Surya Siddhanta

Surya Siddhanta is a Hindu text on astronomy from 4th or 5th-century CE.[1] Above is verse 1.1, which pays homage to Brahma.[2]

The Surya Siddhanta is the name of a Sanskrit treatise in Indian astronomy from late 4th-century or early 5th-century CE.[1][3] The text survives in several versions, was cited and extensively quoted in a 6th-century text, was likely revised for several centuries under the same title.[4][3] It has fourteen chapters.[5] A 12th-century manuscript of the text was translated by Burgess in 1860.[2]

The Surya Siddhanta describes rules to calculate the motions of various planets and the moon relative to various constellations, diameters of various planets, and calculates the orbits of various astronomical bodies.[6][7] The text asserts that earth has a spherical shape.[5] It treats earth as stationary globe around which sun orbits, and makes no mention of Uranus, Neptune or Pluto.[8] It calculates the earth's diameter to be 8,000 miles (modern: 7,928 miles), diameter of moon as 2,400 miles (actual ~2,160) and the distance between moon and earth to be 258,000 miles (actual ~238,000).[6] The text is known for some of earliest known discussion of sexagesimal fractions and trigonometric functions.[1][3][9]

The Surya Siddhanta is one of the several astronomy-related Hindu texts that likely was influenced by ancient pre-Ptolemy Greek ideas. It represents a functional system that made reasonably accurate predictions.[10][11][12] The text was influential on the solar year computations of the luni-solar Hindu calendar.[13]

Textual history

In a work called the Pañca-siddhāntikā composed in the sixth century by Varāhamihira, five astronomical treatises are named and summarised: Paulīśa-siddhānta, Romaka-siddhānta, Vasiṣṭha-siddhānta, Sūrya-siddhānta, and Paitāmaha-siddhānta.[14]:50 The surviving version of the text is placed in the 1st millennium BCE by Markandaya and Srivastava.[5] Most scholars, however, place the text variously from the 4th-century to 5th-century CE.[3][15]

According to John Bowman, the earliest version of the text existed between 350-400 CE wherein it referenced sexagesimal fractions and trigonometric functions, but the text was a living document and revised through about the 10th-century.[3] One of the evidence for the Surya Siddhanta being a living text is the work of medieval Indian scholar Utapala, who cites and then quotes ten verses from a version of Surya Siddhanta, but these ten verses are not found in any surviving manuscripts of the text.[16] According to Plofker, large portions of the more ancient Sūrya-siddhānta was incorporated into the Panca siddhantika text, and a new version of the Surya Siddhanta was likely revised and composed around 800 CE.[4] Some scholars refer to Panca siddhantika as the old Surya Siddhanta and date it to 505 CE.[17]

Vedic influence

The Surya Siddhanta is a text on astronomy and time keeping, an idea that appears much earlier as the field of Jyotisha (Vedanga) of the Vedic period. The field of Jyotisha deals with ascertaining time, particularly forecasting auspicious day and time for Vedic rituals.[18] Max Muller, quoting passages by Garga and others for Vedic sacrifices, states that the ancient Vedic texts describe four measures of time – savana, solar, lunar and sidereal, as well as twenty seven constellations using Taras (stars).[19] The idea of twenty eight constellations and movement of astronomical bodies already appears, states David Pingree – a professor of History of Mathematics and Classics, in the Hindu text Atharvaveda (~1000 BCE).[10] Scholars have speculated that this may have entered India from Mesopotamia. However, states Pingree, this hypothesis has not been proven because no cuneiform tablet or evidence from Mesopotamian antiquity has yet been deciphered that even presents this theory or calculations.[10]

