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

CoverPrinceton University Press (2019)
Hardcover ISBN: 9780691192963
E-book ISBN: 9780691194233

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.


“I greatly enjoyed Richeson’s Tales of Impossibility. It deserves to become a classic and can be highly recommended.”
—Robin Wilson, Times Higher Education

“The historical narrative, the biographical gems, and the enjoyable tangents all make for a highly entertaining read. Nevertheless, it is a scholarly book, richly annotated with endnotes that don’t interrupt the flow of the story.”
—Robert Bradley, Math Horizons

“Even if you never read a single proof through to its conclusion, you’ll enjoy the many entertaining side trips into a geometry far beyond what you learned in high school.”
—Jim Stein, New Books Network

“The whole book, both informative and amusing, is a highly recommended read.”
—Adhemar Bultheel, European Mathematical Society

“By spreading a rather wide net around the history of the Greek geometric construction problems, the author has written a particularly readable history, aimed at a general audience, of a significant part of ancient Greek geometry, as well as of algebra all the way to the first half of the 19th century. . . . The author’s achievement consists in skillfully weaving many other strands into this story.”
—Victor Pambuccian, zbMATH review

“Let me get right to the point: this was hands-down my favorite math book that I read this year. If you don’t already have a copy, you should stop reading this post right now and go buy one! Go on, you’ll thank me.”
—Brent Yorgey, The Math Less Traveled

One of Ben Orlin’s “Books that I loved in 2019.”
—Ben Orlin, Math with Bad Drawings

“The writing style is very easy to read with diagrams throughout and just about the right level of detail so that you can really keep up with every argument, even though not all proofs are laid out in detail. Overall this book was a pleasure to read and I would recommend it for anybody who wants a lovely overview of many areas of the history of mathematics, with a focus on some very easy to understand problems.”
—Jonathan Shock, Mathemafrica

“I have to recommend David Richeson’s Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity. . . . [He] has had success writing an earlier volume on mathematics and its history, and this one is an even more epic story. Being stuck in the house makes finding the time for reading it much less of a task. It is also useful to be reminded that having a feeling that something is doable doesn’t make it so.”
—Thomas Drucker, Here’s What Princeton Alumni and Faculty Are Reading While Staying Home


“This engaging and well-written book covers more ground than previous books on the classical improbability problems. Numerous historical asides add to the enjoyment of this work. Highly recommended!”
—Eli Maor, author of Music by the Numbers

Tales of Impossibility presents an absorbing account of the history and mystery of problems whose infeasibilities are woven into the architecture of mathematics itself. Richeson shows us that what is not possible can be just as inspiring as what is. All math lovers will find gems to mine here.”
—Francis Su, author of Mathematics for Human Flourishing

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

Tales of Impossibility contains mathematics that is interesting and perhaps new to most readers. The book features helpful diagrams and footnotes, quotations that amplify the subject matter, and even funny material here and there.”
—William Dunham, author of The Calculus Gallery

“Richeson has put together a fascinating account of mathematical impossibility, focusing on the ruler and compass problems of the ancient Greeks. This is a story of the problems and the people involved—but even more so of the changes in mathematical thinking that made it possible to prove impossibility.”
—Henry Segerman, Oklahoma State University

“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


p. 10, figure 1.3: “base 1 and height 2” should be “base 2 and height 1”
p. 77, line -1: a/y=x/y=y/b should be a/x=x/y=y/b
p. 251, line 3: 1928? should be 1825
p. 279, line 4: 22 trillion should be 31 trillion