Research in History and Philosophy of Mathematics: The CSHPM 2021 Volume 3031214935, 9783031214936

Table of contents :
Preface
Editorial Board
Contents
Contributors
A Tenth-Century Mathematical Glossary
1 Introduction
2 On Ratios and Definitions: A Title
3 Memorable Introduction to the Art of Arithmetic: Another Title
4 On What Should Be Memorized Before the Arithmetic Book: The Original Title
5 Terms and Definitions
5.1 Number per se (Definitions 1–12)
5.2 Plane and Solid Numbers (Definitions 13–25)
5.3 Number Relations (Definitions 26–36)
5.4 Proportion (37–58)
6 The Book of Arithmetic
7 Conclusion
A.1 Appendix: English Translation of Text and Original Text
References
Euclid in Marāgha: The Age of the Taḥrīr
1 Introduction
2 Essential Background: Transmission of the Elements from Greek to Arabic
3 The Taḥrīr of Naṣīr al-Dīn al-Ṭūsī
3.1 The Author
3.2 General Characteristics of al-Ṭūsī's Taḥrīr
3.3 Historical Influence
3.3.1 Foundation of the Persian Transmission of the Elements
3.3.2 Calcutta School Book Society Edition of Euclid
3.3.3 The Commentary on Book I by Muḥamad Barakat
4 The Taḥrīr of Muḥya al-Dīn al-Maghribī
4.1 The Author
4.2 General Characteristics of the Taḥrīr
4.3 Influence
5 The Pseudo-Ṭūsī Taḥrīr
5.1 The Author
5.2 General Characteristics of the Taḥrīr
5.3 Influence
6 Concluding Thoughts
6.1 Internal Cross-Referencing and Mathematical Pedagogy
6.2 Printing Euclidean Geometry and Ottoman Madrasa Education
6.3 Geometrical Compilations and Mathematical Pedagogy
Manuscripts Consulted
References
The Recreational Problems of Tratado de Prática Darysmetica by Gaspar Nicolas, 1519
1 The Tratado in Its Context
2 The Organisation of the Tratado
3 A Few Selected Problems
3.1 A Man and Three Saints
3.2 Generating Squares
4 A Broken Weight
5 Equal Sales
5.1 Interrupted Game
5.2 Bags
6 Conclusion
References
The Mathematics of Polemic in John Napier's Plaine Discovery
1 Introduction
2 Context
3 Mathematics of Prophecy
4 Seals, Trumpets and Vials
5 Jubilees and the End Times
6 Notes on Table 2
7 Critical Observations
8 Napier's Legacy in Scholarship
9 Closing Remarks
References
Euler's Series for Sine and Cosine. An Interpretation in Nonstandard Analysis
1 Introduction
2 Forerunners
2.1 Ptolemy
2.2 Newton
2.3 Ptolemy–Newton–Euler
3 Setting the Stage: Trigonometry
4 The Crucial Move
4.1 From Finite to Infinite
4.2 Infinite and Infinitesimal Numbers
4.3 Infinite Series vs Hyperfinite Sums
4.4 Extending Trigonometric Functions
4.5 The Revised Proof
5 Nonstandard Analysis
5.1 The Basics of Ordered Field Theory
5.2 Archimedean Axiom
5.3 Real Numbers
5.4 Hyperreals
5.5 Infinitesimals, Infinite Numbers, and Finite Numbers
5.6 Binomial Coefficients
5.7 *Maps
5.8 Hyperfinite Sums
6 Infinitesimals and Infinite Numbers in Institutiones calculi differentialis
6.1 Three Kinds of Quantities: Infinite Numbers
6.2 Infinitesimals
6.3 Two Ways of Comparing Zeros
6.4 Arithmetic of Infinitesimals
6.5 Infinitesimals and Infinite Numbers
6.6 Assignable Numbers and ΩΨ Products
7 Summary
References
Agnesi vs. Colson: Did Location Matter?
1 Introduction
2 Maria Gaetana Agnesi
3 The Reverend John Colson
4 The Witch of Agnesi
5 The Calculus Controversy
6 Comparison of the Italian and English Versions
7 Conclusion
References
The Limits of Understanding and the Understanding of Limits: David Hume's Mathematical Sources
1 The `Paradoxes'
2 The Port Royal Logic
3 Leibniz
4 Malézieu
5 Hume
References
Analysis and Synthesis in Robert Simson's The Elements of Euclid
1 Introduction
2 Robert Simson and The Elements of Euclid
3 Analysis and Synthesis as Methods of Proof
4 Analysis and Synthesis as Mathematical Styles
5 Analysis and Synthesis as Pedagogical Approaches
6 Conclusion
References
Thomas Archer Hirst: Mathematician Xtravagant
1 Introduction
2 Yorkshire
3 Marburg, Germany
4 Göttingen and Berlin
5 Queenwood College, Hampshire
6 France and Italy
7 Return to London
References
A Cambridge Correspondence Course in Arithmetic for Women
1 Introduction
2 Correspondence Courses for Women
3 Hudson's Correspondence Course in Arithmetic
4 Conclusion
Appendix
References
T. E. Peet, a Mathematician Among Egyptologists?
1 Introduction
2 Thomas Eric Peet
3 Peet the Mathematician
4 Correspondence with Mathematicians
5 Ancient Egyptian Mathematicians?
6 Concluding Remarks
Archival Sources
References
Placing a Global Mathematical Literature
1 The Entire World Literature of Mathematics
2 What Literatures Do
3 Placing Review Journals
4 Moving Review Journals
5 Indexing Reviewers
6 Instructing Reviewers
7 Conclusion: Situating Reviews
References
*12ptArchives
*12ptPublished Sources
Latin Squares at Rothamsted in the Time of Fisher and Yates
1 Fisher and Yates at Rothamsted
1.1 People and Place
1.2 Basics of Design of Experiments
2 Latin Squares
2.1 What Is a Latin Square?
2.2 Orthogonal Latin Squares
2.3 Mutually Orthogonal Latin Squares
2.4 A Catalogue of Sets of Mutually Orthogonal Latin Squares
2.5 Randomization
3 Statistical Tables
3.1 The Need for Tables
3.2 What Do the Tables Cover?
3.3 A Catalogue of Latin Squares
3.4 A Catalogue of Complete Sets of Mutually Orthogonal Latin Squares
4 Technical Communication 35
4.1 Brief Overview
4.2 Explanation of Mutually Orthogonal Latin Squares
5 Some Later Developments
5.1 Inequivalent Complete Sets
5.2 Explanation Using Abelian Groups
References
Les équations différentielles ordinaires « raides » et les méthodes robustes : une approche historique
1 Introduction
2 Les premières méthodes numériques à un seul pas : Euler et Cauchy
3 Le triomphe des méthodes à un seul pas et celles à pas multiples
4 La théorie des erreurs de troncature locales et globales
4.1 Le début
4.2 Cauchy et Lipschitz sur la théorie des erreurs locales et globales
5 Existence and unicité
6 Richardson et le contrôle du pas
7 Stabilité, problèmes bien posés, et condition
8 Les équations différentielles « raides » et les méthodes robustes : les outils mathématiques
Références
The Cavendish Computors: The Women Working in Scientific Computing for Radio Astronomy
1 Introduction
2 The Women in Radio Astronomy Computing
3 The First Forays into Scientific Computing
4 Earth Rotation Aperture Synthesis and the Steady State Hypothesis
4.1 The Steady State Hypothesis
5 A Radio Survey of the North Polar Region
6 Discussion
7 Conclusion
References
The Algebra Project, Feature Talk, and the History of Mathematics
1 Algebra as a Civil Rights Issue
2 The Algebra Project as a Curricular Process
3 Feature Talk and the History of Mathematics
References
Corrupt Land Inspectors: Solving Equations with Picture-Language in Ancient Mesopotamia—A Dialogue
1 Introduction and Programme Notes
2 The Play: Solving Equations with Picture-Language
Curtain Rises
References
Entrance into All Obscure Secrets: A Workshop on Bringing Episodes in the History of Mathematics to Life in the Classroom by Means of Theatre, Incorporating a Short Play Set in an Ancient Egyptian Scribal School
1 Introduction
2 Why a Mathematical-Historical Play?
3 Welcome and Synopsis
4 The Play
SLIDE 1
SLIDE 2
SLIDE 3
SLIDE 4
Part 2
SLIDE 5
SLIDE 6
SLIDE 7
SLIDE 8
SLIDE 9
SLIDE 10
5 Discussion
6 Concluding Remarks
References

Citation preview

Annals of the Canadian Society for History and Philosophy of Mathematics Société canadienne d’histoire et de philosophie des mathématiques

Maria Zack David Waszek Editors

Research in History and Philosophy of Mathematics The CSHPM 2021 Volume

Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques Series Editor Maria Zack, Department of Mathematical, Information, and Computer Sciences, Point Loma Nazarene University, San Diego, CA, USA

The books in the series contain selected papers written by members of the Canadian Society for History and Philosophy of Mathematics. Founded in 1974, this society promotes research and teaching in the history and philosophy of mathematics, as well as in the connection between the two. Volumes in this series cover a broad range of topics from a variety of time periods and cultures. They will be accessible to anyone who has had exposure to mathematics at the university level and will appeal to scholars of the history and/or philosophy of mathematics, graduate and undergraduate students undertaking research projects, and anyone with a general interest in mathematics.

Maria Zack • David Waszek Editors

Research in History and Philosophy of Mathematics The CSHPM 2021 Volume

Editors Maria Zack Department of Mathematical Information, and Computer Sciences, Point Loma Nazarene University San Diego, CA, USA

David Waszek Ecole Normale Supérieure Paris, France

ISSN 2662-8503 ISSN 2662-8511 (electronic) Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques ISBN 978-3-031-21493-6 ISBN 978-3-031-21494-3 (eBook) https://doi.org/10.1007/978-3-031-21494-3 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed 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, expressed 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. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

This volume contains 18 papers that have been complied by the Canadian Society for History and Philosophy of Mathematics. These papers provide some interesting insights into contemporary scholarship in the history and philosophy of mathematics. The volume also includes some papers on the teaching of the history of mathematics and concludes with two short plays that can be used to introduce historical topics in the classroom. The volume opens with two papers on Middle Eastern medieval mathematics. In “A Tenth-Century Mathematical Glossary,” Hassan Amini discusses Ab¯u alWaf¯a al-B¯uzj¯an¯ı’s “On What Should Be Memorized Before the Arithmetic Book” which includes four categories of terms: numbers, plane and solid numbers, number relations, and proportion. Amini argues that the terminology in the glossary is a hybrid of Pythagorean and Euclidean arithmetic definitions. In “Euclid in Mar¯agha: The Age of the Tah.r¯ır,” Gregg De Young examines three Arabic editions of Euclid’s Elements from the thirteenth century. De Young’s paper discusses the ways in which these translations point to the transmission and assimilation of the Elements into the Islamic intellectual tradition. The volume continues with papers on sixteenth-century European mathematics. In “The Recreational Problems of Tratado de Prática Darysmetica by Gaspar Nicolas, 1519,” Jorge Nuno Silva and Pedro Jorge Freitas present an overview of the contents of a 1519 Portuguese commercial arithmetic treatise, with a special focus on the recreational problems in the book. In “The Mathematics of Polemic in John Napier’s Plaine Discovery,” Alexander Corrigan examines Napier’s biblical commentary “A Plaine Discovery of the Whole Revelation of Saint John.” Corrigan argues that Napier’s paper is the sixteenth century’s most developed attempt to provide mathematical certainty to religious claims and that Napier’s true intention was not to discover mathematical patterns in history but to impose them where none truly existed. The next two papers focus on eighteenth-century mathematics. In “Euler’s Series for Sine and Cosine: An Interpretation in Nonstandard Analysis,” Piotr Błaszczyk and Anna Petiurenko examine the proof of Euler’s famous formula. Błaszczyk

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and Petiurenko argue that there is there is one implicit lemma underlying Euler’s proof which requires specific techniques from non-standard analysis and they discuss other locations where Euler may have used similar techniques. In “Agnesi vs. Colson: Did Location Matter?,” Cynthia Huffman compares Maria Gaetana Agnesi’s highly regarded algebra and calculus text Instituzioni Analitiche with John Colson’s translation Analytical Institutions. Both the original book and the translation were written not long after the Newton–Leibniz calculus controversy, and Huffman examines whether the geographic location of Angesi (Italy) and Colson (England) made a difference in the mathematical notation each of them used. The focus on the eighteenth century continues with Brendan Lavor’s “The Limits of Understanding and the Understanding of Limits: David Hume’s Mathematical Sources.” In this paper Lavor examines how Hume made use of mathematical examples in his writing and argues that Hume is one of a collection of philosophers who used a small handful of mathematical examples to argue for either fideist or skeptical conclusions. In “Analysis and Synthesis in Robert Simson’s The Elements of Euclid,” Amy Ackerberg-Hastings considers how three different uses of the terms “analysis” and “synthesis” influenced the creators of geometry texts in Western Europe and North America. Ackerberg-Hastings does this by examining Robert Simson’s attempt restore Euclid’s text, The Elements of Euclid, which appeared in 1756 in both English and Latin versions. The volume moves to the nineteenth century with two papers about mathematics in Victorian Britain. Robin Wilson’s “Thomas Archer Hirst: Mathematician Xtravagant” considers the writings of Hirst, a lifelong diarist, who was well-connected in British scientific and social circles. Wilson uses extracts from Hirst’s diaries to discuss the distinguished people that Hirst knew and the varied activities in which he participated. In “A Cambridge Correspondence Course in Arithmetic for Women,” Shawn McMurran and James Tattersall focus on an example of distance learning provided to women in England in the late nineteenth century, when the Cambridge Examination for Women was established to verify the qualifications of women who wished to become teachers. Using a specific example, McMurran and Tattersall discuss the structure of the correspondence courses offered to women who were unable to attend lectures in Cambridge. The volume’s collection of papers on twentieth-century mathematics begins with two papers that examine some surprising pieces of mathematical history. In “T. E. Peet, A Mathematician Among Egyptologists?” Christopher Hollings and Robert Parkinson consider the implications of the fact that before embarking on an archaeological career, Peet studied mathematics at university. Hollings and Parkinson examine three different ways in which Peet employed the word “mathematician,” which illuminates both his own career trajectory and self-presentation, and the way in which different disciplines interacted within the study of ancient mathematical texts. In “Placing a Global Mathematical Literature: Geography, Infrastructure, and Information in the Mid-century Zentralblatt für Mathematik and Mathematical Reviews,” Michael J. Barany discusses how the Zentralblatt für Mathematik und ihre Grenzgebiete and its slightly younger counterpart Mathematical Reviews formed a lynchpin for a widely distributed publishing infrastructure that created a

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meaningfully global mathematical literature. Barany argues that this changed the process for how mathematicians organized their research communities and their ideas. The exploration of twentieth-century mathematics continues with two papers on the history of applied mathematics. In “Latin Squares at Rothamsted in the Time of Fisher and Yates,” Rosemary Bailey describes the work of R. A. Fisher and Frank Yates using Latin squares for agricultural crop experiments at the Rothamsted Experimental Station. From this beginning, Bailey traces the use of Abelian groups for the construction of factorial experiments. In “Les équations différentielles ordinaires « raides » et les méthodes robustes : une approche historique” Roger Godard, John de Boer, and Mark Lewis look at the history of “stiff” differential equations. These are differential equations where the problem of initial values is (extremely) difficult to solve by explicit methods. Godard de Boer and Lewis trace the history of these equations from Cauchy’s explicit and implicit methods to the work of Curtis and Hirchfelder on a specific example (1952). The papers on the twentieth century conclude with two papers that examine events in the 1960s. In “The Cavendish Computers: The Women Working in Scientific Computing for Radio Astronomy,” Verity Allan discusses the advancement of computing techniques at the Cavendish Laboratory at the University of Cambridge. Allen specifically focuses on the many ways that women contributed this endeavor during the 1960s. In “The Algebra Project, Feature Talk, and the History of Mathematics,” Madeline Muntersbjorn discusses the work of Robert P. Moses, an activist and educator who taught mathematics and studied philosophy. Moses is renowned for his efforts to organize the volunteers from across the USA who came to Mississippi to help Black Americans register to vote in the 1960s. This paper describes his less well-known work as an education reformer. Muntersbjorn describes Moses’ role in the Algebra Project and considers the usefulness of what Moses calls “Feature Talk” as opposed to “People Talk.” The book concludes with two short plays that are well suited for use in the classroom. In “Corrupt Land Inspectors: Solving Equations with Picture-Language in Ancient Mesopotamia, a Dialogue,” Gavin Hitchcock provides the text of a play that is set in a scribal school in Mesopotamia in 2000 BCE. In the play, the head scribe uses as motivation his student’s concern over the way tax-collectors are defrauding the poorest farmers by walking the perimeter to arrive at the tax bracket. This is the starting point for basic geometric computations relating perimeter and area. In “Entrance into All Obscure Secrets: A Workshop on Bringing Episodes in the History of Mathematics to Life in the Classroom by Means of Theatre, Incorporating a Short Play Set in an Ancient Egyptian Scribal School,” Gavin Hitchcock describes a workshop where a short play was staged and includes some of the reflections of the participants. The text of the short play is included in the paper and focuses on problems from the Rhind Papyrus (c 1650 BCE). This collection of papers contains several gems from the history and philosophy of mathematics, which will be enjoyed by a wide mathematical audience. These papers were written during the coronavirus pandemic and many of them were presented at virtual conferences. As editors, we are grateful to the authors and the

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referees who persisted in their scholarly work during a complicated period in history and brought this volume into being. San Diego, CA, USA Paris, France

Maria Zack David Waszek

Editorial Board

The editors wish to thank the following people who served on the editorial board for this volume: Amy Ackerberg-Hastings Convergence, Mathematical Association of America Eisso Atzema University of Maine Orono Christopher Baltus State University New York College of Oswego David Bellhouse Western University Moira Chas Stony Brook University Daniel Curtin Northern Kentucky University Thomas Drucker University of Wisconsin, Whitewater Richard Edwards Michigan State University Craig Fraser University of Toronto Cynthia Huffman Pittsburg State University Elaine Landry University of California, Davis

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Editorial Board

Shawnee McMurran California State University, San Bernardino Nicolas Michael Universiteit Utrecht Jean-Pierre Marquis Université de Montréal Michael Saclolo Saint Edwards University Dirk Schlimm McGill University James Tattersall Providence College Glen Van Brummelen Quest University David Waszek École Normale Supérieure, Paris Maria Zack Information, and Computer Sciences, Point Loma Nazarene University

Contents

A Tenth-Century Mathematical Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hassan Amini

1

Euclid in Mar¯agha: The Age of the Tah.r¯ır . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gregg De Young

17

The Recreational Problems of Tratado de Prática Darysmetica by Gaspar Nicolas, 1519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jorge Nuno Silva and Pedro Jorge Freitas The Mathematics of Polemic in John Napier’s Plaine Discovery. . . . . . . . . . . . Alexander Corrigan Euler’s Series for Sine and Cosine. An Interpretation in Nonstandard Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Piotr Błaszczyk and Anna Petiurenko

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Agnesi vs. Colson: Did Location Matter? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 Cynthia J. Huffman The Limits of Understanding and the Understanding of Limits: David Hume’s Mathematical Sources. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 Brendan Larvor Analysis and Synthesis in Robert Simson’s The Elements of Euclid . . . . . . . . 133 Amy Ackerberg-Hastings Thomas Archer Hirst: Mathematician Xtravagant . . . . . . . . . . . . . . . . . . . . . . . . . . 149 Robin Wilson A Cambridge Correspondence Course in Arithmetic for Women . . . . . . . . . . 169 Shawnee L. McMurran and James J. Tattersall T. E. Peet, a Mathematician Among Egyptologists?. . . . . . . . . . . . . . . . . . . . . . . . . . 183 Christopher D. Hollings and R. B. Parkinson

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Placing a Global Mathematical Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199 Michael J. Barany Latin Squares at Rothamsted in the Time of Fisher and Yates . . . . . . . . . . . . . 213 R. A. Bailey Les équations différentielles ordinaires « raides » et les méthodes robustes : une approche historique. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 Roger Godard, John de Boer and Mark Lewis The Cavendish Computors: The Women Working in Scientific Computing for Radio Astronomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 Verity Allan The Algebra Project, Feature Talk, and the History of Mathematics . . . . . . 261 Madeline Muntersbjorn Corrupt Land Inspectors: Solving Equations with Picture-Language in Ancient Mesopotamia—A Dialogue . . . . . . . . . . . . . . . . . . . 277 Gavin Hitchcock Entrance into All Obscure Secrets: A Workshop on Bringing Episodes in the History of Mathematics to Life in the Classroom by Means of Theatre, Incorporating a Short Play Set in an Ancient Egyptian Scribal School . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287 Gavin Hitchcock

Contributors

Amy Ackerberg-Hastings Convergence, Mathematical Association of America, Washington, DC, USA Verity Allan Cavendish Laboratory, University of Cambridge, Cambridge, UK Hassan Amini University of Tehran, Tehran, Iran R. A. Bailey University of St Andrews, St Andrews, Scotland Michael J. Barany University of Edinburgh, Edinburgh, Scotland Piotr Błaszczyk Pedagogical University of Kraków, Kraków, Poland Alexander Corrigan University of Edinburgh, Edinburgh, Scotland John de Boer Royal Military College of Canada, Kingston, ON, Canada Gregg De Young The American University in Cairo, New Cairo, Egypt Pedro Jorge Freitas University of Lisbon, Lisbon, Portugal Roger Godard Royal Military College of Canada, Kingston, ON, Canada Gavin Hitchcock Independent Scholar, London, UK Christopher D. Hollings The Queen’s College, Oxford, UK Cynthia J. Huffman Pittsburg State University, Pittsburg, KS, USA Brendan Larvor University of Hertfordshire, Hertfordshire, UK Mark Lewis Saint Lawrence College, Kingston, ON, Canada Shawnee L. McMurran California San Bernardino, CA, USA

State

University,

San

Bernardino,

Madeline Muntersbjorn University of Toledo, Toledo, OH, USA R. B. Parkinson The Queen’s College, Oxford, UK

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Contributors

Anna Petiurenko Pedagogical University of Kraków, Kraków, Poland Jorge Nuno Silva University of Lisbon, Lisbon, Portugal James J. Tattersall Providence College, Providence, RI, USA Robin Wilson The Open University and Oxford University, Oxford, UK

A Tenth-Century Mathematical Glossary Hassan Amini

Abstract Ab¯u al-Waf¯a’al-B¯uzj¯an¯ı is a tenth-century mathematician, mostly known for his works on practical mathematics. There are manuscripts dedicated to arithmetic among his extant works, notably a treatise, which is a mathematical glossary. In this chapter, the manuscripts of this treatise are introduced, its exact original title is discussed, and its content and references are presented and appended by the edition and translation of the text. The original title is On What Should Be Memorized Before the Arithmetic Book. It includes 58 terms and definitions in four main categories: number per se, plane and solid numbers, number relations, and proportion. A survey of the content and terminology of the treatise shows that it is a hybrid of Pythagorean and Euclidean arithmetic definitions.

1 Introduction The treatise F¯ım¯a yanbagh¯ı an yuh.fizu qabl kit¯ab al-arithm¯a.t¯ıq¯ı (On What Should Be Memorized Before the Arithmetic Book) by Ab¯u al-Waf¯a’ Muh.ammad ibn Muh.ammad ibn Yah.yá ibn Ism¯a‘¯ıl ibn al-‘Abb¯as al-B¯uzj¯an¯ı (940–998 AD) is a particular type of treatise: it is explicitly dedicated to mathematical nomenclature and provides a list of mathematical terms that should be memorized. The idea of memorizing mathematical terms is intriguing and controversial, since the general belief is that mathematics is about understanding rather than memorizing material. Ab¯u al-Waf¯a’ al-B¯uzj¯an¯ı is a tenth-century mathematician. He was born in B¯uzj¯an¯ı, a city in the east of today’s Iran, and he went to Baghdad when he was 20 years old. He had been in correspondence with other scientists of his era, notably al-B¯ır¯un¯ı. In the history of mathematics, al-B¯uzj¯an¯ı is known mainly for

H. Amini () Institute for History of Science, The Faculty of Theology, University of Tehran, Tehran, Iran e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_1

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H. Amini

his achievements in trigonometry. However, his practical attitude in geometry and arithmetic made him the leading source in these two branches for centuries.

2 On Ratios and Definitions: A Title The treatise Ris¯ala f¯ı l-nisab wa I-ta‘r¯ıf¯at (Treatise on Ratios and Definitions) is mentioned among B¯uzj¯an¯ı’s works (Sezgin 1974, p. 324, no. 6; p. 403). The manuscript representing this treatise belongs to the H.asan Nar¯aq¯ı’s collection in Tehran (known as number 9602). However, an image of the manuscript can be found on microfilm in the Central Library of the University of Tehran (number 3564/9). The collection 3564 includes several mathematical and astronomical treatises that were all scribed in 836/1432. The treatise has no title1 and was identified based on its content by Abu al-Q¯asim Qurb¯an¯ı, an Iranian historian of mathematics (Qurb¯an¯ı 1986, p. 164, no. 6) (Fig. 1).

Fig. 1 (a) Number 3564/9 University of Tehran Central Library. (b) Number 4841 National Library of Iran. (Images taken by the author)

1 It could be distinguished by its heading, written in a strange calligraphic form, which should be an artistic way of writing Basmala.

A Tenth-Century Mathematical Glossary

3

A survey of B¯uzj¯an¯ı’s works shows that there are four copies of this treatise in Iranian libraries, as listed below: U: University of Tehran Central Library, no. 3564/9 (132v-133r), dated 836/1432, no title. N: National Library of Iran (Mill¯ı), no. 4841 (56r-57v), dated 1017/1608, no title. M: The Mashhad University, the Faculty of Theology Library, no. 751 (38r-39v), dated 1274/1857, no title. P: The Library of Iranian Parliament (Majlis), no. 6657 (105r-106r), dated thirteenth/nineteenth century, under the title: Ris¯ala f¯ı al-h.is¯ab (A Treatise on Arithmetic). The manuscripts U and N both are recorded as Is.til¯ah.a¯ t al-r¯ıd.iyya (The Mathematical Terms), and the manuscript M is recorded under the title of F¯ım¯a yanbagh¯ı an yuh.fizu qabl arithm¯a.t¯ıq¯ı (On What Should Be Memorized Before the Arithmetic) in corresponding catalogues; it seems these titles are given to them by catalogers. All four manuscripts have the same content with some minuscule variations. As we will see in the next section, these are the copies of B¯uzj¯an¯ı’s work on arithmetic, which has not been widely studied.

3 Memorable Introduction to the Art of Arithmetic: Another Title A further study on B¯uzj¯an¯ı’s works shows that there are three other extant manuscripts on arithmetic in the list of his works: R1: Reza Rampur Library, no. 3773/5 (94b-98b), dated twelfth/eighteenth century, under the title: al-Madkhal al-h.ifz¯ı il¯a s.in¯a‘at al-arithm¯a.t¯ıq¯ı.2 R2: Reza Rampur Library, no. 3773/6 (103b-104b), dated twelfth/eighteenth century, no title. T: Tashkent Institute for Oriental Studies, no. 4750/8 (255b-257b). The manuscript R1 has the title al-Madkhal al-h.ifz¯ı il¯a s.in¯a‘at al-arithm¯a.t¯ıq¯ı (Memorable Introduction to the Art of Arithmetic) in the same handwriting of the text. However, the manuscripts R2 and T, which have no titles, are recorded by catalogers respectively as Ris¯ala f¯ı al-h.is¯ab (A Treatise on Arithmetic) and Ris¯ala f¯ı al-arithm¯a.t¯ıq¯ı (A Treatise on Arithmetic). The manuscripts R2 and T, like manuscripts U, N, M, and P, represent an abridged version of manuscript R1. The manuscript R1 is an extended version with some examples, comments, and additional terms and definitions. It seems that the abridged version is the original treatise because the extended version is more than

2 The text is published in al‘Al¯ı. 1977. “al-Madkhal al-hifz¯ı il¯ . . a s.in¯a‘at al-arithm¯at.¯ıq¯ı.” al-Tur¯ath al-‘ilm¯ı al-‘arab¯ı 1: 18–29.

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could be memorized. The number of copies and their dates also make the extended version more likely to be a later commentary on the treatise. In conclusion, two versions of a treatise by B¯uzj¯an¯ı have survived; several copies of the abridged version and a copy of the extended version are known. The oldest copy of the abridged version is manuscript U. The copies are recorded under various titles, none of which is precisely correct.

4 On What Should Be Memorized Before the Arithmetic Book: The Original Title All of the titles mentioned above are not original; they were invented either by scribes or catalogers. The descriptive titles Is.til¯ah.a¯ t al-r¯ıd.iyya (The Mathematical Terms) and Ris¯ala f¯ı l-nisab wa I-ta‘r¯ıf¯at (Treatise on Ratios and Definitions) were invented by catalogers. Since the text lists some arithmetical terms and definitions, including ratios, these titles could be generally acceptable, but they do not describe the content adequately. The simple titles Ris¯ala f¯ı al-h.is¯ab, or al-arithm¯a.t¯ıq¯ı (A Treatise on Arithmetic) could be the scribe’s choice to introduce the treatise’s genre, even considering that the treatise is not indeed a typical treatise in Arabic arithmetic tradition. There is a subtle nuance between two terms: al-h.is¯ab and al-arithm¯a.t¯ıq¯ı, both commonly translated to arithmetic. The word al-arithm¯a.t¯ıq¯ı is the Arabized version ᾿ of the Greek word αριθμ´ oς, which means the art of numbers; on the other hand, al-h.is¯ab is an Arabic term derived from the root h.asaba, which means to calculate. Over time, al-h.is¯ab, as a mathematical branch, came to include all traditions of mathematics, including Indian numerals and the associated calculating practices, as well as the Greek tradition of geometry and problem-solving, to form a widespread body of knowledge concentrated on practical mathematics. As a result of this assimilation, meaning of the word al-arithm¯a.t¯ıq¯ı became narrower and was confined to problems relating to number theory, also known as ‘ilm al-‘adad. So based on the content of the treatise, al-arithm¯a.t¯ıq¯ı rather than al-h.is¯ab is a more fitting word to refer to the document’s subject. The other titles represent the same idea: a treatise including some arithmetical technical terms that should be memorized in advance. Ibn al-Nad¯ım, in his comprehensive bibliography Kit¯ab al-Fihrist, mentioned the title F¯ım¯a yanbagh¯ı an yuh.fizu qabl kit¯ab al-arithm¯a.t¯ıq¯ı (On What Should Be Memorized Before the Arithmetic Book) for al-B¯uzj¯an¯ı’s work (al-Nad¯ım 1872, p. 283). This title is different from the title attributed to manuscript M in only one word: kit¯ab (book). This word is crucial in the title, determining that the words should be memorized before learning arithmetic, that is, a practical branch of mathematics, or before studying a particular book called the arithmetic book. Kit¯ab al-Fihrist is a reliable source because Ibn al-Nad¯ım was a contemporary of al-B¯uzj¯an¯ı. The next step is to identify the kit¯ab al-

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arithm¯a.t¯ıq¯ı (the Arithmetic Book) mentioned in the title, verifying if the arithmetic terms are in the book; we will return to this topic in Sect. 6. The manuscript R1’s title, which is al-Madkhal al-h.ifz¯ı il¯a s.in¯a‘at al-arithm¯a.t¯ıq¯ı (Memorable Introduction to the Art of Arithmetic) is a mistaken mélange of the titles of two of al-B¯uzj¯an¯ı’s works. Ibn al-Nad¯ım brings the name of a non-extant work of al-B¯uzj¯an¯ı, Kit¯ab al-Madkhal il¯a al-arithm¯a.t¯ıq¯ı (Introduction Book to the Arithmetic), before the original title of the treatise, so the manuscript R1 title is a mixture of these two titles.

5 Terms and Definitions The treatise includes 58 terms and their definitions. Note that the treatise, including an English translation, can be found in the appendix of this chapter. The terms could be categorized into four principal subjects: 1. 2. 3. 4.

Number per se (definitions 1–12) Plane and solid numbers (definitions 13–25) Number relations (definitions 26–36) Proportion (37–58)

The three major Greek works on arithmetic are Nicomachus’ Introduction to the Arithmetic, Theon of Smyrna’s On Mathematical Matters Useful for Reading Plato, and Euclid’s Elements Books VII, VIII, and IX. They are almost about the same subject; however, Nicomachus and Theon’s works have more in common (Nicomachus and Martin Luther 1926, p. 47), representing a Neopythagorean school. This section investigates the relationship between these Greek sources and al-B¯uzj¯an¯ı’s treatise, particularly Nicomachus’ Introduction, the first guess for the Arithmetic Book in the title. On the other hand, this investigation shows us which Greek Arithmetic tradition was more influential on al-B¯uzj¯an¯ı’s attitude toward arithmetic. Two of these works are translated into Arabic in the first centuries of the classical Islamic age. Nicomachus’s Introduction was translated into Arabic twice. The first translation from Syriac by Hazza H.ab¯ıb b. Bahr¯ız (late eighth to early ninth century) is lost (Freudenthal 2021). The second translation was by Ab¯u al-H.asan Th¯abit ibn Qurrah al-H.arr¯an¯ı (836–901 AD), entitled al-Madkhal ila ‘ilm il-‘adad (Introduction to the Knowledge of Numbers). Ibn al-Nad¯ım recorded a book under the title al-Madkhal ila al-arithm¯a.t¯ıq¯ı (Introduction to Arithmetic), in two books, as Nicomachus’s work (al-Nad¯ım 1872, p. 269). Al-Qift.¯ı also mentioned the name of Kit¯ab fi al-arithm¯a.t¯ıq¯ı (Book on Arithmetic) by Nicomachus among the books translated by Th¯abit ibn Qurrah (al-Qift.¯ı 1903, p. 117). All of these titles probably refer to the same work. However, the possibility that there was another translation of Nicomachus’s Introduction under the title mentioned by al-Nad¯ım or Al-Qift.¯ı cannot be excluded.

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The translation of Euclid’s Elements into Arabic is a complicated subject. However, the accepted idea is that there are two main versions of the Arabic translation of Elements: the first version is attributed to Ish.a¯ g ibn H.unayn with the revision of Th¯abit ibn Qurrah and the second version is attributed to al-H.ajj¯aj (De Young 1984). There is no extant Arabic manuscript of Theon’s On Mathematical Matters Useful for Reading Plato. Ibn al-Nad¯ım mentioned Kit¯ab mar¯atib qar¯a’at kutub afl¯a.tu¯ n (The book of the stages of reading Plato) as written by him, which could be the Arabic translation of this book (al-Nad¯ım 1872, p. 255).

5.1 Number per se (Definitions 1–12) The treatise begins with two controversial definitions: The unit is by which an entity is called one; the number is the plurality of units. These are, respectively, definitions VII.1 and VII.2 of Euclid’s Elements (Heath 1956, vol. 2, p. 113). The definition of number is not the same in the two other Greek major arithmetic works. Nicomachus discussed the subject in I.7 (Book I, chapter VII). He presented his definition of number as “limited multitude or a combination of units or a flow of quantity made up of units” (Nicomachus and Martin Luther 1926, p. 190), which is almost the same in its Arabic translation (Nicomachus 1959, p. 19). Theon dedicated a chapter to a discussion about “The one and the monad” (Toulis 1979, p. 12). In this chapter, he concentrated exclusively on the concept of unit, while Nicomachus’s definition is focused on the concept of number as a multitude of units. It seems that defining a unit without generalizing it to a number is incomplete; on the other hand, defining number by units without a preceding definition of a unit is vague. Euclid had already presented each concept in the Elements, Book VII, as a definition to have a complete definition of number. B¯uzj¯an¯ı’s first two definitions are undoubtedly Euclidean, not Neopythagorean. The question arising is which tradition of the Arabic translation of the Elements conveyed these two definitions to B¯uzj¯an¯ı. The history of the translation of the Elements is not clear, but only one tradition has similar definitions. The manuscript FATIH 3439/1, representing the original Ish.a¯ g-Th¯abit translation, has a two-concept definition. Nevertheless, the terminology used in the definitions is not precisely the same as B¯uzj¯an¯ı’s definitions, particularly in the definition of number, where aljam¯a‘at al-murakkaba (compound group) is used in the translation of Elements (Euclid, manuscript, p. 13) while kathrat (plurality) is in al-B¯uzj¯an¯ı’s treatise. Most of the other definitions 1–12 have the same story as the two first definitions. They are the definitions of the Elements, Book VII, and related to the concepts in Introduction, sections I.7-I.16.

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5.2 Plane and Solid Numbers (Definitions 13–25) The definitions in this part are related to Introduction II.7-II.17, and some of them are not included in Elements, Book VII, definitions. For example, two types of numbers, plinth [definition 22] and beam [definition 23] numbers, are not mentioned at all in the Elements. Hence, we can say that this part of the treatise is a combination of two sources.

5.3 Number Relations (Definitions 26–36) The definitions are exclusively related to Introduction, I.12-I.23. This part is absolutely Nichmacusian, which refers mainly to five kinds of numbers, whose idea is totally absent in the Elements.

5.4 Proportion (37–58) This part of the treatise, which is dedicated to introducing different kinds of proportion, is related to Introduction, II.28; it enumerates ten types of proportion, which are not mentioned in the Elements. On the other hand, this part includes some techniques for working with ratios, such as alternate ratio [item 46], which are not referred to in the Introduction but are defined in the Elements, Book V. This part is Nichmacusian with some Euclidean amendments. Thus, the content of al-B¯uzj¯an¯ı’s treatise is a combination of both traditions. The sequence of concepts is also noteworthy, which in al-B¯uzj¯an¯ı’s treatise follows Theon’s On Mathematical Matters Useful for Reading Plato, while in Nichomachus’ Introduction, the concepts in part 3 are before the concepts of part 2. This sequence is far from the arrangement in the Elements, where ratios are introduced before numbers.

6 The Book of Arithmetic In Sect. 5, the comparison between the Greek sources, including al-B¯uzj¯an¯ı’s terms and their definitions, showed that none of them could be The Arithmetic Book referred to in the title of al-B¯uzj¯an¯ı’s treatise; in particular, it is not a good match with Nicomachus’ Introduction, since not only were some of the definitions borrowed from the Elements but also the sequence of concepts was different. In addition, the terminology of al-B¯uzj¯an¯ı’s treatise is not the same as the Arabic translation of the Introduction. Definition 17, ‘adad al-ghayr¯ı (the differing number)

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H. Amini

refers to a heteromecic number (one that is the product of two consecutive numbers), but the term used in the Arabic translation of Introduction for this number is Mukhtalif al-ad.l¯a‘ (with different sides); in definition 18, ‘adad al-mustat.¯ıl (oblong number), which means a promecic number (one that is the product of two numbers that differ by two or more), in the Arabic translation of Introduction is translated to s.ah.¯ıh.at (correct). (Nicomachus and Martin Luther 1926, p. 254; Nicomachus 1959, pp. 84–85). Since the list of terms appears to be a combination of Greek sources, it is possible that The Arithmetic Book is a book which was compiled later. Ibn al-Nad¯ım provides us with a list of the candidates: 1. al-Madkhal il¯a al-arithm¯a.t¯ıq¯ı (Introduction to Arithmetic) by B¯uzj¯an¯ı himself (al-Nad¯ım 1872, p. 283) 2. Kit¯ab al-arithm¯a.t¯ıq¯ı fi al-‘adad wa al-jabr wa al-muq¯abalat (Book on Arithmetic About Numbers and Algebra) by Ah.mad ibn al-T.ayyib al-Sarakhs¯ı (al-Nad¯ım 1872, p. 261) 3. Ris¯alat fi al-Madkhal il¯a al-arithm¯a.t¯ıq¯ı (A Treatise About Book on Arithmetic), in five books by Abu Y¯usuf Ya‘q¯ub ibn ’Ish.a¯ q as.-S.abb¯ah. al-Kind¯ı (al-Nad¯ım 1872, p. 256)3 There is no decisive evidence to single out one of these books as the likely candidate; however, B¯uzj¯an¯ı’s book al-Madkhal il¯a al-arithm¯a.t¯ıq¯ı (Introduction to Arithmetic) could justify why these arithmetic terms should be memorized. The purpose of memorizing arithmetic terms could be either practical or pedagogical. The practical use is already the primary purpose of the well-known al-B¯uzj¯an¯ı’s work, Kit¯ab f¯ı m¯a yah.t¯aju ilayhi al-kutt¯ab wa-al-‘umm¯al wa-ghayruhum min ‘ilm alh.is¯ab (A Book About What Is Necessary for Scribes, Dealers, and Others from the Science of Arithmetic). The intended readership of the treatise is the administrative staff of courts and businessmen who need calculations as a part of their job or for financial tasks. Comparing this book with the terms in On What Should Be Memorized Before the Arithmetic Book shows no similarity between them. However, at the beginning of this book, after a short passage on ratio and its types, al-B¯uzj¯an¯ı informs us that he had written an introduction on the art of arithmetic, which is, in his opinion, sufficient for this subject (al-B¯uzj¯an¯ı 1971, p. 71). This non-extant book could be both this introduction and The Arithmetic Book in the title of the treatise On What Should Be Memorized Before the Arithmetic Book. The meaningful distinction between A Book About What Is Necessary for Scribes, Dealers, and Others from the Science of Arithmetic and On What Should Be Memorized Before the Arithmetic Book represents two different traditions in arithmetic in the tenth century AD. The first book is influenced by the Indian tradition, which was on the verge of general acceptance, regarding the practical capacities of Hindi numerals in calculations; the second treatise belongs to Greek tradition, characteristically showing a more abstract attitude toward number theory.

3 The

book is lost, but arithmetic of al-Kind¯ı is discussed in Brentjes 2021.

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In this framework, al-B¯uzj¯an¯ı’s treatise could have been written for a pedagogical purpose, to teach the students the basic concepts of numbers, so the treatise is a list of definitions of basic arithmetic concepts for students of Mathematics in alB¯uzj¯an¯ı’s time. The learning and teaching of mathematics in Islamic classical age is a broad area of scholarship; this treatise is probably the earliest evidence of a tradition of education in number theory. There are some other instances, for example, alB¯ır¯un¯ı’s (d. after 1048) Kit¯ab al-tafh¯ım li-aw¯a’il s.in¯a‘at al-tanj¯ım (The Book of Instruction in the Elements of the Art of Astrology) which provides a short similar introduction, and the goal of the book is surely educational (Brentjes 2021, pp. 128– 131).

7 Conclusion The F¯ım¯a yanbagh¯ı an yuh.fizu qabl kit¯ab al-arithm¯a.t¯ıq¯ı (On What Should Be Memorized Before the Arithmetic Book) is the title of a work by al-B¯uzj¯an¯ı of which six different manuscripts remain, recorded under various titles. The arithmetic book in the title could refer to another of his works on arithmetic, which has not survived. The treatise’s content is a combination of Neopythagorean concepts and Euclidean definitions.

A.1 Appendix: English Translation of Text and Original Text This is a treatise of Abu al-wafa muhammad ibn muhammad ibn al-Muhandis alhasib He said: [1] The unit is by which an entity is called one. [2] The number is the plurality of units. [3] The even-times-even number is that which divides into halves, and each half of its halves divides into halves until it ends in a unit. [4] An even-times-odd number is that which divides into halves once, and its division does not end in a unit. [5] A prime number is that which is measured only by a unit. [6] A composite number is that which is also measured by another number other than a unit. [7] An odd-times-odd number is that which is measured only by an odd number. [8] Relatively composite numbers are those which share one part, and that part is one common number, measuring both of them. [9] A perfect number is a number that, if its parts are added together, is equal to it. [10] An abundant number is a number that, if its parts are added together, is more than it.

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[11] A deficient number is a number that, if its parts are added together, is less than it. [12] Amicable numbers are those that, if the parts [of one of them] are added, [the result] is equal to the other. [13] Multiplication is the reproducing of one of the two numbers as many times as there are units in the other. [14] A plane number is what is formed by multiplying two numbers one by another. [15] A square number is what is formed by multiplying a number by itself. [16] The side of the square and its root is what has been multiplied by itself. [17] The differing number is what is formed by multiplying two numbers, whose difference is one, in each other. [18] An oblong number is what is formed by multiplying two numbers, whose difference is more than one, in each other. [19] A complementary number is what is formed by multiplying the sides of two squares by each other. [20] A solid number is what is formed by multiplying a plane number by another number. [21] A cube number is what is formed by multiplying a square number by its side. [22] A plinth number is what is formed by multiplying a square number by less than its side. [23] A beam number is what is formed by multiplying a square number by more than its side. [24] A circular number is what, if multiplied by a [number] like it, returns to itself.4 [25] Analogous numbers are those whose ratio to each other is like the ratio of a square number to a square number. [26] An absolute quantity is what is every single number called. [27] A relative quantity is what is formed by the relation of a number to a number. [28] An equal relation is a relation to an equal number. [29] An unequal relation is a relation to an unequal number. [30] A greater difference is what forms by relating a number to a number less than it. [31] A smaller difference is what forms by relating a number to a number more than it. [32] A Multiple is what is formed by relating a number to a number in a way that what relates is a multitude of what is related [33] A superparticular is what is formed by relating a number to a number in a way that what relates is more than what is related, as much as one part of it [34] A superpartient is what is formed by relating a number to a number in a way that what relates is more than what is related [35] A multiple superparticular is what is formed by relating a number to a number in a way that what relates is a multitude of what is related plus one part of it

4 The square number of a number ends with the same digits as the number itself, for example 5.5 = 25.

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[36] A multiple superpartient is what is formed by relating a number to a number in a way that what relates is a multitude of what is related plus parts of it. And when these five are worked out, and the less is related to the more, and a point is also added to its writings, and it is said that it is under the multiple, under the superpartient, under the multiple superparticular, under the multiple superpartient.5 [37] A ratio is the relation of numbers one to another [number]. [38] A proportion is the combination of ratios. [39] Terms are the numbers that some of them are in a ratio to each other. [40] An antecedent term is a number that is in relation [to another number]. [41] A consequent term is a number that [another number] is in relation to it. [42] A continued proportion is which connects one term to two terms in a proportion that is consequent to the first term and antecedent to the second term. [43] A disjunct proportion is in which, after the first term, there is a term that is not antecedent to the third term, and after the third term, there is a term to which the third term is its antecedent. [44] Combination of ratios is the ratio of antecedent plus consequent to consequent. [45] Separation of ratio is the ratio of the excess by which the antecedent exceeds the consequent to consequent. [46] Alternate ratio is the ratio of antecedent to antecedent and consequent to consequent. [47] Inverse ratio is the ratio of consequent to antecedent. [48] Conversion ratio is the ratio of antecedent to the excess by which the antecedent exceeds the consequent. [49] Equality of ratio is the ratio of the extreme terms to each other. [50] Numerical proportion is what is possible for terms to be in the same ratio. [51] The third proportion is that the ratio of the greatest term to the smallest term is like the difference between two greater terms and two smaller terms. [52] The fourth proportion is that the ratio of the greatest term to the smallest term is like the difference between two smaller terms and two greater terms. [53] The fifth proportion is that the ratio of the middle term to the smallest term is like the difference between them and two greater terms. [54] The sixth proportion is that the ratio of the greatest term to the middle term is like the difference between two smaller terms and two greater terms. [55] The seventh proportion is that the ratio of the greatest term to the smallest term is like the difference between two extreme terms and two greater terms.

5 multiples, as, mn : n superparticulars, as n + I : n superpartients, as n + k: n, k > I multiple superparticulars, as mn + I : n, m > I multiple superpartients, as mn + k : n, both m and k being greater than 1.

