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Aryabhata (IAST: Āryabhaṭa; Bahasa Sanskrit: आर्यभट) (476–550 AM) adalah yang pertama dalam turutan ahli matematik-ahli astronomi hebat dari zaman klasik matematik India dan astronomi India. Karya termasyhurnya adalah Aryabhatiya (499 AM, ketika dia berusia 23 tahun) dan Arya-siddhanta.
- 1 Biografi
- 2 Karya
- 3 Matematik
- 4 Astronomi
- 5 Legasi
- 6 Lihat pula
- 7 Rujukan
- 8 External links
Biografi[sunting | sunting sumber]
Nama[sunting | sunting sumber]
Sungguhpun terdapat kecenderungan bagi salah mengeja sebagai "Aryabhatta" menurut analogi dengan nama-nama lain yang mempunyai akhiran "bhatta", namanya secara sesuai dieja Aryabhata: tiap teks astronomi mengeja namanya oleh itu, termasuk rujukan Brahmagupta padanya "dalam lebih daripada seratus tempat mengikut nama". Lebih lanjutnya, dalam kebanyakan contoh "Aryabhatta" tidak muatkan meter juga.
Kelahiran[sunting | sunting sumber]
Aryabhata tidak memberikan maklumat tentang tempat kelahirannya. Satu-satunya maklumat datang dari Bhāskara I, yang menggambarkan wAryabhata sebagai āśmakīya, "seseorang yang datangnya dari negeri aśmaka." Sungguhpun aśmaka pada adalnya terletak di barat laut India, ia dibukti ramai bahawa, ketika zaman Buddha, satu cabang suku Aśmaka menetap di kawasan antara sungai Narmada dan sungai Godavari, di selatan Gujarat–Utara kawasan Maharashtra tengah India. Aryabhata dipercayai dilahirka di sana. Bagaimanapun, teks awal Buddha menggambarkan Ashmaka sebelai lebih jauh keselatan, di dakshinapath atau Deccan, sementara teks lain menggambarkan Ashmakas sebagai menentang Alexander, yang akan meletakkan mereka lebih jauh ke utara.
Karya[sunting | sunting sumber]
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. 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.
Hipotesis Kerala[sunting | sunting sumber]
It has also been suggested that aśmaka (Sanskrit for "stone") might 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"). It is also claimed that the fact that several commentaries on the Aryabhatiya have come from Kerala suggest that it was Aryabhata's main place of life and activity. But K. V. Sarma, the authority on Kerala's astronomical tradition, disagrees and cites many commentaries that have come from outside Kerala and the Aryasiddhanta's being completely unknown in Kerala. In recent (2007) papers, K. Chandra Hari uses a discrepancy in Aryabhata's astronomical values to deduce that he carried out his calculations from a place in Kerala at the same meridian as Ujjayini, possibly Chamravattam (10°N51, 75°E45) in central Kerala. He further hypothesizes that Asmaka was the Jain country surrounding Shravanabelagola, taking its name from the stone monoliths there.
Karya[sunting | sunting sumber]
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.
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.
Aryabhatiya[sunting | sunting sumber]
Direct details of Aryabhata's work are therefore 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:
- 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.
- Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
- 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.
- 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).
Matematik[sunting | sunting sumber]
Menempatkan sistem nilai dan kosong[sunting | sunting sumber]
The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. ; he certainly did not use the symbol, but 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
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.
Pi sebagai tidak rasional[sunting | sunting sumber]
Aryabhata worked on the approximation for Pi (), and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
"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."
This implies that the ratio of the circumference to the diameter is ((4+100)×8+62000)/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).
Mensurasi dan trigonometri[sunting | sunting sumber]
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."
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.
Persamaan tidak tetap[sunting | sunting sumber]
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 Bhaskara'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. 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[sunting | sunting sumber]
Astronomi[sunting | sunting sumber]
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.
Mosi sistem suria[sunting | sunting sumber]
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).  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."
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. 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.
Gerhana[sunting | sunting sumber]
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 1765-08-30 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.
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%.
Tempoh sidereal[sunting | sunting sumber]
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.
Heliosentrisme[sunting | sunting sumber]
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. Thus, it has been suggested that Aryabhata's calculations were based on an underlying heliocentric model, in which the planets orbit the Sun. 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." However, some concede that Aryabhata's system stems from an earlier heliocentric model, of which he was unaware. It has even been claimed that he considered the planet's paths to be elliptical, but no primary evidence for this has been found. Though Aristarchus of Samos (3rd century BCE) and sometimes Heraclides of Pontus (4th century BCE) are usually credited with knowing the heliocentric theory, the version of Greek astronomy known in ancient India as the Paulisa Siddhanta (possibly by a Paul of Alexandria) makes no reference to a heliocentric theory.
Legasi[sunting | sunting sumber]
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 he is mentioned by the 10th century Arabic scholar Al-Biruni, who states that Aryabhata's followers believed that the Earth rotated on its axis.
