Nine Chapters on Mathematical Modernity

Transcultural Research – Heidelberg Studies on Asia and Europe in a Global Context Series editors Madeleine Herren Thomas Maissen Joseph Maran Axel Michaels Barbara Mittler More information about this series at http://www.springer.com/series/8753 Andrea Bréard Nine Chapters on Mathematical Modernity Essays on the Global Historical Entanglements of the Science of Numbers in China 123 Andrea Bréard Faculté des Sciences d’Orsay Université Paris-Sud Orsay, France ISSN 2191-656X ISSN 2191-6578 (electronic) Transcultural Research – Heidelberg Studies on Asia and Europe in a Global Context ISBN 978-3-319-93694-9 ISBN 978-3-319-93695-6 (eBook) https://doi.org/10.1007/978-3-319-93695-6 © Springer International Publishing AG, part of Springer Nature 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Cover illustration: Cover image designed by Britta Eriskat www.eriskat.de. Formula from (Fu and Hua 1896), seal showing the character shu (number) carved by Anne Adam. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Dedicated to Shirley, Clee and Madeley, the spring flowers on green meadows Preface The title of the book and its number of chapters is a pun on the canonical Nine Chapters on Mathematical Procedures (Jiu zhang suan shu 九章算術), compiled during the first century CE. The classic and its commentaries played an important scientific and political role in the mathematical endeavours of nineteenth- and twentieth-century mathematicians working in China, not in isolation but connected to the outside world. The Nine Chapters served as a model for writing mathematics algorithmically; they were the epitome of China’s tradition, comparable to the Euclidean canon, and even became a source of inspiration for alternative proof techniques in the mid-twentieth century. Each of the nine chapters in this book illustrates how Chinese scholars mediated between new mathematical objects and discursive modes, and China’s autochthonous scientific roots. Actors developed diverse strategies to situate themselves within or against the foreign scientific knowledge systems that they encountered in the emerging global setting following the Opium Wars in the mid-nineteenth century. They shaped the “science of numbers” (shuxue 數學), as mathematics had been called in Chinese since the turn of the twentieth century, as a discipline and they gradually loosened, but sometimes also strengthened, its ties to the authority of the past. The nine chapters of this book grew out of a project idea that was sketched out during a train ride from Zürich to Heidelberg in 2015, after my intervention at Prof. Harald Fischer-Tiné’s inspiring research colloquium on Extra-European History and Global History at the ETH. An exciting visit to Zürich turned into this book, which has posed a major challenge not only to the author, but probably also to its readers. The main problem I encountered is that, to date, not a single Chinese mathematical writing is available in its entirety in a foreign language. Most of the research for this book, however, is based on original texts and Chinese secondary sources. To make the material digestible for a non-Chinese public with varying levels of mathematical literacy required a new format of publication. Using the advantages of online publishing, the book is meant to be read by chapter, with or without the technical details and the translations in the appendices. This does not mean that the technical level is equal in all chapters, Chaps. 2, 3 and 5 are certainly more difficult vii viii Preface than others, but I hope that the many illustrations will help to make the technical aspects of my argument more accessible. In bringing this project to fruition, I wish to thank first and foremost my colleague Andrea Hacker, managing editor of open-access publications of the Cluster of Excellence Asia and Europe in a Global Context in Heidelberg, who has welcomed my book with great enthusiasm and has provided much encouragement and logistical support along the way. Without her, it may have never come to an end. I would also like to extend my heartfelt thanks to the numerous colleagues and strangers in the audiences who listened and asked questions when I presented my research over the last years. I am grateful for the generous intellectual and institutional support of Fabio Acerbi, Catherine Jami, Joachim Kurtz, Michael Lackner and Bernard Vitrac who led the projects in which I participated and whose ideas provided tremendous inspiration