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

A book on four unsolvable mathematical problems that have captivated scholars since antiquity. **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—demonstrating 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. "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"--Publisher's description A comprehensive look at four of the most famous problems in mathematics Tales of Impossibility recounts the intriguing story of the so-called 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—demonstrating the impossibility of solving them using only a compass and straightedge—depended upon and resulted in the growth of mathematics. Richeson explores how celebrated luminaries, including Euclid, Archimedes, Viète, Descartes, Newton, and Gauss labored to understand the problems of antiquity, and how many major mathematical discoveries were related to these explorations. Though 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. A little-known mathematician named Pierre Wantzel and Ferdinand von Lindemann, through his work on π, finally determined the problems were impossible to solve. Along the way, Richeson provides entertaining anecdotes connected to the problems, such as how the Indiana legislature passed a bill setting an incorrect value for π, and how Leonardo da Vinci made elegant contributions to the puzzles. Taking readers from the classical period to the present, Tales of Impossibility demonstrates how four unsolvable problems captivated mathematical thinking for centuries. 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. 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, this volume chronicles how four unsolvable problems have captivated mathematical thinking for centuries. --From publisher description 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. Pierre Wantzel 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 David S. Richeson. Includes Bibliographical References (pages 405-428) And Index.