On the hypotenuse BC of an isosceles right triangle ABC, construct, outwardly, a semicircle Ω_1, and construct an arc Ω_2 with center A and radius AB. The lune-like region enclosed between Ω_1 and Ω_2 is what is known as the lune of Hippocrates of Chios (fifth century BCE), who proved that its area equals that of the triangle ABC, thus exhibiting the first example of a curvilinear region that can be squared (in the sense of constructing with straightedge and compass a square of equal area). This is particularly interesting when viewed in the context of attempts at squaring the circle (and having in mind that triangles, and in fact all polygonal regions, can be squared). Hippocrates’ lune is sometimes referred to as the lune of al-Hazen (or ibn al-Haytham) (965-1040 AD) since it appears in his work. Hippocrates’ {it lunes} refer to two lunes in a slightly more involved configuration. A beautiful generalization of Hippocrates’ theorem is contained in Exercise 318 (p. 254) of {it J. Hadamard}’s book [Leçons de géométrie élémentaire. Tome 1. Géométrie plane. Paris: Colin (1898; JFM 29.0428.14)]. The exercise states that if two rays from A meet Ω_1 at P_1, Q_1, and Ω_2 at P_2, Q_2, then the area of the curvilinear quadrilateral P_1P_2Q_2Q_1 is equal to the area of triangle AP^*Q^*, where P^*, Q^* are the orthogonal projections of P_1, Q_1 on BC. This implies Hippocrates’ result (by taking P_1=P_2=P^*=B, Q_1=Q_2=Q^*=C) and also shows that one can divide Hippocrates’ lune (using straightedge and compass) into n equal parts for any n by so dividing the line segment BC. In view of the complexity of the analogous cyclotomy problem of the circle, one finds this quite interesting. The paper under review provides a proof of the aforementioned gem of Hadamard. It is well written and nicely illustrated by figures. Readers interested in knowing about the four other squarable lunes and other aspects of the subject may read Section 9.1 (pp. 137-144) of the delightful book [{it C. Alsina} and {it R. B. Nelsen}. Charming proofs. A journey into elegant mathematics. Washington, DC: The Mathematical Association of America (MAA) (2010; Zbl 1200.00021)], and the relevant references therein. It is also worth mentioning that Hadamard’s book mentioned above has been translated by Mark Saul as [Lessons in geometry. I. Plane geometry. Providence, RI: American Mathematical Society (AMS); Newton, MA: Education Development Center (2008; Zbl 1156.51012)]. (ZDM/Mathdi)