According to Pingree, the influence may have flowed the other way initially, then flowed into India after the arrival of Darius in Indus Valley about 500 BCE. The mathematics and devices for time keeping mentioned in these ancient Sanskrit texts, proposes Pingree, such as the water clock may also have thereafter arrived in India from Mesopotamia. However, Yukio Ohashi considers this proposal as incorrect,[20] suggesting instead that the Vedic timekeeping efforts, for forecasting appropriate time for rituals, must have begun much earlier and the influence may have flowed from India to Mesopotamia.[21] Ohashi states that it is incorrect to assume that the number of civil days in a year equal 365 in both Hindu and Egyptian–Persian year.[22] Further, adds Ohashi, the Mesopotamian formula is different than Indian formula for calculating time, each can only work for their respective latitude, and either would make major errors in predicting time and calendar in the other region.[23]

Kim Plofker states that while a flow of timekeeping ideas from either side is plausible, each may have instead developed independently, because the loan-words typically seen when ideas migrate are missing on both sides as far as words for various time intervals and techniques.[24][25]

Greek influence

It is hypothesized that there were cultural contacts between the Indian and Greek astronomers via cultural contact with Hellenistic Greece, specifically the work of Hipparchus (2nd-century BCE). There were some similarities between Suryasiddhanta and Greek astronomy in Hellenistic period. For example, Suryasiddhanta provides table of sines function which parallel the Hipparchus table of chords, though the Indian calculations are more accurate and detailed.[26] According to Alan Cromer, the Greek influence probably arrived in India by about 100 BCE.[27] The Indians adopted the Hipparchus system, according to Cromer, and it remained that simpler system rather than those made by Ptolemy in the 2nd century.[28]

Astronomical calculations: Estimated time per sidereal revolution[29]
Planet Surya Siddhanta Ptolemy 20th-century
Mangala (Mars) 686 days, 23 hours, 56 mins, 23.5 secs 686 days, 23 hours, 31 mins, 56.1 secs 686 days, 23 hours, 30 mins, 41.4 secs
Budha (Mercury) 87 days, 23 hours, 16 mins, 22.3 secs 87 days, 23 hours, 16 mins, 42.9 secs 87 days, 23 hours, 15 mins, 43.9 secs
Bṛhaspati (Jupiter) 4,332 days, 7 hours, 41 mins, 44.4 secs 4,332 days, 18 hours, 9 mins, 10.5 secs 4,332 days, 14 hours, 2 mins, 8.6 secs
Shukra (Venus) 224 days, 16 hours, 45 mins, 56.2 secs 224 days, 16 hours, 51 mins, 56.8 secs 224 days, 16 hours, 49 mins, 8.0 secs
Shani (Saturn) 10,765 days, 18 hours, 33 mins, 13.6 secs 10,758 days, 17 hours, 48 mins, 14.9 secs 10,759 days, 5 hours, 16 mins, 32.2 secs

The influence of Greek ideas on early medieval era Indian astronomical theories, particularly zodiac symbols (astrology), is broadly accepted by scholars.[26] According to Jayant Narlikar, the Vedic literature lacks astrology, the idea of nine planets and any theory that stars or constellation may affect an individual's destiny. One of the manuscripts of the Surya Siddhanta mentions deva Surya telling asura Maya to go to Rome with this knowledge I give you in the form of Yavana (Greek), states Narlikar.[30] The astrology field likely developed in the centuries after the arrival of Greek astrology with Alexander the Great,[20][31][32] their zodiac signs being nearly identical.[18]

According to Pingree, the 2nd-century CE cave inscriptions of Nasik mention sun, moon and five planets in the same order as found in Babylon, but "there is no hint, however, that the Indian had learned a method of computing planetary positions in this period".[33] In the 2nd-century CE, a scholar named Yavanesvara translated a Greek astrological text, and another unknown individual translated a second Greek text into Sanskrit. Thereafter started the diffusion of Greek and Babylonian ideas on astronomy and astrology into India, states Pingree.[33] The other evidence of European influential on the Indian thought is Romaka Siddhanta, a title of one of the Siddhanta texts contemporary to Surya Siddhanta, a name that betrays its origin and probably was derived from a translation of a European text by Indian scholars in Ujjain, then the capital of an influential central Indian large kingdom.[33]

