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The History of Mathematics: A Source-Based Approach, Volume 2
 
June Barrow-Green The Open University, Milton Keynes, United Kingdom
Jeremy Gray The Open University, Milton Keynes, United Kingdom
Robin Wilson The Open University, Milton Keynes, United Kingdom
The History of Mathematics: A Source-Based Approach, Volume 2
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7299-3
Product Code:  TEXT/61.S
List Price: $89.00
MAA Member Price: $66.75
AMS Member Price: $66.75
eBook ISBN:  978-1-4704-5693-1
Product Code:  TEXT/61.E
List Price: $89.00
MAA Member Price: $66.75
AMS Member Price: $66.75
Softcover ISBN:  978-1-4704-7299-3
eBook: ISBN:  978-1-4704-5693-1
Product Code:  TEXT/61.S.B
List Price: $178.00 $133.50
MAA Member Price: $133.50 $100.13
AMS Member Price: $133.50 $100.13
The History of Mathematics: A Source-Based Approach, Volume 2
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The History of Mathematics: A Source-Based Approach, Volume 2
June Barrow-Green The Open University, Milton Keynes, United Kingdom
Jeremy Gray The Open University, Milton Keynes, United Kingdom
Robin Wilson The Open University, Milton Keynes, United Kingdom
MAA Press: An Imprint of the American Mathematical Society
Softcover ISBN:  978-1-4704-7299-3
Product Code:  TEXT/61.S
List Price: $89.00
MAA Member Price: $66.75
AMS Member Price: $66.75
eBook ISBN:  978-1-4704-5693-1
Product Code:  TEXT/61.E
List Price: $89.00
MAA Member Price: $66.75
AMS Member Price: $66.75
Softcover ISBN:  978-1-4704-7299-3
eBook ISBN:  978-1-4704-5693-1
Product Code:  TEXT/61.S.B
List Price: $178.00 $133.50
MAA Member Price: $133.50 $100.13
AMS Member Price: $133.50 $100.13
  • Book Details
     
     
    AMS/MAA Textbooks
    Volume: 612022; 687 pp
    MSC: Primary 01

    The History of Mathematics: A Source-Based Approach is a comprehensive history of the development of mathematics. This, the second volume of a two-volume set, takes the reader from the invention of the calculus to the beginning of the twentieth century. The initial discoverers of calculus are given thorough investigation, and special attention is also paid to Newton's Principia. The eighteenth century is presented as primarily a period of the development of calculus, particularly in differential equations and applications of mathematics. Mathematics blossomed in the nineteenth century and the book explores progress in geometry, analysis, foundations, algebra, and applied mathematics, especially celestial mechanics. The approach throughout is markedly historiographic: How do we know what we know? How do we read the original documents? What are the institutions supporting mathematics? Who are the people of mathematics? The reader learns not only the history of mathematics, but also how to think like a historian.

    The two-volume set was designed as a textbook for the authors' acclaimed year-long course at the Open University. It is, in addition to being an innovative and insightful textbook, an invaluable resource for students and scholars of the history of mathematics. The authors, each among the most distinguished mathematical historians in the world, have produced over fifty books and earned scholarly and expository prizes from the major mathematical societies of the English-speaking world.

    Readership

    Undergraduate and graduate students and researchers interested in the history of mathematics.