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‫‪[56] The eighth proportion is that the ratio of the greatest term to the smallest term‬‬ ‫‪is like the difference between two extreme terms and two smaller terms.‬‬ ‫‪[57] The ninth proportion is that the ratio of the middle term to the smallest term is‬‬ ‫‪like the difference between two extreme terms and two smaller terms.‬‬ ‫‪[58] The tenth proportion is that the ratio of the middle term to the smallest term is‬‬ ‫‪like the difference between two extreme terms and two greater terms.‬‬ ‫‪The treatise is finished.‬‬

‫بسم الله الرحمن الرحیم‬ ‫هذه ‪ 6‬رسالة البی الوفاء محمد بن محمد المهندس الحاسب‬ ‫قال ]‪ [1‬الوحدة هی التی ‪ 8‬بها یقال علی کل موجود واحد‬ ‫]‪ [2‬العدد هو کثرة الوحدات‬ ‫]‪ [3‬زوج الزوج هو ما ینقسم بنصفین و کل نصف من انصافه ینقسم بنصفین حتی‬ ‫ینتهی إلی الوحدة‬ ‫]‪ [4‬زوج الفرد هو ما ینقسم بنصفین مرة واحدة و الینتهی به القسمة إلی الوحدة‬ ‫]‪ [5‬العدد األول هو الذی یعدّه الوحدة فقط‬ ‫]‪ [6‬العدد المرکب هو الذی یعده مع الوحدة عدد‬ ‫]‪ [7‬فرد الفرد هو الذی یعده عدد فرد فقط‬ ‫]‪ [8‬االعداد المشترکة هی التی یشترک فی جزء واحد و هی التی یوجد لها عدد واحد‬ ‫مشترک یعدها جمیعا‬ ‫]‪ [9‬العدد التام هو الذی یکون اجزاؤه اذا جمعت مساویة له‬ ‫]‪ [10‬العدد الزاید هو الذی یکون اجزاؤه اذا جمعت أکثر منه‬ ‫]‪ [11‬العدد الناقص هو الذی یکون اجزاؤه اذا جمعت أقل منه‬ ‫]‪ [12‬االعداد المتحابه هی التی اذا جمعت أجزاء کل منها کانت متساویة لالخر‬ ‫]‪ [13‬الضرب هو تضعیف احد العددین بقدر ما فی اآلخر من اآلحاد‬ ‫]‪ [14‬العدد المسطح هو ما یکون من ضرب عددین احدهما فی اآلخر‬ ‫]‪ [15‬العدد المربع هو مایکون من ضرب عدد فی مثله‬ ‫]‪ [16‬ضلع المربع و جذره هو العدد الذی ضرب فی مثله‬ ‫]‪ [17‬العدد الغیری ‪ 9‬هو ما یکون من ضرب عددین تفاضلها واحد احد هما فی اآلخر‬ ‫]‪ [18‬العدد المستطیل هو ما یکون من ضرب عددین تفاضلهما اکثر من واحد احدهما‬ ‫فی اآلخر‬ ‫]‪ [19‬العدد المتمم هو مایکون من ضرب ضلعی مربعین احدهما فی اآلخر‬ ‫]‪ [20‬العدد المجسم هو ما یکون من ضرب عدد مسطح فی عدد آخر‬ ‫]‪ [21‬العدد المکعب هو ما یکون من ضرب عدد مربع فی ضلعه‬ ‫]‪ [22‬العدد اللبنی هو ما یکون من ضرب مربع فی أقل من ‪ 10‬ضلعه‬ ‫‪7‬‬

‫هذه‬ ‫افولا‬ ‫ ‪8 M:‬‬‫یتلا‬ ‫‪9 M:‬‬ ‫‪; there is no dots in other manuscripts‬یربعلا‬ ‫ ‪10 M, N, P, U:‬‬‫‪ in margin‬لقا یف ‪۲:‬ض ؛نم لقا‬ ‫‪-‬‬

‫‪6 R2:‬‬ ‫‪7 M:‬‬

‫‪13‬‬

‫‪A Tenth-Century Mathematical Glossary‬‬

‫]‪ [23‬العدد التیری هو ما یکون من ضرب ‪ 11‬مربع فی أکثر من ضلعه‬ ‫]‪ [24‬العدد الدوری هو الذی اذا ضرب فی مثله عاد إلی نفسه‬ ‫]‪ [25‬االعداد المتشابهه هی التی تکون نسبة ‪ 12‬بعضها إلی بعض کنسبة عدد مربع‬ ‫إلی عدد مربع‬ ‫]‪ [26‬الکمیة المفردة هی ما یقال علی کل عدد مفرد‬ ‫]‪ [27‬الکمیة المضافة هی ما یکون من إضافة عدد إلی عدد‬ ‫]‪ [28‬إضافة التساوی هی ما یکون من إضافة ‪ 13‬إلی عدد مساو له‬ ‫]‪ [29‬إضافة االختالف هی ما یکون من إضافة عدد إلی عدد مخالف له‬ ‫]‪ [30‬االختالف األعظم هو ما یکون من إضافة عدد إلی عدد أقل منه‬ ‫]‪ [31‬االختالف األصغر هو ما یکون من إضافة عدد إلی عدد أکثر منه‬ ‫]‪ [32‬ذو األضعاف هو ما یکون من إضافة عدد إلی عدد یکون المضاف أضعاف المضاف‬ ‫الیه‬ ‫‪14‬‬ ‫]‪ [33‬الزاید جزًا‬ ‫هو ما یکون من إضافة عدد إلی عدد یکون المضاف اکثر من‬ ‫مضاف الیه بجزء منه‬ ‫]‪ [34‬الزاید األجزاء هو ما یکون من تضعیف عدد إلی عدد یکون المضاف اکثر من‬ ‫المضاف الیه‬ ‫]‪ [35‬ذواألضعاف و زیادة جزء هو ما یکون من إضافة عدد إلی عدد یکون المضاف‬ ‫أضعاف المضاف الیه و زیادة جزء منه ‪15‬‬ ‫]‪ [36‬ذواألضعاف و زیادة أجزاء هو ما یکون من إضافة عدد إلی عدد یکون المضاف‬ ‫أضعاف المضاف الیه و زیادة أجزاء منه ‪16‬‬ ‫‪17‬‬ ‫و اذا عملت هذه الخمسه و اضیف األقل الی‬ ‫األکثر و اضیف ایضا إلی رسومها نقطة‬ ‫و قیل هی تحت ذی األضعاف‪ ،‬تحت الزاید أجزاء‪ ،‬تحت ذی األضعاف و زیادة جزء‪ ،‬تحت‬ ‫ذی األضعاف ‪ 18‬و زیادة األجزاء‬ ‫]‪ [37‬النسبة هی إضافة عددین احدهما إلی اآلخر‬ ‫]‪ [38‬المناسبة هی جمع النسب‬ ‫]‪ [39‬الحدود هی االعداد التی ینسب بعضها إلی بعض‬ ‫]‪ [40‬الحد المقدم هو العدد المنسوب‬ ‫]‪ [41‬الحد التالی هو العدد المنسوب الیه‬

‫ددع ‪+‬‬ ‫ةبسن ‪-‬‬ ‫‪13 M: +‬‬ ‫ددع یلإ‬ ‫ ‪14 M, N, P, U:‬‬‫نوکی ام وه‬ ‫‪R2: in margin‬‬ ‫ ‪15 M, P:‬‬‫ةدایز و فاعضألاوذ‬ ‫‪11 R2:‬‬ ‫‪12 R2:‬‬

‫;ًازجلا دیازلا هیلا فاضملا فاعضأ فاضملا نوکی ددع یلإ ددع ةفاضإ نم‬

‫فاضملا فاعضأ فاضملا نوکی ددع یلإ ددع ةفاضإ نم نوکی ام وه ءزج‬ ‫هنم ءزج ةدایز و هیلا‬ ‫‪16 R2:‬‬ ‫هیلا فاضملا فاعضأ فاضملا نوکی ددع یلإ ددع ةفاضإ نم نوکی ام وه ءازجأ ةدایز و فاعضألاوذ‬ ‫‪ in margin‬هنم ءازجأ ةدایز و‬ ‫‪17 R2: +‬‬ ‫ددع‬ ‫ ‪18 M:‬‬‫فاعضألا یذ تحت ءزج ةدایز و‬

‫‪H. Amini‬‬

‫‪14‬‬

‫]‪ [42‬المناسبته المتصله هی أن یتصل حد واحد بحدین فی النسبة و یکون عند االول‬ ‫تالیا و عند الثانی مقدما‬ ‫]‪ [43‬المناسبته المنفصله هی أن یکون للحد االول حد یتلوه و الیکون مقدما للثالث‬ ‫و یکون للحد الثالث حد ‪ 19‬یتلوه و هو مقدم له‬ ‫]‪ [44‬ترکیب النسبة هو نسبة المقدم و التالی إلی التالی‬ ‫]‪ [45‬تفصیل النسبة هو نسبة فضل المقدم علی التالی إلی التالی‬ ‫]‪ [46‬ابدال النسبة هو ‪ 20‬نسبة المقدم إلی المقدم و التالی إلی التالی‬ ‫]‪ [47‬خالف النسبة هو نسبة التالی إلی المقدم‬ ‫]‪ [48‬قلب النسبة هو نسبة المقدم إلی فضله علی التالی‬ ‫]‪ [49‬مساواة النسبة هی نسبة اطراف الحدود بعضها إلی بعض‬ ‫]‪ [50‬المناسبة العددیه هی أن یمکن الحدود علی نسبة مساویة‬ ‫]‪ [51‬المناسبة الثالثة هی أن تکون نسبة الحد األعظم إلی الحد األصغر کنسبة تفاضل‬ ‫األعظمین إلی تفاضل األصغرین‬ ‫]‪ [52‬المناسبة الرابعة هی أن تکون نسبة الحد األعظم إلی الحد األصغر کنسبة تفاضل‬ ‫األصغرین إلی تفاضل األعظمین‬ ‫]‪ [53‬المناسبة الخامسة هی أن تکون نسبة الحد االوسط إلی الحد األصغر کنسبة‬ ‫تفاضلهما إلی تفاضل األعظمین‬ ‫]‪ [54‬المناسبة السادسة هی أن تکون نسبة الحد األعظم إلی الحد االوسط کنسبة‬ ‫تفاضل األصغرین إلی تفاضل األعظمین‬ ‫]‪ [55‬المناسبة السابعة هی أن تکون نسبة الحد األعظم إلی الحد األصغر کنسبة تفاضل‬ ‫الطرفین إلی تفاضل األعظمین‬ ‫]‪ [56‬المناسبة ثامنه هی أن تکون نسبة الحد األعظم إلی الحد األصغر کنسبة تفاض‬ ‫ل الطرفین إلی تفاضل األصغرین ‪21‬‬ ‫]‪ [57‬المناسبة التاسعة هی أن تکون نسبة الحد االوسط إلی الحد األصغر کنسبة‬ ‫تفاضل الطرفین إلی تفاضل األصغرین‬ ‫]‪ [58‬المناسبة العاشرة هی أن تکون نسبة الحد االوسط إلی الحد األصغر کنسبة‬ ‫تفاضل الطرفین إلی تفاضل األعظمین‬ ‫تمت الرسالة‪.‬‬

‫‪References‬‬ ‫‪Al-Buzj¯an¯ı. (1971). T¯ar¯ıkh ‘ilm al-h.is¯ab al-‘arab¯ı. Vol. 1: h.is¯ab al-yad (hand calculation), the‬‬ ‫‪edition of Kit¯ab f¯ı m¯a yah.t¯aju ilayhi al-kutt¯ab wa-al-‘umm¯al wa-ghairuhum min ‘ilm al-h.is¯ab‬‬ ‫‪by Ahmad Salim Saidan, Amman.‬‬ ‫دح ثلاثلا دحلل نوکی و ‪-‬‬ ‫نوکی ام ‪+‬‬ ‫ ‪21 M, N, P, U:‬‬‫نیفرطلا لضافت ةبسنک رغصألا دحلا یلإ مظعألا دحلا ةبسن نوکت نأ یه هنماثلا ةبسنلا‬ ‫‪19 R2:‬‬ ‫‪20 R2:‬‬

‫نیرغصألا لضافت یلإ‬

A Tenth-Century Mathematical Glossary

15

Al-Nad¯ım, I., Flügel, G., Roediger, J., & Müller, F. A. (1872). Kit¯ab al-fihrist. Khayats. Al-Qift.¯ı, A. L. H. (1903). T¯ar¯ıkh al-Hukam¯a’. Edition by Lippert. Leipzig. Brentjes, S. (2021). Nicomachean Number Theory in Arabic and Persian Scholarly Literature. In Brill’s Companion to the Reception of Pythagoras and Pythagoreanism in the Middle Ages and the Renaissance (pp. 111–140). Brill. Euclid. Elements. The manuscript no. 3439/1. Fatih library. Turkey. De Young, Gregg. “The Arabic textual traditions of Euclid’s Elements.” Historia Mathematica 11.2 (1984): 147–160. Freudenthal, G. (2021). The Tribulations of the Introduction to Arithmetic from Greek to Hebrew Via Syriac and Arabic. Nicomachus of Gerasa, H.abib Ibn Bahr¯ız, al-Kind¯ı, and Qalonymos ben Qalonymos. In Brill’s Companion to the Reception of Pythagoras and Pythagoreanism in the Middle Ages and the Renaissance (pp. 141–170). Brill. Heath, T. L. (Ed.). (1956). The thirteen books of Euclid’s Elements. Courier Corporation. Nicomachus, G., & Martin Luther, D. O. (1926). Introduction to arithmetic. Macmillan. Nicomachus, of Gerasa (tr. Thâbit ibn Qurra, ed. Wilhelm Kutsch). (1959). Thâbit B. Qurra’s arabische übersetzung der Arithmêtikê eisagôgê des Nikomachos Von Gerasa zum ersten Mal herausgegeben. von Wilhelm Kutsch. Beyrouth: Impr. catholique. Qurb¯an¯ı, A. A. Q. (1986). Zindag¯ı-n¯amah-i riy¯ad¯ı’d¯an¯an dawrah-i Isl¯am¯ı. Tehran: Markaz-i Nashr-i Danishgahi. Sezgin, F. (1974). Geschichte des arabischen Schrifttums: Mathematik bis ca. 430 H (Vol. 5). Leiden: EJ Brill. Toulis, Christos. (1979). Mathematics useful for understanding plato by Theon of Smyrna: Translated from the 1892 Greek/French edition of J. Dupuis by Robert and Deborah Lawlor and edited and annotated by Christos Toulis and others. Secret Doctrine Reference Series. San Diego, CA (Wizards Bookshelf).

Euclid in Mar¯agha: The Age of the Tah.r¯ır Gregg De Young

Abstract This paper examines three Arabic tah.r¯ır or editions of Euclid’s Elements prepared in Mar¯agha during the seventh/thirteenth century by Nas.¯ır al-D¯ın al-T.u¯ s¯ı, Muh.yi al-D¯ın al-Maghrib¯ı, and an unnamed author (whose work was incorrectly attributed to al-T.u¯ s¯ı on its title page when printed in Rome in 1594). These editions represent what might arguably be called the culmination of the process of assimilation for Euclidean geometry into the Islamic intellectual tradition. These tah.r¯ır also reveal the role of geometry as the foundation of education in the mathematical sciences as well as the idiosyncratic mathematical interests of their creators. Finally, these Tah.r¯ır cast an indirect light on the complex issue of the early transmission of the Elements into Arabic and the relations between the various translation or transmission strands as they were known in the seventh/thirteenth century Mar¯agha.

1 Introduction The seventh/thirteenth century1 is often seen by historians of medieval mathematics as one of the high points in the development of the mathematical sciences within the Islamicate world. Frequently the emphasis of these historians is on mathematical

1 Modern Western scholars conventionally indicate dates in the Islamicate world using a dual calendar system. Dating of events and manuscripts are recorded within the Islamic context using the lunar Hijri calendar, which takes its starting point from the flight (hijra) of Muh.ammad from Mecca to Medina in year 622 of the Gregorian calendar. The first date is always the Hijri date, separated from the Gregorian equivalent by a forward slash mark.

G. De Young () Department of Mathematics and Actuarial Science, The American University in Cairo, Cairo, Egypt e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_2

17

18

G. De Young

cosmography (hay’a), which saw the production of several sophisticated nonPtolemaic models2 of the universe along with a mathematical apparatus that gave these models considerable predictive power. But during this period, nearly all the mathematical sciences saw considerable advancement and elaboration. Euclidean geometry was a part of this mathematical flowering. Traditionally, Euclid’s Elements had served as the standard introduction to “higher” mathematical disciplines, leading ultimately to a study of mathematical astronomy in Ptolemy’s Almagest. Seventh/thirteenth-century Maragha saw the composition of three tah.r¯ır based on Euclid’s Elements—by Nas.¯ır al-D¯ın al-T.u¯ s¯ı, Muh.y¯ı al-D¯ın al-Maghrib¯ı, and the Pseudo-T.u¯ s¯ı. Although these three tah.r¯ır differ from one another in important ways, when taken together they reveal important features of the status of Euclidean studies at Mar¯agha and what their authors considered important in Euclidean geometry. The term tah.r¯ır has sometimes been translated as redaction (in the traditional sense of re-editing or revising an existing text). And on one level, this is an adequate description of the genre. These revisions of Euclid’s Elements do include restating the content of the Arabic translations of the Elements in a purer or more flowing Arabic style, usually without creating new technical vocabulary or introducing major changes in the arrangement of the contents. But these tah.r¯ır do more than merely paraphrase Euclid’s geometry in a more elegant Arabic. The authors also tried to enrich the original Euclidean text in various ways, bringing it up-to-date by adding results of more recent scholarship. And so each tah.r¯ır also reflects its author’s view of what was important in earlier discussions of the Euclidean classic. And, for the historian interested in the transmission history of the Elements, these tah.r¯ır offer some tantalizing glimpses into the complex transmission of Euclid’s treatise from Greek into Arabic. They do so both explicitly and implicitly. Some offer explicit testimony through editorial comments scattered at irregular intervals throughout the text reporting differences in structural features that the authors attribute to one or another of the early Arabic translators. They offer implicit evidence of the continuing influence of these translations through their choice of technical vocabulary and structural features incorporated into their respective tah.r¯ır.

2 Essential Background: Transmission of the Elements from Greek to Arabic Although many of us have been taught that a literary drama begins in media res, a historical drama should begin at the beginning. And in this mathematical drama, the beginning is the transmission of Euclid from Greek into Arabic during

2 By non-Ptolemaic models, I mean those models that attempted to avoid use of the system of deferent and epicycle circles—and especially the eccentric equant point—developed by Ptolemy in order to provide a mathematical description of the observed irregularities in the motions of celestial objects in terms of the uniform and circular motions postulated in Aristotelian physics.

Euclid in Mar¯agha

19

the second/eighth century. Our story begins with the often-repeated report of alNad¯ım, a fourth/tenth-century Baghdad bookseller and biobibliographer, in his Fihrist (Al-Nad¯ım 1430/2009, II, 207–210; Dodge 1970, II, 634–636). The first protagonist introduced in his report is al-H.ajj¯aj ibn Y¯usuf ibn Mat.ar. It was he who is credited with producing the first Arabic version of the Elements under the caliph H¯ar¯un al-Rash¯ıd (ruled 170–193/786–809). Later, during the reign of caliph al-Ma’m¯un (ruled 198–218/813–833), he produced a new version. This new version—modern scholars doubt that it was a completely new translation—involved some manipulation of the text in order to correct errors, supply missing features, omit unneeded phrases, and generally reduce the text to a smaller size and make it more user-friendly (Murdoch 1971, 439; Brentjes 2006a, 171). Neither of these two versions is extant in pristine form, although brief quotations preserved in a few Arabic documents provide clues as to what they might have been like (De Young 1991, Brentjes 1994, De Young 2003a; 2014). According to al-Nad¯ım’s report, a second translation was made later in the third/ninth century by Ish.a¯ q ibn H.unayn (son of the famous translator of Greek medical treatises into Arabic, H.unayn ibn Ish.a¯ q). The relation of this translation to the earlier translation of al-H.ajj¯aj is left ambiguous by the biobibliographer, but historians have traditionally assumed that the two translations were relatively independent. Ish.a¯ q’s translation, in turn, was revised or edited by the mathematician Th¯abit ibn Qurra (221–288/836–901), apparently using additional Greek manuscripts. Thus, according to the narrative of al-Nad¯ım, in the course of only a century, no less than four versions of the Elements were produced in Arabic. Until the beginning of this century, it seemed that all that had survived of this complex transmission derived from the Ish.a¯ q–Th¯abit transmission.3 The surviving manuscript tradition is not so monolithic as this brief summary might seem to imply at first glance. Within the so-called Ish.a¯ q–Th¯abit transmission, there are two distinct families of manuscripts. These families have sometimes been designated as Group A and Group B (De Young 1984). The two families differ from one another in the ordering of some definitions and propositions and in the technical vocabulary they employ, as well as the architecture of the diagrams for certain propositions. Recent studies have shown that a small subset of diagrams—I shall use the diagrams for propositions II, 14, VI, 22, and VII, 14—can provide a useful diagnostic test to distinguish between manuscripts in Group A and Group B traditions (Figs. 1, 2, and 3). Nearly two decades ago Brentjes (2006a) identified the manuscript Mumbai, Mull¯a F¯ır¯uz, R.I.6, as a version of the Elements different from what had been previously known in the manuscript tradition.4 Although the manuscript contains no explicit attribution, it incorporates nearly all the structural features that were 3 Even though a few manuscripts carry explicit attributions of some sections or books to al-Hajj¯ . aj, the textual readings are typically so close to those of other manuscripts explicitly attributed to Ish.a¯ q–Th¯abit that most scholars assumed these attributions to al-H.ajj¯aj must be incorrect. 4 De Young (2023) has recently discovered another similar but not identical manuscript (Paris, BULAC, Ara 606) from the same tradition. The Paris manuscript is complete, while the Mumbai manuscript breaks off after book IX.

20

G. De Young

Diagram edited from: Tehran, Malik 3586, folio 77a

Diagram edited from: Oxford, Bodleian Library, Thurston 11, folio 26a

Fig. 1 Diagram for Elements II, 14 associated with Group A (left) and Group B (right)

Diagram edited from: Copenhagen, Kongelige Biblioteket, Mehren LXXXI, folio 27b

Diagram edited from: Cambridge University, ad. 1075, folio 71a

Fig. 2 Diagrams for VI, 22 associated with Group A (left) and Group B (right)

Diagram edited from: Dublin, Chester Beatty, arab. 3035, folio 45a

Diagram edited from: Cambridge University, ad. 1075, folio 80b

Fig. 3 Diagrams for VII, 14 associated with Group A (left) and Group B (right)

Euclid in Mar¯agha

21

explicitly linked to al-H.ajj¯aj in various medieval sources.5 She concluded that this version of the Elements was derived from the translation of al-H.ajj¯aj. In the course of further studies, Brentjes discovered that large sections of the text (at least in books III–IX) were nearly identical to readings in manuscripts explicitly attributed to Ish.a¯ q–Th¯abit. Based on careful analysis of internal philological features, Brentjes has now proposed a radically different understanding of the narrative of al-Nad¯ım in relation to the surviving Arabic manuscripts. It is not the Ish.a¯ q–Th¯abit translation that survives in the Arabic manuscripts, but rather that of al-H.ajj¯aj. The explicit attributions to Ish.a¯ q and Th¯abit arose, she believes, through the process of editing and re-editing the Arabic text (Brentjes 2018b). It was on the foundation of this historically complex transmission of Euclid’s Elements from Greek into Arabic described by al-Nad¯ım that a rich tradition of elaborations, discussions, and interpretations of the Euclidean classic was created over the next four centuries. And it is against the background of this now rapidly shifting historical interpretation that we shall attempt to situate the three tah.r¯ır or editions of the Elements made in seventh/thirteenth-century Mar¯agha.

¯ ı 3 The Tah.r¯ır of Nas.¯ır al-D¯ın al-T.us¯ Because it is chronologically the first of the three tah.r¯ır and because it was the most influential historically, we begin with the treatise of al-T.u¯ s¯ı.

3.1 The Author Ab¯u Ja‘far Muh.ammad b. Muh.ammad b. al-H.asan Nas.¯ır al-D¯ın al-T.u¯ s¯ı, like many scholars of his time, was a polymath and wrote on topics ranging from mathematics to medicine and from prosody to ethics to religion (Ragep 1993, I, 3–23; Djebbar 2003). As is often the case with medieval scholars, we know little of his personal life apart from what he tells us in his “spiritual autobiography” (Al-T.u¯ s¯ı 1999). Born in T.u¯ s, near Mashhad in what is now northeastern Iran in 597/1201, he informs us in his autobiography that he was educated first at home by his father, an important religious scholar of his day, and by an uncle who introduced him to mathematics, especially algebra and geometry. After the death of his father, he traveled to N¯ısh¯ap¯ur, where he studied philosophy with Far¯ıd al-D¯ın al-D¯am¯ad, a follower of Ibn S¯ın¯a’s school, and medicine—primarily the Canon of Ibn S¯ın¯a— under Qut.b al-D¯ın al-Mas.r¯ı, the most outstanding student of Fakhr al-D¯ın al-R¯az¯ı (died 606/1210). He also read mathematics under the tutelage of the mathematician Kam¯al al-D¯ın ibn Y¯unus (550–639/1156–1242). It was at N¯ısh¯ap¯ur that he first began to acquire a reputation as an exceptional mathematician and scholar. 5 Elior

(2020) has identified a medieval Hebrew manuscript that also includes many of the features attributed to al-H.ajj¯aj.

22

G. De Young

Al-T.u¯ s¯ı lived through the political and social turbulence surrounding the collapse of the Abbasid Caliphate after the invasion of the Mongol warlords of Central Asia. Thanks to his reputation, he was invited to join the court of the Ism¯a‘¯ıl¯ı ruler, Nas.¯ır al-D¯ın Muh.tashim, and moved to Quhistan sometime before 1232, the year in which he completed his Akhl¯aq-i n¯as.ir¯ı and dedicated the work to his patron. It was also during this period that he completed his Tah.r¯ır of the Elements (646/1246). By 1256, the Mongol invaders had put an end to the Ism¯a‘¯ıl¯ı state in northern Iran. The Mongol ruler, H¯ulag¯u, who founded the Ilkh¯an dynasty, invited al-T.u¯ s¯ı to join his court. He soon persuaded H¯ul¯ag¯u, who had a keen interest in prognostic astrology, to found a new observatory. Here he directed an observational program that attracted scholars from all parts of the Islamicate world to Mar¯agha. Al-T.u¯ s¯ı spent the remainder of his career in service to the Ilkh¯an administration.

3.2 General Characteristics of al-T.us¯ ¯ ı’s Tah.r¯ır Al-T.u¯ s¯ı’s Tah.r¯ır included Euclid’s original 13 books of the Elements as well as the two appended treatises called books XIV and XV which are attributed to Hypsicles in the Arabic Euclidean transmission.6 This Tah.r¯ır provides a re-edited version of the original Arabic translation, an edition that is more in tune with traditional Arabic diction and style, streamlining the text and abandoning the translator’s attempt to reproduce every element in the Greek text. Another feature of his editing or streamlining of the text is evident in book X, Euclid’s extended discussion of families of incommensurable lines. Where Euclid had lengthy demonstrations, alT.u¯ s¯ı often reduces them to only a few lines, omitting the repetitions in Euclid’s demonstrations. One quickly receives the impression that al-T.u¯ s¯ı had little interest in this aspect of Euclidean mathematics. Although al-T.u¯ s¯ı’s Tah.r¯ır is based on what he understood to be the Ish.a¯ q–Th¯abit Arabic transmission, he was also familiar with the version traditionally attributed to al-H.ajj¯aj.7 This is stated explicitly in the introduction to his Tah.r¯ır, where al-T.u¯ s¯ı explains8

6 Euclid’s original treatise contained only 13 books. Two more (XIV and XV) were added in late antiquity. In the Arabic transmission, these two books are typically ascribed to Hypsicles. Modern scholarship assigns only book XIV to Hypsicles (flourished second-century BCE), who probably used material from a treatise by Apollonius to construct his own treatment of regular solids. Isidore of Miletus (flourished in the first half of the sixth-century CE) is sometimes credited with Book XV (Vitrac and Djebbar 2011). 7 Sezgin (1974, 112) says that al-Tu . ¯ s¯ı followed the transmission deriving from al-H.ajj¯aj. This mistake appears to arise because Sezgin did not initially distinguish clearly between the genuine Tah.r¯ır of al-T.u¯ s¯ı and the later tah.r¯ır incorrectly attributed to him (De Young 2003b; 2012c). By the time he published his facsimile edition of the Pseudo-T.u¯ s¯ı Tah.r¯ır in 1997, he had adopted a more modern view regarding the ascription of this early printed edition. 8 My translation, based on British Library, Add. 23387, folio 2b.

Euclid in Mar¯agha

23

The treatise consists of 15 books including the last two appended [books]. The text of alH.ajj¯aj has 468 propositions, and there are an additional 10 propositions in the text of Th¯abit. In some places, there is also a difference in ordering between the two of them. I record the number of the proposition for Th¯abit in red and that of al-H.ajj¯aj in black when they are different.

The diagrams used by al-T.u¯ s¯ı also suggest that he may have known both the Group A and Group B strands of the Th¯abit transmission. The diagram for proposition II, 14, for example, regularly shows the given figure as a quadrilateral (Fig. 4). In this case, the diagram follows the architecture typical of Group B manuscripts (Fig. 1). In the case of the diagram for proposition VI, 22, however, alT.u¯ s¯ı’s diagram (Fig. 5) follows the pattern of the Group A manuscripts (Fig. 2). The diagram for proposition VII, 14 in his Tah.r¯ır (Fig. 6) likewise follows the pattern of the Group A manuscripts (Fig. 3). Fig. 4 Diagram for II, 14 of al-T.u¯ s¯ı Tah.r¯ır, edited from British Library, Add. 23387, folio 47a

Fig. 5 Diagram for VI, 22 of al-T.u¯ s¯ı Tah.r¯ır, edited from Munich, Bayerische Staatsbibliothek, cod. arab. 2697, folio 69b Fig. 6 Diagram for VII, 14 of al-T.u¯ s¯ı Tah.r¯ır, edited from Tehran, Majlis Sh¯ur¯a 6109, folio 114a

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Al-T.u¯ s¯ı also appended to several propositions brief statements describing how what he called the version of al-H.ajj¯aj differed from what he called the version of Th¯abit. For example: • I, 45—This proposition (construction of a parallelogram with a given angle equal to a given rectilinear figure) is not in the text of al-H.ajj¯aj. • III, 37—This proposition (the inverse of Euclid’s proposition III, 36) is not in the text of al-H.ajj¯aj.9 It is among those added by Th¯abit because there is a need for it in the tenth [proposition] of book IV. • VI, 12—This proposition (to find a fourth line proportional to three given lines) is among those which Th¯abit added.10 • Post VIII, 25—The Arabic tradition includes two propositions, the converse of Elements VIII, 24 and 25. Al-T.u¯ s¯ı reports that these two propositions (the 24th and 25th) are not in the text of al-H.ajj¯aj. Pseudo-T.u¯ s¯ı converts these propositions into corollaries of Euclid’s propositions VIII, 22 and 23, respectively. Al-Nayr¯ız¯ı attributes these two propositions to Heron (Curtze 1899, 194–195). • IX, 12—In the text of al-H.ajj¯aj, this proposition precedes the one before it. • IX, 14—This proposition is the twentieth in the text of al-H.ajj¯aj. Al-T.u¯ s¯ı also incorporated numerous notes from earlier Arabic discussions of Euclid’s classic treatise into his edition. These notes included alternative demonstrations for some of Euclid’s propositions, as well as discussions of mathematical and logical difficulties raised by the Euclidean text. These notes are introduced by the stereotypical phrase aq¯ulu (“I say”), perhaps intended to suggest that they were the original work of al-T.u¯ s¯ı himself, an impression strengthened by the fact that al-T.u¯ s¯ı does not mention any mathematical predecessors by name in his treatise. De Young (2009) has shown, however, that the content of many of these notes was borrowed from the fifth/eleventh-century commentator on Euclid, Ibn al-Haytham. The editorial comments mentioned above, though, seem to be from al-T.u¯ s¯ı himself. In addition to these borrowed notes, al-T.u¯ s¯ı makes two major additions or insertions into the text of Euclid, both occurring in book I. The first is a demonstration of the famous parallel lines postulate of Euclid, inserted following Elements I, 28. The second major insertion is an extended discussion of the various possible placements of the squares on the sides of the right triangle which follows Euclid’s proposition I, 47, each case being worked out in full detail. Al-T.u¯ s¯ı provides two demonstrations of Euclid’s parallel lines postulate. The first consists of seven propositions and is based on an approach also used by alKhayy¯am. It involves replacing the Euclidean postulate with an equivalent statement (already used both by Th¯abit ibn Qurra and Ibn al-Haytham) that if two lines are 9 This proposition is numbered 36 by al-Tu . ¯ s¯ı because proposition III, 12 is routinely omitted by the Arabic authors. Heath (1956, II, 28) regards the proposition as an interpolation attributable to Heron. 10 This proposition is numbered 11 by al-Tu . ¯ s¯ı because the Arabic tradition rearranges several Greek propositions in book VI: Euclid’s propositions 9–10–11–12–13 are re-ordered as 13–11– 12–10–9.

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approaching one another (the distance between the lines is getting progressively smaller in one direction), they will eventually meet in that direction. The key to this first demonstration is the use of the equivalent of a Saccheri quadrilateral. Al-T.u¯ s¯ı follows this with an alternative demonstration in which he replaces propositions six and seven of the first demonstration with two new propositions followed by an eighth proposition which used the same approach to demonstration as that employed by al-Jawhar¯ı (active during the fourth/tenth century).11 The key feature of this alternative demonstration is the argument that if a point is chosen inside an angle and a line parallel to one of the two sides of the angle is constructed to pass through this point, then that line will intersect a line connecting the two endpoints of the sides of the angle (Jaouiche 1986, 99–106).12

3.3 Historical Influence The Tah.r¯ır of al-T.u¯ s¯ı continued to be read and studied for centuries. This is clear already from the large number of surviving manuscript copies, many of them annotated by readers. A possible reason for the continuing popularity of this Tah.r¯ır is that al-T.u¯ s¯ı had prepared similar redactions for a wide range of classical mathematical works, all derived from the Greek mathematical tradition. Since these mathematical treatises were traditionally intermediate between the Elements and the mathematical cosmography developed in the Almagest of Ptolemy, they were called collectively Mutawassit.a¯ t or Intermediate Books.13

11 Al-Tu . ¯ s¯ı’s eight-proposition alternative demonstration is also transmitted in an independent treatise, Ris¯ala al-sh¯afiya ‘an al-shakk f¯ı’l-khut.u¯ .t al-mutaw¯aziya, in which al-T.u¯ s¯ı discusses the status of Euclid’s postulate within the structure of the Elements, then criticizes several earlier attempts to demonstrate the postulate, and finally presents his own demonstration (Al-T.u¯ s¯ı 1359/1940, II, treatise 8). 12 Al-Tu . ¯ s¯ı’s demonstration gave rise to an interesting exchange of letters between Qays.ar ibn Ab¯ı al-Q¯asim (574–649/1178–1251), a lesser-known mathematician, and al-T.u¯ s¯ı in which they discuss Euclid’s parallel lines postulate. This discussion is one of several surviving examples of mathematical correspondence during the later medieval period. It reveals both the intense interest of mathematicians of the day in the problem of the parallel lines postulate and the existence of a vibrant mathematical community that communicated regularly, sometimes across long distances. The correspondence was initiated by Qays.ar, who wrote to al-T.u¯ s¯ı with a critique of the approach to the parallel postulate developed by the Hellenistic commentator, Simplicius, that had been quoted in the commentary of al-Nayr¯ız¯ı (Sabra 1969, 8–13). Al-T.u¯ s¯ı responded with his own brief analysis of the demonstration of Simplicius. In reply, Qays.ar pointed out that al-T.u¯ s¯ı had failed to consider all the possible cases and proposed a more general solution. In his final reply, al-T.u¯ s¯ı not only resolved the problem raised by Qays.ar but also insisted on the necessity to distinguish between geometric and philosophical demonstrations of the parallel postulate. The mathematical content of the entire correspondence has been translated into French by Jaouiche (1986, 227–231). 13 A facsimile edition has been published by Jafar Aghayani-Chavoshi (Al-Tu . ¯ s¯ı 2005, Brentjes 2006b). Although there are numerous manuscript copies, not all collections are exactly alike (Brentjes 2018a, 228–233). The importance of these Mutawassit¯at is evident from the fact that

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The treatise was also influential through its translations and through the printing of commentaries on or editions of the text.

3.3.1

Foundation of the Persian Transmission of the Elements

When Euclid’s Elements was first transmitted to Persian-speaking regions, it was in the form of a Persian translation of al-T.u¯ s¯ı’s Tah.r¯ır by Qut.b al-D¯ın al-Sh¯ır¯az¯ı (d. 710/1311),14 not a Persian translation of the original Arabic primary Euclidean transmission of the Elements (Brentjes 1998). Al-Sh¯ır¯az¯ı introduced several important modifications to al-T.u¯ s¯ı’s text. Perhaps most importantly, he removed nearly all the mathematical notes, alternative demonstrations, and additional cases that had been prominent in the Arabic original (De Young 2007). But although al-Sh¯ır¯az¯ı removed a considerable number of al-T.u¯ s¯ı’s additions—including the “demonstration” of the parallel lines postulate—he inserted several additions of his own devising, including a set of “demonstrations” for several of Euclid’s postulates (Abdeljaouad and De Young 2020). One of his more unusual additions is a short appendix to book I that combines into a single diagram all the diagrams from book I. This diagram is accompanied by a verbal description explaining which lines one needs to choose in order to replicate each diagram of book I (Doostgharin 2012a, De Young 2013). This Persian translation, completed in 682/1282, was dedicated to the vizier Am¯ır Sh¯ah ibn T¯aj al-D¯ın Mu‘tazz ibn T.a¯ hir.15 Presumably this dedication was part of an on-going attempt to get and retain the patronage that would enable al-Sh¯ır¯az¯ı to continue his research and writing.16 Al-Sh¯ır¯az¯ı later re-used this Persian translation (with very little re-editing) as part of his philosophical compendium, Durrat al-t¯aj li-ghurrat al-Dubb¯aj, which he completed in 705/1305. The compendium as a whole was dedicated to Dubb¯aj ibn H.us¯am al-D¯ın F¯ıl-Sh¯ah ibn Sayf al-D¯ın Rustam ibn Dubb¯aj Ish.a¯ q¯awand, ruler of Bayah Pas in G¯ıl¯an province of Iran (Savage-Smith 2005, 67). In this Persian compendium, al-Sh¯ır¯az¯ı’s intent seems to have been the construction of a philosophical compendium comparable to that composed some three centuries earlier by Ibn S¯ın¯a (De Young 2012d), which had included an epitome of Euclid’s Elements in its mathematical component (Ibn S¯ın¯a 1977).

the Ottoman scholar and bibliophile Mü’eyyedzade Abdurrahman Effendi (died 922/1516) owned at least five copies which he personally annotated (Pfeiffer 2020). 14 Doostgharin (2012b) discusses the distinctive features of this translation. 15 Al-Sh¯ır¯ az¯ı also dedicated his Arabic treatise on hay’a (mathematical cosmography), al-Tuh.fa al-sh¯ahiyya f¯ı al-hay’a, completed in 684/1285, to this same vizier (Ragep 2021). 16 The multi-faceted role of patronage in developing mathematical sciences in the Islamicate societies has been described by Brentjes (2009).

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3.3.2

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Calcutta School Book Society Edition of Euclid

In 1824, the Calcutta School Book Society published both an Arabic and a Persian edition of the first six books of Euclid, intended for use in native schools. The Arabic version, edited by the Reverend Thomas T. Thomason (1774–1829), was based on the Tah.r¯ır of al-T.u¯ s¯ı. As editor, Thomason removed the mathematical comments and other material added by al-T.u¯ s¯ı (De Young 2012a, 7). Thomason also altered the text when it suited his purposes to do so. For example Thomason (1824, 1) replaces al-T.u¯ s¯ı’s definition of the straight line (“the straight [line] is such that the placement (location) of any point whatever that may be specified upon it stands exactly facing (tataq¯abila) any other point upon it”17 ) with the definition of Archimedes (“The straight line is the shortest distance between two points”).18 The goal of this editing was to provide the native population of India with a textbook on Euclidean geometry appropriate for a British-style education in a language they could understand. The emphasis on the first six books was consistent with educational practices in Britain at the time—books I–VI were taught in grammar schools, while the stereometric geometry of books XI and XII were taught in universities and not as a compulsory course (Ackerberg-Hastings 2002, 65). The Persian edition was prepared by an Indian colleague of Thomason, Hydar Alee, a member of the CSBS Board of Directors and a native speaker of Persian. The Persian edition appears to be based primarily on the Persian translation of alT.u¯ s¯ı’s Tah.r¯ır, Durrat al-T¯aj, following the same editing principles as the Arabic edition—accretions such as al-Sh¯ır¯az¯ı’s demonstrations of Euclid’s postulates were removed. The text itself was also lightly edited—Hyder Alee, the editor, appears to have altered some verb tenses, for example. The Persian edition often appears to be following the Arabic edition, since (a) the third definition of book I is also replaced in the Persian edition by the Archimedean definition used in the Arabic edition (Alee 1824, 1) and (b) the illustrations of the Euclidean definitions from the Arabic text have been imported into the Persian edition although they were not used in the Durrat al-T¯aj.19

17 Euclid’s

Greek definition is obscure (Heath 1956, I, 165–168). Al-T.u¯ s¯ı’s definition follows the formulation used in the Arabic translation manuscripts. 18 The inspiration for this alteration in the text is unknown. The definition is explicitly attributed to Archimedes in the fourth/tenth-century Arabic commentary of al-Kar¯ab¯ıs¯ı (Brentjes 2000, 32). Ibn al-Haytham (354–422/965–1041) also mentions this alternative form of the definition although without an attribution (Sude 1974, I, 33). 19 The origin of these illustrative diagrams is not clear. They do not appear in the earliest known copy of al-T.u¯ s¯ı’s Tah.r¯ır (British Library, ad. 23387, dated 656/1258), but they are included in the lithograph edition published in Tehran (Al-T.u¯ s¯ı 1298/1881) as well as in some manuscript copies from the later medieval period. It does not appear likely that these diagrams were introduced from or modeled on the Pseudo-T.u¯ s¯ı Tah.r¯ır (1594, 4–8), which includes similar diagrams illustrating the definitions of book I, because they do not match the ordering of the Pseudo-T.u¯ s¯ı.

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3.3.3

G. De Young

The Commentary on Book I by Muh.amad Barakat

In seventeenth-century India, Muh.ammad Barakat wrote an Arabic commentary on book I of the Elements—more precisely on book I of the Tah.r¯ır of al-T.u¯ s¯ı. This commentary was later included in the proposed curricular reform of Niz.a¯ m alD¯ın al-Sihlaw¯ı that came to be known as the Dars-i-Niz.a¯ m¯ı and was associated primarily with the Farangi Mahal educational reform movement in Lucknow (De Young 2012b, 179–183). These curricular reforms apparently produced sufficient demand for the commentary so that it was printed at least twice in India during the last half of the nineteenth century using lithography (De Young 2012a, 10–13). The text was printed in imitation of the style of a traditional Persian commentary, with the commentary notes placed in slanted lines in the margins of the pages and the base text (book I of al-T.u¯ s¯ı’s Tah.r¯ır) occupying the main portion of the page. In the printed text, the commentary is followed by the appendix that Qut.b al-D¯ın alSh¯ır¯az¯ı had inserted into his Persian version of the Tah.r¯ır of al-T.u¯ s¯ı—the composite diagram incorporating all the diagrams from book I (De Young 2012b).

¯ of Muh.ya al-D¯ın al-Maghrib¯ı 4 The Tah.rir The Tah.r¯ır by al-Maghrib¯ı is probably the least known to modern historians. Perhaps this results from the inclusion of fewer notes and discussions in the sections on plane geometry, ratios, or theory of numbers than occur in the other Tah.r¯ır. (It is only in the stereometric books at the end of Euclid’s treatise where al-Maghrib¯ı departs from standard Euclidean content.) And perhaps it occurs because this tah.r¯ır appears to have had less historical impact on medieval mathematical education.

4.1 The Author Yah.y¯a ibn Muh.ammad ibn Ab¯ı’l-Shukr Muh.y¯ı al-D¯ın al-Maghrib¯ı al-Andalus¯ı was, judging from his name, born in Andalusia (southern Spain) early in the seventh/thirteenth century. Although almost nothing is known of his early life or education, he must have migrated to the Eastern Mediterranean and the Abbassid court because he composed an astronomical handbook, T¯aj al-Azw¯aj, in Damascus in 1257. When the Mongols overran Damascus shortly thereafter, al-Maghrib¯ı apparently moved eastward once more. We next encounter him at Mar¯agha, where he apparently composed his Tah.r¯ır—the colophon of Bodleian Library, Or. 448 (folio 175b) reports the date of composition as 659/1260–1261 in Mar¯agha. He died in Mar¯agha about 1283 (Hogendijk 1994, 134; Saliba 1983, 391–2).

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4.2 General Characteristics of the Tah.r¯ır The Tah.r¯ır of al-Maghrib¯ı is known in only three manuscripts, suggesting that it has been less popular over the years than that of al-T.u¯ s¯ı. Of the three tah.r¯ır discussed in this paper, al-Maghrib¯ı’s contains the greatest number of changes to the Euclidean text, although the majority occur at the end of the treatise in books XIII–XV. The material in the first 12 books generally mirrors the content and arrangement to the definitions and propositions in Euclid’s Elements, with the exception of a few brief interpolations, such as a “demonstration” of Euclid’s fifth postulate following proposition 34 of book I (Jaouiche 1986 118–119, 250– 251), a preliminary lemma (muqaddima) to proposition II, 13, and the insertion of definitions of numbers that exceed or fall short of another number in book VII. In book XIII, Maghrib¯ı has placed several propositions extracted from Elements XIII, along with some additional propositions that deal with plane figures. Maghrib¯ı’s book XIV contains propositions from Elements XIII that deal with solid figures and most of what was originally included in Elements XV. Maghrib¯ı’s book XV contains mostly new material devoted to the construction of line segments having to one another the ratios of the edges, the surface areas, the faces, the perpendicular distances between the center and the face, and the volumes of the five regular solids when they are inscribed in a given sphere (Hogendijk 1994, 134–139). The treatise begins with a short introduction in which al-Maghrib¯ı names (and criticizes) three of his predecessors as part of the justification for composing his own Tah.r¯ır:20 As for the Outstanding Sheikh (Ab¯u ‘Al¯ı ibn S¯ın¯a), he omitted the enunciations and many of the lemmas (muqqadim¯at) and he failed to carry out his duty to resolve difficulties.21 Al-Nays¯ab¯ur¯ı did likewise,22 together with the production of additions for which there is no need and the deletion of important matter that is necessary, such as his elaboration concerning positions of the sixth [book?] and other things. And he shortened it in the tenth book since he omitted the demonstrations of the apotomes completely and claimed that they

20 Al-Maghrib¯ı apparently refrained from naming more recent expositions of the Elements by his contemporaries, such as the Is.l¯ah. Us.u¯ l Uql¯ıdis of Ath¯ır al-D¯ın al-Abhar¯ı (died 663/1265) and the Tah.r¯ır al-T.u¯ s¯ı. But if his Tah.r¯ır was intended to support a request for patronage from the Ilkh¯an ruler, H¯ul¯ag¯u, he might have felt it unwise to appear to attack or to criticize the senior mathematician in the court. 21 The reference is apparently to the epitome of the Elements included in Ibn S¯ın¯ a’s encyclopedic philosophical compendium, Kit¯ab al-Shif¯a’, completed early in the fourth/eleventh century. This epitome does indeed omit the enunciations and many of the lemmas found in the Arabic manuscripts (Ibn S¯ın¯a 1977). 22 Al-Maghrib¯ı apparently refers to Ab¯ u al-Q¯asim ‘Al¯ı ibn Ism¯a‘¯ıl al-Nays¯ab¯ur¯ı or al-N¯ıs¯ab¯ur¯ı (active late fifth/tenth and early sixth/eleventh century). His Tah.r¯ır of the Elements was long known in a unique manuscript, Kaysari, Ra¸sit Efendi 1230 (Rosenfeld and Ihsano˘glu 2003, 110). A second nearly complete copy was recently sold by Sotheby’s (25 April 2018, Lot 30). What al-Maghrib¯ı intended by “likewise” is not clear. Al-Nays¯ab¯ur¯ı typically retains the enunciation but omits the setting out and specification components.

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G. De Young were obvious, together with the hiddenness of their situation. And he imagined concerning the correctness of the fourteenth proposition of book XII. As for Ab¯u Ja‘far al-Kh¯azin,23 he included the enunciations and he was better at correcting them. He altered the numbering of the propositions and their ordering since he combined a number of diagrams of the propositions into a single proposition and he condensed it with regard to its expression and made the propositions fewer without solution of any doubt and did not repudiate any propositions.

The Tah.r¯ır of al-Maghrib¯ı appears to be a hybrid, in the sense that it appears to have been influenced both by the transmission attributed by al-T.u¯ s¯ı to al-H.ajj¯aj and the transmission attributed to Th¯abit (Ish.a¯ q). In terms of general structural features, the Tah.r¯ır exhibits mainly features of the transmission attributed to Th¯abit. It does not omit, for example, Elements I, 45 nor Elements VI, 12 (finding a fourth magnitude proportional to three given magnitudes) nor the two propositions following Elements VIII, 24 which al-T.u¯ s¯ı reports were interpolated by Th¯abit. Moreover, Elements III, 35 and 36 are each proved only for a single case, which al-T.u¯ s¯ı reports is the result of a condensation by Th¯abit, although the diagrams (Oxford, Bodleian Library, Or 448, folio 28b; Istanbul, Süleymaniye Library, Aya Sofya 2719, folio 18b; Istanbul, Süleymaniye Library, Mihri Sah ¸ 337, folio 32b) for all three cases proved by al-H.ajj¯aj are included by al-Maghrib¯ı. Several examples of alternative demonstrations for propositions at the beginning of book II suggest that al-Maghrib¯ı’s Tah.r¯ır was, like the Tah.r¯ır of al-T.u¯ s¯ı, influenced by the alternative formulations attributed to Heron in the commentary of al-Nayr¯ız¯ı (Besthorn and Heiberg 1893–1910, II, 14–17; Lo Bello 2009, 23–26). In several of these propositions, al-Maghrib¯ı has set as his primary demonstration the alternative version of al-T.u¯ s¯ı and has placed al-T.u¯ s¯ı’s primary demonstration as an alternate demonstration in his own Tah.r¯ır. In doing so, al-Maghrib¯ı is following the formulation used in the earlier Tah.r¯ır of al-Nays¯ab¯ur¯ı. This intriguing apparent relationship between these two tah.r¯ır, despite al-Maghrib¯ı’s criticism of al-Nays¯ab¯ur¯ı, deserves further study in the future. In terms of diction, though, al-Maghrib¯ı seems closer to the formulation of al-H.ajj¯aj than to that of Th¯abit. For example, the enunciations of the first nine propositions in book II are formulated in terms of the multiplication (d.arb) of lines into lines rather than in terms of squares and rectangles and parallelograms. And in book VII, al-Maghrib¯ı inserts the definition of “evenly even and odd numbers” (zawj al-zawj wa-l-fard), an interpolation attributed to al-H.ajj¯aj (De Young 2003a, 148), which al-T.u¯ s¯ı, who is generally following the Th¯abit version, did not include. In this example as well, al-Maghrib¯ı follows the formulation of al-Nays¯ab¯ur¯ı. The architecture of the test diagrams further suggest that his primary inspiration was probably the Group A tradition of the Th¯abit transmission. The diagram for proposition II, 14, for example, represents the known figure as a triangle, although the problem only specifies a rectilinear figure (Fig. 7), a feature found also the Tah.r¯ır 23 Al-Kh¯ azin was a well-known mathematician, active in the mid-fourth/tenth century. He is credited with a no-longer extant Kit¯ab al-Us.u¯ l al-handasiyya (Rosenfeld and Ihsano˘glu 2003, 82), which may be the treatise to which al-Maghrib¯ı refers.

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Fig. 7 Diagram for II, 14 of al-Maghrib¯ı Tah.r¯ır edited from Oxford, Bodleian Library, Or. 448, folio 19a

Fig. 8 Diagram for VI, 22 of al-Maghrib¯ı Tah.r¯ır edited from Oxford, Bodleian Library, Or. 448, folio 48b. The copyist has omitted from the diagram the two line segments that are usually present (see Fig. 5)

¯ ur ¯ ¯ı.24 Manuscripts from the Group B tradition typically represent the of al-Naysab given figure as a quadrilateral (Fig. 1). In the diagram of proposition VI, 22, alMaghrib¯ı’s Tah.r¯ır (Fig. 8) follows the architectural pattern typical of the Group B transmission manuscripts (Fig. 1).25 The architecture of the diagram for Elements VII, 14 in al-Maghrib¯ı’s Tah.r¯ır labels each line segment with a single letter (Fig. 9). This architecture, typical of Group A manuscripts (Fig. 3), is also found in the Tah.r¯ır of al-Nays¯ab¯ur¯ı. These observations suggest that al-Maghrib¯ı’s diagrams show a mix of characteristics from both the Group A and Group B branches of the Ish.aq–Th¯abit transmission. Whether this represents a conscious choice on the part of al-Maghrib¯ı or simply the adoption of the diagram architecture found in the Tah.r¯ır of al-Nays¯ab¯ur¯ı will require further investigation.

24 In

the Byzantine Greek transmission, the architecture of the diagram regularly represents the given figure as a triangle, although the diagram in Heiberg’s Greek edition (1883–1888, I, 161)— apparently following the diagram in the edition of August (1826, Plate II)—shows the given figure as a quadrilateral. On Heiberg’s use of August’s diagrams, see also Saito and Sidoli (2012, 136– 138). 25 The diagram of al-Nays¯ ab¯ur¯ı also has two triangles and three quadrilaterals, although it also includes two additional line segments and alters the labeling of the figures slightly.