According to John Roche – a professor of Mathematics with publications on the history of measurement, the astronomical and mathematical methods developed by Greeks related arcs to chords of spherical trigonometry.[34] The Indian mathematical astronomers, in their texts such as Surya Siddhanta developed other linear measures of angles, made their calculations differently, "introduced the versine, which is the difference between the radius and cosine, and discovered various trigonometrical identities".[34] For instance, states Roche, "where the Greeks had adopted 60 relative units for the radius, and 360 for circumference", the Indians chose 3,438 units and 60x360 for the circumference thereby calculating the "ratio of circumference to diameter [pi, π] of about 3.1414".[34]

The tradition of Hellenistic astronomy ended in the West after Late Antiquity. According to Cromer, the Surya Siddhanta and other Indian texts reflect the primitive state of Greek science, nevertheless played an important part in the history of science, through its translation in Arabic and stimulating the Arabic sciences.[35] According to a study by Dennis Duke that compares Greek models with Indian models based on the oldest Indian manuscripts such as the Surya Siddhanta with fully described models, the Greek influence on Indian astronomy is strongly likely to be pre-Ptolemaic.[11]

The Surya Siddhanta was one of the two books in Sanskrit translated into Arabic in the later half of the eighth century during the reign of Abbasid caliph Al-Mansur. According to Muzaffar Iqbal, this translation and that of Aryabhatta was of considerable influence on geographic, astronomy and related Islamic scholarship.[36]

Contents

The contents of the Surya Siddhanta is written in classical Indian poetry tradition, where complex ideas are expressed lyrically with a rhyming meter in the form of a terse shloka.[37][38] This method of expressing and sharing knowledge made it easier to remember, recall, transmit and preserve knowledge. However, this method also meant secondary rules of interpretation, because numbers don’t have rhyming synonyms. The creative approach adopted in the Surya Siddhanta was to use symbolic language with double meanings. For example, instead of one, the text uses a word that means moon because there is one moon. To the skilled reader, the word moon means the number one.[37] The entire table of trigonometric functions, sine tables, steps to calculate complex orbits, predict eclipses and keep time are thus provided by the text in a poetic form. This cryptic approach offers greater flexibility for poetic construction.[37][39]

The Surya Siddhanta thus consists of cryptic rules in Sanskrit verse. It is a compendium of astronomy that is easier to remember, transmit and use as reference or aid for the experienced, but does not aim to offer commentary, explanation or proof.[15][38] The text has 14 chapters and 500 shlokas.[38] It is one of the eighteen astronomical siddhanta (treatises), but thirteen of the eighteen are believed to be lost to history. The Surya Siddhanta text has survived since the ancient times, has been the best known and the most referred astronomical text in the Indian tradition.[38][7]

The fourteen chapters of the Surya Siddhanta are as follows, per the much cited Burgess translation:[5][40]

Chapters of Surya Siddhanta
Chapter # Title Reference
1 Of the Mean Motions of the Planets [41]
2 On the True Places of the Planets [42]
3 Of Direction, Place and Time [43]
4 Of Eclipses, and Especially of Lunar Eclipses [44]
5 Of Parallax in a Solar Eclipse [45]
6 The Projection of Eclipses [46]
7 Of Planetary Conjunctions [47]
8 Of the Asterisms [48]
9 Of Heliacal (Sun) Risings and Settings [49]
10 The Moon's Risings and Settings, Her Cusps [50]
11 On Certain Malignant Aspects of the Sun and Moon [51]
12 Cosmogony, Geography, and Dimensions of the Creation [52]
13 Of the Armillary Spehere and other Instruments (Gnomon) [53]
14 Of the Different Modes of Reckoning Time [54]

The methods for computing time using the shadow cast by a gnomon are discussed in both Chapters 3 and 13.

