Mean motions in Indian astronomy

J. Samuel Cornelius (Retired)

Madras Christian College, Chennai – 600059, India

This article is a follow-up of a very important contribution to the understanding of Indian astronomy by Roger Billard who has calculated the mean longitudes of the sun, moon, moon’s apogee and node, and those of the five known planets, from the numerical data in the main astronomical texts. The errors of these longitudes are determined by comparing them with the results of the modern formulae.

Billard uses the methods of mathematical statistics to find the most probable dates of the observations. His graphs, in which the errors of the longitudes are plotted against time from BC 500 to AD 1900, illustrate his most important conclusions. For example, the Aryabhata graph shows that he made accurate observations at about AD 510. There are large errors on earlier and later dates, showing that the rate of change of longitudes, the mean motions, are far from correct.

Another important result of Billard’s, investigation is the dating of the Maha-siddhanta which Billard finds to be in the sixteenth century, contradicting the earlier estimate of Dikshit of near AD 980. Billard’s dating has been strongly supported by an independent study of Raymond Mercier.

Billard finds that amongst the Indian astronomers, Lalla gets a much better accuracy than Ptolemy’s method. Later the sixteenth- century astronomer Nilakantha Somayaji gave computations that were comparable to Lalla’s in accuracy.

In this paper we follow up Billard’s conclusions by investigating mean motions, instead of mean longitudes as was done by Billard.

To get a historical perspective we go back to the Babylonian records. Using a statistical measure of error, the mean values of Babylonian and Ptolemy’s errors are computed and compared. Ptolemy is about three times more accurate.

A similar exercise is done, to calculate the errors of the mean lunar and planetary motions of Ptolemy and the most prominent Indian astronomers, Aryabhata, Brahamagupta, Lalla, Nilakantha Somayaji and seven other Indian Canons. We compare the accuracy of planetary motions of Ptolemy and the Indian astronomers in this paper. We also compare the accuracy of lunar motions of Ptolemy and the Indian astronomers.