for my work. Rui Magone was also crucial in many ways, giving backstage support whenever I needed it most. The insightful comments of anonymous reviewers also strengthened this work and helped to sharpen my arguments; I am most grateful for their vigilance and enthusiasm. John Day kept a close eye on the entire manuscript, and I feel lucky to count him among my best friends. In its final stage, the manuscript greatly benefited from professional editing by Angela Roberts, who proofread the entire manuscript and offered valuable suggestions to improve my English. Any inaccuracies remaining in the final version of this book, however, are entirely my own or are perhaps due to the birds in the trees, whose singing and twirling was often a most welcome distraction. Finally, I would like to express my love and appreciation for my children, MaxEmanuel and Sarah-Lou, who have patiently followed me across many cultures and languages during the formative years of their youth while I conducted the research for what they see as a “cool” but unreadable book. Heidelberg, Germany September 2017 Andrea Bréard Contents 1 Visions of Antiquity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 About This Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2 Saving the Nation Through Mathematics . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3 Mathematics as History . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1 5 7 10 16 2 The Ellipse Seen from Nineteenth-Century China . . .. . . . . . . . . . . . . . . . . . . . 2.1 Xia Luanxiang and Conics . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2 The Global Fate of Conics . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3 Using the Past to Solve the New . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 19 23 31 38 45 3 Filling Euclid’s Gaps.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1 Beyond the First Six Books of Euclid’s Elements . . . . . . . . . . . . . . . . . . . . 3.2 Primality in Chinese Sources . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Fermat’s Little Theorem.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 51 53 62 69 74 4 Negotiating a Linguistic Space In-Between . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 77 4.1 The Translation Enterprise . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 78 4.2 A Proto-Grammatical Symbolism . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 81 4.3 Western and Chinese Algebra .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 84 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 103 5 Discourse Transformed: Changing Modes of Argumentation . . . . . . . . . . 5.1 The Concept(s) of “Comparable Categories” . . . . .. . . . . . . . . . . . . . . . . . . . 5.2 Li Shanlan’s “Comparable Categories”.. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 5.3 “Comparable Categories” in the West . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 107 112 118 134 140 6 Fate Calculation 算命: The Mathematics of Divination . . . . . . . . . . . . . . . . . 143 6.1 Mathematical Problems Before the Qing .. . . . . . . . .. . . . . . . . . . . . . . . . . . . . 144 6.2 Hexagrams as Symbolic Algebra . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 150 ix x Contents 6.3 Proving the Scientificity of Correlative Cosmology . . . . . . . . . . . . . . . . . . 155 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 165 7 Data Management and Knowledge Production in Late Qing Institutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.1 Reform as Context .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.2 Modernizing Statistical Practices . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 7.3 What’s New in a Number? . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 169 170 178 189 191 8 Applied Versus Pure Mathematics . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 8.1 Mathematics Before it Becomes a Discipline . . . . .. . . . . . . . . . . . . . . . . . . . 8.2 Mathematical and Other Approaches to Statistics . . . . . . . . . . . . . . . . . . . . 8.3 Surviving the 1949 (Statistical) Revolution . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 195 197 202 216 221 9 Visions of Modernity.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.1 The Comeback of “National Studies” . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.2 The Case of Zhang Yitang . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 9.3 On “Mathematical Modernity”.