Planets and their characteristics

Earth is a sphere

Thus everywhere on [the surface of] the terrestrial globe,
people suppose their own place higher [than that of others],
yet this globe is in space where there is no above nor below.

—Surya Siddhanta, XII.53
Translator: Scott L. Montgomery, Alok Kumar[7][55]

The text treats earth as a stationary globe around which sun, moon and five planets orbit. It makes no mention of Uranus, Neptune and Pluto.[56] It presents mathematical formulae to calculate the orbits, diameters, predict their future locations and cautions that the minor corrections are necessary over time to the formulae for the various astronomical bodies. However, unlike modern heliocentric model for the solar system, the Surya Siddhanta relies on a geocentric point of view.[56]

The text describes some of its formulae with the use of very large numbers for divya yuga, stating that at the end of this yuga earth and all astronomical bodies return to the same starting point and the cycle of existence repeats again.[57] These very large numbers based on divya-yuga, when divided and converted into decimal numbers for each planet give reasonably accurate sidereal periods when compared to modern era western calculations.[57] For example, the Surya Siddhanta states that the sidereal period of moon is 27.322 which compares to 27.32166 in modern calculations. For Mercury it states the period to be 87.97 (modern W: 87.969), Venus 224.7 (W: 224.701), Mars as 687 (W: 686.98), Jupiter as 4,332.3 (W: 4,332.587) and Saturn to be 10,765.77 days (W: 10,759.202).[57]

The Surya Siddhanta also estimates the diameters of the planets. The estimate for the diameter of Mercury is 3,008 miles, an error of less than 1% from the currently accepted diameter of 3,032 miles. It also estimates the diameter of Saturn as 73,882 miles, which again has an error of less than 1% from the currently accepted diameter of 74,580. Its estimate for the diameter of Mars is 3,772 miles, which has an error within 11% of the currently accepted diameter of 4,218 miles. It also estimated the diameter of Venus as 4,011 miles and Jupiter as 41,624 miles, which are roughly half the currently accepted values, 7,523 miles and 88,748 miles, respectively.[58]

Calendar

The solar part of the luni-solar Hindu calendar is based on the Surya Siddhanta.[59] The various old and new versions of Surya Siddhanta manuscripts yield the same solar calendar.[60] According to J. Gordon Melton, both the Hindu and Buddhist calendars in use in South and Southeast Asia are rooted in this text, but the regional calendars adapted and modified them over time.[61][62]

The Surya Siddhanta calculates the solar year to be 365 days 6 hours 12 minutes and 36.56 seconds.[63][64] On average, according to the text, the lunar month equals 27 days 7 hours 39 minutes 12.63 seconds. It states that the lunar month varies over time, and this needs to be factored in for accurate time keeping.[65]

According to Whitney, the Surya Siddhanta calculations were tolerably accurate and achieved predictive usefulness. In Chapter 1 of Surya Siddhanta, states Whitney, "the Hindu year is too long by nearly three minutes and a half; but the moon's revolution is right within a second; those of Mercury, Venus and Mars within a few minutes; that of Jupiter within six or seven hours; that of Saturn within six days and a half".[66]

Trigonometry

The Surya Siddhanta contains the roots of modern trigonometry. Its trigonometric functions jyā and koti-jyā (reflecting the chords of Hipparchus) are the direct source (via Arabic transmission) of the terms sine and cosine.

It also contains the earliest use of the tangent and secant when discussing the shadow cast by a gnomon in verses 21–22 of Chapter 3:

Of [the sun's meridian zenith distance] find the jya ("base sine") and kojya (cosine or "perpendicular sine"). If then the jya and radius be multiplied respectively by the measure of the gnomon in digits, and divided by the kojya, the results are the shadow and hypotenuse at mid-day.

