https://euro-math-soc.eu/review/tales-impossibility

Squaring the circle (i.e. finding the side of a square whose area equals the area of a circle using only compass and straightedge) is probably the best known problem that became iconic for representing something that is impossible to do. But the classical geometric problems from Greek antiquity, equally impossible to solve with these instruments are doubling the cube, trisection a general angle, and constructing a general regular polygon. For the latter two problems it could be done in some cases, but not for a general one. These problems have attracted many cranks who claimed to have solved the problem until Pierre Wantzel published his theorem in 1837 proving that these problems were impossible to solve (which by the way did not stop cranks to continue spamming editors with their solutions). Richeson describes in this book the history of these problems, and it is quite interesting to read that just trying to solve these problems gave rise to a lot of investigation that resulted in proper theorems. Progress towards the solution was only possible when algebra was introduced and linked to geometry and number theory. But of course the story starts with the definition of the problems and it is illustrated that solving them corresponds to the following problems respectively: given a segment of length 1, construct one with a length $pi, sqrt[3]{2}, cos(theta/3)$ for arbitrary $theta$, or $cos(360^circ/n)$ for arbitrary $n$. There is a lot to tell about the mathematics as it was practised in Greek antiquity. They had natural numbers of course, but they did not consider other numbers as such. In their geometric framework, it were rather ratios of lengths of line segments or of areas of geometric figures. The number $pi$ as the ratio of the circumference of a circle over its diameter is an universally known example. However this did not work so well with the area of a circle and its radius since the dimensions of an area and a length did not match. What made sense was comparing the area of a circle and the area of a square, which explains the problem of squaring the circle. Sticking to these ratios suggests that they considered all quantities as being commensurable, that is multiples of a fundamental unit, so that their number concept is essentially one of rational numbers. However $pi$ for the circle and $sqrt{2}$ in the Pythagoras theorem were known examples of non-rational numbers which they tried to approximate. Richeson skims the most important mathematicians of antiquity to illustrate how they dealt with $pi$ and how they attempted to find quadratures, that is to find squares or rectangles that had the same area as the area of another geometric figure. That could be a circle, but also other ones that showed up in their quest to solve the circle problem, like lunes or some parts of a circle or part of another conic. But it was Archimedes who came up with many formulas and with bounds for $pi$. Since the compass and straightedge were not able to solve the problem, people tried to relax on these restrictions. With neusis constructions or marked straightedge some of the unsolvable problems became solvable. In the centuries that followed, many ingenious instruments were designed to produce all kinds of curves (quadratrix, conchoid, limaçon of Pascal, spirals, carpenter's square curve, ...). On the other hand, one could try to find out what could be done with less than compass and straightedge. For example what if the compass is rusty and has only a fixed angle, or what if we only had a compass. Georg Mohr was the first to prove (1672) that with only a compass one can do everything that can be done with compass and straightedge. He was forgotten and Mascheroni re-discovered this much later. A straightedge alone however can not do the same job. When algebra was introduced in Europe by the Arabs, mathematicians concentrated on solving equations using formulas, rather than by geometric constructions. When people started to represent curves by algebraic formulas (traditionally attributed to Descartes), the idea of constructible numbers was born and the first impossibility claims emerged. More complicated quadratures and better approximations for $pi$ were computed, certainly with the newly invented calculus. With complex numbers new construction methods for regular $n$-gons were produced. They also gave formulas for solving polynomial equations and that paved the way for Pierre Wantzel (1814-1848) to eventually come up with his theorem about the degree of the minimal polynomial of a constructable number from which followed a precise statement about what was possible and what was impossible to produce with compass and straightedge. Richeson is able to bring the story as a popularizing book about mathematical history with a brief characterization or biography of the mathematicians involved and of course the evolution of mathematics from geometrical ideas of antiquity to the algebraic number theory of Wantzel. It is however far from a "hard core" history book. It has many citations (all in English) and there are many notes at the end, but these are all informative and do not disrupt the reading. The impossible problems discussed are easy to understand and have attracted many mathematical hobbyists in the past. So the discussion in this book is easily accessible from a mathematical point of view. Although every chapter is somehow related to the four problems, Richeson takes a broad view and the computation of $pi$ or the constructions of curves with mechanical instruments can hardly be called diversions from the main theme. Most amusing are the intermezzo's that he calls "tangents" after each chapter. These give some diversions of all sorts. The first one about how to recognize a mathematical crank is particularly amusing. There are others like what geometry can be done using toothpicks (line segments with a fixed length), how to compute $pi$ at home, what geometric constructions are possible using square origami paper, there is the story of the Indiana $pi$-bill in which Edwin Goodwin tried to pass his circle squaring by law, etc. The whole book, both informative and amusing, is a highly recommended read.