His definitions of sine (jya), cosine (kojya), versine (ukramajya), and inverse sine (otkram jya) influenced the birth of trigonometry. He was also the first to specify sine and versine (1 - cosx) 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).
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, 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, as is Bacillus aryabhata, a species of bacteria discovered by ISRO scientists in 2009.
Lihat pula[sunting | sunting sumber]
Rujukan[sunting | sunting sumber]
- K. V. Sarma (2001), "Āryabhaṭa: His name, time and provenance", Indian Journal of History of Science 36 (4): 105–115
- 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 Great Britain and Ireland, p. 392
- Ansari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions". Bulletin of the Astronomical Society of India 5 (1): 10–18. Diperoleh pada 2007-07-21.
- Cooke (1997). "The Mathematics of the Hindus". p. 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.Missing or empty
- K. Chandra Hari, "Critical Evidence to Fix the Native Place of Āryabhat̟a-I", Current Science, Vol. 93, Issue 8, 25 October 2007
- K. Chandra Hari, "Alleged Mistake of Āryabhat̟a — Light onto His Place of Observation", Current Science Vol. 93, Issue 12, 25 December 2007, pp. 1870–73.
- K. Chandra Hari, "Āryabhat̟a on the Heliacal Rise and Set of Canopus", Current Science, Vol. 94, Issue 1, 10 January 2008
- Clark 1930, p. 68
- S. Balachandra Rao (2000), Indian Astronomy: An Introduction, Orient Blackswan, p. 82, ISBN 9788173712050: "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: "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: "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
- Ebenezer Burgess; Phanindralal Gangooly (1989), The Surya Siddhanta: A Textbook of Hindu Astronomy, Motilal Banarsidass Publ., p. 46, ISBN 9788120806122
- P. Z. Ingerman, "Panini-Backus form," Communications of the ACM 10 (3)(1967), p.137
- G. Ifrah (1998). A Universal History of Numbers: From Prehistory to the Invention of the Computer. John Wiley & Sons. Unknown parameter
- Dutta, Bibhutibhushan & Avadhesh Narayan Singh (1962), History of Hindu Mathematics, Asia Publishing House, Bombay, ISBN 81-86050-86-8 (reprint)
- S. Balachandra Rao (1994/1998). Indian Mathematics and Astronomy: Some Landmarks. Jnana Deep Publications. ISBN 81-7371-205-0. Unknown parameter
|location=suggested) (bantuan); Check date values in:
- 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.)Check date values in:
- Howard Eves (1990). An Introduction to the History of Mathematics (edisi ke-6). Saunders College Publishing House, New York. p. 237.
- Amartya K Dutta, "Diophantine equations: The Kuttaka", Resonance, October 2002. Also see earlier overview: Mathematics in Ancient India.
- Boyer, Carl B. (1991). "The Mathematics of the Hindus". A History of Mathematics (edisi ke-Second). 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.
- Pingree, David (1996), "Astronomy in India", written at London, in Walker, Christopher, Astronomy before the Telescope, British Museum Press, 123–142, ISBN 0-7141-1746-3 pp. 127–9.
- 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
- Hugh Thurston, Early Astronomy, New York: Springer-Verlag, 1996, pp. 178–189. ISBN 0-387-94822-8
- "JSC NES School Measures Up", NASA, 11th April, 2006, retrieved 24th January, 2008.
- "The Round Earth", NASA, 12th December, 2004, retrieved 24th January, 2008.
- 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.
- 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.
- Noel Swerdlow, "Review: A Lost Monument of Indian Astronomy," Isis, 64 (1973): 239–243.
- 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.
- 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."
- Douglas Harper (2001). "Online Etymology Dictionary". Diperoleh pada 2007-07-14.
- "Omar Khayyam". The Columbia Encyclopedia (edisi ke-6). 2001-05. Diperoleh pada 2007-06-10. Check date values in:
- "Maths can be fun". The Hindu. 2006-02-03. Diperoleh pada 2007-07-06.
- Discovery of New Microorganisms in the Stratosphere. Mar. 16, 2009. ISRO.
Rujukan lain[sunting | sunting sumber]
- Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823.
- Clark (1930), The Āryabhaṭīya of Āryabhaṭa: An Ancient Indian Work on Mathematics and Astronomy, University of Chicago Press; reprint: Kessinger Publishing (2006), ISBN 978-1425485993 Unknown parameter
- Kak, Subhash C. (2000). 'Birth and Early Development of Indian Astronomy'. In Selin, Helaine (2000), Astronomy Across Cultures: The History of Non-Western Astronomy, Boston: Kluwer, ISBN 0-7923-6363-9
- Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.
- Thurston, H. (1994), Early Astronomy, Springer-Verlag, New York, ISBN 0-387-94107-X
- http://www.scribd.com/doc/20912413/The-Aryabhatiya-of-Aryabhata-English-Translation - The Aryabhatiya of Aryabhata English Translation
- John J. O'Connor dan Edmund F. Robertson. Aryabhata di Arkib Sejarah Matematik MacTutor.
- Aryabhata and Diophantus' son, Hindustan Times Storytelling Science column, Nov 2004