In modern notation, this gives the shadow of the gnomon at midday as

and the hypotenuse of the gnomon at midday as

where is the measure of the gnomon, is the radius of the gnomon, is the shadow of the gnomon, and is the hypotenuse of the gnomon.

Calendrical uses

The Indian solar and lunisolar calendars are widely used, with their local variations, in different parts of India. They are important in predicting the dates for the celebration of various festivals, performance of various rites as well as on all astronomical matters. The modern Indian solar and lunisolar calendars are based on close approximations to the true times of the Sun’s entrance into the various rasis.

Conservative "panchang" (almanac) makers still use the formulae and equations found in the Surya Siddhanta to compile and compute their panchangs. The panchang is an annual publication published in all regions and languages in India containing all calendrical information on religious, cultural and astronomical events. It exerts great influence on the religious and social life of the people in India and is found in most Hindu households.

Editions

See also

References

  1. 1 2 3 Menso Folkerts, Craig G. Fraser, Jeremy John Gray, John L. Berggren, Wilbur R. Knorr (2017), Mathematics, Encyclopaedia Britannica, Quote: "(...) its Hindu inventors as discoverers of things more ingenious than those of the Greeks. Earlier, in the late 4th or early 5th century, the anonymous Hindu author of an astronomical handbook, the Surya Siddhanta, had tabulated the sine function (...)"
  2. 1 2 P Gangooly (1935, Editor), Translator: Ebenezzer Burgess (1930), Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 1
  3. 1 2 3 4 5 John Bowman (2005). Columbia Chronologies of Asian History and Culture. Columbia University Press. p. 596. ISBN 978-0-231-50004-3., Quote: "c. 350-400: The Surya Siddhanta, an Indian work on astronomy, now uses sexagesimal fractions. It includes references to trigonometric functions. The work is revised during succeeding centuries, taking its final form in the tenth century."
  4. 1 2 Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 71–72 with footnotes. ISBN 0-691-12067-6.
  5. 1 2 3 4 Markanday, Sucharit; Srivastava, P. S. (1980). "Physical Oceanography in India: An Historical Sketch". Oceanography: The Past. Springer New York. pp. 551–561. ISBN 978-1-4613-8092-4. doi:10.1007/978-1-4613-8090-0_50., Quote: "There are five important Hindu astronomical books known as Siddhantas. Surya Siddhanta is the oldest (about the 6th century BC). According to Al-Biruni it was written by Lata. The book is divided into 14 chapters (Table 1). According to Surya Siddhanta the earth is a sphere."
  6. 1 2 Richard L. Thompson (2007). The Cosmology of the Bhagavata Purana. Motilal Banarsidass. pp. 16, 76–77, 285–294. ISBN 978-81-208-1919-1.
  7. 1 2 3 Scott L. Montgomery; Alok Kumar (2015). A History of Science in World Cultures: Voices of Knowledge. Routledge. pp. 104–105. ISBN 978-1-317-43906-6.
  8. ↑ Richard L. Thompson (2004). Vedic Cosmography and Astronomy. Motilal Banarsidass. p. 10. ISBN 978-81-208-1954-2.
  9. ↑ Brian Evans (2014). The Development of Mathematics Throughout the Centuries: A Brief History in a Cultural Context. Wiley. p. 60. ISBN 978-1-118-85397-9.
  10. 1 2 3 David Pingree (1963), Astronomy and Astrology in India and Iran, Isis, Volume 54, Part 2, No. 176, pages 229-235 with footnotes
  11. 1 2 Duke, Dennis (2005). "The Equant in India: The Mathematical Basis of Ancient Indian Planetary Models". Archive for History of Exact Sciences. Springer Nature. 59 (6): 563–576. doi:10.1007/s00407-005-0096-y.
