Geometry and philosophy of mathematics in the 10th century. Mathematical works of al-Sijzī. Volume II. Translated by the editors (Q6101081)

The volume under review is the second dedicated by the authors to the writings of al-Sijzī. While the initial volume delves into texts of geometry and number theory [\textit{R. Rashed}, Œuvre mathématique d'al-Sijzī. Vol. I. Géométrie des coniques et théorie des nombres au Xe siècle. Louvain: Éditions Peeters (2004; Zbl 1208.01009)](, the second continues to center on geometry but also provides insights into the philosophy of al-Sijzī. Al-Sijzī was a Persian mathematician (also an astronomer and astrologer), probably originated from \textit{Sijistan} (Southeastern Iran), from the latter half of the 10th century. Pinpointing his life's timeline precisely is challenging (the authors take a closer look here), but his mathematical and astrological activities extended from the early 960s to the very beginning of the 11th century; he is a contemporary of al-Bīrūnī, with whom he extensively exchanged ideas. He seems to owe much of his scientific grounding to his father, himself mathematician and astrologer. Al-Sijzī undeniably belongs to the intellectual milieu flourishing under the \textit{Buyid} dynasty (945--1055); he enjoyed the patronage of `Aḍud al-Dawla (936--983). Al-Sijzī demonstrates a deep familiarity with Greek geometry, particularly the works of Euclid, Archimedes, and Apollonius, as well as the geometry developed by his predecessors from Islamic countries such as Thābit ibn Qurra, al-`Alā' ibn Sahl, Abū Sahl al-Qūhī, Abu al-Jūd ibn al-Layth, and al-ḥasan ibn Mūsā (whose treatise on Conics al-Sijzī cites, now is lost). Al-Sijzī also maintained significant correspondence with numerous mathematicians -- whose figure is more that of the enlightened amateur than that of the innovative mathematician (p.~16) --, which, among other things, would partially fuel his scientific production, as is the case with his \textit{Anthology of Problems} (edited in the volume under review, see below), demonstrating the richness of mathematical reflections of his time. Al-Sijzī's scientific production is really abundant; the authors/editors list no fewer than 47 different titles, of which 35 are available today. All these titles are described in the introduction of the book under review (chapter 1, p.~20-32) according to several categories: ``general works on geometry, its objects, and its methods'' (2), ``writings related to the treatises of Euclid, Archimedes, and Apollonius'' (9), ``collections of problems'' (3), ``geometry of Conics'' (11), ``geometry with ruler and compass'' (20), and ``number theory'' (2). The dating and history of those texts follow. The book under review contains precisely the critical edition of several of al-Sijzī's writings in plane and solid geometry, thus significantly enriching research in the history and philosophy of mathematics. The volume is divided into four main chapters and appendices. All chapters are structured in the same way: a brief introduction guiding the reader to thematically contextualize the edited texts' production, the text edition (odd pages) with the French translation (even pages) facing it. In Chapter 2, the editors focus on geometry in general with two texts: \textit{al-madkhal ilā `ilm al-handasa} [Introduction to Science of Geometry] (p.~52-197), in which al-Sijzī explains to beginner researchers in geometry, for example, its objects, principles, extensions, and applications, and the \textit{Risāla [...] fī anna al-ḍil` ghaīr mushārik li-l-quṭr} [Epistle on the Diagonal of the Square] (p.~200-213) concerning the ancient problem of the incommensurability of the diagonal with the side of the square. In reality, this latter text is, for al-Sijzī, a pretext to address the classical opposition between direct proof and proof by contradiction. In Chapter 3, the editors focus on four texts that position al-Sijzī in relation to Euclid's \textit{Elements}. Through those texts, al-Sijzī demonstrates a fine knowledge of the Euclidean corpus, and of the \textit{Elements} in particular. He is very interested in demonstrations (like many contemporary mathematicians and philosophers); for al-Sijzī, this often means seeking alternative proofs to those of Euclid, or developing the Greek mathematician's proof further (for example, using the direct route where Euclid uses absurdity, considering figures neglected by Euclid, changing the order of propositions). This is what we find in the edited texts here. The \textit{Barāhīn kitāb Uqlīdis fī al-uṣūl} [Proofs of Euclid's book on the \textit{Elements}] (p.