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

Many books have been written about the classical Greek geometric construction problems (using a compass and straightedge to square the circle, double the cube, and trisect an angle). I used one, Charles Hadlock’s excellent Field theory and its classical problems, when I taught Galois Theory last fall. David Richeson’s Tales of Impossibility: The 2000-Year Quest to Solve the Mathematical Problems of Antiquity is the latest book to discuss these problems.

Richeson traces the history of these problems, along with the closely related problem of using a compass and straightedge to inscribe a regular \(n\)-sided polygon inside of a given circle. The story begins in ancient Greece in the fifth century BCE and works its way through Egypt, India, the Islamic empire, Italy and France before ending in nineteenth century Germany with Lindemann’s 1882 proof that \( \pi \) is transcendental (which implies that one cannot square the circle). Given the enormous length of time considered, it is perhaps not surprising that the book is in some sense a history of mathematics itself. Early on we learn about Hippasus’ discovery that the side and diagonal of a square are incommensurable magnitudes, a fact that is equivalent to the irrationality of \(\sqrt{2}\). Later on we learn about the development of the Indo-Arabic numeral system, the development of algebra, the discovery of complex numbers and of transcendental numbers, etc. There’s nothing here that will be new to the historian, and little is covered in great depth, but Richeson’s pace is leisurely enough to give one a good idea of what’s going on and whet one’s appetite for a more in-depth study of the topics covered.

In describing some of the most important developments in mathematics Richeson is always careful to include sections on the (often slow) acceptance of these new ideas by the mathematicians of the age. This is one of my favorite aspects of the book. The most memorable instance of this is when, after discussing Descartes’s melding of algebra and geometry, Richeson goes forward in time and paints an Isaac Newton that is truly conflicted about the infusion of algebra into geometry. Newton is as responsible as anyone for the blending of these two disciplines, and algebra was of course a crucial ingredient in his work on calculus. Nevertheless, in his Universal Arithmetick he writes:

The Ancients so assiduously distinguished them one from the other that they never introduced arithmetical terms into geometry; while recent people, by confusing both, have lost the simplicity in which all elegance in geometry consists.

The real workhouses in this book are Chapter 13: The Dawn of Algebra and Chapter 14: Viète’s Analytic Art. These chapters take us from a mathematical landscape dominated by Euclidean geometry to the Renaissance, where Richeson describes the work of Cardano, Ferrari, Tartaglia, Viète and others on solutions of cubic and quartic equations. By the end of chapter 14 we’ve seen the development of the Cardano-Tartaglia formula for the solutions of cubic equations with arbitrary coefficients (having a certain form), as well as some early attempts at using the newly developed algebra to attack the Greek impossibility problems.

Although Viète’s geometric work was noteworthy for its novel use of algebra, it’s really only in chapter 15 that things start to come together. In this chapter we learn about Descartes’s Geometry and his theory of arithmetic for line segments. In particular Descartes showed how one can use a compass and straightedge in order to add, subtract, multiply and divide line segments. This motivated Descartes’s conjecture about what numbers can be constructed by ruler and compass: a real number is constructible if and only if it can be obtained from the integers by means of addition, subtraction, multiplication, division, and the extraction of square roots. Not all real numbers are constructible of course, and if we can show that the numbers \(\sqrt[3]{2}, \cos(\theta/3), \cos(2\pi/n),\pi \) are not constructible then it will follow that one cannot double a cube, trisect a general angle, construct a regular n-gon, or square a circle.

At this point Richeson works towards Wantzel’s theorem, which says that the degree of the minimal polynomial of a constructible number must be a power of \(2\). Using this theorem one can give very short proofs that it is impossible to double a cube, trisect an arbitrary angle and construct a regular \(n\)-gon for all \(n\). For example, to see that one cannot double a cube we simply note that doubling a cube is equivalent to proving that \(\sqrt[3]{2}\) is constructible. But the minimal polynomial of \(\sqrt[3]{2}\) is \(x^3-2\), which has degree \(3\) and therefore does not satisfy the criterion given in Wantzel’s theorem. Pierre Wantzel is probably the least well known of the actors in this story, so it is nice that Richeson devotes an entire chapter to his life and his mathematics.

The final chapter of the book is devoted to Lindemann’s proof that one cannot square the circle. The idea of his proof is that it suffices to prove that \(\pi\) is not a constructible number. By Wantzel’s theorem it suffices to prove that \(\pi\) is transcendental. What Lindemann actually proved was that if \(a\) is a non-zero algebraic number then \(e^a\) is transcendental. To see that this proves that \(\pi\) is transcendental, we suppose that \(\pi\) is algebraic and apply the contrapositive of Lindemann’s result with \(a=\pi i\). Richeson goes on to describe Hilbert’s seventh problem, which asks for a generalization of Lindemann’s result, and culminates with the Gelfond-Schneider theorem.

Tales of Impossibility was a real pleasure to read. The exposition is friendly but not chatty and scholarly but not dry. After every chapter there is a brief “Tangent” section where Richeson devotes a few pages to describing an interesting digression. These tangents range from the academic (philosopher Thomas Hobbes’ work on geometry) to the weird (can you trisect an angle given a compass, straightedge and tomahawk? can you double a cube using origami?) to the just plain nutty (the Indiana legislature’s 1897 bill claiming that \(\pi=3.2\)).

Friendly as the book is, it is definitely a math book. Richeson doesn’t give any technical proofs or assume any background on the part of the reader, but he is clearly very intent on making sure that the reader gets the “big picture” and is able to see how all of the different mathematical results and subfields described interact together and lead to the resolution of the four geometric construction problems. He doesn’t shy away from presenting the reader with dozens of messy yet beautiful Euclidean geometry figures (the ones crammed full of intersecting lines, arcs and triangles) when they’re necessary to describe the work of various mathematicians and isn’t afraid of writing out technical expressions when describing things like formulas for the solutions of certain cubic equations or consequences of Euler’s formula \(e^{i\theta}=\cos(\theta)+i\sin(\theta)\).

Tales of Impossibility is neither a textbook nor a book meant for the general public. But I think that it’s a perfect book for a young math major. It provides the reader with a fascinating glimpse into the 2000 year history of four geometric problems and, along the way, offers snapshots of many highlights from the history of mathematics.

Benjamin Linowitz (benjamin.linowitz@oberline.edu) is an Assistant Professor of Mathematics at Oberlin College. His website can be found at http://www2.oberlin.edu/faculty/blinonowit/.