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 225 226 228 232 235 A A Timeline of Mathematics from the Late Ming to the People’s Republic of China . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 237 B Translation of Li Shanlan’s Methods for Testing Primality (Kao shugen fa 考數根法), 1872 .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 241 C On Conics (Some Technicalities) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.1 Binomial Expansions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.2 The Circle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.3 The Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . C.4 Constructing the Ellipse . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 257 257 259 263 266 270 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 273 List of Figures Fig. 1.1 Fig. 1.2 Needham et al. (1962) p. 331 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Hua Loo-keng 華羅庚, “Mathematics is the discipline my country’s people excel in” Hua (1951) . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 2.1 Fig. 2.2 Fig. 2.3 Fig. 2.4 Fig. 2.5 Fig. 2.6 J. Kepler, Ad Vitellionem Paralipomena . . . (Kepler 1604) .. . . . . . . . Illustration of √ the trigonometric lines for circular arcs . . . . . . . . . . . . . . Expansion of √a + x in Xia (1898a) .. . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Expansion of n a + x in Lu (1902) .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . A banana-shaped field in Qin (1842) . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Diagrams and explanations of magnificent machines from the Far West, 1627 Deng and Wang (1830) . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 2.7 Proof of Propostion 18 in The Meaning of Compared [figures] Inscribed in a Circle (Yuanrong jiaoyi 圜容較義) (Ricci and Li 1614) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 2.8 C. Clavius, Geometria Practica Book IV Prop. V Clavius (1604) p. 129 .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 2.9 Ratio between the surface areas of ellipse and circle in the Essence of Numbers and their Principles (Shuli jingyun 數理精蘊) (Yunzhi 允祉1723) .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 2.10 Dong (1821).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 2.11 Problem of the kudzu vine in the Nine Chapters on Mathematical Procedures (first century AC) . . . .. . . . . . . . . . . . . . . . . . . . Fig. 2.12 Oblique cut of a cylinder compared to the helix (in blue) . . . . . . . . . . Fig. 3.1 Fig. 3.2 Fig. 3.3 Fig. 3.4 Fig. 3.5 Fig. 3.6 A portrait of Li Shanlan from The Chinese Scientific and Industrial Magazine (Gezhi Huibian 格致彙編) (July, 1877) .. . . . . Ricci and Xu (1607) Book I, Prop. 1 . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Title pages of Clavius (1591) and Billingsley (1570) .. . . . . . . . . . . . . . Billingsley (1570) Book IX Prop. 20 . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Chinese translation of Book IX Prop. 20 in Li and Wylie (1865) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Clavius (1591) Book IX Prop. 20 .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3 15 21 24 29 29 32 34 34 37 39 40 41 43 52 54 55 58 59 60 xi xii List of Figures Fig. 3.7 Scholium to Book IX Prop. 20, MS. Dorvillianus 301, fol. 171r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.8 “Method for testing primality by the celestial element [method] and reduction to One” in Li (1872b) . . . . . . . . . . . . . . . . . . . . . . Fig. 3.9 List of all prime numbers (here from 461 to 733) in Hua (1893a) p. 29B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. 3.10 Table of remainders in von Gumpach (1869) p. 153 .. . . . . . . . . . . . . . . Fig. 3.11 “The Converse of Fermat’s Theorem” (Jeans 1898) . . . . . . . . . . . . . . . . 61 64 66 71 74 Fig. 4.1 Equations from Galloway (1839), translated in Fu and Hua (1896) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 83 Fig. 4.2 Symbolic formula in Fu and Hua (1873), scroll 25 . . . . . . . . . . . . . . . . . 85 Fig. 4.3 “Notation and explanation of the signs” as translated in Fu and Hua (1873) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 86 Fig. 4.4 Series expansion of (a + x)1/2 in Xi and Gui (1880) vol. 2, p. 50A–50B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 90 Fig. 4.5 Solution with the traditional “celestial element” (tianyuan 天元) method in Xi and Gui (1880) vol. 2, p. 43A.. . . . . . . . . . . . . . . . . 