Śaṅkaranārāyaṇa (c. 840 – c. 900) was an Indianastronomer and mathematician in the court of King Sthanu Ravi Varman (844- 885 CE) of the Later Cheras in Kerala. He is believed to have established the first astronomical observatory in India at Kodungallur in Kerala. His most famous work was the Laghubhāskarīyavivaraṇa which was a commentary on the Laghubhāskarīya of Bhaskara I which in turn is based on the work of Aryabhata I. The Laghubhāskarīyavivaraṇa was written 869 CE for the author writes in the text that it is written in the Shaka year 791 which translates to a date CE by adding 78.
Śaṅkaranārāyaṇa was a student of the astronomer and mathematician Govindasvami (c. 800 – c. 860).
Information on observatories in India is meager. Many astronomers patronized by kings carried out astronomical observations. The places of these observations could be called as observatories. The first extant reference to a place of observation with some instruments in India is in the treatise Laghubhāskarīyavivaraṇa authored by Śaṅkaranārāyaṇa. In this work, Śaṅkaranārāyaṇa speaks of a place with instruments in the capital city Mahodayapuram of King Sthanu Ravi Varma of the Kulasekhara dynasty in Kerala. Mahodayapuram has been identified with the present day Kodungallur. The observatory was fitted with an armillary sphere which is a model of the celestial sphere. At the directions of Śaṅkaranārāyaṇa, in every 'Kadigai' duration of 34 minutes, bells were sounded at different important centres of the town to announce correct time.
The following is a translation of the verses in Laghubhāskarīyavivaraṇa containing references to the existence of an observatory in Mahodayapura:
"(To the King): Oh Ravivarmadeva, now deign to tell us quickly, reading off from the armillary sphere installed (at the observatory) in Mahodayapura, duly fitted with all the relevant circles and with the sign (-degree-minute) markings, the time of the rising point of the ecliptic (lagna) when the Sun is at 10° in the Sign of Capricorn, and also when the Sun is at the end of the Sign Libra, which I have noted."
Laghubhāskarīyavivaraṇa 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 Indian method involves using the Euclidean algorithm. It is called kuttakara ("pulveriser").
The most unusual feature of the Laghubhāskarīyavivaraṇa is the use of katapayadi system of numeration as well as the place-valueSanskrit numerals which Laghubhāskarīyavivaraṇa frequently uses. Śaṅkaranārāyaṇa is the first author known to use katapayadi system of 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 called varnasamjna was invented by the astronomer Haridatta, and it was explained by him in a text which was written in 684.
The system is based on writing numbers using the letters of the Indian alphabet:
... 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 place-value system with zero. In fact many different "words" could represent the same number and this was highly useful for works written in verse.
^J.B. harley; David Woodward, eds. (1992). The history of cartography : Volume 2 Book 1 : Cartography in the traditional Islamic and South Asian societies. University of Chicago Press. p. 360. ISBN0-226-31635-1.