Na ̄ra ̄yan.a Pan.d.ita wrote, in ad1356, a book for pa ̄t. ̄-ganı .ita,Gan.itakaumudı ̄,
in which he developed, among other things, theories of factorization, of parti-
tioning, of combinatorics, and of magic squares. His work for bı ̄ja-gan.ita,
Bı ̄jagan.ita ̄vatam.sa, seems, judging from its extant portion, to have been modeled
after Bha ̄skara II’sBı ̄jagan.ita.
II.5 The fifteenth to seventeenth centuries – a new wave in
the south
South India in this period produced many talented mathematicians and
astronomers. Particularly important is the academic lineage headed by Ma ̄dhava
of San.gamagra ̄ma (fl. 1380/1420), which is often called the Ma ̄dhava school.
Ma ̄dhava, a resident of San.gamagra ̄ma near Cochin in Kerala, was one of the
most brilliant mathematicians in the world. His name is, and will be, remem-
bered for his discovery of a power series expansion ofpat least and perhaps
also of those of trigonometric functions such as sine, cosine, versed sine, and
arctangent. The verses that state these series are found not in his extant astro-
nomical works but in the works of his successors.
In his commentary on the Lı ̄la ̄vatı ̄,S ́an.kara explicitly ascribes two methods for
calculating the circumference of a circle to Ma ̄dhava. S ́an.kara also cites
Ma ̄dhava’s verse which expresses, in the word-numeral notation (Bhu ̄tasam.khya ̄),
an approximation to pcorrect to 11 decimal places, 2827433388233/(9 · 10^11 ).
Nı ̄lakan.t.ha Somaya ̄ji, son of Ja ̄tavedas, was born ca. 14 June 1444 in
Kun.d.apura near Tirur, Kerala, and studied under Da ̄modara, son of
Parames ́vara, at A ̄lattu ̄ r, Kerala.
He wrote an elaborate commentary on the A ̄ryabhat.ı ̄yain about 1510. It
shows his great talent in mathematics as well as in astronomy. To cite a few
examples, he rediscovered the correct meaning of A ̄ryabhat.a’s rule for sine-
differences. He expressly states the incommensurability of the diameter and the
circumference of a circle, although whether he has proved it or not is not
known. He cites and proves Ma ̄dhava’s formulas for interpolation in the sine
table, and for the sum and difference of sines. He died after 1542.
Indian mathematics thus made unique, remarkable progress up to the sixteenth
century. It was only in the 1720’s that Jaganna ̄tha (1652–1744), at the request
of his patron, Jai Singh Sawai (1688–1744), produced the first Sanskrit version
of Euclid’sElementsunder the title Rekha ̄-gan.ita(Mathematics of Lines) from the
Arabic version of Nas.ı ̄r al-Dı ̄n al-T.u ̄sı ̄.
Bibliography
For more bibliographical information, see Pingree 1970–94 and 1981 and Hayashi 2000.
Amulya Kumar Bag, 1979, Mathematics in Ancient and Medieval India, Varanasi:
Chaukhambha Orientalia, 1979.
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