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Fig. 9 Diagram for VII, 14 of al-Maghrib¯ı Tah.r¯ır edited from Oxford, Bodleian Library, Or. 448, folio 58a

4.3 Influence The historical influence of al-Maghrib¯ı’s Tah.r¯ır has yet to be thoroughly investigated. No direct references to his Euclidean studies have been found in later Arabic discussions, although Langermann and Hogendijk (1984) have noted that a similar collection of propositions on polygonal and polyhedral figures exists in a Hebrew manuscript, Oxford, Bodley Heb. d. 4. These propositions (translated by Kalonymos ben Kalonymos in 1309) are often strikingly similar in phrasing to the Arabic discussion of al-Maghrib¯ı, suggesting perhaps some continuing influence of this Tah.r¯ır. But even if we have so far been able to discern little direct influence, it is clear that its discussion of regular and semi-regular polygons and polyhedra is embedded in a tradition that extended from the time of Euclid until late in the medieval period (Hogendijk 1984, De Young 2008) and culminated in the systematic analysis of Kepler in his Harmonice Mundi (1619).

¯ ı Tah.r¯ır 5 The Pseudo-T.us¯ This was the first Arabic geometry text to be printed. Published by the Medicean Press in Rome in 1594, its status as a printed text has lent it a certain (perhaps undeserved) notoriety over the years because it was readily available to scholars and perhaps because it presented its European readers with fewer legibility issues than did many manuscripts. So when European historians of mathematics wanted to consult a “representative” Arabic version of the Elements, it was to this printed text that they often turned. The incorrect ascription to al-T.u¯ s¯ı on its title page has continued to influence and mislead historians until today (De Young 2003b;

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2012c). The result has been an all-too-frequent misunderstanding of the relationship between this printed edition and the discussions of the Elements available primarily in manuscript form.

5.1 The Author The title page of the Rome 1594 edition incorrectly attributed this Tah.r¯ır to Nas.¯ır al-D¯ın al-T.u¯ s¯ı (Rosenfeld et al. 1966, Sabra 1968, 15). The origin of this mistaken ascription is unclear, but the colophon of the only surviving manuscript (Florence, Biblioteca Medicea Laurenziana, ms orientali 50) does not mention the name of the author, although it gives the date of completion as 698/1298, nearly a quarter of a century after al-T.u¯ s¯ı’s death. Thus it seems certain that he cannot be the author. Therefore, it appears likely that the error was introduced, however unintentionally, at the time of the printing. A number of historians have argued that the author must be S.adr al-D¯ın, al-T.u¯ s¯ı’s older son who succeeded him as director of the observatory at Mar¯agha (Rosenfeld and Ihsano˘glu 2003, 220). These historians point out that he was reputed to be a competent mathematician, although he has left us no extant writings on any mathematical topics. But since the manuscript contains no explicit statement of an author’s name, all such hypotheses must be relegated, pending discovery of further documentary evidence, to the realm of speculation.

5.2 General Characteristics of the Tah.r¯ır The treatise is known today in only one manuscript (Florence, Biblioteca Medicea Laurenziana, ms orientali 50).26 The treatise includes only the genuine 13 books of the Elements, although some copies are reported to end with book XII (Cassinet 1993, 23). The printed edition contains several unique features, but since these are not present in the only medieval manuscript to survive, they appear to be the result of editorial decisions during the printing process. One of the most visually arresting features is the use of columns or rows of dots to represent discrete numbers, which are made up of units, rather than the line segments which commonly occur in manuscripts containing Euclidean diagrams in the arithmetical books of the Elements (Fig. 12). Another unusual visual feature is that the diagram for proposition I, 47 is drawn without overspecification—although the diagram in Florence, Laurenziana, orientali 50 (folio 23a) shows the typical overspecification

26 The reputed second (incomplete) copy (Biblioteca Medicea Laurenziana, ms orientali 20) appears to be a copy of ms orinetali 50 produced probably for use by the printers of the Medicean edition by Patriarch Nehemias, who apparently brought the manuscript to Florence (De Young 2012c, 278–280).

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Diagram edited from Pseudo-T.u¯ s¯ı (1293 / 1881), p. 117.

Diagram edited from Florence, Biblioteca Medicea Laurenziana, orientali 50, folio 31b.

Fig. 10 Diagrams for Elements II, 14. The distinctive shape of figure alif (left) is unique to the print tradition Fig. 11 Diagram for VI, 22 of Pseudo-T.u¯ s¯ı Tah.r¯ır edited from Pseudo-T.u¯ s¯ı (1293/1876), p. 296

found in nearly all medieval manuscripts of the Elements.27 One can only assume that these alterations were introduced into the text diagrams at the time of printing. The diagrams in the surviving manuscript copy, Florence, Biblioteca Medicea Laurenziana, orientali 50, often follow an architectural pattern similar to those found in the Tah.r¯ır of al-T.u¯ s¯ı. For example, the diagram for proposition II, 14 shows the given figure as an irregular quadrilateral (Fig. 10) similar to the style adopted in many manuscripts of al-T.u¯ s¯ı’s Tah.r¯ır (Fig. 4). The diagram for proposition VI, 22 in Florence, Biblioteca Medicea Laurenziana, orientali 50 (Fig. 11) again shares the basic architecture as the diagram of al-T.u¯ s¯ı (Fig. 5). Likewise, the architecture of the diagram for proposition VII, 14 (Fig. 12) appears to repeat the style of al-T.u¯ s¯ı (Fig. 6). It is sometimes suggested that this Tah.r¯ır is only a re-editing of the Tah.r¯ır of al-T.u¯ s¯ı (Babayev and Medzlumbeyova 2015, 2). This hypothesis seems unlikely, though. There are many stylistic differences between the two tah.r¯ır. The differences are visible already on the level of simple diction of the enunciations. For example, consider the enunciation of the first proposition of book I:28

27 “Overspecification”

refers to the tendency of copyists to produce diagrams with more regularity that is required by the verbal text. For example, general triangles are often drawn as isosceles or equilateral (Saito 2006, 82; Saito and Sidoli 2012, 140–143). Acerbi (2017, 244–246) has proposed an explanation for the origins of such overspecification. 28 My translation, from British Library, Add. 23387, folio 4b–5a and Pseudo-Tu . ¯ s¯ı (1594), p. 9.

Euclid in Mar¯agha

Diagram edited from Pseudo-T.u¯ s¯ı (1293 / 1881), p. 344.

35

Diagram edited from Florence, Biblioteca Medicea Laurenziana, Orientali 50, folio 83b.

Fig. 12 Diagrams for Elements VII, 14. The portrayal of the numbers as points (left) rather than line segments is unique to the print tradition

Al-T.u¯ s¯ı: We want to draw an equilateral triangle on a bounded line. Pseudo-T.u¯ s¯ı: We are to construct on any given bounded straight line an equilateral triangle.

Or consider the enunciation of proposition VIII, 9:29 Al-T.u¯ s¯ı: Any pair of mutually incommensurable numbers [such that] there occur between the two of them numbers and they (these numbers) are in continued proportion, then between the unit and each one of the two of them there occur numbers according to that quantity and they are in continued proportion. Pseudo-T.u¯ s¯ı: Any pair of mutually incommensurable numbers [such that] there occur between the two of them numbers, however many [they may be] such that they are in continuous proportion with the two of them, then there occurs between the unit and each one of the two mutually incommensurable numbers [other] numbers according to the quantity of what occurred between the mutually incommensurable numbers and they are, together with the unit and each one of the two of them in a continuous proportion.

In each case, we see that the enunciation of Pseudo-T.u¯ s¯ı is more verbose, more mathematically explicit, and more precise, as though the author is more consciously writing for beginners in the field. The Pseudo-T.u¯ s¯ı Tah.r¯ır also differs from that of al-T.u¯ s¯ı in several other ways. For example, it contains many more corollaries or extensions of Euclidean propositions than are found in the Tah.r¯ır of al-T.u¯ s¯ı. Furthermore, it contains more frequent references to earlier propositions, and it places these references directly in the text, rather than interlinearly as in the case of the Tah.r¯ır of al-T.u¯ s¯ı. Moreover, it omits most of the mathematical notes found in al-T.u¯ s¯ı’s treatise—especially the alternative demonstrations that al-T.u¯ s¯ı had “borrowed” from Ibn al-Haytham. Alternative cases, such as those that accompany 29 My

translation, from British Library, Add. 23387, folio 120b and Pseudo-T.u¯ s¯ı (1594), p. 193.

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proposition I, 2, which had already existed in the literature prior to al-T.u¯ s¯ı’s work, are still present in this Tah.r¯ır. Several of al-T.u¯ s¯ı’s structural notes discussing differences between the text of Th¯abit and that of H.ajj¯aj appear also in the Pseudo-T.u¯ s¯ı but they are written in a different style and often offer either somewhat different reports or more information or more extended explanation than is found in the notes of al-T.u¯ s¯ı. The differences are obvious from the structural note following proposition I, 45:30 al-T.u¯ s¯ı: This proposition is not in the text of al-H.ajj¯aj. Pseudo-T.u¯ s¯ı: Al-H.ajj¯aj does not mention this proposition in his treatise. It is found in the text of Th¯abit and the truth is that there is no need for it after the preceding proposition. That is because the technique of Euclid, in this treatise of his, is that if a proposition or lemma is demonstrable from earlier propositions he does not make it one of the propositions of his treatise. Nor does he draw out the lemma from potentiality to actuality but rather he does not mention anything concerning it, relying on the minds of those who attempt the mastery of this book of his, because there is [already] mentioned in the text its correctness and the branches are without end. And I have also omitted it from the text of the treatise and I have made it a corollary to the previous proposition. And I have mentioned it in actuality because my technique in this treatise demands that.

Like the other tah.r¯ır already discussed, that composed by the Pseudo-T.u¯ s¯ı contains various additions. For example, the author introduces several “demonstrations” into the premises (muqaddim¯at) or axioms of book I. These demonstrations are not unique to this Tah.r¯ır—most are known already from the Greek transmission (De Young 2007, 31–40; Abdeljaouad and De Young 2020). These “demonstrations” are not rigorous mathematical proofs. Rather, they seem intended to convince beginning students that these axioms or principles can be believed. The author also inserts, in the premises of book V, paraphrases of the ziy¯ad¯at by al-Jawhar¯ı, three “demonstrations” added to the Elements with the intent to explain more completely what Euclid meant by “being in the same ratio” (Brentjes 1997, De Young 1997). Like the other two major tah.r¯ır, that of the Pseudo-T.u¯ s¯ı includes a demonstration of Euclid’s parallel lines postulate. In this Tah.r¯ır, the demonstration, preceded by three preliminary lemmas (muqaddima, is inserted between propositions 28 and 29 of book I (Jaouiche 1986, 109–111). And, like the Tah.r¯ır of al-T.u¯ s¯ı, book I ends with a discussion of the various placements of the squares on the sides of the right triangle, although the discussion is much shorter.

5.3 Influence The Pseudo-T.u¯ s¯ı Tah.r¯ır appears to have influenced medieval Europe primarily by way of the Hebrew transmission. Rabbi Levi ben Gershon (1288–1344) (known in Latin as Gersonides) wrote a small treatise on Euclid’s fifth postulate (Hibbur Hokhmat ha-Tishboret) and another more general treatise on the premises of Euclid (Hagahot le-Sefer Euqlidus). Both these treatises reveal some influence from the Tah.r¯ır of Pseudo-T.u¯ s¯ı (Lévy 1992a;b). 30 My

translations, based on British Library, Add. 23387, folio 27b and Pseudo-T.u¯ s¯ı (1594), p. 44.

Euclid in Mar¯agha

37

The mathematical influence of the treatise is difficult to estimate, though. In Europe, the printed Arabic edition seems to have enjoyed some popularity. Its “demonstration” of Euclid’s fifth postulate was discussed, for example, by John Wallis after being translated from the Arabic by Edward Pococke (Cassinet 1993, 12–17). Its primary influence appears to be the perpetuation of historical misinformation concerning the Tah.r¯ır of al-T.u¯ s¯ı, although most of those who spread this incorrect information do so in good faith, accepting the attribution to al-T.u¯ s¯ı on the title page as genuine. The printed edition appears also to have had some influence in the Ottoman Empire. The report of the Abbé Toderini describing his own observations of geometry education in Istanbul at the end of the eighteenth century suggests that the Pseudo-T.u¯ s¯ı Tah.r¯ır may have been used in Ottoman madrasas:31 Geometry falls under the group of Turkish studies. In academies (madrasa), there are professors (mudarris) for teaching it [geometry] to young people. The period between mathematics and rhetoric classes is allocated to this mathematical branch. . . This science is taught in a special manner. I have been to the Valide Madrasa twice, during which time students had gathered to listen to the geometry class. They used an Arabic translation of Euclid. There are many versions as well as commentaries of this book. Nas.¯ır al-D¯ın alT.u¯ s¯ı’s commentary, which is regarded as the best of these, has become popular thanks to the Medicis Publishing House. This copy contains a copy of the Turkish license granted by Sultan Murad III (ruled 1574–1595) in Istanbul in 1587. He has granted permission for the sale of this book without any tax or liability within the entire Ottoman territory . . .

Although some copies of the printed treatise circulated in the East, the market seems to have been much less than anticipated by the Press.32 The publishers had invested in a fairly large print run and found themselves with more than half the copies still unsold decades later (Jones 1994, 108). Although we know of only one Arabic manuscript produced after the text was printed in Rome, this manuscript was copied from the printed treatise in 1101/1690 (De Young 2012c, 281–3). Given this apparent lack of historical influence, it is somewhat surprising that when a printer in Fez decided to issue a Euclidean geometry text by lithography in 1293/1876, he chose as his model the edition published in Rome almost three centuries earlier (De Young 2012c, 283–4).33

31 As

quoted by Ayduz (2011, 26) from Toderini and L’Abbé (1789, I, 100–105). The translation is presumably by Ayduz. 32 Antoine Galland (1646–1715) visited Istanbul in 1672-1673 and reported that printed copies circulated primarily among Christian readers and European missionaries. The majority of Muslims, he wrote, usually chose manuscript copies rather than printed editions, even though manuscripts were considerably more expensive (Pekta¸s 2015, 5–6). 33 Because the text was published by lithography, the entire text had to be rewritten. In this process, several typographical errors in the Medici edition were corrected.

38

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6 Concluding Thoughts These three tah.r¯ır, although each is somewhat different in focus, share some common features. Like the earlier Tah.r¯ır of al-Nays¯ab¯ur¯ı, they contain an edition of the entire treatise of Euclid but add to or modify the text in different ways. The tah.r¯ır genre reached its high point with the group of mathematicians in seventh/thirteenth century Mar¯agha. The historical influence of these tah.r¯ır was sometimes far beyond that envisioned by their original authors. Although these tah.r¯ır do not mark a bold new beginning in mathematics, the repeated editions of the Elements reveal a continuing fascination with Euclid’s classic treatise and with the tradition of geometrical thought that had grown up around it.

6.1 Internal Cross-Referencing and Mathematical Pedagogy Another thread uniting these tah.r¯ır is their use of internal cross-referencing, apparently intended to remind the reader of the justification for the major steps in the demonstrations. Some of these justifications are related to claims of equality or inequality or similarity. Others justify constructions undertaken in the Euclidean demonstrations by referring back to the propositions in which the constructions were first demonstrated. On one level, the use of a common system of cross-referencing is hardly surprising since all tah.r¯ır are derived from Euclid’s Elements and so all follow the same mathematical argumentation. But the fact that this cross-referencing occurs so consistently in each tah.r¯ır suggests that the authors considered this feature a vital component of the genre. In fact, over time this cross-referencing component of the tah.r¯ır seems to become increasingly important, moving from the margins to interlinear spaces, then into the text itself. Many copies of the Tah.r¯ır of al-T.u¯ s¯ı contain cross-references to earlier propositions. Typically such justifications are placed in red lettering below the line of the text of the manuscript.34 These references include the number of the proposition followed by the number of the book in which the proposition occurs, both numbers being in the standard alphanumeric (abjad) form. We do not know whether these cross-references were added by al-T.u¯ s¯ı, but they are already present in the earliest known copy (London, British Library, ad. 23387, dated 656/1258), where they are placed in the margins in red ink, with red ink insertion marks in the text to indicate what the note refers to. The same set of cross-references appears also in the lithographed edition produced in Tehran (Al-T.u¯ s¯ı 1298/1881), where they are placed below the line of the text, and in the Istanbul printed edition (Al-T.u¯ s¯ı 1216/1801), where they are placed in parentheses within the line of the text.35 The Tah.r¯ır of al-Maghrib¯ı regularly places the cross-references to earlier propositions directly into the text, although set off from the text by use of red ink. They 34 A similar system of cross-referencing also appears in the Isl¯ . ah. Uql¯ıdis by Ath¯ır al-D¯ın al-Abhar¯ı (died 661/1263). 35 The same system of cross-referencing is carried over into manuscripts of the Persian versions of al-T.u¯ s¯ı’s Tah.r¯ır.

Euclid in Mar¯agha

39

are typically written in an abbreviated form “[proposition] X from [book] Z”— both numerals written in the usual alphanumeric form—and are often more frequent than the cross-references in al-T.u¯ s¯ı’s Tah.r¯ır because they include references to the postulates and axioms, which are not always included in al-T.u¯ s¯ı’s treatise. As in the case of al-T.u¯ s¯ı’s Tah.r¯ır, the same cross-references appear in all copies, suggesting that they may have been present in the original text. These cross-references may imply a more explicit pedagogical intent on the part of the author. Their insertion directly into the text suggests that the author may have envisioned a somewhat less sophisticated readership who would need regular reminders of the internal logic of the Elements. The Pseudo-T.u¯ s¯ı Tah.r¯ır also contains these cross-references to earlier propositions. They are incorporated into and are verbally integrated with the text (“by proposition X from book Z”). This formulation may simply be the personal preference of the author, or it may indicate that he envisioned an audience of beginners who might misinterpret the more condensed style of cross-references used by al-T.u¯ s¯ı and al-Maghrib¯ı. Whatever the style adopted for the cross-referencing, though, it seems likely that the references were included with a pedagogical intent because readers already familiar with the structure and argumentation of the Elements should not require such assistance.

6.2 Printing Euclidean Geometry and Ottoman Madrasa Education One measure indicating the continuing importance of al-T.u¯ s¯ı’s Tah.r¯ır is the number of surviving manuscript copies—the treatise continued to be copied regularly until well into the nineteenth century, often with accretions of annotations by owners and readers (Brentjes 2019). A second indicator of its importance comes from the fact that this Tah.r¯ır was printed at least twice during the nineteenth century—Istanbul (1216/1801) and Tehran (1298/1880). The printers must have expected to be able to recoup their economic investment in preparing these printed editions. Thus it seems reasonable to assume that al-T.u¯ s¯ı’s Tah.r¯ır continued to function as an introductory textbook in traditional madrasa education (De Young 2012a). The work of al-T.u¯ s¯ı formed an important foundation for education in the mathematical sciences in madrasas throughout the Ottoman Empire. One evidence supporting this claim is the statement in the widely cited Kevâkib-i seb’a, a report on Ottoman education drawn up at the request of the French ambassador to the Sublime Porte, the Marquis de Villeneuve, in 1742. This report includes mention of the basic textbooks used to teach geometrical subjects:36

36 As

˙ quoted by Ihsano˘glu (2004, 14). The English translation is presumably by Ihsano˘ glu.

40

G. De Young Geometry and arithmetic are easily apprehendable subjects, and because they do not require much deep thought are not studied as separate subjects. . . . There is a book titled Eskâl Te’sis in geometry at the iktisar level that they would read. Following that, they would read Euclid with its proofs at the istiksa level.37

This report reminds us that, despite its obvious popularity, al-T.u¯ s¯ı’s Tah.r¯ır was not the only geometry textbook used in mathematics education. Both the Ashk¯al alTa’s¯ıs of al-Samarqand¯ı (De Young 2001) and its commentary by Q¯ad.¯ız¯ade al-R¯um¯ı (Souissi 1984) were also texts commonly used in the mathematics curriculum of the madrasas (Brentjes 2018a, 237–238). Q¯ad.¯ız¯ade al-R¯um¯ı’s commentary was also printed in Istanbul during the nineteenth century (De Young 2012a, 13–16), offering another indication—in addition to the remarks of the Marquis de Villeneuve quoted above—of the continuing importance and popularity of this short introduction of Euclidean geometry. Still another indicator of the continued pedagogical importance of al-Samarqand¯ı’s treatise is the production of a summary written in rhymed prose ¯ by al-H.urr al-Amil¯ ı during the eleventh/seventeenth century (De Young 2016).

6.3 Geometrical Compilations and Mathematical Pedagogy Another facet of the pedagogical popularity of al-T.u¯ s¯ı’s Tah.r¯ır comes from the collections of texts related to Euclidean geometry that were compiled repeatedly during the early modern period. Usually copied by a single copyist, these collections typically begin with a longer treatise—most often al-T.u¯ s¯ı’s Tah.r¯ır—followed by several shorter, more focused treatises. And although the membership of these ancillary treatises fluctuates somewhat, there seems to be a common core that were frequently included (Abdeljaouad and De Young 2020, 70–77).38 Brentjes (2019) has recently discussed one example of this mathematical genre— Munich, Bayerische Staatsbibliothek, cod. arab. 2697—as an example of a mathematical textbook. A colophon gives the date of completion as 1142/1729, during the Ottoman period, although other examples of the genre date from Timurid Iran, ¯ The contents nearly five centuries earlier. The codex opens with al-T.u¯ s¯ı’s Tah.rir. reveal that it was “a textbook compiled for teaching the higher levels of Euclidean geometry” (Brentjes 2019, 446–447). The copious marginalia reveal “the continued interest in philosophical themes among madrasa circles” as well as a continuing “commitment of the teachers to the methodology of debate” (Brentjes 2019, 446– 447). Thus even though their original authors may have envisioned their tah.r¯ır as

37 As Fazlio˘ glu (2008, 23) explains, “the courses were divided into three main levels: for beginners, abridgments (ikhtis.a¯ r); for the intermediate students, the middle works (iqtis.a¯ d); and, for the advanced, detailed works (istiqs.a¯ ’). 38 Rashed (2002, 713–714) has also called attention to the importance of such collections, but his focus seems more codicological than pedagogical.

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a handbook for mature scholars, at least two of these tah.r¯ır were later adopted into the context of madrasa education. Acknowledgments The diagrams included in this chapter have been edited using DRaFT, a free, Java-based software utility created by Ken Saito. The software can be downloaded from Saito’s webpage: https://www.greekmath.org/draft/draft_index.html. I am very grateful to Mahdi Abdeljaouad for assistance in locating some primary source materials. I am also grateful to the anonymous referees for their many helpful suggestions.

Manuscripts Consulted • Cambridge University Library. Ad. 9-1075. Arabic translation manuscript. Undated. • Copenhagen. Kongelige Biblioteket, LXXXI. Arabic translation manuscript. Undated. • Florence. Biblioteca Medicea Laurenziana, orientali 20. Pseudo-T.u¯ s¯ı Tah.r¯ır. Copied from orientali 50. Undated. • Florence. Biblioteca Medicea Laurenziana, orientali 50. Pseudo-T.u¯ s¯ı Tah.r¯ır. Undated. • Istanbul, Süleymaniye Kütüphanesı, Aya Sofya 2719. Tah.r¯ır of al-Maghrib¯ı. Dated 914/1509. • Istanbul, Süleymaniye Kütüphanesı, Mihri Sah ¸ 337. Tah.r¯ır of al-Maghrib¯ı. Dated 1159/1746. • Kaysari, Ra¸sit Efendi Kütüphanesı 1230. Tah.r¯ır of al-Nays¯ab¯ur¯ı. Dated 528/1133 or 1134. • London, British Library, Add. 23387. Al-T.u¯ s¯ı Tah.r¯ır. Dated 656/1258. • Munich. Bayerische Staatsbibliothek, cod. arab. 2697. Al-T.u¯ s¯ı Tah.r¯ır. Dated 1142/1729. • Dublin. Chester Beatty Library, arab. 3035. Arabic translation manuscript. Undated. • Oxford University. Bodleian Library, Bodley Heb. d.4. Anonymous Hebrew treatise on polygonal and polyhedral figures. • Oxford University. Bodleian Library, Thurston 11. Arabic translation manuscript. Undated. • Oxford University. Bodleian Library, Or. 448. Al-Maghrib¯ı Tah.r¯ır. Dated 659/1260. • Tehran. Majlis Sh¯ur¯a Library, 6109. Al-T.u¯ s¯ı Tah.r¯ır. Dated 1019/1610. • Tehran. Malik Library, 3586. Arabic translation manuscript. Dated 343/954 or 955.

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The Recreational Problems of Tratado de Prática Darysmetica by Gaspar Nicolas, 1519 Jorge Nuno Silva and Pedro Jorge Freitas

Abstract In this paper, we present the contents of a 1519 Portuguese treatise of commercial arithmetic, the Tratado de Prática Darysmetica, by Gaspar Nicolas, with a special focus on the recreational problems that accompany the algorithms and exercises meant for commercial practice.

1 The Tratado in Its Context By the end of the fifteenth and early sixteenth centuries, commercial activity in Europe had become very intense, accompanying the movements of expansion and interconnection, promoting the transition from a feudal economic system to another in which trade took centre stage. This preeminence of trade and travel called for a more pragmatic and reliable use of mathematics that had not been seen before. In this context, and following the example of what was happening in the rest of Europe, two types of mathematical publications appeared in Portugal: those supporting navigation and those dealing with problems related to trade. As an example of publications of the first type, we have Abraão Zacuto’s Almanach Perpetuum, published in 1496, originally in Hebrew, later translated into Latin and Castilian by José Vizinho, with astronomical tables from the year 1473, which were used at that time in Portuguese and foreign navigation. Gaspar Nicolas produced the Quadrennial Tables of declination for 1517–1520 (Almeida 1994, p. 82).

J. N. Silva Ludus Association, CIUHCT, University of Lisbon, Lisbon, Portugal e-mail: [email protected] P. J. Freitas () CIUHCT, FCUL, University of Lisbon, Lisbon, Portugal e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_3

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There were about 40 arithmetic manuals published in Europe between 1472 and 1519 (Almeida 1994, p. 25). This proliferation may have accompanied the establishment of abacus schools, especially in Italy, after the publication of Fibonacci’s Liber Abaci in 1202, which introduced the use of Indo-Arabic numerals and related algorithms. These schools, practical in nature and geared to the needs of merchants, promoted the introduction of this system in everyday calculations and accounting. At the beginning of the sixteenth century, three arithmetical treatises were printed in Portugal. The first one was Tratado de Prática Darysmetica by Gaspar Nicolas, which will be the focus of this paper. The book was first published in 1519 and had 12 editions, the last one in 1716. The other two treatises were Prática Darismética by Rui Mendes in 1540 and Tratado da Arte de Arismética by Bento Fernandes in 1555. They all follow a similar structure, starting with the organised presentation of arithmetical procedures, either abstractly or immediately applied to practical cases. All of them use, from the beginning, Indo-Arabic numerals, completely abandoning Roman numerals and the operations performed with them, even though these can still be found in some contemporary treatises. The positional notation, typical of the Indo-Arabic system, greatly facilitates written algorithms which, unlike abacuses, keep the intermediate steps visible at all times and available for inspection. We analyse this structure in more detail in the next section.

2 The Organisation of the Tratado The book is organised in several thematic sections, which we describe here, with reference to the folios occupied by each section. 1. 2. 3. 4. 5. 6. 7.

Arithmetic Practice (Prática aritmética): F1–F27 Opposition (Oposição): F28–F38 Numbers (Números): F29–F45 Questions (Perguntas): F46–F77 Roots (Tirar raízes): F78–79 Geometry (Geometria): F80–F94 Silver alloys (Liga da prata): Fi–Fxxiii

The sections “Arithmetic Practice” and “Roots” are mostly concerned with the four basic arithmetical operations and the extraction of roots, which are presented in a practical fashion, followed by exercises of increasing difficulty. The only algorithm that is considerably different from the one used today is that of division: the book uses galley division (Swetz and Smith 1987, p. 90). After the description of the four operations, the book presents calculation rules, such as the rule of three, which occupies several sections, and is presented with several variants (which can be considered implementations of the compound rule of three, nowadays abandoned because it can be reduced to an iterated use of the rule of three). There are several sections devoted to fractions and to the extension of the calculation rules to cases

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where the numbers are rational. The last section, on silver alloys, is just a long list of problems on this specific topic (the folios for this section have an independent numbering). The section “Opposition” contains mostly word problems that lead to linear equations. Most of these are solved using the “rule of opposition”, which is the name given by the author to the rule of double false position, a method that is used systematically to solve linear problems. This rule gives the correct solution to a linear equation from the results obtained by giving two arbitrary values to the unknown. Its use was widespread in Europe at the time, being progressively abandoned as algebraic methods took over. The section “Numbers” contains abstract word problems that involve many different methods of solution. Many of them lead to linear equations or linear systems. Sometimes, the solution can be obtained using standard methods presented earlier in the book, whereas some other problems need a specific argument for the solution. The section “Questions”, the longest one in the book, starts off with a list of commerce-related problems, most of which are solved using the rule of three. After that, the problems have a more recreational nature, including some famous problems such as the division of wine with unmarked bottles, or a variant of the problem of points, which is at the origin of the theory of probability. Most of these problems appear in collections by authors like Alcuin (Hadley and Singmaster 1992), Fibonacci (Sigler 2002), and Pacioli (Pacioli ca. 1509), which are often considered to be important early sources of recreational mathematics. Even when their resolutions follow a well-known method, the questions aim to entertain and educate simultaneously. Finally, the section “Geometry” presents problems on squares, rectangles, circles, cubes, areas, volumes, and other topics in geometry. After the initial exercises for practice, some recreational problems are introduced. Contrary to the pedagogical principles to which we are accustomed today, the solutions to these problems are mostly presented without explanation—the author begins the resolution with the expression “Do it this way” and describes the method for solving the problem (in many cases it is not immediately clear why the given resolution actually solves the problem). The motivation, clearly, was to mechanise these methods so that the reader could put them into practice expediently in the daily problems of commerce. Alongside these pragmatic considerations, we find a long collection of problems of a recreational nature. It was a mediaeval tradition to accompany mathematical textbook exercises with lists of problems intended to develop reasoning and to entertain. Gaspar Nicolas explicitly refers to Luca Pacioli’s Summa as a source for these problems, but some of them come from older traditions, both mediaeval European and classical Greek. Some of these problems can be solved using the methods presented earlier in the book, but others need an ad hoc reasoning, or, as the author says, can only be solved “by fantasy”, that is, by thinking of a resolution specifically for the given problem.

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Some well-known collections of problems were searched for previous occurrences of the problems Nicolas used; among them were Metrodorus’ Greek Anthology (Paton 1980), Alcuinus’ Propositiones ad Acuendos Juvenes (Hadley and Singmaster 1992), Fibonacci’s Liber Abaci (Sigler 2002), Treviso’s Arithmetic (Swetz and Smith 1987), Pamiers’ Arithmetic (Sesiano 2018), Chuquet’s Le Treviso (Chuquet 1881, Chuquet and Marre 1881), Summa (Pacioli 1494) and De viribus quantitatis (Pacioli ca. 1509) by Luca Pacioli, and Conpusicion de la arte de la arismetica y Juntamente de geometria by Juan Ortega (Ortega 1512). Recreational mathematics, often immersed in works of another nature, has often been little noticed, even decried, by scholars. However, it is increasingly becoming unavoidable in historical research. Its roots are thousands of years old, and it can no longer be denied that recreational motivation is present in the evolution of mathematics (Singmaster 2017). Of the three arithmetic books we mentioned, it is the one by Gaspar Nicolas that devotes the most space to topics of recreational mathematics, about a third of the book. The treatise by Rui Mendes has no significant references to problems of this type, and Bento Fernandes devotes about one-sixth of his book to these problems, taking up many of those used by Gaspar Nicolas. The mixed style of these sixteenth-century arithmetic books, containing both strictly arithmetic and recreational topics, was kept in other later treatises of a similar nature, such as Gaspar Cardoso de Sequeira’s Thesouro de Prudentes, first published in 1612, also reprinted several times. The third of its four sections is dedicated to arithmetic, and it has a section entirely dedicated to illusionism (using both numbers and playing cards).

3 A Few Selected Problems 3.1 A Man and Three Saints I say that a man walked into a church and we don’t know how much money he was carrying. He told the first saint to double the money he was carrying and that he would give him 12 reais. The saint doubled it, and he gave him 12 reais and he still had money. Then he went to another saint, asking him to double the money that was left, and he would give him 12 reais. The saint doubled the money and the man gave him 12 reais and he still had money. And he went to another saint, asked him to double what was left, and he would give him 12 reais, and the saint doubled it, and the man gave him 12 reais, and he had nothing left. Now I demand how much money this good man carried. You will do it by opposition. [...] But you can do it as a rule, without opposition. And if you remember, I told you to take the half of what you spend and then take the half of the half, and then take the half of the other half again, i.e., divide by two as many times as he says he gives money. Well then, take the half of 12, which is 6. Now take the half of 6, which is 3. Now take the half of 3, which is one and a half, and add all these 3 parts, i.e., 6 and 3 plus and one and a half, and it’s 10 and a half. The man had this much money.

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This is a problem of the type “Monkey and coconuts”. This type of problems abounds in mediaeval problem collections, namely, in Fibonacci (Sigler 2002, pp. 372–383, 397–8, 439–445, 460–1), Chuquet (Chuquet and Marre 1881, p. 423, XXX–XXXII; p. 424, XXXIII), and Pacioli (Pacioli 1494, F105v, 20, 22; F187, 8), and (Pacioli ca. 1509, F120r–120v, 111r–111v, LXVII). For the first solution (which we omitted), a standard rule of double false position is used. The second method consists of a summary of a solution by retrograde analysis, that is, thinking about the gains and expenses from the end to the beginning. If in the end the man ran out of money, he had 12 reais before giving it to the third saint, and as these are twice what he had before, then he had 6 when he met the third saint. Likewise, we can conclude that he had 9 reais when he met the second saint, and 10.5 reais when he entered the church. Observing the amounts that are obtained successively by this process, we see that they are 6 + 6 + 3, and 6 + 3 + 1.5, which are the summands obtained by the method presented.

3.2 Generating Squares Give me a number such that, if you take away 11 from it, it becomes a square, and if you put 10 on it, it also becomes a square. This is the method: join these quantities that you want to take and put, and from that sum always take one. Divide in the middle what remains, and that half always multiply in itself. To this multiplication you will add that amount that you want to take out, and after all this, you’ll get the number you were asked for. In this example: add 10 and 11, you get 21. Take 1 and 20 remain, take half of that, which is 10. These I say multiply in themselves, and you will make 100. To these you must add the amount you want to take out, and you get 111. And this is the number such that if you take 11 from it, you get a hundred, which is a square, whose root is 10, and if you add 10 you will also get a square number, which is 121 whose root is 11.

The solution above, by Nicolas, is the particular implementation of the general procedure we now describe. We are looking for a number x such that  .

x − 11 =  x + 10 = 

where  represents a generic perfect square. The procedure given by the author leads to the solution x = 111 (111 − 11 = 102 , 111 + 10 = 112 ). In general, given two integers a and b, with odd sum s, one is asked to calculate  .

 s−1 2 +a 2

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Naturally, subtracting a, the result is a perfect square. If we add b to it, we get  .

   s−1 2 s+1 2 s 2 + 2s + 1 +s = = 2 4 2

The perfect squares obtained by this method are always consecutive. Other solutions can be obtained, noting that 1 + 3 + 5 + · · · + (2n − 1) = n2 , and therefore the difference of two squares, possibly non-consecutive, is always the sum of consecutive odd numbers. In this case, we can write 21 = 5 + 7 + 9, so 21 = 1 + · · · + 9 − (1 + 3) = 25 − 4, and therefore 4 + 11 = 15 would be another solution. As 21 cannot otherwise be expressed as the sum of consecutive odd numbers, these are the only solutions on the integers.

4 A Broken Weight A man had a stone that weighed 40 arráteis, it hit the ground and was broken in four pieces. With these 4 pieces he produced any number of arráteis as he was asked for, from one to 40. Now I demand how much each of them weighed. Know that this one has no rule [. . .] it is made of fantasy.1 Know that one of the pieces has one arrátel another has three and the other has 9, which is the square of three, and the other has 27, whose cubic root is three. But this rule is not general, it is by fantasy. So you will say that from the four numbers 1, 3, 9, and 27 the four pieces were made.

This problem appears already in Fibonacci (Sigler 2002, p. 420), Chuquet (Chuquet and Marre 1881, p. 451, CXLII), and Pacioli (Pacioli 1494, F97, 34). However, its origin is more remote; Tropfke references a Persian occurrence in the XI century (see Tropfke et al. 1980, p. 633). Presumably the problem concerns the use of a balance scale, with two plates, in each of which you can place either merchandise or weights. Thus, we look for four positive integers, x1 , x2 , x3 , x4 , whose sum is 40 and such that any natural n, 1  n  40, can be expressed as a linear combination n = α1 x1 + α2 x2 + α3 x3 + α4 x4

.

where the coefficients αi can take the values −1, 0, 1. The author presents the solution, justifying it with “fantasy”. Chuquet states that the sequence of weights starts with 1 and then each weight is one unit plus twice the sum of those that precede it. Pacioli starts with 1 and successively multiplies by 3, which is equivalent. Fibonacci mentions both procedures. Nicolas seems to have followed Pacioli.

1 Here, “fantasy” must be understood as reasoning without application of any standardised procedure.

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There is an interesting detail in the way the author poses the problem: in no other source are the four weights the result of a larger object being broken. This particular setup tells us that probably Nicolas had access to another source, possibly not as well-known as the ones to which we refer.

5 Equal Sales There were three women that came to sell oranges to the square, and one of them brought 50 oranges, and the other brought 30 oranges, and the other brought 10 oranges. Each one sold the same number of oranges for one real, and all left with the same amount of money. Now I demand how many oranges each of them sold for one real and how much money each one made. Know that these things do not have a general rule, only by your fantasy can you solve them. Know that each one gave 7 oranges for one real, and then the ones that were left over, that didn’t reach seven, they gave each one of those for 3 reais. Now you know that out of 50, giving seven for one real, you get 7 reais and one orange is left over. You sell this one for 3 reais, and with the 7 that I had already had, you get 10. Now consider the woman who had 30, which also sold 7 for one real. She got four reais and had two oranges left, and these two leftover oranges she sold each for 3 reais, as stated above, so that six reais are added to the 4, getting 10 reais. Now consider the one who had 10, she also sold 7 for one real, and there were three left over. She sold each of these three for 3 reais, which are 9 reais, which, added to one real that she had already had, make 10 reais. That’s how each one sold the same number of oranges for one real as any other, and ended up with as much money as any other.

This is another problem solved by “fantasy”. Other versions appear, with two sellers, already in Fibonacci (Sigler 2002, p. 421), but with three only in Pamiers Arithmetic (Sesiano 2018, p. 209, C112), Chuquet (Chuquet and Marre 1881, p. 453, CXLV ), and Pacioli (Pacioli ca. 1509, p. F119, LXV ). To fulfil the condition of selling at the same price, the author uses two prices: 1/7 reais per orange and 3 reais per orange. Whoever had 50, sold 7 at the first price and 1 at the second; the one who had 30 sold 28 at 1/7 and 2 oranges at 3 reais each; the one with 10 sold 7 at 1/7 and 3 at 3 reais. Each had 10 reais in the end. In Pamiers Arithmetic and Chuquet, whose versions of this problem are similar, there is one more solution, involving three prices (each obtaining 5 reais). For a mathematical analysis of this problem, see Singmaster 1999.

5.1 Interrupted Game Three men will play the crossbow, to see who first fires 5 shots, and they bet among all of them 12 cruzados. When the first one has 4 shots, the second one has three and the third one has 2, they don’t want to play anymore. I demand how the said 12 cruzados will be distributed.

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J. N. Silva and P. J. Freitas Do it this way. See how many shots they need to make among all of them. And you will see it this way: say that all three, each one of them, had four shots, which are 12 among all, and so that one of them should reach five, there would be 13 shots on the whole. Therefore the one who had 4 has 4 thirteenths of the money and the one who had 3 shots would have three thirteenths of the money and the one who had two shots would have two thirteenths of the money. [...]

This problem is phrased as the famous Problem of the points, which Pacioli presents (Pacioli 1494, F197v, 51) and which was at the genesis of Probability Theory (Katz 2009, p. 489). The author’s resolution follows that of Fra Luca, offering two methods. In the first, the author calculates the maximum number of turns that the game can have, in this case 13. With the points each one has—4, 3, and 2—we obtain the proportions 4/13, 3/13, and 2/13 that define the parts of the prize, 12, to be distributed to each one. These parts do not exhaust the prize, so the remainder must be divided. For this, the author resorts to the “rule of companies” (defined earlier in the book), which is equivalent to making a distribution using the proportions 4/9, 3/9, and 2/9. Adding the two parts, we get the prize for each player. The second method corresponds to applying, from the beginning, the proportions 4/9, 3/9, and 2/9 to the amount to be divided, 12. The results are the same, regardless of the method. Note that, in the first part of the first method, using denominator 13, or any other, leads to the same final result. To clarify this point, we resort to more general language. Suppose we want to divide the amount K among three players whose scores are a, b, and c. The second method corresponds to dividing K into three summands .

a K a+b+c

b K a+b+c

c K a+b+c

In the first, we first calculate the proportions of K corresponding to a/n, b/n, and c/n, for a certain n (in our case, it was n = 13), and we add the result of dividing what is left over by the rule of companies. That is, in the case of player a,   a b c a K− K− K− K . a+b+c n n n The total is   a a b c a a K− K− K− K = K . K + n a+b+c n n n a+b+c with similar results for players b, and c.

The Recreational Problems of Tratado de Prática Darysmetica by Gaspar Nicolas, 1519

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5.2 Bags There are two bags, one holds 8 alqueires of wheat and the other holds two. Now, I unsew them and make a bag that’s as tall as they were before. I ask how many alqueires the big sack holds. Do it this way. Combine 2 with 8 and that’s 10, save these. Now multiply the bags against each other, meaning 2 times 8, which is 16, take the root which is 4, double it, and it is 8. These 8 together with the 10 that I ordered you to save and are 18, and these many alqueires will the big sack hold.

Let’s assume that sacks are obtained from two overlapping rectangles of fabric sewn along three sides, like the so-called burlap sacks. It is natural to assume that the volume of each bag is proportional to the square of the base seam. As the height is constant in this problem, we can suppose it to be unitary and, being the measures of the base seams c1 and c2 , the volumes will be 8 = c12 k, 2 = c22 k, for some constant k. Joining the pieces of cloth, we obtain a sack with a seam at the base measuring c1 + c2 . The volume will then be  (c1 + c2 )2 k = c12 k + 2c1 c2 k + c22 k = c12 k + 2 c12 kc22 k + c22 k = 8 + 8 + 2 = 18

.

as in the text. Alternatively, if we consider the sacks as the side surfaces of cylinders, assuming unit height, the problem is reduced to determining the area of the circle whose perimeter is the sum of the perimeters of two circles of areas 8 and 2. The result is, again, 18.

6 Conclusion The Tratado de Prática Darysmetica can be seen as a source for understanding the commercial mathematics of the sixteenth century, crucial for Portugal and relevant for the rest of Europe. In the teaching tradition of practical mathematics, Nicolas’ text shines as a sophisticated pedagogical book. Including both pragmatic exercises, solved by application of general rules, together with recreational problems, which entertain with their appeal to imagination and “fantasy”, the author offers the students in abacus schools (and, more generally, those training for commercial matters) a glimpse of abstract rigorous thought in a ludic context. Gaspar Nicolas was not the first to understand that the teaching of mathematics is a difficult enterprise, but that, on the other hand, this science is the fitting realm where people should enjoy the training of creative and rigorous thought. Most of his problems have no practical use, they are just recreational. As Singmaster shows (Singmaster 2017), this pedagogical function of mathematical recreations is as old as teaching itself. We believe that the Tratado, being the first book on mathematics printed in Portugal, contributed decisively to the dissemination of mathematical teaching, now aided

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by the printing press, which permitted a much faster and more secure expansion of the subject. Its popularity can be gauged by the number of editions of the book (up until the eighteenth century), meaning that it became a source of mathematical learning for several generations of Portuguese merchants, accountants, and likely also people interested in a good problem to wrap their heads around.

References Almeida, António Marques de (1994). Aritmética como descrição do real, 1519–1679: contributos para a formação da mentalidade moderna em Portugal. 2 vols. Comissão Nacional para as Comemorações dos Descobrimentos Portugueses, INCM–Imprensa Nacional Casa da Moeda. ISBN : 9722705237. Chuquet, N. (1881). Le Triparty en la Science des Nombres, par maistre Nicolas Chuquet, Parisien, publié d’après le manuscrit fonds français n. 1346 de la Bibliothèque nationale de Paris et précédé d’une notice par M. Aristide Marre. . . Impr. des Sciences mathématiques et physiques. Chuquet, N. and Aristide Marre (1881). “Appendice au Triparty en la Science des Nombres de Nicolas Chuquet, Parisien”. In: Bullettino di bibliografia e di storia delle science matematiche e fisiche v. 14. Luglio, pp. 413–460. Hadley, John and David Singmaster (1992). “Problems to Sharpen the Young”. In: The Mathematical Gazette 76.475, pp. 102–126. ISSN: 00255572. Katz, Victor J. (2009). A History of Mathematics. 3rd ed. Pearson. ISBN: 0321387007. Ortega, J. de (1512). Conpusicion de la arte de la arismetica y Juntamente de geometria. Lyon: Casa de Maistro Nicolau de Benedictis. Pacioli, Luca (1494). Summa de Arithmetica geometria proportioni : et proportionalita. . . Paganino de Paganini. — — — (ca. 1509). “De Viribus Quantitatis”. Biblioteca Universitaria Bologna codice nr. 250. Paton, W.R. (1980). The Greek Anthology, Books 13–16. Loeb classical library. Book XIV. Harvard University Press. ISBN: 9780674990951. Sesiano, J. (2018). L’arithmétique de Pamiers: Traité mathématique en langue d’oc du XVe siècle. Presses Polytechniques et Universitaires Romandes. ISBN: 9782889152421. Sigler, L. (2002). Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation. Sources and Studies in the History of Mathematics and Physical Sciences. Springer New York. ISBN: 9781461300793. Singmaster, David (1999). “Some Diophantine recreations”. In: The Mathemagician and Pied Puzzler: A Collection in Tribute to Martin Gardner. Ed. by E.R. Berlekamp and T. Rodgers. Natick, Massachusetts: A K Peters, pp. 219–235. ISBN: 156881075X. — — — (2017). “The Utility of Recreational Mathematics”. In: Proceedings of the Recreational Mathematics Colloquium V. Ed. by Jorge Nuno Silva. Lisbon: Ludus, pp. 1–44. ISBN: 9789899950627. Swetz, F.J. and D.E. Smith (1987). Capitalism and Arithmetic: The New Math of the 15th Century, Including the Full Text of the Treviso Arithmetic of 1478. Open Court. ISBN: 9780812690149. Tropfke, J. et al. (1980). Geschichte der Elementarmathematik: Arithmetik und Algebra. B: Veröffentlichungen des Forstungsinstituts des Deutschen Museums für die Geschichte der Naturwissenschaften und der Technik. Walter de Gruyter. ISBN: 9783110048933.

The Mathematics of Polemic in John Napier’s Plaine Discovery Alexander Corrigan

Abstract John Napier of Merchiston (1550–1617) is famous as the inventor of logarithms and one of Scotland’s most revered mathematicians. Previous research has regarded his theological and mathematical endeavours as either distinct or linked in ways which do not withstand scrutiny. This chapter provides an outline of the mathematical content of Napier’s biblical commentary A Plaine Discovery of the Whole Revelation of Saint John, identifying that work as the sixteenth century’s most developed attempt to provide mathematical certainty to religious claims. Napier’s work emerged from an intellectual context in which thinkers increasingly identified mathematics with exactitude. Nevertheless, this chapter demonstrates that Napier’s true intention was not to discover mathematical patterns in history but to impose them where none truly existed. Napier’s commentary employed mathematical and historical enquiry only to reinforce polemical arguments. Indeed, Napier saw no inherent value in history and reduced the past to a mathematically precise pattern which he believed would allow him to predict the future. Napier’s methodology and results have hitherto been widely misunderstood, which has led to misconceptions about their legacy. The following seeks to correct those errors and further the scholarly understanding of Napier’s work, its impact and place within late-sixteenth–century intellectual traditions.

1 Introduction John Napier of Merchiston (1550–1617) is one of Scotland’s most famous and important Mathematicians. The impact of his invention of logarithms on seafaring

A. Corrigan () University of Edinburgh, Edinburgh, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_4

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and astronomy is widely known.1 Less well understood are the mathematics which underpinned his religious thought. This chapter argues that his 1593/4 work, A Plaine Discovery of the Whole Revelation of Saint John (Napier 2017a, pp. 97– 390), is crucial for understanding his worldview. Whilst Napier’s theological and mathematical works were distinct, they emerged from a coherent ideological position, which prioritised mathematics as the best tool for understanding humanity’s place in the universe and God’s predestined plan. The greatest part of the Plaine Discovery was a commentary on the biblical Book of Revelation, the final and most controversial book of the Christian canon. The Revelation is widely agreed to be largely allegorical and describes a series of visions whose meaning is not obviously clear. Scholars have debated its interpretation, and whether it belongs in the Bible at all, for centuries. Whether ‘The Revelation to John’ was written by John the Apostle had been doubted since at least the third century and questioned by the great Renaissance scholar Erasmus in works of 1516 and 1522 (Backus 1998, pp. 651–653). So influential were Erasmus’s claims that subsequent scholars, including Napier, were forced to defend the canonicity of the Revelation. The Revelation was also controversial because it was linked to Chiliasm, the belief in a future millennium in which Christ would reign over the earth, which was often denounced in Napier’s lifetime. Nevertheless, the text so obviously lent itself to denunciations of Rome that Protestant scholars tended to embrace it as biblical evidence for the evils of the Roman Catholic Church, which they saw as their enemy. For example, Revelation 17:9 refers to the Whore of Babylon sitting on seven hills, geographical features commonly associated with Rome. The great reformer Martin Luther came to accept its place in the Bible because it was intuitively predisposed to attacks on Rome (McGinn 1994, p. 205). Several prophecies in the Revelation have entered popular culture, such as the four horsemen of the Apocalypse and the number of the beast 666. Napier was especially concerned with the seven seals in Chapters 6 and 8, seven trumpets in chapters 8, 9 and 11, “thundering angel’s jubilees” in chapter 14 (see Table 2). Napier was following the historicist method of interpreting most of the Revelation’s prophecies as having been fulfilled in the past. Above all, the Plaine Discovery was an attempt to impose mathematical order onto and detect patterns in history, and thereby predict the future. It reveals the context of late-sixteenth–century intellectual thought, which prioritised mathematics as providing certainty to a range of claims. Moreover, Napier’s influence and legacy have been widely misunderstood because historians have made errors in interpreting the mathematical methodology which

1 The logarithms were published as Mirifici Logarithmorum Canonis Descriptio (Edinburgh 1614) and the posthumously published Mirifici Logarithmorum Canonis Constructio (Edinburgh 1619) expanded on the work and explained the processes by which Napier developed the logarithms. English translations are available in Brian Rice, Enrique González-Velasco and Alexander Corrigan (eds.) The Life and Works of John Napier, ed. (Springer, New York, 2017), pp. 475–751 and 752–808 respectively. For a discussion of the mathematical works and their legacy, see Enrique González-Velasco, ‘Mathematical Introduction in ibid., pp. 391–474 and Julian Havil, John Napier: Life, Logarithms and Legacy, Princeton University Press, Princeton, 2014, Ch. 3–7.