    This item is also available as part of a set:
  • Table of Contents
     
     
    • Front Cover
    • Title page
    • Copyright
    • Contents
    • Acknowledgments
    • Permissions & Acknowledgments
    • Introduction
    • Part I. The 17th and 18th centuries
    • Chapter 1. Introduction: The 17th and 18th centuries
    • Chapter 2. The Invention of the Calculus
    • Introduction
    • 2.1. Tangents, maxima, and minima
    • 2.2. Area and volume problems
    • 2.3. The situation mid-century
    • 2.4. Further reading
    • Chapter 3. Newton and Leibniz
    • Introduction
    • 3.1. Newton
    • 3.2. Newton and his calculus
    • 3.3. Leibniz and his calculus
    • 3.4. Further reading
    • Chapter 4. The Development of the Calculus
    • Introduction
    • 4.1. Inverse tangent problems
    • 4.2. Newton’s calculus and inverse tangent problems
    • 4.3. Newton’s mature calculus
    • 4.4. Leibniz’s mature calculus
    • 4.5. A comparison
    • 4.6. Further reading
    • Chapter 5. Newton’s Principia Mathematica
    • Introduction
    • 5.1. The creation of Newton’s Principia
    • 5.2. The content of the Principia
    • 5.3. Responses to the Principia
    • 5.4. Newton’s final years
    • 5.5. Further reading
    • Chapter 6. The Spread of the Calculus
    • Introduction
    • 6.1. The next generation
    • 6.2. The calculus, 1690–1730
    • 6.3. The Continental reception of the Principia
    • 6.4. Further reading
    • Chapter 7. The 18th century
    • Introduction
    • 7.1. Euler
    • 7.2. D’Alembert and Lagrange
    • 7.3. Algebra
    • 7.4. Further reading
    • Chapter 8. 18th-century Number Theory and Geometry
    • Introduction
    • 8.1. Number theory
    • 8.2. Infinite series
    • 8.3. Euler and geometry
    • 8.4. The study of curves
    • 8.5. Further reading
    • Chapter 9. Euler, Lagrange, and 18th-century Calculus
    • Introduction
    • 9.1. Early critiques of the calculus
    • 9.2. Euler’s calculus
    • 9.3. Differential equations
    • 9.4. The foundations of the calculus
    • 9.5. Further reading
    • Chapter 10. 18th-century Applied Mathematics
    • Introduction
    • 10.1. The vibrating string
    • 10.2. Euler’s vision of mechanics
    • 10.3. Further reading
    • Chapter 11. 18th-century Celestial Mechanics
    • Introduction
    • 11.1. Testing the Principia
    • 11.2. Academy prizes
    • 11.3. Laplace
    • 11.4. The stability of the solar system
    • 11.5. Jupiter and Saturn
    • 11.6. Further reading
    • Part II. The 19th Century
    • Chapter 12. Introduction: The 19th Century
    • Chapter 13. The Profession of Mathematics
    • Introduction
    • 13.1. The social context
    • 13.2. Mathematics in France
    • 13.3. Mathematics in Germany
    • 13.4. Journals and publishing
    • 13.5. The later 19th century
    • 13.6. Further reading
    • Chapter 14. Non-Euclidean Geometry
    • Introduction
    • 14.1. The first Western attempts
    • 14.2. Lobachevskii and Bolyai
    • 14.3. The reformulation of metrical geometry
    • 14.4. Further reading
    • Chapter 15. Projective Geometry and the Axiomatisation of Mathematics
    • Introduction
    • 15.1. The rediscovery of projective geometry in France
    • 15.2. Projective geometry in Germany
    • 15.3. The establishment of projective geometry
    • 15.4. The re-unification of geometry
    • 15.5. The axiomatisation of geometry
    • 15.6. Further reading
    • Chapter 16. The Rigorisation of Analysis
    • Introduction
    • 16.1. Bolzano, Cauchy, and continuity
    • 16.2. Cauchy’s mistake
    • 16.3. Cauchy on differentiation and integration
    • 16.4. Conclusion
    • 16.5. Further reading
    • Chapter 17. The Foundations of Mathematics
    • Introduction
    • 17.1. Dedekind’s definition of the real numbers
    • 17.2. Cantor, sets, and the infinite
    • 17.3. Foundational questions
    • 17.4. The philosophy of mathematics
    • 17.5. Set theory and logic
    • 17.6. Further reading
    • Chapter 18. Algebra and Number Theory
    • Introduction
    • 18.1. Number theory
    • 18.2. Prime numbers
    • 18.3. Complex numbers and quaternions
    • 18.4. Vectors
    • 18.5. Further reading
    • Chapter 19. Group Theory
    • Introduction
    • 19.1. Solving polynomial equations
    • 19.2. Galois and Galois theory
    • 19.3. Impossibility theorems
    • 19.4. Galois’s theory of groups and equations
    • 19.5. Group theory
    • 19.6. Further reading
    • Chapter 20. Applied Mathematics
    • Introduction
    • 20.1. The uses of Fourier series
    • 20.2. Potential theory
    • 20.3. Transatlantic cables
    • 20.4. Further reading
    • Chapter 21. Poincaré and Celestial Mechanics
    • Introduction
    • 21.1. Late 19th-century cel
    • 21.2. Henri Poincaré
    • 21.3. Poincaré and differential equations
    • 21.4. Poincaré and cel
    • 21.5. Poincaré’s memoir
    • 21.6. Poincaré’s later work in celestial mechanics
    • 21.7. Conclusion
    • 21.8. Further reading
    • Chapter 22. Coda
    • Introduction
    • 22.1. The international community of mathematicians
    • 22.2. Further reading
    • Chapter 23. Exercises
    • Advice on tackling the exercises
    • Exercises: Part A
    • Exercises: Part B
    • Exercises: Part C
    • Bibliography
    • Index
    • Back Cover
  • Reviews
     
     
    • This volume picks up where the authors left off in their first volume (2019), from about 1650 to the start of the 20th century. The intent of these books is to ask: Who did the mathematics, and why? How was the work disseminated (or not)? How did it emerge from the culture of the time, and why is it still relevant today? The approach is to use extensive quotations from original sources as the best way to answer some of those questions. The book concludes uniquely with dozens of suggested essay exercises that are "firmly historical, rather than primarily mathematical;" some call for supporting or contesting claims about mathematical discoveries. The two volumes were designed for a year-long course, and they provide excellent material for a senior-level course to help students survey the mathematics that they have learned and put it into cultural and scientific context.

