For his explicit mention of the relativity of motion, he also qualifies as a major early physicist.[8]
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,[9] includingBrahmagupta's references to him "in more than a hundred places by name".[1] Furthermore, in most instances "Aryabhatta" would not fit the metre either.[9]
Time and place of birth
Aryabhata mentions in theAryabhatiya that he was 23 years old 3,600 years into theKali Yuga, but this is not to mean that the text was composed at that time. This mentioned year corresponds to 499 CE, and implies that he was born in 476.[6] Aryabhata called himself a native of Kusumapura orPataliputra (near present dayPatna,Bihar).[1]
Other hypothesis
Bhāskara I describes Aryabhata asāśmakīya, "one belonging to theAśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in the region between theNarmada andGodavari rivers in central India.[9][10]
It has been claimed that theaśmaka (Sanskrit for "stone") where Aryabhata originated may be the present dayKodungallur which was the historical capital city ofThiruvanchikkulam of ancient Kerala.[11] This is based on the belief that Koṭuṅṅallūr 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 has been 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.[9] K. Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence.[12]
Aryabhata mentions "Lanka" on several occasions in theAryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as hisUjjayini.[13]
Education
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time.[14] Both Hindu and Buddhist tradition, as well asBhāskara I (CE 629), identify Kusumapura asPāṭaliputra, modernPatna.[9] A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university ofNalanda was in Pataliputra at the time, it is speculated that Aryabhata might have been the head of the Nalanda university as well.[9] However Aryabhata's connection with Nalanda has been termed as "speculation" as he himself never declared Nalanda to be his patron in his writings which given the prestige of Nalanda at the time, would have made his connection to it unlikely.[15] The claim that Nalanda used to be located in Kusamapura/Pataliputra also cannot be proven.[16] Aryabhata is also reputed to have set up an observatory at the Sun temple inTaregana, Bihar.[17]
TheArya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary,Varahamihira, and later mathematicians and commentators, includingBrahmagupta andBhaskara I. This work appears to be based on the olderSurya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise inAryabhatiya.[10] It also contained a description of several astronomical instruments: thegnomon (shanku-yantra), a shadow instrument (chhaya-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra /chakra-yantra), a cylindrical stickyasti-yantra, an umbrella-shaped device called thechhatra-yantra, andwater clocks of at least two types, bow-shaped and cylindrical.[10]
A third text, which may have survived in theArabic translation, isAl ntf orAl-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 thePersian scholar and chronicler of India,Abū Rayhān al-Bīrūnī.[10]
Direct details of Aryabhata's work are known only from theAryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name.[8] His discipleBhaskara I calls itAshmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to asArya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text.[20][8] It is written in the very terse style typical ofsutra 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 fourpādas or chapters:
Gitikapada: (13 verses): large units of time—kalpa,manvantra, andyuga—which present a cosmology different from earlier texts such as Lagadha'sVedanga Jyotisha (c. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during amahayuga is given as 4.32 million years.
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.[19]
TheAryabhatiya 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, c. 600 CE) and byNilakantha Somayaji in hisAryabhatiya Bhasya (1465 CE).[20][19]
Aryabhatiya is also well-known for his description of relativity of motion. He expressed this relativity thus: "Just as a man in a boat moving forward sees the stationary objects (on the shore) as moving backward, just so are the stationary stars seen by the people on earth as moving exactly towards the west."[8]
However, Aryabhata did not use the Brahmi numerals. Continuing theSanskritic tradition fromVedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in amnemonic form.[22]
Approximation ofπ
Aryabhata worked on the approximation forpi (π), and may have come to the conclusion that π is irrational. In the second part of theAryabhatiyam (gaṇitapāda 10), he writes:
"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."[23]
This implies that for a circle whose diameter is 20000, the circumference will be 62832
i.e, = =, which is accurate to two parts in one million.[24]
It is speculated that Aryabhata used the wordāsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (orirrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi (π) was proved in Europe only in 1761 byLambert.[25]
After Aryabhatiya was translated intoArabic (c. 820 CE); this approximation was mentioned inAl-Khwarizmi's book on algebra.[10]
Trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as