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undergirded his religious claims. This chapter presents an analysis of the key aspects of that mathematical system to illuminate Napier’s ideas and intellectual context. This chapter was presented as part of a symposium on mathematical antiquaries, but Napier does not fit comfortably into that category of intellectuals. As will be demonstrated below, his view of the past was overwhelmingly negative, being typified by corruption in and cruelty of the Roman Catholic Church, which Napier sought to prove was the false church of Antichrist. As such, his historical scheme was constructed solely to support his polemical claims. History was, for Napier, akin to mathematics as a tool for understanding the world rather than an area of enquiry with intrinsic value. Moreover, his polemical focus necessitated a negative view of the past as marked by evil and corruption, which meant that Napier almost entirely lacked the antiquary’s interest in material culture.

2 Context The fact that most Europeans in the early modern period saw all human activity through a lens of Christian belief is vital to understanding Napier’s context. There were also important developments in philosophical and religious perceptions of mathematics in the late-sixteenth century. Grafton has shown that the recognised need for reform of the Julian Calendar in the sixteenth century, and the triumph of mathematical research culminating in the Gregorian reform of 1582, prompted Europeans to think differently about mathematics (Grafton 1993, pp. 4–7).2 If calculations could add certainty to humanity’s understanding of the motions of the heavens, why should they not do the same for claims about the Bible or divine providence? The Plaine Discovery represents the most developed expression of this mindset. Previous works in the Anglophone tradition, such as John Bale’s Image of Both Churches, had an explicit polemical agenda, aiming to show that the pope was Antichrist and the doctrine of the Roman Catholic Church was corrupted and evil (Minton 2013). Bale implied a sense of chronological development, with events in the Revelation fulfilled in approximate historical periods. European thinkers like Joseph Scaliger had constructed more precise chronologies, verifying biblical claims with calculations based on historical patterns. They were concerned with the entire Bible and viewed sacred chronology as an inherently sacred endeavour, in which arguments against those with opposing religious views were not as pronounced as in works like Bale’s Image. Napier’s great innovation was to combine Bale’s use of the Revelation for polemic against the Roman Catholic Church with the mathematical precision of the emerging European tradition. As a result, Napier’s work was unmatched in its polemical impact.

2 Oosterhoff

(2020, p. 160) has also reflected on the extent of concern for calendrical reform amongst Napier’s contemporaries.

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Napier’s ideas reflect the fears of Protestants in the British Isles following the Spanish Armada of 1588. He and his compatriots had good reason to suspect another invasion would follow and might succeed (Yellowlees 2003, pp. 118–122). A committed Presbyterian, who sat on the General Assembly of the Scottish Kirk, Napier was fearful about the activities of Jesuits in Scotland and believed the best hope for the British Isles was for England and Scotland to unite under Reformed Protestantism and force out all Roman Catholics. He felt that predicting the Second Coming of Christ, Last Judgement and end of the world in around a century was the best way of convincing his English and Scots readers to get their affairs in order. The theme of Anglo-Scottish union within an eschatological context was shared by Napier’s king, James VI (1588). The Plaine Discovery aimed to court the monarch by endorsing his designs on the throne of England and his hope that Scots and English would unite and fight against European Catholic forces. Napier and James interpreted that hypothetical war, which appeared inevitable as the political borders of Europe were increasingly drawn along confessional lines, as part of the final battle between good and evil foretold in the Revelation (James VI 1588, sig. B3vB4v; Napier 2017b, pp. 945–7). Napier went further than any of his contemporaries. Believing that only mathematical exactitude could give that claim the urgency and impact it required, he set about trying to prove that the world would end between 1688 and 1700. Napier’s methodology rested on the assumption that discovering mathematical patterns in God’s historical providence meant that those patterns would yield predictions about the future. Commenting on Revelation 11:1, Napier wrote: And there was power and knowledge given me straightlie, as with a metwand, to measure the estate of thinges to come: so Christ the great Angell of the covenant assisting me, commanded me to arise from all earthlie affections, and to prophecie now the precise measure of times, that God hath carefully appointed over his true inward and invisible Church his holie religion, and all the true professours thereof. (ibid., pp. 247–8)

In Napier’s lifetime, and for centuries preceding it, a metewand or meteyard represented the benchmark of accuracy in measuring physical space. Napier’s words reflect the perception of mathematics and mathematical instruments as applicable to a range of naturalistic and religious areas of enquiry. He believed his mathematical interpretation could measure time in the past, present and future with the accuracy of the revered metewand.3 Moreover, Napier interpreted instances in the Revelation in which its author claimed to be “seeing” visions as “understanding”. He thus prioritised cognisance and intellectual faculties above sensory experience, further emphasising the importance of his calculations, which he claimed were divinely inspired (ibid., p. 355).

3 Metewands

appeared several times in the Bible and were discussed by John Bale (Minton 2013, p. 177). They represented judiciousness and were often buried alongside corpses, a tradition which survived the Reformation in some areas of England.

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3 Mathematics of Prophecy The mathematics of Napier’s system rested on several foundational concepts. The mystical number seven was the most fundamental integer in Napier’s scheme. It had immense cultural significance and was the number of seals, trumpets and vials described in the Book of Revelation. Napier followed the common practice of interpreting “days” when discussed in prophetic texts as meaning “years”. Accordingly, he extrapolated that a “year” in the biblical text must mean 360 years, based on the 360-day Hebrew calendar. This he combined with the concept of shmita, the sabbatical interpretation of years established in the Old Testament, which ascribes significance to every seventh year and greater importance to every 49th year, or “jubilee”. The numbers 1260 and 490 were also important for Napier, the former being typically interpreted as the period for which Antichrist would reign over the earth. Napier discerned many allusions to 1260 in the two most important prophetic books of the Bible, Daniel and Revelation. Revelation 12:6 had “And the woman fled into the wildernesse, where shee hath a place prepared of God, that they should feed her there a thousand two hundredth and threescore daies” (ibid., pp. 258–9, Napier’s translation). Napier interpreted those days as years (ibid.). Daniel 12.7 stated: And I heard the man that was clothed in linen, that stood upon the waters of the river, when he had lifted up his right hand, and his left hand to heaven, and had sworn by him that liveth for ever, that it should be unto a time, and times, and half a time. And when the scattering of the band of the holy people shall be accomplished, all these things shall be finished.4

Napier took “a time” to mean one year in the ancient 360-day calendar. Therefore “time, and times, and half a time” meant 360 + (360 × 2) + (360 ÷ 2) = 1260 (ibid., p. 129). 490 derived from the 70 weeks in Daniel 9, which were believed to have predicted the nativity, an interpretation Napier probably derived from John Calvin’s commentary on Daniel (Meyers 2021, pp. 189–94; Firth 1979, pp. 35–6). This interpretation allowed Napier to increase the rhetorical impact of his argument, claiming that those who might doubt his mathematical system were denying the Bible and Christ: upon necessitie of salvation, all Christians must confesse, in the seventie weekes of DANIEL, a day to be taken for a yeare, extending in the whole to 490 yeares, otherwise that prophecie of the Messias comming, would not fall upon the just time of Christs comming, as necessarily it ought to doe. (Napier 2017a, p. 107)5 .

4 Since Napier primarily relied upon the 1560 Geneva Bible, quotations from beyond the Revelation, which Napier produced in full, are from the facsimile version: Berry, L. E. (ed.), The Geneva Bible: A Facsimile of the 1560 Edition, Peabody, Hendrickson Publishers, 2007. 5 Daniel 9:24 has: “Seventy weeks are shortened upon thy people, and upon thy holy city, that transgression may be finished, and sin may have an end, and iniquity may be abolished; and everlasting justice may be brought; and vision and prophecy may be fulfilled; and the Saint of saints may be anointed.” 70 × 7 = 490.

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Drawing on Johann Carion’s Chronicle, Napier presented 490 years as the “fatall period of Empires” (ibid., p. 113). In his providential, mathematical view of history, this was the amount of time God permitted an empire to persist before falling apart, to prevent it becoming too powerful and corrupt, and entirely annihilating God’s chosen people.

4 Seals, Trumpets and Vials Napier’s chronology was based primarily on the seals in Revelation 6 and 8; trumpets in chapters 8, 9 and 11; vials in chapter 16; and “thundering angels” in Chapter 14. The vials repeated and reinforced the trumpets, in a technique known as “recapitulation”.6 The seals were opened every 7 years. 490 years was too great a number to be fitted into Napier’s chronology seven times, so he halved it and had his trumpets and vials occurring at junctures of 245 years. The “thundering angels” blew trumpets every 49 years, on jubilees. The first seal was opened in 29 AD, when Napier claimed Christ was baptised and the Gospel of Matthew was written. Napier’s contemporaries commonly believed Matthew was the first gospel, but the modern scholarly consensus is that Mark was the earliest of the canonical gospels (Barclay 2001, p. 1). The seals dealt with the authorship of the gospels and persecution of the early church by Roman emperors. Nero was given special attention, but the shortcomings of Napier’s system are clear from 57 AD, the year of the fifth seal. Napier claimed that in that year “arose the tyrant Nero” but 57 AD had no major significance to that emperor’s reign (Napier 2017a, p. 218). That was the first of many instances in which the mathematical rigidity of Napier’s system was ironically its downfall. The trumpets and vials began in 71 AD, which, to ensure no gaps, was also the date of the seventh seal. They occurred at seven junctures of 245 years, occupying the longest and most negative period in Napier’s chronology. It was characterised by the rise to the status of Antichrist by the papacy and increasing corruption in and tyranny of the Roman Church. Recapitulation was a deliberate device to increase the polemical potency of the chronology; the most negative period was repeated to reinforce in the Protestant reader a sense that the Roman Catholic Church and its adherents were their enemies and urge them to resist the temptations of the Whore of Babylon, which Napier believed was that church. The most important turning point for the Roman Catholic Church was Constantine’s move of the imperial capital from Rome to Constantinople, which Napier claimed happened in 316 but occurred gradually from 306 to 330, when the city was consecrated.

6 Recapitulation was proposed by Victorinus of Pettau in the third century and popularised by scholars including Antoine du Pinet, whose work influenced English religious exiles in Europe. Their work was a key influence on Napier. For more information, see Backus 2000, pp. xiii & 135 and Corrigan 2014, Ch. 9.

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Napier championed the authenticity of the Donation of Constantine, despite Lorenzo Valla’s masterful refutation of the document’s authenticity being well known by the time of writing (Bowersock 2008). Napier hoped to show that the transfer of authority over the Western Empire by the emperor caused the papacy to become Antichrist by dominating the secular and ecclesiastical arenas. That idea that the papacy’s status as the Antichrist foretold in the Book of Revelation was linked to its usurping of the power rightly reserved for secular princes was a common Protestant trope. John Bale, who influenced Napier profoundly, had employed it, but Napier imbued it with more force and urgency by identifying the precise moment in history when the transformation into Antichrist occurred (for example: Bale 1574, sig. 101r). This is one of the most contentious aspects of Napier’s legacy, since Arthur Williamson has argued that Napier was undermining James VI’s imperial ambitions by subverting Constantine the great as a model for godly kingship (Williamson 1994, p. 197). This matter has been discussed elsewhere (Corrigan 2014, pp. 25– 8). Nevertheless, Napier tacitly endorsed James’s claim to the English throne and some form of Anglo-Scottish union (ibid., pp. 71–7). Moreover, Napier’s views on Constantine were more nuanced than Williamson has allowed. Constantine was, for Napier, a godly ruler whose adoption of Christianity as the state religion of the Roman Empire began the 1000-year binding of Satan in Revelation 20:2 (ibid., pp. 321–2). Far from having evil intentions, his actions unwittingly yielded evil, longterm results.

5 Jubilees and the End Times Beginning in 1541, the angels’ jubilees occurred every 49 years until 1688. This period was different from those that preceded it, most obviously because the third and fourth jubilees were in the future. Napier’s historicist interpretation warned people of a millennium of corruption in the Roman Church, but it also had the more ambitious aim of finding a pattern so “plainly” true that it must extend into the future. The final period was also the most “positive” for Napier, since it was typified by the Protestant Reformation, which would lead to the fall of Rome, since the angels’ task was to pronounce God’s vengeance on Babylon. Nevertheless, Napier’s tendency towards pessimism expressed itself in his insistence that the age in which he lived was sinful and typified by laxity in religion, drunkenness and prostitution. This stance appears contradictory but is consistent with his preoccupation with the end of the world. If the world was not sinful, why ought it not persist far into the future and why should his readers improve their behaviour or unite to drive out Catholic influence? One of the more challenging aspects of Napier’s chronology was what might appear a discrepancy between the final trumpet and the terminal date of the final jubilee. If the trumpets lasted 245 years and the final one was blown in 1541, surely the world should persist until 1786 and not 1688 to 1700 as Napier insisted? This has confused many historians and created a widespread scholarly misunderstanding concerning Napier’s influence.

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Table 1 The end of the world Biblical reference Daniel 6:1

Biblical event Wait 1335 days after the abolition of daily sacrifice

Napier’s interpretation World will end 1335 years after the destruction of the Temple of Jerusalem

Year (actual) 363

Year (Napier) 365

Napier’s calculation 365 + 1335 = 1700

Napier repeatedly argued that the world would end before 1786, referring to Matthew 24:22: “except those dayes shulde be shortened, there shulde no flesh be saved: but for the electes sake those dayes shalbe shortened” (e.g. Napier 2017a, pp. 117 & 256). 1688 was simply the date of the fourth jubilee, whereas the year 1700 was derived from numbers found in the Book of Daniel. Daniel 12:11–12 had, “And from the time that the dayly sacrifice shal cease, & the abhomination put to desolation shal be 1290 daies. Blessed is he that waiteth and commeth to the thousand, thre hundreth and five and thirtie dayes” (Geneva Bible). Napier read the abomination and sacrifice as the rebuilding of the temple of Jerusalem under Emperor Julian “the Apostate” in 365 AD, which was “put to desolation” by an earthquake. Assuming “days” meant “years”, he added 1335 to make 1700. The earthquakes and abandonment of the temple’s rebuilding actually occurred in 363 AD and Napier appears to have deliberately misrepresented the date to yield the round figure of 1700 (Table 1). To understand Napier and the Plaine Discovery, one must understand the controversial nature of his actions in predicting even approximately when the Second Coming would occur. Doing so had been blamed for the revolt at Münster in 1534–1535, when a congregation of Anabaptists declared the Second Coming was imminent and Christ would reign over the earth from Münster as the New Jerusalem, for 1000 years. The protracted siege ended in brutal violence and was rumoured to have compelled the city’s inhabitants to resort to cannibalism (Gribben 2000, p. 35). Europeans looked on the events in horror for many years and Protestants sought to underscore their religious orthodoxy and distance themselves from millenarianism and apocalyptic imminence. Even despite those events, scholars accepted that the Bible forbad attempts to predict when the Second Coming would occur. On this issue, Napier abandoned his usually uncompromising tone. Understanding that he had to justify his actions, he employed an idiosyncratic reading of scripture. He interpreted literally Mark 13:32: “But that day and houre knoweth no man, no, not the Angels which are in heaven, nether the Sonne him self, save the Father” (Geneva Bible). Napier claimed that scriptural verse only proscribed predicting the exact time and his approximate period of 1688–1700 was therefore acceptable: “let none be so base, of judgement as to conclude thereby, that the yeare or age thereof, is also unknowne to Christ, or unable to be known any waies to his servants” (Napier 2017a, p. 121). Napier even implied that to doubt that Christ had some awareness of the date was blasphemous. On Revelation 12:12 he wrote, “seeing that the Devill

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hath great wrath in these latter daies, and doth know his time is short, shal we say, that Christ shall be ignorant of that, which the Devill doeth know” (ibid., p. 123). Most revealing of his mindset was Napier’s contention that his lifetime was special because of its proximity to the end times and the rules imposed upon former generations, including the disciples, no longer applied. Acts 1:7 reported Jesus telling the disciples, “It is not for you to know the times, or the seasons, which the Father hathe put in his owne power” (Geneva Bible). Napier responded with: “God hath hitherto concealed these misteries from them whom the knowledge thereof might have endammaged: yet that prooveth not, that the same shall be hidde from us, to whom the knowledge thereof might bring repentance and amendement” (Napier 2017a, p. 121). To bolster his claim, Napier cited Daniel 12:4, “Seale the book till the appointed time, manie shall gae to & fro, and knowledge shal be encreased” (ibid. & Geneva Bible). He believed that time had come and was manifested in God’s revealing the final mysteries of scripture to prophets like him.

6 Notes on Table 2 Table 2 summarises Napier’s chronology. Napier’s place within the historicist school of scholarship on the Book of Revelation is clear; most events occurred before 1593/4, when Napier wrote the Plaine Discovery. The seals occurred at junctures of 7 years. The trumpets and vials, which referred to the same historical events, occurred every 245 years and the thundering angels’ jubilees were every 49 years. Note that the seventh seal and first trumpet/vial occur in the same year, as do the seventh trumpet/vial and first jubilee, to ensure there were no gaps. The 1260 years of Antichrist’s reign and millennium of Satan’s binding are further included. The modern reader will note that most of the dates are incorrect, but this should not simply be ascribed to dishonesty on Napier’s part. Historical information is now more accurate and easily verified than in Napier’s lifetime. The precise years in which the gospels were written are impossible to know but Napier was being faithful to scholarly consensus of his age. The Scottish Reformation was finalised in 1560 and the accuracy of that date illustrates its importance for Napier and his intended audience. As noted above, Napier’s dating which allowed him to predict the end of the world in 1700 may have been deliberately incorrect. Below is an explanation of abbreviations and some pertinent ideas relating to them. “Sim.”—Simplification. Some of the historical events, such as Constantinople becoming the Imperial Capital, were too complex to be distilled to a single date. Doing so was misleading. “c.”—“Circa”. The precise dating of the Gospels is not known. “r.”—“Reigned”. “Prediction”. Writing in 1593/4 Napier moved to predictions about the future in the final three jubilees of his chronology.

Year (Napier) 29 36 43 50 57 64 71 71

c. 300 300–316

316 561 806

1051 1296 1300

1541 1541

20:2 20:4

8:8–9/16:13 8:10–11/16:4–7 8:10–11/16:8–9

9:1/16:8–9 9:13/16:12–13 20:7

11:15/16:17–21 14:6

Satan bound Antichrist’s reign began 2nd trumpet/vial 3rd trumpet/vial 4th trumpet/vial

5th trumpet/vial 6th trumpet/vial Satan released

7th trumpet/vial 1st Jubilee

Scriptural event 1st seal 2nd seal 3rd seal 4th seal 5th seal 6th seal 7th seal 1st trumpet/vial

Chapter and verse (Rev.) 6:1 6:3 6:5 6:7 6:9 6:12 8:1 8:7/16:2

Table 2 Napier’s chronology

306–330 Sim. Sim./historically misleading Sim. Ottoman Empire: 1299 Correct/Postdates crusades Sim. Sim.

Year (actual) c. 27/70–90 c. 54–70 c. 26/85 c. 90–100 r. 54–68 Approximately accurate Vespasian r. 69–79 Named emperors r. 81–324 306–313/312 315–17 (Purported)

Cessation of Christian persecutions/Conversion of Constantine. Donation of Constantine. (Exposed as eighth-century forgery in fifteenth century.) Constantine shifted imperial seat from Rome to Constantinople Rise of Islam and apostasy of Christians in Near East Church corrupted by Islam in East & Papacy in West/Charlemagne divided Holy Roman Empire between sons. Rising power of Islam Unification of Islamic peoples Boniface VIII introduced Jubilees for remission of sins/Warfare between Popes and Islamic peoples as Gog and Magog. Protestant Reformers active and successful Protestant Reformers bringing “truth” of Gospels to light

Napier’s historical application Baptism of Christ/Gospel of Matthew. Christians persecuted/Gospel of Mark. Global famine. Gospel of Luke Gospel of John Nero’s temporal power increasing Nero persecuted Christians, committed incest and matricide Persecution of Christians suspended by Flavian Dynasty. Effeminate, then fierce, Roman Emperors.

66 A. Corrigan

1590

14:8

14:9–10 14:14

3rd Jubilee 4th Jubilee

1639 1688– 1700

1560

11:11

Antichrist’s reign ended 2nd Jubilee Prediction Prediction; 1700 based on flawed dating

Sim.

Date correct Fall of Rome as new Babylon. Military success of Protestant vs. Catholic states Final defeat of Rome Second Coming of Christ, Day of Judgement, destruction of world, creation of new heaven and new earth

Scottish Reformation showed the power of papal Antichrist waning

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“Purported”. The Donation of Constantine is a proven forgery, but this was the date its proponents claimed it was written. Thus, the table shows that the precise mathematical interpretation of history was less important for Napier than the idea of mathematics as able to provide the illusion of absolute precision when interpreting the past, which could extend to predictions about the future. The mathematical pattern appears to have been devised relating to certain key events, such as the baptism of Christ and Scottish Reformation, and then fitted to other events to the best of Napier’s ability. Mathematical precision was a powerful façade to impose onto history, which would be memorable to his audience, even if elsewhere in the text he had to concede that some of the dating was more approximate or vague than he would have liked the reader to imagine.

7 Critical Observations Napier lived at a time of fear, warfare and political intrigue. Like many apocalyptic works, his Plaine Discovery hoped to provide psychological comfort by making sense of traumatic events. Its historicist system was ostensibly concerned with finding mathematically precise patterns in history which might extend into the future. It also hoped to impose order onto events which seemed random and chaotic, but the elegance of the system was an illusion. The years 57 and 64 were not especially important in the rise of Nero or his persecution of Christians. Constantine did not shift the imperial capital east in 316. There is an irony to the fact that Napier’s resolute championing of mathematics as adding certainty to interpretations of the Bible compelled him to falsify or oversimplify some of the most important numbers. Nevertheless, at a time when historical dates were less easily verified than today, Napier would have believed that his ends of reforming society and increasing the safety of the British Isles justified the means. His contemporaries would probably have seen years being incorrect by a small margin as minor errors; they went unmentioned in the seventeenth-century anglophone works which drew on or criticised the Plaine Discovery (Corrigan 2020). Napier’s interpretation of the past as almost entirely negative is at odds with what modern observers might expect from such a work. Even the millennium during which Satan was supposedly bound in Revelation 20:2–3, instigated by Constantine’s adoption of Christianity and ending in the papacy of Boniface VIII, was a significantly negative time. Due to God’s mercy, it coincided with the 1260-year reign of Antichrist, “least on both sides, Gods Church were utterly extinguished” (Napier 2017a, p. 177). Napier similarly lacked interest in the material past, famously dismissing a Roman monument which once stood in Musselburgh, and coins found in Scotland as merely evidence of Roman hubris and blasphemy to defame the Roman Catholic Church, which he saw as a continuation of the pagan Roman Empire (Napier 2017a, p. 303; McGinnis and Williamson n.d., p. 2). Napier elevated to cosmic significance Scotland and England, casting them as horns of the beast in Revelation 13, which turned against the Whore of Babylon to bring about her destruction. He

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integrated 1560, the year of Scotland’s Protestant Reformation into his scheme as the terminal date of Antichrist’s temporal dominion (ibid., p. 178). Nevertheless, he never sought to imbue his country with a glorious sense of its own past to forge a robust contemporary identity in the same way as scholars such as George Buchanan. Implicit in the works of Buchanan was a sense that Scotland should have a glorious future as a country distinct from England, but Napier’s apocalyptic urgency overrode any such concerns. Nor was Napier focused on didactic or devotional interests in the same way as James Ussher’s famous commentary of 1650. Indeed, Ussher and his anglophone contemporaries might be regarded as closer to the European tradition. They were more interested in the Old Testament than Napier; and although Ussher regarded mathematics as a useful tool for better understanding the Bible and creation, he did so as an inherently sacred endeavour. While Napier’s prophetic focus on the Day of Judgement and compelling his readers to prepare for it was the raison d’être of the Plaine Discovery, it was not a focus that was present in the works of Buchanan or Ussher.

8 Napier’s Legacy in Scholarship Napier’s chronology was unprecedented in both its mathematical precision and its complexity, which has led many scholars to misunderstand it and its legacy. In her classic study of 1979, Katherine Firth asserted that Napier had predicted the world would end in 1786, which, as shown above, is incorrect (Firth 1979, p. 164). Correcting this error, which has been repeated as recently as 2014, is no mere case of academic pedantry but is vital for understanding Napier’s influence (Drinnon 2013, p. 30). Firth (1979, p. 144) asserted that the time between the fall of Rome in 1639 and the end of the period of the seventh trumpet in 1786 represented a 147year “golden age”, which broke the taboo surrounding millennialism and allowed Thomas Brightman and Joseph Mede to propose a coming 1000-year reign of Christ on earth, which in turn became the dominant millenarian position in England. The period of 49–61 years Napier predicted was far from the millennium foreseen by Brightman and Mede. Rather than a period of Christ’s earthly reign, it would be a time of preaching unrestrained by the by-then defeated Roman Catholic Church, which would see as much of the world as possible converted to Protestantism before the last judgement. Moreover, several scholars have missed the point that Napier understood the millennia in Revelation 20:2 and 20:6 as separate events (Jue 2006, p. 93; Crome 2014, p. 78). The millennium of Satan’s binding was in around 300–1300 AD but the “millennium” of Christ’s reign was to be eternal (Napier 2017a, pp. 323 & 328). Those misunderstandings combined with the widely accepted assertion that Napier’s insistence on an imminent Second Coming was not influential has diminished the reputation of the Plaine Discovery in general. In fact, Napier influenced Brightman by proposing a future, figurative millennium after the destruction of the world by fire. Brightman simply moved that millennium to before the destruction. Moreover, Napier introduced a new level of mathematical

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precision into the eschatological tradition of the British Isles. It may be seen in the complex diagrams of Joseph Mede’s 1627 Clavis Apocalyptica. Indeed, specific events from Napier’s chronology were adopted by scholars like Brightman (1611, p. 707) and Richard Bernard (1617, p. 312), who both concurred that Constantine’s adoption of Christianity caused the binding of Satan, who was released during the papacy of Boniface VIII. Moreover, after Napier’s work, the anglophone historicist school especially saw an increase in ascribing precise dates to events which fulfilled prophecies in the Book of Revelation (Corrigan 2020). How should scholars regard the relationship between Napier’s theological and mathematical works? Since the 1960s, many have promoted the “unity of thought” approach to early modern scholars, which was originally developed as a postmodern response to the various activities of Isaac Newton.7 A degree of unity is evident in Napier’s endeavours, which might appear as separate areas of enquiries to modern observers. He produced a manuscript in 1596 which proposed several fantastical weapons of war (Napier 2017c, pp. 952–4). Like the Plaine Discovery, the manuscript referred to the “Iland” of Great Britain, which Napier hoped to protect from “enemies of God’s truth and religion” (ibid., p. 952). Both that manuscript and the Plaine Discovery belong to the context of the post-Armada crisis, and Gladstone-Millar’s (2008, pp. 28–33) claim that they formed in Napier’s mind a “two-pronged attack” against Spain is persuasive. Napier devised the logarithms to assist in the complex calculations needed for accurate navigation at sea. His contemporaries knew a solution for simplifying those calculations must exist and Napier endeavoured for decades to find it. Similarly, the Plaine Discovery was an attempt to use mathematics to address issues Napier felt were urgently pressing, especially the perceived threat of domestic and European Catholics and societal ills like drunkenness and prostitution. However, the limitations of the unity of thought approach have been made clear in recent years (Iliffe 2017, p. 14). Although it remains a useful paradigm, some associated thinking suggests a naïve or flippant approach to the evidence. Hugh Trevor-Roper’s (1965, p. 60) casual assertion that Napier devised the logarithms to calculate the meaning of the Number of the Beast has no factual merit.8 Napier was satisfied with that calculation two decades before the logarithms were published. He made no alterations to the mathematics of his system in the 1611 edition, when logarithmic calculation was almost fully developed. Indeed, Logarithms had no specific relevance to any calculations in the Plaine Discovery. The logarithms and Plaine Discovery were not different aspects of a single enquiry but were “unified” by the belief that mathematics provided the best problem-solving tool at Napier’s disposal and the key to unlock divine mysteries, relating to both scripture and the material world.

7 See 8 For

Corrigan 2014, p. 14, for more. Napier on 666, see Corrigan (2020, pp. 131–2).

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9 Closing Remarks Napier’s view of history was shaped by the need to produce order from chaos and convince the reader to follow his polemical message. This necessitated a balance between making the past seem frightening and finding reassurance in God’s mathematically precise providential pattern. The lengths to which he went may appear questionable to the modern observer, but Napier would have seen any manipulation of dates and figures as unimportant; the accuracy of the mathematics was secondary to their polemical message, which he earnestly believed was for the good of his country. His lack of interest in Scotland’s material culture might appear surprising but made sense within his apocalyptic worldview, and he was not entirely negative about the past. All events were predestined by God and Napier believed that Protestants were the successors to the Israelites as God’s chosen people. That belief was predicated on the existence of a minority “true” church, which was often hidden but was exemplified in the acts of certain godly figures throughout history. This line of thinking provides a glimpse of Napier’s rare positive opinions of the past; Napier appears to have self-fashioned after Archimedes. He derived the word “logarithm” from Archimedes’ The Sand Reckoner, patented a device for draining coal mines based on the “Archimedes Screw” and hoped to recreate the legendary “burning mirror”, which supposedly laid waste the Roman invasion fleet at Syracuse in 212 BC. Along with reverence for the Old Testament, these fragments provide the only evidence that Napier was anything approaching a “mathematical antiquary”.

References Backus, Irena. 2000. Reformation Readings of the Apocalypse: Geneva, Zurich and Wittenburg. Oxford: Oxford University Press. Backus, Irena. 1998. The Church Fathers and the Canonicity of the Apocalypse in Sixteenth Century: Erasmus, Frans Titelmans, and Theodore Beza. Sixteenth Century Journal, 29(3), 651–666. Bale, John. 2013. The Image of Both Churches, ed. G.E. Minton. Dordrecht: Springer. Bale, John. 1574. The pageant of popes. London: Thomas Marshe. Barclay, William. 2001. The Gospel of Mark. Edinburgh: Saint Andrew Press. Valla, Lorenzo. 2008. On the Donation of Constantine. Trans. G.W. Bowersock. Cambridge: Harvard University Press. Bernard, Richard. 1617. A Key of Knowledge for the Opening of the Secret Mysteries of St Iohns Mysticall Revelation. London: Felix Kingston. Brightman, Thomas. A Revelation of the Apocalypse of S. Iohn. 1611. Amsterdam: Jodocus Hondius and Hendrik Laurenszoon Spiegel. Corrigan, Alexander. 2014. John Napier of Merchiston’s Plaine Discovery: A Challenge to the Sixteenth Century Apocalyptic Tradition. Dissertation. The University of Edinburgh. Corrigan, Alexander. 2020. John Napier’s Influence on Seventeenth-Century Apocalyptic Thinking in England. Reformation and Renaissance Review 22:2: 126–147. Crome, Andrew. 2014. The Restoration of the Jews: Early Modern Hermeneutics, Eschatology, and National Identity in the Works of Thomas Brightman. Cham: Springer.

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Drinnon, David A. 2013. The Apocalyptic Tradition in Scotland, 1588–1688. Dissertation. University of St Andrews. Iliffe, Rob. 2017. Priest of Nature: The Religious Worlds of Isaac Newton. Oxford: Oxford University Press Grafton, Anthony. 1993. Joseph Scaliger: A Study in the History of Classical Scholarship: II Historical Chronology. Oxford: Clarendon Press. Jue, Jeffrey K. 2006. Heaven Upon Earth: Joseph Mede (1586–1638) and the Legacy of Millenarianism. Dordrecht: Springer. Firth, Katherine. 1979. The Apocalyptic Tradition. Oxford: Oxford University Press. Gladstone-Millar, Lynne. 2008. Logarithm John. Edinburgh: NMS Publishing Ltd. Gribben, Crawford. 2000. The Puritan Millennuim: Literature & Theology 1550–1682. Portland: Four Courts Press. King James VI of Scotland. 1588. Ane Fruitfull Meditatioun contening ane Plane and Facill Expositioun of the 7. 8. 9. and 10 versis of ye 10 Chap. of the Revelatioun in forme of ane sermone. Edinburgh: Henry Charteris. Mede, Joseph. 1627. Clavis Apocalyptica. Cambridge. Minton, Gretchen E. (ed.). 2013. John Bale’s The Image of Both Churches. Dordrecht: Springer. Meyers, T. (trans.), Calvin, John. Commentaries on the Book of the Prophet Daniel, Vol. II. Grand Rapids: CCEL. http://www.ccel.org/ccel/calvin/calcom25.pdf. Accessed 28 March 2021. McGinn, Bernard. 1994. Antichrist: Two Thousand Years of the Human Fascination with Evil. San Francisco: Harper Collins McGinnis, Paul J. and Williamson Arthur H. n.d. Politics, Prophecy, Poetry: The Melvillian Moment, 1589–96, and its Aftermath. The Scottish Historical Review LXXXIX, 227(1), 1–18. Napier, John. 2017a. A Plaine Discovery of the Whole Revelation of Saint John. In The Life and Works of John Napier, ed. B. Rice, E. González-Velasco and A. Corrigan, 97–390. New York: Springer. Napier, John. 2017b. Secrett Inventionis. In The Life and Works of John Napier, ed. B. Rice, E. González-Velasco and A. Corrigan, 952–954. New York: Springer. Oosterhoff, Richard J. 2020 ‘Tutor, Antiquarian, and Almost a Practitioner: Brian Twyne’s Readings of Mathematics. In Reading Mathematics in Early Modern Europe: Studies in the Production, Collection, and Use of Mathematical Books, ed Philip Beeley, Yelda Nasifoglu and Benjamin Wardhaugh, 151–166. New York: Routledge. Trevor-Roper, Hugh. 1965. The General Crisis of the Seventeenth Century. In Crisis in Europe, 1560–1660: Essays from Past and Present, ed. T. Aston, 59–95. London: Routledge. Ussher, James. 1650. Annales Veteris Testamenti. London. Williamson, Arthur H. 1994. Number and national consciousness: the Edinburgh mathematicians and Scottish Political culture at the union of the crowns. In Scots and Britons: Scottish political thought and the union of 1603, ed. Roger A. Mason, 187–212. Cambridge: Cambridge University Press. Yellowlees, Michael J. 2003. ‘So Strange a Monster as a Jesuite’: The Society of Jesus in SixteenthCentury Scotland. Colonsay: House of Lochar.

Euler’s Series for Sine and Cosine: An Interpretation in Nonstandard Analysis Piotr Błaszczyk and Anna Petiurenko

Abstract In chapter VIII of Introductio in analysin infinitorum, Euler derives a series for sine, cosine, and the formula .eiv = cos v + i sin v. His arguments employ infinitesimal and infinitely large numbers and some strange equalities. We interpret these seemingly inconsistent objects within the field of hyperreal numbers. We show that any non-Archimedean field provides a framework for such an interpretation. Yet, there is one implicit lemma underlying Euler’s proof, which requires specific techniques from nonstandard analysis. Analyzing chapter III of Institutiones calculi differentialis reveals Euler’s appeal to the rules of an ordered field which includes infinitesimals—the same ones that he applies to deriving the series for .sin v, .cos v, and .ev .

1 Introduction In chapter VII of Introductio in analysin infinitorum (§§114–125), Euler introduces the famous number e and derives a series for .ez . This paper will focus on Euler’s technique applied to infinitesimals and infinitely large numbers that he refers to in his arguments. The first paragraph of chapter VII of Institutiones calculi differentialis reads: Since .a 0 = 1, when the exponent on a increases, the power itself increases, provided a is greater than 1. It follows that if the exponent is infinitely small and positive, then the power also exceeds 1 by an infinitely small number. Let .ω be an infinitely small number, or a fraction so small that, although not equal to zero, still .a ω = 1 + ψ, where .ψ is also an infinitely small number. From the preceding chapter we know that unless .ψ were infinitely small then neither would .ω be infinitely small. It follows that .ψ = ω, or .ψ > ω, or .ψ < ω. Which of these is true depends on the value of a, which is not now known, so let

P. Błaszczyk () · A. Petiurenko Pedagogical University of Kraków, Kraków, Polska e-mail: [email protected]; [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_5

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P. Błaszczyk and A. Petiurenko .ψ .ω

= kω. Then we have .a ω = 1 + kω, and with a as the base for the logarithms, we have = log(1 + kω). (Euler 1988, 92)

The infinitesimals referred to in this section are used in mathematical processes in the same way as standard quantities. They are subject to the trichotomy law, .ψ = ω, or .ψ > ω, or .ψ < ω. They are also terms of operations such as exponents or logarithms. Euler continues: Since .a ω = 1 + kω, we have .a j ω = (1 + kω)j , whatever value we assign to j . It follows that .a

jω

=1+

1

j

kω +

j (j − 1) 2 2 j (j − 1)(j − 2) 3 3 k ω + k ω + &c. 1·2 1·2·3

If now we let .j = ωz , where z denotes any finite number, since .ω is infinitely small, then j is infinitely large. Then we have .ω = jz , where .ω is represented by a fraction with an infinite denominator, so that .ω is infinitely small, as it should. When we substitute . jz for .ω then a z = (1 + .

kz j j )

=1+

1 1(j − 1) 2 2 1(j − 1)(j − 2) 3 3 kz + k z + k z + 1 1 · 2j 1 · 2j · 3j

+

1(j − 1)(j − 2)(j − 3) 4 4 k z + &c. 1 · 2j · 3j · 4j

This equation is true provided an infinitely large number is substituted for j , but then k is a finite number depending on a, as we have just seen. (Euler 1988, 93)1

In discussing infinitesimals, infinite numbers, and finite numbers, the above provides some useful relationships: the product . ωz is infinite, given z is finite and z .ω infinitesimal, or . j is infinitesimal, given j is infinite. Moreover, it considers sums with infinitely many terms. Note also that Euler processes fractions involving infinitesimals and infinite numbers as if they were typical fractions: .ω = jz , given z .j = ω . In the next sections, Euler puts .z = 1. Consequently, the reciprocal of an infinite number is infinitesimal, and the reciprocal of an infinitesimal is an infinite number. In section 116 of Euler’s text, we find some strange arithmetic terms involving infinite numbers. Since j is infinitely large, . j −j 1 = 1, and the larger the number we substitute for j , the closer the value of the fraction comes to 1. Therefore, if j is a number larger than any assignable number, then . j −j 1 is equal to 1. For the same reason . j −j 2 = 1, . j −j 3 = 1, and so forth. It follows that . j2−j2 = 12 , . j3−j3 = 31 , . j4−j4 = 14 , and so forth. (Euler 1988, 93–94)

In this paragraph, infinite numbers, i.e., those greater than any assignable 1 is closer number, satisfy a mystifying equality . j −j 1 = 1. Yet it is also true that . j1j− 1

1 In

Blanton’s translation (Euler 1988), instead of the original sign .&c., there are three dots .. . . . In section 4.3 below, we comment on that.

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75

to 1 than . j2j−2 1 , if .j1 > j2 . Clearly, the equality sign in . j −j 1 = 1 does not stand for strict equality. Since . 1j is infinitesimal, . j −j 1 or .1 − 1j is infinitely close to 1 rather than equal to it. −k By substituting . 1k for . jkj in the series for .a z , Euler gets az = 1 +

.

kz k 2 z2 k 3 z3 k 4 z4 + + + + &c. 1 1·2 1·2·3 1·2·3·4

Assuming .z = 1, he finds the following series for a, k3 k4 k2 k + + + &c. + 1 1·2 1·2·3 1·2·3·4

a =1+

.

Taking .k = 1, he can define the number e as follows e =1+

.

1 1 1 1 + + + &c. in infinitum. + 1 1·2 1·2·3 1·2·3·4

Going through that argument again using the number e instead of a, we get the following: eω = 1 + ω,

.

ej ω = (1 + ω)j , ez = (1 + jz )j , where j =

z . ω

(1)

j  1 1 z Finally, assuming . m j m = m! , from (1), the series for .e follows ez = 1 +

.

z2 z3 z4 z + + + + &c. in infinitum. 1 1·2 1·2·3 1·2·3·4

Euler’s argument for the series expansion of .ez contains numerical examples of 1 some terms, such as the following. When .a = 10 and .kω = , then from 1000000 logarithmic tables Euler derives the value of .ω as follows .

 log 1 +

 1000001 1 = 0.00000043429 = ω. = log 1000000 1000000

As a result, .k = 2.30258. These calculations aim to illustrate that k depends on a. We assume that in this case, .ω is “a fraction so small" rather than infinitesimal.

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Euler also estimates the number e, showing that is equal to 2.71828182845904523536028, or 2.718281828459&c.

.

In sections VII and VIII, finite numbers are usually given in decimal form. Products of infinitesimals and infinite numbers, such as .j ω, also represent finite numbers. Euler does not give examples of infinitesimals and infinitely large numbers, instead they seem to constitute a technique for proving some results concerning finite numbers. Indeed, the implicit procedure is this: when a finite number z is given, take any infinite number j —more precisely, a nonstandard z integer j —then, z and j determine the infinitesimal .ω = ω(z, j ) = , and the j formula (1) follows. Although the series for .ez consists of infinitely many terms, to estimate .ez , Euler considers only finitely many of them and does not examine the rest. In chapter VIII, Euler applies the same technique to derive series for sine and cosine starting from the formulas .

.

cos nz =

(cos z + i sin z)n + (cos z − i sin z)n , 2

sin nz =

(cos z + i sin z)n − (cos z − i sin z)n . 2i

He reaches these results by using trigonometric properties described in terms of complex numbers. At the end of that chapter, he combines these results and the formula (1). Once again, a reference to infinitesimals, infinite numbers, and a specific understanding of equality substantially contributes to his argument. Section 138 reads: let z be an infinitely small arc and let n be an infinitely large number j , so that j z has a v v finite value v. Now we have .nz = v and .z = , so that .sin z = , and .cos z = 1. With j j these substitutions  . cos v

=

1+

iv j  iv j + 1− j j 2

 , and sin v =

1+

iv j  iv j − 1− j j . 2i

In the preceding chapter we saw that .(1 + z/j )j = ez where e is the base of the natural logarithms. When we let .z = iv and then .z = −iv we obtain . cos v

=

eiv + e−iv 2

, and sin v =

eiv − e−iv . 2i

From these equations we understand how complex exponentials can be expressed by real sines and cosines, since .eiv = cos v + i sin v, and .e−iv = cos v − i sin v. (Euler 1988, 111– 112)

So Euler’s mathematical toolbox includes:

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77

1. The multiplicative inverse of infinitesimal is infinitely large, 2. The multiplicative inverse of infinitely large is infinitesimal, 3. For a finite number v and infinite number j , there is an infinitesimal .ω such that .j ω = v, j  1 −m 1 1 4. Infinite numbers .j, m satisfy the equality . jmj =m , and . m j m = m! , and   5. The binomial theorem applies to an infinite number j , i.e., .(1 + x)j = 1 + 1j x + j  2 2 x + ... . Chapter VIII of Introductio in analysin infinitorum adds two other assumptions to this list: 6. That .sin z = z and .cos z = 1, for infinitesimal z, and 7. A sum of infinitely many infinitesimals is infinitesimal. In this paper, we interpret these assumptions within the context of nonstandard analysis and focus on Euler’s series expansion for sine and cosine which is developed in chapter VIII of Introductio in analysin infinitorum. In the middle of the eighteenth century, the number e and series for .ez were new topics. Series for the sine and cosine enabled Euler’s novel technique to be applied to ancient mathematical problems. Accordingly, in Sect. 2, we briefly describe the way Ptolemy and Newton used the sine. In Sect. 3, we review the basics of Euler’s trigonometry. Section 4 includes a detailed analysis of how Euler expanded sine and cosine into series and what kind of series these are. In Sect. 5, we introduce some basic concepts from nonstandard analysis and use those to interpret assumptions (1) through (6) listed above. Thesis (7) requires more advanced techniques of nonstandard analysis, and we will not discuss it in this paper. In Sect. 6, we support our interpretation by analyzing chapter III of Institutiones calculi differentialis.

2 Forerunners The mathematical concept of sine has differed through the ages. In this section, we focus on two vital contributions: Ptolemy’s Table of Chords and Newton’s series of sine.

2.1 Ptolemy Ptolemy identified the sine with a chord of the circle, regarding the arc it subtends, and the associated central angle. For an example, see the chord AB, arc AB, and the angle AOB in Fig. 1. In the Almagest, he calculates arcs and chords using the formula arc : 360 :: chord:120.

.

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Fig. 1 Ptolemy, Almagest, p. 51 (the center O and the dashed line added)

Fig. 2 Newton, Analysis of Equations, p. 336

Ptolemy considered the division of the circle into 720 parts, i.e., each .1/2◦ , and managed to determine the chord for every arc from .1/2◦ to .180◦ . In that process, he combined Euclid’s theory of similar figures with Babylonian arithmetic.2 Euclid’s proportion theory, which is the foundation of his computations for similar figures, presupposes that ratios concern magnitudes of the same kind. Consequently, in Greek mathematics, ratios such as .arc : chord were not legitimate objects. Newton and Euler employed novel techniques, which enabled them to relate arcs and line segments. A profound change in the theory of proportion paved the way to their series for the sine. In modern terms, operations in ordered fields replaced the ancient technique of transforming ratios.3

2.2 Newton Analysis by Equations of an Infinite Number of Terms is another treatise in which Newton demonstrates his technique of using infinite series including calculations related to the sine and its arc.4 In Fig. 2, line T D is the tangent to the semicircle ADLE and the chord AD is the sine of the angle ACD. Newton seeks to determine the arc AD in terms of AB, given .AB = x. To this end, he introduces the “indefinitely small rectangle” BGH K and sets .AC = 1/2. The point H lies on T D, but strangely enough, is also a vertex of the triangle H DG, and lies on the semicircle. Following Descartes’ technique,5 Newton finds that BD equals 2 See:

(Ptolemy 1984), Book I, §§ 10–11. Sect. 6.1 below. 4 See (Newton 1745), pp. 336–338, (Guicciardini 2009), ch. III. 5 On how Descartes interpreted the Pythagorean theorem, see (Błaszczyk et al. 2020, §§ 6–7). 3 See

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√

.

x − x 2 . From the similarity of triangles .T DB, .H DG, and .CDB, he derives proportions BK : H D :: BT : DT , BT : DT :: BD : DC,

.

√ and gets .BK : H D :: x − x 2 : 1/2. Ratio .H D : KB represents “the moment of the arc AD” to “the moment of the base AB.” Once again, due to Descartes’ technique,6 the proportion .H D : BK :: DC : BD becomes .

1 HD = √ . BK 2 x − x2

Taking .BK = 1, he gets7 1 HD = √ . 2 x − x2

.

By the binomial theorem, Newton expands .H D into a series, .

1 3 5 63 9/2 1 1 x , &c. = x −1/2 + x 1/2 + x 3/2 + x 5/2 + √ 2 4 16 32 512 2 2 x−x

Via term-wise integration, he gets that “the length of the arc is” 3 5 3 35 4 63 5 1 x + x + x , &c.). x 1/2 (1 + x + x 2 + 6 40 112 1152 2816

.

Then, Newton continues: “After the same manner by supposing CB to be x, the radius CA to be 1, you will find the arc LD to be” 1 5 7 3 x + &c. x + x3 + x5 + 40 112 6

.

(2)

That is, given .BC = sin  DCL = x, .CA = 1, the arc DL equals the series (2). In modern terms .

6 See

1 5 7 3 x + ... . arcsin x = x + x 3 + x 5 + 40 112 6

(Błaszczyk 2022). explains that move as follows: “Neither I am afraid to speak of unity in points, or lines infinitely small, since geometries are wont now to consider proportions even in such a case, when they make use of the method of indivisibles” (Newton 1745, 336). meaning, each kind of objects lines, planes, solids, infinitesimals - have their own unity. 7 Newton

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Fig. 3 Newton, Analysis of Equations, p. 338

In the next section, Newton seeks to determine the sine in terms of its arc. In Fig. 3, the line AB is the sine of the angle .αAD. Given the arc .αD equals z, .AB = x, .Aα = 1, using (2), he obtains 5 7 3 1 x , &c. z = x + x3 + x5 + 40 112 6

.

Using his ingenious technique of finding inverse series, Newton determines the relation between z and x as follows 1 1 5 1 7 x = z − z3 + z − z , &c. 6 120 5040

.

In modern terms, it is the series for sine, namely AB = sinz = z −

.

1 1 1 3 z + z5 − z7 + ... . 5! 7! 3!

2.3 Ptolemy–Newton–Euler Newton’s analysis of sine involves the tangent to a circle. As a result, he considers the sine of half of the relevant angle, that is, the Ptolemy angle. Euler, as we will see, settles Newton’s sine and the tangent line on the unit circle in a way a modern reader takes as normal. At the same time, the tangent gets a new meaning: by definition, it sin z is a ratio . cos z. Since both Newton and Euler employ infinitesimals, their techniques are commonly considered intuitive versions of modern calculus. While Newton’s infinitesimals rely on intuition, Euler defines these seemingly strange objects. In chapter VIII of Introductio in analysin infinitorum (Euler 1748), he applies them to derive the

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sine and cosine series. In the following sections, we show how to interpret Euler’s proof via calculus that uses infinitesimals, that is, using nonstandard analysis.8

3 Setting the Stage: Trigonometry Euler’s Introductio in analysin infinitorum (Euler 1748) is considered the beginning of modern trigonometry. Here is the fundamental assumption of chapter VIII (§§ 126–142): We let the radius, or total sine, of a circle be equal to 1, then it is clear enough that the circumference of the circle cannot be expressed exactly as a rational number. An approximation of half of the circumference of this circle is .3.141592653589793238462643383279502884197169399375105820974944592

3078164062862089986280348253421170679821480865132723066470938446 + .

