Aryabhata

Statue of Aryabhata on the grounds of IUCAA, Pune. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.

Aryabhata (IAST: Āryabhaṭa; Sanskrit: आर्यभटः) (476–550 CE) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499 CE, when he was 23 years old) and the Arya-siddhanta.

Contents

Biography

Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus,[1] including Brahmagupta's references to him "in more than a hundred places by name".[2] Furthermore, in most instances "Aryabhatta" does not fit the metre either.[1]

Birth

Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476 CE.[1]

Aryabhata provides no information about his place of birth. The only information comes from Bhāskara I, who describes Aryabhata as āśmakīya, "one belonging to the aśmaka country." While aśmaka was originally situated in the northwest of India, it is widely attested that, during the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers, in the South Gujarat–North Maharashtra region of central India. Aryabhata is believed to have been born there.[1][3] However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north.[3]

Work

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he lived there for some time.[4] Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.[1] A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well.[1] Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.[5]

Other hypotheses

It was suggested that Aryabhata may have been from Kerala, but K. V. Sarma, an authority on Kerala's astronomical tradition, disagreed[1] and pointed out several errors in this hypothesis.[6]

Aryabhata mentions "Lanka" on several occasions in the Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his Ujjayini.[7]

Works

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.[3]

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Abū Rayhān al-Bīrūnī.[3]

Aryabhatiya

Direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four pādas or chapters:

  1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha(ca. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
  2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
  3. Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.
  4. Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, ca. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE).

Mathematics

Place value system and zero

The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients[8]

However, Aryabhata did not use the brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.[9]

Approximation of pi

Aryabhata worked on the approximation for pi (\pi), and may have come to the conclusion that \pi is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
 ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
 "Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."[10]

This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.

It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert).[11]

After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.[3]

Mensuration and trigonometry

In Ganitapada 6, Aryabhata gives the area of a triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."[12]

Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means "half-chord". For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jiab, meaning "cove" or "bay." (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jiab with its Latin counterpart, sinus, which means "cove" or "bay". And after that, the sinus became sine in English.[13]

Indeterminate equations

A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7

That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems is called the kuṭṭaka (कुट्टक) method. Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm.[14] The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras.

Algebra

In Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes:[15]

1^2 + 2^2 + \cdots + n^2 = {n(n + 1)(2n + 1) \over 6}

and

1^3 + 2^3 + \cdots + n^3 = (1 + 2 + \cdots + n)^2

Astronomy

Aryabhata's system of astronomy was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of his later writings on astronomy, which apparently proposed a second model (or ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in Brahmagupta's khanDakhAdyaka. In some texts, he seems to ascribe the apparent motions of the heavens to the Earth's rotation. He also treated the planet's orbits as elliptical rather than circular.[16][17]

Motions of the solar system

Aryabhata appears to have believed that the earth rotates about its axis. This is indicated in the statement, referring to Lanka , which describes the movement of the stars as a relative motion caused by the rotation of the earth:

"Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in Lanka (or on the equator) as moving exactly towards the west." [achalAni bhAni samapashchimagAni – golapAda.9]

But the next verse describes the motion of the stars and planets as real movements: "The cause of their rising and setting is due to the fact that the circle of the asterisms, together with the planets driven by the provector wind, constantly moves westwards at Lanka."

As mentioned above, Lanka (lit. Sri Lanka) is here a reference point on the equator, which was the equivalent of the reference meridian for astronomical calculations.