92 Fig. 4.6 Solution with algebraic methods in Xi and Gui (1880) vol. 2, p. 45A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 95 Fig. 4.7 Arabic numerals explained in the Curriculum of Western Studies in the Shanghai Polytechnic Institute (Fu 1895) .. . . . . . . . . . . 99 Fig. 4.8 An expression involving fractions in Fu (1895) p. 12A . . . . . . . . . . . . 100 Fig. 4.9 Textbook, compiled by the Education Department (Xuewuchu 1907) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 Fig. 4.10 Double checking the numbers in statistical tables from the fur and leather storage (Piku tongji biao 皮庫統計表), Archives of the Imperial Household Department (1909) . . . . . . . . . . . 102 Fig. 5.1 Fig. 5.2 Fig. 5.3 Fig. 5.4 Fig. 5.5 Fig. 5.6 Fig. 5.7 Fig. 5.8 Josef Kaucký, Remarks on a work by Turán, Matematicko-Fyzikálny Časopis (1962) . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Squared Arithmetic Triangle in Li (1867), scroll 3, where each cell is (Ckn )2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Pebble diagrams related to the squared arithmetic triangle in Li (1867) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Formulation of the “Li Shanlan Identity” in Li (1867) . . . . . . . . . . . . . Collected Essentials of Mathematical Principles (Shuli jingyun 數理精蘊) Yunzhi 允祉 (1723) . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Yang Hui’s Detailed Explanations of the Nine Chapters on Mathematical Methods Yang (1261) .. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Two chutong of comparable category in Yang (1261) . . . . . . . . . . . . . . Yang Hui’s Fast methods of multiplication and division compared to [various] categories of fields and [their] measures Yang (1842) .. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 108 109 110 111 114 115 116 117 List of Figures Fig. 5.9 Fig. 5.10 Fig. 5.11 Fig. 5.12 Fig. 5.13 Fig. 5.14 Fig. 5.15 Fig. 5.16 Fig. 5.17 Fig. 5.18 Fig. 5.19 Fig. 5.20 Fig. 5.21 Fig. 5.22 Fig. 6.1 Fig. 6.2 Fig. 6.3 Fig. 6.4 Fig. 6.5 Fig. 6.6 Fig. 6.7 TOC of Wu Jing’s Great compendium of Comparable Categories to the Nine Chapters on Mathematical Methods Wu (1450) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Arithmetic Triangle, where each cell corresponds to the combinations Ckn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Li Shanlan, Comparable Categories of Discrete Accumulations (1867), scroll 1 .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . “Diagrammatic explanation of the partial number of sequential combinations for ten objects” in Wang Lai (1854) .. . . . . Illustration of C110 = C910 = 1 + 1 + 1 + · · · + 1 = 10 . . . . . . . . . . . . Illustration of C210 = C810 = 1 + 2 + 3 + 4 + · · · + 9 . . . . . . . . . . . . . . Illustration of C310 = C710 = 1 + 3 + 6 + 10 + · · · + 36 . . . . . . . . . . . Illustration of C410 = C610 = 1 + (1 + 3) + (1 + 3 + 6) + · · ·+ (1 + 3 + 6 + 10 + 15 + 21 + 28) . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Coefficient table for squared piles . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Technical vocabulary used for deductive patterns in Li Shanlan’s Comparable Categories of Discrete Accumulations (Li 1867) . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Technical vocabulary used for deductive patterns in Li Shanlan’s Comparable Categories of Discrete Accumulations (Li 1867) (cont.) . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . The Arithmetic Triangle in Pascal’s Traité du Triangle Arithmétique (1665) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Corollary 8 in the Traité du Triangle Arithmétique (1665) . . . . . . . . . Corollary 12 in the Traité du Triangle Arithmétique (1665) .. . . . . . . Physiognomy and Geometry on one page in Anonymous (1644) scroll 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Rhymes for determining if child will be a girl or boy by divination (right) and by calculation (left) in Anonymous (1612) scroll 27 “On Childbirth”.. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Some hexagrams as drawn in Jiao Xun’s Précis of Diagrams in the Changes (1974) . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . “The ancient diagram of the origins of root extraction” in Jiao Xun’s Explanation of Addition, Subtraction, Multiplication and Division (1799) .