Aryabhata (476–550 CE) was the first in the line of great
mathematicianA mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....

astronomerAn astronomer is a scientist who studies celestial bodies such as planets, stars and galaxies.Historically, astronomy was more concerned with the classification and description of phenomena in the sky, while astrophysics attempted to explain these phenomena and the differences between them using...
s from the classical age of
Indian mathematicsIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
and Indian astronomy. His most famous works are the
ĀryabhaṭīyaĀryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa. Structure and style:...
(499 CE, when he was 23 years old) and the
AryasiddhantaSiddhanta, a Sanskrit term, roughly translates as the Doctrine or the Tradition. It denotes the established and accepted view of a particular school within Indian philosophy.Hindu philosophy:...
.
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
BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
's references to him "in more than a hundred places by name". Furthermore, in most instances "Aryabhatta" does not fit the metre either.
Time and Place of birth
Aryabhata mentions in the
Aryabhatiya that it was composed 3,630 years into the
Kali YugaKali Yuga is the last of the four stages that the world goes through as part of the cycle of yugas described in the Indian scriptures. The other ages are Satya Yuga, Treta Yuga and Dvapara Yuga...
, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476 CE.
Aryabhata provides no information about his place of birth. The only information comes from
Bhāskara IBhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the HinduArabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
, who describes Aryabhata as
āśmakīya, "one belonging to the
aśmaka country." The Asmaka were one of the 16 of Ancient Indian
MahajanapadasMahājanapadas , literally "great realms", were ancient Indian kingdoms or countries...
, and the only one situated south of the Vindhyas.
It is widely attested that during the midfirst millennium BCE, a branch of the Aśmaka people settled in the region between the
NarmadaThe Narmada , also called Rewa is a river in central India and the fifth largest river in the Indian subcontinent. It is the third largest river that completely flows within India after Ganges and Godavari...
and Godavari rivers in central India, and it is possible Aryabhata was born there. 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.
Many are of the view that he was born in the south of India in
Keralaor Keralam is an Indian state located on the Malabar coast of southwest India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
and lived in
MagadhaMagadha formed one of the sixteen Mahājanapadas or kingdoms in ancient India. The core of the kingdom was the area of Bihar south of the Ganga; its first capital was Rajagriha then Pataliputra...
at the time of the Gupta rulers.
Education
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 IBhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the HinduArabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
(CE 629), identify Kusumapura as Pāṭaliputra, modern
PatnaPaṭnā , is the capital of the Indian state of Bihar and the second largest city in Eastern India . Patna is one of the oldest continuously inhabited places in the world...
. A verse mentions that Aryabhata was the head of an institution (
) at Kusumapura, and, because the university of
NalandaNālandā is the name of an ancient center of higher learning in Bihar, India.The site of Nalanda is located in the Indian state of Bihar, about 55 miles south east of Patna, and was a Buddhist center of learning from the fifth or sixth century CE to 1197 CE. It has been called "one of the...
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. Aryabhata is also reputed to have set up an observatory at the Sun temple in
TareganaTaregana or Taregna , is a small town in Bihar, India, about from Patna.History:In Taregna Aryabhata set up an Astronomical Observatory in the Sun Temple 6th century. It is believed that here he proposed the Heliocentric Model, and suggested for the first time in history that Earth revolves...
, Bihar.
Other hypotheses
Some archeological evidences suggests that Aryabhata could have originated from the present day
KodungallurKodungallur is a municipality in Thrissur District, in the state of Kerala, India on the Malabar Coast. Kodungallur is located about 29 km northwest of Kochi city and 38 km Southwest of Thrissur, on National Highway 17 . Muziris the ancient seaport at the mouth of the Periyar River was...
in
Keralaor Keralam is an Indian state located on the Malabar coast of southwest India. It was created on 1 November 1956 by the States Reorganisation Act by combining various Malayalam speaking regions....
region. 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 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 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.
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.
Works
Aryabhata is the author of several treatises on
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
and
astronomyAstronomy is a natural science that deals with the study of celestial objects and phenomena that originate outside the atmosphere of Earth...
, 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
arithmeticArithmetic or arithmetics is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple daytoday counting to advanced science and business calculations. It involves the study of quantity, especially as the result of combining numbers...
,
algebraAlgebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
,
plane trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
, and
spherical trigonometrySpherical trigonometry is a branch of spherical geometry which deals with polygons on the sphere and the relationships between the sides and the angles...
. It also contains continued
fractionsA fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, we specify how many parts of a certain size there are, for example, onehalf, fiveeighths and threequarters.A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists...
