Surya Siddhanta
Sūrya Siddhānta is an ancient Indian treatise in Astronomy. Like many classical Indian works, the Sūrya Siddhānta is a poem in Sanskrit language. It has fourteen chapter and 500 verses. It is composed in śloka metrical style of Sanskrit. It contain works on Indian sine tables, cosmology, eclipses, planetary motions, conjunctions, star positions, geography, instrumentation, concepts of time and mathematics. Unlike conventional books Sūrya Siddhānta contains advanced calculation and methods which are not easily comprehensible for a rank beginner.^{[1]} The recorded calculation of Earth's obliquity (axial tilt) of 23.975° and the presence of both North celestial Pole star and South celestial pole star recorded within the text places it in the epoch of 3000 BC. The last update made to the text
Sūrya Siddhānta contains many interesting observation which had been used to date the treatise or date the addition of that particular observation into the text. One such calculation is in chapter 2 for the obliquity (tilt) of Earth's axis to be precisely 23.975°. In Chapter 12, it mentions the presence of a North Pole Star and a South Pole Star which can be seen along the horizon at the equatorial region. In the same chapter, Sūrya Siddhānta again indirectly provides the obliquity of exact 24°. In combination these astronomical event are found to have occurred around 3000 BC.^{[2]} ^{[3]} Another observation recorded in the Sūrya Siddhānta is the Sun's equation of centre which also provides an epoch of around 3000 BC. In terms of the epoch another observation is found in Satapatha Brahmana that Krittika Nakshatra rises exactly in the East (point of equinox) which also occurred around 3000 BC and now it rises between East and NorthEast.^{[4]} Where as the longitudinal data in Surya Siddhanta indicates the epoch of 580 CE.^{[5]}
The current values of the Surya Siddhanta’s pulsating epicycle parameters for the Sun appear to have been set in the 50005500 BC timeframe.^{[6]} It can be seen that Sūrya Siddhānta has gone through several updates in the past over a period of several thousand years from the 6th millennium BC to the 1st millennium CE.
History[edit]
Sūrya Siddhānta is well known, most referred and most esteemed. The original author of Sūrya Siddhānta is Mayasura as described in the story in the first chapter that Mayasura obtained his knowledge from Sūrya (the Sun). Siddhānta in Sanskrit means treatise and it usually has author's name prefixed to it. There were several other works on Astronomy in ancient India, many of which have since been lost.
Surya Siddhānta  Brahma Siddhānta  Soma Siddhānta 
Vyasa Siddhānta  Vashishtha Siddhānta  Atri Siddhānta 
Parashira Siddhānta  Kashyap Siddhānta  Nārad Siddhānta 
Garga Siddhānta  Marici Siddhānta  Manu Siddhānta 
Angiras Siddhānta  Lomasha Siddhānta  Pulisha Siddhānta 
Cyavana Siddhānta  Yavana Siddhānta  Bhrigu Siddhānta 
Content[edit]
Sūrya Siddhānta contains 14 chapters and 500 verses. The chapters contain observations, methods, instruments and calculations of various astronomical phenomenas. There is a scarcity of scientific analysis done on the text of Surya Siddhanta. Majority western work is based on Indology dates which in itself is controversial and based on their biased opinion of granting the origin of any science or mathematics to the ancient Greek or babylonians despite of immense textual evidence pointing otherwise. Their analysis of Surya Siddhanta primarily avoids the study of actual data and observations recorded within the Surya Siddhanta.
Indian origin of seconds, minutes and degrees[edit]
Surya Siddhanta in chapter 2 describes the units of seconds, minutes and degrees. These units of measurement are primary basis of the calculations of earth's obliquity and sine tables of Surya Siddhanta. It is reasonable to think that these units or concepts had been in existence prior to other calculations and observation made in the epoch of 6th millennium BC as discussed in this article. The descriptions are
Modern SI units  Surya Siddhanta units  Value 

