Āryabhaṭa



Statue of Aryabhata at the IUCAA, Pune (although there is no historical record of his appearance)


Born  476 CE (Unclear)
Kusumapura, Pataliputra (present day Patna, Bihar)
or, Asmaka (Modernday MaharashtraAndhra Pradesh) 
Died  550 CE
Pataliputra, Gupta Empire (modernday Patna, India)

Academic background  
Influences  Surya Siddhanta 
Academic work  
Era  Gupta era 
Main interests  Mathematics, astronomy 
Notable works  Āryabhaṭīya, Aryasiddhanta 
Notable ideas  Explanation of lunar eclipse and solar eclipse, rotation of Earth on its axis, reflection of light by moon, sinusoidal functions, solution of single variable quadratic equation, value of π correct to 4 decimal places, diameter of Earth, calculation of the length of sidereal year 
Influenced  Lalla, Bhaskara I, Brahmagupta, Varahamihira, Kerala school of astronomy and mathematics, Islamic Astronomy and Mathematics 
Aryabhata Photo on Wall of Patna Junction
Aryabhata (ISO: Āryabhaṭa) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of Indian mathematics and Indian astronomy He flourished in the Gupta Era and produced works such as the Aryabhatiya (which mentions that in 3600 Kali Yuga, 499 CE, he was 23 years old) and the Aryasiddhanta
Aryabhata created a system of phonemic number notation in which numbers were represented by consonantvowel monosyllables Later commentators such as Brahmagupta divide his work into Ganita (“Mathematics”), Kalakriya (“Calculations on Time”) and Golapada (“Spherical Astronomy”) His pure mathematics discusses topics such as determination of square and cube roots, geometrical figures with their properties and mensuration, arithmetric progression problems on the shadow of the gnomon, quadratic equations, linear and indeterminate equations Aryabhata calculated the value of pi (π) to the fourth decimal digit and was likely aware that pi (π) is an irrational number, around 1300 years before Lambert proved the same Aryabhata’s sine table and his work on trignometry were extremely influential on the Islamic Golden Age; his works were translated into Arabic and influenced AlKhwarizmi and AlZarqali In his spherical astronomy, he applied plane trigonometry to spherical geometry and gave calculations on solar, lunar eclipses He discovered that the apparent westward motion of stars is due to the spherical Earth’s rotation about its own axis Aryabhata also noted that the luminosity of the Moon and other planets is due to reflected sunlight
Table of 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, including Brahmagupta’s references to him “in more than a hundred places by name” Furthermore, in most instances “Aryabhatta” would not fit the metre either
Time and place of birth
Aryabhata mentions in the Aryabhatiya that he was 23 years old 3,600 years into the Kali 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 Aryabhata called himself a native of Kusumapura or Pataliputra (present day Patna, Bihar)
Other hypothesis
Bhāskara I describes Aryabhata as āśmakīya, “one belonging to the Aśmaka country” During the Buddha’s time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India
It has been claimed that the aśmaka (Sanskrit for “stone”) where Aryabhata originated may be the present day Kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala This is based on the belief that Koṭuṅṅallūr was earlier known as KoṭumKallūr (“city of hard stones”); however, old records show that the city was actually Koṭumkolūr (“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 K Chandra Hari has argued for the Kerala hypothesis on the basis of astronomical evidence
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
Education
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and 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 near 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 Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar
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, sumsofpower series, and a table of sines
The Aryasiddhanta, 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 midnightday reckoning, as opposed to sunrise in Aryabhatiya It also contained a description of several astronomical instruments: the gnomon (shankuyantra), a shadow instrument (chhAyAyantra), possibly anglemeasuring devices, semicircular and circular (dhanuryantra / chakrayantra), a cylindrical stick yastiyantra, an umbrellashaped device called the chhatrayantra, and water clocks of at least two types, bowshaped and cylindrical
A third text, which may have survived in the Arabic translation, is Al ntf or Alnanf 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 alBīrūnī
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 AryashatasaShTa (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 (c 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 432 million years
 Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku–chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuṭṭaka)
 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), kShayatithis, and a sevenday 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, c 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE)
Mathematics
Place value system and zero
The placevalue system, first seen in the 3rdcentury 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 placevalue 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
Approximation of π
Aryabhata worked on the approximation for pi (π), and may have come to the conclusion that π is irrational In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
This implies that for a circle whose diameter is 20000, the circumference will be 62832
ie,
$\pi =\frac{62832}{20000}=31416$, which is accurate to three decimal places
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 true, it is quite a sophisticated insight because the irrationality of pi (π) was proved in Europe only in 1761 by Lambert
After Aryabhatiya was translated into Arabic (c 820 CE) this approximation was mentioned in AlKhwarizmi’s book on algebra
Trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as
 tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ
that translates to: “for a triangle, the result of a perpendicular with the halfside is the area”
Aryabhata discussed the concept of sine in his work by the name of ardhajya, which literally means “halfchord” 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 jaib, meaning “pocket” or “fold (in a garment)” (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 jaib with its Latin counterpart, sinus, which means “cove” or “bay”; thence comes the English word sine
Indeterminate equations
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to Diophantine equations that have the form ax + by = c (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as the Chinese remainder theorem) 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, elaborated by Bhaskara in 621 CE, is called the kuṭṭaka (कुट्टक) method Kuṭṭaka means “pulverising” 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 firstorder diophantine equations in Indian mathematics, and initially the whole subject of algebra was called kuṭṭakagaṇita or simply kuṭṭaka
Algebra
In Aryabhatiya, Aryabhata provided elegant results for the summation of series of squares and cubes:
 ${1}^{2}+{2}^{2}+\cdots +{n}^{2}=\frac{n(n+1)(2n+1)}{6}$
and

