Surya Siddhanta

Verse 1.1 (prayer to Brahman)

The Surya Siddhanta (IAST: Sūrya Siddhānta; lit. 'Sun Treatise') is a Sanskrit treatise in Indian astronomy dated to 4th to 5th century,[1][2] in fourteen chapters.[3][4][5] 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 is known from a 15th-century CE palm-leaf manuscript, and several newer manuscripts.[8] It was composed or revised probably c. 800 CE from an earlier text also called the Surya Siddhanta.[5] The Surya Siddhanta text is composed of verses made up of two lines, each broken into two halves, or pãds, of eight syllables each.[3]

As per al-Biruni, the 11th-century Persian scholar and polymath, a text named the Surya Siddhanta was written by Lāṭadeva, a student of Aryabhatta I.[8][9] The second verse of the first chapter of the Surya Siddhanta attributes the words to an emissary of the solar deity of Hindu mythology, Surya, as recounted to an asura called Maya at the end of Satya Yuga, the first golden age from Hindu texts, around two million years ago.[8][10]

The text asserts, according to Markanday and Srivatsava, that the Earth is of a spherical shape.[4] It treats Earth as stationary globe around which Sun orbits, and makes no mention of Uranus, Neptune and Pluto.[11] It calculates the Earth's diameter to be 8,000 miles (modern: 7,928 miles),[6] the diameter of the Moon as 2,400 miles (actual ~2,160)[6] and the distance between the Moon and the Earth to be 258,000 miles[6] (now known to vary: 221,500–252,700 miles (356,500–406,700 kilometres).[12] The text is known for some of the earliest known discussions of fractions and trigonometric functions.[1][2][13]

The Surya Siddhanta is one of several astronomy-related Hindu texts. It represents a functional system that made reasonably accurate predictions.[14][15][16] The text was influential on the solar year computations of the luni-solar Hindu calendar.[17] The text was translated into Arabic and was influential in medieval Islamic geography.[18] The Surya Siddhanta has the largest number of commentators among all the astronomical texts written in India. It includes information about the mean orbital parameters of the planets, such as the number of mean revolutions per Mahayuga, the longitudinal changes of the orbits, and also includes supporting evidence and calculation methods.[3]

Textual history

Further information: Jyotisha

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.: 50  Most scholars place the surviving version of the text variously from the 4th-century to 5th-century CE,[19][20] although it is dated to about the 6th-century BCE by Markandaya and Srivastava.[21]

According to John Bowman, the version of the text existed between 350 and 400 CE wherein it referenced fractions and trigonometric functions, but the text was a living document and revised through about the 10th-century.[19] One of the evidence for the Surya Siddhanta being a living text is the work of medieval 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.[22] According to Kim 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 probably composed around 800 CE.[5] Some scholars refer to Panca siddhantika as the old Surya Siddhanta and date it to 505 CE.[23]

Based on a study of the longitude variation data from the text, Indian scientist Anil Narayanan (2010) concludes that the text has been updated several times in the past, with the last update around 580 CE. Narayan obtained a match for the nakshatra latitudinal data in the period 7300-7800 BCE based on a computer simulation.[24]

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 dates and times for Vedic rituals.[25] Vedic sacrifices state that the ancient Vedic texts describe four measures of time – savana, solar, lunar and sidereal, as well as twenty seven constellations using Taras (stars).[26] According to mathematician and classicist David Pingree, in the Hindu text Atharvaveda (~1000 BCE or older) the idea already appears of twenty eight constellations and movement of astronomical bodies.[14]

According to Pingree, the influence may have flowed the other way initially, then flowed into India after the arrival of Darius and the Achaemenid conquest of the 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 Ôhashi considers this proposal as incorrect,[27] 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.[28] Ôhashi states that it is incorrect to assume that the number of civil days in a year equal 365 in both Indian (Hindu) and Egyptian–Persian year.[29] Further, adds Ôhashi, 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.[30]

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.[31][32]