,
quadratic equationIn mathematics, a quadratic equation is a univariate polynomial equation of the second degree. A general quadratic equation can be written in the formax^2+bx+c=0,\,...
s, sumsofpower series, and a
table of sinesĀryabhaṭa's sine table is a set of twentyfour of numbers given in the astronomical treatise Āryabhaṭiya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa , for the computation of the halfchords of certain set of arcs of a circle...
.
The
Aryasiddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary,
VarahamihiraVarāhamihira , also called Varaha or Mihira, was an Indian astronomer, mathematician, and astrologer who lived in Ujjain...
, and later mathematicians and commentators, including
BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
and
Bhaskara IBhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the HinduArabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
. This work appears to be based on the older
Surya SiddhantaThe Surya Siddhanta is one of the earliest siddhanta in archeoastronomy of the Hindus by an unknown author. It describes the archeoastronomy theories, principles and methods of the ancient Hindus. This siddhanta is supposed to be the knowledge that the Sun god gave to an Asura called Maya. Asuras...
and uses the midnightday reckoning, as opposed to sunrise in
Aryabhatiya. It also contained a description of several astronomical instruments: the
gnomonThe gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....
(
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 clockA water clock or clepsydra is any timepiece in which time is measured by the regulated flow of liquid into or out from a vessel where the amount is then measured.Water clocks, along with sundials, are likely to be the oldest timemeasuring instruments, with the only exceptions...
s of at least two types, bowshaped and cylindrical.
A third text, which may have survived in the
ArabicArabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...
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
PersianThe Persian people are part of the Iranian peoples who speak the modern Persian language and closely akin Iranian dialects and languages. The origin of the ethnic Iranian/Persian peoples are traced to the Ancient Iranian peoples, who were part of the ancient IndoIranians and themselves part of...
scholar and chronicler of India, Abū Rayhān alBīrūnī.
Aryabhatiya
Direct details of Aryabhata's work are known only from the
AryabhatiyaĀryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa. Structure and style:...
. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple
Bhaskara IBhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the HinduArabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
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
sutraSūtra is an aphorism or a collection of such aphorisms in the form of a manual. Literally it means a thread or line that holds things together and is derived from the verbal root siv, meaning to sew , as does the medical term...
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
The ' is an Indian text on Jyotisha, redacted by Lagadha .The text is foundational to the Vedanga discipline of Jyotisha. It is dated to the final centuries BCE...
(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 4.32 million years.
 Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon
The gnomon is the part of a sundial that casts the shadow. Gnomon is an ancient Greek word meaning "indicator", "one who discerns," or "that which reveals."It has come to be used for a variety of purposes in mathematics and other fields....
/ shadows (shankuchhAyA), simple, quadratic, simultaneousIn mathematics, simultaneous equations are a set of equations containing multiple variables. This set is often referred to as a system of equations. A solution to a system of equations is a particular specification of the values of all variables that simultaneously satisfies all of the equations...
, 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), kShayatithis, and a sevenday week with names for the days of week.
 Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere
In astronomy and navigation, the celestial sphere is an imaginary sphere of arbitrarily large radius, concentric with the Earth and rotating upon the same axis. All objects in the sky can be thought of as projected upon the celestial sphere. Projected upward from Earth's equator and poles are the...
, features of the eclipticThe ecliptic is the plane of the earth's orbit around the sun. In more accurate terms, it is the intersection of the celestial sphere with the ecliptic plane, which is the geometric plane containing the mean orbit of the Earth around the Sun...
, celestial equatorThe celestial equator is a great circle on the imaginary celestial sphere, in the same plane as the Earth's equator. In other words, it is a projection of the terrestrial equator out into space...
, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophonIn publishing, a colophon is either:* A brief description of publication or production notes relevant to the edition, in modern books usually located at the reverse of the title page, but can also sometimes be located at the end of the book, or...
s 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 SomayajiKelallur Nilakantha Somayaji was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehensive astronomical treatise Tantrasamgraha completed in 1501...
in his
Aryabhatiya Bhasya, (1465 CE).
Place value system and zero
The placevalue system, first seen in the 3rd century
Bakhshali ManuscriptThe Bakhshali Manuscript is an Ancient Indian mathematical manuscript written on birch bark which was found near the village of Bakhshali in 1881 in what was then the NorthWest Frontier Province of British India...
, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician
Georges IfrahGeorges Ifrah is a French author and historian of mathematics, especially numerals. He was formerly a teacher of mathematics....
explains that knowledge of zero was implicit in Aryabhata's placevalue system as a place holder for the powers of ten with
nullIn computing:* Null , a special marker and keyword in SQL* Null character, the zerovalued ASCII character, also designated by NUL, often used as a terminator, separator or filler* Null device, a special computer file that discards all data written to it...
coefficients
However, Aryabhata did not use the Brahmi numerals. Continuing the
SanskritSanskrit , is a historical IndoAryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
ic tradition from
Vedic timesThe Vedic period was a period in history during which the Vedas, the oldest scriptures of Hinduism, were composed. The time span of the period is uncertain. Philological and linguistic evidence indicates that the Rigveda, the oldest of the Vedas, was composed roughly between 1700–1100 BCE, also...
, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a
mnemonicA mnemonic , or mnemonic device, is any learning technique that aids memory. To improve long term memory, mnemonic systems are used to make memorization easier. Commonly encountered mnemonics are often verbal, such as a very short poem or a special word used to help a person remember something,...
form.
Approximation of π
Aryabhata worked on the approximation for
pi' is a mathematical constant that is the ratio of any circle's circumference to its diameter. is approximately equal to 3.14. Many formulae in mathematics, science, and engineering involve , which makes it one of the most important mathematical constants...
(
), and may have come to the conclusion that
is irrational. In the second part of the
Aryabhatiyam ( 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."
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 figuresThe significant figures of a number are those digits that carry meaning contributing to its precision. This includes all digits except:...
.
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
LambertJohann Heinrich Lambert was a Swiss mathematician, physicist, philosopher and astronomer.Asteroid 187 Lamberta was named in his honour.Biography:...
.
After Aryabhatiya was translated into
ArabicArabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...
(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 phalashariram samadalakoti bhujardhasamvargah
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. Literally, it means "halfchord". For simplicity, people started calling it
jya. When Arabic writers translated his works from
SanskritSanskrit , is a historical IndoAryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
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". And after that, the
sinus became
sine in English.
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 + by = c, a topic that has come to be known as diophantine equations. This is an example from
BhāskaraBhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the HinduArabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
'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 SutrasThe Shulba Sutras or Śulbasūtras are sutra texts belonging to the Śrauta ritual and containing geometry related to firealtar construction. Purpose and origins :...
, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems is called the
(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 firstorder diophantine equations and is often referred to as the Aryabhata algorithm. The diophantine equations are of interest in cryptology, and the
RSA ConferenceThe RSA Conference is a cryptography and information securityrelated conference held annually in the San Francisco Bay Area.The RSA Conference started in 1991 as a forum for cryptographers to gather and share the latest knowledge and advancements in the area of Internet security...
, 2006, focused on the
kuttaka method and earlier work in the Sulbasutras.
Algebra
In
Aryabhatiya Aryabhata provided elegant results for the summation of
seriesA series is the sum of the terms of a sequence. Finite sequences and series have defined first and last terms, whereas infinite sequences and series continue indefinitely....
of squares and cubes:
and
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
BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
'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
ellipticalIn geometry, an ellipse is a plane curve that results from the intersection of a cone by a plane in a way that produces a closed curve. Circles are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis...
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 in other parts of the world, 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
MoonThe Moon is Earth's only known natural satellite,There are a number of nearEarth asteroids including 3753 Cruithne that are coorbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasisatellites and not true moons. For more...
,
MercuryMercury is the innermost and smallest planet in the Solar System, orbiting the Sun once every 87.969 Earth days. The orbit of Mercury has the highest eccentricity of all the Solar System planets, and it has the smallest axial tilt. It completes three rotations about its axis for every two orbits...
,
VenusVenus is the second planet from the Sun, orbiting it every 224.7 Earth days. The planet is named after Venus, the Roman goddess of love and beauty. After the Moon, it is the brightest natural object in the night sky, reaching an apparent magnitude of −4.6, bright enough to cast shadows...
, the
SunThe Sun is the star at the center of the Solar System. It is almost perfectly spherical and consists of hot plasma interwoven with magnetic fields...
,
MarsMars is the fourth planet from the Sun in the Solar System. The planet is named after the Roman god of war, Mars. It is often described as the "Red Planet", as the iron oxide prevalent on its surface gives it a reddish appearance...
,
JupiterJupiter is the fifth planet from the Sun and the largest planet within the Solar System. It is a gas giant with mass onethousandth that of the Sun but is two and a half times the mass of all the other planets in our Solar System combined. Jupiter is classified as a gas giant along with Saturn,...
,
SaturnSaturn is the sixth planet from the Sun and the second largest planet in the Solar System, after Jupiter. Saturn is named after the Roman god Saturn, equated to the Greek Cronus , the Babylonian Ninurta and the Hindu Shani. Saturn's astronomical symbol represents the Roman god's sickle.Saturn,...
, and the
asterismsIn astronomy, an asterism is a pattern of stars recognized on Earth's night sky. It may form part of an official constellation, or be composed of stars from more than one. Like constellations, asterisms are in most cases composed of stars which, while they are visible in the same general direction,...
