The Àryabhatiya of Àryabhata: The oldest exact astronomical constant?
2012.08.09 - In 1997, when this research article was accomplished, formulas for astronomy constants were not as refined as today. Now, given new formulations and application of the distinction between atomic and ephemeris time, results will be slightly different. Rather than overwrite thisl step in examination of ancient astronomy, I'm leaving the article intact with the caveat that researchers should utilize the latest methods to examine the questions raised herein and should expect the new results to vary. For the latest astronomical constant formulas and applet downloads, go to my Astronomy Pages.
In the work The Àryabhatiya of Àryabhata, An Ancient Indian Work on Mathematics and Astronomy, translated by William Eugene Clark, Professor of Sanskrit at Harvard University (The University of Chicago Press, Chicago, Illinois. 1930), I found the following to be written:
As can be seen from the Clark 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 for two fundamental cosmic motions (1,582,237,500 / 57,753,336 = 27.3964693572).
Jan. 1, 2000 astronomical constants and given the present day formulas
to temporally adjust the astronomical constants, I calculated that Àryabhata's
ratio was exact for 1604 BC.* The resulting data is presented in Table
1. The temporal variation formulas used can be obtained from my Astronomy
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 Àryabhatiya was probably written in A.D 498.
His sources remain obscure.
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. Additionally, lunar orbit and earth rotation are two of the three actual fundamental cosmic motions, rather than the apparent day/night or lunar phase cycles.
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 Siddanta. 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 above. This is
supportive of the suggestion that the information derives from an accurate
Àryabhata wrote the Àryabhatiya 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 Àryabhatiya 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?
* This date result is dependent on the accuracy of the obliquity of the ecliptic formula used. Modern, temporal-change-of-obliquity formulas merit closer analysis before reliance on their precision when date-reaching mechanisms are employed.
At this rewriting, Jan. 1, 2001, 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 astronomical constants or published indications of modern writers noticing the accuracy of the data discovered in the Indian sources.
The WWW is interactive. You can contribute to this niche of knowledge. If you can comment on or answer any of the questions posed please e-mail me your data: Contact. If your contribution is used to update this material, you will be credited. I do not read Sanskrit. If you do, and you have read the original works, your contributions will be especially appreciated.
2009.07.07. Article links:
2009.12.21 - In response to the questions above, I have received few replies. I recently found a more accurate ratio expressed by Aristarchus' "Great Year" when interpreted as 4,876 solar orbits. While comparing accuracy I noted Kay's Àryabhata interval, 57,753,339 lunar orbits (rather than Clarke's 57,753,336), precisely equates to a integer of 4,320,027.0 orbits. The solar orbits to lunar orbits ratio is a more precise ratio than Àryabhata's ratio of rotations to lunar orbit.
Since this article posted in 1997, "Kay notes 57,753,339 lunar orbits" has gone unnoticed as a solar orbit integer, as a clue to the more accurate constant. Given Kay's lunar orbits interpretation is a precise integer of solar orbits and an even more accurate second "great ratio" Kay's number has support as an intended interval, likely with 4,320,027 solar orbits. Half of Aristarchus' "Great Year," the 2,438 solar orbits equaling 892,933 rotations interval, expresses an astronomy ratio with greater accuracy than Àryabhata's ratios. I now seek other citations of 'great year' intervals in ancient literature.
2011.10.05 - In response to the Clark quote, Sunil Bhattacharjya provided the following Sanskrit translation of the Surya Siddhanta:
Along with the following comments:
Clark, William Eugene, The Àryabhatiya 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. India, 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.
Motions and Longitudes in Indian Astronomy
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.
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Published Feb. 1, 1998. Cite as http://www.jqjacobs.net/astro/aryabhata.html