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.

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

"Tales of Impossibility is the story of a mathematical treasure hunt, and it's a treasure chest in its own right. Inside are nifty proofs, historical surprises, tasty miscellany, and most of all, the rich mathematical narrative of a quest that has consumed geniuses and eccentrics alike. This is the history of math's greatest tease—and it is immensely satisfying."—Ben Orlin, author of Math with Bad Drawings

"Tying together Lincoln, Napoleon, dramatic duels, and amazing intellectual achievements spanning more than two millennia, Tales of Impossibility presents a terrific story. Even while unfolding some of the oldest and most familiar logical challenges, Richeson uncovers intriguing ideas and details that will be new to all readers, even the most mathematically experienced."—Glen Whitney, founder of the National Museum of Mathematics