Tales of Impossibility

The 2000-Year Quest to Solve the Mathematical Problems of Antiquity

David S. Richeson

• Publisher: Princeton University Press
• Year: 2019
• Print ISBN:
  9780691218724
  9780691192963
• eBook ISBN:
  9780691194233
• Topic: Mathematics
• Subtopic: Games in Mathematics
• Index: Mathematics

A comprehensive look at four of the most famous problems in mathematics

Tales of Impossibility recounts the intriguing story of the renowned problems of antiquity, four of the most famous and studied questions in the history of mathematics. First posed by the ancient Greeks, these compass and straightedge problems—squaring the circle, trisecting an angle, doubling the cube, and inscribing regular polygons in a circle—have served as ever-present muses for mathematicians for more than two millennia. David Richeson follows the trail of these problems to show that ultimately their proofs—which demonstrated the impossibility of solving them using only a compass and straightedge—depended on and resulted in the growth of mathematics.

Richeson investigates how celebrated luminaries, including Euclid, Archimedes, Viète, Descartes, Newton, and Gauss, labored to understand these problems and how many major mathematical discoveries were related to their explorations. Although the problems were based in geometry, their resolutions were not, and had to wait until the nineteenth century, when mathematicians had developed the theory of real and complex numbers, analytic geometry, algebra, and calculus. Pierre Wantzel, a little-known mathematician, and Ferdinand von Lindemann, through his work on pi, finally determined the problems were impossible to solve. Along the way, Richeson provides entertaining anecdotes connected to the problems, such as how the Indiana state legislature passed a bill setting an incorrect value for pi and how Leonardo da Vinci made elegant contributions in his own study of these problems.

Taking readers from the classical period to the present, Tales of Impossibility chronicles how four unsolvable problems have captivated mathematical thinking for centuries.

Table of contents

• Cover
• Contents
• Preface
• Introduction
• CHAPTER 1. The Four Problems
• Tangent: Cranks
• CHAPTER 2. Proving the Impossible
• Tangent: Nine Impossibility Theorems
• CHAPTER 3. Compass-and-Straightedge Constructions
• Tangent: The Tomahawk
• CHAPTER 4. The First Mathematical Crisis
• Tangent: Toothpick Constructions
• CHAPTER 5. Doubling the Cube
• Tangent: Eratosthenes’s Mesolabe
• CHAPTER 6. The Early History of π
• Tangent: The Great Pyramid
• CHAPTER 7. Quadratures
• Tangent: Leonardo da Vinci’s Lunes
• CHAPTER 8. Archimedes’s Number
• Tangent: Computing π at Home
• CHAPTER 9. The Heptagon, the Nonagon, and the Other Regular Polygons
• Tangent: It Takes Time to Trisect an Angle
• CHAPTER 10. Neusis Constructions
• Tangent: Crockett Johnson’s Heptagon
• CHAPTER 11. Curves
• Tangent: Carpenter’s Squares
• CHAPTER 12. Getting By with Less
• Tangent: Origami
• CHAPTER 13. The Dawn of Algebra
• Tangent: Nicholas of Cusa
• CHAPTER 14. Viète’s Analytic Art
• Tangent: Galileo’s Compass
• CHAPTER 15. Descartes’s Compass-and-Straightedge Arithmetic
• Tangent: Legislating π
• CHAPTER 16. Descartes and the Problems of Antiquity
• Tangent: Hobbes,Wallis, and the New Algebra
• CHAPTER 17. Seventeenth-Century Quadratures of the Circle
• Tangent: Digit Hunters
• CHAPTER 18. Complex Numbers
• Tangent: The τ Revolution
• CHAPTER 19. Gauss’s 17-gon
• Tangent: Mirrors
• CHAPTER 20. PierreWantzel
• Tangent: What Can We Construct with Other Tools?
• CHAPTER 21. Irrational and Transcendental Numbers
• Tangent: Top 10 Transcendental Numbers
• Epilogue: Sirens or Muses?
• Notes
• References
• Index
• Blank Page