Gregory's (1632 AD) Series For Inverse Tangent

istajya-trijyayorghathath kotyaptam prathamam phalam jyavargam

gunakam kritva kotivargam cha haarakam pratha maadiphalebhyo

atha neya phalakrtir muhu: eka-tryaady-ojasankhyabhirabhakteshveteshv

anukramaat ojanam samyutesthyaktva yugmayogam dhanur bhavet

doh-kotyor alpameveha kalpaniyam iha smrtam labdhinam

avasanam syanna thathaapi muhu: krte

Obtain the first result of multiplying the jya (R sine ) by the trijya (radius) and dividing the product by koti (R cos ). Multiply this result by the square of the jya and divide the square by the koti. Thus we obtain a second result a sequence of the further results by repeatedly multiply by the square of the jya and dividing by the square of the koti. Divide the terms of the sequence in order by the odd numbers 1,3,5,…; after this, add all the odd terms and subtract from them all the even terms (without disturbing the order of the terms). Thus is obtained the dhanus whose two elements are the given jya and koti. (Here the smaller of the two elements should be taken as the jya, since other wise the series obtained will be non finite) (use of Tangent)]]>

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