  12. ↑ Pingree, David (1971). "On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle". Journal for the History of Astronomy. SAGE Publications. 2 (2): 80–85. doi:10.1177/002182867100200202.
  13. ↑ Roshen Dalal (2010). Hinduism: An Alphabetical Guide. Penguin Books. p. 89. ISBN 978-0-14-341421-6., Quote: "The solar calendar is based on the Surya Siddhanta, a text of around 400 CE."
  14. ↑ Kim Plofker (2009). Mathematics In India. Princeton University Press. ISBN 978-0-691-12067-6.
  15. 1 2 Carl B. Boyer; Uta C. Merzbach (2011). A History of Mathematics. John Wiley & Sons. p. 188. ISBN 978-0-470-63056-3.
  16. ↑ Romesh Chunder Dutt, A History of Civilization in Ancient India, Based on Sanscrit Literature, vol. 3, ISBN 0-543-92939-6 p. 208.
  17. ↑ George Abraham (2008). Helaine Selin, ed. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science. pp. 1035–1037, 1806, 1937–1938. ISBN 978-1-4020-4559-2.
  18. 1 2 James Lochtefeld (2002), "Jyotisha" in The Illustrated Encyclopedia of Hinduism, Vol. 1: A–M, Rosen Publishing, ISBN 0-8239-2287-1, pages 326–327
  19. ↑ Friedrich Max Müller (1862). On Ancient Hindu Astronomy and Chronology. Oxford University Press. pp. 37–60 with footnotes.
  20. 1 2 Yukio Ohashi 1999, pp. 719–721.
  21. ↑ Yukio Ohashi 1993, pp. 185–251.
  22. ↑ Yukio Ohashi 1999, pp. 719–720.
  23. ↑ Yukio Ohashi (2013). S.M. Ansari, ed. History of Oriental Astronomy. Springer Science. pp. 75–82. ISBN 978-94-015-9862-0.
  24. ↑ Kim Plofker 2009, pp. 41–42.
  25. ↑ Sarma, Nataraja (2000). "Diffusion of astronomy in the ancient world". Endeavour. Elsevier. 24 (4): 157–164. doi:10.1016/s0160-9327(00)01327-2.
  26. 1 2 "There are many evident indications of a direct contact of Hindu astronomy with Hellenistic tradition, e.g. the use of epicycles or the use of tables of chords which were transformed by the Hindus into tables of sines. The same mixture of elliptic arcs and declination circles is found with Hipparchus and in the early Siddhantas (note: [...] In the Surya Siddhanta, the zodiacal signs are used in similar fashion to denote arcs on any great circle. Otto Neugebauer, The Exact Sciences in Antiquity, vol. 9 of Acta historica scientiarum naturalium et medicinalium, Courier Dover Publications, 1969, p. 186.
  27. ↑ "The table must be of Greek origin, though written in the Indian number system and in Indian units. It was probably calculated around 100 B.C. by an Indian mathematicisn familiar with the work of Hipparchus." Alan Cromer, Uncommon Sense : The Heretical Nature of Science, Oxford University Press, 1993, p. 111.
  28. ↑ "The epicyclic model in the Siddnahta Surya is much simpler than Ptolemy's and supports the hypothesis that the Indians learned the original system of Hipparchus when they had contact with the West." Alan Cromer, Uncommon Sense : The Heretical Nature of Science, Oxford University Press, 1993, p. 111.
  29. ↑ Ebenezer Burgess (1989). P Ganguly, P Sengupta, ed. Sûrya-Siddhânta: A Text-book of Hindu Astronomy. Motilal Banarsidass (Reprint), Original: Yale University Press, American Oriental Society. pp. 26–27. ISBN 978-81-208-0612-2.
  30. ↑ Jayant V. Narlikar, Vedic Astrology or Jyotirvigyan: Neither Vedic nor Vigyan, EPW, Vol. 36, No. 24 (Jun. 16-22, 2001), pp. 2113-2115
  31. ↑ Pingree 1973, pp. 2–3.
  32. ↑ Erik Gregersen (2011). The Britannica Guide to the History of Mathematics. The Rosen Publishing Group. p. 187. ISBN 978-1-61530-127-0.
  33. 1 2 3 David Pingree (1963), Astronomy and Astrology in India and Iran, Isis, Volume 54, Part 2, No. 176, pages 233-238 with footnotes