Greek influence

It is hypothesized that contacts between the ancient Indian scholarly tradition and Hellenistic Greece via the Indo-Greek Kingdom after the Indian campaign of Alexander the Great, specifically regarding the work of Hipparchus (2nd-century BCE), explain some similarities between Surya Siddhanta and Greek astronomy in the Hellenistic period. For example, Surya Siddhanta provides table of sines function which parallel the Hipparchian table of chords, though the Indian calculations are more accurate and detailed.[33] According to Alan Cromer, the Greek influence most likely arrived in India by about 100 BCE.[34] 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.[35]

The influence of Greek ideas on early medieval era Indian astronomical theories, particularly zodiac symbols (astrology), is broadly accepted by the Western scholars.[33] 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".[36] 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.[36] 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.[36]

According to mathematician and historian of measurement John Roche, the astronomical and mathematical methods developed by Greeks related arcs to chords of spherical trigonometry.[37] The Indian mathematical astronomers, in their texts such as the 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".[37] For instance "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".[37] The Surya Siddhanta was one of the two books in Sanskrit that were translated into Arabic in the later half of the eighth century during the reign of Abbasid caliph Al-Mansur.[citation needed]

Importance in history of science

Astronomical calculations: Estimated time per sidereal revolution[3]: 26–27 </ref>
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 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.[38][39] 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.[15]

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.[40]

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.[41] 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.[41] 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.[41][42]

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

The fourteen chapters of the Surya Siddhanta are as follows, per the much cited Burgess translation:[4][43]

  1. Of the Mean Motions of the Planets[3]
  2. On the True Places of the Planets[3]: 53 
  3. Of Direction, Place and Time[3]: 108 
  4. Of Eclipses, and Especially of Lunar Eclipses[3]: 143 
  5. Of Parallax in a Solar Eclipse[3]: 161 
  6. The Projection of Eclipses[3]: 178 
  7. Of Planetary Conjunctions[3]: 187 
  8. Of the Asterisms[3]: 202 
  9. Of Heliacal (Sun) Risings and Settings[3]: 255 
  10. The Moon's Risings and Settings, Her Cusps[3]: 262 
  11. On Certain Malignant Aspects of the Sun and Moon[3]: 273 
  12. Cosmogony, Geography, and Dimensions of the Creation[3]: 281 
  13. Of the Armillary Sphere and other Instruments[3]: 298 
  14. Of the Different Modes of Reckoning Time[3]: 310 

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

Description of Time

The author of Surya Siddhanta defines time as of two types: the first which is continuous and endless, destroys all animate and inanimate objects and second is time which can be known. This latter type is further defined as having two types: the first is Murta (Measureable) and Amurta (immeasureable because it is too small or too big). The time Amurta is a time that begins with an infinitesimal portion of time (Truti) and Murta is a time that begins with 4-second time pulses called Prana as described in the table below. The further description of Amurta time is found in Puranas where as Surya Siddhanta sticks with measurable time.[44]

Time described in Surya Siddhanta[44]
Type Surya Siddhanta Units Description Value in modern units of time
Amurta Truti 1/33750 seconds 29.6296 micro seconds
Murta Prana - 4 seconds
Murta Pala 6 Pranas 24 seconds
Murta Ghatika 60 Palas 24 minutes
Murta Nakshatra Ahotra 60 Ghatikas One Sidereal day

The text measures a savana day from sunrise to sunrise. Thirty of these savana days make a savana month. A solar (saura) month starts with the entrance of the sun into a zodiac sign, thus twelve months make a year.[44]

The text further states there are nine modes of measuring time. "Of four modes, namely solar, lunar, sidereal, and civil time, practical use is made among men; by that of Jupiter is to be determined the year of the cycle of sixty years; of the rest, no use is ever made".[45]

North pole star and South pole star

Surya Siddhanta asserts that there are two pole stars, one each at north and south celestial pole. Surya Siddhanta chapter 12 verse 43 description is as following:

मेरोरुभयतो मध्ये ध्रुवतारे नभ:स्थिते। निरक्षदेशसंस्थानामुभये क्षितिजाश्रिये॥१२:४३॥

This translates as "On both sides of the Meru (i.e. the north and south poles of the earth) the two polar stars are situated in the heaven at their zenith. These two stars are in the horizon of the cities situated on the equinoctial regions".[46]