${1}^{3}+{2}^{3}+\cdots +{n}^{3}=(1+2+\cdots +n{)}^{2}$
(see squared triangular number)
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 ardharAtrikA, 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 may have believed that the planet’s orbits as elliptical rather than circular
Motions of the solar system
Aryabhata correctly insisted that the earth 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 thenprevailing view, that the sky rotated This is indicated in the first chapter of the Aryabhatiya, where he gives the number of rotations of the earth in a yuga, and made more explicit in his gola chapter:
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 (c 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 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 twoepicycle model reflects elements of prePtolemaic 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
Eclipses
Solar and lunar eclipses were scientifically explained by Aryabhata He states that the Moon and planets shine by reflected sunlight Instead of the prevailing cosmogony in which eclipses were caused by Rahu and Ketu (identified as the pseudoplanetary lunar 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 gola37) He discusses at length the size and extent of the Earth’s shadow (verses gola38–48) 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 18thcentury 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
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 41 seconds; the modern value is 23:56:4091 Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (36525858 days) is an error of 3 minutes and 20 seconds over the length of a year (36525636 days)
Heliocentrism
As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis His model also gave corrections (the śīgra 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 underlying heliocentric model, in which the planets orbit the Sun, though this has been rebutted It has also been suggested that aspects of Aryabhata’s system may have been derived from an earlier, likely prePtolemaic Greek, heliocentric model of which Indian astronomers were unaware, though the evidence is scant 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 late Babylonian astronomical texts), and that Aryabhata’s system was not explicitly heliocentric
Legacy
India’s first satellite named after Aryabhata
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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 (c 820 CE), was particularly influential Some of his results are cited by AlKhwarizmi and in the 10th century AlBiruni stated that Aryabhata’s followers believed that the Earth rotated on its axis
His definitions of sine (jya), cosine (kojya), versine (utkramajya), 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 375° 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 AlZarqali (11th century) were translated into Latin as the Tables of Toledo (12th century) 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
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
India’s first satellite Aryabhata and the lunar crater Aryabhata are both named in his honour, the Aryabhata satellite also featured on the reverse of the Indian 2rupee note An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Institute of Observational Sciences (ARIES) near Nainital, India The interschool Aryabhatta Maths Competition is also named after him, as is Bacillus aryabhata, a species of bacteria discovered in the stratosphere by ISRO scientists in 2009