Second  Vikala   
Minute  Kala  60 seconds 
Degree  Ansh  60 minutes 
Zodiac Sign  Rashi  30 degrees 
Revolution  Bhagan  12 zodiac signs 
These units are used in several calculations done through out the text of Surya Siddhanta. In the sine tables of Surya Siddhanta the first sine or Jyā is described as the value equal to 1/8th of the number of minutes (Kalas) in a zodiac sign (Rashi).
Indian standard circle[edit]
The Surya Siddhanta is using the Indian standard circle in various calculations through out the text. This standard circle is based on radius of 3,438 minutes. The significance lies in the precision of 1/3438 that the ancient Indian astronomers were able to work with. It is evident from the calculation of obliquity of the earth's axis in chapter 2 where 1397 units is the measured Rsine value. Another interesting outcome of this radius of 3,438 minutes is that the circumference of the standard Indian circle is calculated as 21,600 minutes using the formula of Pi multiply by diameter (twice the radius).
Nakshatra (Asterism) System[edit]
The Surya Siddhanta uses the 27 Nakshatra system throughout the text. The Nakshatra is a smaller constellation typically consisting of 1 to 5 stars. The brightest star is called as Yogtara. Each Nakshatra spans 13° 20' on the ecliptic. Each Nakshatra has its own primary star which is usually the junction star but not always.
Longitudinal updates  580 AD[edit]
Chapter 8 of Surya Siddhanta primarily focuses on the stellar data. It provides the longitudinal data for the Asterisms. In comparison to the present day longitudinal values of these stars and the data of Surya Siddhanta, it becomes clear that this update to Surya Siddhanta was made around 580 AD. THe longitude of the stars change by 1° in every 71 years. From the data it is clear that the data does not represent observation but rather is obtained by adding precessional increment to each of the previously calculated data.
Obliquity (tilt) of the Earth's axis  3000 BC[edit]
Obliquity or the axial tilt of earth is the angle which the earth's axis of rotation makes with the perpendicular of orbital plane. This angle varies between 22.1° and 24.5° and it is cyclic phenomena over a period of 41,000 years. Currently the obliquity is 23.4 degrees.^{[8]} Sūrya Siddhānta in two different chapters calculate and provide the value of obliquity.
Chapter 2, verse 28 translates as
The sine of the greatest declination is 1397 units; Multiply the sine by the said sine 1397; Divide the product by the radius 3438 units; Find the arc whose sine is equal to the quotient. This arc is the mean declination of the planet^{[9]}
This way we obtain the obliquity as Sin^{1}(1397/3438) = 23.975°
Chapter 12, verse 68 translates as
At the distance of the fifteenth part of the Earth's circumference (from the equator) in the regions of the Gods or the Asuras (i.e. at the north and south terrestrial tropic) the sun passes through the zenith when it arrives at the north or south solstitial point (respectively)''^{[10]}
It essentially provides information to calculate the axial tilt of earth which in this case can be calculated as 360°/15 = 24°.
The significance of these verses is that they pin points the exact time when the obliquity calculations were made by ancient Indian astronomers and added into the Sūrya Siddhānta. The epoch this obliquity calculation provides is around 3000BC.^{[11]}
North Pole Star and South Pole Star  3000 BC[edit]
Surya Siddhanta contains an observation of the presence of pole stars at both north celestial pole and south celestial pole. Because of the precession of the earth's axis it is known that the pole star changes over a period of time which is normally more than thousand years. In present times our North Pole star is Polaris.^{[12]} This observation is recorded in chapter 12, verse 4344 and translates as
There are two pole stars, one each, near North Celestial Pole (NCP) and near South Celestial Pole (SCP). From equatorial regions, these stars are seen along the horizon. The pole stars are seen along the horizon, thus the place latitude is close to zero, while declination of NCP and SCP is 90 degrees.