that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."[26]
Aryabhata discussed the concept ofsine in his work by the name ofardha-jya, which literally means "half-chord". For simplicity, people started calling itjya. When Arabic writers translated his works fromSanskrit into Arabic, they referred it asjiba. However, in Arabic writings, vowels are omitted, and it was abbreviated asjb. Later writers substituted it withjaib, meaning "pocket" or "fold (in a garment)". (In Arabic,jiba is a meaningless word.) Later in the 12th century, whenGherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabicjaib with its Latin counterpart,sinus, which means "cove" or "bay"; thence comes the English wordsine.[27]
Indeterminate equations
A problem of great interest toIndian mathematicians since ancient times has been to find integer solutions toDiophantine equations that have the form ax + by = c.[28] (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as theChinese remainder theorem.) This is an example fromBhāskara'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 textSulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems, elaborated by Bhaskara in 621 CE, is called thekuṭṭaka (कुट्टक) method.Kuṭṭaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, and initially the whole subject of algebra was calledkuṭṭaka-gaṇita or simplykuṭṭaka.[29]
Algebra
InAryabhatiya, Aryabhata provided elegant results for the summation ofseries of squares and cubes:[30]
Aryabhata's system of astronomy was called theaudAyaka system, in which days are reckoned fromuday, dawn atlanka or "equator". Some of his later writings on astronomy, which apparently proposed a second model (orardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion inBrahmagupta'sKhandakhadyaka. In some texts, he seems to ascribe the apparent motions of the heavens to theEarth's rotation. He may have believed that the planet's orbits areelliptical rather than circular.[31][32]
Motions of the Solar System
Aryabhata correctly stated in Aryabhatiyam that the Earth is round and it rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the Earth, contrary to the then-prevailing view, that the sky rotated.[24][33]: 30 This is indicated in the first chapter of theAryabhatiya, where he gives the number of rotations of the Earth in ayuga,[34] and made more explicit in hisgola chapter:[35]
In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at the equator, constantly pushed by thecosmic wind.
Aryabhata described ageocentric model of the Solar System, in which theSun and Moon are each carried byepicycles. They in turn revolve around the Earth. In this model, which is also found in thePaitāmahasiddhānta (c. 425 CE), the motions of the planets are each governed by two epicycles, a smallermanda (slow) and a largerśīghra (fast).[36] The order of the planets in terms of distance from earth is taken as: theMoon,Mercury,Venus, theSun,Mars,Jupiter,Saturn, and theasterisms.[10]
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 mean speed as the 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-PtolemaicGreek astronomy.[37] Another element in Aryabhata's model, theśīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlyingheliocentric model.[38]
Eclipses
Solar and lunar eclipses were scientifically explained by Aryabhata. He states that theMoon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused byRahu andKetu (identified as the pseudo-planetarylunar nodes), 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 scientistGuillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of thelunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.[10]
Sidereal periods
Considered in modern English units of time, Aryabhata calculated thesidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds;[39] the modern value is 23:56:4.091. Similarly, his value for the length of thesidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days)[40] is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).[41]
Heliocentrism
As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (theśīgra anomaly) for the speeds of the planets in the sky in terms of the mean speed of the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an underlyingheliocentric model, in which the planets orbit the Sun,[42][43][44] though this has been rebutted.[45] It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-PtolemaicGreek, heliocentric model of which Indian astronomers were unaware,[46] though the evidence is scant.[47] The general consensus is that a synodic anomaly (depending on the position of the Sun) does not imply a physically heliocentric orbit (such corrections being also present in lateBabylonian astronomical texts), and that Aryabhata's system was not explicitly heliocentric.[48][49]
Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. TheArabic translation during theIslamic Golden Age (c. 820 CE), was particularly influential. Some of his results are cited byAl-Khwarizmi and in the 10th centuryAl-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.
His definitions ofsine (jya), cosine (kojya), versine (utkrama-jya),and inverse sine (otkram jya) influenced the birth oftrigonometry. He was also the first to specify sine andversine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.