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles. They in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. CE 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) and a larger śīghra (fast). [18] The order of the planets in terms of distance from earth is taken as: the Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms."[3]

The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same speed as the mean Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac. Most historians of astronomy consider that this two-epicycle model reflects elements of pre-Ptolemaic Greek astronomy.[19] Another element in Aryabhata's model, the śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.[20]

Eclipses

Aryabhata states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth's shadow (verse gola.37). He discusses at length the size and extent of the Earth's shadow (verses gola.38–48) and then provides the computation and the size of the eclipsed part during an eclipse. Later Indian astronomers improved on the calculations, but Aryabhata's methods provided the core. His computational paradigm was so accurate that 18th century scientist Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of the lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[3]

Aryabhata's computation of the Earth's circumference as 39,968.0582 kilometres was only 0.2% smaller than the actual value of 40,075.0167 kilometres. This approximation was a significant improvement over the computation by Greek mathematician Eratosthenes (c. 200 BCE), whose exact computation is not known in modern units but his estimate had an error of around 5–10%.[21][22]

Sidereal periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds is an error of 3 minutes and 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate of the period.

Heliocentrism

As mentioned, Aryabhata claimed that the Earth turns on its own axis, and some elements of his planetary epicyclic models rotate at the same speed as the motion of the Earth around the Sun. The planetary orbits were also given with respect to the Sun and he also states: "Whoever knows this Dasagitika Sutra which describes the movements of the Earth and the planets in the sphere of the asterisms passes through the paths of the planets and asterisms and goes to the higher Brahman." Thus, it has been suggested that Aryabhata's calculations were based on an underlying heliocentric model, in which the planets orbit the Sun.[23][24][25] A detailed rebuttal to this heliocentric interpretation is in a review that describes B. L. van der Waerden's book as "show[ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Aryabhata's description."[26] However, some concede that Aryabhata's system stems from an earlier heliocentric model, of which he was unaware.[27] Though Aristarchus of Samos (3rd century BCE) is credited with holding an heliocentric theory, the version of Greek astronomy known in ancient India as the Paulisa Siddhanta makes no reference to such a theory.

Legacy

Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The Arabic translation during the Islamic Golden Age (ca. 820 CE), was particularly influential. Some of his results are cited by Al-Khwarizmi and in the 10th century Al-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.

His definitions of sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth of trigonometry. He was also the first to specify sine and versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, modern names "sine" and "cosine" are mistranscriptions of the words jya and kojya as introduced by Aryabhata. As mentioned, they were translated as jiba and kojiba in Arabic and then misunderstood by Gerard of Cremona while translating an Arabic geometry text to Latin. He assumed that jiba was the Arabic word jaib, which means "fold in a garment", L. sinus (c.1150).[28]

Aryabhata's astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world and used to compute many Arabic astronomical tables (zijes). In particular, the astronomical tables in the work of the Arabic Spain scientist Al-Zarqali (11th century) were translated into Latin as the Tables of Toledo (12th c.) and remained the most accurate ephemeris used in Europe for centuries.

Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing the Panchangam (the Hindu calendar). In the Islamic world, they formed the basis of the Jalali calendar introduced in 1073 CE by a group of astronomers including Omar Khayyam,[29] versions of which (modified in 1925) are the national calendars in use in Iran and Afghanistan today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlier Siddhanta calendars. This type of calendar requires an ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the Gregorian calendar.

India's first satellite Aryabhata and the lunar crater Aryabhata are named in his honour. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute of Observational Sciences (ARIES) near Nainital, India. The inter-school Aryabhata Maths Competition is also named after him,[30] as is Bacillus aryabhata, a species of bacteria discovered by ISRO scientists in 2009.[31]

