Thread on Sine and Cosine Series indirectly / directly discovered by Madhwa 200 years before Newton . Is he given the credit ? Do all people know this ?
No !
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No !
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Indian Scholar Madhwa (1350-1410) gave a table of almost accurate values of half sine chords for twenty four arcs drawn at equal intervals . This thread discusses the basis of arriving such accurate values and will show that Madhwa established the following sine and cosine series
before Newton , De Moivre and Euler used these relations to compute his table . These are
He has given 24 half sine chords for twenty four arcs in a quarter of circle drawn at equal intervals of 225' , viz. 225' , 450' , 675' , .... 5400'
The corresponding sine values give by him are as follows
The corresponding sine values give by him are as follows
These values are correct to more or less eight to nine places of decimal . How Madhwa arrived at such accurate values of sine table will be shown in this thread .
The following passage found in Tantra Sangrah has left distinct hints that the result contained in the lines were of Madhwa . The verses run as follows
The English Translation :
Further ,
These values when substituted in eqn(1) containing terms from t1 to t5 jivā comes out to be 3437'44"48"' , the 24th sine value given in the table of Madhwa (here s = 5400').Similarly if s is replaced gradually by 225' , 450', 675'.. Madhwa's Sine table is obtained
P.S Eqn 1 is
P.S Eqn 1 is
Proceeding in a similar way and substituting values in second equation value of cosine is obtained.This evidently shows that Madhwa followed by the authors of Tantrasamgraha and Karanapaddhati used the two equations for computation of sine and cosine tables.
Second equation is
Second equation is
How Madhwa arrived at the two equations is not yet definitely known . .
The Tantrasamgraha (chapter 2 verse 12½) of Nilakantha and Karanapaddhati ( chapter 6 verse 19) have both given that for small arc jivā = s-s³/ 3! r² ( approximately )
The Tantrasamgraha (chapter 2 verse 12½) of Nilakantha and Karanapaddhati ( chapter 6 verse 19) have both given that for small arc jivā = s-s³/ 3! r² ( approximately )
The Yuktibhasha has given the complete rational of the two equations . Its author Jyeshthadeva ( 1500-1600) in an effort to find an expression for the difference between any arc and its sine chord , divided the circumference of the quarter of a circle into n divisions and
considered the first and second sine difference . He then found the sum of the first n sine differences and cosine differences by considering all sine chords to be equal to corresponding arc and the small unit of circumference to equal to one unit which evidently
gives jivā = s- s³/ 3! r² and Sara = s²r / 2!r²
Since sine values are not actually equal to its arc length further correction was applied ad-infinitum to each of the terms of the value obtained for jivā and sāra which ultimately give rise to equation 1 and 2
Since sine values are not actually equal to its arc length further correction was applied ad-infinitum to each of the terms of the value obtained for jivā and sāra which ultimately give rise to equation 1 and 2
It would not be quite unlikely presume that the rational was first established by Madhwa before Jyeshthadeva could make use of it.
In Western Mathematic Newton is often give the credit for the expansion of sine and cosine series . The result was established later algebraically on a solid foundation by De Moviere and Euler .
It is clear from the thread and discussion that the Indian Scholar Madhwa (1350-1410) used and possibly established the series of course in finite form before Newton , De Moviere and Euler laid the foundation of his sine table .
Research Paper used of A.K Bag . Indian National Science Academy .
