[Authors and titles are listed at the end of the review]
The volume presents five contributions focusing on ancient mathematical commentaries, divided into two sections: “commentators at work” and “comparing commentaries”. They include sources from China, Greece, India, and Mesopotamia.
As the editors point out in the introduction, although mathematical commentaries are found in the scholarly literature of various mathematical cultures, they have not yet been the topic of comparative analysis. Studying commentaries might not only provide more detailed insights into the concepts found in their respective mathematical texts but also give information about the practitioners of these texts, their procedures, and how the practitioners used them. Until recently, studying commentaries in order to better understand the base text has been the predominant, if not the only, attention that mathematical commentaries have received. However, in 2016, a conference on ancient Greek and Arabic mathematical commentaries was held at the Humboldt University of Berlin. Papers of that conference were published in Historia Mathematica 47 (2019). The current volume takes up this research and broadens the geographic range of studies.
The first contribution (chapter 2) by Orna Harari analyzes the commentary on the First Book of Euclid’s Elements by the late antique Greek philosopher Proclus (c. 412-485 CE). Her chapter opens by addressing the link between the pedagogical and philosophical aims of this commentary (“Teaching and Philosophy”) and then explores the role of the study of canonical texts in the tradition of Greek commentaries on philosophical texts (“Canonical and Authoritative Texts”). She concludes that in Proclus’ commentary on Euclid’s Elements “its pedagogical aim is not distinct from its philosophical aim, and its base text and other canonical texts serve therein as a means of further philosophical inquiry” (p. 54). Chapter 3 by Agathe Keller explores the Sanskrit “Commentary with Explanations” by the ancient commentator Pṛthūdaka on a seventh-century CE astronomical treatise by the astronomer and mathematician Brahmagupta (c. 568-c. 668 CE). She raises the question whether Sanskrit mathematical commentaries constituted a specific genre that was separate from other scholarly disciplines and delineates the limits of what can be answered and what will remain speculation if no further sources are found. The last contribution of the first section (chapter 4) takes the reader to China of the second century CE. Daniel Patrick Morgan studies an argument from a commentary by the philosopher, politician and writer Zheng Xuan (127-200 CE), focusing on a specific mistake therein, its further tradition in further commentaries and the motivation behind this.
These three contributions offer various paths for exploring ancient commentaries that might (in future projects) be continued with other sources of the same origin, but also raise the possibility of comparing commentaries of different origins.
The second section opens with a comparison of two Akkadian commentaries on astral sciences (chapter 5). Zackary Wainer and John Steele analyze and contrast two atypical commentaries to the series Enūma Anu Enlil (“When (the gods) Anu and Enlil…”), namely Enūma Anu Enlil 14 and Šumma Sîn Tāmartišu (“If the moon in its appearance”). The comparison reveals the different traditions of these two texts and their underlying motivations. The final chapter by Karine Chemla and Zhu Yiwen compares commentaries (and subcommentaries) on Chinese texts from the mathematical canon and texts from the Confucian canon. The comparison reveals, as was expected, that both attest to practices of explanation that were connected. In addition, a connection is revealed between the two canonical corpora that originates from their usage in the context of learning. One can easily see how the type of analysis carried out in these two contributions could be fruitfully used on other sources as well. In addition, a comparison of two (or more) commentaries from different origins but on similar texts might yield further interesting insights.
What makes the book particularly valuable are the excerpts from sources attached as an appendix with translation and commentary to each chapter and followed by individual glossaries of the most important technical or conceptual terms. The challenging typesetting of texts from four different backgrounds with distinctive fonts is very well mastered. The volume exemplifies what can be done with this genre of scientific text and thereby provides a stimulating point of departure for further studies.
Authors and Titles
Introduction: Why Study Mathematical Commentaries? – Karine Chemla & Glenn W. Most
Philosophical Commentaries on Mathematical Texts: The Case of Proclus’ Commentary on the First Book of Euclid’s Elements – Orna Harari
Characterizing a Sanskrit Mathematical Commentary: An Exploration of Pṛthūdaka’s Vāsanābhāṣya on Progressions – Agathe Keller
Calling out Zheng Xuan (127-200 CE) at the Crossroads of Ritual, Mathematics, Sport, and Classical Commentary – Daniel Patrick Morgan
Astral Commentaries within the Mesopotamian Received Tradition: The Commentary to Enūma Anu Enlil 14 and Šumma Sîn ina Tāmartišu – Zackary Wainer & John Steele
Contrasting Commentaries and Contrasting Subcommentaries on Mathematical and Confucian Canons. Intentions and Mathematical Practices – Karine Chemla & Zhu Yiwen