  34. 1 2 3 John J. Roche (1998). The Mathematics of Measurement: A Critical History. Springer Science. p. 48. ISBN 978-0-387-91581-4.
  35. ↑ Alan Cromer (1993), Uncommon Sense : The Heretical Nature of Science, Oxford University Press, pp. 111-112.
  36. ↑ Muzaffar Iqbal (2007). Science and Islam. Greenwood Publishing. pp. 36–38. ISBN 978-0-313-33576-1.
  37. 1 2 3 Arthur Gittleman (1975). History of mathematics. Merrill. pp. 104–105. ISBN 978-0-675-08784-1.
  38. 1 2 3 4 Anil Narayan (2010), Dating the Surya Siddhanta using Computational Simulation of Proper Motions and Ecliptic Variations, Indian Journal of History of Science, Vol. 45, Number 4, pages 455-476
  39. ↑ Raymond Mercier (2004). Studies on the Transmission of Medieval Mathematical Astronomy. Ashgate. p. 53. ISBN 978-0-86078-949-9.
  40. ↑ Enrique A. González-Velasco (2011). Journey through Mathematics: Creative Episodes in Its History. Springer Science. pp. 27–28 footnote 24. ISBN 978-0-387-92154-9.
  41. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 1
  42. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 54
  43. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 108
  44. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 143
  45. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 161
  46. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 1
  47. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 187
  48. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 202
  49. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 255
  50. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 262
  51. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 273
  52. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 281
  53. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 298
  54. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 310
  55. ↑ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 289 verse 53
  56. 1 2 Richard L. Thompson (2004). Vedic Cosmography and Astronomy. Motilal Banarsidass. pp. 10–11. ISBN 978-81-208-1954-2.
  57. 1 2 3 Richard L. Thompson (2004). Vedic Cosmography and Astronomy. Motilal Banarsidass. pp. 12–14 with Table 3. ISBN 978-81-208-1954-2.
  58. ↑ Richard Thompson (1997), "Planetary Diameters in the Surya-Siddhanta" (PDF), Journal of Scientific Exploration, 11 (2): 193–200 [196], Archived from the original on January 7, 2010
  59. ↑ Roshen Dalal (2010). The Religions of India: A Concise Guide to Nine Major Faiths. Penguin Books. p. 145. ISBN 978-0-14-341517-6.
  60. ↑ Robert Sewell; Śaṅkara Bālakr̥shṇa Dīkshita. The Indian Calendar. S. Sonnenschein & Company. pp. 53–54.
  61. ↑ J. Gordon Melton (2011). Religious Celebrations: An Encyclopedia of Holidays, Festivals, Solemn Observances, and Spiritual Commemorations. ABC-CLIO. pp. 161–162. ISBN 978-1-59884-205-0.
  62. ↑ Yukio Ohashi (2008). Helaine Selin, ed. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Springer Science. pp. 354–356. ISBN 978-1-4020-4559-2.
  63. ↑ Lionel D. Barnett (1999). Antiquities of India. Atlantic. p. 193. ISBN 978-81-7156-442-2.
  64. ↑ V. Lakshmikantham; S. Leela; J. Vasundhara Devi (2005). The Origin and History of Mathematics. Cambridge Scientific Publishers. pp. 41–42. ISBN 978-1-904868-47-7.
  65. ↑ Robert Sewell; Śaṅkara Bālakr̥shṇa Dīkshita (1995). The Indian Calendar. Motilal Banarsidass. pp. 21 with footnote, cxii–cxv.
  66. ↑ William Dwight Whitney (1874). Oriental and Linguistic Studies. Scribner, Armstrong. p. 368.

Bibliography

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