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Math Through the Ages: A Gentle History for Teachers and Others

William P. Berlinghoff and Fernando Q. Gouvêa
MAA Press/Oxton House Publishers
Publication Date: 
Number of Pages: 
MAA Press Textbooks
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

See our reviews of the Expanded First Edition and of the (non-expanded) Second Edition. This book continues to exist in two states: a reader’s edition (published by Oxton House) and this expanded edition (jointly published by Oxton House and the Mathematical Association of America) that includes questions and projects for classroom use. There is an Instructor’s Guide with answers and comments available to those who adopt the book.

In comparison to the Expanded First Edition, the major change here is the inclusion of five new “historical sketches”:

  • Out of the Shadows: The Tangent Function
  • Counting Ratios: Logarithms
  • Any Way You Slice It: Conic Sections
  • Beyond the Pale: Irrational Numbers
  • Barely Touching: From Tangents to Derivatives

The older material has been revised, expanded in places, and corrected when necessary. The corrections sometimes reflect recent scholarship, for example new translations of the original sources. The bibliography and references to the literature have been completely revised to reflect the scholarship produced during the intervening years. The historical overview “in a large nutshell” has grown just a little bit. The section on what to read next was completely rewritten. A new section entitled When they Lived collects birth and death dates for all significant persons mentioned in the book.

The Instructor’s Guide contains answers to the “questions” and comments on the “projects,” usually fairly short but sometimes extensive and pointing to additional references. In at least one place, the authors put in the Guide an extra problem that wouldn’t fit in the text.

Fernando Q. Gouvêa is editor of MAA Reviews and co-author of Math through the Ages.

   to the Second Edition  vii
   to the First Edition  ix

History in the Mathematics Classroom  1

The History of Mathematics in a Large Nutshell  5
   Beginnings  6
   Greek Mathematics  15
   Meanwhile, in India  25
   Arabic Mathematics  29
   Medieval Europe  33
   The 15th and 16th Centuries  35
   Algebra Comes of Age  37
   Calculus and Applied Mathematics  43
   Rigor and Professionalism  48
   Abstraction, Computers, and New Applications  54
   Mathematics Today  59


  1. Keeping Count: Writing Whole Numbers  67
  2. Reading and Writing Arithmetic: The Basic Symbols  75
  3. Nothing Becomes a Number: The Story of Zero  81
  4. Broken Numbers: Writing Fractions  87
  5. Less Than Nothing?: Negative Numbers  95
  6. By Tens and Tenths: Metric Measurement  103
  7. Measuring the Circle: The Story of \(\pi\)  109
  8. The Cossic Art: Writing Algebra with Symbols  115
  9. Linear Thinking: Solving First Degree Equations  123
  10. A Square and Things: Quadratic Equations  129
  11. Intrigue in Renaissance Italy: Solving Cubic Equations  135
  12. A Cheerful Fact: The Pythagorean Theorem  141
  13. A Marvelous Proof: Fermat's Last Theorem  149
  14. On Beauty Bare: Euclid's Plane Geometry  157
  15. In Perfect Shape: The Platonic Solids  165
  16. Shapes by the Numbers: Coordinate Geometry  171
  17. Impossible, Imaginary, Useful: Complex Numbers  179
  18. Half Is Better: Sine and Cosine  187
  19. Strange New Worlds: The Non-Euclidean Geometries  195
  20. In the Eye of the Beholder: Projective Geometry  203
  21. What's in a Game?: The Start of Probability Theory  209
  22. Making Sense of Data: Statistics Becomes a Science  217
  23. Machines that Think?: Electronic Computers  225
  24. The Arithmetic of Reasoning: Boolean Algebra  233
  25. Beyond Counting: Infinity and the Theory of Sets  239
  26. Out of the Shadows: The Tangent Function  247
  27. Counting Ratios: Logarithms  255
  28. Any Way You Slice It: Conic Sections  263
  29. Beyond the Pale: Irrational Numbers  271
  30. Barely Touching: From Tangents to Derivatives  279

What to Read Next  287
   The Reference Shelf  287
   Twelve Historical Books You Ought to Read  290
   History Online  292

When They Lived  295

Bibliography  301

Index  319

About the Authors