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Mathematical Time Capsules Historical Modules for the
Mathematics Classroom Dick Jardine Digital Instant
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Author(s): Dick Jardine, Amy Shell-Gellasch (eds.)
ISBN(s): 9780883859841, 088385984X
Edition: UKed
File Details: PDF, 5.89 MB
Year: 2011
Language: english

Mathematical Time Capsules
Historical Modules for the Mathematics Classroom

c 2011 by
The Mathematical Association of America (Incorporated)
Library of Congress Control Number 2011926092
Print ISBN 978-0-88385-187-6
Electronic ISBN 978-0-88385-984-1
Printed in the United States of America
Current Printing (last digit):
10 9 8 7 6 5 4 3 2 1

Mathematical Time Capsules
Historical Modules for the Mathematics Classroom
Edited by
Dick Jardine
Keene State College
and
Amy Shell-Gellasch
Beloit College
Published and Distributed by
The Mathematical Association of America

The MAA Notes Series, started in 1982, addresses a broad range of topics and themes of interest to all who are involved with
undergraduate mathematics. The volumes in this series are readable, informative, and useful, and help the mathematical community
keep up with developments of importance to mathematics.
Committee on Books
Gerald Bryce, Chair
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Stephen B Maurer, Editor
Deborah J. Bergstrand Thomas P. Dence
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14. Mathematical Writing, by Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts.
16. Using Writing to Teach Mathematics, Andrew Sterrett, Editor.
17. Priming the Calculus Pump: Innovations and Resources, Committee on Calculus Reform and the First Two Years, a subcomit-
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18. Models for Undergraduate Research in Mathematics, Lester Senechal, Editor.
19. Visualization in Teaching and Learning Mathematics, Committee on Computers in Mathematics Education, Steve Cunning-
ham and Walter S. Zimmermann, Editors.
20. The Laboratory Approach to Teaching Calculus, L. Carl Leinbach et al., Editors.
21. Perspectives on Contemporary Statistics, David C. Hoaglin and David S. Moore, Editors.
22. Heeding the Call for Change: Suggestions for Curricular Action, Lynn A. Steen, Editor.
24. Symbolic Computation in Undergraduate Mathematics Education, Zaven A. Karian, Editor.
25. The Concept of Function: Aspects of Epistemology and Pedagogy,Guershon Harel and Ed Dubinsky, Editors.
26. Statistics for the Twenty-First Century, Florence and Sheldon Gordon, Editors.
27. Resources for Calculus Collection, Volume 1: Learning by Discovery: A Lab Manual for Calculus, Anita E. Solow, Editor.
28. Resources for Calculus Collection, Volume 2: Calculus Problems for a New Century, Robert Fraga, Editor.
29. Resources for Calculus Collection, Volume 3: Applications of Calculus, Philip Straffin, Editor.
30. Resources for Calculus Collection, Volume 4: Problems for Student Investigation, Michael B. Jackson and John R. Ramsay,
Editors.
31. Resources for Calculus Collection, Volume 5: Readings for Calculus, Underwood Dudley, Editor.
32. Essays in Humanistic Mathematics, Alvin White, Editor.
33. ResearchIssues in Undergraduate Mathematics Learning: Preliminary Analyses and Results, JamesJ. Kaput and Ed Dubinsky,
Editors.
34. In Eves Circles, Joby Milo Anthony, Editor.
35. Youre the Professor, What Next? Ideas and Resources for Preparing College Teachers, The Committee on Preparation for
College Teaching, Bettye Anne Case, Editor.
36. Preparing for a New Calculus: Conference Proceedings, Anita E. Solow, Editor.
37. A Practical Guide to Cooperative Learning in Collegiate Mathematics, Nancy L. Hagelgans, Barbara E. Reynolds, SDS, Keith
Schwingendorf, Draga Vidakovic, Ed Dubinsky, Mazen Shahin, G. Joseph Wimbish, Jr.
38. Models That Work: Case Studies in Effective Undergraduate Mathematics Programs, Alan C. Tucker, Editor.
39. Calculus: The Dynamics of Change, CUPM Subcommittee on Calculus Reform and the First Two Years, A. Wayne Roberts,
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40. Vita Mathematica: Historical Research and Integration with Teaching, Ronald Calinger, Editor.
41. Geometry Turned On: Dynamic Software in Learning, Teaching, and Research, James R. King and Doris Schattschneider,
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42. Resources for Teaching Linear Algebra, David Carlson, Charles R. Johnson, David C. Lay, A. Duane Porter, Ann E. Watkins,
William Watkins, Editors.
43. Student Assessmentin Calculus: A Report of the NSF Working Group on Assessment in Calculus, Alan Schoenfeld, Editor.