Such phenomena was last seen around 3000 BC when Thuban was the North Pole Star and Alpha Hydri was the South Pole star.^{[13]} ^{[14]}
The Pulsating Indian Epicycle of the Sun  50005500 BC[edit]
For determining the Sun’s longitude, the pulsating Indian epicycle is far more accurate than the Greek eccentricepicycle model. The pulsating Indian epicycle for the Sun becomes progressively more accurate as one goes back in time. Peak accuracy, of about 1 minute of arc, is reached around 5200 BC. The current values of the Surya Siddhanta’s pulsating epicycle parameters for the Sun appear to have been set in the 50005500 BC timeframe.^{[15]}
The Latitudinal data  73007500BC[edit]
Using computer simulation of nakshatra latitudinal data by varying ecliptic obliquity, eclipticnodelocation and eclipticsink together with proper motion, a match for the Surya Siddhanta latitudinal data was obtained in the timeframe 73007800 BC.^{[16]} Although the author notes that a major assumption made in this investigation is that star proper motion is fairly constant over several thousands of years. The results may be adversely affected if this were found untrue for the star set under consideration. It should also be noted that this time frame matches with the establishment of the oldest archaeological site of Bhirrana found along the Saraswati river paleochannel. In the 8th millennium BC this site shows that the people were living in the dwelling pits.^{[17]} This stands in contrast with the above time frame, the question arises whether people could be that scientifically advanced while they were inhabiting the dwelling pits. Although this view is subject to change given older more advanced archaeological sites are found within the Indian subcontinent.
Surya Siddhanta sine table[edit]
The Surya Siddhanta provides methods to calculate the sine value in chapter 2. It is among the earliest form of Indian sine tables. The sine tables had been improved upon by many ancient Indian mathematicians. Surya Siddhanta uses an Indian standard circle of radius 3438 minutes. It divides the quadrant into 24 equal segments with each segment sweeping an angle of 3.75° and an arc length of 225 minutes. The verse 1516 translates as
The eighth part of the number of minutes contained in a zodiac sign (Rashi) (i.e. 1800) is the first sine (Jya). Divide the first sine by itself, subtract the quotient by that sine and add the remainder to that sine: the sum will be the second sine. In this manner divide successively the sines by the first sine, subtract the quotient from the divisor and add the remainder to the sine last found and the sum will be next sine. Thus you get twenty four sines (in a quadrant of a circle whose radius is 3438 minutes)^{[18]}
The verse 1722 translates as
The Twenty four sines are 225, 449, 671, 890, 1105, 1315, 1520, 1719, 1910, 2093, 2267, 2431, 2585, 2728, 2859, 2978, 3084, 3177, 3256, 3321, 3372, 3409, 3431, 3438.
Subtract the sines separately from 3438 in the inverse order, the remainders are the versed sines. ^{[19]}
The verse 2327 translates as
The versed sines in a quadrant are 7, 29, 66, 117, 182, 261, 354, 460, 579, 710, 853, 1007, 1171, 1345, 1528, 1719, 1918, 2123, 2333, 2548, 2767, 2989, 3213, 3438.^{[20]}
The Surya Siddhanta derived Sin(θ) or Sine values show astonishing precision of 3 to 4 decimal places in comparison to the modern Sine values. The 1st order difference is the value by which each successive sine increases from the previous and similarly 2nd order difference is the increment in the 1st order difference values. Burgess notes that it is remarkable to see that the 2nd order differences increase as the sines and each, in fact, is about 1/225th part of the corresponding sine.^{[21]}
Sl. No  Angle (in degrees, arcminutes)  Surya Siddhanta value of Jyā (R.sine)  Surya Siddhanta versed sines Utkramājyā (R  R.cosine)  Modern value of Jyā (R.sine)  SS derived sine values (Jyā / 3438)  Modern sine values 

