Sankara Narayana 

Born: about 840 in India Died: about
900 in India Sankara Narayana (or
Shankaranarayana) was an Indian astronomer and mathematician. He was
a disciple of the astronomer and mathematician Govindasvami. His most
famous work was the Laghubhaskariyavivarana which was a commentary on
the Laghubhaskariya of Bhaskara I which in turn is based on the work
of Aryabhata I. The Laghubhaskariyavivarana
was written by Sankara Narayana in 869 AD for the author writes in
the text that it is written in the Shaka year 791 which translates to
a date AD by adding 78. It is a text which covers the standard
mathematical methods of Aryabhata I such as the solution of the
indeterminate equation by = ax c (a, b, c integers) in integers which
is then applied to astronomical problems. The standard Indian method
involves using the Euclidean algorithm. It is called kuttakara
("pulveriser") but the term eventually came to have a more general
meaning like "algebra". The paper [2] examines this method. The
reader who is wondering what the determination of "mati" means in the
title of the paper [2] then it refers to the optional number in a
guessed solution and it is a feature which differs from the original
method as presented by Bhaskara I. Perhaps
the most unusual feature of the Laghubhaskariyavivarana is the use of
katapayadi numeration as well as the placevalue Sanskrit numerals
which Sankara Narayana frequently uses. Sankara Narayana is the first
author known to use katapayadi numeration with this name but he did
not invent it for it appears to be identical to a system invented
earlier which was called varnasamjna. The numeration system
varnasamjna was almost certainly invented by the astronomer
Haridatta, and it was explained by him in a text which many
historians believe was written in 684 but this would contradict what
Sankara Narayana himself writes. This point is discussed below. First
we should explain ideas behind Sankara Narayana's katapayadi
numeration. The system is based on writing
numbers using the letters of the Indian alphabet. Let us quote from
[1]: ... the numerical attribution of
syllables corresponds to the following rule, according to the regular
order of succession of the letters of the Indian alphabet: the first
nine letters represent the numbers 1 to 9 while the tenth corresponds
to zero; the following nine letters also receive the values 1 to 9
whilst the following letter has the value zero; the next five
represent the first five units; and the last eight represent the
numbers 1 to 8. Under this system 1 to 5 are
represented by four different letters. For example 1 is represented
by the letters ka, ta, pa, ya which give the system its name (ka, ta,
pa, ya becomes katapaya). Then 6, 7, 8 are represented by three
letters and finally nine and zero are represented by two
letters. The system was a spoken one in the
sense that consonants and vowels which are not vocalised have no
numerical value. The system is a placevalue system with zero but one
may reasonably ask why such an apparently complicated numeral system
might ever come to be invented. Well the answer must be that it lead
to easily remembered mnemonics. In fact many different "words" could
represent the same number and this was highly useful for works
written in verse as the Indian texts tended to be.
Let us return to the interesting point about the date of Haridatta.
Very unusually for an Indian text, Sankara Narayana expresses his
thanks to those who have gone before him and developed the ideas
about which he is writing. This in itself is not so unusual but the
surprise here is that Sankara Narayana claims to give the list in
chronological order. His list is Aryabhata I
Varahamihira Bhaskara I Govindasvami
Haridatta [Note that we have written
Bhaskara I where Sankara Narayana simply wrote Bhaskara. The more
famous Bhaskara II lived nearly 300 years after Sankara
Narayana.] The chronological order in the
list agrees with the dates we have for the first four of these
mathematicians. However, putting Haridatta after Govindasvami would
seem an unlikely mistake for Sankara Narayana to make if Haridatta
really did write his text in 684 since Sankara Narayana was himself a
disciple of Govindasvami. If the dating given by Sankara Narayana is
correct then katapayadi numeration had been invented only a few years
before he wrote his text. Article
by: J J O'Connor and E F Robertson
Source:www.history.mcs.standrews.ac.uk/Mathematicians