See also

References

  1. ↑ 1.0 1.1 1.2 1.3 1.4 1.5 1.6 K. V. Sarma (2001). "Āryabhaṭa: His name, time and provenance". Indian Journal of History of Science 36 (4): 105–115. http://www.new.dli.ernet.in/rawdataupload/upload/insa/INSA_1/20005b67_105.pdf. 
  2. ↑ Bhau Daji (1865). "Brief Notes on the Age and Authenticity of the Works of Aryabhata, Varahamihira, Brahmagupta, Bhattotpala, and Bhaskaracharya". Journal of the Royal Asiatic Society of Greatb Britain and Ireland. p. 392. http://books.google.com/?id=fAsFAAAAMAAJ&pg=PA392&dq=aryabhata. 
  3. ↑ 3.0 3.1 3.2 3.3 3.4 3.5 3.6 Ansari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions". Bulletin of the Astronomical Society of India 5 (1): 10–18. http://hdl.handle.net/2248/502. Retrieved 2007-07-21. 
  4. ↑ Cooke (1997). "The Mathematics of the Hindus". pp. 204. "Aryabhata himself (one of at least two mathematicians bearing that name) lived in the late fifth and the early sixth centuries at Kusumapura (Pataliutra, a village near the city of Patna) and wrote a book called Aryabhatiya." 
  5. ↑ "Get ready for solar eclipe". National Council of Science Museums, Ministry of Culture, Government of India. http://ncsm.gov.in/docs/Get%20ready%20for%20Solar%20eclipse.pdf. Retrieved 9 December 2009. 
  6. ↑ For instance, one hypothesis was that aśmaka (Sanskrit for "stone") may be the region in Kerala that is now known as Koṭuṅṅallūr, based on the belief that it was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala were used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala, and the Aryasiddhanta was completely unknown in Kerala. See Sarma for details.
  7. ↑ See:
    *Clark 1930
    *S. Balachandra Rao (2000). Indian Astronomy: An Introduction. Orient Blackswan. p. 82. ISBN 9788173712050. http://books.google.com/?id=N3DE3GAyqcEC&pg=PA82&dq=lanka. : "In Indian astronomy, the prime meridian is the great circle of the Earth passing through the north and south poles, Ujjayinī and Laṅkā, where Laṅkā was assumed to be on the Earth's equator."
    *L. Satpathy (2003). Ancient Indian Astronomy. Alpha Science Int'l Ltd.. p. 200. ISBN 9788173194320. http://books.google.com/?id=nh6jgEEqqkkC&pg=PA200&dq=lanka. : "Seven cardinal points are then defined on the equator, one of them called Laṅkā, at the intersection of the equator with the meridional line through Ujjaini. This Laṅkā is, of course, a fanciful name and has nothing to do with the island of Sri Laṅkā."
    *Ernst Wilhelm. Classical Muhurta. Kala Occult Publishers. p. 44. ISBN 9780970963628. http://books.google.com/?id=3zMPFJy6YygC&pg=PA44&dq=lanka. : "The point on the equator that is below the city of Ujjain is known, according to the Siddhantas, as Lanka. (This is not the Lanka that is now known as Sri Lanka; Aryabhata is very clear in stating that Lanka is 23 degrees south of Ujjain.)"
    *R.M. Pujari; Pradeep Kolhe; N. R. Kumar (2006). Pride of India: A Glimpse into India's Scientific Heritage. SAMSKRITA BHARATI. p. 63. ISBN 9788187276272. http://books.google.com/?id=sEX11ZyjLpYC&pg=PA63&dq=lanka. 
    *Ebenezer Burgess; Phanindralal Gangooly (1989). The Surya Siddhanta: A Textbook of Hindu Astronomy. Motilal Banarsidass Publ.. p. 46. ISBN 9788120806122. http://books.google.com/?id=W0Uo_-_iizwC&pg=PA46&dq=lanka. 
  8. ↑ George. Ifrah (1998). A Universal History of Numbers: From Prehistory to the Invention of the Computer. John Wiley & Sons. 
  9. ↑ Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962). History of Hindu Mathematics. Asia Publishing House, Bombay. ISBN 81-86050-86-8 (reprint) 
  10. ↑ Jacobs, Harold R. (2003). Geometry: Seeing, Doing, Understanding (Third Edition). New York: W.H. Freeman and Company. p. 70. 