For the sake of brevity we will use the symbol .π for this number. We say, then, that half of the circumference of a unit circle is .π , or that the length of an arc of 180 degrees is .π . (Euler 1988, 101)

The phrase total sine refers to the radius of the circle. Euler’s account is novel in many respects. First, the unit circle sets a new standard. We always assume that the radius of the circle is 1 and let z be an arc of this circle. We are especially interested in the sine and cosine of this arc z. Henceforth we will signify the sine of the arc z by .sin z. Likewise, for the cosine of the arc z we will write .cos z. Since .π is an arc of 180.◦ , .sin 0π = 0 and .cos 0π = 1. Also .sin π/2 = 1, .cos π/2 = 0, .sin π = 0, .cos π = −1, .sin 3π/2 = −1, .cos 3π/2 = 0, .sin 2π = 0, and .cos 2π = 1. Every sine and cosine lies between +1 and -1. Further, we have .cos z = sin(π/2 − z), 2 2 .sin z = cos(π/2 − z). We also have .sin z + cos z = 1. Besides these notations we mention also that .tan z indicates the tangent of the arc z, .cot z for the cotangent of arc z. We agree cos z sin z 1 that .tan z = cos z and .cot z = sin z = tan z , all of which is known from trigonometry. (Euler 1988, 102)

Through subsequent sections, Euler surveys standard identities such as the sine and cosine sum and difference laws and the half-angle formulas. In Fig. 4, we represent his model of the unit circle with sine, cosine, and tangent, although Euler does not include such a diagram in his paper. Euler also does not define sine and cosine, but the definitions are implicit in his discussion. Note that in Fig. 4, sine is parallel to tangent. Thus, in Fig. 2, it would be a line parallel to T D, or perpendicular to DC, passing through A, and equal to DB. In section 133, Euler introduces another revolutionary trick. Starting with the trigonometric version of the Pythagorean Theorem: .

8 Regarding

sin2 z + cos2 z = 1,

the idea of calculus without the concept of limit see (Błaszczyk and Major 2014).

(3)

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he factors it in the field of complex numbers, and gets the following equality (cos z + i sin z)(cos z − i sin z) = 1.

.

(4)

This seemingly simple computation aims to justify the appeal of using complex numbers in trigonometric computations. Euler comments on it as follows: “Although these factors are complex, still they are quite useful in combining and multiplying arcs.”9 Using complex numbers, Euler derives the formula currently named after de Moivre. Comparing the real and imaginary parts of products of complex numbers using trigonometric functions, he shows that (cos x ± i sin x)(cos y ± i sin y) = cos(x + y) ± i sin(x + y),

.

and (cos x ± i sin x)(cos y ± i sin y)(cos z ± i sin z) = cos(x + y + z) ± i sin(x + y + z).

.

On these grounds, he reaches the general conclusion (cos z ± i sin z)n = (cos nz ± i sin nz).

.

(5)

Glen Van Brummelen, an expert in the history of trigonometry, comments on the introductory sections of chapter VIII as follows: With the adoption of the unit circle, for the first time the sine and cosine are considered to be ratios of line segments rather than their lengths. Indeed, at the end of the following sin z cos z paragraph, .tan z and .cot z are introduced directly as . cos z and . sin z respectively. (Van Brummelen 2021, 166)

Fig. 4 Unit circle

9 Rather

√ than the modern symbol i, (Euler 1748) uses . −1.

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Indeed, for Ptolemy and Newton, the sine was a specific line segment. Euler explicitly shows that his sine and cosine take negative values. Therefore, they should be considered coordinates of a point on the unit circle rather than ratios of line segments. On other occasions, Euler identifies them as ratios and numbers, thus the actual meaning of the term ratio in the context of his trigonometry is the quotient. Further, Van Brummelen adds: “Euler does not define the sine and cosine at all in this text; the primitive notions remain geometrical.” Whether on purpose or not, the absence of a definition enables Euler to employ various techniques like Euclidean geometry, and analysis in terms of Cartesian coordinates and complex numbers. Under the line segment interpretation of .sin z and .cos z, the equality (3) means the Pythagorean theorem, where the number 1 in Fig. 4 stands for the line OP (the radius of the circle). One can also interpret complex numbers in (4) geometrically. In that case, in Fig. 4, the number 1 stands for the line segment OB. Although the transition of the formula (3) into (4) is correct, the geometrical meaning is different. Euler’s trigonometry takes advantage of such equivocations.

4 The Crucial Move Euler expands the sine and cosine into series in a seemingly self-evident way through a few lines of sections 133–134. To show his argument, let us write down the plus and minus versions of (5) separately .

(cos z + i sin z)n = cos nz + i sin nz, (cos z − i sin z)n = cos nz − i sin nz.

(6)

He explicitly considers the formulas that follow from (6), namely10 (cos z + i sin z)n + (cos z − i sin z)n , 2 (cos z + i sin z)n − (cos z − i sin z)n sin nz = . 2i

cos nz = .

(7)

Then he writes: “Expanding the binomials, we obtain the following series”:11

10 There

is a typo in Blanton’s translation (Euler 1988): the formula for .sin nz is missing i in the denominator.   11 (1) Euler introduces binomials . n in the explicit fractional form. We employ modern notation, k to get a more compact formula. (2) Instead of Euler’s original sign .&c. ending the series, Blanton’s translation (Euler 1988) applies three dots ...., i.e., the symbol marking infinite series in real analysis. As we will see in section 4.3 below, Euler’s infinite series do have the last term.

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    n n n−2 2 cos nz = cos z − cos z sin z + cosn−4 z sin4 z− 2 4 .   n − cosn−6 z sin6 z + &c. 6 n

(8)

    n n n−1 sin nz = cos z sin z − cosn−3 z sin3 z+ 1 3 .   n + cosn−5 z sin5 z − &c. 5

(9)

Section 134 opens with a passage that encourages our nonstandard analysis interpretation. Let the arc z be infinitely small, then .sin z = z and .cos z = 1. If n is an infinitely large number, so that nz is a finite number, say .nz = v, then, since .sin z = z = nv , we have . cos v

. sin v

=1−

=v−

v2 1·2

v3 1·2·3

+

+

v4 1·2·3·4

−

v5 1·2·3·4·5

v6 1·2·3·4·5·6

−

+ &c,

v7 1·2·3·4·5·6·7

+ &c.

It follows that if v is a given arc, by means of these series, the sine and cosine can be found. (Euler 1988, 107)

  k In the series (8), (9), Euler replaces every term . nk cosn−k z sink z with . vk! , given v .cos z = 1 and .sin z = n . In this process, he assumes the following equalities:     k n n v vk cosn−k z sink z = 1 k = . k k! k n

.

(10)

In the next sections, we will discuss this proof in detail.

4.1 From Finite to Infinite Using the binomial theorem, formulas (8) and (9) follow from (6) and (7). Thus initially n is assumed to be a finite number. In section 134, Euler explicitly considers n infinitely large. In nonstandard analysis, formulas   (8) and (9) are valid whether n is a finite or infinite number, specifically terms . nk make sense for finite and infinite .n, k. More precisely, when the finite case is valid, the infinite is valid too. The same applies to the binomial theorem: one can apply it to terms .(cos z ± i sin z)n whether n is finite or infinite.

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In order to allow the use of an infinite n in the formulas (8) and (9), we need to interpret sums made of infinitely many terms. Before that, we need to discuss infinitesimals.

4.2 Infinite and Infinitesimal Numbers In Introductio in analysin infinitorum (Euler 1748), Euler expressly refers to infinitesimals. On other occasions (Euler 1755), he calls them zeros. He also introduces two different ways of comparing infinitesimals: arithmetic and geometric. The former means a difference and allows interpretation in terms of a relation is infinitely close, the latter is a quotient. Nonstandard analysis provides an obvious explanation for the relation is infinitely close, .x ≈ y, meaning .x − y is infinitesimal. Thus, assuming x and y are infinitesimals, the difference .x − y is still infinitesimal. x The quotient . , however, can be infinitesimal, finite, or infinite. y For Euler (Euler 1755), the inverse of an infinitesimal proves to be an infinite number and vice verse, the inverse of an infinite number is infinitesimal. Euler also shows that for each infinitesimal, there is an infinite number such that their product is finite. He also expressly claims that there are infinitesimals and infinite numbers such that their products are infinite (Euler 1748 and Euler 1755). Therefore we interpret the passage that opens section 134 (described above) as: Let the arc z be infinitely small, then .sin z = z and .cos z = 1. If n is an infinitely large number, so that nz is a finite number, say .nz = v, then z ≈ 0 ⇒ sin z ≈ z, cos z ≈ 1,

.

and for an infinitesimal z there is an infinite number N such that N z is finite.

4.3 Infinite Series vs Hyperfinite Sums Instead of the original term .&c. at the end of equations, Blanton’s translation (Euler 1988) employs three dots to indicate infinite sums, such as this one . cos nz

= cos z− n

  n 2

n−2

cos

2

z sin z+

  n 4

n−4

cos

4

z sin z−

  n 6

cosn−6 z sin6 z+... .

This notation suggests using the real analysis approach to computing the value through finding the limit of the series. However, Euler’s infinite sums allow another reading that these series contain the very last term. This can be seen in section 107 of Euler’s Institutiones calculi differentialis (Euler 1755). Below we cite Blanton’s translation of this section in extenso.

86

P. Błaszczyk and A. Petiurenko From this we see that he who would say that when this same series is continued to infinity, that is, .1

+ x + x2 + x3 + . . . + x∞,

and that the sum is .1/(1 − x), then his error would be .x ∞+1 /(1 − x), and if .x > 1, then the error is indeed infinite. At the same time, however, this same argument shows why the series .1 +x +x 2 +x 3 +x 4 +. . ., continued to infinity, has a true sum of .1/(1 −x), provided that x is a fraction less than 1. In this case the error .x ∞+1 is infinitely small and hence equal to zero, so that it can safely be neglected. Thus if we let .x = 12 , then in truth .1

1 1 1 1 1 + ··· = + + + 4 8 16 2 1−

+

1 2

= 2.

In a similar way, the rest of the series in which x is a fraction less than 1 will have a true sum in the way we have indicated. (Euler 2000, 60)

Instead of using the three dots extension, 1 + x + x 2 + x 3 + x 4 + . . . and 1 +

.

1 1 1 1 + + + + ... , 2 4 8 16

Euler (Euler 1755, § 107) presents infinite sums in the following form 1 + x + x 2 + x 3 + x 4 + &c. and 1 +

.

1 1 1 1 + + + + &c. 2 4 8 16

Thus, the symbol 1 + x + x 2 + x 3 + x 4 + &c.

.

stands for 1 + x + x2 + x3 + · · · + x∞,

.

rather than 1 + x + x2 + x3 + . . . .

.

In nonstandard analysis, infinite sums containing last terms are legitimate objects that are called hyperfinite sums. Given N is an infinite number and .|a| < 1, the following equation 1 + a1 + a2 + a3 + . . . + aN =

.

1 − a N +1 , 1−a

N+1 is can be easily justified within the nonstandard framework. Moreover, . 1−a 1−a 1 infinitely close to . 1−a ,

Euler’s Series for Sine and Cosine: An Interpretation in Nonstandard Analysis

.

87

1 1 − a N +1 ≈ . 1−a 1−a

1 and . 1−a equals . |a|1−a . N+1 Since .|a| < 1, the number .|a|N +1 is infinitesimal. Therefore the product . |a|1−a is also infinitesimal.12 Euler’s original formula

Indeed, the absolute value of the difference of . 1−a 1−a

N+1

1+

.

N+1

1 1 1 1 1 + &c. = + + + = 2, 2 4 8 16 1 − 12

finds an obvious interpretation in nonstandard analysis, namely 1+

.

1 − ( 12 )N +1 1 1 1 1 1 1 ≈ = 2. + + + + ··· + N = 1 2 4 8 16 2 1− 2 1 − 12

Using the hyperfinite sum interpretation of the equation, our only intervention in Euler’s text concerns the infinity sign; instead of .∞ or .∞ + 1, we employ specific infinite numbers (nonstandard natural numbers), like N , or .N + 1.13

4.4 Extending Trigonometric Functions Nonstandard analysis allows the study of trigonometric functions such as sin∗ x, cos∗ x, which are extensions of the real .sin x and .cos x. Moreover, these ∗ . functions obey all the same identities as their real counterparts .sin x and .cos x. Thus, in our interpretation, formula (7) takes the following form: .

.

.

cos∗ nz =

(cos∗ z + i sin∗ z)n + (cos∗ z − i sin∗ z)n , 2

sin∗ nz =

(cos∗ z + i sin∗ z)n − (cos∗ z − i sin∗ z)n , 2i

whether n is a finite or an infinite number. The functional interpretation of Euler’s sine and cosine is prone to an allegation of being anachronistic. Nevertheless, the .∗ extension of .sin x and .cos x works for all

12 See

rule (19) below.

13 Up to his latest study (Ferraro 2022), Ferraro adopts three dots when ending series instead of the

original .&c. and does not consider an alternative interpretation of Euler’s infinite series at all.

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ranges of x whether it is a function or some other object. As long as .sin x and .cos x are given, so are .sin∗ x and .cos∗ x. In what follows, we omit that .∗ superscript.

4.5 The Revised Proof In this section, we make use of our interpretation described in sections 4.1–4.4. Let z be infinitesimal, that is, .z ≈ 0. Then .sin z ≈ z and .cos z ≈ 1. Let N be an infinite number such that Nz is finite and set .Nz = v. Then formulas (8), (9) take the form of hyperfinite sums,14 .

    N N cosN −2 z sin2 z + cosN −4 z sin4 z − . . . + sinN z. 2 4 (11)     N N . sin N z = cosN −1 z sin z − cosn−3 z sin3 z + . . . + sinN z. (12) 1 3

cos Nz = cosN z −

Then, for formula (10), we get   N vk . cosN −k z sink z ≈ . k k!

(13)

Formula (13) is valid whether N and k are finite or infinite. From (11), (12), and (13), relationships (14) and (15) follow, .

.

cos v ≈ 1 −

v6 vN v4 v2 − + ··· + , + 4! 6! N! 2!

(14)

sin v ≈ v −

v5 v7 vN v3 + − + ··· + . 3! 5! 7! N!

(15) k

To get these results, in formulas (11) and (12), we substitute . vk! for N  N −k . z sink z. However, these substitutions involve infinitely many terms. k cos There is no obvious guarantee that .

cosN z −

    N N cosN −2 z sin2 z + cosN −4 z sin4 z − . . . + sinN z 2 4

is infinitely close to

14 We skip a discussion of whether N is even or odd because these cases do not affect the general picture.

Euler’s Series for Sine and Cosine: An Interpretation in Nonstandard Analysis

1−

.

89

v4 v6 vN v2 + − + ··· + , 2! 4! 6! N!

even though for every k, finite or infinite,   N vk cosN −k z sink z ≈ . k k!

.

Indeed, a product of an infinite and an infinitesimal number can be infinite, finite, or infinitesimal. MacKinzie and Tuckey (McKinzie and Tuckey 2001) showed that in this case the hyperfinite sums in formulas (11) and (12) are infinitely close to the hyperfinite sums in formulas (14) and (15), respectively. Their proof requires sophisticated techniques. In this paper, we are assuming their result. Throughout sections 4.1–4.4, we reiterated our belief that the concepts related to infinitesimals and infinite numbers that Euler uses in chapter VIII (Euler 1748) can be given a strict meaning in nonstandard analysis. In the next section, we show how to define in modern terms infinitesimal, infinite number, the relation is infinitely  close, hyperfinite sum, the binomial coefficient . Nk for infinite N and k, .sin∗ x, and .cos∗ x. Any non-Archimedean field provides a framework for most of these definitions. In section 6, through an analysis of (Euler 1755), we will show that Euler explicitly derives some rules of a non-Archimedean field, even though he does not employ the concept of an ordered field.15 Therefore, although nonstandard analysis assumes the theory of real numbers, we do not need to refer to real numbers. There is one exception, however. The just mentioned lemma of MacKenzie and Tuckey applies specific techniques of nonstandard analysis. In this topic only, our analysis may be considered anachronistic.

5 Nonstandard Analysis In the brief introduction to the nonstandard analysis that follows, we focus on concepts crucial to our interpretation of chapter VIII of (Euler 1748).16

5.1 The Basics of Ordered Field Theory A commutative field .(F, +, ·, 0, 1) together with a total ordering .< forms an ordered field when the field operations are compatible with the order, that is x < y ⇒ x + z < y + z, x < y, z > 0 ⇒ xz < yz.

.

15 To

be clear, only Hilbert (Hilbert 1899; 1900) introduced the concept of an ordered field. (Błaszczyk and Major 2014), (Błaszczyk 2016), (Błaszczyk 2021). The section on hyperfinite sums follows (Goldblatt 1998, ch. III).

16 See

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The field of rational numbers is the smallest ordered field, meaning that every ordered field includes fractions . m n , for .m, n ∈ N. In every ordered field, one can define the absolute value  |x| =

.

x, if x ≥ 0, −x, if x < 0,

and the following subsets of .F: L = {x : (∃n ∈ N)(|x| < n)},

.

A = {x : (∃n ∈ N)( 1n < |x| < n)}, Ψ = {x : (∀n ∈ N)(|x| > n)}, Ω = {x : (∀n ∈ N)(|x| < 1n )}. The elements of these sets we call finite numbers, assignable numbers, infinitely large numbers, and infinitely small numbers (or infinitesimals), respectively. Infinitesimals and infinite numbers can be also defined via assignable numbers, Ω = {x : (∀a ∈ A)(|x| < |a|)},

(16)

Ψ = {x : (∀a ∈ A)(|x| > |a|)}.

(17)

.

.

Here are some obvious relationships between these elements, which we will call ΩΨ rules,

.

.

(∀x, y ∈ Ω)(x + y ∈ Ω, xy ∈ Ω), .

(18)

(∀x ∈ Ω)(∀y ∈ A)(xy ∈ Ω), .

(19)

(∀x)(x ∈ A ⇒ x −1 ∈ A), .

(20)

(∀x = 0)(x ∈ Ω ⇔ x −1 ∈ Ψ ), .

(21)

(∀x ∈ Ψ )(∀y ∈ A)(∃z ∈ Ω)(xz = y).

(22)

The set .Ω enables to define an equivalence relation as follows x ≈ y ⇔ x − y ∈ Ω, for x, y ∈ F.

.

We say that x is infinitely close to y, when the relation .x ≈ y holds. We will show in section 6 below that Euler (Euler 1755) explicitly discusses rules equivalent to .ΩΨ rules.

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5.2 Archimedean Axiom Three equivalent forms of the Archimedean axiom are given below. (A1) .(∀x, y ∈ F)(∃n ∈ N)(0 < x < y ⇒ nx > y), (A2) .(∀x ∈ F)(∃n ∈ N)(n > x), (A3) .Ω = {0}. A1 and A2 are well-known versions both in mathematical and historical contexts. Version A3 is useful in our discussion of Euler’s work. A3 states that in a nonArchimedean field, the set of infinitesimals contains at least one non-zero element, say, .ε. Then, . nε , as well as .nε, are also infinitesimals. Furthermore, by .ΩΨ rules, 1 , are infinitely large numbers. reciprocals for these elements, i.e., . nε , . nε In what follows, let .Ω0 stand for non-zero infinitesimals, i.e. Ω0 = Ω \ {0}.

.

5.3 Real Numbers The field of real numbers is an ordered field in which every Dedekind cut .(L, U ) of (F, k. In terms of the elements of the ultrafilter, .{n ∈ N : n > k} ∈ U , or {n ∈ N : 1n < r} ∈ U .

.

Due to the definition of the total ordering in .R∗ , this means that [(1, 21 , 13 , ...)] < [(r, r, r, ...)].

.

Thus .ε ∈ Ω. The hyperreal ε2 = [(1, 212 , 312 , ...)]

.

is another infinitesimal. Generally, if .(rn ) is a null sequence, then .[(rn )] is infinitesimal. Since .ε ∈ Ω, then .ε−1 = [(1, 2, 3, ...)] is an infinitely large number. Indeed, .N = [(1, 2, 3, ..)] is an example of an infinite number. Other infinite numbers are N + 1 = [(2, 3, 4, ...)], N 2 = [(1, 4, 9, ...)], N ! = [(1!, 2!, 3!, ..)].

.

18 Indeed,

it is yet another form of the Archimedean axiom.

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5.6 Binomial Coefficients   Let the binomial coefficients . Nk for infinite N, k, with .N, k ∈ N∗ , be defined as follows. Start with the case where .N = [(n1 , n2 , n3 , ...)] is infinite and k is finite. We define         

N n2 n3 n1 . = , , , ... . k k k k When .K = [(k1 , k2 , k3 , ...)] is an infinite number, we define         

N n1 n2 n3 = , , , ... . . K k1 k2 k3   These new binomial coefficients satisfy the standard identities such as . nk =  n n−k . For an infinite N and a finite or infinite k the implicit assumptions of Euler’s arguments are given in the following form 

.

N −k 1 ≈ , kN k

  1 N 1 ≈ . k k! k N

5.7 *Maps Let f be a real map, that is .f : R → R. By .f ∗ we mean a map .f ∗ : R∗ → R∗ defined by f ∗ ([(rn )]) = [(f (r1 ), f (r2 ), ...)].

.

If .r ∈ R, then .f ∗ (r) = [(f (r), f (r), ...)]. Since we identify the real number r with the hyperreal .[(r, r, ...)], we obtain the equality .f ∗ ([(r, r, ...)]) = f (r). Under this definition, .

sin∗ [(rn )] = [(sin r1 , sin r2 , ...)], cos∗ [(rn )] = [(cos r1 , cos r2 , ...)].

Since for every n the identity .sin2 rn + cos2 rn = 1 holds, we have (sin∗ x)2 + (cos∗ x)2 = 1.

.

Similarly, every trigonometric identity can be changed into an identity involving the maps .sin∗ and .cos∗ . This definition applies also to the exponential map .a x . Thus, when .a ∈ R+ , then

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a [(rn )] = [(a r1 , a r2 , ...)].

.

When .a = [(a1 , a2 , ...)] is hyperreal, then a [(rn )] = [(a1 r1 , a2 r2 , ...)].

.

Then, for .a, b, x, y ∈ R∗+ , we have .(ab)x = a x bx , .a x a y = a x+y , and other similar identities. Let us return to Euler’s supposition that .sin z ≈ z, given z is infinitesimal. Clearly, .| sin∗ z| ≤ 1, since .sin∗ z is assignable or infinitesimal. To get a contradiction, suppose .sin∗ z is an assignable number for some infinitesimal z. Since the standard sine function takes the segment .[0, π2 ] onto the segment of real numbers ∗ .[0, 1], .sin is also a surjective map. As a result, for some assignable number x, π ∗ ∗ .sin z = sin x. On the segment .[0, 2 ], the standard sine function is one-to-one, ∗ therefore .sin is one-to-one on the segment of hyperreals .[0, π2 ]. Hence, .z = x, an infinitesimal which has been shown to be equal to an assignable number. This is impossible. Another way to reach this result is to refer to the continuity of the real sine map. Its continuity at the point 0 translates into the condition .sin z ≈ 0, given .z ≈ 0. Generally, one has to assume some characteristics of the real map f to make claims about .f ∗ . Euler, however, did not define a sine. Thus, we are trapped and have to speculate on what grounds he assumed .sin z ≈ z. The assumption .cos∗ z ≈ 1, given .z ≈ 0, follows from the Pythagorean identity for .sin∗ and .cos∗ , and .sin∗ z ≈ 0.

5.8 Hyperfinite Sums Let .N = [(n1 , n2 , n3 , ...)] be an infinite number, .N ∈ N∗ , and let a be a real number. We set N  .

aj =

n1  

j =0

ai ,

i=0

n2 

ai ,

i=0

n3 



a i , ... .

i=0

Thus 1 + a 1 + a 2 + a 3 + ... + a N

.

is another symbol for the hyperreal number n1   .

i=0

ai ,

n2  i=0

ai ,

n3  i=0



a i , ... .

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Since for every finite n n  .

ai =

i=0

1 − a n+1 , 1−a

we can show N  .

aj =

j =0

 1 − a n1 +1 1 − a n2 +1 1 − a n3 +1 

, , ... , 1−a 1−a 1−a

=

1 − a N +1 . 1−a

When a is a hyperreal number, .a = [(a1 , a2 , a3 , ...)], by setting N  .

n1  

aj =

j =0

i=0

a1i ,

n2  i=0

a2i ,

n3  i=0



a3i , ... ,

we see the same result, namely N  .

j =0

aj =

1 − a N +1 . 1−a

6 Infinitesimals and Infinite Numbers in Institutiones calculi differentialis Chapter III (§§74–111) of Institutiones calculi differentialis (Euler 1755) begins with philosophical considerations such as whether matter consists of indivisible or infinitely divisible parts, or whether the supposed ultimate parts of matter are extended or not. Euler finds these debates inconclusive and turns to mathematics by saying: Even if someone denies that infinite numbers really exist in this world, still in mathematical speculations there arise questions to which answers cannot be given unless we admit an infinite number. (Euler 2000, 50)

6.1 Three Kinds of Quantities: Infinite Numbers In section 82 of the text, Euler offers a definition of infinite numbers:

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[...] this quantity is so large that it is greater than any finite quantity and cannot not be infinite. To designate a quantity of this kind we use the symbol .∞, by which we mean a quantity greater than any finite or assignable quantity. (Euler 2000, 50)

Thus if .A stands for assignable numbers: N is an infinite number ⇔ (∀a ∈ A)(N > a).

.

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The word quantitas (quantity) is reiterated over and over again throughout the discussed chapter. It refers to infinite, assignable, and infinitesimals numbers. Quantitas is the Latin translation of the Greek term μέγεθος (magnitude). In Euclid’s Elements, this general term covers line segments, triangles, convex polygons, circles, angles, arcs of circles, and solids. Magnitudes of the same kind (line segments being of one kind, triangles of another, etc.) are compared in terms of greater–lesser. Transitivity and the law of trichotomy constitute the mathematical sense of the greater than relation. Greeks also considered addition and subtraction (lesser from greater) of magnitudes. The additive and ordinal structure of magnitudes enabled Euclid to introduce proportion theory. Book V of the Elements develops it systematically. This was the basic ancient Greek technique for comparing magnitudes. The Archimedean axiom was an essential part of that theory and significantly restricted the concept of magnitude. Descartes (Descartes 1637) introduced a new operation on line segments: the product. He employed it in such a way that the resulting arithmetic satisfied the rules of an ordered field. While processing formulas, he substitutes equality of quotients for proportions: instead of proportions of line segments, such as .a : b :: c : d, he puts . ab = dc .19 As a result, the Archimedean axiom lost its importance and for a long time was not discussed. It was Stolz (Stolz 1885) who re-introduced this axiom to modern mathematics and re-established its role in foundational studies. Euler, as we have already seen, processes formulas according to the rules of an ordered field. The above genealogy provides a rationale for his manner of naming quotients “geometrical ratios” (see the next sections). He even uses the symbol .a : b, used for the proportion of magnitudes through the seventeenth and eighteenth centuries, though he processes these objects as actual fractions . ab . The term Archimedean axiom never occurs in Euler’s writings. Yet, due to the explicit negation of the axiom A3, his implicit ordered field is non-Archimedean.

6.2 Infinitesimals In section 83, Euler provides a definition of an infinitesimal number:

19 See

(Błaszczyk and Petiurenko 2019), (Błaszczyk 2022).

98

P. Błaszczyk and A. Petiurenko There is also a definition of the infinitely small quantity as that which is less than any assignable quantity. If a quantity is so small that it is less than any assignable quantity, then it cannot not be 0, since unless it is equal to 0 a quantity can be assigned equal to it, and this contradicts our hypothesis. To anyone who asks what an infinitely small quantity in mathematics is, we can respond that it really is equal to 0. (Euler 2000, 51)

In symbols ε is infinitesimal ⇔ (∀a ∈ A)(0 < |ε| < |a|).

.

(24)

Clearly, the above formalization does not comply with Euler’s words “cannot not be 0.” However, his concept of equality is ambiguous: in some contexts, it is strict equality, in others it also means is infinitely close. The latter term comes from nonstandard analysis. We substantiate this in the sections that follow.

6.3 Two Ways of Comparing Zeros Euler explicitly claims that in analysis, the other name for zero is “infinitely small.” If we accept the notation used in the analysis of the infinite, then dx indicates the quantity that is infinitely small, so that both .dx = 0 and .adx = 0, where a is any finite quantity. Despite this, the geometric ratio .adx : dx is finite, namely .a : 1. For this reason these two infinitely small quantities dx and adx, both being equal to 0, cannot be confused when we consider their ratio. (Euler 2000, 51–52)

If we adopt the interpretation of the formula .dx = 0 as .dx ≈ 0, Euler’s two ways of comparing zeros have an obvious meaning. Infinitesimals, say .ε, δ, can be

compared as .ε − δ or . . The first term is always infinitesimal, while the second is δ an infinitesimal, finite, or infinite number. The following passage clarifies his view: [1] Although two zeros are equal to each other, so that there is no difference between them, nevertheless, since we have two ways to compare them, either arithmetic or geometric, let us look at quotients of quantities to be compared in order to see the difference. [2] The arithmetic ratio between any two zeros is an equality. This is not the case with a geometric ratio. [3] We can easily see this from this geometric proportion .2 : 1 = 0 : 0, in which the fourth term is equal to 0, as is the third. From the nature of the proportion, since the first term is twice the second, it is necessary that the third is twice the fourth. (Euler 2000, 51; numerals in square brackets added).

In [1], the “equality of zeros” means the infinitely close relation, .ε ≈ δ. In [2] “Arithmetic ratio” means .ε − δ. In [3], the “geometric ratio” means the quotient . δ . Substituting . for 0 in Euler’s formula .2 : 1 = 0 : 0, we get .2 : 1 = 2 ε . In Greek mathematics, a ratio such as .a : b makes sense only as a part of proportion, say, .a : b :: c : d. Euler, in a manner similar to Newton, defines number as a ratio. Consequently, terms such as .adx : dx are numbers. The phrase “geometric ratio” refers to Euclid’s proportion of magnitudes. However, when looking at actual practice, Euler processes numbers according to the rules of an ordered field, rather than according to the propositions of Euclid’s Book V.

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6.4 Arithmetic of Infinitesimals Euler says the following about infinitely small quantities: Since the infinitely small is actually nothing, it is clear that a finite quantity can neither be increased nor decreased by adding or subtracting an infinitely small quantity. Let a be a finite quantity and let dx be infinitely small. Then .a + dx and .a − dx, or, more generally, .a ±ndx, are equal to a. [...] On the other hand, the geometric ratio is clearly of equals, since a±ndx = 1. (Euler 2000, 52) . a

Given a is a finite number, Euler’s claims obviously translate into a ± ndx ≈ a,

.

a ± ndx ≈ 1. a

Regarding the products of infinitesimals, Euler writes Since the infinitely small quantity dx is actually equal to 0, its square .dx 2 , cube .dx 3 , and any other .dx n , where n is a positive exponent, will be equal to 0, and hence in comparison to a finite quantity will vanish. However, even the infinitely small quantity .dx 2 will vanish when compared to dx. The ratio of .dx ± dx 2 to dx is that of equals, whether the comparison is arithmetic or geometric. There is no doubt about the arithmetic; in the dx ± dx 2 geometric comparison .dx ± dx 2 : dx = = 1 ± dx = 1. (Euler 2000, 52) dx

In our interpretation dx + dx n ≈ 1. dx √ √ Euler also considers infinitesimals of the form . dx and claims that . dx = dx. √ In our interpretation, it means . dx ≈ dx. dx n ≈ dx,

.

6.5 Infinitesimals and Infinite Numbers Euler says the following about infinitesimals and infinite numbers: [1] It should be noted that the fraction .1/z becomes greater the smaller the denominator z becomes. [2] Hence, if z becomes a quantity less than any assignable quantity, that is, infinitely small, then it is necessary that the value of the fraction .1/z becomes greater than any assignable quantity and hence infinite. For this reason, if 1 or any other finite quantity is divided by something infinitely small or 0, the quotient will be infinitely large, and thus an infinite quantity. Since the symbol .∞ stands for an infinitely large quantity, we have the a a equation . dx = dx = 0. (Euler = ∞. [3] The truth of this is clear also when we invert . ∞ 2000, 53; numerals in square brackets added)

The first sentence of this passage [1] explains rule of ordered fields, namely: 0 1, and includes a Graeco-Latin square of order 10.

3.4 A Catalogue of Complete Sets of Mutually Orthogonal Latin Squares Table XVI gives complete sets of mutually orthogonal Latin squares of order 3, 4, 5, 7, 8, and 9. The Introduction explains that, for those of prime order, square .Λj is made by shifting the letters in each row j places to the left of the previous row. The shift is cyclic, so that column n is regarded as one place to the left of column 1. For .n = 4, the complete set in Table XVI is reproduced in Fig. 8. This is not explained in the Introduction. However, similar notation is used for the complete set

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Fig. 8 Complete set of mutually orthogonal Latin squares of order four

Fig. 9 Steps in the construction of Square I of order four

Fig. 10 Labelling of the eight mutually orthogonal Latin squares of order nine

of orthogonal Latin squares of order 8, and the explanation given for these enables us to work out what is happening for order 4. Example 2 The steps in the construction of Square I are shown from left to right in Fig. 9. In the first step, the rows of the square are labelled 1, a, b, and ab in order; and the columns are labelled 1, c, d, and cd in order. Thus, each of the 16 cells has a label of the form .a r bs ct d u for r, s, t, .u ∈ {0, 1}. In the second step, AC calculates .r + t modulo 2, thus giving a partition of the 16 cells into two parts; it is orthogonal to both rows and columns. Likewise, in the third step, BD calculates .s + u modulo 2, giving another such partition. In the fourth step, intersecting the parts of AC and BD gives four parts of size four, which give the letters of Square I. Square II is made in a similar manner, using ACD in Step 2 to calculate .r + t + u modulo 2 and using BC in Step 3 to calculate .s +t modulo 2. Similarly, Square III is made by using AD to calculate .r +u modulo 2 and using BCD to calculate .s +t +u modulo 2. For order 9, Table XVI labels the eight squares as shown in Fig. 10. However, the Introduction does not really explain this notation. It simply refers to “the notation developed by Yates” in [72]. We examine this in the next section.

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4 Technical Communication 35 4.1 Brief Overview The design and analysis of factorial experiments by F. Yates was published by the Imperial Bureau of Soil Science, Harpenden, as Technical Communication No. 35 in 1937. Rothamsted statisticians always call it “TC35”. Because TC35 was an independent booklet, not part of a journal, and also because its methodology for calculation is out of date, it is not easily available online. I am grateful to Chris Brien (University of Adelaide) for sending an electronic copy to me. TC35 is 95 pages long. It was written for practising experimenters, not for mathematicians, so it says “do this” rather than “this is correct because . . . ”. A large proportion of it is devoted to doing the correct arithmetic (and checking it) once the data are available. There is no mention of groups or of fields (in the algebraic sense). Factorial designs with factors with two or three levels are explained in terms of orthogonal partitions (but without that vocabulary).

4.2 Explanation of Mutually Orthogonal Latin Squares The explanation in TC35 of the confounding used in the mutually orthogonal .9 × 9 Latin squares is somewhat similar to the explanation for .4 × 4 squares given in Example 2 and shown in Fig. 9. The rows of the .9 × 9 square are labelled by all combinations of the levels of A and B, and the columns of the square are labelled by all combinations of the levels of C and D. Each of these four factors has levels 0, 1, and 2. Then the mysterious partitions such as .AC(J ) and .ABD(W ), each with three parts of size 27, are combined to give a partition with nine parts of size nine, orthogonal to both rows and columns. Page 40 of TC35 explains I and J for two factors as “diagonals”. See Fig. 11. The three down-sloping diagonals of the left-hand square give the partition I . When there are more than two factors, this must be written as .AC(I ) for clarity. Likewise, the partition .AC(J ) is given by the up-sloping diagonals in the right-hand square. Squares I, II, V, and VI in Fig. 10 are now made by using two of the “diagonal” partitions and combining them, as indicated by the .× sign, following the four steps used for the .4 × 4 square in Fig. 9. Fig. 11 Diagonal partitions I and J

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Fig. 12 Steps in the construction of Square II of order nine Table 1 Table 43 from TC35 Combination of first and second factors 1 00 2 10 3 20 4 01 5 11 6 21 7 02 8 12 9 22

.X1 .W2 .W3 Level of third factor 0 2 1 0 1 0 2 2 2 1 0 1 2 1 0 1 0 2 1 0 1 0 2 2 1 0 2 2 2 1 0 1 0 2 1 0

.W1

.X2

.X3

.Y1

.Y2

.Y3

.Z1

. Z2

.Z3

1 0 2 2 1 0 0 2 1

2 1 0 0 2 1 1 0 2

0 1 2 1 2 0 2 0 1

2 0 1 0 1 2 1 2 0

1 2 0 2 0 1 0 1 2

0 2 1 2 1 0 1 0 2

1 0 2 0 2 1 2 1 0

2 1 0 1 0 2 0 2 1

Example 3 These steps are shown for Square II in Fig. 12. Step 1 labels the rows and the columns. In Step 2, we show the parts of .BC(J ), following Fig. 11. In Step 3, we show the parts of .AD(I ), following Fig. 11. In Step 4, we intersect the parts of .BC(J ) and .AD(I ) to obtain a partition with nine parts, which are labelled 1–9 so that those in the first row come in natural order. Page 42 of TC35 gives the table reproduced in Table 1. Somewhat confusingly, the structure is different from that in Fig. 11. In Fig. 11, the labels of the rows and columns depend on the cells in the set, while the entries in the array show how these cells are partitioned into subsets. In Table 1, the levels of two factors identify the rows, the parts (subsets) of the four partitions identify the columns, and the entries in the table show the level of the third factor. Figure 13 shows the partitions .ABC(W ) and .ABD(Z) in a way that is compatible with Fig. 11. It is clear from Table 1 that, on a set defined by the 27 combinations of levels of three factors, each level of each of W , X, Y , and Z occurs exactly once with each combination of levels of the first and second factors. It takes a little checking to find

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Fig. 13 Two partitions used in the construction of Square VIII of order nine

Fig. 14 Steps in the construction of Square VIII of order nine

out that the same is true for each combination of levels of the first and third factors and for each combination of levels of the second and third factors. Furthermore, the partitions W , X, Y , and Z are all orthogonal to each other. Each of Squares III, IV, VII, and VIII in Fig. 10 uses four partitions of the type W , X, Y , or Z. Combining any two gives the partition corresponding to the nine letters of the Latin square. Example 4 Figure 14 shows how to do this for Square VIII, using .ABC(W ) in Step 2 and .ABD(Z) in Step 3. The partitions named in Fig. 10 are all orthogonal to rows and columns, as well as to each other. It may well be that Yates discovered these by stubbornly trying lots of possibilities. The other solutions for factorial designs in TC35 certainly suggest that this is how Yates worked. Or maybe he was just adapting the approach given in the first edition (1935) of Fisher’s Design of Experiments [31] for decomposing 3factor interactions, using W , X, Y , and Z in place of Fisher’s notation I, II, III, and IV. It is certainly plausible that neither Fisher nor Yates was familiar with group theory at this time.

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5 Some Later Developments 5.1 Inequivalent Complete Sets 1935 saw the publication of the first edition of [31], Fisher’s influential textbook on design of experiments. Six further editions followed, with new material in each one. In Section 35 of the 1937 edition, Fisher presented a complete set of mutually orthogonal Latin squares of order 9 and claimed that this was the set given in [39]. In [49], Norton pointed out that this was incorrect. In a 1942 paper [34] in Annals of Eugenics, Fisher wrote H. W. Norton has suggested that the complete square given in the Design of Experiments (Fisher, 1935) and the set of orthogonal .9 × 9 squares given in Statistical Tables (Fisher & Yates, 1938) belong to different species.

He then claimed to prove Norton wrong. (It is worth noting here that Fisher was the editor of Annals of Eugenics and strongly disapproved of the normal editorial custom of sending submitted papers to independent referees and asking for their opinions: see [19].) A few months later, Fisher published a retraction [36] in the same journal. Citing [16], he said that . . . Bose and Nair have examined these cases more fully (1941), from the point of view of the corresponding algebras. They note that the first set is Desarguesian, while the second corresponds with a Dicksonian algebra. No doubt the same difference may be viewed under many different aspects.

There seems to be no evidence that he knew what the words “Desarguesian” and “Dicksonian” meant. Some more recent discussions of these differences can be found in [7, 23, 50, 51].

5.2 Explanation Using Abelian Groups In yet another 1942 paper [35] in Annals of Eugenics, Fisher showed how the methods of confounding in factorial experiments could be explained through the use of elementary Abelian groups. Almost all of this was devoted to the case where all factors have two levels. In his collected papers [13], he added a foreword, including the following. Group properties are so abstract that the language in which the ideas of the theory are expressed requires the utmost care if the ideas are to be conveyed to the reader’s mind.

In 1945 he generalized this to arbitrary finite elementary Abelian groups in [37]. This paper is hard to follow. The notation is clumsy. It makes no reference to the notation I, II, III, and IV or W , X, Y , and Z used previously. It does not refer to any paper other than [35].

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Table 2 Correspondence between the two sorts of notation Old AC(I) AC(J) AD(I) AD(J) BC(I) BC(J) BD(I) BD(J)

New 2 C AC 2 .A D AD 2 .B C BC 2 .B D BD

.A

Old

New

.ABC(W )

.AB

.ABC(X) .ABC(Y ) .ABC(Z) .ABD(W ) .ABD(X) .ABD(Y ) .ABD(Z)

2

C2 2 .AB C 2 .ABC ABC 2 2 .AB D 2 .AB D 2 .ABD ABD

Old

New

.ACD(W )

.AC

.ACD(X) .ACD(Y ) .ACD(Z) .BCD(W ) .BCD(X) .BCD(Y ) .BCD(Z)

2

D2 .AC D 2 .ACD ACD 2 2 .BC D 2 .BC D 2 .BCD BCD 2

In a 1947 paper [47] in Biometrika, Oscar Kempthorne (in the Statistics Department at Rothamsted from 1941 to 1946) introduced the simpler notation in use today and gave good explanations of Fisher and Yates’ notation. This simpler notation had already been introduced by David Finney (in the Statistics Department at Rothamsted from 1939 to 1945) in [25] for the cases where all treatment factors have two levels or all treatment factors have three levels. However, the main purpose of the latter paper was to introduce the idea of fractional factorial experiments, in which not all combinations of levels of the different treatment factors are used, but important effects can still be estimated. Finney concludes this paper with My thanks are due to Dr F. Yates, who first suggested to me the possibility of using fractional replication and to Mr O. Kempthorne for help at intervals over a long period in discussing its problems.

Kempthorne cites [25] for fractional replication but not for the new factorial notation. He also deals with treatment factors having p levels, where p is an arbitrary prime. The match between the notation used by Fisher and Yates and that is used nowadays is also given in [26]. Although Yates was not a group theorist, it is a little surprising that he did not proceed with the .9 × 9 squares as for the .4 × 4 squares in Sect. 3.4 but doing arithmetic modulo 3 rather than arithmetic modulo 2. Each cell can be uniquely labelled .(r, s, t, u), where r, s, t, and u are the levels of A, B, C, and D, respectively, all in .{0, 1, 2}. If we subtract 1 from the labels for .AC(I ) and .AC(J ) given in Fig. 11, we see that the level of .AC(I ) is .2r + t mod 3 and the level of .AC(J ) is .r + t mod 3. The standard notation for these partitions now is .A2 C and AC, respectively. Similarly, the levels of .ABC(W ), .ABC(X), .ABC(Y ), and .ABC(Z) (after subtracting 1) are .r + 2s + 2t mod 3, .r + 2s + t mod 3, .r + s + 2t mod 3, and 2 2 2 .r +s +t mod 3, respectively. The standard notation for these now is .AB C , .AB C, 2 .ABC , and ABC. Table 2 shows the correspondence between the notation in Fig. 10 and that introduced in [25, 47]. Desmond Patterson worked in the Statistics Department at Rothamsted from 1947 to 1967. Using the approach in [25, 47], he introduced an algorithm called the

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design key in [53, 54] for constructing factorial designs of various sorts. This begins by labelling the experimental units by all combinations of levels of some factors, each of which has the same prime number of levels, and then using combinations of factors like those just described to assign treatment combinations to experimental units. A group theorist looking at Table 1 would probably notice that the combinations of levels (0, 1, and 2) of the first, second, and third factors give an elementary Abelian group of order 27 when addition is done modulo 3. From this viewpoint, it is clear that the subset .W1 is a subgroup and that .W2 and .W3 are cosets of .W1 . Thus W is simply a coset partition. The same is true for X, Y , and Z. It was shown in [1, 2, 8, 55] that the approach of Finney and Kempthorne effectively uses what group theorists call the dual of an elementary Abelian group. Furthermore, as noted in [3], this can be generalized to arbitrary finite Abelian groups. Acknowledgments I should like to thank Joshua Paik, who, while he was a final-year undergraduate at the University of St Andrews, pointed out the error in what I had said in [5, 6]. I should also like to thank Chris Brien for sending me an electronic copy of TC35 when I could not access the hard copy in my office because of Covid lockdown. I am grateful to Stephen Senn for drawing my attention to [59].

References 1. Bailey, R. A.: Patterns of confounding in factorial designs. Biometrika 64:597–603 (1977). 2. Bailey, R. A.: Dual Abelian groups in the design of experiments. In: Schultz, P., Praeger, C. E., Sullivan, R. P. (eds.) Algebraic Structures and Applications, pp. 45–54. Marcel Dekker, New York (1982). 3. Bailey, R. A.: Factorial design and Abelian groups. Linear Algebra and its Applications 70:349–368 (1985). 4. Bailey, R. A.: Orthogonal partitions in designed experiments. Designs, Codes and Cryptography 8:45–77 (1996). 5. Bailey, R. A.: Some history of Latin squares in experiments. Paper presented at the British Society for the History of Mathematics meeting on History of Statistics at Gresham College, London, 30 October 2014. 6. Bailey, R. A.: Latin squares: some history, with an emphasis on their use in designed experiments. Paper presented at the British Society for the History of Mathematics meeting during the British Mathematical Colloquium, University of St Andrews, 13 June 2018. 7. Bailey, R. A., Cameron, P. J., Kinyon, M., Praeger, C. E.: Diagonal groups and arcs over groups. Designs, Codes and Cryptography 90:2069–2080 (2022). https://doi.org/10.1007/s10623-02100907-2 8. Bailey, R. A., Gilchrist, F. H. L., Patterson, H. D.: Identification of effects and confounding patterns in factorial designs. Biometrika 64:347–354 (1977). 9. Bailey, R. A., Greenwood, J. J. D.: Effects of neonicotonoids on bees: an invalid experiment. Ecotoxicology 27:1–7 (2018). 10. Bailey, R. A., Kunert, J., Martin, R. J.: Some comments on gerechte designs. I. Analysis for uncorrelated errors. Journal of Agronomy & Crop Science 165:121–130 (1990).

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11. Bailey, R. A., Kunert, J., Martin, R. J.: Some comments on gerechte designs. II. Randomization analysis, and other methods that allow for inter-plot dependence. Journal of Agronomy & Crop Science 166:101–111 (1991). 12. Behrens, W. U.: Feldversuchsanordnungen mit verbessertem Ausgleich der Bodenunterschiede. Zeitschrift für Landwirtschaftliches Versuchs- und Untersuchungswesen 2:176–193 (1956). 13. Bennett, J. H. (ed.): Collected Papers of R. A. Fisher, Volumes I–V. The University of Adelaide, Adelaide (1971–1974). 14. Bennett, J. H. (ed.): Statistical Inference and Analysis. Selected Correspondence of R. A. Fisher. Oxford University Press, Oxford (1990). 15. Bose, R. C.: On the application of the properties of Galois fields to the problem of construction of hyper-Graeco-Latin squares. Sankhy¯a 3:323–338 (1938). 16. Bose, R. C., Nair, K. R.: On complete sets of Latin squares. Sankhy¯a 5:361–382 (1941). 17. Bose, R. C., Shrikhande, S. S.: On the falsity of Euler’s conjecture about the non-existence of two orthogonal Latin squares of order 4t + 2. Proceedings of the National Academy of Sciences of the United States of America 45:734–737 (1959). 18. Bose, R. C., Shrikhande, S. S., Parker, E. T.: Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler’s conjecture. Canadian Journal of Mathematics 12:189–203 (1960). 19. Box, J. F.: R. A. Fisher. The Life of a Scientist. John Wiley & Sons, New York (1978). 20. Cayley, A.: On latin squares. Messenger of Mathematics 19:135–137 (1890). 21. Charlesworth, B.: Fisher’s historic 1922 paper On the dominance ratio. Genetics 220: iyac006 (2022). https://doi.org/10.1093/genetics/iyac006 22. Dénes, J., Keedwell, A. D.: Latin Squares and Their Applications. Akadémiai Kiadó, Budapest (1974). 23. Egan, J., Wanless, I. M.: Enumeration of MOLS of small order. Mathematics of Computation 85:799–824 (2016). 24. Euler, L.: Recherches sur une nouvelle espèce de quarrés magiques. Verhandelingen uitgegeven door het Zeeuwsch Genootschap der Wetenschappen te Vlissingen 9:85–239 (1782). 25. Finney, D. J.: The fractional replication of factorial arrangements. Annals of Eugenics 12:291– 301 (1945). 26. Finney, D. J.: An Introduction to the Theory of Experimental Design. University of Chicago Press, Chicago (1960). 27. Finney, D. J., Yates, F.: Statistics and computing in agricultural research. In: Cooke, G. W. (ed.) Agricultural Research 1931–1981, pp. 219–236. Agricultural Research Council, London (1981). 28. Fisher, R. A.: On the dominance ratio. Proceedings of the Royal Society of Edinburgh 42:321– 341 (1922). 29. Fisher, R. A.: Statistical Methods for Research Workers. Oliver and Boyd, Edinburgh (first edition 1925, 14th edition 1970). 30. Fisher, R. A.: The arrangement of field experiments. Journal of the Ministry of Agriculture of Great Britain 33:503–513 (1926). 31. Fisher, R. A.: The Design of Experiments. Oliver and Boyd, Edinburgh (first edition 1935, second edition 1937, eighth edition 1960). 32. Fisher, R. A.: Contribution to discussion of “Statistical problems in agricultural experimentation” by J. Neyman. Journal of the Royal Statistical Society, Supplement 2:154–157 (1935). 33. Fisher, R. A.: An examination of the different possible solutions of a problem in incomplete blocks. Annals of Eugenics 10:52–75 (1940). 34. Fisher, R. A.: New cyclic solutions to problems in incomplete blocks. Annals of Eugenics 11:290–299 (1942). 35. Fisher, R. A.: The theory of confounding in factorial experiments in relation to the theory of groups. Annals of Eugenics 11:341–353 (1942). 36. Fisher, R. A.: Completely orthogonal 9 × 9 squares. A correction. Annals of Eugenics 11:402– 403 (1942).

