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Surya Siddhanta

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Surya Siddhanta is a Hindu text on astronomy. Above is verse 1.1, which pays homage to Brahma.[1]

The Surya Siddhanta is the name of a Sanskrit treatise in Indian astronomy. The text has been updated several times in the past and the earliest update was found to be made in 8th millennium BCE. Using computer simulation, a match for the Surya Siddhanta latitudinal data was obtained in the time frame of 7300-7800 BCE.[2] The last update took place in the vicinity of 580 CE when Nakshatra data appears to have been updated by adding a fixed precessional increment to all longitudes.[3] Narayanan (2011) showed that for determining the Sun’s longitude, the pulsating Indian epicycle is far more accurate than the Greek eccentric-epicycle model and that 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 BCE.[4] This led him to the timing of 5000-5500BCE when the current values of the Sūrya-siddhānta’s pulsating epicycle parameters for the Sun appear to have been set. [4] As per the second verse of the chapter 1 of Surya Siddhanta, Maya Asura is the original author of the text. It has fourteen chapters.[5] [1]

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, according to Markanday and Srivatsava, that the earth is of 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. [9][10][11]

The Surya Siddhanta is one of the several astronomy-related Hindu texts. It represents a functional system that made reasonably accurate predictions.[12][13][14] The text was influential on the solar year computations of the luni-solar Hindu calendar.[15]

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.[16]: 50  The surviving version of the text is dated to about the 6th-century BCE by Markandaya and Srivastava.[5] Most scholars, however, had placed the text variously from the 4th-century to 5th-century CE.[10][17] But this was the period when latest update to Surya Siddhanta was made with one of the earliest update being made in 8th millennium BCE.[2]

According to John Bowman, another version of the text existed wherein it referenced sexagesimal fractions and trigonometric functions, but the text was a living document and revised through about the 10th-century.[10] One of the evidence for the Surya Siddhanta being a living text is the work of Indian scholar Utpala, 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.[18] According to Kim Plofker, large portions of the more ancient Sūrya-siddhānta was incorporated into the Panca siddhantika text.[19][10] Some scholars refer to Panca siddhantika as the old Surya Siddhanta.[20]

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.[21] Max Muller, quoting passages by Garga and others, 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).[22] According to Pingree, the idea of twenty eight constellations and movement of astronomical bodies already appears in the Hindu text Atharvaveda.[12]

Similarities with Greek astronomy

It is hypothesized that there were cultural contacts between the Indian and Greek astronomers via cultural contact with Hellenistic Greece, specifically regarding the work of Hipparchus (2nd-century BCE). There were some similarities between Surya Siddhanta and Greek astronomy in Hellenistic period. For example, Surya Siddhanta provides table of sines function which parallel the Hipparchus table of chords, though the Indian calculations are more accurate and detailed.[23] According to Alan Cromer, the knowledge share with Greeks may have occurred by about 100 BCE.[24]

Astronomical calculations: Estimated time per sidereal revolution[25]
Planet Surya Siddhanta (7300-7500 BCE)[2] Ptolemy (2nd century CE) 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

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.[26] The astrology field likely developed in the centuries after the arrival of Greek astrology with Alexander the Great,[27][28][29] their zodiac signs being nearly identical.[21]

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.[30] 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".[30] 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".[30]

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.[31]

Contents

The mean (circular) motion of planets according to the Surya Siddhantha.
The variation of the true position of Mercury around its mean position according to the Surya Siddhantha.

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.[32][3] 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.[32] 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.[32][33]

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.[17][3] The text has 14 chapters and 500 shlokas.[3] 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.[3][7]

The fourteen chapters of the Surya Siddhanta are as follows, per the much cited Burgess translation:[5][34]

Chapters of Surya Siddhanta
Chapter # Title Reference
1 Of the Mean Motions of the Planets [35]
2 On the True Places of the Planets [36]
3 Of Direction, Place and Time [37]
4 Of Eclipses, and Especially of Lunar Eclipses [38]
5 Of Parallax in a Solar Eclipse [39]
6 The Projection of Eclipses [40]
7 Of Planetary Conjunctions [41]
8 Of the Asterisms [42]
9 Of Heliacal (Sun) Risings and Settings [43]
10 The Moon's Risings and Settings, Her Cusps [44]
11 On Certain Malignant Aspects of the Sun and Moon [45]
12 Cosmogony, Geography, and Dimensions of the Creation [46]
13 Of the Armillary Sphere and other Instruments [47]
14 Of the Different Modes of Reckoning Time [48]

