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In the work The Àryabhatíya of Àryabhata, An Ancient Indian Work on Mathematics and Astronomy, translated by William Eugene Clark, Professor of Sanskrit in Harvard University (The University of Chicago Press, Chicago, Illinois. 1930), I found the following to be written: "In a yuga the revolutions of the Sun are 4,320,000, of the Moon 57,753,336, of the Earth eastward 1,582,237,500, of Saturn 146,564, of Jupiter 364,224, of Mars 2,296,824 . . . " (page 9). As can be seen from the Clarke's translation Àryabhata wrote that 1,582,237,500 rotations of the Earth equal 57,753,336 lunar orbits. (These same two numbers are also presented by G. R. Kay in his appendices, where they are attributed to Àryabhata and Pusíla.) This is an extremely accurate ratio (1,582,237,500 / 57,753,336 = 27.3964693572).

Given Jan. 1, 2000 astronomic constants and given the present day formulas to temporally adjust the astronomic constants I have calculated that Àryabhata's ratio would have been exact in 1604 BC. My data is presented in the table below. The temporal variation formulas used can be obtained from my Astronomy Formulas page. The date AD 500 is the approximate epoch in which Àryabhata wrote. Àryabhata was born in 476 in Patna, India and died in 550. His Àryabhatíya was probably written in A.D 498.

While the majority of the ratios presented by Àryabhata are not equally precise, it is difficult to believe that the earth rotations to lunar orbits ratio, given such very large numbers, could be so precise by coincidence. The odds of that being the case are astronomical. This is particularly so given that the data derives from an era when it was more exactive than today. If it derived from an ancient Vedic source, it was even more exactive when it originated.

According to G. R. Kay, Àryabhata and the Paulisa Siddanta present the values below for the lunar periods. Kay's table of durations of sidereal and synodic months also quotes another ancient Indian authority of the era, Paulisa Siddhanta. Obviously the accuracy of the ancient Indian astronomical data is not just coincidence. Note that the lunar orbit period of 27.321668 is accurate for the same epoch as the lunar orbit to earth rotations ratio quoted. This is supportive of the suggestion that the information derives from an accurate ancient source.

Àryabhata wrote the Àryabhatíya in four chapters.
The first chapter presents the astronomical constants and sine tables.
Chapter II is mathematics required for computation.
Chapter III discusses time and the longitudes of the planets.
Chapter IV includes rules of trigonometry and rules for eclipse computations.
Àryabhata's work in effect started a new school of astronomy in South India.

Àryabhata is the first known astronomer to have initiated a continuous counting of solar days, designating each day with a number. This 'count of days' is termed the 'ahargana.' His epoch began at the beginning of the Mahayuga. To avoid excessively large numbers later astronomers changed the beginning of the epoch to the Kali era, commencing at midnight of 17-18 February of 3102 B.C.

The Àryabhatíya is a summary of Hindu mathematics up to his time, including astronomy, spherical trigonometry, arithmetic, algebra and plane trigonometry. Some of his formulas are correct, others not. The first appearance of the sine of an angle appears in the work of Àryabhata. He gave tables of half chords (sine tables).

To the best of my knowledge, Àryabhata's ratio represents the earliest known recorded astronomic ratio with such incredible accuracy. It surprises me that this fact has gone unnoticed to this date (to the best of my knowledge). I suspect that this oversight is due to our present day emphasis on days and years, rather than rotations and orbits. Few readers today would recognize the ratio of rotations of the earth per lunar orbit. Other author's have commented on the accuracy of ancient Indian astronomy, though typically the ratios were assessed in relation to the duration of the Mahayuga (4,320,000 years). It does not surprise me that such an accurate astronomic ratio may have been known to other cultures in earlier eras.

Àryabhata wrote that the apparent motion of the heavens was due to the axial rotation of our planet. Àryabhata taught that the earth is a sphere and rotates on its axis, and that eclipses resulted from the shadows of the moon and earth. Àryabhata's innovations were opposed by Hindu teachers. His teachings were not in accordance with the religious views of his era.

Àryabhata wrote, according to Clarke, "In a yuga the revolutions of the Sun are 4,320,000, of the Moon 57,753,336, of the Earth eastward 1,582,237,500, . . ." Given Àryabhata's value of 27.321668 days per lunar orbit period, the 57,753,336 lunar orbits represent 4,320,027.33 solar orbits (in AD 500), not 4,320,000. Why? Perhaps because the numbers are divisible by 60 and 6. The ancient Indians employed base 60 math. I have no certain answer for this question. Perhaps religious dogma had an influence in this matter. The accuracy of the ratios presented should be considered valid, even though they do not match the exact time intervals considered significant in Hindu cosmology. This inaccuracy poses a question regarding the planetary numbers. Should they be compared to the 4,320,000 years number or to the rotations and lunar orbits numbers?

Here follows a comparative chart of the astronomical numbers presented by the ancient Indian authorities and sources. The Surya Siddhanta is dated to approximately AD 1100.

*Kay notes 57,753,339 lunar orbits rather than 57,753,336 per Clarke.

BIBLIOGRAPHY

Clark,William Eugene, The Àryabhatíya of Àryabhata, An Ancient Indian Work on Mathematics and Astronomy, The University of Chicago Press, Chicago, Illinois. 1930.

Kay, G. R., Hindu Astronomy, Ancient Science of the Hindus, Cosmo Publications, New Dehli. Indi, 1981.

Pingree, David, Jyotihsastra, Astral and Mathematical Literature, Otto Harrassowitz, Weisbaden, 1981.

Sastri, Pundit Bapu Deva, and Lancelot Wilkinson, The Surya Siddhánta, or An Ancient System of Hindu Astronomy, Philo Press, Amsterdam, 1974.

Sen, S. N., and K. S. Shukla, History of Astronomy in India, Indian National Science Academy, New Dehli, 1985.

ACKNOWLEDGEMENTS

Since I first published this page several people have offered advice and encouragement. Some have asked for further information. In particular I wish to thank Dr. Vijay Bedekar and David B. Kelley for encouraging further research. This updated and expanded page has resulted. Thanks to Ramana Bhamidipati for his input and suggestions.

SOME QUESTIONS:

At this writing, May 26, 1999, I still await some answers to the questions posed below. This may be indicative of the answers. To date I have found no indication of older accurate astronomic constants or published indications of modern writers noticing the accuracy of the data discovered in the Indian sources.

Do you know of any source previously noticing and publishing the accuracy of Aryabhata's ratio?

Do you know of any older record reflecting such an accurate astronomic ratio? From India? In Sanskrit? From other parts of the world?

Do you know of any astronomic record reflecting such an accurate astronomic ratio prior to the last two centuries?

When did modern astronomers first arrive at an astronomic ratio of comparable accuracy?

The WWW is interactive. You can contribute to this niche of knowledge. If you can answer or comment on any of the questions posed please send me your data at aegeo@yahoo.com. Your contribution may be used to update this material. I do not read sanskrit. If you do, and you have read the original works, your contributions will be especially appreciated.