The Sine table

Further information: Āryabhaṭa's sine table

The Surya Siddhanta provides methods of calculating the sine values in chapter 2. It divides the quadrant of a circle with radius 3438 into 24 equal segments or sines as described in the table. In modern-day terms, each of these 24 segments has angle of 3.75°.[47]

Table of Sines [3]: 115 </ref>
No. Sine 1st order

differences

2nd order

differences

No. Sine 1st order

differences

2nd order

differences

0 0 - - 13 2585 154 10
1 225 225 1 14 2728 143 11
2 449 224 2 15 2859 131 12
3 671 222 3 16 2978 119 12
4 890 219 4 17 3084 106 13
5 1105 215 5 18 3177 93 13
6 1315 210 5 19 3256 79 14
7 1520 205 6 20 3321 65 14
8 1719 199 8 21 3372 51 14
9 1910 191 8 22 3409 37 14
10 2093 183 9 23 3431 22 15
11 2267 174 10 24 3438 7 15
12 2431 164 10

The 1st order difference is the value by which each successive sine increases from the previous and similarly the 2nd order difference is the increment in the 1st order difference values. Burgess says, 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.[3]

Calculation of tilt of Earth's axis (Obliquity)

The tilt of the ecliptic varies between 22.1° to 24.5° and is currently 23.5°.[48] Following the sine tables and methods of calculating the sines, Surya Siddhanta also attempts to calculate the Earth's tilt of contemporary times as described in chapter 2 and verse 28, the obliquity of the Earth's axis, the verse says "The sine of greatest declination is 1397; by this multiply any sine, and divide by radius; the arc corresponding to the result is said to be the declination".[3]: 65  The greatest declination is the inclination of the plane of the ecliptic. With radius of 3438 and sine of 1397, the corresponding angle is 23.975° or 23° 58' 30.65" which is approximated to be 24°.[3]: 118 

Planets and their characteristics

Question: How Can the Earth Be a Sphere?

Thus everywhere on the terrestrial globe (bhūgola),
people suppose their own place higher,
yet this globe (gola) is in space where there is no above nor below.

—Surya Siddhanta, XII.53
Translator: Scott L. Montgomery, Alok Kumar[7][3]: 289, verse 53 

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.[49] 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.[3]

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.[50] 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.[50]

Sidereal Periods[50]
Surya Siddhanta Modern values
Moon 27.322 days 27.32166 days
Mercury 87.97 days 87.969 days
Mars 687 days 686.98 days
Venus 224.7 days 224.701 days
Jupiter 4,332.3 days 4,332.587 days
Saturn 10,765.77 days 10,759.202 days

Calendar

See also: Astronomical basis of the Hindu calendar

The solar part of the luni-solar Hindu calendar is based on the Surya Siddhanta.[51] The various old and new versions of Surya Siddhanta manuscripts yield the same solar calendar.[52] According to J. Gordon Melton, both the Hindu and Buddhist calendars that are in use in South and Southeast Asia are rooted in this text, but the regional calendars adapted and modified them over time.[53][54]

The Surya Siddhanta calculates the solar year to be 365 days 6 hours 12 minutes and 36.56 seconds.[55][56] 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.[57]

According to Whitney, the Surya Siddhanta calculations were tolerably accurate and achieved predictive usefulness. In Chapter 1 of Surya Siddhanta, "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".[58]

The Surya Siddhanta was one of the two books in Sanskrit translated into Arabic during the reign of 'Abbasid caliph al-Mansur (r. 754–775 CE). According to Muzaffar Iqbal, this translation and that of Aryabhata was of considerable influence on geographic, astronomy and related Islamic scholarship.[40]

Editions

  • The Súrya-Siddhánta, an antient system of Hindu astronomy ed. FitzEdward Hall and Bápú Deva Śástrin (1859).
  • 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 translated 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.

