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Product Code:  TEXT/61.S 
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eBook ISBN:  9781470456931 
Product Code:  TEXT/61.E 
List Price:  $89.00 
MAA Member Price:  $66.75 
AMS Member Price:  $66.75 
Softcover ISBN:  9781470472993 
eBook: ISBN:  9781470456931 
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 
Softcover ISBN:  9781470472993 
Product Code:  TEXT/61.S 
List Price:  $89.00 
MAA Member Price:  $66.75 
AMS Member Price:  $66.75 
eBook ISBN:  9781470456931 
Product Code:  TEXT/61.E 
List Price:  $89.00 
MAA Member Price:  $66.75 
AMS Member Price:  $66.75 
Softcover ISBN:  9781470472993 
eBook ISBN:  9781470456931 
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 DetailsAMS/MAA TextbooksVolume: 61; 2022; 687 ppMSC: Primary 01
The History of Mathematics: A SourceBased Approach is a comprehensive history of the development of mathematics. This, the second volume of a twovolume 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 twovolume set was designed as a textbook for the authors' acclaimed yearlong 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 Englishspeaking world.
ReadershipUndergraduate 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 midcentury

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. 18thcentury 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 18thcentury 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. 18thcentury Applied Mathematics

Introduction

10.1. The vibrating string

10.2. Euler’s vision of mechanics

10.3. Further reading

Chapter 11. 18thcentury 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. NonEuclidean 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 reunification 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 19thcentury 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


Additional Material

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 yearlong course, and they provide excellent material for a seniorlevel 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 onesemester 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 
There is an increasing interest in using history in the teaching of mathematics, and these volumes are a major resource.
Albert C. Lewis, MathSciNet


RequestsReview Copy – for publishers of book reviewsDesk Copy – for instructors who have adopted an AMS textbook for a courseExamination Copy – for faculty considering an AMS textbook for a courseAccessibility – to request an alternate format of an AMS title
 Book Details
 Table of Contents
 Additional Material
 Reviews
 Requests
The History of Mathematics: A SourceBased Approach is a comprehensive history of the development of mathematics. This, the second volume of a twovolume 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 twovolume set was designed as a textbook for the authors' acclaimed yearlong 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 Englishspeaking world.
Undergraduate and graduate students and researchers interested in the history of mathematics.

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 midcentury

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. 18thcentury 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 18thcentury 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. 18thcentury Applied Mathematics

Introduction

10.1. The vibrating string

10.2. Euler’s vision of mechanics

10.3. Further reading

Chapter 11. 18thcentury 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. NonEuclidean 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 reunification 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 19thcentury 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 yearlong course, and they provide excellent material for a seniorlevel 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 onesemester 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 
There is an increasing interest in using history in the teaching of mathematics, and these volumes are a major resource.
Albert C. Lewis, MathSciNet