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Symbolic interpretation of “The ancient diagram of the origins of root extraction” in Jiao Xun’s Explanation of Addition, Subtraction, Multiplication and Division (1799) .. . . . . . . . Yuan Shushan’s Account of the Theories of Divination and Astrology (Shu bushi xingxiang xue 述卜筮星相學), 1929 in Yuan (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . John Fryer’s and Hua Hengfang’s translation of the Algebra article from the Encyclopaedia Britannica (Fu and Hua 1873) . . . . xiii 118 121 122 122 123 123 124 124 127 130 131 136 137 138 144 149 151 152 153 158 159 xiv List of Figures Fig. 6.8 The complex plane with complex numbers as points on the unit circle .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 160 Fig. 6.9 Arrangement of the Five Elements by division of the unit circle into five equal segments . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 162 Fig. 6.10 Arrangement of the Ten Heavenly Stems by division of the unit circle into ten equal segments .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 163 Fig. 7.1 Fig. 7.2 Fig. 7.3 Fig. 7.4 Fig. 7.5 Fig. 7.6 Fig. 8.1 Fig. 8.2 Fig. 8.3 Fig. 8.4 Fig. 8.5 Fig. 8.6 Fig. 8.7 Fig. 8.8 Fig. 8.9 Zhang Zhidong et al., Memorial to Fix the Regulations for New Schools (1904).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Anonymous, Explanation of Statistics in Japan (ca. 1907) .. . . . . . . . Growing number of students in schools (Board of Education, First Statistical Tables on Education, 1907) Board of Education, Department of General Affairs (1973) . . . . . . . . . . . . . . . . . . Statistical table showing the total number of students in Shuntian Prefecture for the year 1910 . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Statistical table showing grain prices for Yunnan Province (1911) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Title page of Meng Sen’s translation of Yokoyama Masao’s Tōkei tsuron (1909) .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fryer and Hua (trans.), A Treatise on Probability (1898 ed.) .. . . . . . A problem from the Nine Chapters solved algebraically with logarithms in Xi and Gui (1880) . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Content of Thomas Galloway’s A Treatise on Probability (Galloway 1839) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Statistical landscape as portrayed in Liu (1942) p. 13 . . . . . . . . . . . . . . Statistical landscape as portrayed in Liu (1942) p. 13 . . . . . . . . . . . . . . The Diagram of the Supreme Ultimate as an example of a statistical chart in Historical Statistics (Wei 1934) p. 157 .. . . . . . . . . Table of contents of Chen (1934) . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Newly collected papers on statistics (Zhonghua quanguo zong gonghui tongji chu 1950) .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Main sources of Jin (1934) . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 174 177 180 182 185 187 199 203 207 213 214 215 217 217 219 Fig. 9.1 One Hundred Thousand Whys (1962) and Xia Daoxing’s π and e (1964).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 230 Fig. C.1 Elliptical compass (upper left) and gardener construction (upper right) in F. Verbiest’s Records on the newly built astronomical instruments from the Beijing Observatory (1674).. . . Later Volumes of the Thorough Investigation of Calendrical Astronomy Imperially Composed (1738) . . . . . . . .. . . . . . . . . . . . . . . . . . . . Construction of the ellipse as described in the Later Volumes of the Thorough Investigation of Calendrical Astronomy Imperially Composed (1738) .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Claude François Milliet Dechales, De sectionibus conicis (Dechales 1674).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Fig. C.2 Fig. C.3 Fig. C.4 267 268 269 270 List of Tables Table 1.1 Entries on Mathematics (Suanxue 算學) in the Index to Essays on National Studies 國學論文索引(Liu 1936) . . . . . . . . . . . . . Table 2.1 Table of Content of Xia Luanxiang’s 夏鸞翔Diagrammatic Explanations [of Procedures] for Curves (Zhiqu tujie 致曲 圖解), 1861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 2.2 Overview of writings and translations on conic sections and the ellipse in particular . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Table 2.3 Overview of writings and translations on conic sections and the ellipse in particular (cont.) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 12 25 27 28 Table 8.1 Li Shanlan’s 李善蘭mathematical examination problems in Xi and Gui (1880) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 200 Table 8.2 Chinese translations and compilations of writings on statistics from ca. 1900 to 1940 .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 210 xv