JYOTPATTI

Bhaskaracharya ,(Bhaskar II, 1114- 1185 AD),  was one of the great mathematicians in India. His text “ Sindhant Shiromani ” (SS) was treated as the base of the further research oriented results by most of all the mathematicians after him. SS contains of two parts:, Goladhyaya and Grahaganit . Jyotpatti is the last chapter in Goladhyaya.  Jyotpatti consists of 25 Shloka (stanzas), all in Sanskrit language. It is a general impression that SS contains Lilavat i and Beejganit also, but that is not so. Jyotpatti deals with trigonometry. This was a milestone in developing geometry in India. Jya means sine and Utapatti means creation.  Hence the name Jyotpatti ( jya + upapatti ). Jya and Kojya( or kotijya) stand for the Rsine and Rcosine ratios respectively. The trigonometry developed by Bhaskara II is based on a circle of radius R, and not on a right-angled triangle as taught in the schools. After defining Jya, Kotijya and Utkrama ( verse jya) etc, Bhaskara  obtains these ratios for the standards angles of 30,45, 60, 36 and 28,(all in degrees) by inscribing a regular polygon in a circle of radius R. Bhaskara called these angles as Panchajyaka. Not only this, Bhaskara developed   these results for addition and subtraction of two angles. This result was further developed for the similar results, for the multiple angles. Bhaskara compares jya and kotijya with the longitude –latitude of earth and those with lateral threads of a cloth. Contents in Jyotpatti (Only a few mentioned here) (1) R jya 45 =R, and other similar R jya values. (All in degrees) (2)  R jya 36 = 0.5878 approx. (3) Sn = side of a regular polygon of n sides = D sin (π/n), D is the diameter of circle in which polygon is inscribed. (4) Derivation of formulae for sin (θ + ϕ) and cosine (θ + ϕ) called as samas bhavana and antar bhavana . (5) Concept of derivatives, that is, δ(sin θ) = (cos θ) δθ etc. Which is Rolle’s Theorem. Indian mathematicians developed trigonometry in different way than that of western mathematicians. Though Jyotpatti is a small text, it is a landmark in development of ancient and medieval trigonometry.