The Àryabhatiya of Àryabhata, the oldest precise 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 this 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 new results to vary only slightly. For the latest astronomical constant formulas and applet downloads, go to the Astronomy Pages.

The Àryabhatiya of Àryabhata: The oldest exact astronomical constant?
©1997 by James Q. Jacobs.

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:

"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 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).

Given 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 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 Àryabhatiya was probably written in A.D 498. His sources remain obscure.

Table 1. Comparison of The Àryabhatiya of Àryabhata and Astronomic values.
Astronomy Constants AD 2000.0
AD 500
1604 BC
Rotations per solar orbit 366.25636031
Days per solar orbit
Days per lunar orbit
Rotations per lunar orbit 27.39646289

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 ancient source.

Table 2. Comparison of astronomic periods and historical sources.
Lunar orbit
Lunar synodic
AD 2000.0
AD 498
Paulisa Siddanta
1604 BC

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?

Table 3. Comparison of the astronomical numbers presented by the ancient Indian sources. The Surya Siddanta is dated to approximately AD 1100.
Surya Siddanta
Years in Cycle
Lunar Orbits
Kay notes 57,753,339 lunar orbits rather than 57,753,336 per Clark.
Synodic Months

* 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.

  • Do you know of any source previously noticing and publishing the accuracy of Àryabhata'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 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:

Verse No. 12
Sixty Nadis is known as Nakshatra Ahoratra (ie. day and night together). Thirty of that (ie. Ahoratra) become one Masa (ie. Nakshatra Masa). Like that thirty Savana days with thirty Arkodaya or Sunrises is one Savana Masa or Savana month.

Verse No. 13
The same way the Tithis also (ie. thirty Tithis make one Saumya or Chandra Masa). Like that one Sankranti to the other Sankranti (ie. from the entry of the Sun into one Rashi to the entry of the Sun in to the next Rashi) is called Saura Masa. Twelve of this (Saura Masa) is called Divya Varsha.

Verse No. 14
What is Deva's day is Asura's night and what is Asura's day is Deva's night. Six times sixty of that (i.e. 360 of the day and night) make one Divya Varsha and Asura Varsha as well.

Verse No. 15
12,000 of that Suryabda (ie. Divya / Asura Varsha) is Chaturyuga (or Mahayuga). Numerically it (Chaturyugas) is 4,320,000 (Ahoratras or days and night together).

Along with the following comments:

"The Sanskrit verses of the Suryasiddhanta are the same. ... In the 19th century CE some scholars ... started interpreting the span of the Mahayuga as 4,320,000 years whereas it is only 4,320,000 days."

Watching Eclipses, Counting Orbits PowerPoint with AeGeo code.

2012.06.12 - "New evidence implies Maya astronomers possessed precise astronomy as early as 800 C.E. ... the astronomical tables found in an excavated room in Xutlun, Guatemala, "Many of these hieroglyphs are calendrical in nature and relate astronomical computations..." One numerical array presents several large numbers that imply precise astronomical constants. ... Not only does the interval 341,640 days equal an integer number of full moons (11,519.022), interpreting the number as solar orbits also presents an integer number of full moons, and at a ratio with far greater accuracy." More: Precise Maya Astronomical Knowledge at Xultun


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.

Mean Motions and Longitudes in Indian Astronomy
Dennis W. Duke, 2008, has additional current references.


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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