1  03° 45′

225′

7'

224.8560

0.06544503

0.06540313

2  07° 30′

449′

29'

448.7490

0.13059919

0.13052619

3  11° 15′

671′

66'

670.7205

0.19517161

0.19509032

4  15° 00′

890′

117′

889.8199

0.25887144

0.25881905

5  18° 45′

1105′

182′

1105.1089

0.3212078

0.32143947

6  22° 30′

1315′

261′

1315.6656

0.38248982

0.38268343

7  26° 15′

1520′

354′

1520.5885

0.44211751

0.44228869

8  30° 00′

1719′

460′

1719.0000

0.50000000

0.50000000

9  33° 45′

1910′

579′

1910.0505

0.55555556

0.55557023

10  37° 30′

2093′

710′

2092.9218

0.60878418

0.60876143

11  41° 15′

2267′

853′

2266.8309

0.65939500

0.65934582

12  45° 00′

2431′

1007′

2431.0331

0.70709715

0.70710678

13  48° 45′

2585′

1171′

2584.8253

0.75189063

0.75183981

14  52° 30′

2728′

1345′

2727.5488

0.79348458

0.79335334

15  56° 15′

2859′

1528′

2858.5925

0.83158813

0.83146961

16  60° 00′

2978′

1719′

2977.3953

0.86620128

0.86602540

17  63° 45′

3084′

1918′

3083.4485

0.89703316

0.89687274

18  67° 30′

3177′

2123′

3176.2978

0.92408377

0.92387953

19  71° 15′

3256′

2333′

3255.5458

0.94706225

0.94693013

20  75° 00′

3321′

2548′

3320.8530

0.96596859

0.96592583

21  78° 45′

3372′

2767′

3371.9398

0.98080279

0.98078528

22  82° 30′

3409′

2989′

3408.5874

0.99156486

0.99144486

23  86° 15′

3431′

3213′

3430.6390

0.99796393

0.99785892

24  90° 00′

3438′

3438′

3438.0000

1.00000000

1.00000000

See Also[edit]
References and notes[edit]
 ↑ Anil Narayanan, "Dating the Surya Siddhanta using Computer simulation of Proper Motions and Ecliptic variations", Indian Journal of History of Science, Volume 45, issue 4, p1, 23 March 2010.
 ↑ Anil Narayanan, "Dating the Surya Siddhanta using Computer simulation of Proper Motions and Ecliptic variations", Indian Journal of History of Science, Volume 45, issue 4, p2021, 23 March 2010. Accessible at: https://insa.nic.in/writereaddata/UpLoadedFiles/IJHS/Vol45_4_1_ANarayan.pdf
 ↑ Nilesh Oak and Rupa Bhatty,"Ancient updates to Sūryasiddhānta", "India Facts", 19 March 2019, Accessible at: http://indiafacts.org/ancientupdatestosuryasiddhanta/
 ↑ Anil Narayanan, "Wonders, Mysteries and Misconceptions in Indian Astronomy – I", 09 Sept 2019, Accessible at: http://indiafacts.org/wondersmysteriesandmisconceptionsinindianastronomyi/
 ↑ Anil Narayanan, "Wonders, Mysteries and Misconceptions in Indian Astronomy – I", 09 Sept 2019, Accessible at: http://indiafacts.org/wondersmysteriesandmisconceptionsinindianastronomyi/
 ↑ Anil Narayanan, "The Pulsating Indian Epicycle of the Sun", Indian Journal of History of Science, Volume 46, issue 3, p15, 30 June 2011.
 ↑ Pundit Bapu Deva Shastri, "English Translation of Surya Siddhanta",p11, 1861
 ↑ Alan Buis, "Milankovitch (Orbital) Cycles and Their Role in Earth's Climate", "NASA's Jet Propulsion Laboratory" https://climate.nasa.gov/news/2948/milankovitchorbitalcyclesandtheirroleinearthsclimate/
 ↑ E. Burgess, "Translation of Surya Siddhanta", p26, Accessible at https://www.jstor.org/stable/pdf/592174.pdf
 ↑ Pundit Bapu Deva Shastri, "Translation of Surya Siddhanta", "Baptist Mission Press", 1861, Accessible at https://www.wilbourhall.org/pdfs/suryaEnglish.pdf
 ↑ Anil Narayanan, "Dating the Surya Siddhanta using Computer simulation of Proper Motions and Ecliptic variations", Indian Journal of History of Science, Volume 45, issue 4, 23 March 2010.
 ↑ Bruce McClure, "Polaris is the North Pole Star", "Earthsky", 21 May 2019, Accessible at https://earthsky.org/brighteststars/polaristhepresentdaynorthstar
 ↑ Nilesh N Oak and Rupa Bhatty, "Ancient Updates to Surya Siddhanta", 09 March 2019, "India Facts" Accessible at http://indiafacts.org/ancientupdatestosuryasiddhanta/
 ↑ Anil Narayanan, "Wonders, Mysteries and Misconceptions in Indian Astronomy – I", 'India facts", 09 Sept 2019, Accessible at http://indiafacts.org/wondersmysteriesandmisconceptionsinindianastronomyi/
 ↑ Anil Narayanan, "The Pulsating Indian Epicycle of the Sun", Indian Journal of History of Science, Volume 46, issue 3, p15, 30 June 2011.
 ↑ Anil Narayanan, "Dating the Surya Siddhanta using Computer simulation of Proper Motions and Ecliptic variations", Indian Journal of History of Science, Volume 45, issue 4, p21, 23 March 2010.
 ↑ Bhirrana, "Archaeological Survey of India", http://excnagasi.in/excavation_bhirrana.html
 ↑ Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.
 ↑ Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.
 ↑ Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 16.
 ↑ Burgess, Rev. Ebenezer (1860). Translation of the Surya Siddhanta. p. 115.