**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 place-value 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 numbers1to9while the tenth corresponds to zero; the following nine letters also receive the values1to9whilst the following letter has the value zero; the next five represent the first five units; and the last eight represent the numbers1to8.

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 place-value 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*