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

North pole star and South pole star

One of the most interesting observation made in Surya Siddhanta is the observation of two pole stars, one each at north and south celestial pole. Surya Siddhanta chapter 12 verse 42 description is as following:

मेरोरुभयतो मध्ये ध्रुवतारे नभ:स्थिते।

निरक्षदेशसंस्थानामुभये क्षितिजाश्रिये॥१२:४३॥

This translates as "There are two pole stars, one each, near North celestial pole and South celestial pole. From equatorial regions, these stars are seen along the horizon".[49] Currently our North Pole star is Polaris. It is subject to investigation to find out when this astronomical phenomenon occurred in the past to date the addition of this particular update to Surya Siddhanta.

Calculation of Earth's Obliquity (3000 BCE)

In Surya Siddhanta chapter 2 and verse 28, it calculated the obliquity of the Earth's axis. The verse says "The sine of greatest declination(obliquity) is 1397.....", which means that R-sine is 1397 where R is 3438[50]. To obtain the obliquity in the unit of degree, we have to take the inverse of Sine of the ratio (1397/3438), which gives us 23.975182 degrees and this tilt indicates a period of 3000 BCE[51]. It can be noted that this update was made during 3000 BCE to the Surya Siddhanta.

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][52]

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.[53] 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.[53]

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.[54] 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.[54] 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).[54]

Calendar

The solar part of the luni-solar Hindu calendar is based on the Surya Siddhanta.[55] The various old and new versions of Surya Siddhanta manuscripts yield the same solar calendar.[56] 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.[57][58]

The Surya Siddhanta calculates the solar year to be 365 days 6 hours 12 minutes and 36.56 seconds.[59][60] 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.[61]

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".[62]

Editions

  • Translation of the Sûrya-Siddhânta: A text-book of Hindu astronomy, with notes and an appendix by Ebenezer Burgess Originally published: Journal of the American Oriental Society 6 (1860) 141–498. Commentary by Burgess is much larger than his translation.
  • Surya-Siddhanta: A Text Book of Hindu Astronomy by Ebenezer Burgess, ed. Phanindralal Gangooly (1989/1997) with a 45-page commentary by P. C. Sengupta (1935).
  • Translation of the Surya Siddhanta by Bapu Deva Sastri (1861) ISBN 3-7648-1334-2, ISBN 978-3-7648-1334-5. Only a few notes. Translation of Surya Siddhanta occupies first 100 pages; rest is a translation of the Siddhanta Siromani by Lancelot Wilkinson.

