. • During this time, two professors supported him, and even helped him obtain funds for travel to Denmark to meet other mathematicians. • During this trip he met his future fiancée Crelly Kemp; they were engaged in 1824. Abel • He wrote a proof that the quintic was not solvable by radicals in 1823. Understanding its importance, he wrote a very brief version of the proof and used it as a sort of “business card” to send to mathematicians. • The one he sent to Gauss was never opened, and was found among Gauss’s papers after his death. -
View This Volume's Front and Back Matter
i i “IrvingBook” — 2013/5/22 — 15:39 — page i — #1 i i 10.1090/clrm/043 Beyond the Quadratic Formula i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page ii — #2 i i c 2013 by the Mathematical Association of America, Inc. Library of Congress Catalog Card Number 2013940989 Print edition ISBN 978-0-88385-783-0 Electronic edition ISBN 978-1-61444-112-0 Printed in the United States of America Current Printing (last digit): 10987654321 i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page iii — #3 i i Beyond the Quadratic Formula Ron Irving University of Washington Published and Distributed by The Mathematical Association of America i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page iv — #4 i i Council on Publications and Communications Frank Farris, Chair Committee on Books Gerald M. Bryce, Chair Classroom Resource Materials Editorial Board Gerald M. Bryce, Editor Michael Bardzell Jennifer Bergner Diane L. Herrmann Paul R. Klingsberg Mary Morley Philip P. Mummert Mark Parker Barbara E. Reynolds Susan G. Staples Philip D. Straffin Cynthia J Woodburn i i i i i i “IrvingBook” — 2013/5/22 — 15:39 — page v — #5 i i CLASSROOM RESOURCE MATERIALS Classroom Resource Materials is intended to provide supplementary class- room material for students—laboratory exercises, projects, historical in- formation, textbooks with unusual approaches for presenting mathematical ideas, career information, etc. 101 Careers in Mathematics, 2nd edition edited by Andrew Sterrett Archimedes: What Did He Do Besides Cry Eureka?, Sherman Stein Beyond the Quadratic Formula, Ronald S. -
The Frederick Mccormick Korean Collection This Inventory Is Created Based on the Following Two Sources
The Frederick McCormick Korean Collection This inventory is created based on the following two sources. The numeric numbers follow the list numbers in Judith Boltz’ 1986 inventory. 1. 1986 inventory of the McCormick collection, by Judith Boltz. 2. Frederick McCormick Collection in the Special Collections in the Claremont Colleges Library, the Overseas Korean Cultural Heritage Foundation, 2015. I. CHINESE CLASSICS, EDITIONS AND STUDIES A: THE BOOK OF HISTORY 1. (NLK Catalog #: 1) 서전대전 | Sŏjŏn taejŏn | 書傳大全 Complete Commentaries on the Book of History 1820, 10 vols., Woodblock Part of Complete Collection of the Five Classics (五經大全) complied by Hu Guang at the order of Emperor Yongle (永樂, r. 1402-1424) of China’s Ming Dynasty. This is a collection of scholastic discourses based on Collected Commentaries on the Book of History compiled by Cai Shen. (蔡沈, 1167-1230) 2. (NLK Catalog #: 2) 서전언해 | Sŏjŏn ŏnhae | 書傳諺解 Commentary on the Book of History 1820, 5 vols., Woodblock A vernacular translation of the Book of History (書經) that provides pronunciations and syntax in Korean throughout the original Chinese text. This is part of Vernacular Translations of Four Books and Three Classics (書經諺解) published by the order of King Seonjo (宣祖) of Joseon. B. THE BOOK OF ODES 3. (NLK Catalog #: 3) 시전대전 | Sijŏn taejŏn | 詩傳大全 Complete Commentaries on the Book of odes 1820, 10 vols., Woodblock Part of Complete Collection of the Five Classics (五經大全) that complies commentaries based on Collected Commentaries on the Book of Odes (詩集傳) published by Zhu Xi (朱熹, 1130-1200) of China’s Song Dynasty. 4. (NLK Catalog #: 4) 시경언해 | Sigyŏng ŏnhae | 詩經諺解 Commentary on the Book of Odes 1820, 7 vols., Woodblock A vernacular translation of the Book of Odes (詩經) that provides pronunciation and syntax in Korean throughout the original Chinese text. -
Ancient Chinese Mathematics (A Very Short Summary)
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. -
One Quadratic Equation, Different Understandings: the 13Th Century Interpretations by Li Ye and Later Commentaries in the 18Th and 19Th Centuries
Journal for History of Mathematics http://dx.doi.org/10.14477/jhm.2017.30.3.137 Vol. 30 No. 3 (Jun. 2017), 137–162 One Quadratic Equation, Different Understandings: the 13th Century Interpretations by Li Ye and Later Commentaries in the 18th and 19th Centuries Charlotte Pollet Ying Jia-Ming* The Chinese algebraic method, the tian yuan shu, was developed during Song pe- riod (960–1279), of which Li Ye’s works contain the earliest testimony. Two 18th century editors commentated on his works: the editor of the Siku quanshu and Li Rui, the latter responding to the former. Korean scholar Nam Byeong-gil added an- other response in 1855. Differences can be found in the way these commentators considered mathematical objects and procedures. The conflicting nature of these commentaries shows that the same object, the quadratic equation, can beget differ- ent interpretations, either a procedure or an assertion of equality. Textual elements in this paper help modern readers reconstruct different authors’ understandings and reconsider the evolution of the definition of the object we now call ‘equation’. Keywords: algebra, quadratic equation, Song China, Qing China, Joseon Korea MSC: 01A13, 01A25, 01A35, 01A50, 01A55, 12–03 1 Introduction It is well-known to sinologists that our present knowledge of Song dynasty (960– 1279) mathematics derives from reconstructions made by Qing dynasty editors [24]. This time period is known for its evidential studies particularly in the fields of philol- ogy, phonology and exegesis, and for the editions of Classics in order to republish ancient Chinese texts. Some editors, convinced of the Chinese origin of algebra, used philological techniques to recover lost materials and restore the roots of ‘Chi- nese mathematics’ [15, 30]. -