44. Readings in Cooperative Learning for Undergraduate Mathematics, Ed Dubinsky, David Mathews, and Barbara E. Reynolds,
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45. Confronting the Core Curriculum: Considering Change in the Undergraduate Mathematics Major, John A. Dossey, Editor.
46. Women in Mathematics: Scaling the Heights, Deborah Nolan, Editor.
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48. Writing in the Teaching and Learning of Mathematics, John Meier and Thomas Rishel.
49. Assessment Practices in Undergraduate Mathematics, Bonnie Gold, Editor.
50. Revolutions in Differential Equations: Exploring ODEs with Modern Technology, Michael J. Kallaher, Editor.
51. Using History to Teach Mathematics: An International Perspective, Victor J. Katz, Editor.
52. Teaching Statistics: Resources for Undergraduate Instructors, Thomas L. Moore, Editor.
53. Geometry at Work: Papers in Applied Geometry, Catherine A. Gorini, Editor.
54. Teaching First: A Guide for New Mathematicians, Thomas W. Rishel.
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Barbara E. Reynolds, Neil A. Davidson, and Anthony D. Thomas, Editors.
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59. Linear Algebra Gems: Assets for Undergraduate Mathematics, David Carlson, Charles R. Johnson, David C. Lay, and A.
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60. Innovations in Teaching Abstract Algebra, Allen C. Hibbard and Ellen J. Maycock, Editors.
61. Changing Core Mathematics, Chris Arney and Donald Small, Editors.
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67. Innovative Approaches to Undergraduate Mathematics Courses Beyond Calculus, Richard J. Maher, Editor.
68. From Calculus to Computers: Using the last 200 years of mathematics history in the classroom, Amy Shell-Gellasch and Dick
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69. A Fresh Start for Collegiate Mathematics: Rethinking the Courses below Calculus, Nancy Baxter Hastings, Editor.
70. Current Practices in Quantitative Literacy, Rick Gillman, Editor.
71. War Stories from Applied Math: Undergraduate Consultancy Projects, Robert Fraga, Editor.
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74. Resources for Teaching Discrete Mathematics: Classroom Projects, History Modules, and Articles, Brian Hopkins, Editor.
75. The Moore Method: A Pathway to Learner-Centered Instruction, Charles A. Coppin, W. Ted Mahavier, E. Lee May, and G.
Edgar Parker.
76. The Beauty of Fractals: Six Different Views, Denny Gulick and Jon Scott, Editors.
77. Mathematical Time Capsules: Historical Modules for the Mathematics Classroom, Dick Jardine and Amy Shell-Gellasch,
Editors.
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Preface
Mathematical Time Capsules offers teachers historical modules for immediate use in the mathematics classroom.
Relevant history-based activities for a wide range of undergraduate and secondary mathematics courses are included.
The genesis of this volume was a Contributed Papers Session on Using History of Mathematics in Your Mathematics
Courses, organized by the editors at the Joint Mathematics Meetings, San Antonio, Texas, in January of 2006. That
session was very well attended, which prompted Andrew Sterrett from MAA publications to suggest that we put
together our second volume for the MAA Notes series.
Purpose
For a wide variety of reasons, instructors are looking for ways to include the history of mathematics in their courses.
It is not uncommon to see requests for “how to” posted to the History of Mathematics Special Interest Group of the
MAA (www.homsigmaa.org) email list, such as this 2008 posting:
...I am a newcomer to HOM. Where and how should a newcomer begin? Right now, I would liketo include
HOM in a meaningful way in the courses that we teach. Weteach courses from college arithmetic to linear
algebra.
In response to such inquiries, we hope to serve the broader mathematical community by offering practical suggestions
on how to use the history of mathematics quickly and easily in the mathematics classroom.
A time capsule can be defined as a container preserving articles and records from the past for scholars of the
future. Of course our volume does not fit that precise definition, but readers who open this book will find articles
and activities from mathematics history that enhance the learning of topics typically associated with undergraduate
or secondary mathematics curricula. Each capsule presents one topic or perhaps a few related topics, or a historical
thread that can be used throughout a course. The capsules were written by experienced practitioners to provide other
teachers with the historical background, suggested classroom activities, and further references and resources on the
chapter subject. An instructor reading a capsule will have increased confidence in engaging students with at least one
activity rich in the history of mathematics that will enhance student learning of the mathematical content of the course.
Most of the historical topics contained in a capsule can be implemented in one class period with minimal additional
preparation on the part of the teacher.
How to use Mathematical Time Capsules
Teaching styles have been categorized along a spectrum from lecture-oriented practices at one extreme to student-
centered approaches in which the teacher guides student work in the classroom. Mathematical Time Capsules respects
the diversity of teaching styles which individual teachers adopt. Some of the capsules are, in some sense, ready-
made lectures the instructor can adopt and adapt as appropriate. Examples of those include Victor Katz’s “Copernican
Trigonometry,” Roger Cooke’s “Numerical Solution of Equations,” or Jim Tattersall’s “Finding the Greatest Common
Divisor and More... .” Other capsules clearly engage the students more actively, such as Vicky Klima’s “A Different
Sort of Calculus Debate.” But the capsules should not be categorized as appropriate for one pedagogical approach or
the other. For example, “Finding the Greatest Common Divisor and More...” could be adapted for use as a student
project to be presented by the student(s) in class after the Euclidean algorithm is covered.
The reader may note that the authors of the capsules demonstrate a variety of approaches to integrating history. The
differences are consistent with the nature of this volume, created with respect for the diversity of our authors and our
vii

