Menelaus’ Spherics was written in the 2nd century CE, addressing and extending earlier work in spherical geometry, but it was probably never seriously studied in its entirety in the ancient period and only fragments of the Greek text survive, preserved in later authors.1 The treatise can be divided into three topics. The first treats the geometrical properties of spherical triangles by developing analogies between these and the properties of plane triangles developed in Euclid’s Elements. The second shows how certain arcs of spherical triangles can be treated using the lengths of chords related to these and based on a theorem known as the Sector Theorem (Menelaus Theorem). The third topic develops these methods for application to problems in spherical astronomy—a field that investigated issues such as the length of daylight and night-time, and the rising times of stars or arcs of the ecliptic.
The book under review is a valuable contribution to our understanding of the history of Menelaus’ Spherics in the medieval period, as well as the mathematics developed in the treatise. The first part deals with the various medieval versions of, and witnesses to, Menelaus’ treatise; the second part provides mathematical commentaries, including commentaries and studies by medieval scholars; the third part gives critical editions of a fragment (breaking off in Proposition 36) of an early Arabic translation ( A, pp. 408-483) and of the al-Māhānī/al-Harawī version ( M/H, pp. 500-777), along with English translations. There is also a postface on spherical geometry and its history. The mathematical commentaries are useful for understanding the text and the critical editions, and the many editions and translations of medieval sources are an extremely valuable contribution to our knowledge of this text.
The M/H version of the Spherics, edited and translated along with A, in “Part III: Text and translation,” is historically quite interesting, but al-Harawī’s many interventions, along with his failure to grasp some of the mathematical details, introduce nearly as many problems as they resolve.2 Al-Harawī has added two historical and philosophical prefaces to the text (pp. 500-505, 684-685); inserted a number of lemmas (pp. 686-695), one of which is mathematically incorrect (pp. 692-695); rewritten some propositions, sometimes incorrectly; and introduced some terminological innovations, which cause more confusion then help and are not used by any other medieval scholar (pp. 688-691). Hence, this version of the treatise cannot be taken as a reader’s text, and Naṣr Manṣūr ibn ‘Irāq’s version, N, edited by Krause, and the revision by Naṣīr al-Dīn al-Ṭūsī, available in the Hyderabad series, must still be consulted in order to understand the mathematics involved.3
Another welcome contribution of Rashed and Papadopoulos’s book is “Part II: Mathematical commentary,” which explains the mathematical details of the text and gives commentary to each proposition, including the relevant scholarship of both Ibn ‘Irāq and al-Ṭūsī. Hence, this section of the book provides a fairly clear picture of the mathematical issues involved.
In “Part I,” Rashed and Papadopoulos give an introduction to Menelaus and his work, and then discuss the text history of the Spherics in the medieval period. Here, they follow the scholarship of Krause and Hogendijk,4 although they provide some new evidence to support the findings of these scholars—namely, that the source translation for M/H and N differ, and that the source used by Ibn Hūd has the same characteristics as that used for the Latin translation by Gerhard of Cremona. They also propose that the new fragment they found and edited, A, is not the source translation for M/H, which is not convincing; and that the source translation for N is that attributed in some scholia to Abū ‘Uthmān al-Dimashqī, which is possible but not proven.5
One disappointing aspect of this book is Rashed and Papadopoulos’s lack of any attempt to situate their work in the context of previous scholarship. This means that the only readers who will be able to appreciate what is new and what was already known are those who have previously read all of the literature on the subject. Three examples will make this case: (a.) Rashed and Papadopoulos argue at length that the first part of the Latin version is based on al-Māhānī’s version, while the second part is based on the same source as Ibn ‘Irāq’s edition (pp. 26-71). The way that they express themselves makes it sound like they have discovered this—but this was also argued for at length by Krause. Rashed and Papadopoulos do give further evidence beyond that presented by Krause, for which we are grateful, but their work serves to confirm Krause’s findings. (b.) In the section on Ibn Hūd, Rashed and Papadopoulos claim that the question of his source has “not been correctly addressed until now” (p. 74), and then proceed to address the question using the same methodology as Hogendijk and come to the same conclusion—namely that the first part is from al-Māhānī’s version and the second part from the same source translation as N (pp. 73-121). Again, Rashed and Papadopoulos’s actual contribution is to give further evidence, including edited texts, which help to confirm Hogendijk’s previously established position. (c.) Finally, Rashed and Papadopoulos caution against believing that there was ever a full Syriac copy of the Spherics, while noting the Syriac influence on A (p. 486), a position already argued for by Sidoli and Kusuba.6
We should be grateful to Rashed and Papadopoulos for their work in producing two new editions of the Spherics ( A and M/H), in providing the original sources for much of the medieval scholarship on this important work, and in commenting on the overall mathematical development of the treatise. As noted above, however, we cannot simply read M/H as Menelaus’ Spherics, because it is a highly edited version of the treatise. In our current state of knowledge, it remains that we must read M/H along with N and al-Ṭūsī’s texts in order to assess Menelaus’ work, and we still await critical editions of the Latin and Hebrew versions before we can hope to fully understand the medieval transmission of the text.
1. The Greek fragments are collected and studied by A.A. Bjørnbo, Studien über Menelaos’ Sphärik (Leipzig, 1902): 22-25 and F. Acerbi, “Traces of Menelaus’ Sphaerica in Greek Scholia to the Almagest,” SCIAMVS 16 (2015): 91-124.
2. For an overview of the al-Harawī version of the text, see N. Sidoli and T. Kusuba, “Al-Harawī’s Version of Menelaus’ Spherics,” Suhayl 13 (2014): 149-212.
3. M. Krause, Die Sphärik von Menelaos aus Alexandrien in der Verbesserung von Abū Naṣr Manṣūr b. ‘Alī b. ‘Irāq (Berlin: Weidmannsche Buchhandlung, 1936); Naṣr al-Dīn al-Ṭūsī, Kitāb Mānālāwus, Taḥrīr (Hyderabad: Osmania Oriental Publications Bureau 1359AH/1940CE).
4. For Krause, see note 3; J. Hogendijk, “Which Version of Menelaus’ Spherics was Used by Al-Mu’taman ibn Hūd in his Istikmāl ?” in: M. Folkerts (ed.), Mathematische Probleme im Mittelalter (Wiesbaden, 1996): 17–44.
5. I will discuss the details of these transmission issues in a longer review to appear in Aestimatio.
6. See note 2, pp. 191-192.