JYOTPATTI

Abstract

<p>Bhaskaracharya ,(Bhaskar II, 1114- 1185 AD), was one of the great mathematicians in India. His text “<em>Sindhant Shiromani</em>” (SS) was treated as the base of the further research oriented results by most of all the mathematicians after him.<br> SS contains of two parts:, <em>Goladhyaya and Grahaganit</em>. Jyotpatti is the last chapter in <em>Goladhyaya. Jyotpatti</em> consists of 25 <em>Shloka</em> (stanzas), all in Sanskrit language. It is a general impression that SS contains <em>Lilavat</em>i and <em>Beejganit</em> also, but that is not so.<br> <em>Jyotpatti</em> deals with trigonometry. This was a milestone in developing geometry in India. <em>Jya</em> means sine and <em>Utapatti</em> means creation. Hence the name <em>Jyotpatti ( jya + upapatti</em>). <em>Jya and Kojya( or kotijya)</em> 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.<br> After defining <em>Jya, Kotijya and Utkrama</em> (<em>verse jya) </em>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 <em>Panchajyaka. </em>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 <em>jya and kotijya</em> with the longitude –latitude of earth and those with lateral threads of a cloth.<br> Contents in Jyotpatti (Only a few mentioned here)<br> (1) R jya 45 =R, and other similar R jya values. (All in degrees)<br> (2) R jya 36 = 0.5878 approx.<br> (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.<br> (4) Derivation of formulae for sin (θ + ϕ) and cosine (θ + ϕ) called as <em>samas bhavana</em> and <em>antar bhavana</em>.<br> (5) Concept of derivatives, that is, δ(sin θ) = (cos θ) δθ etc. Which is Rolle’s Theorem.<br> Indian mathematicians developed trigonometry in different way than that of western mathematicians. Though <em>Jyotpatti</em> is a small text, it is a landmark in development of ancient and medieval trigonometry.</p>