viii Preface
readers. We acknowledge that many teachers prefer to develop their own course materials, and we encourage readers
to modify the offerings of the authors.
Mathematical Time Capsules is organized in three sections. The first capsules have as their target mathematical
topics that are usually addressed in courses taught in secondary school, at two-year colleges, or during the first two
years of the undergraduate mathematics curriculum. These courses are not often taken by mathematics majors, and
include, for example, algebra, geometry, mathematics for elementary teachers, trigonometry, or precalculus.
The third section of capsules address topics included in courses traditionally taken by mathematics majors, such as
calculus, differential equations, number theory, abstract algebra, differential equations, and analysis.
As an interlude between the first and third, we offer some ideas that can be applied to a wide variety of courses
throughout the undergraduate (including two-year college) or secondary curriculum. These interlude capsules (15, 16,
and 17) are of a general pedagogical nature, not mathematical, and could be adapted for use in any course.
We could have arranged the capsules differently, and we ask that the reader not limit investigating the offerings in
this book thinking that lower-level material is in the front and upper-level material at the back. For example, a teacher
interested in historical ideas for a numerical methods course should consider Roger Cooke’s “Numerical Solution
of Equations,” Randy Schwartz’s “Rule of Double False Position,” Clemency Montelle’s “Roots, Rocks, and Newton-
Raphson Algorithms for Approximating
p
2 3000 Years Apart,” and Dick Jardine’s “Euler’s Method in Euler’s Words.”
Some teachers are interested in having students read original sources in the history of mathematics, and Montelle’s
“Amo, Amas, Amat! What’s the Sum of That?” and Jardine’s “Euler’s Method in Euler’s Words” provide opportunities
to do just that with brief excerpts in the words of the originators.
Where possible, we grouped the capsules within the sections according to mathematical subject area. As an example,
there are three capsules that address Pythagorean triples, but each of those capsules approaches the topic in a very
different way and from the perspective of a different era of the history of mathematics. That latter notion is significant,
as it is important for students to see the evolution of a mathematical idea and how it is viewed through a different
lens depending on how mathematics was done at various times in history. One of the goals of Mathematical Time
Capsules is to provide a vehicle for teachers to help their students learn the historical context for a mathematical
development while they are learning the specific mathematical concept. Learning the history of an idea promotes
deeper understanding of the idea.
Additional purposes—assessment and teacher certification
Beyond a teacher’s personal interest in using the history of mathematics in teaching, accrediting agencies and state
certifying agencies now require pre-service teachers to be well-versed in the history of mathematics in specific content
areas. The requirement that school teachers demonstrate understanding of the connection between a mathematics topic
and the history of that specific topic is explicitly documented in state and national standards. It is becoming an imper-
ative that many college teachers introduce the history of mathematic in their teaching of mathematics in order to pass
muster for state and national certification of teacher education programs. The National Council for Accreditation of
Teacher Education (NCATE) and the National Council of Teachers of Mathematics (NCTM) have published standards
which list within the content areas a requirement that candidates for teacher certification demonstrate their knowledge
of the historical development of that content area. For example, the current content Standard 10 is on the subject
of algebra. For certification, in addition to demonstrating expertise in the usual algebra concepts, candidates must
“Demonstrate knowledge of the historical development of algebra including contributions from diverse cultures” [2].
In this era of outcomes-based assessment, to satisfy state and national evaluators it is not sufficient to show that
history was included in the syllabus or to claim that history was included in a lecture. Actual student work (interestingly
for us interested in history, the student work is called an artifact in current assessment jargon) must be presented
to the accrediting agency or state department of education evaluators. The material presented herein will provide
teachers with actual activities that students can do. The products of these activities become the artifacts necessary
to validate that students are engaged in learning the historical development of the mathematical topic. Mathematical
Time Capsules provides materials enhancing student understanding and interest in mathematics, always keeping the
learning of mathematics clearly at the center of each capsule’s focus.