      Mathematics Magazine
    • ...this volume (and the series as a whole) is an outstanding addition to the body of history of mathematics texts now available to instructors and students, providing a wonderfully rich treasure trove of primary source material. While few may choose this book as a text for a one-semester survey course on the history of mathematics, it is certainly an excellent option for those who wish to focus solely on the modern era from a professional historical perspective.

      Calvin Jongsma, Dordt University
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 612022; 687 pp
MSC: Primary 01

The History of Mathematics: A Source-Based Approach is a comprehensive history of the development of mathematics. This, the second volume of a two-volume set, takes the reader from the invention of the calculus to the beginning of the twentieth century. The initial discoverers of calculus are given thorough investigation, and special attention is also paid to Newton's Principia. The eighteenth century is presented as primarily a period of the development of calculus, particularly in differential equations and applications of mathematics. Mathematics blossomed in the nineteenth century and the book explores progress in geometry, analysis, foundations, algebra, and applied mathematics, especially celestial mechanics. The approach throughout is markedly historiographic: How do we know what we know? How do we read the original documents? What are the institutions supporting mathematics? Who are the people of mathematics? The reader learns not only the history of mathematics, but also how to think like a historian.

The two-volume set was designed as a textbook for the authors' acclaimed year-long course at the Open University. It is, in addition to being an innovative and insightful textbook, an invaluable resource for students and scholars of the history of mathematics. The authors, each among the most distinguished mathematical historians in the world, have produced over fifty books and earned scholarly and expository prizes from the major mathematical societies of the English-speaking world.

Readership

Undergraduate and graduate students and researchers interested in the history of mathematics.

