Tales of Impossibility
The 2000-Year Quest to Solve the Mathematical Problems of Antiquity
David S. Richeson
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.
David S. Richeson is professor of mathematics at Dickinson College and editor of Math Horizons. He is the author of Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Princeton). Twitter @divbyzero