Preface ix
Acknowledgements
We could not have taken on this project without the interest and enthusiasm of the many contributors from campuses
around the world. We are indebted to the successful leadership efforts of V. Frederick Rickey and Victor Katz in
bringing together a community of educators and encouraging our passion for improving student learning through the
use of the history of mathematics. The editors of this volume met as participants in the NSF funded Institute for the
History of Mathematics and its use in Teaching, organized by Fred and Victor, which provided a sound foundation for
our work and a network of like-mindedcolleagues and friends. We also offer special thanks to Chris Arney, who led the
way as Department Head of the Department of Mathematical Sciences at the United States Military Academy at West
Point in not only documenting that the history of mathematics is a significant learning outcome at the program level in
undergraduate mathematics departments but also provided an exemplar of effective implementation. We both worked
for Chris and he made a real difference by encouraging and supporting our early use of the history of mathematics to
deepen student learning of our discipline. Finally, and perhaps most importantly, we also acknowledge the interest and
enthusiasm of our students, who let us know each semester that we teach that learning the history of mathematics is
important to them.
References
1. Amy Shell-Gellasch and D. Jardine, ed., From Calculus to Computers: Using the Last 200 Years of Mathematics
History in the Classroom, Mathematical Association of America, Washington, 2005.
2. NCATE/NCTM Program Standards (2003), Programs for Initial Preparation of Mathematics Teachers, Standards
for Secondary Mathematics Teachers
www.nctm.org/uploadedFiles/Math_Standards/,
accessed May 12, 2009.

Contents
Preface vii
1 The Sources of Algebra, Roger Cooke 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Egyptian problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Mesopotamian problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.4 “Algebra” in Euclid’s geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.5 Chinese problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.6 An Arabic problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.7 A Japanese problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.8 Teaching note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.9 Problems and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.10 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2 How to Measure the Earth, Lawrence D’Antonio 7
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Historical Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Taking it Further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3 Numerical solution of equations, Roger Cooke 17
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.2 The ancient Chinese method of solving a polynomial equation . . . . . . . . . . . . . . . . . . . . . 18
3.3 Non-integer solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 The cubic equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.5 Problems and questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.6 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Completing the Square through the Millennia, Dick Jardine 23
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.2 Historical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.3 Student activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.4 Summary and conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Appendix: Student activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
xi

xii Contents
5 Adapting the Medieval “Rule of Double False Position” to the Modern Classroom, Randy K. Schwartz 29
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
5.4 Taking It Further . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
6 Complex Numbers, Cubic Equations, and Sixteenth-Century Italy, Daniel J. Curtin 39
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
6.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
6.4 Rafael Bombelli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
7 Shearing with Euclid, Davida Fischman and Shawnee McMurran 45
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
7.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8 The Mathematics of Measuring Time, Kim Plofker 55
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
8.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
9 Clear Sailing with Trigonometry, Glen Van Brummelen 63
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
9.3 Navigating with Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
9.4 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
9.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
Appendix: Michael of Rhodes: Did He Know the Law of Sines? . . . . . . . . . . . . . . . . . . . . . . . 70
10 Copernican Trigonometry, Victor J. Katz 73
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
10.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