Commentaries

The historical popularity of Surya Siddhanta is attested by the existence of at least 26 commentaries, plus another 8 anonymous commentaries.[59] Some of the Sanskrit-language commentaries include the following; nearly all the commentators have re-arranged and modified the text:[60]

  • Surya-siddhanta-tika (1178) by Mallikarjuna Suri
  • Surya-siddhanta-bhashya (1185) by Chandeshvara, a Maithila Brahmana
  • Vasanarnava (c. 1375–1400) by Maharajadhiraja Madana-pala of Taka family
  • Surya-siddhanta-vivarana (1432) by Parameshvara of Kerala
  • Kalpa-valli (1472) by Yallaya of Andhra-desha
  • Subodhini (1472) by Ramakrishna Aradhya
  • Surya-siddhanta-vivarana (1572) by Bhudhara of Kampilya
  • Kamadogdhri (1599) by Tamma Yajvan of Paragipuri
  • Gudhartha-prakashaka (1603) by Ranganatha of Kashi
  • Saura-bhashya (1611) by Nrsimha of Kashi
  • Gahanartha-prakasha (IAST: Gūḍhārthaprakāśaka, 1628) by Vishvanatha of Kashi
  • Saura-vasana (after 1658) by Kamalakara of Kashi
  • Kiranavali (1719) by Dadabhai, a Chittpavana Brahmana
  • Surya-siddhanta-tika (date unknown) by Kama-bhatta of southern India
  • Ganakopakarini (date unknown) by Chola Vipashchit of southern India
  • Gurukataksha (date unknown) by Bhuti-vishnu of southern India

Mallikarjuna Suri had written a Telugu language commentary on the text before composing the Sanskrit-language Surya-siddhanta-tika in 1178.[60] Kalpakurti Allanarya-suri wrote another Telugu language commentary on the text, known from a manuscript copied in 1869.[61]

See also

References

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  30. ^ Yukio Ôhashi (2013). S.M. Ansari (ed.). History of Oriental Astronomy. Springer Science. pp. 75–82. ISBN 978-94-015-9862-0.
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  33. ^ a b "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.
  34. ^ "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.
  35. ^ "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.
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  38. ^ "Surya Siddhanta the basis of space studies, says Governor". The Hindu. 2020-01-24. ISSN 0971-751X. Retrieved 2021-09-02.
  39. ^ Alan Cromer (1993), Uncommon Sense : The Heretical Nature of Science, Oxford University Press, pp. 111-112.
  40. ^ a b Muzaffar Iqbal (2007). Science and Islam. Greenwood Publishing. pp. 36–38. ISBN 978-0-313-33576-1.
  41. ^ a b c Arthur Gittleman (1975). History of mathematics. Merrill. pp. 104–105. ISBN 978-0-675-08784-1.
  42. ^ Raymond Mercier (2004). Studies on the Transmission of Medieval Mathematical Astronomy. Ashgate. p. 53. ISBN 978-0-86078-949-9.
  43. ^ 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.
  44. ^ a b c Deva Shastri, Pandit Bapu. Translation of the Surya Siddhanta. pp. 2–3.
  45. ^ Burgess, Ebenezer (1860). TRANSLATION OF THE SURYA-SIDDHANTA (1935 ed.). UNIVERSITY OF CALCUTTA. p. 310.
  46. ^ Deva Sastri, Pundit Bapu (1861). The Translation of Surya Siddhanta (PDF). Calcutta: Baptist Mission Press. pp. 80–81.
  47. ^ Deva Shastri, Pundit Bapu (1861). Translation of the Surya Siddhanta. pp. 15–16.
  48. ^ "Milutin Milankovitch". earthobservatory.nasa.gov. 2000-03-24. Retrieved 2020-08-15.
  49. ^ Richard L. Thompson (2004). Vedic Cosmography and Astronomy. Motilal Banarsidass. pp. 10–11. ISBN 978-81-208-1954-2.
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  51. ^ Roshen Dalal (2010). The Religions of India: A Concise Guide to Nine Major Faiths. Penguin Books. p. 145. ISBN 978-0-14-341517-6.
  52. ^ Robert Sewell; Śaṅkara Bālakr̥shṇa Dīkshita (1896). The Indian Calendar. S. Sonnenschein & Company. pp. 53–54.
  53. ^ 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.
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  56. ^ 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.
  57. ^ Robert Sewell; Śaṅkara Bālakr̥shṇa Dīkshita (1995). The Indian Calendar. Motilal Banarsidass. pp. 21 with footnote, cxii–cxv. ISBN 9788120812079.
  58. ^ William Dwight Whitney (1874). Oriental and Linguistic Studies. Scribner, Armstrong. p. 368.
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Bibliography

Further reading

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