The mathematics of India in classical times was so wrapped up with astronomy that many of its most widely-known texts are mathematical sections within siddhantas (comprehensive works on mathematical astronomy), or — like Bhaskara's famous Lilavati and Bijaganita — were commonly treated as such. Hence a present-day mathematician desiring to encounter the texts themselves, either in their original language or in translation, should have at his or her disposal some reference-material for the key terms in Indian astronomy, and the methods employed by the Indians in astronomical calculation. The new translation, by K. Ramasubramanian of the Cell for Indian Science and Technology in Sanskrit at the Indian Institute of Technology and M.S. Sriram of the Department of the Department of Theoretical Physics at the University of Madras, of the important astronomical treatise Tantrasangraha (literally, “Summary of the System”) by the 16th-century Keralese astronomer Nilakantha Somayaji, can serve as a such a reference.
Nilankantha (the name means “blue-throated” and is a common epithet for the god Shiva), was a prominent representative of the Kerala School of mathematics in Southern India, which flourished from about 1400 to 1600 CE. Mathematicians nowadays mostly remember the Kerala School for several remarkable contributions to mathematical analysis — including infinite series for π and for the sine, cosine and arctangent functions — that appear to originate with Madhava of Sangamagrama (~1340–~1420 CE), the “founder“ of the School and its most celebrated member. However the mathematicians of the Keralese tradition also made considerable advances in astronomy proper, especially in planetary theory, which they based on extensive new observations of the heavens. Nilakantha (1444–~1545 CE) is known to have proposed a planetary scheme in which only the so-called “exterior” planets orbit the Earth, whereas the “interior” planets Mercury and Venus orbit the Sun, which in turn orbits the Earth. (A similar system was proposed about a century later in Europe by the Danish astronomer Tycho Brahe.)
The focus of the Tantrasangraha is squarely on computational methods in astronomy: computation of the mean and true longitude of planets, beginning, middle and ending time for eclipses, etc., so much of the mathematics involves spherical trigonometry. There is essentially no direct use of the Kerala School’s results on infinite series, although these might have been employed in the construction of sine-tables. (The Tantrasangraha does give methods for constructing quite accurate tables from which sines and arcs can be computed, and the discussion of these methods indicates use of low-order approximations such as sin(α) ≈ α – α3/3! for small values of α.) Even Nilakantha’s innovative planetary model is not discussed explicitly in the treatise, although it may be inferred from the computational procedures he sets forth. (He does discuss offer an explicit geometrical description of his planetary model in another work, his massive and currently-untranslated commentary on the Aryabhatiya of Aryabhata.)
The presentation of the text and the translator’s explanations is satisfyingly complete. For each passage, the translators provide the original text in Devanagari and in Romanized script, along with a translation, followed by a detailed mathematical explanation that relates Sanskrit terms to modern mathematical notation. Quite often the translators give illuminating quotations from later Keralese works, such as the Laghuvivrti and the Yuktidipika, which were written in part to explain the methods of the Tantrasangraha. Ample examples of computation are also provided. Useful resources at the end of the book include treatments of spherical trigonometry, coordinate systems in astronomy, and planetary models in Indian astronomy. The book can thus serve as a reasonably self-contained introduction to mathematical astronomy as practiced in classical India.
This volume will be of interest to general historians of mathematics, and also to those mathematicians who have some knowledge of Sanskrit and who would like to read Indian mathematical texts in the original, but who currently lack the technical vocabulary and the knowledge of observational astronomy that are required to do so independently.
Homer White is Professor of Mathematics at Georgetown College, in Kentucky. Although primarily a Jack-of-All-Trades small-college mathematician, he especially enjoys the teaching of statistics at all levels, and his interests in the history of mathematics include the geometrical works of Leonhard Euler and the mathematics of classical India.