See also

References

  1. ^ a b P Gangooly (1935, Editor), Translator: Ebenezzer Burgess (1930), Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 1
  2. ^ a b c Naranan, Anil (23 March 2010). "Dating the SU— Rya Siddha— Nta Using Computational Simulation of Proper Motions and Ecliptic Variations" (PDF). Indian Journal of history of Science. 45 (4): 455–476.
  3. ^ a b c d e 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
  4. ^ a b Narayanan, Anil (17 February 2011). "The Pulsating Indian Epicycle of the Sun" (PDF). Indian Journal of History of Science. 46 (3): 411–425.
  5. ^ a b c d Markanday, Sucharit; Srivastava, P. S. (1980). "Physical Oceanography in India: An Historical Sketch". Oceanography: The Past. Springer New York. pp. 551–561. doi:10.1007/978-1-4613-8090-0_50. ISBN 978-1-4613-8092-4., Quote: "According to Surya Siddhanta the earth is a sphere."
  6. ^ a b 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. ^ a b c 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. ^ 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 (...)"
  10. ^ a b c d John Bowman (2000). 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."
  11. ^ 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.
  12. ^ a b David Pingree (1963), Astronomy and Astrology in India and Iran, Isis, Volume 54, Part 2, No. 176, pages 229-235 with footnotes
  13. ^ Duke, Dennis (2005). "The Equant in India: The Mathematical Basis of Ancient Indian Planetary Models". Archive for History of Exact Sciences. 59 (6). Springer Nature: 563–576. doi:10.1007/s00407-005-0096-y.
  14. ^ Pingree, David (1971). "On the Greek Origin of the Indian Planetary Model Employing a Double Epicycle". Journal for the History of Astronomy. 2 (2). SAGE Publications: 80–85. Bibcode:1971JHA.....2...80P. doi:10.1177/002182867100200202.
  15. ^ 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."
  16. ^ Kim Plofker (2009). Mathematics In India. Princeton University Press. ISBN 978-0-691-12067-6.
  17. ^ a b Carl B. Boyer; Uta C. Merzbach (2011). A History of Mathematics. John Wiley & Sons. p. 188. ISBN 978-0-470-63056-3.
  18. ^ Romesh Chunder Dutt, A History of Civilization in Ancient India, Based on Sanscrit Literature, vol. 3, ISBN 0-543-92939-6 p. 208.
  19. ^ Kim Plofker (2009). Mathematics in India. Princeton University Press. pp. 71–72 with footnotes. ISBN 0-691-12067-6.
  20. ^ 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.
  21. ^ a b James Lochtefeld (2002), "Jyotisha" in The Illustrated Encyclopedia of Hinduism, Vol. 1: A–M, Rosen Publishing, ISBN 0-8239-2287-1, pages 326–327
  22. ^ Friedrich Max Müller (1862). On Ancient Hindu Astronomy and Chronology. Oxford University Press. pp. 37–60 with footnotes.
  23. ^ "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.
  24. ^ "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.
  25. ^ 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.
  26. ^ Jayant V. Narlikar, Vedic Astrology or Jyotirvigyan: Neither Vedic nor Vigyan, EPW, Vol. 36, No. 24 (Jun. 16-22, 2001), pp. 2113-2115
  27. ^ Yukio Ohashi 1999, pp. 719–721.
  28. ^ Pingree 1973, pp. 2–3.
  29. ^ Erik Gregersen (2011). The Britannica Guide to the History of Mathematics. The Rosen Publishing Group. p. 187. ISBN 978-1-61530-127-0.
  30. ^ a b c John J. Roche (1998). The Mathematics of Measurement: A Critical History. Springer Science. p. 48. ISBN 978-0-387-91581-4.
  31. ^ Muzaffar Iqbal (2007). Science and Islam. Greenwood Publishing. pp. 36–38. ISBN 978-0-313-33576-1.
  32. ^ a b c Arthur Gittleman (1975). History of mathematics. Merrill. pp. 104–105. ISBN 978-0-675-08784-1.
  33. ^ Raymond Mercier (2004). Studies on the Transmission of Medieval Mathematical Astronomy. Ashgate. p. 53. ISBN 978-0-86078-949-9.
  34. ^ 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.
  35. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 1
  36. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 54
  37. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 108
  38. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 143
  39. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 161
  40. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 1
  41. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 187
  42. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 202
  43. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 255
  44. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 262
  45. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 273
  46. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 281
  47. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 298
  48. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 310
  49. ^ Deva Sastri, Pundit Bapu (1861). The Translation of Surya Siddhanta (PDF). Calcutta: Baptist Mission Press. pp. 80–81.
  50. ^ 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. p. 65. ISBN 978-81-208-0612-2.
  51. ^ Narayanan, Anil (2019-09-09). "Wonders, Mysteries and Misconceptions in Indian Astronomy - I". IndiaFacts. Retrieved 2020-06-17.
  52. ^ P Gangooly (1935, Editor), Translator: Ebenezzer Burgess, Translation of Surya Siddhanta: A Textbook of Hindu Astronomy, University of Calcutta, page 289 verse 53
  53. ^ a b Richard L. Thompson (2004). Vedic Cosmography and Astronomy. Motilal Banarsidass. pp. 10–11. ISBN 978-81-208-1954-2.
  54. ^ a b c Richard L. Thompson (2004). Vedic Cosmography and Astronomy. Motilal Banarsidass. pp. 12-14 with Table 3. ISBN 978-81-208-1954-2.
  55. ^ Roshen Dalal (2010). The Religions of India: A Concise Guide to Nine Major Faiths. Penguin Books. p. 145. ISBN 978-0-14-341517-6.
  56. ^ Robert Sewell; Śaṅkara Bālakr̥shṇa Dīkshita. The Indian Calendar. S. Sonnenschein & Company. pp. 53–54.
  57. ^ 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.
  58. ^ 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.
  59. ^ Lionel D. Barnett (1999). Antiquities of India. Atlantic. p. 193. ISBN 978-81-7156-442-2.
  60. ^ 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.
  61. ^ Robert Sewell; Śaṅkara Bālakr̥shṇa Dīkshita (1995). The Indian Calendar. Motilal Banarsidass. pp. 21 with footnote, cxii–cxv.
  62. ^ William Dwight Whitney (1874). Oriental and Linguistic Studies. Scribner, Armstrong. p. 368.

Bibliography

Further reading

  • Victor J. Katz. A History of Mathematics: An Introduction, 1998.
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