The Kerala School · Chapter 1 of 2

#The Kerala astronomers

Source. This chapter follows one directly-read authority: David Pingree, Jyotihshastra: Astral and Mathematical Literature (A History of Indian Literature VI.4, Wiesbaden 1981), the sections on south Indian astronomy and on mathematics. Nothing here is taken from popular summaries.

#Three systems

Pingree describes the astronomy of the south, and of Kerala above all, as insular: "some of the most brilliant work ever to be done in India in mathematics was produced in the fourteenth and fifteenth centuries without having any effect whatsoever on the north." Three related systems grew up there:

• the Parahita and the vakya systems, which follow the school of Aryabhata (Aryapaksha);
• the drigganita system, which follows the school of the Surya Siddhanta (Saurapaksha).

Kerala also cultivated a kind of book called a tantra: not a complete system, but practical solutions to chosen problems, and many short essays on single questions.

#Haridatta and the Parahita system

The Parahita system was first set out in the Grahacaranibandhana of Haridatta. By tradition the system began in 683; since Haridatta is already quoted as an authority by Govindasvamin in the first half of the ninth century, Pingree finds 683 "not an unreasonable date".

The book versifies the planets' mean daily motions according to Aryabhata's school, tables of the sines of their equations at intervals of 3°45′ (which Haridatta calls vakyas), and the planets' mean longitudes on 20 March 355 — all of the numbers written in the katapayadi system, in which syllables stand for digits. In another work, probably the lost Mahamarganibandhana, Haridatta gave corrections (bija) to the planets' yearly motions for the years after 522. (Edited by K. V. Sarma, Madras 1954.)

#The Moon-sentences

The vakyas ("sentences") are tables built on the planets' synodic periods and, for the Moon, on cycles of its anomalistic month (the time from one slowest point to the next). The Moon-sentences, the Candravakyani, are attributed to Vararuci, and use the relation

9 anomalistic months = 248 days,

with larger cycles of 110 anomalistic months = 3,031 days and 449 = 12,372 days. The two smaller cycles were already in the Vasishtha Siddhanta that Varahamihira summarised (Panchasiddhantika 3.8–9). The earliest attested epoch of the lunar vakyas is 1184.

Several Sanskrit poems of Kerala carry these sentences inside their verses by the katapayadi code. The most notable is the Haricarita of Parameshvara (fifteenth century), in which the life of Krishna is the surface meaning and the Moon tables lie beneath. The vakyas were still in use in south India in modern times; John Warren described them in 1825.

The planetary vakyas are given in an anonymous Vakyakarana, whose lunar epoch is 22 May 1282; it seems to have been composed about 1300, probably near Kanchi. Its commentator Sundararaja (late fifteenth or early sixteenth century) corresponded with Nilakantha Somayajin.

#Madhava of Sangamagrama

The most important school of the south was founded by Madhava of Sangamagrama (near Cochin), in the late fourteenth century. Pingree writes that he "evidently discovered" the series for π now called Gregory's, and the power series for sine and cosine associated with Newton and Leibniz. His surviving astronomical works refine the vakya system: the Sphutacandrapti; the Venvaroha (epoch 1403), which uses the 248-day cycle to find the Moon's longitude at nine equal times in a day; and a new set of Moon-sentences giving daily longitudes correct to the second sexagesimal place (Vararuci's go to the first). He also wrote on the planets (the Aganitagrahacara, apparently 1418) and on the ascendant.

#Parameshvara

Madhava's most distinguished pupil was Parameshvara of Asvatthagrama, about 1380–1460 in Pingree's dating. He commented on Aryabhata, Bhaskara, the Surya Siddhanta and others, and made a series of observations of eclipses of the Sun and Moon between 1393 and 1432. His drigganita system (Drgganita, 1431) took its mean motions from the Surya Siddhanta's school with new values for other elements. (W. E. Clark, in his 1930 Aryabhatiya, called Parameshvara a sixteenth-century commentator; Pingree's later dating rests on Parameshvara's own dated works.)

#Nilakantha Somayajin and his successors

Nilakantha Somayajin, a pupil of Parameshvara's son Damodara, was born about 14 June 1444 at Kundapura near Tirur. In his commentary on the Aryabhatiya he refers to his own observations of solar eclipses in 1467 and 1501. He set out new parameters in the Tantrasangraha (1501), and in the Jyotirmimamsa (1504) he argued that astronomical parameters must constantly be corrected by observation. The commentaries on the Tantrasangraha by his pupil Sankara, about 1550, are important for the transmission of Madhava's series.

Another pupil of Damodara, Jyeshthadeva, wrote the Malayalam Yuktibhasa; and among his pupils was Acyuta Pisarati, who died on 7 July 1621, a prolific author on astronomy. The series became known in Europe through C. M. Whish's paper of 1830 on the Tantrasangraha, the Yuktibhasa and other Kerala books.

#Summary

• Kerala had three systems: Parahita and vakya (Aryabhata's school) and drigganita (the Surya Siddhanta's).
• Haridatta's Grahacaranibandhana (traditionally 683) wrote all its numbers in katapayadi.
• The Moon-sentences use 9 anomalistic months = 248 days; the Vakyakarana's epoch is 1282.
• Madhava (late 14th century) evidently discovered the series for π, sine and cosine; Parameshvara observed eclipses 1393–1432 and founded drigganita; Nilakantha wrote the Tantrasangraha (1501).

#Exercises

1. Which two Kerala systems follow Aryabhata's school, and which follows the Surya Siddhanta's?
2. How are the numbers written in Haridatta's book?
3. What cycle do the Moon-sentences use?
4. Why does Nilakantha's Jyotirmimamsa matter?

Answers: 1. The Parahita and vakya systems; the drigganita system. 2. In the katapayadi system, in which syllables stand for digits. 3. 9 anomalistic months = 248 days (with larger cycles of 3,031 and 12,372 days). 4. It argues that astronomical parameters must constantly be corrected by observation.