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37. Fisher, R. A.: A system of confounding for factors with more than two alternatives, giving completely orthogonal cubes and higher powers. Annals of Eugenics 12:283–290 (1945). 38. Fisher, R. A., Yates, F.: The six by six Latin squares. Proceedings of the Cambridge Philosophical Society 30:492–507 (1934). 39. Fisher, R. A., Yates, F.: Statistical Tables for Biological, Agricultural and Medical Research. Oliver and Boyd, Edinburgh (first edition 1938, sixth edition 1963). 40. Frolov, M.: Recherches sur les permutations carrées. Journal des Mathematiques Spéciales 4:8–11 (1890). 41. Frolov, M.: Recherches sur les permutations carrées. Journal des Mathematiques Spéciales 4:25–30 (1890). 42. Grundy, P. M., Healy, M. J. R.: Restricted randomization and quasi-Latin squares. Journal of the Royal Statistical Society, Series B 12:286–291 (1950). 43. Henderson, W.: British Agricultural Research and the Agricultural Research Council; a personal historical account. In: Cooke, G. W. (ed.) Agricultural Research 1931–1981, pp. 1– 113. Agricultural Research Council, London (1981). 44. Hurlbert, S. H.: Pseudoreplication and the design of ecological field experiments. Ecological Monographs 54:187–211 (1984). 45. Hurlbert, S. H.: The ancient black art and transdisciplinary extent of pseudoreplication. Journal of Comparative Psychology 123:446–443 (2009). 46. Jones, H. L.: Inadmissible samples and confidence limits. Journal of the American Statistical Association 53:482–490 (1958). 47. Kempthorne, O.: A simple approach to confounding and fractional replication in factorial experiments. Biometrika 34:255–272 (1947). 48. MacMahon, P. A.: Combinatory Analysis. Cambridge University Press, Cambridge (1915). 49. Norton, H. W.: The 7 × 7 squares. Annals of Eugenics 9:269–307 (1939). 50. Owens, P. J., Preece, D. A.: Complete sets of pairwise orthogonal Latin squares of order 9. Journal of Combinatorial Mathematics and Combinatorial Computing 18:83–96 (1995). 51. Owens, P. J., Preece, D. A.: Aspects of complete sets of 9×9 pairwise orthogonal Latin squares. Discrete Mathematics 167/168:519–525 (1997). 52. Parker, E. T.: Orthogonal Latin squares. Proceedings of the National Academy of Sciences of the United States of America 45:859–862 (1959). 53. Patterson, H. D.: The factorial combination of treatments in rotation experiments. Journal of Agricultural Science 65:171–182 (1965). 54. Patterson, H. D.: Generation of factorial designs. Journal of the Royal Statistical Society, Series B 38:175–179 (1976). 55. Patterson, H. D., Bailey, R. A.: Design keys for factorial experiments. Journal of the Royal Statistical Society, Series C 27:335–343 (1978). 56. Preece, D. A., Bailey, R. A., Patterson, H. D.: A randomization problem in forming designs with superimposed treatments. Australian Journal of Statistics 20:111–125 (1978). 57. Sade, A.: Énumération des carrés latins. Applications au 7e ordre. Conjecture pour les Ordres Supérieurs. Published by the author, Marseille (1949). 58. Sade, A.: An omission in Norton’s list of 7 × 7 squares. Annals of Mathematical Statistics 22:306–307 (1951). 59. Savage, L. J.: On rereading R. A. Fisher, Annals of Statistics, 4:441–500 (1976). (Text completed by J. W. Pratt from a draft after Savage died.) 60. Savage, L. J., Bartlett, M. S., Barnard, G. A., Cox, D. R., Pearson, E. S., Smith, C. A. B.: The Foundations of Statistical Inference. A Discussion. Methuen, London (1962). 61. Schönhardt, E.: Über lateinische Quadrate und Unionen. Journal für die Reine und Angewandte Mathematik 163:183–229 (1930). 62. Singer, B. H., Pincus, S.: Irregular arrays and randomization. Proceedings of the National Academy of Sciences 95:1363–1368 (1998). 63. Sparks, T. H., Bailey, R. A., Elston, D. A.: Pseudoreplication: common (mal)practice. SETAC News 7(3): 12–13 (1997).

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64. Stevens, W. L.: The completely orthogonalized Latin square. Annals of Eugenics 9:82–93 (1939). 65. Tarry, G.: Les permutations carrées de base 6. Memoires de la Société Royale des Sciences de Liège, Série 3, 2, mémoire No. 7 (1900). 66. Tedin, O.: The influence of systematic plot arrangement upon the estimate of error in field experiments. Journal of Agricultural Science 21:191–208 (1931). 67. Tjur, T.: Analysis of variance models in orthogonal designs. International Statistical Review 52:33–81 (1984). 68. Yates, F.: The principles of orthogonality and confounding in replicated experiments. Journal of Agricultural Science 23:108–145 (1933). 69. Yates, F.: The formation of Latin squares for use in field experiments. Empire Journal of Experimental Agriculture 1:235–244 (1933). 70. Yates, F.: Complex experiments. Journal of the Royal Statistical Society, Supplement 2:181– 247 (1935). 71. Yates, F.: Incomplete randomized blocks. Annals of Eugenics 7:121–140 (1936). 72. Yates, F.: The Design and Analysis of Factorial Experiments. Technical Communication 35. Imperial Bureau of Soil Science, Harpenden (1937). 73. Yates, F.: Bases logiques de la planification des expériences. Annales de l’Institut Henri Poincaré 12:97–112 (1951). 74. Yates, F.: Quelques developpements modernes dans la planification des expériences. Annales de l’Institut Henri Poincaré 12:113–130 (1951). 75. Yates, F.: A fresh look at the basic principles of the design and analysis of experiments. In: Le Cam, L. M., Neyman J. (eds.) Proceedings of the Fifth Berkeley Symposium on Mathematics and Probability, Volume 4: Biology and Problems of Health, pp. 777–790. University of California Press (1967).

Les équations différentielles ordinaires « raides » et les méthodes robustes : une approche historique Roger Godard, John de Boer and Mark Lewis

Résumé Les équations différentielles ordinaires (édo) ont été bien étudiées par les historiens des mathématiques. Cependant l’étude des équations différentielles « raides » est relativement récente, et n’apparait pas dans cette littérature. En 1952, Curtiss and Hirschfelder, de l’université du Wisconsin, ont donné un exemple d’édo raide. Ils ont aussi introduit les concepts de méthodes de solutions numériques « explicites » ou « implicites ». L’équation est implicite si l’inconnue apparaît à gauche et à droite de l’équation différentielle. Un problème de valeurs initiales est dit raide (stiff en anglais) s’il est (extrêmement) difficile de le résoudre par des méthodes explicites. Ces problèmes existent souvent en théorie du contrôle, dans l’étude des réactions chimiques, etc. Le principe de cette définition est différent de la question de savoir si le problème mathématique est bien posé, ou sous quelles conditions il l’est. Il est lié à la robustesse des algorithmes numériques. Tout d’abord, on passe brièvement en revue l’histoire des méthodes explicites et des méthodes implicites, notamment le travail original et éclairant de Cauchy en 1824 sur la méthode implicite dite d’Euler et la méthode du trapèze. On présente ensuite les méthodes à un seul pas (l’école allemande) et les méthodes à pas multiples (l’école anglo-saxonne). On examine ensuite le problème des erreurs de troncature locales et globales pour un problème de valeurs initiales; un problème déjà esquissé par Euler, celui du contrôle du pas d’intégration; celui de la stabilité et de la condition d’un problème; puis l’exemple de Curtis et Hirchfelder en 1952 d’équation différentielle raide. On finit par l’époque moderne et le concept de stabilité A.

R. Godard () · J. de Boer Royal Military College of Canada, Kingston, ON, Canada e-mail: [email protected]; [email protected] M. Lewis Saint Lawrence College, Kingston, ON, Canada e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_14

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1 Introduction Les équations différentielles ordinaires commencent à apparaitre au XVIIe siècle avec I. Newton et G. W. Leibniz (Hairer et Wanner 1987; Tournès 1996). Les solutions numériques aux équations différentielles ordinaires (édo) émergent avec A. Clairaut en 1759 et 1760 (Tournès 1996, 1998), tandis que L. Euler publie sa méthode en 1768 (Euler 1768; Chabert et al. 1993: 417–420). C’est une méthode populaire qui démarre toute seule, mais qui manque de précision. On l’appelle parfois méthode avant d’Euler (forward en anglais), méthode explicite d’Euler ou méthode de la tangente. Nous étudierons cette méthode et la méthode du trapèze d’après les travaux de A. L. Cauchy en (1824), ainsi que le calcul des erreurs, qui est fondamental. Puis nous étudierons les méthodes numériques en les séparant entre méthodes explicites et méthodes implicites. Ceci sera la pierre d’angle de ce travail. Nous commenterons aussi brièvement les problèmes d’existence d’une solution, de sa stabilité, de sa condition, et finalement la stabilité A pour des équations différentielles raides.

2 Les premières méthodes numériques à un seul pas : Euler et Cauchy Soit le problème de valeur initiale : dy = f (x, y) ; y (x0 ) = y0 . dx

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On cherche une solution numérique pour X = x à l’équation différentielle. On divise alors l’intervalle [x0 , X] en intervalles partiels x, que Cauchy appelle en (1824) « elements », et que nous désignons par le pas d’intégration h. Euler propose en (1768) une « équation aux différences » : y = f (x, y) x.

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On calcule alors yn au moyen des équations successives y1 − y0 = f (x0 , y0 ) x, y2 − y1 = f (x1 , y1 ) x, . . . yn − yn−1 = f (xn−1 , yn−1 ) x,

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Les équations différentielles ordinaires « raides » et les méthodes robustes. . .

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où x1 , x2 , ..., xn − 1 sont appelés les « quantités intermédiaires ». Ici, on utilise le vocabulaire mathématique de Cauchy (1824) et de Coriolis (1837), et non celui d’Euler, par souci d’uniformité. Si on pose x = h, on a yi+1 = yi + hf (xi , yi ) ; i = 0, . . . , n − 1.

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Cette méthode est appelée méthode explicite d’Euler car toute l’information nécessaire est contenue dans l’équation 4. C’est une méthode qui démarre toute seule. On l’appelle aussi la méthode de la tangente. On verra que c’est une méthode peu précise et instable. Il faut reconnaitre qu’Euler n’est pas le premier à vouloir résoudre numériquement des équations différentielles ordinaires. En astronomie, tout au long du XVIIIe et du XIXe siècle, les mathématiciens s’intéressent aux méthodes des perturbations spéciales, età des systèmes différentiels du sixième ordre (Tournès 1996, 1998). Il est important citer les apports de A. Clairaut sur la comète de Halley, de J.-M. Legendre et de l’équipe C.-F. Gauss-J. F. Encke sur des astéroïdes. Dans la septième leçon du Résumé donné à l’École royale polytechnique en (1824), Cauchy établit le théorème suivant pour le problème de valeur initiale : dy = f (x, y) ; y (x0 ) = y0 , pour l’intervalle [x0 , X]. dx Il pose y(X) = Y qui est à déterminer. Cauchy suppose que la fonction f (x, y) est continue par rapport aux variables x et y et demeure comprise entre les limites ±A. Alors la valeur de Y peut être représentée sous la forme : Y = y0 + (X − x0 ) f (x0 + θ (X − x0 ) , y0 + A (X − x0 )) ,

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avec θ et  inférieurs à un. Si on fait θ = 0 et  = 0, on retrouve la méthode explicite d’Euler. Les méthodes nouvelles de Cauchy sont exposées dans la douzième leçon qu’il intitule « Méthodes diverses qui peuvent être employées au calcul numérique . . . ». Il ajoute : La méthode que nous avons développée [méthode explicite d’Euler] n’est pas la seule . . . plusieurs autres méthodes, que nous allons connaître, peuvent être employées au même usage; et en général elles méritent d’être préférées, parce qu’elles resserrent les limites entre lesquelles les valeurs des inconnues se trouvent comprises.

Tout d’abord, Cauchy pose y = F(x). Ensuite, il va écrire la solution intégrale de l’équation différentielle comme :  F (X) = F (x0 ) +

X

f (x, F (x)) dx

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x0

Ceci est une équation intégrale non linéaire qui ne peut être résolue que par approximations successives. Il va alors resserrer les bornes de l’équation 5 et pose : F (X) = y0 + (X − x0 ) f (x0 + θ (X − x0 ) , y0 +  (F (X) − y0 )) .

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Il prend alors le cas limite θ =  = 1. Supposons alors que l’intervalle [x0 , X] soit découpé en petits pas h, on aura alors au pas xi + 1 : yi+1 = yi + hf (xi+1 , yi+1 ) .

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Cette équation est appelée improprement la méthode implicite d’Euler. On dit que l’équation est implicite car l’inconnue yi + 1 se trouve aussi dans le membre de droite dans l’équation.1 Cette méthode devrait en fait s’appeler la méthode de Cauchy. Il va alors résoudre l’exemple suivant par approximations :  dy = cos

x+y 5

 dx; y(0) = 0.

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Cauchy veut trouver la solution pour X = 1. Il part, comme première itération, de la solution donnée par la méthode explicite d’Euler Y(k = 0) = cos (0) = 1. Il procède ensuite par la méthode du point fixe : (k+1)

yi+1

  (k) = yi + hf xi+1 , yi+1 .

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Ici, k représente une itération. On voit que la méthode de Cauchy ne démarre pas toute seule. Elle a besoin d’une méthode explicite pour démarrer. Dans son exemple, Cauchy fait les calculs pour un seul pas X − x0 = 1. Il trouve après quatre itérations Y = 0.926. Pour améliorer la précision, il utilise la méthode du trapèze qu’il appelle la « demi-somme » des calculs : yi+1 = yi +

  h f (xi , yi ) + f xi+1 , yi+1 . 2

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Cauchy trouve alors .Y = 12 [1 + 0.926] = 0.963. Puis, pour augmenter la précision des résultats, il recommence les calculs pour un pas h = 0.2. Avec le logiciel ode23t de Matlab, nous avons trouvé comme solution à cette équation Y = 0.9738. En fait, Cauchy a ainsi posé la base des méthodes robustes pour résoudre les équations différentielles raides. La robustesse est un mot fréquemment utilisé en statistiques pour désigner les méthodes parvenant à une solution finale fiable mais sans se soucier de la rapidité. Nous allons étudier les équations différentielles raides (stiff en anglais) dans le paragraphe 8 de ce travail. La raideur provient d’une analogie avec la raideur d’un ressort ou d’un système mécanique. Dans son

1 Déjà en 1748, dans son livre Introduction à l’analyse infinitésimale, Euler fait la distinction entre les fonctions explicites et les fonctions implicites. Pour lui, une fonction est explicite lorsqu’on peut la calculer au moyen d’un nombre fini d’opérations élémentaires portant sur les variables ou les constantes. Elle est dite implicite dans le cas contraire.

Les équations différentielles ordinaires « raides » et les méthodes robustes. . .

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chapitre treize, Cauchy généralise ses méthodes pour intégrer par approximations des systèmes d’équations différentielles ordinaires du premier ordre.

3 Le triomphe des méthodes à un seul pas et celles à pas multiples La méthode explicite d’Euler ou les méthodes implicites de Cauchy à un seul pas ne sont pas très précises. Dans la deuxième partie du XIXe siècle, et au début du XXe siècle, on assiste à une explosion d’idées. Tournès (1996, 1998) a distingué deux écoles, l’une anglo-saxonne et l’autre allemande. Les méthodes explicites à un seul pas furent retravaillées en Allemagne avec C. T. Runge en 1895, K. Heun en 1900, et W. Kutta en 1901 (Goldstine 1977; Chabert et al. 1993: 427–441). Ces méthodes devinrent très populaires sous de nom de méthodes de Runge-Kutta. Ce sont des méthodes très précises avec une erreur de troncature locale proportionnelle à h5 , où h est le pas d’intégration. On les appelle parfois méthodes à pas libre car elles sont suffisamment souples pour pouvoir contrôler et modifier les pas d’intégration h sans trop de difficultés. On verra cependant que ces méthodes explicites ne sont pas adaptées aux équations différentielles raides. Les premières méthodes implicites de Runge-Kutta datent de 1955 avec le travail de Hammer et Hollingsworth (Hairer et Wanner 1987: ch II.7). La clé vient de la solution numérique de l’intégrale de l’équation (6). Si on utilise la méthode de Radau (1880) pour sa solution, on obtient : k1 = f (x   i , yi ) , k2 = f xi + 23 h, yi + 3h (k1 + k2 ) , yi+1 = yi + 4h (k1 + 3k2 ) .

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Notons que l’équation donnant k2 est implicite. Bien sûr, il existe d’autres méthodes plus précises que celle indiquée. La méthode de Radau consiste à inclure les extrémités de l’intervalle d’intégration dans la quadrature de Gauss. Tournès (1996, 1998) a étudié en détails l’histoire des techniques de l’école anglo-saxonne avec les méthodes à pas liés, appelées aussi méthodes à pas multiples. Soit le problème de valeur initiale, dit problème de Cauchy : dy = f (x, y) ; y (x0 ) = y0 . dx

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La solution est donnée d’après l’éq. (6) par :  yi+1 = yi +

xi+1

f (x, y(x)) dx. xi

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Fig. 1 Interprétation géométrique de la méthode à pas liés

On ne connait pas f (x, y(x)), mais on peut utiliser les valeurs précédentes fi − k + 1 , .., fi − 1 , fi pour trouver un polynôme d’interpolation tel que ceci est indiqué sur la Fig. 1, et extrapoler pour prédire p∗ et trouver yi + 1 explicitement en intégrant directement le polynôme d’interpolation entre xi et xi + 1 . La méthode implicite consiste à ne pas extrapoler, mais à inclure dans l’interpolation le point xi + 1 , et une valeur d’essai pour p∗ . Il faudra ensuite la réajuster par approximations successives et résoudre l’équation 16. La méthode est implicite, et ne démarre pas toute seule. Parmi les nombreux travaux de l’école anglo-saxonne, il faut citer les contributions de l’astronome américain G. P. O. Bond en 1849, les travaux sur la capillarité de F. Bashforth d’après les méthodes de J. C. Adams en 1883 (Chabert et al. 1993: 441–446; Tournès 1996, 1998), et les autres méthodes à pas multiples de G. H. Darwin en 1897, W. F. Sheppard en 1899, etc. Toutes ces méthodes sont liées aux polynômes d’interpolation de J. Gregory et I. Newton, et aux différences finies. Parmi les acquis de cette période, il faut souligner les efforts originaux et audacieux de C. Störmer en Norvège. Ses efforts concernent des équations du second ordre. Il écrivit en (1921) : En 1904, j’eus besoin d’une pareille méthode pour calculer les trajectoires des corpuscules électrisés dans un champ magnétique, et en essayant diverses méthodes déjà connues, mais sans les trouver assez commodes pour mon but, je fus conduit moi-même à élaborer une méthode assez simple, dont je me suis servi ensuite. À l’aide de cette méthode, toute une série de trajectoires fut calculée par un travail numérique de mes assistants et de moi-même, qui monta jusqu’à 5000 heures de travail et qui dura pendant plusieurs années.

Il ajouta que les calculs comprenaient 3000 pages in-folio et 358 grandes planches. Sa simulation consiste à intégrer les équations du mouvement en trois dimensions (Störmer 1907) dans un tube magnétique. La méthode explicite de Störmer revient à remplacer des dérivées secondes par des différences centrales, et à résoudre des systèmes du type : xi+1 − 2xi + xi−1 = h2 fi

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Sur la Fig. 2, nous avons reproduit sur ordinateur quelques trajectoires spectaculaires de Störmer avec ses valeurs initiales :

Les équations différentielles ordinaires « raides » et les méthodes robustes. . .

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Fig. 2 Simulations numériques de trajectoires de Störmer, superposées de sa fig. 7. Les valeurs des angles u dans sa figure ne sont pas spécifiées, alors les trajectoires en couleur sont en degrés entiers qui les approximent. Aussi, nos contours de couleur rose sont des lignes de champ magnétique dipolaire, quoique ces contours dénotent des valeurs du sinus de l’angle θ, défini dans son éq. 3 p. 119 et sa Fig. 4; voir aussi ses éq. 17 et VII p. 137 (Störmer 1907)

La version implicite fut préconisée par B. Numerov en 1924 (Hairer et Wanner 1987: 464–465).

4 La théorie des erreurs de troncature locales et globales 4.1 Le début Euler, dans son texte de (1768) sur l’intégration des équations différentielles par approximations, est bien conscient du problème du contrôle des erreurs. Ceci est crucial pour contrôler la précision des résultats numériques et les erreurs de troncature. Il écrit dans son « corollaire 2 » : Plus les intervalles dont les valeurs de x sont supposées progresser sont petits, plus les valeurs prises une à une seront obtenues avec précision. Ce faisant, cependant, les erreurs commises sur chaque valeur, même si elles sont beaucoup plus petites, vont s’accumuler du fait de leur multitude.

Dans un compte rendu à l’Académie, J. I. Chevilliet (1874) cite Euler pour la solution de l’intégrale : X . x0 f (x)dx par la méthode du trapèze. Malheureusement, il ne donne pas la référence exacte. Le raisonnement d’Euler est important car il aboutit à une méthode

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pour estimer l’erreur globale E pour une intégrale (ou l’équation différentielle dy linéaire équivalente . dx = f (x); y (x0 ) = y0 ) sur tout l’intervalle de calcul. Soit : .

  h2 f (X) − f  (x0 ) E = − 12

Ainsi on pouvait choisir le pas d’intégration et contrôler les erreurs.

4.2 Cauchy et Lipschitz sur la théorie des erreurs locales et globales La neuvième leçon du cours de Cauchy de (1824) concerne les « Limites des erreurs que l’on peut commettre en se servant . . . » de la méthode explicite d’Euler, et le processus d’accumulation des erreurs. Il part à nouveau de l’équation différentielle non linéaire : dy = f (x, y) dx; y (x0 ) = y0 ,

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et il cherche la valeur calculée yn , et la différence Y − yn pour x = X. Il dit expressément : Pour passer de la quantité Y à la valeur de yn , il suffit de subdiviser la différence X − x0 en éléments infiniment petits . . .

Dans sa preuve qui est un peu longue, Cauchy va à nouveau utiliser l’équation ∂f (5). Il aura besoin que les fonctions .f (x, y) , ∂f ∂x , ∂y soient finies et continues. R. Lipschitz en (1876: 155) donnera une preuve moins restrictive pour estimer l’erreur globale pour la méthode explicite d’Euler. Nous l’écrivons avec nos notations: | Y (X) − yn (X) |≤

 hM  L(X−x0 ) e −1 . 2L

(17)

Ici, . max y  (x) ≤ M, et la fonction f (x, y) obéit à une condition de Lipschitz: x∈[x0 ,X] f (x, y1 ) − f (x, y2 ≤ L y1 − y2 . L est appelée constante de Lipschitz avec L > 0.

5 Existence and unicité Au XVIIe siècle et au XVIIIe siècle, les mathématiciens s’intéressent aux méthodes pour résoudre des équations différentielles ordinaires. Cependant, vers les années 1820–1821, Cauchy est le premier à étudier les problèmes d’existence d’une solution pour un problème de valeur initiale d’une équation différentielle du premier ordre, et ses Résumés des leçons données à l’École Royale Polytechnique constituent un trésor d’idées. Les treize premières leçons furent redécouvertes par C. Gilain à

Les équations différentielles ordinaires « raides » et les méthodes robustes. . .

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la bibliothèque de l’Institut. Gilain en a écrit l’introduction. Cauchy va utiliser la méthode explicite d’Euler pour prouver son théorème d’existence. On sait que ses conditions étaient trop restrictives, et que ceci fut levé par Lipschitz en 1876, et que E. Picard, dans un mémoire de (1890) donne une démonstration d’existence définitive (Chabert et al. 1993: 421–425). Concernant l’unicité d’une solution, dès 1734, Clairaut étudie des équations du type: y − xy' + f (y') = 0 qui possèdent deux types de solutions: y = Cx − f (C), où C est un paramètre, mais aussi l’enveloppe des solutions précédentes. D’Alembert en 1748, Euler en 1768, Lagrange en 1774, puis Cauchy trouvent aussi des équations différentielles ordinaires singulières. G. Peano en (1890) écrira : La condition de l’existence et la continuité de la dérivée de f (x, y) par rapport à y, ou au ' (x,y) , suffisante pour déduire moins d’une limite supérieure finie pour le rapport . f (x,yy)−f ' −y l’existence d’une solution unique qui a une valeur initiale donnée, . . . n’est pas nécessaire.

6 Richardson et le contrôle du pas L’article de Richardson en (1910) marque un tournant numérique pour la solution d’équations différentielles partielles et le contrôle des erreurs de troncature dues aux différences finies. La règle est de diminuer le pas d’intégration et d’examiner la stabilité des résultats à chaque nœud de la grille de calcul. Richardson est surtout connu pour ses techniques d’extrapolation qui permettent de gagner en précision sans trop d’efforts (Godard 2002a). Ses techniques sont maintenant couramment utilisées pour la solution approchée d’équations différentielles ordinaires ou partielles. Aux travaux de Richardson sur le contrôle du pas d’intégration et la stabilité des résultats, il faut associer bien sûr les conditions nécessaires de Courant-Friedrichs et Lewy en 1928 pour la stabilité de problèmes dépendant du temps, les études de J. Crank et A. Nicholson en 1947 pour les problèmes paraboliques, et celles de J. von Neumann et R. D. Richmyer aussi en 1947, où il faut lier ensemble le pas temporel au pas spatial h (Godard 2002a).

7 Stabilité, problèmes bien posés, et condition Spijker (1996) a défini trois questions concernant le problème de valeur initiale : (1) la convergence (c’est-à-dire le comportement de yn − Y(xn ) lorsque hn → 0); (2) la stabilité (c’est-à-dire l’étude des perturbations); (3) les méthodes implicites et la raideur. À la page 80 du livre de Hairer et Wanner (1987), on lit dans le chapitre concernant la stabilité d’un système :

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Ce qui est désiré est une condition correspondante [sur la stabilité] nous permettant de décider quand un mouvement dynamiquement possible d’un système est tel que « s’il est dérangé légèrement, le mouvement continuera en ne s’écartant que légèrement . . . » (« The Examiners » dans E. J. Routh, 1877).

En d’autres mots, on recherche des systèmes qui, s’ils sont peu perturbés dans leurs valeurs initiales, sont peu perturbés dans les résultats. Cela peut être mathématiquement ou numériquement vrai. Il faut souligner aussi les travaux de A. Liapounov (1907) sur la stabilité d’un mouvement.2 Rappelons aussi les travaux de J. Hadamard sur les équations différentielles partielles et le concept d’un problème bien posé (Maz’ya et Shaposhnikova 2005) : 1. La solution existe 2. La solution est unique 3. La solution dépend continûment des données. Bien que l’art du calcul ait des racines profondes dans l’histoire, de nouveaux problèmes émergent avec l’ordinateur numérique, notamment avec la théorie des erreurs de chute. A. Turing fut le premier mathématicien à définir des matrices

2 Lyapounov

prouve les questions de stabilité par deux méthodes distinctes (Lyapounov 1907; Parks 1992; Leine 2010). Dans la première méthode, dite méthode indirecte, il utilise un processus de linéarisation. La deuxième méthode, dite méthode directe, est plus générale. Elle est liée à un théorème de Lagrange-Dirichlet. Dans son mémoire, Lyapounov définit son exposant caractéristique. Si deux points y1 (t) et y2 (t) d’un système dynamique sont initialement proches, c’est-à-dire que ||y1 (0) – y2 (0)|| est petit, alors l’exposant caractéristique est :

y (t) − y (t) 1 1 2

lim ln (1)

y (0) − y (0) t→+∞ t 1 2 Dans la première méthode, Lyapounov se réfère en particulier à l’analyse de la stabilité locale des équilibres du système : y˙ = f (y, t) .

(2)

On peut considérer la dynamique de (2) au voisinage de l’origine comme une perturbation du système linéaire : y˙ = J(t)y; J(t) =

∂f . ∂y y=0

(3)

Si toutes les valeurs propres du Jacobien ont une partie réelle négative, alors y = 0 est asymptotiquement stable, et si une valeur propre a une partie réelle positive, elle est instable. Cependant aucune conclusion ne peut être tirée dans le cas de valeurs propres ayant une partie réelle nulle. Pour résoudre la question de stabilité lorsque (2) ne se prête pas à la linéarisation, Lyapounov a conçu sa deuxième méthode. À la place de l’énergie totale d’un système mécanique apparaît une fonction de Lyapounov V(y, t). Si sa dérivée temporelle le long des trajectoires de (2) est semidéfinie négative, alors y = 0 correspond à un équilibre stable. Si la dérivée est définie négative, la stabilité est asymptotique. Voir Lyapounov (1907), Parks (1992) et Leine (2010).

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ainsi que des systèmes d’équations mal conditionnés (Turing 1948; Godard 2002b). Un problème mal conditionné correspond au cas où de petites perturbations dans les données initiales, par exemple dans les coefficients d’une matrice dans un système d’équations linéaires, conduisent à de grandes variations dans les résultats. Un problème peut être mal conditionné en raison du problème mathématique, et dans ce cas le problème n’est pas sûr, ou de l’algorithme utilisé, ou des deux. Le problème que posent les équations différentielles raides n’est pas lié à leur condition, mais à l’algorithme choisi. Dans le cas du problème de valeur initiale de Cauchy, la condition de l’équation est directement liée à la théorie des erreurs globales. Supposons que f (x, y) soit Lipschitz, alors si les données d’entrée y(x0 ) sont perturbées et changées en z(x0 ), on aura pour les changements en sortie en X (Sauer 2006: 292–293) : | y(X) − z(X) |≤ eL(X−x0 ) | y (x0 ) − z (x0 ) | .

(18)

8 Les équations différentielles « raides » et les méthodes robustes : les outils mathématiques C. F. Curtis and J. O. Hirchfelder de l’Université du Wisconsin publient en (1952) un article sur l’intégration des équations différentielles raides. Leur article commence ainsi : Dans l’étude de réactions chimiques, de la théorie des circuits électriques et des problèmes de guidage de missiles, une classe d’équations différentielles émerge qu’il est excessivement difficile de résoudre par des procédés numériques ordinaires . . . un exemple typique d’équation raide est l’équation représentant la formation de radicaux libres dans des réactions chimiques complexes . . .

Ce sont souvent des problèmes ayant deux (ou plusieurs) constantes de temps très différentes. Les auteurs disent avoir aussi bénéficié de l’aide de J. Tuckey et de son expertise sur les équations différentielles raides. Ce sont eux qui introduisent les termes de méthodes « explicites » et « implicites ». Ils prennent l’exemple suivant : dy = [y − G(x)] /a (x, y) . dx

(19)

Si on choisit la méthode explicite d’Euler, on a l’équation aux différences suivante :   a (xi yi ) G (xi ) − a(x ,y ) . yi+1 = yi 1 + (20) i i x x

On voit que le problème est instable si . a(x,y) x > 1. min |Re (λ)|

(24)

Examinons la méthode explicite d’Euler pour l’équation (23). Par différences finies et par induction mathématique, on a : yi+1 = yi + hλyi = y0 (1 + hλ)i+1 .

(25)

z = hλ, une variable complexe, avec Re(z) < 0. La solution sera stable si Posons z - (-1) ≤ 1 qui est un disque de rayon 1 et de centre −1. On aura pour la méthode implicite de Cauchy : yi+1 = yi + hλy i+1 =

y0 yi = . 1 − hλ (1 − hλ)i+1

(26)

Cette fois, le domaine de stabilité est beaucoup plus grand et correspond à l’extérieur du disque de rayon 1 et de centre 1. Examinons maintenant la méthode implicite du trapèze pour la même équation différentielle :   1 + hλ/2 i+1 1 yi+1 = yi + hλ (yi + yi+1 ) = y0 . 1 − hλ/2 2

(27)

Alors le domaine de stabilité correspond au demi-plan gauche de l’axe des z. On a reproduit en bleu sur la Fig. 4 les différentes régions de stabilité pour la

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y

y

x

x

1 2 3

1 2 3

a) méthode explicite d´Euler

b) méthode implicite de Cauchy

x 1 2 3

c) méthode implicite du trapèze

Fig. 4 Les différentes régions de stabilité pour l’équation test de Dahlquist

méthode explicite d’Euler, pour la méthode implicite de Cauchy et pour la méthode du trapèze. On voit que les méthodes de Cauchy qui furent longtemps délaissées retrouvent maintenant leur importance fondamentale. Nous disons simplement qu’une méthode est considérée comme ayant la stabilité A si sa région de stabilité inclut tout le demiplan gauche des z. L’art de programmer sera de détecter automatiquement le degré de raideur de l’équation différentielle ordinaire ou d’un système (Hairer et Wanner 1987). L’avènement des langages formels comme Maple, bien ignoré par J. C. Butcher dans le chapitre sur les équations différentielles du livre édité par Brezenski (Butcher 2001), a aussi permis une révolution dans la programmation pour résoudre formellement des systèmes d’équations différentielles raides mais linéaires. C’est en fait la meilleure méthode pour résoudre des systèmes d’équations chimiques ou des problèmes de contrôle. On n’a plus de problèmes de stabilité d’une méthode. Pour conclure, dès 1960 environ, Dahlquist souligne que le monde est plein de problèmes raides ! Et depuis 1952, des centaines d’articles ont été publiés sur ce sujet, contenant soit la construction d’algorithmes performants, soit des analyses théoriques de ces algorithmes. Le sujet n’est pas clos.

Références Butcher J C (2001) Numerical methods for ordinary differential equations in the 20th century. Numerical Analysis: Historical Developments in the 20th Century. C Brezinski and L Wuytack Eds. Elsevier Amsterdam: 449–477. Cauchy A L (1824) Résumé des Leçons données à l’École Royale Polytechnique, Suite du calcul Infinitésimal. Équations différentielles ordinaires, cours inédit. Introduction par C Gilain. Études vivantes et Johnson Reprint Corporation, Paris. Chabert J-L et al. (1993) Histoire d’algorithmes. Belin éditeur, Paris. Chevilliet 1874 Sur le degré d’exactitude de la formule de Simpson relative à l’évaluation approchée des aires. C R Acad Sci Paris, v 78:1841–1843. Coriolis G (1837) Mémoire sur le degré d’approximation qu’on obtient pour les valeurs numériques d’une variable qui satisfait à une équation différentielle, en employant pour calculer ces valeurs diverses équations aux différences plus ou moins approchées. J Math pures et appliquées, 2: 229–244. Curtis C F and Hirschfelder J O (1952) Integration of stiff equations. Proceedings of the National Academy of Sciences of US 38: 235–243.

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Euler L (1768) Concerning the approximate integration of differential equations. Instituonum Calculi integralis. Vol. 1, part 1, section II, ch. 7, problem 85. Translated and annotated by Ian Bruce. Dahlquist G (1963) A special stability problem for linear multistep methods. BIT Numerical mathematics. 3: 27–43. Godard R (2002a) Numerical PDE: An historical sketch. Conference proceedings of the 28th annual meeting of CSHPM, Toronto 15: 73–88. Godard R (2002b) The Art of Computing. Actes de la Conférence Internationale d’Histoire de l’Informatique, Grenoble, France. Goldstine H (1977) A history of numerical analysis from the 16th century through the 19th century. Springer, New York. Hairer E and Wanner G (1987–1991) Solving Ordinary Differential Eqs. I and II. Springer, New York. Leine R I (2010) The historical development of classical stability concepts: Lagrange, Poisson and Lyapunov stability. Nonlinear Dyn 59: 173–182. Liapounov A M (1907) Problème général de la stabilité du mouvement. Annales fac. sci. Toulouse, 2e série tome 9: 203–474. Lipschitz R (1876) Sur la possibilité d’intégrer complètement un système donné d’équations différentielles. Bulletin sci. math. et astro. 10: 149–159. Maz’ya V and Shaposhnikova T (2005) Jacques Hadamard un mathématicien universel. EDP sciences, Paris. Parks P C (1992) A.M. Lyapunov’s stability theory-100 years on. IMA J of Mathematical Control & Information, 9: 275–303. Peano G (1890) Démonstration de l’intégrabilité des équations différentielles ordinaires. Math Ann 37:182–228. Picard E (1890) Mémoire sur la théorie des équations aux dérivées partielles et la méthode des approximations successives. J de Math Pures et Appl, 4e série 6:145–210. Radau R (1880) Étude sur les formules d’approximation qui servent à calculer la valeur numérique d’une intégrale définie. J Math Pures et Appl, 3e série 6: 283–336. Richardson L F (1910) The approximate arithmetical solution by finite differences of physical problems involving differential equations, with an application to the stresses in a masonry dam. Phi Trans of the Royal Soc of London A: 304–357. Sauer T (2006) Numerical Analysis. Pearson, Toronto. Spijker M N (1996) Stiffness in numerical-value problems. J. Comp. and Applied Math 72:393– 406. Störmer C (1907) Sur les trajectoires des corpuscules électrisés dans l’espace sous l’action du magnétisme terrestre avec application aux aurores boréales. Archives sci. phys. et natur. (Genève), 24: 5–28, 113–158, 221–247. Störmer C (1921) Méthode d’intégration numérique des équations différentielles ordinaires. C R Congr Inter Math Strasbourg. Tournès D (1996) L’intégration approchée des équations différentielles ordinaires (1671–1914). Thèse de doctorat de l’Université Paris 7 Denis Diderot. eDisponible par Internet. Tournès D (1998) L’origine des méthodes multipas pour l’intégration numérique des équations différentielles ordinaires. Revue d’histoire des Mathématiques, 1: 5–72. Turing A M (1948) Rounding-off errors in matrix processes. Quart J Mech. 1: 287–308.

The Cavendish Computors: The Women Working in Scientific Computing for Radio Astronomy Verity Allan

Abstract The decades after the Second World War is an important period in the history of scientific computing for Radio Astronomy in the Cavendish Laboratory of the University of Cambridge. The development of the aperture synthesis technique for Radio Astronomy required using the new computing technology developed by the University’s Mathematical Laboratory: the EDSAC, EDSAC 2 and Titan computers. This article looks at the scientific advances made by the Radio Astronomy group was in assembling the evidence which contradicted the Steady State hypothesis. It also examines the software advances that allowed bigger telescopes to be built: the fast Fourier transform (FFT) and the degridding algorithm. This paper focuses on the work of the women in the Cavendish. From the diagrams they drew for scientific publications, through programming and operating computers, to writing scientific papers, the contribution of women is examined.

1 Introduction The years following the Second World War were a time of massive scientific and technological innovation in the University of Cambridge. The University’s Mathematical Laboratory was set up prior to the War to investigate numerical methods, and after the war, Maurice Wilkes became its director. Having visited the United States and seen the ENIAC (Electronic Numerical Integrator and Computer) computer, he started building EDSAC (Electronic Delay Storage Automatic Calculator), putting Cambridge at the forefront of the computer revolution (Ahmed 2013, p. 41). In the same post-War years, the Cavendish Radio Astronomy group was assembled, with Martin Ryle at its head. The Cavendish Laboratory of the University of Cambridge

V. Allan () Battcock Centre for Experimental Astrophysics, Cavendish Laboratory, University of Cambridge, Cambridge, UK e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_15

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had been doing fundamental physics research for decades, and staff members had won Nobel Prizes (Cavendish Laboratory 2021, Cited 30 Sep 2021). The Radio Astronomy group would add to this collection, as Ryle and Tony Hewish won the Nobel Prize for Physics in 1974 for their innovative telescope design and the discovery of pulsars—work that relied on the computers of the Mathematical Laboratory (Nobel Media AB 1974 Apr 7 [cited 2021 Apr 7]). This reliance on computers is exemplified by Ryle’s request for a bigger computer: One day, Ryle came to me to say that he was planning the erection of a much larger telescope and to ask whether the Mathematical Laboratory could undertake to provide the computing support required. (Wilkes 1985, p. 193)

This paper will explain why Radio Astronomy needed a bigger computer and will elucidate the role of other members of the Radio Astronomy group and the Mathematical Laboratory beyond the relatively well-known names of Ryle, Hewish and Wilkes. It has become more widely known in recent years that a lot of early computing relied on women—thanks to work by Abbate (2012), Hicks (2018) and Shetterly (2016). Women were clearly involved in the early years of computing at Cambridge, as can be seen from the archives of the Department of Computer Science and Technology, which is a successor to the Mathematical Laboratory. A photograph of staff from 1949 shows 20 staff, of whom 8 were women (Computer Laboratory 1999). In this paper, we will uncover some of the contributions of women to science and scientific programming in Radio Astronomy in the years from 1950 to 1963. The University computers that the Radio Astronomy group used during this period were EDSAC, the first electronic digital stored program computer in the world; EDSAC 2, which led to many innovations in microprogramming; and Titan, which led to advances in computer graphics and computer aided design (CAD) (Ahmed 2013, p. 72). Titan was the computer requested by Ryle, which was needed for the One-Mile telescope.

2 The Women in Radio Astronomy Computing The Radio Astronomy group became reliant on computing, and by 1965, over 40% of all papers produced by the group had required the use of a computer to process data (Leedham and Allan 2022, p. 14). Women were extensively involved in this endeavour. Around 50% of people acknowledged in papers for their contribution to the programming and operation of computers were women (Leedham and Allan 2022, p. 14). These women were employed in a variety of roles.

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Some were hired as Computors in the 1950s and 1960s,1 initially to do calculations and then later to assist with programming. Computor or Computer as a job description was common in the 1940s, first to describe someone who was hired to do calculations by hand or with a mechanical or electro-mechanical computer and then to describe a worker who used a computer in some capacity (Light 1999). The Cavendish records also show that people (often women) were hired as scanners for High Energy Physics: their job was to analyse photographic plates from cloud chamber experiments. Women were also hired as Research Assistants: Elizabeth Waldram was hired as such (Ahmed 2013, p. 63). Judy Bailey, who later became Deputy Director of the University Computing Service (another successor to the Mathematical Laboratory), worked for Radio Astronomy as a Technical Officer (University of Cambridge 2008 [cited 2021 Apr 7]). Women were also hired to prepare diagrams for scientific papers, drawing graphs and contour diagrams, as it was not yet possible to prepare these with computers. They were known as the girls in the attic (Clifford, F. E. 1964, p. viii). Some of the women involved in scientific computing were PhD students and researchers themselves. Ann Neville, one of Ryle’s PhD students, did a considerable amount of programming. She had to program for EDSAC 2, and to save space, many constants had to be combined into one constant, just to shave a few bytes off the size of the program.2

3 The First Forays into Scientific Computing While Charles Wynn-Williams was a pioneer of electro-mechanical computing for the Cavendish (Wynn-Williams 1931), the Radio Astronomy group did not leap straight to digital computing for all their papers. We can trace the use of computers and the contributions of programmers primarily through the acknowledgements section of papers. Their first paper using EDSAC was published in 1953, and use of computing through the 1950s was fairly sporadic (Leedham and Allan 2022, p. 14). The first women we encounter in the acknowledgements probably were the computers. In 1951, a Mrs A. C. Hollis-Hallett is credited as having analysed a large proportion of the records (Smith 1951, p. 963). It’s not possible to verify this, as no records survive. Similarly, a Mrs Moore is credited in a paper by Scheuer and Ryle from 1953, for help in the computation of results (Scheuer and Ryle 1953, p. 17). It’s likely that Mrs Moore was the computer, not EDSAC, as EDSAC was not listed in the acknowledgements (which was the common practice at the time). However, the development of aperture synthesis, a new technique in Radio Astronomy, was going to change the group’s relationship with computers.

1 Source: 2 Donald

Cavendish Laboratory Archive Personnel Records. Wilson, Personal Communication, 22 April 2021.

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The basic principles were sketched out by Ryle in a 1957 paper (Ryle 1957) and implemented by John Blythe. Using aperture synthesis, a scientist can use two or more smaller telescopes, to obtain the resolution (but not the sensitivity) of a larger telescope with diameter equivalent to the longest spacing between antennas.3 However, when measuring a signal with a particular pair of antennas, only part of the signal is observed—the measurement is of a Fourier component of the signal (Christiansen and Högbom 1985, Chapter 8). To reconstruct the complete signal, the other Fourier components must be measured, which can be achieved by moving the antennas around to fill in the aperture. The Fourier components then need to be combined and transformed into a useful measurement of sky brightness. Thus the Fourier Transform must be calculated for aperture synthesis telescopes, and it is not feasible to do this by hand. So the Radio Astronomy group was forced to turn to computers. Blythe’s papers from 1957, where he implemented this technique, used EDSAC for the calculations (Blythe 1957, p. 651 and p. 659). One of these papers used EDSAC to perform over two million operations in 15 h. The data output was in a form suitable for making contour maps (Blythe 1957). In both papers, Blythe credited a Mrs J. Sandos who performed most of the analysis. No-one in the Radio Astronomy group can recall Mrs Sandos (who may have been a member of the Mathematical Laboratory), but she undoubtedly helped with the complex data preparation process: error-free punched tapes were required as input, and all the data had to be arranged in a particular way in order to compensate for EDSAC’s lack of storage space. By 1960, EDSAC 2 appears in paper acknowledgements. However, the workflow still retains a number of manual steps, as one paper notes: It would be possible to record such information directly in digital form, but it has been found relatively easy to transfer the present records to punched tape in a record reader and manual perforator (Costain and Smith 1960, p. 412).

The acknowledgements are silent about who did this, but it’s quite likely to be either PhD students or Computors.

4 Earth Rotation Aperture Synthesis and the Steady State Hypothesis In 1961, an extension of the aperture synthesis technique was developed: using the rotation of the Earth to help fill in the aperture of a telescope, thus performing Earth Rotation Aperture Synthesis. This technique was prototyped using the Cambridge 178 MHz telescope and EDSAC 2—the new, more powerful, computer in the

3 The distance between a given pair of antennas is known as a baseline: the length of the longest baseline dictates the resolution achievable by the telescope.

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Mathematical Laboratory. This telescope and others were used to collect data that seriously undermined the Steady State hypothesis. In 1961, Scott, Ryle and Hewish performed earth rotation aperture synthesis for the first time (Scott et al. 1961). This paper says: The processes of filtering and convolution are basically equivalent and in this case where the output of the receiver was already recorded in digital form it was convenient to carry out a direct convolution with a time function whose Fourier transform provides the required polar diagram. (Scott et al. 1961, p. 100)

This is an early use of a digital filter rather than a piece of electronics; the work is moving from hardware to software. Note also that the digital form is paper tape! To record the data, a sample and hold electronic circuit was implemented electromechanically, using photo-electric cells, to output two columns of 5-hole tape for each data point (giving ten bits of precision).4 Each tape started at the same sidereal time: at the same time with respect to the fixed stars in the sky, which is most useful to astronomers, so non-local astronomical bodies can be located easily. The sidereal time arrangement meant that the data could be efficiently processed—while EDSAC 2 had more storage space than EDSAC, programmers still could not afford to be profligate. During the data input phase, the data were convolved with a function which allowed for error correction and which also allowed the extraction of the sine and cosine components which were required for the main computation. The data were then sorted into a more convenient order before being written onto magnetic tape. This magnetic tape was used as the input data for the main calculation, in which the data were put through the Fourier transform. The final outputs were effectively a two-dimensional map of the sky, sampled at discrete intervals. The probability distribution of the measured intensity was also calculated and output. The final values for the sky map needed to be interpolated, but this was done manually, possibly by the girls in the attic or junior members of the group. As well as being the first practical demonstration of aperture synthesis using the earth to fill in the aperture, this paper also was important for providing evidence against the Steady State hypothesis. The authors note: there is no evidence for a reduction in the density of radio sources . . . this observation implies an isotropy which extends to at least 10,000 sources steradian. (Scott et al. 1961, p. 106).

This isotropy in extra-galactic faint sources conflicted with predictions from the Steady State model, which predicted that there should be many fewer faint sources, and that there should be no difference between galactic and extra-galactic sources. Once again we find a woman tucked away in the acknowledgements: Mrs R. Feinstein. Ruth Feinstein was employed by the Mathematical Laboratory (Of Cambridge Computer Laboratory 1999, [cited 2021 Apr 7]) and was an assistant to Dr Lucy Slater, who became Head of Computing in the University’s Department of Applied Economics (Andrews 2013, cited 2021 Apr 7, p. 6). Ruth Feinstein

4 Thanks

to B.J. Harris for spotting that they were building a sample and hold circuit.

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assisted her husband, Charles, with calculating and typing for his PhD thesis (Offer 2008 Jun, [cited 2021 Apr 7], p. 4). Unfortunately, she left the Mathematical Laboratory shortly after marriage (Of Cambridge Computer Laboratory 1999, [cited 2021 Apr 7]), and she seems to play no further part in computing for Radio Astronomy. She was, along with Dr David Wheeler, responsible both for the running and design of the programme (Scott et al. 1961, p. 111).5

4.1 The Steady State Hypothesis The Steady State hypothesis states that the universe is expanding, but has no beginning and no end, and that its appearance does not change over time. In contrast, the Big Bang Theory states that the universe expanded from an initial state of high density and temperature, and thus its appearance changes over time. Ryle’s group was building up to attack the Steady State hypothesis. There are two papers from Harriet Tumner and Patricia Leslie, both PhD researchers in the group (Leslie 1961, Tunmer 1960). Neither paper mentioned using EDSAC 2, though both used data from the Cambridge surveys that were processed using that computer. These papers look at the brightness, density, size, and clustering of extra-galactic radio sources—where the Steady State and Big Bang hypotheses generated different predictions. A 1961 paper from Paul Scott and Martin Ryle (Scott and Ryle 1961) looks at the relationship between the number of radio sources and their flux density (essentially their brightness). The Steady State hypothesis assumes that there is a particular relationship between the brightness of the source and how many sources of that brightness there are. The acknowledgements in this paper read: We should like to thank Dr Hewish and Miss Patricia Leslie for useful discoveries, and Miss Ann Neville and Mr DR Marks for assistance in the reduction of observations. (Scott and Ryle 1961, p. 397)

Ann Neville was one of Ryle’s PhD students, and she appeared as an author on later papers. In a later paper, Hewish modelled the source number–flux density relationship and used EDSAC 2 to do it. This is one of the earliest uses of the Monte Carlo method in Radio Astronomy: successors to this method are being used today as a key modelling technique (Feroz et al. 2019). Hewish doesn’t credit anyone else, so he most likely did a lot of the programming himself. This contrasts with Ryle, who didn’t program at all, even though Ryle was well aware of how essential computers were for aperture synthesis.6

5 Note 6 Dr

that at this time, the UK still used programme to refer to computer programs. Elizabeth Waldram, personal communication 4 May 2021.

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Finally, there is a 1961 Ryle and Clarke paper in which they showed that extragalactic radio sources have no variation in density in different parts of the sky— the only variation comes when observing the Milky Way galaxy, which is nearby in astronomical terms (Ryle and Clarke 1961). The observed source number–flux density relationship doesn’t match the Steady State hypothesis. They conclude: These observations do, however, appear to provide conclusive evidence against the steady state model.

Any last hopes that the Steady State theorists had of salvaging their model was blown away conclusively in 1964 when Penzias and Wilson detected the Cosmic Microwave Background (Penzias and Wilson 1965).