KUIL, Vol. 26, 2003, 429-474 the HISTORY of CHINESE
KUIL, vol. 26, 2003, 429-474 THE HISTORY OF CHINESE MATHEMATICS: THE PAST 25 YEARS ANDREA EBERHARD-BRÉARD REHSEIS (CNRS), France • Ph. D. Program in History • The Graduate Center, CUNY JOSEPH W. DAUBEN Ph. D. Program in History • The Graduate Center, CUNY XU YIBAO Ph.D. Program in History • The Graduate Center, CUNY RESUMEN ABSTRACT El presente artículo presenta los más This paper presents the major accom- importantes logros de la investigacián plishments of research over the past quar- sobre historia de las matemáticas chinas en ter century in the field of Chinese mathe- el ŭltimo cuarto del siglo XX. Comienza matics and its history. We begin with a con una breve panordmica sobre el estado brief overview of the progress of our de conocimientos y las principales figuras knowledge of that history, and the major que realizaron las prirneras contribuciones figures who contributed the fundamental fundamentales antes de 1975 para después early works prior to 1975, and then exam- examinar más detenidamente las aporta- ine more carefully the achievements of the ciones Ilevadas a cabo durante el ŭltimo past quarter century, beginning with a cuarto de siglo: autores europeos, publica- general overview of the subject before ciones en inglés y, finalmente, la extraor- surveying in more detail the contributions dinaria produccián en chino, en su mayor made in the past twenty-five years, first parte tras la Revolución Cultural. by Europeans, then by scholars publishing in English, after which this survey turns to examine the extraordinary production of scholarship written in Chinese, most of it since the end of the Cultural Revolution. -
A Case Study of the Duoji Method and Its Development
EASTM 20 (2003): 45-72 The Westernization of Chinese Mathematics: A Case Study of the duoji Method and its Development Tian Miao /Tian Miao is Associate Professor of History of Science at the Institute for the History of Natural Sciences, Chinese Academy of Science. Her research covers the history of mathematics in seventeenth- to nineteenth-century China. Recent publications include "Qingmo shuxue jiaoyu dui Zhongguo shuxue zhiyehua de yinxiang" m5K Wl + #1!.. ~ 31:1 c:p 00 Wl + ~R ~ 1t El~ :irJ Ufa] (The Impact of the Development of Mathematics Education on the Professionalization of Chi nese Mathematicians in Late Qing China) (Ziran kex.ueshi yanjiu El ~ 'f4 + _§e_ jiff ~ (Studies in the History of Natural Sciences), /998), "Qingmo shuxue jiaoshi de goucheng de tedian" m5K Wl + #1!.. jffi ITT ,tt] fflG ~ ,9- (A Study on 1he Formation of Mathematical Teachers in the Late Qing Dynasty) (Zhongguo keji shiliao c:p 00 N t:5Z _§e_ fl (Historical Materials of Science and Technology), 1998), "Jiegenfang, Tian yuan and Daishu: Algebra in Qing China" (Historia Scientiarum, /999), "Siyuan yujian de Qingdai banben Ji Jialing sicao de jiao /.:an yanjiu" iz:g 5t .=E ~ ITT mft fJ§_ * Ez ® 1;- iz:g "J/i- ITT ~ WI iiff Ji: (Textual Criticism Research on the Different Versions of the Siyuan Yujian and the /'roof~ for the Jialing sicao during the Qing Dynasty) (Ziran kexueshi yanjiu, /999).J * * * The duoji ±~ fl (lit., "summing piles") method of calculating the sum of a given pile is a major subject in traditional Chinese mathematics. After the sixteenth n.:ntury, Chinese mathematics failed to keep pace with Western mathematics, but Ilic duoji method is one in which the late Qing mathematicians made advances over their Western colleagues. -
History of Algebra
History of Algebra The term algebra usually denotes various kinds of mathematical ideas and techniques, more or less directly associated with formal manipulation of abstract symbols and/or with finding the solutions of an equation. The notion that in mathematics there is such a sepa- rate sub-discipline, as well as the very use of the term “algebra” to denote it, are them- selves the outcome of historical evolution of ideas. The ideas to be discussed in this article are sometimes put under the same heading due to historical circumstances no less than to any “essential” mathematical reason. Part I. The long way towards the idea of “equation” Simple and natural as the notion of “equation” may appear now, it involves a great amount of mutually interacting, individual mathematical notions, each of which was the outcome of a long and intricate historical process. Not before the work of Viète, in the late sixteenth century, do we actually find a fully consolidated idea of an equation in the sense of a sin- gle mathematical entity comprising two sides on which operations can be simultaneously performed. By performing such operations the equation itself remains unchanged, but we are led to discovering the value of the unknown quantities appearing in it. Three main threads in the process leading to this consolidation deserve special attention here: (1) attempts to deal with problems devoted to finding the values of one or more unknown quantities. In Part I, the word “equation” is used in this context as a short-hand to denote all such problems,