Tales of impossibility. The 2000-year quest to solve the mathematical problems of antiquity.

*(English)*Zbl 1429.01001
Princeton, NJ: Princeton University Press (ISBN 978-0-691-19296-3/hbk; 978-0-691-19423-3/ebook). xii, 436 p. (2019).

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 story of the ancient geometrical constructions problems can be told in a few words, by stating what they asked for, the successful attempts in antiquity to solve them with means surpassing ruler and compass, followed by Gauss’ discovery of more constructible regular \(n\)-gons, P. Wantzel’s 1837 paper proving the impossibility of cube duplication and angle trisection, as well as the proof that there are no other constructible regular \(n\)-gons, besides those mentioned by Gauss, to be followed by Lindemann’s 1882 proof that \(\pi\) is a transcendental number. Add to this Hippocrates’s squarable moons problem, with three squarable moons found by Hippocrates himself, two additional squarable moons discovered by D. Wijnquist in 1766 (to be rediscovered by Euler and Clausen), and the proof that there are no other moons with rational ratio of the angles formed between the center of each circle and the two points of intersection of the two circles forming the moon, spread out over the work of Landau (1902), Tschakaloff (1929), Chebotarëv (1935), and Dorodnov (1947). A theorem of A. Baker (1966) implies, as pointed out by Girstmair (2003), that there can be no other squarable moons, even if the ratio of the angles mentioned above is not assumed to be rational.

The author’s achievement consists in skillfully weaving many other strands into this story. The reader finds out about the discovery of incommensurable magnitudes, as well as Theodorus’ proof of irrationalities of the square roots of odd non-squares up to \(17\), a fairly comprehensive history of \(\pi\), down to the latest fast-converging series (and thus convenient for computing many exact digits of \(\pi\)), as well as interesting asides surrounding \(\pi\), such as its connection with Buffon’s needle or with the probability that two numbers picked at random are relativley prime, a visit to Archimedes’ workshop, the neusis construction, the quadratrix, the conchoid, the limaçon of Pascal, the spiral of Archimedes, compass-only constructions and the Mohr-Mascheroni theorem, constructions with a rusty compass, the Poncelet-Steiner theorem, origami, a history of algebra from Diophantus to Viète, Descartes’s Géométrie, the history of complex numbers and of transcendental numbers.

The reviewer found it odd that the author states that there are (only) four ancient Greek construction problems. Since he mentions on pages 99–103 Hippocrates’ squarable moon problem and provides the complete story of its solution (partly in footnotes), he must consider it part of the squaring of the circle problem, which it is not, given that the solution in 1882 of the latter did not solve the former, which took much longer (1966) to be settled. Had he considered that problem as a self-standing one (even though its origin lies in the squaring of the circle), he would have stated that there are five major ancient construction problems.

When mentioning constructions with a “rusty compass”, the author could have stated that all ruler-and-compass constructions can be performed with a rusty compass alone, as proved by J. Zhang et al. [Geom. Dedicata 38, No. 2, 137–150 (1991; Zbl 0722.51017)]. When mentioning Cauer’s work on straightedge-only constructions, he could have mentioned the correction brought by C. Gram [Math. Scand. 4, 157–160 (1956; Zbl 0070.16102)] ([A. Akopyan and R. Fedorov, Proc. Am. Math. Soc. 147, No. 1, 91–102 (2019; Zbl 1407.51023)] appeared too late to be included). Given his extraordinarily good bibliography, J. Schönbeck [Centaurus 46, No. 3, 208–229 (2004; Zbl 1073.01008)] and C. Lathrop and L. Stemkoski [The MAA Tercentenary Euler Celebration 5, 217–225 (2007; Zbl 1172.01305)] should be added to it and referred to when mentioning Clausen’s rediscovery of additional squarable moons. Regarding the whole squarable moons story, it is worth pointing out that C. J. Scriba [Mitt. Math. Ges. Hamb. 11, No. 5, 517–539 (1988; Zbl 0639.01001)] presents the complete story. Finally, when mentioning on page 96 a (modern) method for turning a rectangle into a square of the same area, one could also mention the ancient way of accomplishing that feat. A. Seidenberg [Arch. Hist. Exact Sci. 18, 301–342 (1978; Zbl 0392.01002)] has pointed out that the ancient method for doing so is, rather mysteriously, the same in Euclid’s Elements II.5 (together with the Pythagorean theorem) and in the Śulvasūtras. These should be taken as suggestions for a second edition, not as criticism of this book.

The author’s achievement consists in skillfully weaving many other strands into this story. The reader finds out about the discovery of incommensurable magnitudes, as well as Theodorus’ proof of irrationalities of the square roots of odd non-squares up to \(17\), a fairly comprehensive history of \(\pi\), down to the latest fast-converging series (and thus convenient for computing many exact digits of \(\pi\)), as well as interesting asides surrounding \(\pi\), such as its connection with Buffon’s needle or with the probability that two numbers picked at random are relativley prime, a visit to Archimedes’ workshop, the neusis construction, the quadratrix, the conchoid, the limaçon of Pascal, the spiral of Archimedes, compass-only constructions and the Mohr-Mascheroni theorem, constructions with a rusty compass, the Poncelet-Steiner theorem, origami, a history of algebra from Diophantus to Viète, Descartes’s Géométrie, the history of complex numbers and of transcendental numbers.

The reviewer found it odd that the author states that there are (only) four ancient Greek construction problems. Since he mentions on pages 99–103 Hippocrates’ squarable moon problem and provides the complete story of its solution (partly in footnotes), he must consider it part of the squaring of the circle problem, which it is not, given that the solution in 1882 of the latter did not solve the former, which took much longer (1966) to be settled. Had he considered that problem as a self-standing one (even though its origin lies in the squaring of the circle), he would have stated that there are five major ancient construction problems.

When mentioning constructions with a “rusty compass”, the author could have stated that all ruler-and-compass constructions can be performed with a rusty compass alone, as proved by J. Zhang et al. [Geom. Dedicata 38, No. 2, 137–150 (1991; Zbl 0722.51017)]. When mentioning Cauer’s work on straightedge-only constructions, he could have mentioned the correction brought by C. Gram [Math. Scand. 4, 157–160 (1956; Zbl 0070.16102)] ([A. Akopyan and R. Fedorov, Proc. Am. Math. Soc. 147, No. 1, 91–102 (2019; Zbl 1407.51023)] appeared too late to be included). Given his extraordinarily good bibliography, J. Schönbeck [Centaurus 46, No. 3, 208–229 (2004; Zbl 1073.01008)] and C. Lathrop and L. Stemkoski [The MAA Tercentenary Euler Celebration 5, 217–225 (2007; Zbl 1172.01305)] should be added to it and referred to when mentioning Clausen’s rediscovery of additional squarable moons. Regarding the whole squarable moons story, it is worth pointing out that C. J. Scriba [Mitt. Math. Ges. Hamb. 11, No. 5, 517–539 (1988; Zbl 0639.01001)] presents the complete story. Finally, when mentioning on page 96 a (modern) method for turning a rectangle into a square of the same area, one could also mention the ancient way of accomplishing that feat. A. Seidenberg [Arch. Hist. Exact Sci. 18, 301–342 (1978; Zbl 0392.01002)] has pointed out that the ancient method for doing so is, rather mysteriously, the same in Euclid’s Elements II.5 (together with the Pythagorean theorem) and in the Śulvasūtras. These should be taken as suggestions for a second edition, not as criticism of this book.

Reviewer: Victor V. Pambuccian (Glendale)