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The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook

Victor J. Katz, editor
Princeton University Press
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

This is an essential book for anyone interested in the early history of mathematics. Go thou and buy thyself a copy.

It is also an impressive editorial achievement. Victor Katz has put together five experts: Annette Imhausen on Egypt, Eleanor Robson on Mesopotamia, Joe Dauben on China, Kim Plofker on India, and Len Berggren on Islam. These are all well-known historians, and several of them are writing or have written books on the mathematics of these cultures. They have done a wonderful job of selecting, annotating, and contextualizing sources.

Apart from the Greek mathematical tradition, these five are the best-documented and most impressive pre-modern mathematical cultures. (Well, one could argue that one is missing: the Medieval European tradition, which has also been too little studied, as Menso Folkerts points out.) At least a few translations of primary sources for the Greek tradition are available, including several sourcebooks. That is not the case for Egypt, Mesopotamia, India, China, and Islam: a few items have been published here and there, but this is the first systematic collection of such translations; in fact, several of the sources presented here have been newly translated. The editors include detailed introductions emphasizing the current state of knowledge about each area and period.

There are two ways to introduce readers to a new mathematical tradition: the expert can act as a tour guide, pointing out the sights at every point, or can give us an overall idea of the layout of the terrain, and then allow us to go out an explore on your own. It is the second approach that characterizes a "sourcebook": after some general orientation, we are left to study the sources on our own.

Mostly, this is what we get here. In the case of Egypt, the sources are so scanty that we end up getting a mixed approach. In the other sections, however, we get a real chance to explore for ourselves, which leads to a much deeper experience. By reading extended texts, one can come to a better feel for how each culture "did" mathematics.

To help distinguish what is source material and what is commentary, the publisher has used different typefaces: translated sources are printed in sans serif, the modern historian's commentary in a serifed font. This works, but it does require the reader to pay attention and to have sharp eyes. The book is a well-made, sturdy hardcover. At over 680 large pages, you want it in hardcover. (If Princeton ever decides to make a paper edition, I'd actually suggest five little volumes, one for each tradition, instead of a big paperback that will inevitably fall apart.) Given the overall package, the price seems quite reasonable.

There is much here that we who teach history can use in class, and there is much more that will greatly enrich our presentation of these mathematical traditions. I wouldn't use the book as a text, at least not for a first course in the history of mathematics. (On the other hand, it would be a spectacular resource for a student wanting to do independent study on one of these traditions.) I'm sure Princeton University Press would be happy to authorize reproduction of certain sections for use in class (for an appropriate fee, of course). I plan to take advantage of this next time I teach the course.

Examining the non-Greek pre-modern mathematical traditions is both fascinating and important. This book is now the best place to start.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College and the co-author, with William P. Berlinghoff, of Math through the Ages.

Preface ix
Permissions xi
Introduction 1

Chapter 1: Egyptian Mathematics
   Annette Imhausen
Preliminary Remarks 7
I. Introduction 9
a. Invention of writing and number systems 13
b. Arithmetic 14
c.Metrology 17
II. Hieratic Mathematical Texts 17
a. Table texts 18
b. Problem texts 24
III. Mathematics in Administrative Texts 40
a. Middle Kingdom texts: The Reisner papyri 40
b. New Kingdom texts: Ostraca from Deir el Medina 44
IV. Mathematics in the Graeco-Roman Period 46
a. Context 46
b. Table texts 47
c. Problem texts 48
V. Appendices 52
a. Glossary of Egyptian terms 52
b. Sources 52
c. References 54

Chapter 2: Mesopotamian Mathematics
   Eleanor Robson
I. Introduction 58
a. Mesopotamian mathematics through Western eyes 58
b.Mathematics and scribal culture in ancient Iraq 62
c. From tablet to translation 65
d. Explananda 68
II. The Long Third Millennium, c. 3200-2000 BCE 73
a. Uruk in the late fourth millennium 73
b. Shuruppag in the mid-third millennium 74
c. Nippur and Girsu in the twenty-fourth century BCE 76
d. Umma and Girsu in the twenty-first century BCE 78
III. The Old Babylonian Period, c. 2000-1600 BCE 82
a. Arithmetical and metrological tables 82
b. Mathematical problems 92
c. Rough work and reference lists 142
IV. Later Mesopotamia, c. 1400-150 BCE 154
V. Appendices 180
a. Sources 180
b. References 181

Chapter 3: Chinese Mathematics
   Joseph W. Dauben
Preliminary Remarks 187
I. China: The Historical and Social Context 187
II. Methods and Procedures: Counting Rods, The "Out-In" Principle 194
III. Recent Archaeological Discoveries: The Earliest Yet-Known Bamboo Text 201
IV. Mathematics and Astronomy: The Zhou bi suan jing and Right Triangles (The Gou-gu or "Pythagorean" Theorem) 213
V. The Chinese "Euclid", Liu Hui 226
a. The Nine Chapters 227
b. The Sea Island Mathematical Classic 288
VI. The "Ten Classics" of Ancient Chinese Mathematics 293
a. Numbers and arithmetic: The Mathematical Classic of Master Sun 295
b. The Mathematical Classic of Zhang Qiujian 302
VII. Outstanding Achievements of the Song and Yuan Dynasties
(960-1368 CE) 308
a. Qin Jiushao 309
b. Li Zhi (Li Ye) 323
c. Yang Hui 329
d. Zhu Shijie 343
VIII. Matteo Ricci and Xu Guangxi, "Prefaces" to the First Chinese
Edition of Euclid's Elements (1607) 366
IX. Conclusion 375
X. Appendices 379
a. Sources 379
b. Bibliographic guides 379
c. References 380

Chapter 4: Mathematics in India
   Kim Plofker
I. Introduction: Origins of Indian Mathematics 385
II. Mathematical Texts in Ancient India 386
a. The Vedas 386
b. The S´ulbasutras 387
c. Mathematics in other ancient texts 393
d. Number systems and numerals 395
III. Evolution of Mathematics in Medieval India 398
a.Mathematics chapters in Siddhanta texts 398
b. Transmission of mathematical ideas to the Islamic world 434
c. Textbooks on mathematics as a separate subject 435
d. The audience for mathematics education 477
e. Specialized mathematics: Astronomical and
cosmological problems 478
IV. The Kerala School 480
a. Madhava, his work, and his school 480
b. Infinite series and the role of demonstrations 481
c. Other mathematical interests in the Kerala school 493
V. Continuity and Transition in the
Second Millennium 498
a. The ongoing development of Sanskrit mathematics 498
b. Scientific exchanges at the courts of Delhi and Jaipur 504
c. Assimilation of ideas from Islam; mathematical table texts 506
VI. Encounters with Modern Western Mathematics 507
a. Early exchanges with European mathematics 507
b. European versus "native" mathematics education in British India 508
c. Assimilation into modern global mathematics 510
VII. Appendices 511
a. Sources 511
b. References 512

Chapter 5: Mathematics in Medieval Islam
   J. Lennart Berggren
I. Introduction 515
II. Appropriation of the Ancient Heritage 520
III. Arithmetic 525
IV. Algebra 542
V. Number Theory 560
VI. Geometry 564
a. Theoretical geometry 564
b. Practical geometry 610
VII. Trigonometry 621
VIII. Combinatorics 658
IX. On mathematics 666
X. Appendices 671
a. Sources 671
b. References 674

Contributors 677
Index 681