~224-415) present 119 demonstrations for about fifty propositions of plane geometry (Books I, II, III, IV, VI, and XIII). The \textit{Risāla [...] fī ḥall al-shakk alladhī fī al-shakl al-thālith wa-l-`ishrīn min al-maqāla al-ḥādiyya `shara min kitāb Uqlīdis fī al-uṣūl} [Epistle to resolve the doubt concerning proposition XI-23 of the \textit{Elements}] (p.~418-437) concerning the solid angle, and the \textit{Istidrāk wa-shakk fī al-shakl al-rābi` `ashar min al-maqāla al-thāniyya `ashara min kitāb al-uṣūl} [Amendment and doubt concerning proposition XII-14 of the \textit{Elements}] (p.~440-447) -- this is XII-17 in the Greek tradition -- are in the same spirit as the preceding text: al-Sijzī demonstrated a deep knowledge of the Euclidean treatise and wished to deepen it without challenging it. Finally, the \textit{Risāla [...] fī jawāb mas'ala `an kitāb Yūḥannā ibn Yūsuf} [Epistle on Answer to a Problem from the Book of Yūḥannā ibn Yūsuf] (p.~450-457) concerning proposition I.10 (division of a straight line into two equal parts) in a treatise by Yūḥannā ibn Yūsuf, now lost. Chapter 4 is dedicated to plane geometry. Four texts are edited. The first, the most quantitatively important in the book under review, is \textit{al-masā'il al-mukhtāra} [Anthology of Problems] (p.~466-737) also known as \textit{Ta`līqāt handasiyya} [Geometrical Comments]. It is a miscellaneous collection, without an established plan, of 115 problems; it is the text that al-Sijzī cites most in his works. The authors/editors of the book under review see it as a text in which al-Sijzī records, as he exchanges and reflects on his own, ``the developments he is tempted to give them in a set to which he assigns sufficient value to make it a treatise likely to be referred to later'' (p.~461). The second, strongly linked to the first, is the \textit{Kitāb [...] fī taḥṣīl al-qawānīn al-handasiyya al-maḥdūda} [Book on the acquisition of determined geometric laws] (p.~740-761) which consists of 12 statements whose aim is the search for invariants. The next text, the \textit{Qawl [...] fī khawāṣ murabba` quṭr al-dā'ira} [Remarks on the properties of the diameter square of the circle] (p.~764-775), is a collection of about ten propositions thematically grouped (possibly for didactic purposes). Finally, the last text of this chapter is the \textit{Risāla fī rasm al-musaddas fī al-murabba` wa-l-murabba` fī al-musaddas [...]} [Epistle on Drawing a Hexagon in a Square and a Square in a Hexagon] (p.~778-793). It concerns both construction problems and an isoperimetric study on the square and hexagon. The last chapter concerns the division of plane figures that we are familiar with (see, in particular, [\textit{M. Moyon}, La géométrie de la mesure dans les traductions arabo-latines médiévales (Arabic). Turnhout: Brepols Publishers (2017; Zbl 1376.01003)]) with only one text: his \textit{Kitāb [...] fī al-ajwiba `an masā'il sa'alaha `anhu ba`ḍ muhandisī Shīrāz} [Book on Answers to Problems Asked to him by a Mathematician from Shiraz] (p.~800-839) which complements the Arabic version of Euclid's \textit{On Divisions} attributed to al-Sijzī (see, in particular, [\textit{J. P. Hogendijk}, in: Vestigia mathematica. Studies in medieval and early modern mathematics in honour of H. L. L. Busard on the occasion of his 70th birthday. Amsterdam: Editions Rodopi B. V.. 143--162 (1993; Zbl 0810.01001)]) and some propositions from his \textit{Anthology} on the same subject. It is again a contribution from al-Sijzī that shows his deep understanding of Euclid's \textit{Elements} that the Persian mathematician respects and extends. Finally, the book under review concludes with four appendices: the first two provide the edition of demonstrations (by a reader according to a marginal note, or by al-Sijzī himself but alternative) of results used in the \textit{Anthology of Problems}, the last two describe lists of al-Sijzī's works known by two well-known manuscript witnesses (the most important is in Dublin, the other is in Lahore). The glossary of geometric concepts, the indexes of proper names, cited treaties, and manuscripts complement the already very rich introductions of the different chapters and the very successful editions/translations of this new volume of the beautiful serie ``Scientia graeco-arabica''. In conclusion, the book under review is a fine work of erudition, both in its content and editorial form, absolutely comprehensive and useful to any historian of mathematical sciences, whether or not they are from Islamic countries or medieval period. It is new geometric and philosophical contributions from al-Sijzī that P.~Crozet and R.~Rashed offer us here, with direct access to the Arabic text, for which we thank them.