#### Overview

Dunham places each theorem within its historical context and explores the very human and often turbulent life of the creator — from Archimedes, the absentminded theoretician whose absorption in his work often precluded eating or bathing, to Gerolamo Cardano, the sixteenth-century mathematician whose accomplishments flourished despite a bizarre array of misadventures, to the paranoid genius of modern times, Georg Cantor. He also provides step-by-step proofs for the theorems, each easily accessible to readers with no more than a knowledge of high school mathematics.

A rare combination of the historical, biographical, and mathematical, **Journey Through Genius** is a fascinating introduction to a neglected field of human creativity.

#### Praise

—

**Los Angeles Times**

“It is mathematics presented as a series of works of art; a fascinating lingering over individual examples of ingenuity and insight. It is mathematics by lightning flash.”

— Isaac Asimov

“Dunham deftly guides the reader through the verbal and logical intricacies of major mathematical questions, conveying a splendid sense of how the greatest mathematicians from ancient to modern times presented their arguments.”

—Ivars Peterson, author of **The Mathematical Tourist**

#### Table of Contents

Acknowledgments

**Chapter 1. Hippocrates’ Quadrature of the Lune (ca. 440 B.C.)**

The Appearance of Demonstrative Mathematics

Some Remarks on Quadrature

Great Theorem

Epilogue

**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

Epilogue

**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*

Epilogue

**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*

Epilogue

**Chapter 5. Heron’s Formula for Triangular Area (ca. A.D. 75)**

Classical Mathematics after Archimedes

Great Theorem

Epilogue

**Chapter 6. Cardano and the Solution of the Cubic (1545)**

A Horatio Algebra Story

Great Theorem

Further Topics on Solving Equations

Epilogue

**Chapter 7. A Gem from Isaac Newton (Late 1660s)**

Mathematics of the Heroic Century

A Mind Unleashed

Newton’s Binomial Theorem

Great Theorem

Epilogue

**Chapter 8. The Bernoullis and the Harmonic Series (1689)**

The Contributions of Leibniz

The Brothers Bernoulli

Great Theorem

The Challenge of the Brachistochrone

Epilogue

**Chapter 9. The Extraordinary Sums of Leonhard Euler (1734)**

The Master of All Mathematical Trades

Great Theorem

Epilogue

**Chapter 10. A Sampler of Euler’s Number Theory (1736)**

The Legacy of Fermat

Great Theorem

Epilogue

**Chapter 11. The Non-Denumerability of the Continuum (1874)**

Mathematics of the Nineteenth Century

Cantor and the Challenge of the Infinite

Great Theorem

Epilogue

**Chapter 12. Cantor and the Transfinite Realm (1891)**

The Nature of Infinite Cardinals

Great Theorem

Epilogue

Afterword

Chapter Notes

References

Index