Kriyakramakari is an elaborate commentary in Sanskrit written by Sankara Variar and Narayana, two astronomer-mathematicians belonging to the Kerala school of astronomy and mathematics, on Bhaskara II's well-known textbook on mathematics Lilavati. Kriyakramakari, along with Yuktibhasa of Jyeshthadeva, is one of the main sources of information about the work and contributions of Sangamagrama Madhava, the founder of Kerala school of astronomy and mathematics. Also the quotations given in this treatise throw much light on the contributions of several mathematicians and astronomers who had flourished in an earlier era. There are several quotations ascribed to Govindasvami a 9th-century astronomer from Kerala. Sankara Variar, the first author of Kriyakramakari, was a pupil of Nilakantha Somayaji and a temple-assistant by profession. He was a prominent member of the Kerala school of astronomy and mathematics. His works include Yukti-dipika an extensive commentary on Tantrasangraha by Nilakantha Somayaji. Narayana, the second author, was a NamputiriBrahmin belonging to the Mahishamangalam family in Puruvanagrama. Sankara Variar wrote his commentary of Lilavati up to stanza 199. Variar completed this by about 1540 when he stopped writing due to other preoccupations. Sometimes after his death, Narayana completed the commentary on the remaining stanzas in Lilavati.
On the computation of π
As per K.V. Sarma's critical edition of Lilavati based on Kriyakramakari, stanza 199 of Lilavati reads as follows : This could be translated as follows; Taking this verse as a starting point and commenting on it, Sanakara Variar in his Kriyakrakari explicated the full details of the contributions of Sangamagrama Madhava towards obtaining accurate values of π. Sankara Variar commented like this: Sankara Variar then cites a set of four verses by Madhava that prescribe a geometric method for computing the value of the circumference of a circle. This technique involves calculating the perimeters of successive regular circumscribed polygons, beginning with a square.
Sankara Variar then describes an easier method due to Madhava to compute the value of π. To translate these verses into modern mathematical notations, let C be the circumference and D the diameter of a circle. Then Madhava's easier method to find C reduces to the following expression for C: This is essentially the series known as the Gregory-Leibniz series for π. After stating this series, Sankara Variar follows it up with a description of an elaborate geometrical rationale for the derivation of the series.
The theory is further developed in Kriyakramakari. It takes up the problem of deriving a similar series for the computation of an arbitrary arc of a circle. This yields the infinite series expansion of the arctangent function. This result is also ascribed to Madhava. The above formulas state that if for an arbitrary arc θ of a circle of radius R the sine and cosine are known and if we assume that sinθ < cos θ, then we have:
