Aryabhateeya was composed in 499 CE by Aryabhata in Kusumapura, which is identified with Pataliputra (essentially modern Patna). This text heralded the advent of siddhaantic astronomy in India, which gives a systematic mathematical treatment of astronomical problems. Aryabhateeya is very brief and has only 121 stanzas. It has four parts, one of which is devoted to mathematics. It is in this part that the computation of the sine function (jyaa in India) is discussed.
The jyaa is far more convenient than the Greek chord for astronomical computations, and is essentially the modern ‘sin theta’. Aryabhateeya states: “The globe of the Earth stands (supportless) at the centre of the circular frame of asterisms surrounded by the orbits (of the planets); it is made up of water, earth, fire and air and is spherical.” It is one of the earliest texts anywhere to discuss the rotation of the earth: “Just as a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by people at Lanka (on the equator), as moving exactly towards the west.”
Aryabhateeya introduces the concept of a mahaayuga of 43,20,000 years, in which all the planets (the Sun and the Moon are also considered as planets) make an integral number of revolutions around the earth. The number of revolutions in a mahaayuga is specified for each planet.
From this, the ‘mean’ position of the planet at any time can be calculated. Then, ‘epicycle’ models are used to calculate the true positions of the planets. Aryabhata has his own version of the epicycle theory, which differs in details from the Greek versions. Thanks to the jyaa, the true positions can be found from simple formulae, in contrast to the method of Ptolemy, the most important Greek astronomer who lived in the second century in Alexandria. Ptolemy’s methods for calculating the true positions of planets and other astronomical quantities are far more complicated, involving cumbersome geometrical reasoning and use of several tables.
Apart from the planetary positions, Aryabhateeya gives algorithms for finding solutions for problems associated with the daily movements of the planets and the stars on the celestial sphere, relation between the time and the shadow of a gnomon, parallax, lunar and solar eclipses, and so on. Aryabhateeya laid the framework for the future development of mathematical astronomy or the siddhaantas in India. Aryabhateeya is very cryptic, whereas the latter texts are far more detailed. However, they are also mainly algorithmic. There would be commentaries on the main texts, which would give detailed explanations, and derivations and proofs too at times.
Some of the most important texts in the siddhaantic tradition after Aryabhateeya were Pancasiddhaantikaa of Varaahamihira (around 520 CE), Mahaabhaaskareeya and Aryabhateeyabhaashya (629 CE) of Bhaskara-I, Braahmasphutasiddhaanta of Brahmagupta (628 CE), Siddhaantasiromani of Bhaskara-II (1150 CE), and the Kerala works that we will discuss shortly. Siddhaantasiromani is a landmark in the history of Indian astronomy, with a marvellous auto-commentary that explains all the concepts with proofs. One can clearly discern a continuous evolution of ideas and improvements in calculational procedures in the siddhaantic tradition.
Apart from the theoretical treatises, there were karana texts that excelled in giving simplified procedures for calculations using tables and elementary arithmetical manipulations, without giving up accuracy. These made it possible for a variety of pancaangas (almanacs) to be prepared throughout India. There is also the vaakya system where the positions of the Sun, the Moon and the planets can be determined at any time, using tables of vaakyas (words and phrases that can be translated into numbers), and some bare arithmetical calculations.
Madhava (late 14th century), Parameshvara, Neelakantha Somayaaji, Jyeshthadeva, Sankara Variar, Acyuta Pisaarati, and Putumana Somayaaji are some of the important figures in the ‘Kerala school’, which made very important contributions to mathematics and astronomy during the 14th to 17th centuries. In an earlier article, we mentioned the first significant steps in calculus taken by Madhava and the astronomer-mathematicians who followed him in Kerala.
In astronomy, Neelakantha Somayaaji (circa 15th-16th century CE) made important innovations in planetary models and spherical trigonometry. He made a detailed scientific analysis of the earlier theories for the motions of the planets Mercury, Venus, Mars, Jupiter and Saturn, especially the first two. From this, he came to the conclusion that the planets move in ‘eccentric orbits’ around the Sun, which itself moves around the Earth. This model is implicit in his Tantrasangraha (1500 CE), but described in detail in his famous commentary on Aryabhateeya, and other works like Siddhaantadarpana and Grahasphutaanayane Vikshepavaasanaa. His model (in the illustration above) was before the famous heliocentric model of Copernicus (1542 CE).
The Copernican model has some flaws that are absent in Neelakantha's model, which is very similar to Tycho Brahe’s one (1580 CE). Candrasekhara Simha Saamanta (1835-1904) from Orissa made extensive observations of the Moon and the planets using simple instruments designed by himself. Based on these, he made many innovations within the siddhaantic tradition. These are described in his monumental treatise named Siddhaantadarpana. His planetary model is similar to that of Neelakantha and Tycho Brahe. (This is the fifth article in the series on India’s contributions to science and technology)
M S Sriram
Theoretical Physicist & President, Prof. K.V. Sarma Research Foundation
(sriram.physics@gmail.com)