This item is also available as part of a set:
  • Front Cover
  • Title page
  • Copyright
  • Contents
  • Acknowledgments
  • Permissions & Acknowledgments
  • Introduction
  • Part I. The 17th and 18th centuries
  • Chapter 1. Introduction: The 17th and 18th centuries
  • Chapter 2. The Invention of the Calculus
  • Introduction
  • 2.1. Tangents, maxima, and minima
  • 2.2. Area and volume problems
  • 2.3. The situation mid-century
  • 2.4. Further reading
  • Chapter 3. Newton and Leibniz
  • Introduction
  • 3.1. Newton
  • 3.2. Newton and his calculus
  • 3.3. Leibniz and his calculus
  • 3.4. Further reading
  • Chapter 4. The Development of the Calculus
  • Introduction
  • 4.1. Inverse tangent problems
  • 4.2. Newton’s calculus and inverse tangent problems
  • 4.3. Newton’s mature calculus
  • 4.4. Leibniz’s mature calculus
  • 4.5. A comparison
  • 4.6. Further reading
  • Chapter 5. Newton’s Principia Mathematica
  • Introduction
  • 5.1. The creation of Newton’s Principia
  • 5.2. The content of the Principia
  • 5.3. Responses to the Principia
  • 5.4. Newton’s final years
  • 5.5. Further reading
  • Chapter 6. The Spread of the Calculus
  • Introduction
  • 6.1. The next generation
  • 6.2. The calculus, 1690–1730
  • 6.3. The Continental reception of the Principia
  • 6.4. Further reading
  • Chapter 7. The 18th century
  • Introduction
  • 7.1. Euler
  • 7.2. D’Alembert and Lagrange
  • 7.3. Algebra
  • 7.4. Further reading
  • Chapter 8. 18th-century Number Theory and Geometry
  • Introduction
  • 8.1. Number theory
  • 8.2. Infinite series
  • 8.3. Euler and geometry
  • 8.4. The study of curves
  • 8.5. Further reading
  • Chapter 9. Euler, Lagrange, and 18th-century Calculus
  • Introduction
  • 9.1. Early critiques of the calculus
  • 9.2. Euler’s calculus
  • 9.3. Differential equations
  • 9.4. The foundations of the calculus
  • 9.5. Further reading
  • Chapter 10. 18th-century Applied Mathematics
  • Introduction
  • 10.1. The vibrating string
  • 10.2. Euler’s vision of mechanics
  • 10.3. Further reading
  • Chapter 11. 18th-century Celestial Mechanics
  • Introduction
  • 11.1. Testing the Principia
  • 11.2. Academy prizes
  • 11.3. Laplace
  • 11.4. The stability of the solar system
  • 11.5. Jupiter and Saturn
  • 11.6. Further reading
  • Part II. The 19th Century
  • Chapter 12. Introduction: The 19th Century
  • Chapter 13. The Profession of Mathematics
  • Introduction
  • 13.1. The social context
  • 13.2. Mathematics in France
  • 13.3. Mathematics in Germany
  • 13.4. Journals and publishing
  • 13.5. The later 19th century
  • 13.6. Further reading
  • Chapter 14. Non-Euclidean Geometry
  • Introduction
  • 14.1. The first Western attempts
  • 14.2. Lobachevskii and Bolyai
  • 14.3. The reformulation of metrical geometry
  • 14.4. Further reading
  • Chapter 15. Projective Geometry and the Axiomatisation of Mathematics
  • Introduction
  • 15.1. The rediscovery of projective geometry in France
  • 15.2. Projective geometry in Germany
  • 15.3. The establishment of projective geometry
  • 15.4. The re-unification of geometry
  • 15.5. The axiomatisation of geometry
  • 15.6. Further reading
  • Chapter 16. The Rigorisation of Analysis
  • Introduction
  • 16.1. Bolzano, Cauchy, and continuity
  • 16.2. Cauchy’s mistake
  • 16.3. Cauchy on differentiation and integration
  • 16.4. Conclusion
  • 16.5. Further reading
  • Chapter 17. The Foundations of Mathematics
  • Introduction
  • 17.1. Dedekind’s definition of the real numbers
  • 17.2. Cantor, sets, and the infinite
  • 17.3. Foundational questions
  • 17.4. The philosophy of mathematics
  • 17.5. Set theory and logic
  • 17.6. Further reading
  • Chapter 18. Algebra and Number Theory
  • Introduction
  • 18.1. Number theory
  • 18.2. Prime numbers
  • 18.3. Complex numbers and quaternions
  • 18.4. Vectors
  • 18.5. Further reading
  • Chapter 19. Group Theory
  • Introduction
  • 19.1. Solving polynomial equations
  • 19.2. Galois and Galois theory
  • 19.3. Impossibility theorems
  • 19.4. Galois’s theory of groups and equations
  • 19.5. Group theory
  • 19.6. Further reading
  • Chapter 20. Applied Mathematics
  • Introduction
  • 20.1. The uses of Fourier series
  • 20.2. Potential theory
  • 20.3. Transatlantic cables
  • 20.4. Further reading
  • Chapter 21. Poincaré and Celestial Mechanics
  • Introduction
  • 21.1. Late 19th-century cel
  • 21.2. Henri Poincaré
  • 21.3. Poincaré and differential equations
  • 21.4. Poincaré and cel
  • 21.5. Poincaré’s memoir
  • 21.6. Poincaré’s later work in celestial mechanics
  • 21.7. Conclusion
  • 21.8. Further reading
  • Chapter 22. Coda
  • Introduction
  • 22.1. The international community of mathematicians
  • 22.2. Further reading
  • Chapter 23. Exercises
  • Advice on tackling the exercises
  • Exercises: Part A
  • Exercises: Part B
  • Exercises: Part C
  • Bibliography
  • Index
  • Back Cover
  • This volume picks up where the authors left off in their first volume (2019), from about 1650 to the start of the 20th century. The intent of these books is to ask: Who did the mathematics, and why? How was the work disseminated (or not)? How did it emerge from the culture of the time, and why is it still relevant today? The approach is to use extensive quotations from original sources as the best way to answer some of those questions. The book concludes uniquely with dozens of suggested essay exercises that are "firmly historical, rather than primarily mathematical;" some call for supporting or contesting claims about mathematical discoveries. The two volumes were designed for a year-long course, and they provide excellent material for a senior-level course to help students survey the mathematics that they have learned and put it into cultural and scientific context.

    Mathematics Magazine
  • ...this volume (and the series as a whole) is an outstanding addition to the body of history of mathematics texts now available to instructors and students, providing a wonderfully rich treasure trove of primary source material. While few may choose this book as a text for a one-semester survey course on the history of mathematics, it is certainly an excellent option for those who wish to focus solely on the modern era from a professional historical perspective.

    Calvin Jongsma, Dordt University
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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