Contents xiii
11 Cusps: Horns and Beaks, Robert E. Bradley 89
11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
11.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
11.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
11.5 Notes on Classroom Use . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
12 The Latitude of Forms, Area, and Velocity, Daniel J. Curtin 101
12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
12.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
12.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
12.6 Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
13 Descartes’ Approach to Tangents, Daniel J. Curtin 107
13.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
13.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
14 Integration à la Fermat, Amy Shell-Gellasch 111
14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
14.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
14.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
15 Sharing the Fun: Student Presentations, Amy Shell-Gellasch and Dick Jardine 117
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
15.2 Getting Started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
15.3 Presentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
15.4 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
15.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
16 Digging up History on the Internet: Discovery Worksheets, Betty Mayfield 123
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
16.2 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
16.3 Learning about History — and about the Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
16.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
Appendix 1: Who Was Gauss? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix 2: History of Matrices and Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Appendix 3: Stephen Smale, a contemporary mathematician . . . . . . . . . . . . . . . . . . . . . . . . . 126
Appendix 4: Nancy Kopell, a female mathematician . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

xiv Contents
17 Newton vs. Leibniz in One Hour!, Betty Mayfield 127
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
17.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
17.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Appendix 1: Instructions to the class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
18 Connections between Newton, Leibniz, and Calculus I, Andrew B. Perry 133
18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
18.3 Newton’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
18.4 Leibniz’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
18.5 Berkeley’s Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
18.6 Later Developments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
18.7 In The Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
18.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
19 A Different Sort of Calculus Debate, Vicky Williams Klima 139
19.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
19.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
19.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Appendix A: Fermat’s Method Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
Appendix B: Barrow’s Theorem Worksheet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Appendix C: The Debates: Roles, Structure, Hints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
20 A ‘Symbolic’ History of the Derivative, Clemency Montelle 151
20.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
20.2 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
20.3 Isaac Newton (1643–1727): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
20.4 Gottfried Wilhelm von Leibniz (1646–1716): . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
20.5 Joseph-Louis Lagrange (1736–1813) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
20.6 Louis François Antoine Arbogast (1759-1803) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
20.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
20.8 Reflective Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
21 Leibniz’s Calculus (Real Retro Calc.), Robert Rogers 159
21.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
21.2 Differential Calculus (Rules of Differences) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
21.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

Contents xv
22 An “Impossible” Problem, Courtesy of Leonhard Euler, Homer S. White 169
22.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
22.2 Historical Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
22.3 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
22.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Appendix: Remarks on Selected Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
23 Multiple Representations of Functions in the History of Mathematics, Robert Rogers 179
23.1 Introduction (A Funny Thing Happened on the Way to Calculus) . . . . . . . . . . . . . . . . . . . . 179
23.2 Area of a Circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
23.3 Ptolemy’s Table . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
23.4 From Geometry to Analysis to Set Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
23.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
24 The Unity of all Science: Karl Pearson, the Mean and the Standard Deviation, Joe Albree 189
24.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
24.2 Karl Pearson: Historical preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
24.3 A data set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
24.4 The Mean and the First Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
24.5 The Second Moment and the Standard Deviation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
24.6 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
24.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
24.8 Activities and Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
25 Finding the Greatest Common Divisor, J.J. Tattersall 199
25.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
25.3 More Historical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
25.4 In The classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
25.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
26 Two-Way Numbers and an Alternate Technique for Multiplying Two Numbers, J.J. Tattersall 203
26.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
26.3 In The Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
26.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207
27 The Origins of Integrating Factors, Dick Jardine 209
27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
27.3 Mathematical preliminaries: Integrating factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
27.4 Bernoulli’s and Euler’s use of integrating factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
27.5 Student activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
Appendix: Student activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

