The contribution of Kerala (on the south west coast of India) to mathematics, especially from the last quarter of the 14\(^{th}\) century CE to the end of the 16\(^{th}\) is slowly getting worldwide recognition and appreciation. Three names are of special importance among the contributors. The first is Mādhavan (Mādhava in the anglicized form), the second one is Nīlakanţhan (Nīlakanţha Somayāji) and the third one is Jyeshţhadevan who wrote what can be called the first textbook of calculus, Yuktibhāşā or YB in short for the rest of this review (The name Jyeshţhadevan is attributed to the author of YB by many historians including the current author; however, the reviewer and many others do not agree to this view, possibly the name of the author of YB was Brahmadatha or something else). YB is a comprehensive account, in its last two chapters, of the fundamental principles of integral and differential calculus (in that natural order in the Kerala approach to calculus), and also the relationship between them, and their use in the study of trigonometric functions. Tantrasamgraham of Nīlakanţhan was studied and quoted until recently as the text for studying the long tradition of Indian mathematics and Astronomy that the Kerala School represented.
One reason was that YB is in the language Malayalam (that too old style) – not easily accessible to outsiders. An English translation of YB by K.V. Sarma, has been published in 2008 [Ganita - Yuktibhāşā of Jyeshţhadevan, Vol 1, Hindusthan Book Agency, New Delhi, (2008; Zbl 1160.01022); Springer, Berlin (2008; Zbl 1160.01022)]. To those who know Malayalam, the commentary and explanation of YB part 1 by Rama Varma (Maru) Thampuran and A.R. Akhileswara Iyer [Mangalodayam Ltd., Thrissur, (1948)] is the best book to understand the YB. “YB gives us a portrait of the mathematical culture of its time as few other Indian texts of any period do. And since it also cites all developments relevant to its concerns, it is an ideal vantage point from which to survey the entire landscape of Indian mathematical thought of which the work of the Kerala School forms the final summit.”
“This article is about what we can learn from YB and its recapitulations of earlier material about one such widely favoured general procedure, that of recursion. Three broadly distinct manifestations of the general ideas of recursion find significant use in YB: recursive description, recursive computation and construction and recursive proof.” Section 1 of the article is an overview of the entire material. “What they do have in common is the constant presence of a certain spirit, that of the decimal place-value number system”. Section 2 is devoted to a retracing of this frequently traversed path, ahead of the technically more demanding mathematical aspects in the subsequent sections. Varieties of recursion are discussed with technical details in the later sections - the Arc tangent series in section 3, the sine/cosine series in section 4. The last section is titled Two Footnotes - one on Decimal Numbers and Polynomials and the other on The Roots of Samskāram, Roots by Samskāram.
On the whole, this long scholarly study of YB is a must for anyone who studies the history of mathematics and who should therefore be interested in the work done in Kerala between the last quarter of 14\(^{ th}\) century CE and the end of the 16\(^{th}\).
For the entire collection see [Zbl 1195.00068].