Abstract
At barely 200 pages and under 40 euros, this is a slim, affordable volume with a very broad title that nonetheless accurately represents the scope and richness of its content. In fact, the chronological range of the subject matter is even more extensive than the non-Indologist reader might infer from the term “ancient,” as it covers the period from the late first millennium of the previous era to the seventeenth century of the present one.
A short preface (in French) by the editor introduces and summarizes each of its seven articles (three in French and four in English), originally presented in 2009 at the Université libre de Bruxelles. They address various topics in traditional Classical Sanskrit sciences or śāstras that incorporate mathematical methods:
R. Mercier, “The reality of Indian astronomy”;
J. M. Delire, “Entre astronomie et mathématiques, les découvertes indiennes en trigonométrie: la construction des tables de sinus et quelques méthodes d’interpolation”;
P. -S. Filliozat, “Mathématiques et scolastique dans l’Inde médiévale, l’exemple du Haricarita de Parameśvara Bhaṭṭa”;
S. R. Sarma, “Yantrarāja for Dāmodara: The earliest extant Sanskrit astrolabe”;
K. Ramasubramanian, “Indian planetary models: Āryabhaṭa to Nīlakaṇṭha”;
F. Patte, “Rythmes et algorithmes: le génie mathématique indien”;
K. Mahesh, Venketeswara Pai R. and K. Ramasubramanian, “Mādhava series for pi and its fast convergent approximations.”
Only (7) is drawn from a textual context that we would consider strictly “mathematics,” namely, approximations to infinite series for the constant pi preserved in a highly sophisticated commentary on a section of a standard twelfth-century arithmetic/geometrical treatise dealing with circles. Most of the other articles explore trigonometry, calendrics, and related subjects as they apply directly to medieval Indian mathematical astronomy.
Each of the chapters treats specialized research but can also serve non-Indologists as an accessible entry point to much informative and interesting material on Indian mathematical sciences. Most of the quoted Sanskrit is transliterated, except in (5), and the numerous Sanskrit technical terms are helpfully explained, although some Sanskrit titles of works are not. A common theme in several of the chapters is the seminal work of Āryabhaṭa (b. 476) in mathematical astronomy. For example, (1) provides an updated and expanded discussion of the statistical analysis techniques originally proposed by R. Billard for investigating the parameters in Indian treatises on spherical astronomy beginning with Āryabhaṭa’s. Likewise, (2) and (5) focus on the trigonometric and physical structure of Indian orbital models first fully attested in Āryabhaṭa’s work. Both of the latter articles as well as (7) emphasize the innovations in infinite series and planetary astronomy of the so-called “Mādhava school” or “Kerala school” flourishing around 1350–1600 in south India, sometimes also called the “Āryabhaṭa school” for their adherence to Āryabhaṭa’s parameters. A slightly earlier development in south Indian astronomy is addressed in (3), namely, the remarkable Haricarita attributed to the philosopher Parameśvara, in which alphabetically encoded lunar longitude values (or “sentences”) are recast as a devotional verse paean to Lord Kṛṣṇa.
Interactions between Indian and Islamic astronomy are discussed in (1) and in the detailed description of Indian astrolabes in (4), which primarily investigates the early seventeenth-century instrument now revealed as the earliest surviving Sanskrit-language astrolabe. The author in footnote 20 comments on the unexplained numbering from 1 to 12 in groups of three, with every other group inverted, on the sine graph on the astrolabe’s back. I would tentatively propose that this is intended as a visual guide for reducing an arc ending in any of the 12 thirty-degree “duodecants” of the circle to the arc with the same sine in the first quadrant. Thus, for example, the sine of an arc in the fourth duodecant between 90° and 120° equals the sine of its supplementary arc in the third. So the user reads the sines of fourth-duodecant arcs off the graph like those of third-duodecant arcs but “backwards,” i.e., with the excess over 90° measured from the vertical rather than the horizontal.
Finally, (6) examines a different set of mathematical methods in a very different context. It expounds the combinatorial rules, found in a fourteenth-century treatise on music theory, for enumerating musical rhythms composed of beats of varying lengths, involving what are now called partitions of integers.
Each article is formatted individually, so features such as citation style, references, and section numbering vary from one article to the next. A more uniform format might have been more convenient, particularly for non-Indologists unfamiliar with most of the literature. For instance, the inconsistently alphabetized references in (7) do not always clearly distinguish between given names and surnames (and, moreover, are abruptly and inexplicably truncated in the middle of a citation). Especially, since the authors and editor have taken pains to make their topics accessible to non-specialists, it would have been helpful in some cases to sample prior and recent research more widely. For example, the references in (7) do not include relevant publications by Hayashi et al. (1990) “The correction of the Mādhava series for the circumference of a circle,” Centaurus 33 and Gupta (1992) “On the remainder term in the Mādhava-Leibniz series,” Gaṇita Bhāratī 14. nor does the bibliography of (6) mention Sridharan et al. (2010) “Combinatorial methods in Indian music,” in: Seshadri (ed.) Studies in the History of Indian Mathematics. And (2) does not refer to Hayashi (1997),” Āryabhaṭa’s rule and table for sine-differences,” Historia Mathematica 24. But overall, the volume is a very readable and instructive addition to the literature on Indian mathematical and astral sciences.