  11. ↑ S. Balachandra Rao (1994/1998). Indian Mathematics and Astronomy: Some Landmarks. Jnana Deep Publications. ISBN 81-7371-205-0. 
  12. ↑ Roger Cooke (1997.). "The Mathematics of the Hindus". History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823. "Aryabhata gave the correct rule for the area of a triangle and an incorrect rule for the volume of a pyramid. (He claimed that the volume was half the height times the area of the base.)" 
  13. ↑ Howard Eves (1990). An Introduction to the History of Mathematics (6 ed.). Saunders College Publishing House, New York. p. 237. 
  14. ↑ Amartya K Dutta, "Diophantine equations: The Kuttaka", Resonance, October 2002. Also see earlier overview: Mathematics in Ancient India.
  15. ↑ Boyer, Carl B. (1991). "The Mathematics of the Hindus". A History of Mathematics (Second ed.). John Wiley & Sons, Inc.. p. 207. ISBN 0471543977. ""He gave more elegant rules for the sum of the squares and cubes of an initial segment of the positive integers. The sixth part of the product of three quantities consisting of the number of terms, the number of terms plus one, and twice the number of terms plus one is the sum of the squares. The square of the sum of the series is the sum of the cubes."" 
  16. ↑ J. J. O'Connor and E. F. Robertson, Aryabhata the Elder, MacTutor History of Mathematics archive:
    "He believes that the Moon and planets shine by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses."
  17. ↑ Hayashi (2008), Aryabhata I
  18. ↑ Pingree, David (1996). "Astronomy in India". In Walker, Christopher. Astronomy before the Telescope. London: British Museum Press. pp. 123–142. ISBN 0-7141-1746-3  pp. 127–9.
  19. ↑ Otto Neugebauer, "The Transmission of Planetary Theories in Ancient and Medieval Astronomy," Scripta Mathematica, 22 (1956), pp. 165–192; reprinted in Otto Neugebauer, Astronomy and History: Selected Essays, New York: Springer-Verlag, 1983, pp. 129–156. ISBN 0-387-90844-7
  20. ↑ Hugh Thurston, Early Astronomy, New York: Springer-Verlag, 1996, pp. 178–189. ISBN 0-387-94822-8
  21. ↑ "JSC NES School Measures Up", NASA, 11th April, 2006, retrieved 24th November, 2009.
  22. ↑ "The Round Earth", NASA, 12th December, 2004, retrieved 24th January, 2008.
  23. ↑ The concept of Indian heliocentrism has been advocated by B. L. van der Waerden, Das heliozentrische System in der griechischen, persischen und indischen Astronomie. Naturforschenden Gesellschaft in Zürich. Zürich:Kommissionsverlag Leeman AG, 1970.
  24. ↑ B.L. van der Waerden, "The Heliocentric System in Greek, Persian and Hindu Astronomy", in David A. King and George Saliba, ed., From Deferent to Equant: A Volume of Studies in the History of Science in the Ancient and Medieval Near East in Honor of E. S. Kennedy, Annals of the New York Academy of Science, 500 (1987), pp. 529–534.
  25. ↑ Hugh Thurston (1996). Early Astronomy. Springer. p. 188. ISBN 0387948228 
  26. ↑ Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy," Isis, 64 (1973): 239–243.
  27. ↑ Dennis Duke, "The Equant in India: The Mathematical Basis of Ancient Indian Planetary Models." Archive for History of Exact Sciences 59 (2005): 563–576, n. 4 [1].
  28. ↑ Douglas Harper (2001). "Online Etymology Dictionary". http://www.etymonline.com/. Retrieved 2007-07-14. 
  29. ↑ "Omar Khayyam". The Columbia Encyclopedia (6 ed.). 2001-05. http://www.bartleby.com/65/om/OmarKhay.html. Retrieved 2007-06-10. 
  30. ↑ "Maths can be fun". The Hindu. 3 February 2006. http://www.hindu.com/yw/2006/02/03/stories/2006020304520600.htm. Retrieved 2007-07-06. 
  31. ↑ Discovery of New Microorganisms in the Stratosphere. Mar. 16, 2009. ISRO.

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