xvi Contents
28 Euler’s Method in Euler’s Words, Dick Jardine 215
28.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
28.3 Euler’s description of the method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
28.4 Student Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
Appendix A: Student Assignments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Appendix B: Original source translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
29 Newton’s Differential Equation
P
y
P
x
D 1 3x C y C xx C xy, Hüseyin Koçak 223
29.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
29.2 Newton’s differential equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
29.3 Newton’s solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
29.4 Phaser simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
29.5 Remarks: Newton, Leibniz, and Euler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226
29.6 Suggested Explorations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
30 Roots, Rocks, and Newton-Raphson Algorithms for Approximating
p
2 3000 Years Apart, Clemency
Montelle 229
30.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
30.2 The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
30.3 Solving
p
2 — Second Millennium C.E. Style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
30.4 Time Warp: Solving
p
2 — Second Millennium B.C.E. Style . . . . . . . . . . . . . . . . . . . . . . 232
30.5 Taking it Further: Final Reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
30.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
31 Plimpton 322: The Pythagorean Theorem, More than a Thousand Years before Pythagoras, Daniel E.
Otero 241
31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
31.3 Reading the Tablet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
31.4 Sexagesimal numeration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
31.5 So What Does It All Mean? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
31.6 Why Tabulate These Numbers? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
31.7 Plimpton 322 in the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
31.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
32 Thomas Harriot’s Pythagorean Triples: Could He List Them All?, Janet L. Beery 251
32.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
32.2 Mathematical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
32.4 In the Classroom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
32.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Appendix: Who Was Thomas Harriot? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

Contents xvii
33 Amo, Amas, Amat! What’s the sum of that?, Clemency Montelle 261
33.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
33.2 The Harmonic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
33.3 Bernoulli and the Harmonic Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
33.4 The Mathematical Explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
33.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267
34 The Harmonic Series: A Primer, Adrian Rice 269
34.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
34.3 Introducing the harmonic series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
34.4 A “prime” piece of mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273
34.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
35 Learning to Move with Dedekind, Fernando Q. Gouvêa 277
35.1 Historical Background: What Dedekind Did . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
35.2 In the Classroom 1: Transition to Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
35.3 In the Classroom 2: Abstract Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
35.4 Moving with Dedekind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283
About the Editors 285

1
The Sources of Algebra
Roger Cooke
University of Vermont
1.1 Introduction
Nowadays we recognize written algebra by the presence of letters (called variables) standing for unspecified numbers,
and especially by the presence of equations involving those letters. These two features—letters and equations—reveal
the techniques of algebra, but algebra itself is not these techniques. Rather, algebra consists of problems in which the
goal is to find a number knowing certain indirect information about it. If you were told to multiply 7 by 3, then add
26 to the product, you would be doing arithmetic, that is, you would be given not only the data, but also told which
operations you must perform (multiplication followed by addition). But if you were asked for a number having the
property that if it is multiplied by 3 and 26 is added to the product, the result is 57, you would be facing an algebra
problem. In an algebra problem, the operations and some of the data are given to you, but these operations are not for
you to perform. Rather, you assume someone else has performed them, and you need to find the number(s) on which
they were performed. Using this definition, we can recognize algebra problems in very ancient texts that contain no
equations at all.
But how do such problems arise? Why were people interested in solving them? Those are questions that any student
who looks beyond the horizon of tomorrow’s homework assignment is bound to ask. In the following paragraphs,
we shall look at some examples and see if we can answer such questions. In this article, we are going to state a
number of problems taken from “classic” textbooks, paraphrased and reformulated in plain English and in a style
reflective of contemporary textbooks. Students who have had at least one year of algebra can study these problems in
the language of xs and ys and solve them using modern methods. At the same time, students are encouraged to use
their imaginations in order to think of the motivation that led each author to believe it was worthwhile to write about
problems of this sort.
1.2 Egyptian problems
Several problems from the Rhind Mathematical Papyrus, which was written some 3500 years ago, make use of the
notion of pesu, which measures the amount to which grain is “stretched” or diluted in making bread or beer. If you get
three (standard-size) loaves of bread per hekat of grain, the pesu of that bread is 3. Obviously a high pesu means very
thin or light bread (or a very small loaf) or very weak beer. Here is Problem 73 from the Rhind Papyrus: One hundred
loaves of pesu 10 are to be traded for loaves of pesu 15. How many of the latter will there be?
1

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Transcriber’s Note:
Hyphenation has been retained as it appeared
in the original publication; punctuation has been
standardised.
Page 59
exclaimed Nancy, increduously
changed to
exclaimed Nancy, incredulously
Page 81
“That one! ‘Busted’ badly!” She
mocked changed to
“That one! ‘Busted’ badly!” she mocked
Page 171
on the bedisde light changed to
on the bedside light
Page 214
repeated Nancy, increduously changed
to
repeated Nancy, incredulously
Page 241
They may be picnicing changed to
They may be picnicking
Page 278
were picnicing and frolicing around
changed to
were picnicking and frolicking around

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