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Śaṅkaranārāyaṇa

Sankara Narayana (c. 840 – c. 900 AD) was an Indian astronomer-mathematician in the court of Ravi Kulasekhara (c. 844 – c. 883 AD) of the Chera Perumal kingdomofKerala.^[1][2] He is best known as the author of Laghu Bhaskariya VivaranaorVyakha (869/870 AD), a detailed commentary on treatise Laghu Bhaskariya by 7th century mathematician Bhaskara I (which in turn was based on the works of the 5th century polymath Aryabhata).^[3][4] Sankara Narayana is known to have established an astronomical observatory at the port of Kodungallur in central Kerala.^[2][5]

Laghu Bhaskariya Vivarana (Chapter VII), produced in the court of king Ravi Kulasekhara at Kodungallur, explicitly states that it was composed in Saka Year 791 (=869/70 AD).^[4][1] It is also mentions that the year was the 25th regnal year of king Ravi Kulasekhara.^[6] In the second verse of the vivarana Narayana remembers five major precessors in the field of mathematics (Aryabhata, Varahamihira, Bhaskara I, Govinda and Haridatta), including his possible master Govinda (c. 800 – c. 860 AD).^[6]

• There are references to an instrument called "Rashi Chakra" marked by a "Yanthra Valaya" in the vivarana. This instrument might be the same as the Gola Yanthra/Chakra Yanthra mentioned by famous polymath Aryabhata. The Chakra Yanthra was developed further and called Phalaka Yanthra by Bhaskara I.^[7]

"Oh [king] Ravi Varma Deva, 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."^[8]

• At the directions of Sankara Narayana, in every 'katikai' (= 34 minutes), bells were sounded at different important centres of Mahodayapura to announce correct time.

• Laghu Bhaskariya Vivarana 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").^[4]
• The most unusual feature of the Laghubhāskarīyavivaraṇa is the use of katapayadi systemofnumeration as well as the place-value Sanskrit numerals which Laghubhāskarīyavivaraṇa frequently uses.^[4]

• The opening verse of Laghu Bhaskariya Vyakha gives an indirect invocation to the lord called "Sthanu" (carefully composed to be applicable to god Siva and the ruling king).^[6]

"Sa Sthanurjayati trirupasahito lingepi lokarcitah".

— Sankara Narayana, Laghu Bhaskariya Vyakha, Chapter I (c. 870 AD)

• Sankara Narayana also mentions that the full name of his king as "Ravi Varma Kulasekhara".^[6]
• Laghu Bhaskariya Vivarana was composed in the 25th regnal year of king Kulasekhara.^[6]

• "Angartvambara nanda devamanubhir yate dinanam gane"
  • Anga = 6, Rtu = 6, Ambara = 0, Nanda = 9, Veda = 4, and Manu = 14
  • Order - 6609414
  • Reverse Order - 1449066
• Kali Date - 3967 years and 86 days = 25 Mithuna, Kollam Era 41 = 870 AD
• "Evam Sakabdah punariha candra randhramuni sankhyaya asambhiravagatah"
  • Candra = 1, Randhra = 9, and Muni = 7
  • Order - 197
  • Reverse Order - 791 (Saka Year) = 870 AD

"Angartvambara nanda devamanubhir yate dinanam gane
Graste tigma mayukhamalinitamobhute parahne divi
Prsta praggrahanad dvitiyaghatika grasa pramanam raver
Bharta sri Kulasekharena vilasad velavrtaya bhuva".

— Sankara Narayana, Laghu Bhaskariya Vyakha, Chapter IV (c. 870 AD)

"Evam Sakabdah punariha candra randhramuni sankhyaya asambhiravagatah".

— Sankara Narayana, Laghu Bhaskariya Vyakha, Chapter 1 (c. 870 AD)

"Capapravista guru sauri samatva kalam
Yamyottaram gamanamantaratah pramanam
Acaksvya sarvamavagamya bhatoktamargad
Ityuktavan ravirasena nrpabhivandya".
"Tada pancavimsati Varsanyatitani devasya".

— Sankara Narayana, Laghu Bhaskariya Vyakha, Chapter VII (c. 870 AD)

• Meeting of Guru (=Jupiter) and Sauri (=Saturn) in Capa (Dhanu) = 25th regnal year of the king = 870 AD

• Indian mathematics
• History of mathematics
• List of astronomers and mathematicians of the Kerala school