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Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity
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.
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.
456 pages, Hardcover
First published October 8, 2019
About the author
David S. Richeson
2 books25 followersDavid Richeson is a professor of mathematics at Dickinson College and the editor of Math Horizons, the undergraduate magazine of the Mathematical Association of America. He received his undergraduate degree from Hamilton College and his masters and PhD from Northwestern University. He lives with his wife and two children in Carlisle, Pennsylvania.
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Displaying 1 - 7 of 7 reviews
January 1, 2024
This is the first math history book I’ve ever read (gift from Max). This book clarified and unified the field of mathematics in my mind so eloquently and even inspired me to buy a set of basic geometry tools (compass and straightedge).
The content is rewarding while not being too proof-heavy, and for one with limited [pure] mathematics background such as myself, I can say that I merely had to digest the material more slowly than a modern-day Euclid, Archimedes, or Gauss might. In summary, this book gives a robust history of the field of synthetic geometry, and the necessary algebra and additionals fields of math required to prove the impossibility of the following 4 problems of antiquity:
1. Squaring the Circle
2. Doubling the Cube
3. Trisecting the Angle
4. Constructing all regular n-gons
These problems are indeed impossible to perform using the classical Euclidean construction tools. In spite of this fact, they have continuously captured and inspired the minds of countless individuals in the past and I count myself lucky to count myself among them.
The content is rewarding while not being too proof-heavy, and for one with limited [pure] mathematics background such as myself, I can say that I merely had to digest the material more slowly than a modern-day Euclid, Archimedes, or Gauss might. In summary, this book gives a robust history of the field of synthetic geometry, and the necessary algebra and additionals fields of math required to prove the impossibility of the following 4 problems of antiquity:
1. Squaring the Circle
2. Doubling the Cube
3. Trisecting the Angle
4. Constructing all regular n-gons
These problems are indeed impossible to perform using the classical Euclidean construction tools. In spite of this fact, they have continuously captured and inspired the minds of countless individuals in the past and I count myself lucky to count myself among them.
February 19, 2020
I'm not big brain enough to understand this
June 11, 2020
This is a really excellent book. Its starting point are the classical geometrical problems of Antiquity: the squaring of the circle, the trisection of a general angle, the doubling of the cube, and the construction of a general n-gon (a polygon with sides) using only a compass and an unmarked straight edge. The problems are stated with precision (and the reader is alerted to the crucial importance to state problems with precision!), and then the wonderful history of the effort of countless persons, some of the major genius of humankind among them, throughout more than two millennia to solve them and, in the end, to prove them impossible! Actually, part of this fascinating story is that, in the process of trying to solve them as the ancient Greeks stated them (with compass and straight edge) a whole new set of techniques were developed, and constructions using conics other than the circle, or using marked straight edge, or origami type techniques, were proved successful for some of the problems. Other proved a really harder nut to crack, such as the construction of the general n-gon (which required the intervention of complex numbers and was only completely settled by Pierre Wantzel in 1837) and, of course, the squaring of the circle, which boiled out to the proof of the transcendence of pi, achieved in the famous 1882 paper by Ferdinand von Lindemann, who used a slight adaptation of an earlier argument by Charles Hermite. In this long, two millennia process, we see some of the great human minds struggle with problems, invent techniques, develop concepts, creating new branches of Mathematics (such as Algebra and Analysis), refining and greatly expanding others (such as Geometry and Number Theory) and ultimately achieving the point where we have a sophisticated enough machinery to settle the problem for good! This is really a wonderful book telling a tremendous story! An example that sometimes the problem is really much more important than the solution...
April 12, 2020
More than you ever wanted to know about the three/four classic problems of antiquity. Touches on many other areas of mathematics.
October 16, 2020
Decent book, but it did not captivate me like Euler's Gem.
July 21, 2020
I enjoyed this book because I think the most beautiful and exceptional area of Mathematics is proofs. However, I’m personally not very emotionally fond of Euclidean Geometry, which takes a good part of the book. Anyhow, it’s still a great read about the development of impossible problems, the history of Mathematics and the advancements that were made trying to solve these problems.
March 9, 2020
math is good
This entire review has been hidden because of spoilers.
Displaying 1 - 7 of 7 reviews