Can you square the circle? At school, we all learned how to divide a given angle into two equal parts by using just an unmarked ruler (no measuring allowed!) and a compass. The ancient Greeks described this and many other constructions, but found themselves unable to carry out any of the following tasks:
Trisecting an angle: dividing a given angle into three equal parts.
Doubling a cube: constructing a cube with twice the volume of a given cube.
Squaring the circle: constructing a square whose area equals that of a given circle.
The difficulty of the last of these led to the use of the phrase “squaring the circle” for a task that seemed impossible to carry out. But was it really impossible or just exceedingly difficult?
Over the next 2,000 years, mathematicians and cranks alike sought ruler-and-compass constructions for these problems or attempted to show why no such constructions can be achieved. In the 17th century in particular, René Descartes, John Wallis and Thomas Hobbes failed to provide answers, and it was not until the 19th century that a relatively unknown figure, Pierre Wantzel, proved definitively that the first two constructions cannot be implemented. Later, in 1882, Ferdinand von Lindemann proved the impossibility of squaring the circle.
There was yet another problem that defeated the Greeks. They described how to construct equilateral triangles, squares and regular pentagons, but were unable to construct regular polygons with seven or nine sides. This led to the following question: which regular polygons can be constructed by ruler and compass?
This problem also had to wait some 2,000 years for a solution, when Carl Friedrich Gauss showed how to construct a regular 17-sided polygon and gave a necessary condition (involving certain prime numbers) for deciding when such regular polygons can be constructed – a condition that was later proved by Wantzel to be sufficient.
This area of geometry lends itself to popular exposition, and it seems surprising that more books on the subject have not appeared over the years. But this one has been worth waiting for. The author is already well known for his book, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (2012), on Leonhard Euler’s assertion that the numbers of faces and corners of any polyhedron add together to two more than the number of edges. For example, a cube has six faces, eight corners and twelve edges, and 6 + 8 = 12 + 2. David Richeson’s latest book is equally well written and presents a detailed historical account of the four problems listed above, with welcome diversions into such topics as geometrical constructions; the meaning of impossibility when applied to mathematical proofs and constructions; the history of the circle number; the parallel development of algebra; the nature of irrational and transcendental numbers; and much else besides. He also gives pointed advice to the many cranks who still try to solve the four problems without the necessary background knowledge or an understanding of the futility of their task.
I greatly enjoyed Richeson’s Tales of Impossibility. It deserves to become a classic and can be highly recommended.
Robin Wilson is professor of pure mathematics at The Open University. His books include Lewis Carroll in Numberland: His Fantastical Mathematical Logical Life (2009) and Euler’s Pioneering Equation: The Most Beautiful Theorem in Mathematics (2018).
Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity
By David S. Richeson
Princeton University Press, 456pp, £25.00
ISBN 9780691192963
Published 27 November 2019
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Print headline: Right angles and wrong ones
