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Journey Through Genius: The Great Theorems of Mathematics

William Dunham
Penguin Books
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The Basic Library List Committee considers this book essential for undergraduate mathematics libraries.

[Reviewed by
Charles Traina
, on

The book by William Dunham has become a classic text covering some of the history of mathematics. In the preface Dunham writes:

For disciplines as diverse as literature , music, and art, there is a tradition of examining masterpieces- the “great novels,” the “great symphonies,” the “great paintings”- as the finest and most illuminating objects of study. Books are written and courses are taught on precisely these topics to acquaint us with some of the creative milestones of the discipline and with the men and women who produced them.

The present book offers an analogous approach to mathematics, where the creative unit is not the novel or symphony, but the theorem.

This is exactly what Dunham does. He introduces the reader to some theorems which have withstood the test of time and are recognized as important milestones and masterpieces in mathematics.

Dunham writes in a clear and easy to read style. There is enough history about the mathematician and what led to his or her theorem to set the stage and background, without becoming a lesson on history. The mathematics is clearly presented with clear notation. There is enough of an explanation for the reader to understand the mathematics and invite the reader to research further the topic.

I have used the book several times when teaching Senior Seminar, the capstone course for mathematics majors. The book brings together topics that are covered in the undergraduate program and shows the nature of mathematical research and invites the student to research a topic further.

The theorems include propositions from Euclid, Archimedes’ determination of the area of a circle, Heron’s Formula for the area of a triangle, something which is not always covered, the work of Newton, and more.

In summary, Dunham’s book is a classic in the history of mathematics. The reader will learn not only mathematics but gain insight to the personalities of the mathematicians responsible.

Charles Traina is Professor of Mathematics at St. John’s University in New York.



Chapter 1. Hippocrates' Quadrature of the Lune (ca. 440 B.C.)
The Appearance of Demonstrative Mathematics
Some Remarks on Quadrature
Great Theorem

Chapter 2. Euclid's Proof of the Pythagorean Theorem (ca. 300 B.C.)
The Elements of Euclid
Book I: Preliminaries
Book I: The Early Propositions
Book I: Parallelism and Related Topics
Great Theorem

Chapter 3. Euclid and the Infinitude of Primes (ca. 300 B.C.)
The Elements, Books II-VI
Number Theory in Euclid
Great Theorem
The Final Books of the Elements

Chapter 4. Archimedes' Determination of Circular Area (ca. 225 B.C.)
The Life of Archimedes
Great Theorem
Archimedes' Masterpiece: On the Sphere and the Cylinder

Chapter 5. Heron's Formula for Triangular Area (ca. A.D. 75)
Classical Mathematics after Archimedes
Great Theorem

Chapter 6. Cardano and the Solution of the Cubic (1545)
A Horatio Algebra Story
Great Theorem
Further Topics on Solving Equations

Chapter 7. A Gem from Isaac Newton (Late 1660s)
Mathematics of the Heroic Century
A Mind Unleashed
Newton's Binomial Theorem
Great Theorem

Chapter 8. The Bernoullis and the Harmonic Series (1689)
The Contributions of Leibniz
The Brothers Bernoulli
Great Theorem
The Challenge of the Brachistochrone

Chapter 9. The Extraordinary Sums of Leonhard Euler (1734)
The Master of All Mathematical Trades
Great Theorem

Chapter 10. A Sampler of Euler's Number Theory (1736)
The Legacy of Fermat
Great Theorem

Chapter 11. The Non-Denumerability of the Continuum (1874)
Mathematics of the Nineteenth Century
Cantor and the Challenge of the Infinite
Great Theorem

Chapter 12. Cantor and the Transfinite Realm (1891)
The Nature of Infinite Cardinals
Great Theorem

Chapter Notes


akirak's picture

This book gives a thorough treatment of the history of some important mathematical results. There are a number of interesting mathematical examples set in an historical context which makes the book very joyful to read. The author has hit the right balance between the mathematics (such as proofs of theorems) and the history behind each theorem. It is good to see the author does not shy away from producing proofs of results which many popular writers tend to eschew so that they can increase their sales. The result can be challenging at times for the reader, but these parts can be skipped without losing the flow. Dunham has a fantastic writing style which keeps the reader hooked and intrigued.

Another great asset of the book is that it is portable and reasonably cheap at around £10. I managed to read most of it in Starbucks, with pen and paper of course. However I have following reservations:

  • The font size is too small and it is particularly difficult to read some of the fractions.
  • I found two typographical errors: On page 170 the fraction should be 5/128 and not 5/12; on page 238 the factorization should be over 2a and not just a.
  • On page 235 the first Fermat prime 3 is missing.

A less serious issue is that once the author has covered a particular concept he expects the reader to have fully digested it. Dunham has tackled this by signposting his earlier results, but I think it would have been more readable for the layman to see the statement of the result again.

This is a book for anybody interested in history of mathematics or mathematics in general. You do not need to be a mathematician to appreciate this book. Overall I would say this is an excellent book and would recommend anybody interested in mathematics to purchase this.