In fact, the modern terms "sine" and "cosine" are mistranscriptions of the wordsjya andkojya as introduced by Aryabhata. As mentioned, they were translated asjiba andkojiba in Arabic and then misunderstood byGerard of Cremona while translating an Arabic geometry text toLatin. He assumed thatjiba was the Arabic wordjaib, which means "fold in a garment", L.sinus (c. 1150).[50]
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 theArabic Spain scientistAl-Zarqali (11th century) were translated into Latin as theTables of Toledo (12th century) and remained the most accurateephemeris 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 thePanchangam (theHindu calendar). In the Islamic world, they formed the basis of theJalali calendar introduced in 1073 CE by a group of astronomers includingOmar Khayyam,[51] versions of which (modified in 1925) are the national calendars in use inIran andAfghanistan today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlierSiddhanta 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 theGregorian calendar.[citation needed]
Aryabhatta Knowledge University (AKU), Patna has been established by Government of Bihar for the development and management of educational infrastructure related to technical, medical, management and allied professional education in his honour. The university is governed by Bihar State University Act 2008.
^O'Connor, J J; Robertson, E F."Aryabhata the Elder". www-history.mcs.st-andrews.ac.uk.Archived from the original on 11 July 2015. Retrieved18 July 2012.
^abcdefgAnsari, S.M.R. (March 1977). "Aryabhata I, His Life and His Contributions".Bulletin of the Astronomical Society of India.5 (1):10–18.Bibcode:1977BASI....5...10A.hdl:2248/502.
^See: *Clark 1930 *S. Balachandra Rao (2000).Indian Astronomy: An Introduction. Orient Blackswan. p. 82.ISBN978-81-7371-205-0.: "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.ISBN978-81-7319-432-0.: "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.ISBN978-0-9709636-2-8.: "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.ISBN978-81-87276-27-2. *Ebenezer Burgess; Phanindralal Gangooly (1989).The Surya Siddhanta: A Textbook of Hindu Astronomy. Motilal Banarsidass Publ. p. 46.ISBN978-81-208-0612-2.
^Cooke (1997). "The Mathematics of the Hindus".History of Mathematics: A Brief Course. Wiley. p. 204.ISBN9780471180821.Aryabhata himself (one of at least two mathematicians bearing that name) lived in the late 5th and the early 6th centuries atKusumapura (Pataliutra, a village near the city of Patna) and wrote a book calledAryabhatiya.
^"Get ready for solar eclipse"(PDF). National Council of Science Museums, Ministry of Culture, Government of India. Archived fromthe original(PDF) on 21 July 2011. Retrieved9 December 2009.
^George. Ifrah (1998).A Universal History of Numbers: From Prehistory to the Invention of the Computer. London: John Wiley & Sons.
^Dutta, Bibhutibhushan; Singh, Avadhesh Narayan (1962).History of Hindu Mathematics. Asia Publishing House, Bombay.ISBN81-86050-86-8.{{cite book}}:ISBN / Date incompatibility (help)
^Jacobs, Harold R. (2003).Geometry: Seeing, Doing, Understanding (Third ed.). New York: W.H. Freeman and Company. p. 70.ISBN0-7167-4361-2.
^S. Balachandra Rao (1998) [First published 1994].Indian Mathematics and Astronomy: Some Landmarks. Bangalore: Jnana Deep Publications.ISBN81-7371-205-0.
^Howard Eves (1990).An Introduction to the History of Mathematics (6 ed.). Saunders College Publishing House, New York. p. 237.
^Christianidis, J. (1994). On the History of Indeterminate problems of the first degree in Greek Mathematics. Trends in the Historiography of Science, 237-247.
^Boyer, Carl B. (1991)."The Mathematics of the Hindus".A History of Mathematics (Second ed.). John Wiley & Sons, Inc. p. 207.ISBN0-471-54397-7.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.
^[achalAni bhAni samapashchimagAni ... – golapAda.9–10]. Translation from K. S. Shukla and K.V. Sarma, K. V.Āryabhaṭīya of Āryabhaṭa, New Delhi: Indian National Science Academy, 1976. Quoted in Plofker 2009.
^Pingree, David (1996). "Astronomy in India". In Walker, Christopher (ed.).Astronomy before the Telescope. London: British Museum Press. pp. 123–142.ISBN0-7141-1746-3. 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.ISBN0-387-90844-7
^Hugh Thurston,Early Astronomy, New York: Springer-Verlag, 1996, pp. 178–189.ISBN0-387-94822-8
^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.
^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"Archived copy"(PDF).Archived(PDF) from the original on 18 March 2009. Retrieved8 February 2016.{{cite web}}: CS1 maint: archived copy as title (link).