5 A Radio Survey of the North Polar Region A 1962 paper by Ryle and Neville (1962) discussed two major computational innovations for use in Radio Astronomy. The first is Wheeler’s fast Fourier transform (FFT) algorithm, tersely described in a single paragraph, plus an equation. From this, we can glean that the data from the sky needs to be placed onto a Cartesian grid with .(x, y) co-ordinates. This means that data must be converted from the spherical co-ordinates that are most naturally used for observing objects on the sky into Cartesian co-ordinates. While the mathematician Gauss developed the theory of FFT, this practical implementation pre-dates the one by Cooley and Tukey in 1965, which is usually credited as the first practical implementation (Heideman et al. 1984). The second programming innovation was the map degridding algorithm, which enables conversion from Cartesian co-ordinates into spherical co-ordinates, so that the radio sources are correctly located on the sky. This process produced some of the world’s first computer graphics. The output from the computer was plotted on a CRT (Cathode Ray Tube) monitor, where it lasted for a few seconds, just long enough to be photographed (Ahmed 2013, p. 63). This work was done by Elizabeth Waldram, a then Research Assistant in the group. This degridding algorithm and its successors were used for all subsequent interferometers built by the group.7 With the FFT and the degridding algorithm, it was possible to build a map of the northern radio sky. It is very likely that the final construction of the map from the photographic plates was made by the girls in the attic. This kind of mapping was a large part of their job. They were not working directly with computers themselves, but they were a vital part of the scientific workflow, often alongside secretarial staff who would type up manuscripts. A rare acknowledgment for the women doing such work can be seen in a 1962 paper:

7 Dr

Elizabeth Waldram, personal communication 04 May 2021.

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We are grateful to Miss C. Dunn and Mrs M. Warren for helping with the analysis and the preparation of the contour maps (Turtle and Baldwin 1962).

The contour maps were drawn from EDSAC 2 output (Turtle and Baldwin 1962, p. 476). With this paper, we have the foundations for modern radio astronomy computing. Though Wheeler’s FFT was essential to the paper, he was not a co-author. Instead, he was recognised in the acknowledgements. It is clear from this that that the exclusion of women programmers from authorship wasn’t purely a result of sexism. Sexism may still have played a part, as a majority of those acknowledged for programming in Radio Astronomy papers were women (Leedham and Allan 2022, p. 14). Mar Hicks has shown how work done by women, especially in the computing field, was devalued, which may also have been a contributing factor (Hicks 2018, passim). Wheeler was acknowledged in Ryle’s Nobel Prize lecture for his fast Fourier transform algorithm (Ryle 1992); none of the women got mentioned.

6 Discussion It’s clear from the preceding sections that women were deeply involved in the production of scientific papers in the 1950s and 1960s and that they were also involved in doing much of the programming and a good deal more of the science than is usually recognised. The programming also required good mathematical skills and an understanding of Radio Astronomy, so that programmers could work with sidereal time and Right Ascension and Declination (the spherical co-ordinate system used by astronomers) and their conversion into Cartesian co-ordinates for use in the FFT—something that mathematics undergraduates often struggle with. Perhaps the girls in the attic didn’t require such advanced understanding, but they were expected to work to the highest standards of accuracy. It is extraordinarily difficulty to get a picture of many of the computational methods used. Programs were not routinely recorded at the time, though there are some contemporaneous program booklets stored in the archive of the Department of Computer Science and Technology.8 However, none of these survive for Radio Astronomy. The information presented in papers is minimal—usually just enough to allow for some mathematical recreation, but not enough to understand the algorithms. This may have been the result of the culture of the research group at the time; Ryle may have maintained more of a war-time culture of secrecy than we find desirable as scientists today.9 It is also worth noting that, while women may have left the Mathematical Laboratory on marriage, this wasn’t true for members of the Radio Astronomy 8 See, for example, shelfmark V75-58, which lists programs used by the Department of Geology on Titan. 9 Donald Wilson, personal communication 22 April 2021.

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group. Indeed, programmers were in such high demand that they could work parttime and flexibly, many years ahead of this becoming widely available. Dr Waldram recalls working 10–15 h a week in the early 1960s, to accommodate her childcare needs.10

7 Conclusion The acknowledgements to papers provide one of the few ways of finding out about the work done in early scientific computing, as they often indicate that work was done using a computer, and who did that work. We can get tantalising glimpses of what that work was, and the people behind it. This could usefully be supplemented by more archival work within the University, once this is again possible. Acknowledgments My thanks go to Dr Elizabeth Waldram, one of the Cavendish’s computing pioneers, who has given her time very generously, to Dr Donald Wilson, who provided many valuable insights into this period, to Dr David Green, who helped me with my LATEX, to J. Nevins, who helped proof-read, and to Professor Malcolm Longair, who has supported this project.

References Abbate J (2012) Recoding gender: women’s changing participation in computing / Janet Abbate. History of computing, MIT Press, Cambridge, Mass. Ahmed H (2013) Cambridge Computing: The First 75 Years. Third Millennium Publishing, London, UK Andrews GE (2013, cited 2021 Apr 7) The Discovery of Ramanujan’s Lost Notebook: In Honor of the 125th Birthday of Ramanujan. URL www.personal.psu.edu/lxv1/alg/antheory/preprint/ andrews/295.pdf Blythe JH (1957) A new type of pencil beam aerial for radio astronomy. MNRAS117:644–659, DOI https://doi.org/10.1093/mnras/117.6.644 Of Cambridge Computer Laboratory U (1999, [cited 2021 Apr 7]) EDSAC 1 and after a compilation of personal reminiscences. URL https://www.cl.cam.ac.uk/events/EDSAC99/ reminiscences/ Cavendish Laboratory (2021, Cited 30 Sep 2021) Nobel Prizes. URL https://www.phy.cam.ac.uk/ history/nobel Christiansen WN, Högbom JA (1985) Radiotelescopes. Cambridge University Press Clifford, F E (1964) The formation of the iron peak elements. PhD thesis, University of Cambridge, Cambridge, UK Computer Laboratory UoC (1999) Archive photos. URL https://www.cl.cam.ac.uk/relics/archive_ photos.html Costain CH, Smith FG (1960) The radio telescope for 7.9 metres wavelength at the Mullard Observatory. MNRAS121:405–412, DOI https://doi.org/10.1093/mnras/121.4.405

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The Algebra Project, Feature Talk, and the History of Mathematics Madeline Muntersbjorn

Abstract Robert P. Moses (1935–2021) was an activist and educator who taught mathematics and studied philosophy. Bob Moses is renowned for his efforts to organize volunteers from around the country who came to Mississippi to help black Americans register to vote. Moses is less widely known as an education reformer, whose philosophy of mathematics shaped the Algebra Project, a curricular approach designed to teach algebra to adolescents. This paper considers what Moses calls “Feature Talk” as opposed to “People Talk.” People talk is any natural language. In contrast, feature talk is what results from the mathematization of natural languages when restrictions are placed on how language is used in problem-solving settings to make mathematical relations more tractable. Feature talk is an intermediary between ordinary languages, as commonly spoken, and formal notations, as written by mathematicians and scientists. Feature talk imposes rules that regiment ordinary language, as proposed by Quine. Moses’ concept of feature talk is useful for historians of mathematics, especially historians of algebra, who show us how past mathematicians regimented ordinary language in extraordinary ways before the development of algebraic symbolism. Feature talk highlights part of a powerful pedagogical strategy and provides insight into how mathematical practices are cultivated over time.

1 Algebra as a Civil Rights Issue Bob Moses was a legendary civil rights activist, famous for his efforts organizing the Freedom Summer Project and the Mississippi Freedom Democratic Party (MFDP) in 1964. Never one to put himself in the spotlight, Moses preferred to shine light on his students, whether they were in his Algebra Project classrooms or in the Freedom

M. Muntersbjorn () University of Toledo, Toledo, OH, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_16

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Schools set up along with voter registration drives in Mississippi. While less famous than the sit-ins, the Freedom Schools not only helped adults learn to read and register to vote but also taught local students everything from history to citizenship, poetry to French (Rothschild 1982). Folks who have heard of Moses, but not the Algebra Project, might presume Moses was a civil rights activist who developed an interest in education later in life. A brief biographical sketch shows Moses was always “all of the above:” mathematics teacher, philosopher of mathematics, and community organizer. Robert P. Moses was born in Harlem and graduated from Stuyvesant High School in 1952, a selective public school in Manhattan, known for excellence in science and math, that continues to offer tuition-free accelerated academics to city residents, as it has since the early twentieth century. Moses earned a bachelor’s degree from Hamilton College in 1956, where he majored in philosophy. Moses earned a master’s degree in philosophy from Harvard in 1957. From 1958 to 1960, Moses taught mathematics at Horace Mann, an elite private school in the Bronx. His studies at Harvard included mathematical logic, but he did not have the certification a job teaching mathematics in the public school system required. At this time, Moses’ attention was drawn to the Southern United States where he was to undertake the organizational efforts for which he is justly celebrated. Under the tutelage of Ella Baker, Moses became a field secretary for SNCC, the Student Nonviolent Coordinating Committee (pronounced “snick”). His initial forays into Mississippi in the early 1960s, to help blacks register to vote, were met with violent resistance but Moses persisted. In a 1961 letter from jail for disturbing the peace, Moses referred to Mississippi as “the middle of the iceberg” wherein white people in positions of power saw their state as a nation unto itself, not a member of a union wherein the right to vote is federally protected (SNCC Digital Gateway). Under an umbrella organization known as COFO, or the Council of Federated Organizations, Moses coordinated the Freedom Summer Project in 1964 wherein college students from around the country came to Mississippi to enlarge upon the work SNCC had begun. Violent resistance increased even as consensus among diverse groups was built around the idea that the United States Constitution guarantees “One Man, One Vote.” In the mid-1960s, Moses turned his attention away from SNCC and toward protesting the Vietnam War. In 1966, despite being 31 years old, Moses was drafted, prompting him to leave the country. Moses settled first in Canada and then, from 1969 to 1976, he taught math in Tanzania. Moses returned to the USA in 1976 and resumed his doctoral studies in the philosophy of mathematics at Harvard. Two events coincided in 1982 that led to the Algebra Project. Moses’ eldest daughter went to eighth grade at a public school that did not offer algebra to eighth graders; at the same time, Moses won a MacArthur Fellowship to honor his achievements in the Civil Rights era. This “genius award” gave him the freedom to devote himself to teaching middle school algebra at his daughter’s school (Moses 1995). Moses not only taught algebra at his kids’ school from 1982 to 1987 but also began to promote the idea that access to algebra was the key to mathematical literacy and, ultimately, full citizenship in the twenty-first century. The Algebra Project, Inc. was formally launched in 1990 and has been an active agent of education reform ever since,

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growing from a handful of students in one school in Cambridge, Massachusetts, to thousands of students in hundreds of schools across the USA. The argument Moses makes in support of the view that access to algebra is a civil rights issue is based on an analogy. Unregistered voters in the south in the early 1960s were said to be “disinterested” in exercising their right to vote as well as “illiterate” and thus not qualified to vote. When Moses speaks of those days, he recalls the motivation behind the Freedom Schools: Education is a basic American value, and it was fundamentally unfair to deny people access to education and to use their lack of education to deny them political access (Moses 2009, p. 377).

Community consensus was built around the idea of one person, one vote as disenfranchised residents of Mississippi learned to read, write, speak, and vote on their behalf. As Moses reminds us: . . . it was not radical in the 60s to do voter registration, but it was radical to do it in the Mississippi Delta among people who were said to be apathetic. Similarly, it is not radical today to teach math using games and enjoyable experiences. What is radical is to teach math to students at the bottom in a way that allows them to rise to the middle and top (Moses et al. 2009, p. 242).

For decades, Moses worked to promote the idea that students who currently score in the bottom quartile on standardized tests deserve an education that prepares them for high school, college, and beyond. The Algebra Project seeks to build consensus around the idea that, “if we can teach algebra to all public-school students then we should.” This is a radical proposal, not because it makes mathematics more fun, but because it seeks to make mathematics more accessible to a wider variety of students, especially those who score poorly on state-sponsored assessments. The objective of the Algebra Project is universal mathematical literacy. Moses suggests we focus on the bottom quartile for whom, evidently, current educational strategies are inadequate. Participants in Algebra Project initiatives are students most at risk of being tracked into non-college preparatory mathematics courses or dropping out of school altogether. What Silva wrote in the project’s early days is still true: The conviction of the Algebra Project is that all children can learn algebra. The Algebra Project challenges the assumptions and practices that deny access to algebra and virtually guarantee failure in higher mathematics and science as a result (Silva et al. 1990, p. 375). [emphasis in original]

The Algebra Project is built on the belief that there are potential scientists, engineers, and financiers in all quartiles, residing in all postal codes. Children who score in the bottom quartile are denied access to the kind of mathematics education they need to succeed in these fields at the college level. The Algebra Project directly challenges the “ability model” that restricts early access to algebra to those who are “gifted” or otherwise “inclined” on the grounds that it is possible to “raise the floor without capping the ceiling,” to invoke a useful building metaphor for student achievement (Jetter 1993, p. 28).

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In Moses and Cobb, we observe that when folks are asked, “should your child be taught algebra?” the universal reply is affirmative but when asked, “should all children be taught algebra?” answers varied: All parents thought their child should do algebra, but not all parents thought that every child should do algebra (Moses and Cobb 2001, p. 98).

Moses continues: The Algebra Project seeks to develop a demand for math literacy in those most affected by its absence – the young people themselves. This approach, which places a high value on the importance of peer culture, is an outgrowth of experience in the civil rights movement of the 1960s, as well as the emergence of Algebra Project graduates into a group with their own perspectives and initiatives (Moses et al. 2009, p. 242).

For example, The Young People’s Project (TYPP) grew out of an instantiation of the Algebra Project in Baltimore, Maryland. TYPP offers training to young math literacy workers who teach math to elementary school students at after-school programs and summer camps. Since its inception in 1996, TYPP has spread to ten locations across the USA and has trained thousands of teens and college students to be math literacy workers to assist over 10,000 primary school students with the skills they need to do well in math in secondary school (The Young People’s Project 2021). The Algebra Project is inspired by the political philosophy of American activist Ella Baker (1903–1986) who taught that sustained social change cannot be established from the top down but must be organized from the bottom up through initiatives that make the beneficiaries of social change the architects and enablers of those changes. In her lecture “The Black Woman in the Civil Rights Struggle,” Baker famously said: In order for us as poor and oppressed people to become part of a society that is meaningful, the system under which we now exist has to be radically changed. This means that we are going to have to learn to think in radical terms. I use the term radical in its original meaning – getting down to and understanding the root cause. It means facing a system that does not lend itself to your needs and devising means by which you change that system (Grant 1998, p. 230).

The lecture from which this quote is taken was delivered at the Institute for the Black World meeting in Atlanta in 1969, the first year Stuyvesant High School in NYC let female students sit their entrance exams. The system that does not lend itself to our needs, that the Algebra Project seeks to change, is a stratified state-bystate patchwork system of US public schools that perpetuate a caste system wherein the best jobs are only career options for the mathematically literate (Moses 2021). Too many poor students are so ill-served by their educations, they can graduate from high school yet struggle to pass remedial mathematics courses at the postsecondary level, significantly, without the prospect of earning college credit. In a short video for the Children’s Defense Fund, Moses (2013) cites three public institutions, one each in California, Maryland, and Florida, where as many as 70%– 80% of all first-year students take developmental mathematics for which they earn

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no college credit because, according to their entrance test scores, they are not ready to take college-level mathematics: . . . these are kids who, in terms of the high schools, are achieving. They are graduating. They have scored whatever the state says they need to score, right, on the state tests. They have scored whatever SAT or ACT says you need to score to get into college, but they get there, and they can’t do it. The Algebra Project has been working on a piece of that problem. (See also Chen and Simone 2016).

What do Algebra Project students do? Briefly, Algebra Project students take more math than the minimum required to graduate. While the project originally started with a focus on preparing middle school students to do elementary algebra, the project grew such that to be a part of the project today may entail a more substantial commitment on the part of secondary students. Moses writes, With funding from the National Science Foundation, the Algebra Project works within individual classrooms, where we get students to agree to do ninety minutes of math with us every day for four years of high school. They work to catch up their deficits and jump through the country’s three hoops: the state hoop, the ACT/SAT hoop, and the university hoop (Moses 2009, p. 379).

Note that the Algebra Project is not about dismantling any hoops. It is not that we must stop assessing students’ mathematical skills; rather, we must increase access to mathematical skills. The focus is on getting as many students as possible through as many hoops as possible so that when they graduate from high school, their ambitions will not be curtailed by their lack of mathematical literacy. The vote and algebra are both tools for progressive social change. Because the Algebra Project is a grassroots project, it involves participation not just from teachers, administrators, and education experts, but from parents and students who must invest time and effort. Sometimes this effort involves catching up on arithmetic skills over the summer; other times it involves attending Saturday academies where parents can get help with their math skills also. Conjoining these interested parties around the shared goal of mathematical literacy makes other positive change possible. As Wahman writes, . . . while algebra and the vote may not appear to be drastic or dramatic enough factors to fundamentally alter the structures of oppression, they are as radical as sit-ins and marches – even more so in that they are likely to achieve long-standing results. . . . Moses speaks of them as tools that enable people to use rather than be used by the socioeconomic system and to become persons with the power to continually transform their own communities as new needs and problems arise (Wahman 2009, p. 8).

In the same way that SNCC was part of COFO, today the Algebra Project is part of a larger network, We the People, which brings together organizations such as the Young People’s Project. We the People was started to promote consensus among diverse groups around the idea that access to a quality mathematics education is a constitutionally guaranteed right insofar as math literacy is necessary for full citizenship in the Information Age, just as reading literacy was necessary in the Industrial Age.

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In his 1993 lecture, “Algebra, The New Civil Right,” Moses addressed assembled mathematicians, mathematics educators, and historians of mathematics: We are not just talking about professionals and jobs, we really are talking about democracy and citizenship. What is the content of the democracy going to be in this country? These are heavy issues to lay on a group of mathematicians (Moses 1995, p. 58).

On Moses’ view, we need to build consensus around the idea that, if we can teach algebra to all adolescents then we should. Admittedly, this idea is more complex than, one person, one vote. Significantly, communities do not have to buy any specific textbook or purchase any software packages to participate in the Algebra Project. Top-down curricular reform is not the goal. Deciding which course materials to use should involve extensive discussion at the local level. As Moses notes, One of the central things we’re saying is that the ongoing struggle for citizenship and equality for minority people is now linked to an issue of math and science literacy. This idea is the background of everything we do in the Algebra Project. . . . It’s important to make it clear that even the development of some sterling new curriculum – a real breakthrough – would not make us happy if it did not deeply and seriously address the issue of access to literacy for everyone (Moses 1994, p. 107).

In the service of this goal, Moses and colleagues have given considerable thought to how people learn mathematics, especially algebra, and have developed a curricular process that can be implemented in a wide variety of mathematics classrooms.

2 The Algebra Project as a Curricular Process In an early account of the Algebra Project, Moses (Moses et al. 1989, p. 433) presents a five-step curricular process based on their classroom experiences: 1. Physical event 2. Picture or model of this event 3. Intuitive (idiomatic) language description of this event 4. A description of this event in regimented English 5. Symbolic representation of the event

The process starts from shared experiences of physical events. Moses proposed that many students learn mathematics best by approaching the subject along a “complex, concrete” path as opposed to a “simple, abstract” path: That is, if you think of simple as opposed to complex, and abstract as opposed to concrete, we are looking to develop curriculum which finds the right level of complexity in concrete events. We are not building the curriculum around concepts which are simple and abstract. We are trying to build a curriculum around events which are concrete and complex (Moses 1995, p. 65).

The challenge is to manage the complexity by providing rich experiences that can be regimented into the artificial languages of mathematics. Moses is not saying that

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one could not teach mathematics by leading students along a simple, abstract, and axiomatic path. In Moses (1995, p. 66), he recounts teaching secondary students along this axis at Horace Mann the year after Sputnik launched and set-theoretic approaches to mathematics education became popular. Instead, the Algebra Project seeks to cultivate skills in those students who need to test the mathematics they are learning against their own experiences. Shared events dispel the air of mystery around the mathematics classrooms where teachers have all the answers, and students are empty vessels to be filled with so many detached facts. Instead, student voices are centered. As Moses said in a podcast, So the math class is traditionally a class where the students themselves are disempowered and they sit and listen to an expert, who is the teacher or someone. And so what we set about in the Algebra Project was developing a curricular process to flip that around. . . (Moses 2020)

For example, to help students learn that mathematicians need to know how to find answers to not only “How many?” questions but also “Which way?” questions, Moses would organize a field trip on public transport (step 1). After the experience, students are encouraged to draw their own pictures of the journey (step 2) and describe these drawings in their own words to one another (step 3). In recounting their journey later, students learn not only to count stops between stations but also to note what direction the stops were in, inbound or outbound. By making their idiomatic descriptions of events more precise and uniform (step 4), students are better able to understand a formal number line, with positive and negative integers (step 5). On Moses’ view, students learn how mathematical practices come into being by actively participating in this process. The group dynamic is critical to Algebra Project pedagogy. Algebra Project secondary school students sign up to be classmates for years at a time for peer support. Together students learn that mathematics is a performative group activity they can do as they practice presenting their thoughts to one another. Because formal symbols come into play so late in the learning process, traditional teachers unfamiliar with Algebra Project pedagogy may not recognize these linguistic activities as mathematics at first. As Payne (2001, p. 249) notes, Moses “believes that it takes a year of exposure before a teacher can make a truly informed decision about whether to buy in.” The Algebra Project is not a “quick fix” as lasting mathematical literacy takes considerable patience. According to Moses, the philosophy behind Algebra Project pedagogy combines two different philosophical insights. The first insight is the cycle of experiential learning, as inspired by Dewey, Piaget, and Lewin. Experiential learning theory is grounded in the countless cyclical experiences in which people try something (experience), then think about what they did (reflection), then make improvements (abstract conceptualization), then practice their improvements (application) (Moses et al. 2009, p. 246).

The second philosophical insight is how, precisely, one should think about “abstract conceptualization” as a stage in the experiential cycle. Language-based exercises

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encourage students to speak about their mathematical experiences, first in their native language, then in formal and regimented language, and finally in mathematical symbols. Step 4 in this process, a description of this event in regimented English, is what came to be known as feature talk in subsequent presentations of the project’s curricular process. When we compare the earlier presentation of the five-step curricular process with versions that came later, we see how the five-step curricular process became more economical. Consider this example of the five-step process: When Moses began to work with teachers, he explained how math could be taught as a ‘fivestep curricular process’ in which students should (1) engage in a physical experience, which they could then (2) represent in their own words and pictures, then in structured language, including (3) everyday language (‘people talk’), (4) ‘feature talk,’ and (5) conventional mathematical notation (Davis et al. 2007, p. 71).

Moses notes that feature talk is a conceptual language that no one speaks ordinarily; it is an example of what Quine refers to as regimented languages. Moses and Cobb observe: But this is Quine’s insight: the symbolic representations in all of mathematics and science are representations of a conceptual language that no people on earth speak as a natural tongue. We came, in time, to call [it] “feature talk,” an example of what Quine calls “regimented language” (Moses and Cobb 2001, p. 201). [emphasis in original]

In people talk, students may say, “Jon is taller than Amy.” But in feature talk, students learn to extract content from the comparative relation “taller than” and say, “the height of Jon is greater than the height of Amy,” which can more easily be translated into algebraic symbols. Moses points out that his studies in both philosophy and mathematics gave him insight into how mathematics is cultivated over time. Moses explains: I did my work in philosophy of math and I don’t think I would have come across thinking like this if I had done my work just in math. And at Harvard in the 1950s, Willard Van Orman Quine [said] . . . that elementary arithmetic, elementary set theory, and elementary logic get off the ground by what he called the regimentation of ordinary speech. And that this regimented language was really nobody’s spoken language. It was just a form of artificial speak, which was an intermediary between the spoken languages and the actual symbols that are found in the math and science textbooks. . . . Now the feature talk, the height of Jon is greater than the height of Amy, can go right over into symbols. All you need is a symbol for height, a symbol for greater than, a symbol for your names, right. . . . But the crux of the matter is the move from what Quine would call ordinary language to regimented language (Moses 2020).

According to Ryg, Moses’ “Algebraic Pragmatism” is American pragmatism in the tradition of Dewey and Quine, though Ryg is careful to point out, The philosophy of Bob Moses is more heavily indebted to Ella Baker than it is to John Dewey. The philosophy of the Algebra Project has been more profoundly influenced by Bob Moses than it has been by Quine (Ryg 2014, p. 291).

Even so, it is worth considering Quine’s influence further, especially as we seek to articulate a more precise account of the “crux of the matter,” namely, feature talk.

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W.V.O. Quine (1908–2000) was influenced by the logical positivism of the Vienna Circle and their careful analyses of the languages of mathematics and science. With respect to his philosophy of mathematics, Quine urged that some, but not necessarily all, mathematical things exist but only in relation to the linguistic structures we rely upon to make them manifest. On Quine’s view, how we choose to talk about specific mathematical abstracta will reflect our ontological commitments so we must learn to speak with the utmost clarity if we want to avoid confusion. Further, there is value in cultivating diverse ways of talking about, and thereby committing to, different mathematical ontologies. According to Gregory (2020), while Quine is known as an empirical, naturalist, and pragmatic philosopher, Quine can also be read as a deflationary structuralist about mathematical objects. Gregory observes, Regimentation and a critical philosophical eye show us that the domain of objects we are interested in depends somewhat on the questions we are asking (Gregory 2020, p. 111).

Gregory calls our attention to the introduction to section 12 of Quine’s Pursuit of Truth, “Indifference of Ontology” wherein Quine writes, True sentences, observational and theoretical, are the alpha and omega of the scientific. They are related by structure, and objects figure as mere nodes of the structure. (Quine 1992, p. 31 as quoted by Gregory 2020, p. 110)

Mathematical languages are used to transform scientific sentences about observations into theoretical sentences about the logical relations between scientific sentences. These sentences may refer to objects as things in the world, but those specific particulars may be of less interest to us as mathematicians than the more general classes they belong to and the structural relations into which they enter. Quine’s complex answer to the perennial question, “do mathematical entities exist?” has been widely discussed. Less attention has been paid to Quine’s answer to the related question, “how does mathematics come into being?” Quine’s answer is featured in his essay, “Success and Limits of Mathematization” (Quine 1981, pp. 148–155). On Quine’s view, the origins of mathematics are found in the solutions to everyday problems. These solutions often require that ordinary language be regimented or rendered more rigid in its terms and more formulaic in its rules: I already urged that set theory, arithmetic, and symbolic logic are all of them products of the straightforward mathematization of ordinary interpreted discourse – mathematization in situ. Set-theoretic laws come of regimenting the ways of reasoning about classes or properties, ways of reasoning that already prevailed more or less tacitly in natural science and ordinary discourse (Quine 1981, p. 150).

Ryg notes that, in Quine’s view, mathematization “is continuous with the growth of precision, and it blossoms at last into algorithms and proof procedures” (Ryg 2014, p. 23). In a posthumously published essay, Moses (2021) argued for a “federal civil rights bill for education, akin to the Civil Rights Act of 1957.” The Constitution of the USA must guarantee all citizens an education that allows them to become mathematically literate in an era wherein “algorithms and proof procedures” are

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ubiquitous. When it comes to mathematics education, federal resources have been invested in assessment and accountability. What must happen so that more resources are invested in access and opportunity? Both We the People and anyone who sees mathematical literacy as necessary for twenty-first-century citizenry face a monumental task. In his lecture at the same conference that Moses spoke at in 1993, historian Victor Katz reminded the audience: In fact, throughout most of recorded history, algebra was only taught to the “few,” not the “many.” So the concept of “algebra for all” is a very recent one that will take a lot of work to make into reality (Katz 1995, p. 17).

3 Feature Talk and the History of Mathematics Feature talk, as promoted by the Algebra Project, highlights part of a powerful pedagogical strategy and provides insight into how mathematical practices are cultivated. The possibility that the history of algebra yields valuable insights into the teaching of algebra has been explored by Katz (2001) and Katz (2007). For example, Katz avers, There are many lessons that history provides for the teaching of algebra, both at the school and university level. First, the subject must have a clear focus, the solving of equations (Katz 2001, p. 358).

One question asked of Algebra Project advocates is, how can an algebra class focus on solving equations and follow the five-step curricular process? Worksheets covered in equations may not be the best way to focus on solving equations. Especially at the introductory level, students struggle with preliminary equations such as .a − b = a + (−b). The Algebra Project replies by posing this equation as a why-question when educating communities of parents, teachers, administrators, and others, about Algebra Project pedagogy. Moses observes, Posing such a question helps project members learn how people actually think about some of the mathematics that our children must learn, and is simultaneously a way to engage stakeholders in the subject (Moses et al. 2009, p. 239).

By doing this mathematical activity themselves, as a conversational then performative practice, stakeholders experience Algebra Project pedagogy firsthand. Many are surprised to hear professional mathematicians disagree as to why, precisely, .a − b = a + (−b). These experiences teach communities how to build consensus around shared mathematical meanings as well as educational goals. Katz continues the list of lessons to be drawn from the history of algebra for those who teach algebra: Second, symbolism should be introduced gradually, as it is needed. Students will eventually discover for themselves the advantages of symbols over words in the context of problem solving, particularly when the verbal procedures become very complicated (Katz 2001, p. 359).

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This second lesson from Katz’s list underscores the importance of feature talk as an intermediary between ordinary language and formal symbolism. For it is feature talk where verbal statements become complex as precision frequently requires making implicit features explicit, resulting in greater verbosity. This increase in statement length is exactly what good notation abbreviates. It makes no sense to present symbols to students before they are needed to translate long feature talk sentences into more economical equations. The gap from people talk to feature talk must be traversed first before subsequent transition to symbolism will make sense to most students. The next lesson from Katz is, Third, geometrical ideas are always part of algebra and should be used whenever possible, both in the development of algebraic procedures and in developing the notions of analytic geometry (Katz 2001, p. 359).

This lesson reminds us that both algorithms and coordinate systems have geometrical roots. This historical lesson gives credence to the Algebra Project premise that, no matter what concept needs to be taught, one can devise a concrete experience with the right amount of complexity to make that concept accessible to students. Of course, this work of the Algebra Project is never done, and new ideas for classroom activities that exhibit abstract mathematical relations are always in development. The last lesson from Katz is, . . . algebra teaching needs a constant emphasis on the techniques of problem solving, even if many of the problems are artificial (Katz 2001, p. 359).

This lesson may seem to be where the history of algebra and Algebra Project pedagogy diverge, insofar as the latter prefers “real” problems. However, if the fundamental technique of problem-solving is sharing our thoughts with others while striving to be more rigid in our terminology, then the divergence is superficial. One Algebra Project activity, the Winding Game, teaches students about dividends and remainders using the Chinese zodiac. This game is more artificial, in a sense, than the activity where students learn about ratios by mixing different parts lemon juice, water, and sugar to make lemonade. The goal in both cases is to equip students with mathematical experiences that give meaning to the equations that, after regimentation, represent the relevant magnitudes and relations. Concrete experiences may need to be part of the pedagogical process: To the extent that learning theories are correct about the need for concrete experiences, then the generalizations inherent in drawing and reading maps, building structures, experiencing thrown and falling objects, categorizing characteristics of things, folding paper, and sketching and painting are important components of mathematics education (Katz 2007, p. 199).

These activities may resemble activities outside the math classroom, like cooking, taking public transport, or simply “making do” with what one has on hand. But these experiences may also be “artificial” problems that do not resemble activities outside the math classroom, such as the Winding Game.

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Bill Crombie, Director of Professional Learning at the Algebra Project, Inc., described his first reaction to seeing the five-step curricular process in action as gathered stakeholders discussed “making do” stories: It felt initially like an English Language Arts exercise. You wrote the stories and compared them, discussed the similarities and differences. And then it was like pulling a rabbit out of a hat; the features that make a make do story are very much the same features of equivalence. It was actually a discussion about mathematics that the middle school teachers were engaged in, about the difference between equality and equivalence (McCallum and Umland 2021).

While it may seem at first like the symbolic equations students learn at the end of the five-step process result from prestidigitation, of course the rabbit was in the hat all along. The mathematical features the teachers want to focus on are built into specific activities from the outset, making the (admittedly daunting) task of regimenting the speech of so many idiosyncratic individuals easier. Thus far we have considered Algebra Project pedagogy in light of the history of algebra. In conclusion, I consider what lessons historians and philosophers of mathematics can learn from the Algebra Project. The importance of intermediate languages, between ordinary discourse and formal symbolism, has not escaped the notice of astute historians of mathematics. What Moses calls “feature talk,” Chemla (2006) calls “artificial languages” in her paper, “Artificial Languages in the Mathematics of Ancient China.” In her introduction Chemla writes, The example presented suggests how artificial languages might result from specific elaborations within, and on the basis of, the usual written or oral language (Chemla 2006, p. 31).

Recall how Moses and Quine portray how mathematics gets started. First, experiences are translated into original sentences in ordinary languages. Then, artificial constraints are imposed upon these sentences so they can function as more precise tools when talking about shared experiences in groups, especially when we seek to abstract the universal and isolate the quantifiable features of these experiences. This regimentation of the ordinary language precedes formal symbolization in the classroom and in the history of mathematics. Chemla concludes by generalizing from her thirteenth-century case study: In conclusion, apparently for the sake of mathematical practice, an artificial language, or a sublanguage, was designed within the usual written language. However, its artificial features are not signaled by the use of special symbols. They appear only if we observe the syntax of the expressions thus formed . . . . Although this conclusion was reached on the basis of a particular example – that of a book written in 13th century China – the phenomenon brought to light seems to me to have a much greater validity: mathematics is always, and has always been, carried out with artificial languages especially designed for carrying out mathematical activity. All these languages must be studied if we are to understand the full implication of such phenomena (Chemla 2006, p. 54).

While the “full implication” of such phenomena may not fit into a single essay, there are some preliminary insights we can draw. The transition from people talk to feature talk is as important as the transition from artificial languages to symbolic formulas. While historians should pay close attention to increased reliance

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on symbolism in the history of algebra, we also need to better understand those mathematical practices wherein ordinary languages are regimented in extraordinary ways in problem-solving settings. This is not to say that the history of algebra, as a whole, can be cleaved into discrete developmental stages. Rather, Algebra Project pedagogy offers insight into the dynamic dimensions of the growth of mathematics as a group activity. Individual experiences are not translated directly into shared symbols but indirectly via intermediate languages of increasing precision. Consider the perennial question, is mathematics created or discovered? This question poses a false dilemma insofar as it does not exhaust all possible origin stories for mathematics. If mathematics is purely a social construction, a formal game of signs we make up as we go along, it is hard to explain the success of mathematics in modelling the natural world. Quine’s indispensability argument for the reality of at least some mathematical abstracta hinges precisely on this difficulty. If mathematics is eternal, necessary, and out there waiting to be discovered, it is hard to explain how we can know anything about it and, at the same time, explain why we cannot see the whole at once. That is, there are two questions surrounding our epistemological access to timeless and necessary truths facing those who think mathematics is discovered. First, how can we know anything about this eternal necessary realm as finite contingent beings? Second, once we grant that contact is possible somehow, why does it take so long to expand the scope of our mathematical purview? Instead of being able to deduce the whole from a few well-chosen axioms, we must be content to learn mathematics through piecemeal, painstaking, and sometimes disappointing efforts. Both Moses (1995) and Moses (2012) reference the Frege-Hilbert Controversy surrounding the relations between truth and consistency. In Moses’ retelling in Moses (2012), Frege wrote to Hilbert that, “the axioms of geometry are consistent because they are true” while Hilbert wrote back to say, “the axioms of geometry are true because they are consistent.” In the decades that followed, “. . . the battle between Hilbert and Frege really was won by Hilbert.” As a result Moses concludes, . . . what we got in the 20th Century was the elaboration of Hilbert’s idea in the teaching of math about consistency, . . . so you have certain axioms for school mathematics, and they hang together (Moses 2012).

While Hilbert’s formal approach may work for some students, for too many students it does not work. As Moses explains: The purpose of the five steps is to avert student frustration in “the game of signs,” or the misapprehension that mathematics is the manipulation of a collection of mysterious symbols and signs (Moses et al. 1989, p. 433).

Even if not all mathematics students need to be able to trace symbolic procedures back to lived experiences, all students benefit from experiences that help them better understand how mathematics comes into being as abstract content conveyed by artificial but universal languages anyone can learn to speak, no matter what their native language is.

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Moses sides with Frege’s realism over Hilbert’s formalism for pedagogical reasons. But he also acknowledges that Quine’s realism is very different from Frege’s. As Moses notes, Quine doesn’t believe truth as Frege saw it. He has his own version of truth, but he is saying that meaning is attached to language and discourse and so forth (Moses 1995, p. 66).

Perhaps mathematics is neither created nor discovered but cultivated through linguistic processes analogous to cultivation in the agrarian arts. On my view, “wild” mathematical relations are “domesticated” by formal systems of signification that are cultivated in response to practical problems. The regimentation of informal discourse and its translation into more formal feature talk is key to understanding where mathematics comes from. Yes, the regimentation process is artificial. But, like artificial selection, the mathematics that results from our regimentation efforts are not arbitrary. Mathematics comes into being slowly and reflects interactions between our problem-solving ambitions and natural-world constraints. When thinking about so-called pure mathematics, and whether it is an exception or challenge to this empirical model, it is important to recall the cyclical nature of experiential learning and note there is not one final form that all artificial feature talks must take. Talking about mathematics in precise terms creates new shared experiences that can themselves be discussed, first in ordinary languages, then in more regimented languages, and ultimately in new short-hand symbols and notations that express the new regimented languages more efficiently and explicitly. In this way, what we think of as pure mathematics, detached from lived experiences and real-world applications, may be understood as what results from regimenting how we talk about mathematical experiences to make what is shared and universal in those experiences more salient and manipulable. From this dynamic perspective, all mathematics is applied; what we call pure mathematics is mathematics applied to itself (cf. Steiner 2008, p. 360).

References Chemla, K (2006) Artificial Languages in the Mathematics of Ancient China. J Indian Philos 34: 31–56 Chen X, Simone, S (2016) Remedial Coursetaking at U.S. Public 2- and 4-Year Institutions: Scope, Experiences, and Outcomes. US DOE https://files.eric.ed.gov/fulltext/ED568682.pdf Cited 14 Nov 2021 Davis, F et al (2007) Transactions of Mathematical Knowledge in the Algebra Project. In: Nasir, NS, Cobb, P (eds) Improving Access to Mathematics: Diversity and Equity in the Classroom. Teachers College, New York: p 69–88 Grant, J (1998) Ella Baker: Freedom Bound. Wiley, New York Gregory, PA (2020) Quine’s Deflationary Structuralism. In: Janssen-Lauret, L (ed) Quine, Structure, and Ontology. Oxford University, Oxford: p 96–116 Jetter, A (1993) Mississippi Learning. NYT Magazine. February 21, Section 6: 28 Katz, V (1995) The Development of Algebra and Algebra Education. In: Lacampagne, CB et alia (eds) The Algebra Initiative Colloquium. vol 1. US DOE: 15–32

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Katz, V (2001) Using the History of Algebra in Teaching Algebra. In: Chick, H et al (eds) The Future of the Teaching and Learning of Algebra. vol 2. UMP, Melbourne, p 353–359 Katz, V (2007) Stages in the History of Algebra with Implications for Teaching. Educ Stud Math 66: 185–201 McCallum, W, Umland, K (2021) Remembering Bob Moses. Illustrative Mathematics. 3 Nov 2021. https://illustrativemathematics.blog/2021/11/03/remembering-bob-moses/ Cited 14 Nov 2021 Moses, RP (1994) Remarks on the Struggle for Citizenship and Math/Science Literacy. J Math Behav 13: 107–111 Moses, RP (1995) Algebra, The New Civil Right. In: Lacampagne, CB., et al (eds)The Algebra Initiative Colloquium. vol 2. US DOE: 53–68 Moses, RP (2009) An Earned Insurgency: Quality Education as a Constitutional Right. Harvard Educ Rev 79 (2): 370–381 Moses, RP (2012) The Algebra Project, Theory and Practice. Video interview for NYU Steinhardt: https://www.youtube.com/watch?v=cnyVIXmUq1s. Cited 14 Nov 2021 Moses, RP (2013) Bob Moses: The Nation’s Dirty Secret. Video interview for Children’s Defense Fund: https://www.youtube.com/watch?v=A4HADIb_XQA Cited 14 Nov 2021 Moses, RP (2020) The Algebra Project: Bob Moses on math literacy as a civil right – Part 1. Interview with Moscow, J, Halpern-Laff, A: https://ethicalschools.org/2020/02/the-algebraproject-bob-moses-on-math-literacy-as-a-civil-right/ Cited 14 Nov 2021 Moses, RP (2021) Returning to ‘Normal’ in Education is Not Good Enough. The Imprint Youth and Family News. 8/24/21: https://imprintnews.org/opinion/returning-to-normal-in-educationis-not-good-enough Cited 14 Nov 2021 Moses, RP, Cobb, CE (2001) Radical Equations. Beacon: Boston Moses, RP et al (1989) The Algebra Project: Organizing in the Spirit of Ella. Harvard Educ Rev 59 (4): 423–443 Moses, RP et al (2009) Culturally Responsive Mathematics Education in the Algebra Project. In: Greer, B et al (eds) Culturally Responsive Mathematics Education. Routledge, New York, p 239–256 Payne, CM (2001) So Much Reform, So Little Change: Building-Level Obstacles to Urban School Reform. In: Joseph, L (ed) Education Policy for the 21st Century: Challenges and Opportunities in Standards-Based Reform. University of Chicago, Chicago, p 239–278 Quine, WVO (1981) Theories and Things. Harvard UP, Cambridge Quine, WVO (1992) Pursuit of Truth. Rev Ed. Harvard UP, Cambridge Rothschild, MA (1982) The Volunteers and the Freedom Schools: Education for Social Change in Mississippi. Hist Educ Quart 22 (4): 401–420 Ryg, M (2014) Radical Equations as American Philosophy: Dewey’s Experience, Quine in a Straight Jacket, and Algebra from Moses. Cont Pragmatism 11 (2): 19–32 Silva, CM et al (1990) The Algebra Project: Making Middle School Mathematics Count. J Negro Educ 59 (3): 375–391 SNCC Digital Gateway. https://snccdigital.org/events/sncc-leaves-mccomb/ Cited 14 Nov 2021 Steiner, M (2008) Teaching Elementary Arithmetic through Applications. In: Curren, R (ed) A Companion to the Philosophy of Education. Wiley, New York, p 354–364 Wahman, JT (2009) ‘Fleshing Out Consensus’: Radical Pragmatism, Civil Rights, and the Algebra Project. Educ and Culture 25 (1): 7–16 The Young People’s Project. www.typp.org Cited 14 Nov 2021

Corrupt Land Inspectors: Solving Equations with Picture-Language in Ancient Mesopotamia—A Dialogue Gavin Hitchcock

Abstract This paper incorporates a scripted one-act play involving two actors, which can be used in the classroom. The dialogue aims to be an entertaining piece of mathematical theatre displaying ancient Mesopotamian cultural perceptions of mathematics, while illustrating some fundamental aspects of mathematical practice. It also aims to demonstrate the power of theatre (and history of mathematics) to motivate learners in algebra. The play takes place in a Mesopotamian scribal school, nearly four thousand years ago. The original verbal approach, sand-tray pictures and cut-and-paste geometrical methods are presented as authentically as possible, and a transcription is given in modern mathematical notation of the procedures, which we would think of as solving quadratic equations. The Head Scribe uses as motivation his student’s concern over the way tax collectors are defrauding the poorest farmers by walking the perimeter to arrive at the tax bracket. Two kinds of problems are discussed, including ‘An unknown square is the same as a given rectangle on the unknown square-side and a given area’. Different methods are shown for picturing and solving such problems. The Head Scribe forecasts a great future for his student and celebrates the scribes’ role in seeking justice, by reading an old praise-poem.

1 Introduction and Programme Notes We present the script of a dialogue between a Head Scribe and a student scribe in an ancient Mesopotamian scribal school. This short play is suitable for use in the classroom, using some slides or flipcharts to illustrate the mathematics. During the course of the dialogue, a transcription is given in modern mathematical notation of the procedures carried out by the scribes. Transcripts are also provided of the diagrams the scribes are drawing in sand-trays. The dialogue combines three aims. Firstly, it may be taken as an entertaining piece of mathematical theatre showing

G. Hitchcock () Independent Scholar, London, UK © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_17

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mathematics as a creative human activity. Early discoveries related to the solving of quadratic equations are portrayed so as to present mathematics not only as arising out of practical needs but also as being pursued and developed for sheer love of the subject. Secondly, the dialogue is a dramatic evocation of ancient Mesopotamian mathematical practice and cultural perceptions of mathematics, drawing from primary sources and scholarship in ancient Babylonian mathematics and attempting an unusual authenticity in treating their numeral system and their algorithms for solving equations. Thirdly, the dialogue is presented as an illustration of the power of theatre (and history of mathematics) to motivate learners in algebra. For the way these ancient scribes visualise their methods, using concrete pictures and a sort of cut-and-paste geometry to arrive at algorithms, can be inspirational in teaching today. The dialogue aims to show how their approach to problem-solving went far beyond merely following algorithmic routines. Our scribes will combine a passion for real-world solutions with a delight in mathematical form. The student scribe, Ea-shar-ili, is deeply concerned about the way tax collectors are defrauding the poorest farmers by using, as tax-bracket standard, the length of fence around a field, this being easier to estimate than the area. The Head Scribe, Kuningal, has previously used this to motivate a lesson on finding the length and width of a rectangular field, for any given semi-perimeter and area. Now, in a follow-up lesson, they review this form of problem and go on to another form, ‘An unknown square is the same as a given rectangle on the unknown square-side and a given area’, exemplified by a problem. Ea-shar-ili asks to see a picture demonstration first, then the procedure is deduced, and the solution to the problem is found. Ku-ningal demonstrates another way of picturing and solving such problems. At the close, Kuningal commends Ea-shar-ili for grasping the meaning of the mathematics and for his concern for justice. Over four thousand years ago, this geometrical picture-language was used to pose and solve a variety of problems in the context and language of land surveying—problems which we would see as solving quadratic equations. Words and mathematics were written in cuneiform, wedge-shaped script, using a reed stylus to imprint wedges on moist clay tablets. Many of these tablets were recycled in the scribal schools by soaking in bitumen-lined bins or were used in brickmaking. But the most important tablets were baked hard and preserved. Thousands of such mathematical tablets have been discovered since the 1840s, and the task of digging, sorting, deciphering and analysing goes on. In recent decades, scholars have aimed to interpret the contents of mathematical tablets in the context of the cultural, social and material circumstances in which they were written. For example, the association between scribal mathematics and concern for justice is implied in the praise-poem with which the play concludes; see also the last two footnotes with references. Relating the mathematics to the non-mathematical tablets unearthed in similar locations in the same periods involves archaeologists, cultural historians and cuneiform experts, as well as historians of mathematics. Few diagrams have been preserved on clay tablets. But, by analysing the language in which methods are described, scholars can make informed guesses at the sand-tray pictures the scribes used to arrive at and teach numerical procedures.

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Fig. 1 Babylonian sexagesimal numerals

14; 30, 15 = 14 +

15 30 + 60 602

14, 30; 15 = (14 × 60) + 30 +

1 15 = 870 60 4

Fig. 2 Babylonian tablet Plimpton 322. (Source: George Arthur Plimpton Collection, Rare Book and Manuscript Library, Columbia University, courtesy of Wikimedia Commons)

They used a place-value numeral system with base sixty. To them we owe our base60 time and angle measurement. With just two wedge-shaped cyphers, they could write any number, as big as required. Remarkably, the place-value was extended to fractional parts, so they could and did achieve incredible levels of accuracy, far beyond practical needs. In the play, the base 60 numbers are expressed using modern numerals, with a semicolon for the sexagesimal place, which the ancient Babylonians did not indicate. Here is a number arising in the play, expressed in cuneiform notation, with vertical wedges for units and horizontal wedges for tens (Fig. 1). It can be helpful for us to think: 14 h, 30 min and 15 s. The number represented depends on where we guess the sexagesimal point is. It could be just over 14 and a half (as if counting hours), but in the context of land measurement it’s taken to be 870 and a quarter (as if counting minutes). Most of the unearthed mathematical tablets are administrative records, but some were once mathematical tables, student texts and workbooks in the scribal schools for educating the elite, the priests and the bureaucrats (Fig. 2). This is inferred from external evidence of the context in which the tablets are found, but also from the internal evidence of the contents of the tablet. The numbers involved and the extraordinary accuracy demanded are appropriate to stylised problems and

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challenging exercises. This point emerges in the dialogue. The most important tablets escaped recycling. Many were broken and have been pieced together by scholars from fragments. The adventure of interpretation still faces mysteries. But it is clear that here, in the schools of the Old Babylonian period, 2000 to 1600 BCE, the scribal teachers turned mathematics, originally invented for practical purposes, into an art and a subject in its own right.

2 The Play: Solving Equations with Picture-Language Curtain Rises SCENE: A Mesopotamian Scribal School; the Head Scribe KU-NINGAL sits in his office, and the student scribe EA-SHAR-ILI enters for another lesson. There is a sand tray for rough diagrams, and a number of clay tablets on shelves, both hard-baked ones covered with wedge-shaped cuneiform marks, and also newly-made, blank, moist ones for imprinting with wedge-shaped styluses. There is a bitumen-lined bin for recycling the less important tablets that are not for baking. EA-SHAR-ILI: [rushing in out of breath] Good morning, Mr Ku-ningal, Sir! KU-NINGAL: Good morning, Ea-shar-ili! You are late – have you been following the land-inspectors and tax collectors around again? EA-SHAR-ILI: Well, umm, yes, Sir. I want to find out if they are really cheating people by charging them too much tax. And I want to see if they know any mathematics. They did not like me interrupting them. They really believe that walking the perimeter of a man’s field is a fair measure of how to tax him. [laughs] I think they are very ignorant – KU-NINGAL: [laughing] Young Ea-shar-ili! – where you see ignorance, I see dishonesty! Do you think you are ready to challenge them? EA-SHAR-ILI: [passionately] Yes, Sir! – I am not going to stand by and watch the poorest people with the thin strips of land being taxed so heavily! KU-NINGAL: Well, you showed in our last lesson that you are a quick learner. One day you will be a great solver of problems, and a divider and measurer of plots, too. [strikes a pose, with affected voice and dramatic arm gestures] With you, the art of the tablets will come together with tape and rod, and field pegs, and you will be a bringer of peace and justice. [resumes brisk teaching voice] But first, you have much to learn. I shall be glad to channel your enthusiasm for justice into some serious learning of beautiful mathematics! Yesterday, we looked at this problem: Given the area and the semi-perimeter of a rectangle, to find the length and width. So, we know the sum and the product of the two unknowns, and we have to find them. EA-SHAR-ILI: [enthusiastically] Which makes it obvious that you can have a very big perimeter and yet a very small crop-growing area!

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given semiperimeter 6 square on given area breadth

square on length

Square on length, appended to given area, is rectangle: length-by-semiperimeter. Square on breadth, appended to given area, is rectangle: breadth-by-semiperimeter.

?

In modern style: x + y = s, xy = A; gives x2 + A = xs, and y2 + A = ys. Fig. 3 Yesterday’s problem

KU-NINGAL: That’s right – down with the corrupt inspectors! Now, there are different kinds of these problems about finding square-sides [roots]. And for each kind there is a different method. We used one method yesterday – do you remember the diagram we started from? Here it is, still in this sand-tray (Fig. 3). KU-NINGAL: We started with a field of known area, and appended a square on the length and another square on the breadth. Do you remember what we did with this figure? EA-SHAR-ILI: Yes, I think so – the two legs of this figure each have the same length – the known semiperimeter, and we break off [bisect] each to get a square on the quarter-fence. Then we mark one rectangle and move it down to make a little square on the bottom right corner, and then – KU-NINGAL: Well done, Ea-shar-ili! It is not easy to see how this method works, and it’s obvious you went over it for yourself afterwards. So, do you feel you are now equipped to apply this method to another problem? EA-SHAR-ILI: I think so, Sir. Will you give me another example of this kind to practise on my own? KU-NINGAL: Hmm . . . ‘a square and a given quantity is the same as a given rectangle on the square-side’ . . . I know there’s a good problem somewhere on this tablet . . . . . . Ah, I have found one: ‘An unknown number when added to its reciprocal becomes two, none, none, thirty-three, none. What is this number?’ 1 There will be two solutions (Fig. 4). EA-SHAR-ILI: [jotting it down with stylus on a small tablet cradled in his hand] Hmm . . . the given number could be a tiny bit bigger than two units or it might be sixty times bigger . . . This is obviously not about a real field measurement,

1 Boyer

and Merzbach 1991, p. 33.

282 Fig. 4 A takeaway problem

G. Hitchcock Unknown number added to its reciprocal is 2; 0, 0, 33, 0. What is this number? In modern style: Setting y = 1/x, hence

Fig. 5 A new kind of problem

x + y = s, xy = 1. x + 1/x = s x2 + 1 = xs; find x.

In modern style : xy = A, x − y = b; find x and y. This gives hence

x2 − bx = A, x2 = bx + A; find x.

I took away my (unknown) square-side from inside the area, so that it was 14, 30

I guess it was invented to give students trouble! [both laugh] I will assume it’s two units. As the numbers will be reciprocals of each other, their product will be unity, so I can see how this problem can be solved, in the same way as before. What other problems are there? Will you demonstrate another kind? KU-NINGAL: Let’s look at the form that arises when we are given the difference of two quantities as well as their product. We can express it like this: ‘An unknown square is the same as a given rectangle on the unknown square-side and a given area’. And here is a problem of this form (Fig. 5): ‘I took away my (unknown) square-side from inside the area, so that it was fourteen, thirty.’ 2 EA-SHAR-ILI: Fourteen, thirty – the size of a nice barley-field. Square-side taken away from area – how do we picture that? KU-NINGAL: This means you have drawn an unknown square and have broken off the projection of the side – a rectangle of unit width. What remains is fourteen, thirty. You are to find the unknown square-side. Here is the picture (Fig. 6). Shall we go through the standard procedure of calculating the unknown first, or would you like the picture demonstration first? EA-SHAR-ILI: Oh, let me see the demonstration first. I like to see what I am doing and why the procedure works. KU-NINGAL: Ha! You have passed the test! Understanding before memorising – that is the way of a true scribe! Here we go (Fig. 7): We take away the projection also from this adjacent right-hand side of the square, like this . . . Then we break off part of the projection, and move it to the bottom, like this . . . and we mark each with the stylus, like this . . . EA-SHAR-ILI: So the two rectangles you have marked are equal . . . 2 BM 13901, Problem 2, in Katz and Parshall 2014, pp. 25–26; Problem (ii) in Katz 2007, pp. 102– 104; see also Boyer and Merzbach 1991, p. 31, and Fauvel and Gray 1987, p. 31.

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Form of problem: An unknown square is the same as a given rectangle on the unknown square-side and a given area. Modern expression: x2 = bx + A

remaining area 14, 30

6 unit ? unknown side

I took away my square-side from inside the area, so it was 14, 30.

Modern expression: x2 − x = 870.

Fig. 6 Picturing the new problem

Unknown square take away square-side is 14, 30. In modern expression, x2 − x = 870.

given area 14, 30

6 unit ? unknown side

v v v v vv v vv v vv

vv vv vv vv v vv v

Fig. 7 Picture demonstration of the method

KU-NINGAL: Yes. Can you see what that means for the area of the dashed line square? Can we find it? EA-SHAR-ILI: Umm . . . it extends below the rectangle by that one marked piece, but it is also less than the rectangle by the other one on the right side – so the difference from the rectangle is just the little corner square – we’ve got it! The dashed line square has the given area increased by the known area of the little square. So, we can find its square-side [square root], and then we add the squareside of the little square to it. Then we have found the unknown square-side we wanted.

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29; 30

6 29; 30

v v v v v ?v v v v v v v ?0; 30 6

vv vv vv vv v v v v

Half of unity is 0; 30. Squaring: 0; 15. Adding 14, 30 gives 14, 30; 15. Its square-side is 29; 30. Adding 0; 30 gives answer 30. Modern expression: x2 − x = 870, x=



(0.5)2 + 870 + 0.5 = 29.5 + 0.5 = 30.

Fig. 8 Applying the method

KU-NINGAL: Right! Let me see you apply the method to find the unknown! (Fig. 8) EA-SHAR-ILI: I am looking at that little square – to get its side I must take off half of unity, which is thirty [0; 30]. I combine this with itself to find its square, and it is a quarter, or fifteen [0; 15]. I add this to the given area to get fourteen, thirty, fifteen [14, 30; 15]. Now what is the square-side? May I use that tablet with the square-side tables? [Consults it] Umm . . . it’s the square on side twenty-nine, thirty [29; 30] – twenty-nine and a half. I add the half-unit [0; 30] to this, and I have thirty [30], the side required. KU-NINGAL: You have the answer, well done! Now, to conclude, I want to show you another way we find helpful for picturing certain problems. Let’s look at this problem again: we have this square whose side is to be found; the projection on one side – a rectangle of unit width – is taken from it to leave the remaining rectangle, which is given to be fourteen, thirty [14, 30] (Fig. 9). KU-NINGAL: Now, start with a unit square area, like this, and add four of the remainder rectangles onto it, like this. Now what is the total area of the resulting large square?3 EA-SHAR-ILI: It’s a cross and four squares, whose areas we don’t know yet . . . KU-NINGAL: Good. Now can you see another way of subdividing the large square? EA-SHAR-ILI: Oh, yes! It’s the central unit square with four rectangles appended, each of area fourteen, thirty [14, 30]. So it has area unity added to fourteen, thirty [14, 30] taken four times.

3 See

Katz and Parshall 2014, p. 28 for a similar method applied by the scribe in BM 13901, Problem 23.

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Area 14, 30

Total area: 1 + (4 × 14, 30)

1 4 (area

of large square)

Dashed square is 0; 15 + 14, 30 = 14, 30; 15 Fig. 9 Another way of looking at the problem

KU-NINGAL: Excellent! – and one quarter of this large square is cut off by the dashed lines I have drawn. Can you see that? EA-SHAR-ILI: Yes, I see. We now have exactly the dashed line square we had before – with a quarter the area of the large square, which is fifteen added to fourteen, thirty [14, 30], which is fourteen, thirty, fifteen [14, 30; 15]. So, as before, we find the square-side and add a half-unit. KU-NINGAL: Very well done, Ea-shar-ili! I am proud of you! You have grasped the meaning of what we have been doing. One day you will able to divide plots of land and apportion pieces of fields so that when wronged people have a quarrel you will be able to soothe their hearts, and brother will live at peace with brother.4 You may take the rest of the day off. I expect you will want to be following those wicked tax collectors around the farms! EA-SHAR-ILI: I will indeed! May I wear my novice-scribe sash to impress them? KU-NINGAL: Ea-shar-ili, you are a true follower of the great king Lipit-Eshtar. His learning was as great as his just rule, if we are to believe the tablets that praise him, and the old scribes that glorify him. Here is an old praise poem in his honour . . . [takes down a tablet] I will use it to compose a blessing for you, before you leave . . . [strikes a pose, and reads in ringing tones] I call on the great goddess Nisaba, the woman radiant with joy, the true woman, the scribe,

4 Inspired by an existing cuneiform dialogue (or ‘rumbustious slanging match’) apparently between an advanced student and an incompetent younger student, in Robson and Stedall 2010, pp. 217– 218. Robson interprets the interchange as illustrating the importance given to accurate land surveys for legal reasons, e.g. inheritance, sales and harvest contracts. The advanced student’s point is that ‘if the surveyor cannot provide his services effectively he will unwittingly cause disputes or prevent them being settled peacefully’.

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the lady who knows everything – I call on Nisaba to guide your fingers on the clay, to make your fingers put beautiful wedges on the tablets and adorn them with a golden stylus. May Nisaba generously bestow on you the measuring rod, the surveyor’s gleaming line, the yardstick, and the tablets which confer wisdom. Ea-shar-ili, son [or daughter] of Enlil, you have begun to pursue justice and righteousness. In you shall the true scribal vocation be manifest: the mighty will not commit robbery and the strong will abuse the weak no more! The Land will be content!5 EA-SHAR-ILI: [bowing] Thank you, Master! KU-NINGAL: Here’s your sash – put it on. You may even quote my authority! Why don’t you take this tablet along in a padded bag? That will impress the inspectors even more, although the chances are they won’t understand any of it. You can pretend to consult it! But do be very careful with that valuable tablet, won’t you? And make sure you get the mathematics right! CURTAIN FALLS Acknowledgments This dialogue was enacted online by Duncan Melville and Nicholas Scoville when I recorded my talk. I wish here to express my gratitude to them for their engaging performance and creative use of the challenging virtual environment.

References Boyer CB, Merzbach UC (1991) History of Mathematics. Wiley, New York Fauvel J, Gray, J (eds) (1987) The History of Mathematics: A Reader. Macmillan Education, London Katz VJ (ed) (2007) The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, Princeton Katz VJ, Parshall, KH (2014) Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press, Princeton Robson E, Stedall J (eds) (2010) The Oxford Handbook of the History of Mathematics, paperback edn. Oxford University Press, Oxford

5 This is drawn, with a little editing, from a praise poem to the Mesopotamian king Lipit-Eshtar who ruled in Isin, 1934–1924 BCE; in Katz 2007, pp. 91–92, and in slightly different translation in Robson and Stedall 2010, p. 218.

Entrance into All Obscure Secrets: A Workshop on Bringing Episodes in the History of Mathematics to Life in the Classroom by Means of Theatre, Incorporating a Short Play Set in an Ancient Egyptian Scribal School Gavin Hitchcock

Abstract This workshop provided an opportunity to experience and reflect on the ways that the devices of narrative, dialogue, drama and theatre can bring mathematical ideas and history to life in the classroom. We aimed to make the case and be inspirational too, by demonstrating theatre in action, involving as many participants as possible in the production and enactment of a short pre-scripted play, followed by discussion and feedback. The scene is set in a scribal school in ancient Egypt, with two scribes solving and recording solutions to problems based on those in the Rhind Mathematical Papyrus, dating from about 1800 BCE. In the play, the problems are related to the cultural context of the time, and the methods include multiplication and division in the Egyptian style, algebraic manipulation and the rule of false position. A narrator and a modern mathematics teacher introduce and interpret what the scribes are doing, and eight mathematicians from different ages and cultures appear in cameo, giving their names for ‘the unknown’ in equationsolving, while the narrator explains the meaning of the terms involved. The dialogue is lively and accessible, with emotions and humour, aiming to stimulate an interest in authentic contextual history of algebra as well as motivate the learning of algebra.

1 Introduction In order to communicate the flavour of a live workshop, the second and third sections of this paper record almost verbatim what was spoken by way of introduction for the workshop participants. About 15 min were spent introducing the workshop, the play and the cast—pre-casting was necessary for a virtual workshop. The presentation of

G. Hitchcock () Independent Scholar, London, UK © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 M. Zack, D. Waszek (eds.), Research in History and Philosophy of Mathematics, Annals of the Canadian Society for History and Philosophy of Mathematics/ Société canadienne d’histoire et de philosophie des mathématiques, https://doi.org/10.1007/978-3-031-21494-3_18

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the play followed, taking about half an hour. The rest of the workshop was spent in sharing feedback, previous experiences of harnessing the power of theatre to inspire and teach and ideas on how this play might be used (and improved) for education purposes at various levels. A challenge was issued to consider constructing short, accessible plays, preferably based on primary sources and staying as true as possible to the history and context. The play itself is divided into two parts, with an optional break for classroom presentation. The dialogue takes place in an ancient Egyptian scribal school, around 1800 BCE. The Head scribe A’H-MOSE orders the junior scribe AMINHOTEP to copy a list of problems and solutions for the students from an old papyrus roll onto a new one. His first task is to start copying out a reference table expressing as sums of unit fractions the quotients of 2 by odd numbers. This is the opening Recto of the Rhind Mathematical Papyrus (RMP) on which the mathematical content of this play is based. There are entertaining interchanges over such issues as the rather grand introduction, the importance of showing reverence for the old pharaohs, the sacred nature of the task, the apparently unchanging form of the mathematics over centuries, the importance of copying correctly for posterity and the rules about whose name gets recorded on the roll—not the junior copier! Problems worked by AMINHOTEP under the supervision of A’H-MOSE are as follows: (1) How to divide six loaves between the ten men; (2) The unknown added to one fourth of the unknown gives fifteen; what is it? These are the RMP Problems 3 and 26. In the play we suggest some cultural context, with the ten men of Problem 3 working in a brewery, and Problem 26 introduced as arising from an everyday problem about men hired to work on an irrigation canal being given beer for their lunch. During the course of the play, elementary algebraic manipulation is demonstrated and explained: calculating with the unknown number directly until it stands alone equal to a number. The Rule of False Position is also explained, and the play concludes with the solution of a third problem arising from practical needs: the cost of filling a grain bin which has been partially depleted by millstone workers, offset by repayment of a loan to the supervisor of the quarry diggers. There are 12 parts to play, most very small, and memorisation is not expected. A narrator, MARIA, introduces the play and gives commentary. The main dialogue is between A’H-MOSE and AMINHOTEP; but a modern mathematics teacher, EMMY, interprets and explains in our terms what they are doing, with slides displaying modern notation and hieroglyphic notation (not hieratic script, for simplicity). At one point, when the scribes talk about unknowns, MARIA introduces eight representative scribes or mathematicians from different cultures, who make cameo appearances, giving the names for ‘the unknown’ (displayed on a slide) used in equation-solving in their time and context. MARIA explains the meaning of the terms involved.

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2 Why a Mathematical-Historical Play? The aim of this workshop is to explore how creative re-construction of historical events in theatrical form may be used in practice in mathematics teaching contexts with minimal fuss and maximum fun! Can we put on a play, almost from scratch— and can we do it virtually? In a way, we were forced to cheat, by casting in advance. It may be hard to believe, unless you see it happen in a live environment, that casting, stage-setting and rehearsing can be achieved in a mere 20 min. I can only assure you that it can be done, with actors reading scripts, but practising (under directors’ guidance) voice projection and body language, for a few selected lines. What we want to do is experience and reflect on ways that theatre can bring mathematical history to life. The idea of using dialogue or theatre as a communication or polemical tool has been around for thousands of years and has long been used for communicating mathematical, scientific and philosophical ideas, by Plato, Galileo and many more. But I think live dialogue form is sadly neglected in current curriculum-driven, time-constrained educational systems. There are films with mathematical, historical and biographical themes, and these are great. But there’s something fundamental missing! If you have observed the effect of spontaneous drama, mime and improvisation on young people of various ages, you might easily share my conviction: positively involving learners in the live re-enactment of historical episodes is a powerful tool for getting their attention and engaging their hearts and their minds. This workshop will make the case and (I hope) be inspirational too, by demonstrating theatre in action, involving as many of you as possible in the play and the discussion. I will invite you to share ideas and any similar experiences and discuss in groups how the power of theatre can be harnessed in our work and teaching. Now I am not primarily a playwright or a historian, but a mathematician. It’s true I have had plays enacted in whole or in part at a number of European Summer Universities, History and Pedagogy of Mathematics (HPM) meetings and other conferences, in many places.1 And some of these involved teacher trainees, and some of my plays have been used in classrooms or mathematics camps elsewhere by others. But—here’s the disclaimer—I am an academic; though passionate about using history and theatre for enrichment of mathematics teaching in schools and universities, I have had limited opportunity to mount plays under real classroom constraints. I am not the expert! I am hoping some of you will pick up the torch, and run with it, or encourage others in this direction. It’s not so daunting to put on a play! To convince you of this is a major goal of this workshop. We’ll put this play on now with minimal props and minimal rehearsal, and I hope it will be lots of fun!

1 See Hitchcock (1996), Hitchcock (1997), Hitchcock (1998), Hitchcock (2000), Hitchcock (2004),

Hitchcock (2012), Hitchcock (2016), Hitchcock (2017), and Hitchcock (2019).

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3 Welcome and Synopsis Welcome to our virtual workshop experiment! Let’s see if it works. We will be mounting a play (written by me) intended as an example of how an interactive theatrical experience can be both engaging and mathematically inspiring, both authentic and accessible. But the logistics of virtual presentation is very tricky, and we couldn’t manage without the help of two people from St Andrews: Calum and Kate. Good to have you onboard, you two! I have never staged this particular play before—it’s a world premiere! Look around you—do you see any people with strange virtual backgrounds, unusual clothes and weird names? For this online workshop I had to ask for volunteers in advance and send them the script. So the cast are here among you. Here’s a reminder to them—please make sure your video NAME is your character name in capitals! And to all participants: click on Speaker View for the whole workshop. The point of a workshop is to involve everybody, and we will still aim to do this in the virtual environment, by planning general discussion and feedback, and breakout groups with group feedback. But if this was a physical workshop, here’s what we’d have done. I would be casting all 12 speaking parts on the spot as well as directors, by listing the roles and calling for volunteers. I would present you with colourcoded placards to hang around your necks, with roles or stage names, giving each actor and director a highlighted script. Actors would be provided with headdress or other simple, appropriate costume item; or else you would be given a basket of appropriate costume items to choose from. I would also recruit stage crew to manage some simple props, and maybe also sound crew and lighting crew. That would take 10 min. A short pep-talk from me would suffice—just to urge everyone to relax and have fun, while (importantly) aiming to enact historical characters in a manner that is sensitive and respectful to the diverse cultures, while preserving as much authenticity as possible. Then (if we were physically in a classroom) our front stage presenters EMMY & MARIA would go to one corner with their director (colourcoded placards help here); the two Egyptian scribes, A’HMOSE & AMINHOTEP would go to another corner with their director; and the MATHEMATICIANS from eight different periods and cultures would be rehearsed in a third corner by their director. Meanwhile the stage and sound and lighting crews would be making plans, with all participants invited to help with direction or staging. This can be done in 10 min, and all preliminaries can be achieved in half an hour. I have done this successfully with as many as 17 players, plus the directors and crew, involving about 25 people. But we are virtual today, so we will not be casting or rehearsing, which will save us some time. The cast have already been alerted to the need for sensitivity and respect in their portrayal.

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4 The Play [Introductory music plays]

SLIDE 1 Entrance into All Obscure Secrets CHARACTERS Narrator, MARIA Teacher, EMMY Head Scribe, A’H-MOSE Junior Scribe, AMINHOTEP BABYLONIAN SCRIBE (c. 2000 BCE) DIOPHANTUS OF ALEXANDRIA (3rd century) ARABIC MATHEMATICIAN (9th century) INDIAN MATHEMATICIAN (12th century) CHINESE MATHEMATICIAN (13th century) ITALIAN ABBACIST (14th century) EUROPEAN MATHEMATICIAN (15th century) GERMAN COSSIST (16th century) [SLIDE 1 goes down, music fades, MARIA & EMMY appear side by side] MARIA: Hello! My name is Maria, and I am your storyteller. I want to invite you to come with me to Egypt, about four thousand years ago. And this is Emmy, a mathematics teacher, who will come with us and help us relate what we see the ancient scribes doing to what we might call doing algebra today. EMMY: Hello! I am very excited to be travelling back in time with you! Algebra looks very different now, to what it once was. We think of algebra as being about ‘variables’ that we write as x or y, and expressions containing powers of these, combined in equations. But what’s common in the ancient ways that we will look at is the idea of ‘unknown’, even if it is not given a symbolic name, and the idea of an equation to be solved. We will see everything described in words, where now we use symbols. Also, algebra was once about everyday things, where now it can become quite abstract. But for solving simple equations we still use essentially the same methods that were discovered so long ago. MARIA: Many thousands of years ago, humans developed the art of counting things, and then they learnt to write, and to measure lengths, areas and other quantities very accurately. People who were good at calculating with numbers

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and keeping records became very important, especially as societies grew more complex and more highly organised. These people were called scribes, and the trainee scribes were probably chosen for their writing and calculating skills from among the elite—the priesthood or the ruling class. Then they were trained in special schools, where the subject we call mathematics was born. For in these scribal schools arose, from as early as 5000 years ago, the theory and practice, for their own sake, of ideas that had emerged originally out of the needs of the society. [MARIA & EMMY disappear, MUSIC plays, in ancient Egyptian style]

SLIDE 2 SCENE: Egyptian Scribal School, about 1800 BCE Head Scribe A’H-MOSE and Junior Scribe AMINHOTEP are seated at a table, each holding quill-pens. There are papyrus scrolls all around, on shelves and tables. They dip the quills periodically into an inkwell and write on papyrus paper. [SLIDE 2 goes down, A’H-MOSE & AMINHOTEP appear side by side with the same virtual background, music fades . . . A’H-MOSE is seated, AMINHOTEP has just come in, and bows respectfully.] A’H-MOSE: Good morning, Aminhotep! AMINHOTEP: Good morning, Chief Scribe, Sir! [he takes his seat] A’H-MOSE: Aminhotep, we must copy another list of problems and solutions for our students, from this old scroll here. You copy it on a new papyrus – a nice long one, say . . . six paces long when unrolled . . . Yes that will do! Now, for a good impressive introduction . . . take this down: [dictates in deep sepulchral tone, while AMINHOTEP writes on papyrus roll]2 Accurate Calculation. The entrance into the knowledge of all existing things and all obscure secrets –

2 Much of what the Egyptian scribes do in this scene is based on the contents of the Rhind Mathematical Papyrus (RMP ca. 1800 BCE), sometimes called the A’h-mose Papyrus, which is about 18 feet long and 13 inches wide. It was rolled up as a scroll, and they wrote on it with quill-pen, from right to left. More details may be found in Chapter 1 of Katz 2007 and in Chace 1979.

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AMINHOTEP: Excuse me, Chief Scribe A’h-mose, isn’t that a bit . . . erm . . . over the top? I know mathematics is called our most powerful instrument of administration, but still! A’H-MOSE: You’ve got to impress these students, you know! They are an irreverent lot these days. They don’t know how lucky they are to be selected for initiation into the sacred mysteries of the priesthood. Let’s continue . . . [dictates again, still in pompous tones but in more business-like fashion, Aminhotep scribbles frantically] This book was copied in the year 33, in the 4th month of the inundation season, under the majesty of the king of the Upper and Lower Egypt, ‘Aweserr¯e’ Ap¯opi, – AMINHOTEP: Whoa! Hold on! Not so fast! ‘ – the king of the Upper and Lower Egypt, ‘Aweserr¯e’ Ap¯opi . . . ’ A’H-MOSE: [glares at his subordinate and sighs] – endowed with life, in likeness to writings of old made in the time of the king of the Upper and Lower Egypt, Nema’r¯e’ Ammenem¯es the Third. It is the scribe A’h-mose who copies this writing. AMINHOTEP: Wait! How do you write the name of old Ammenem¯es? Is this right? A’H-MOSE: [checking Aminhotep’s writing] Aminhotep, be careful! Speak only reverently of the pharoahs! Yes, that is correct. AMINHOTEP: How long ago was this papyrus made? A’H-MOSE: Ammenem¯es the Third (may Osiris bless him on his journey!) was king over all Egypt about, let me see . . . only two hundred years ago. AMINHOTEP: Two hundred years! But, Chief Scribe, haven’t we changed our administration methods or our mathematics since then? Is this old scroll still relevant? A’H-MOSE: [annoyed] You are very presumptuous in your questioning, Aminhotep! I have never known a scribe like you! Of course nothing of note has changed in two hundred years. Why should it? That is a very short time in our long history. Is what was excellent for our forefathers not excellent also for us? Are you ready to proceed? Where was I? Ah, yes: It is the scribe A’h-mose who copies this writing – AMINHOTEP: Erm . . . what about writing my name too? A’H-MOSE: No, no, mine will do. It is not our custom to record the names of junior scribes. Your time will come. AMINHOTEP: [looks sulky and mutters aside to the audience] And there’ll be some changes made then, I can tell you! A’H-MOSE: Now, where were we? . . . We’ll copy some problems later; first you must copy the Recto table from that old papyrus, – it should give expressions, as sums of parts of unity, for two-divided-by-five, two-divided-by-seven, twodivided-by-nine, and so on, for all the odd divisors up to one-hundred-and-one. AMINHOTEP: [goes into furious scribbling mode] Right, Chief! A’H-MOSE: [sternly] And make sure you get it right – no mistakes! These decompositions took many generations of priests to calculate and to select the best choices out of many possible ways of combining parts of unity. If you make one mistake it will be foul sacrilege! It will cause future generations of priests

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and scribes to get many wrong answers. As you know, this table is consulted in the solution of almost every problem.3

SLIDE 3 [AMINHOTEP’S work is displayed as he speaks, while EMMY may spotlight the relevant lines] Recto Table

Modern expression

Divisor 2

÷

3

is

2/3 (or 1/1 12 )

2

÷

5

is

1/3 and 1/15

2

÷

7

is

1/4 and 1/28

2

÷

9

is

1/6 and 1/18

AMINHOTEP: [copying with great concentration, tongue protruding from the corner of his mouth] Two divided by three is two-thirds – now this one we accept as a proper quantity by itself, with this special mark . . . Next, two divided by five is a third and a fifteenth. Two divided by seven is a quarter and a twenty-eighth. Two divided by nine is a sixth and an eighteenth . . . . [SLIDE 3 goes down, A’H-MOSE & AMINHOTEP disappear, while MARIA & EMMY appear side by side] MARIA: We will leave A’h-mose and Aminhotep working on their school trainingmanual, while I tell you a bit more about scribes. They were priests, and as such were the custodians of all the inherited wisdom of the past. The temple was at the very heart of the community and its social, political and business life, and priests were also the chief administrators. They naturally became very important and powerful when cultures grew more sophisticated and kings ruled over large domains. I will continue to show you what these two scribes are doing, but not in the shorthand hieratic script they would have used in their day, for you would have to learn too many new symbols. Instead, as in the first diagram, I will show you transcriptions

3 The Rhind Mathematical Papyrus begins with this table, which occupies a great deal of its space. A full transcription of the table appears in Katz 2007, p. 20.

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into the earlier hieroglyphic form, which is simpler but still allows us to see something of their style. And when they use unit fractions, or parts of unity, my transcription will follow the standard ‘overbar’ notation used by historians today, also to let you see something of the ancient style. Thus, for the Egyptians, a half would be written as a two underneath a symbol, which we will replace by the overbar.4 EMMY: And I will continue to explain what they are doing, in terms of our HinduArabic numerals and standard fractional notation. You will notice that where we write from left to right, the Egyptian scribes are writing from right to left. MARIA: The scribes were in charge of land distribution, taxation and census, hiring of labourers and organisation of slaves, estimation, sourcing and transporting of building materials. They had to know how many bricks or quarried stones would be needed for building a ramp or a granary, and how many men were needed to transport large stones a certain distance or to shift a certain amount of sand in the required time. They were also responsible, on a day-to-day basis, for the production and distribution of food for the workers. They had to keep track of the growing, harvesting and storage of grain in the granaries, and the subsequent cleaning, grinding and dough-making, and the baking of bread. They had to record the amounts of barley, dates and water that went into the brewing of beer, and the subsequent fermentation in vats and final pouring into smaller containers for transportation and delivery.5 Part of the scribes’ daily business, then, was finding unknown numbers which it would be very useful to know: [MARIA & EMMY disappear, A’H-MOSE & AMINHOTEP appear side by side] A’H-MOSE: How to divide six loaves between the ten men working in the brewery today.6 AMINHOTEP: Hmm . . . ten men, six loaves . . . I can see how to do this quickly: If there were only five loaves, we’d divide each loaf in half, and give one half to each man. So now we have six loaves, we just need to divide the sixth loaf into ten pieces and give one piece to each man as well. A’H-MOSE: You have it, Aminhotep. That’s good enough for practical purposes. Those numbers are simple. But we want to give students practice at the correct methods of division for any numbers, so you must record on the papyrus how to divide six into ten parts by the proper method.

4 These conventions follow standard expositions of ancient Egyptian mathematics such as given by Annette Imhausen in Katz 2007. It is optional, in any classroom presentation, to focus on the modern expressions in Slides 3 and 4 and to simplify Slides 4 and 8 by replacing the overbar 1 transcription in .2, 10, 4, with the modern . 12 , 10 , 41 . 5 See Rossi 2010 for more on the way ancient Egyptian mathematics related to the material culture and activities of daily life—agriculture and food production, building and quarrying, metal-work, etc. 6 This is RMP Problem 3. That they were brewery workers is my addition. It should be noted that the problems of the mathematical papyri are not always of any obvious practical use. The mathematics was motivated originally by everyday needs but evolved well beyond these; the extant texts are all school exercises for training purposes.

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AMINHOTEP: Right. I must calculate with ten to get six somehow . . . I will do it on a scrap papyrus first.

SLIDE 4 [display while AMINHOTEP writes and A’H-MOSE speaks; EMMY may spotlight relevant lines] To divide six loaves among ten men. Calculate with 10 to get 6 ... (In translating, we use 2, 10 for unit fractions 1/2, 1/10 )

2

1

10

2

5

10

1

10

6

r

Total

Total

A’H-MOSE: There, you have it! Write it out now. That’s right – half of ten is five, a tenth of ten is one. Now, you have marked those two, because six is five and one. Good. Then the answer is – they each get one half loaf and one tenth of a loaf. Well done, Aminhotep. AMINHOTEP: Thank you, Sir. But I will also check directly, as you have taught me. Ten times one half is five, and ten times one tenth is one, and five and one together is indeed six! This is it! A’H-MOSE: Good. So, as you saw at the start, five loaves must be cut in halves, and the sixth loaf cut into into tenths. Aminhotep, write that down and make sure the servant girl is told to do that. And then record this problem and the proper solution very carefully on the big papyrus. AMINHOTEP: Yes, Chief Scribe, Sir! [Scribes disappear, SLIDE remains up, and EMMY’s video is turned on for her speech] EMMY: Hello, this is Emmy again. Do you remember seeing on the previous slide that special hieroglyphic symbol for two-thirds? Now we can see another symbol for one-half. But unit fractions like one-tenth are expressed by placing the ten underneath a sort of flattened rugby ball hieroglyph meaning ‘part’. We now call a unit fraction a ‘part of unity’. You can see how they mark their unit fractions to be added – half of ten is five, and a tenth of ten is one, and five added to one gives six; so the answer is one half and one-tenth. Is there anything else you’d like explained on this slide? . . .

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MARIA: Okay, let’s have a break; then we will go back to see what the scribes will do next. [SLIDE 4 goes down, MARIA & EMMY EXIT]

Part 2

[MUSIC plays briefly, in ancient Egyptian style] [SLIDE 2 goes back up] SCENE: Egyptian Scribal School, about 1800 BCE Head Scribe A’H-MOSE and Junior Scribe AMINHOTEP are seated at a table, each holding quill-pens. There are papyrus scrolls all around, on shelves and tables. They dip the quills periodically into an inkwell and write on papyrus paper. AMINHOTEP: Chief Scribe A’h-mose, Sir – we have another problem. The unknown added to one fourth of the unknown gives fifteen; what is it? A’H-MOSE: Where did you find that problem, Aminhotep? AMINHOTEP: Remember when we hired those four men to work on the irrigation canal, we promised to pay out of our own pockets for a jug of beer for their lunch, to a value that matches whatever wage the Chief Priest decides to give them. But we don’t want them to get more than fifteen pennies each, do we?7 A’H-MOSE: If they did, the other workers will certainly complain! So, you want to know how much to claim from the Chief Priest . . . [both mutter, heads as close together as virtually possible, writing on papyrus] A’H-MOSE: Let the unknown wage for each man be called Aha!8 [Scribes disappear, MARIA appears large for her speech. She is then replaced in turn by each Scribe in the following cameo appearances. Each has culture and time displayed as their Zoom video name in capitals, and each turns mic and video on before speech, and off after speech]

7 Wages

were in grains and shekels, but we use pennies for simplicity. See Rossi 2010 for more on measurement units. 8 Egyptologists use special conventions and letters for transcribing ancient Egyptian script, although the ancient pronunciation cannot be known for certain. I have used the standard, simple word ‘Aha’, which is a reasonable guess at how they might have pronounced their word for the unknown – especially if the ‘h’ is pronounced as the Arabic ‘h’. The Egyptian technical term for ˙ quantity would be written today as aha. ˙

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MARIA: Did you hear what the Chief Scribe said? ‘Let the unknown be called Aha! Today, we often call the unknown ‘x’, and our name for the art of finding such unknowns is ‘Algebra’. It has been developing for a long time, with contributions from many cultures. Whatever culture a scribe came from, the unknown would be given a special name. Let’s hear from a few! BABYLONIAN SCRIBE (1800 BCE): The unknown shall be called Line. DIOPHANTUS OF ALEXANDRIA (3rd century): I say let the unknown number be called Arithmos. ARABIC MATHEMATICIAN (9th century): The unknown shall be called shei, and the unknown square shall be called M¯al [pronounced to rhyme with first syllable of ‘starling’]. INDIAN MATHEMATICIAN (12th century): In calculation the unknowns are the b¯ıja; I call the measure of an unknown quantity, y¯avatt¯avat; or, if there is more than one unknown number to be found, I call them by the names of colours – and I write a letter, for short. CHINESE MATHEMATICIAN (13th century): The name of the unknown element is Yuan; or if four unknowns are present we call them Tian, Di, Ren and Wu. ITALIAN ABBACIST (14th century): The unknown is called Cosa! EUROPEAN MATHEMATICIAN (15th century): That which we seek shall be called res, or radix. GERMAN COSSIST (16th century): We shall call the unknown Coss!

SLIDE 5

THE UNKNOWN 2000 BCE Aha 1800 BCE Line

Egyptian Babylonian Diophantus (Egyptian/Greek) Arab Indian

3rd c. 9th c. 12th c.

Arithmos M¯al B¯ıja Y¯avatt¯avat

Chinese

13th c.

Yuan Tian, Di, Ren, Wu

Italian Abbacist European German Cossist

14th c. 15th c. 16th c.

Cosa Res, Radix Coss

Heap, Pile Geometric line The Number Treasure Seeds As much as so much Elemental source Heaven, Earth, Man, Matter The Thing Thing, Root from the Italian

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[EMMY turns her video on, SLIDE stays up, and MARIA remains] MARIA: In the language of A’h-mose, the Egyptian scribe, Aha means Heap, or Pile – an unknown pile of whatever – pennies, grains, shekels. The Babylonians did everything with geometrical diagrams, so their word for unknown means ‘line’. Diophantus’s word Arithmos simply means ‘the number’. The Arabic word M¯al means ‘treasure’, or valued property. The Indian word b¯ıja means seeds – things yet to germinate in the dark earth and emerge into the light! The word y¯avatt¯avat means ‘as much as so much’; the letters spoken of stand for three colours, where we use letters x, y, z for two or three unknowns. Their way is to use the first letters from three colour names, like the r, the b, or the y, for red, blue, yellow. Of course, they would use colour names from their own language and letters from their own script. The Chinese word yuan for an unknown, means something like elemental source, and those names for four unknowns, which I would rather not pronounce, mean Heaven, Earth, Man and Matter. For centuries, European scholars wrote in Latin, and the words res and radix mean ‘thing’, and ‘root’. The Italian word Cosa means ‘The Thing’, and the fourteenth century Italian experts at the art of calculating with Hindu-Arabic numerals were called maestri d’abbaco, or abbacists, meaning ‘master-calculators’. The German word Coss is derived from the Italian word, and in sixteenth-century Germany algebra was called the Art of the Coss, and so its practitioners were known as Cossists. [SLIDE 5 goes down; MARIA and EMMY appear side by side] What do we call the unknown today, Emmy? EMMY: Well, algebra is mostly symbols now. This helps for economy and elegance, but, just like a foreign language, it can be tough at first, for learners. If there’s one unknown, we tend to call it x, ever since René Descartes, the 17th century French mathematician. MARIA: It is possible that there are ancient origins in various x-words. The medieval Arabic word, shei, for an unknown thing was translated into Greek as xei: X, E, I. And the Greek word for an unknown person is xenos, from which we get ‘xenophobia’ – an irrational fear of outsiders, foreigners. EMMY: I like the fact that when the German physicist Wilhelm Röntgen discovered a wonderful, mysterious, and previously unknown form of radiation which could help us ‘see’ inside the human body, he naturally named the rays X-strahlen, which we still call X-rays, though Germans call them Röntgenstrahlen. And then there is the X-factor – made famous by a certain television programme seeking talented people with that mysterious, undefinable thing that makes celebrities out of unknowns . . . [both laugh] MARIA: The quest to discover unknowns is everywhere! Now let’s go back to ancient Egypt, where the numerical unknown is called Aha, meaning Heap, or Pile. We will watch the scribes solving their problem by using the Rule of False Position, or False Supposition. Emmy, please explain what this means.

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EMMY: This is a method used in earlier times, before symbolic variables like x made their entrance. It is a way of turning a difficult problem into a simpler one, and then using proportions to adjust the simple solution. You will see the scribe giving the unknown number a ‘false’ value, four, designed to make the calculation easy. His calculation then gives five instead of the required fifteen. So he takes this proportion fifteen-to-five of his false value to get the true value of the unknown. [A’H-MOSE & AMINHOTEP appear again side by side, poring over the problem, while MARIA and EMMY disappear] AMINHOTEP: The unknown wage, Aha, must be added to one fourth the same amount, which we will spend on the beer for four men, and the total wage each receives must be fifteen . . . So Aha and a quarter Aha is fifteen. We want to find Aha. A’H-MOSE: How have I taught you to solve such a problem, Aminhotep? AMINHOTEP: I know – I work out the answer for a simple case, and then adjust the situation to turn the false total into the true total. Assume that Aha is something easy – say four, so it is very easy to find a fourth of it . . . A’H-MOSE: Write it out in rough!

SLIDE 6

The Method of False Position a quarter Aha gives

Aha

and

fifteen

4

and

1/4 of 4

is

4 and 1, which is 5

15

divided by

5

is

3

3

times

4

is

12

Aha

is

12

AMINHOTEP: So, I will assume that Aha is 4 pennies, because a fourth of that is just 1. Then Aha and 1/4 Aha is 4 and 1, so the false total is five. This is not fifteen, so we divide fifteen by it, to get . . . erm . . . 3. That means 15 is 3 times 5, so we must multiply 3 times 4 to get our result, which is 12 pennies. A’H-MOSE: Good. But there’s another correct procedure for this type of problem, harder for beginners, but often quicker when you know how. [SLIDE 6 is replaced by SLIDE 7]

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SLIDE 7 Simplifying an Equation: Another Method Aha

and 1/4 Aha Aha 5 Aha Aha

4 Aha and

is is is is

15 60 60 12

A’H-MOSE: Aha added to one fourth Aha gives fifteen, and so four Ahas added to one Aha must give four times fifteen, which is sixty, so five Ahas is sixty, therefore Aha is sixty divided by five – twelve pennies. AMINHOTEP: Just as I found! So we do not need to check this one. [SLIDE 7 goes down; A’H-MOSE and AMINHOTEP appear side by side] A’H-MOSE: But note this carefully, Aminhotep – when you teach students, it is better to use the standard procedure of false position, and include the full details of division and multiplication the first few times, until they have learnt the procedure. You must train them also to verify at the end that the answer really works, for even when we have drilled them in the correct procedures they always make mistakes. Write that problem out now on a spare papyrus, as you would use it to instruct a student – then show me. If you do it well, it can be copied into a problem papyrus for use in the school. And don’t omit to use red ink for the statement and the conclusion of the problem. AMINHOTEP: Yes, Sir! A’H-MOSE: Oh! – before I forget, we must ask the Chief Priest to give us twelve pennies for each of those canal workers, that’s four times twelve, or forty eight pennies we must claim! See to it, Aminhotep! AMINHOTEP: Yes, Sir! And we must put twelve pennies worth of beer in their jug. I will organise that as soon as I have written out this problem properly. [takes a small scrap piece of papyrus and begins to write]

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SLIDE 8

Writing out Correct Procedure A quantity is given and its 4 is added to it, so that 15 results Calculate with 4. (Solution, using You shall calculate its 4 as 1. Total 5. Rule of False Position) Divide 15 by 5. • 5 10 2 (15) 3 3 shall result. Multiply 3 times 4. • 3 6 2 4 12 12 shall result •

12 3 4 The quantity 12 its 4 is 3, total 15

(Verifying)

(Conclusion statement)

[A’H-MOSE & AMINHOTEP disappear, while MARIA & EMMY appear side by side] MARIA: Here, expressed in our terms, is what Aminhotep wrote out9 – and A’hmose said he had got it right, and it was copied into the problem text he was working on, where it became Problem 26, for the instruction of trainee scribes. Emmy, will you take us quickly through the steps? EMMY: Okay – I will point out the steps on the slide and anyone can ask questions . . . [she outlines the method and calculation, and discussion may take place] MARIA: Thank you, Emmy! Now will you tell us how such problems are approached today? [SLIDE 8 goes down]

9 Problem

26 in the RMP. The transcription is by Annette Imhausen and appears in Katz 2007, p. 26. The words in boldface are written in red ink in the original, and the words and numbers in brackets are added by me.

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EMMY: The first method the scribes used was that Rule of False Position, which we don’t often use to solve equations now – we would see it as a problem in proportion.10 And this is the method Aminhotep was ordered to record on the papyrus for students to learn from. But they also knew this other method, which is actually what we use today, except that we use a symbol like x for the unknown. The Egyptian scribes calculate directly with their unknown number, named Aha, until it stands alone equal to a number, and therefore no longer unknown! Such a procedure became known eventually as ‘solving an equation’, because a problem like this always boils down, when you express it as simply as possible, to something (involving the unknown) being equal to something else (possibly also involving the unknown). This means you have an equation saying something about the unknown. And one way to solve the problem was to do things to this equation until you had isolated – or exposed, or unwrapped, or disentangled – the unknown. We would do that same problem today as follows:

SLIDE 9 The modern way of solving the problem Given the equation:

.

multiply both sides by 4,

1 x + x = 15, 4 4x + x = 60,

that is,

5x = 60,

hence

x = 12.

MARIA: And that’s just the way the Chief Scribe A’h-mose did that problem, after Aminhotep had solved it by the method of false position. Except of course his unknown was a word, Aha! EMMY: Right! You start with an equation – two sides being equal to one another – and you do exactly the same things to each side, like adding or subtracting equal amounts from both sides, or multiplying each side by the same amount. You will still have a true equation, with two sides equal to one another. And so people learnt that if you had some amount being subtracted (or taken away) from one of those sides, you could move it to the other side as an added amount, without stopping the equation being true; this is because what you have done is just to add that amount to each side.

10 This is true of most of what we would call ‘algebra’ in extant ancient Egyptian mathematical papyri. See Chapter 2 in Katz and Parshall 2014 for more on this.

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[Slide 9 goes down. A’H-MOSE and AMINHOTEP appear side by side, while MARIA and EMMY disappear] AMINHOTEP: And another problem, Sir: we want these ten men to re-fill the grain container today, so we’ll send them after lunch. According to my records here, we know it holds twelve hekats of grain, and yesterday it was written that the millstone workers used it all up. And they were each issued with only one hekat jar each to refill the container, and there were only seven of them. We already have two hekats promised for today by the supervisor of the quarry diggers to repay our loan last week. How many hekats should each of the ten men carry today? A’H-MOSE: The accepted manner of approach is this: we take Aha as the unknown amount – AMINHOTEP: Then the number of hekats to go into the granary will be ten times Aha, plus the two hekats we have been promised! And what’s needed to fill it is twelve hekats, except that there should already be seven hekats in there. So . . .

SLIDE 10 Filling the Grain Bin Ten Aha and two is twelve subtract seven hekats; ten Aha is (twelve subtract seven) subtract two, which is five subtract two; ten Aha is three, Aha is three divided by ten.

Calculating with ten to get three, we have one-fifth and one-tenth AMINHOTEP: Ten Aha and two must be equal to twelve subtract seven . . . Therefore, ten Aha is equal to twelve subtract seven, then subtract two: which is five subtract two. That is three. So, Aha is three divided by ten. Calculating with ten to get three, it is easy – one-fifth of a hekat and one-tenth of a hekat. [SLIDE 10 goes down] A’H-MOSE: You have found it. At lunchtime please ensure that each of our men is issued with two grain scoops – a one-fifth scoop, and a one-tenth scoop. AMINHOTEP: Yes, Sir. Lunchtime, well . . . actually it is just about lunchtime, Sir – shall I go and do it right away? A’H-MOSE: Such devotion to duty, Aminhotep, is very touching . . . or is it devotion to your own stomach that calls you? Bring me some beer and a loaf of bread with some fish, as soon as you have carried out that task. AMINHOTEP: Yes, Chief Scribe, Sir! [EXITS]

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CURTAIN FALLS [A’H-MOSE & AMINHOTEP, MARIA & EMMY appear together for applause, led by GAVIN, who also appears. Then all the other actor-scribes join them onscreen for a final round of applause]

5 Discussion After the play, there was plenary discussion and feedback, reflecting on what we had been part of and how it might be used in classrooms. Participants were asked for their ‘gut responses’ to the play, and critique and questions were invited. A lively conversation took place, prompted by questions such as the following: • What works? What got through to you? • What could be changed? • Was the dialogue engaging and accessible, with appropriate emotions and humour? • What was surprising or instructive in the ancient Egyptian manner of calculating and solving problems? • How might this play be used, and improved, for educational purposes at various levels? • Could it help to motivate the learning of algebra and stimulate an interest in authentic contextual history of algebra? • Can personal involvement – actually enacting a character from another world – bring a new dimension into our learning and motivation? • How can such experiences be encouraged under the pressures of the curriculum? The discussion was wide-ranging and positive and worked so well online that participants suggested we should not curtail the flow by going into the planned breakout groups. There is no space here to attempt more than a very brief summary. Surprise was expressed at how little effort and preparation was needed to produce an effective piece of theatre, and discussion focused on the issue of making readily available for teachers more resources that are relevant and accessible to learners. Some found the ancient use of unit fractions (parts of unity) surprising, and its benefits were discussed. The difficulties of finding a place for theatre in education were raised. Ways of persuading educators to encourage (and make space for) such activities were pondered, noting that the curriculum machine can wear teachers down with administration, targets and tests. It was agreed that physical participation in a play is an excellent way of ‘getting into the heads’ of old mathematicians and feeling something of their emotions.

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6 Concluding Remarks A live workshop would have offered each participant the opportunity to act, help direct and rehearse the actors or set up stage props, lighting or sound. The virtual environment had to rely on advance casting, losing some of the excitement of teamwork, but there were compensations. The actors had more time to work on their lines and provide some appropriate dress and background, and there was more time for discussion and feedback. For comparison, further reading and resources, see previous articles, talks and workshops by the author, including dialogues and plays, in the references (Hitchcock 1992–2019). The workshop closed by issuing a challenge: We need more theatre, and we need more resources. Would you consider being co-creators of an exciting educational tool and art-form? Run with the ball, catch a vision, translate your own love of mathematics and its history into magic theatrical moments in your own classrooms and share your experiences with others! Acknowledgments Thanks to the conference organizers and, in particular, to Isobel Falconer who encouraged me in many ways and assisted with the casting. Thanks also to Calum and Kate for technical help in mounting a live virtual workshop. I would like to record here my gratitude to each of the cast members for volunteering and giving time and creative energy to engage with their roles and rehearse. The four main parts were spoken by Kristín Bjarnadóttir, Moira Chas, Tony Gardiner, Adam Fletcher; and the smaller parts were spoken by Pilar Gil, Yashashwi Singhania, Troy Astarte, Maria Zack, Emma Baxter, Emmanuel Olowe, Fenny Smith and Kenneth Falconer. Thanks, also, to all workshop participants for feedback during a lively discussion.

References Chace AB (1979) [1927–29] The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. Classics in Mathematics Education. Vol. 8. 2 vols (Reston: National Council of Teachers of Mathematics Reprinted ed.). Mathematical Association of America, Oberlin. Hitchcock AG (1996) Dramatizing the Formation of Mathematical Concepts: Two Dialogues. In: Calinger R (ed) Vita Mathematica: Historical Research and Integration with Teaching. Mathematical Association of America Notes No. 40. 27–41 Hitchcock AG (1997) Teaching the Negatives 1870–1970: A Medley of Models. FLM 17.1: 17–25, 42 Hitchcock AG (1998) Entertaining Strangers: a dialogue between Galileo and Descartes, Comparative Criticism (Cambridge University Press) 20: 63–85 Hitchcock AG (2000) A Window on the World of Mathematics 1870: Reminiscences of Augustus De Morgan. In: Katz V (ed) Using History to Teach Mathematics: An International Perspective. Mathematical Association of America Notes No. 51: 225–230 Hitchcock AG (2004) Alien Encounter: Learning to Live with the Negative Numbers: A twoact dramatic reconstruction of the struggles of European mathematicians of the 17th and 18th centuries to come to terms with new number concepts. Convergence, online journal of the Mathematical Association of America

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Hitchcock AG (2012) Discovering History by Dialogue. In: T. Archibald (ed) Proc of the 37th Annual Meeting of the CSHPM and 5th Joint Conference with the BSHM, Trinity College Dublin, 15–17 July 2011:47–67 Hitchcock AG (2016) Remarkable Similarities: A dialogue between Boole and De Morgan. In: Zack M, Landry E (eds) Research in History and Philosophy of Mathematics: The CSHPM 2015 Annual Meeting in Washington DC: 69–82. Birkhäuser, Springer International Publishing Switzerland Hitchcock AG (2017) Breaking News, November 1813: A Dramatic Presentation Celebrating Joseph Gergonne. Proc. HPM 9, Montpellier (18–22 July, 2016) Hitchcock AG (2019) Niels Abel: ‘So Many Ideas’. A workshop on using theatre to bring episodes in the history of mathematics to life in the classroom. In: É Barbin, U. T. Jankvist, T. H. Kjeldsen, B. Smestad & C. Tzanakis (eds) Proceedings of the Eighth European Summer University on History and Epistemology in mathematics Education (ESU-8) (Skriftserie 2019, nr 11). Oslo: Oslo Metropolitan University (ISBN 978-82-8364-211-7 (printed version), ISBN 978-82-8364-212-4 (online version)) Katz VJ (ed) (2007) The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press, Princeton Katz VJ, Parshall KH (2014) Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press, Princeton Rossi C (2010) Mixing, building, and feeding: mathematics and technology in ancient Egypt. In: E. Robson and J. Stedall (eds) The Oxford Handbook of the History of Mathematics, paperback edn. Oxford University Press, Oxford: 407–428