."
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. Aryabhata states that the
MoonThe Moon is Earth's only known natural satellite,There are a number of nearEarth asteroids including 3753 Cruithne that are coorbital with Earth: their orbits bring them close to Earth for periods of time but then alter in the long term . These are quasisatellites and not true moons. For more...
and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by pseudoplanetary nodes
RahuIn Hindu mythology, Rahu is a cutoff head of an asura, that swallows the sun or the moon causing eclipses. He is depicted in art as a serpent with no body riding a chariot drawn by eight black horses. Rahu is one of the navagrahas in Vedic astrology...
and
KetuKetu is the descending lunar node. 'Ketu' is said to be the body of Rahu, after the head of the asura was cut off by God Vishnu. In Hindu mythology, Ketu is generally referred to as a "shadow" planet. It is believed to have a tremendous impact on human lives and also the whole creation...
, 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 GentilGuillaume Joseph Hyacinthe JeanBaptiste Le Gentil de la Galaisière was a French astronomer.Biography:...
, during a visit to Pondicherry, India, found the Indian computations of the duration of the
lunar eclipseA lunar eclipse occurs when the Moon passes behind the Earth so that the Earth blocks the Sun's rays from striking the Moon. This can occur only when the Sun, Earth, and Moon are aligned exactly, or very closely so, with the Earth in the middle. Hence, a lunar eclipse can only occur the night of a...
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 4.1 seconds; the modern value is 23:56:4.091. Similarly, his value for the length of the
sidereal yearA sidereal year is the time taken by the Earth to orbit the Sun once with respect to the fixed stars. Hence it is also the time taken for the Sun to return to the same position with respect to the fixed stars after apparently travelling once around the ecliptic. It was equal to at noon 1 January...
at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days) is an error of 3 minutes and 20 seconds over the length of a year (365.25636 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
heliocentricHeliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...
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
GreekGreek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the ancient Greek, Hellenistic, GrecoRoman, and Late Antiquity eras. It is not limited geographically to Greece or to ethnic Greeks, as the Greek language had become the...
, 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
Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The
ArabicArabic is a name applied to the descendants of the Classical Arabic language of the 6th century AD, used most prominently in the Quran, the Islamic Holy Book...
translation during the
Islamic Golden AgeDuring the Islamic Golden Age philosophers, scientists and engineers of the Islamic world contributed enormously to technology and culture, both by preserving earlier traditions and by adding their own inventions and innovations...
(c. 820 CE), was particularly influential. Some of his results are cited by AlKhwarizmi and in the 10th century
AlBiruniAbū alRayḥān Muḥammad ibn Aḥmad alBīrūnīArabic spelling. . The intermediate form Abū Rayḥān alBīrūnī is often used in academic literature...
stated that Aryabhata's followers believed that the Earth rotated on its axis.
His definitions of
sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
(
jya), cosine (
kojya), versine (
utkramajya),
and inverse sine (
otkram jya) influenced the birth of
trigonometryTrigonometry is a branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves...
. He was also the first to specify sine and
versineThe versine or versed sine, versin, is a trigonometric function equal to and 2sin2. It appeared in some of the earliest trigonometric tables and was once widespread, but it is now littleused...
(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 CremonaGerard of Cremona was an Italian translator of Arabic scientific works found in the abandoned Arab libraries of Toledo, Spain....
while translating an Arabic geometry text to
LatinLatin is an Italic language originally spoken in Latium and Ancient Rome. It, along with most European languages, is a descendant of the ancient ProtoIndoEuropean language. Although it is considered a dead language, a number of scholars and members of the Christian clergy speak it fluently, and...
. 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 (
zijZīj is the generic name applied to Islamic astronomical books that tabulate parameters used for astronomical calculations of the positions of the Sun, Moon, stars, and planets. The name is derived from the Middle Persian term zih or zīg, meaning cord...
es). In particular, the astronomical tables in the work of the
Arabic SpainAlAndalus was the Arabic name given to a nation and territorial region also commonly referred to as Moorish Iberia. The name describes parts of the Iberian Peninsula and Septimania governed by Muslims , at various times in the period between 711 and 1492, although the territorial boundaries...
scientist AlZarqali (11th century) were translated into Latin as the
Tables of ToledoThe Toledan Tables, or Tables of Toledo, were astronomical tables which were used to predict the movements of the Sun, Moon and planets relative to the fixed stars...
(12th c.) and remained the most accurate
ephemerisAn ephemeris is a table of values that gives the positions of astronomical objects in the sky at a given time or times. Different kinds of ephemerides are used for astronomy and astrology...
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
PanchangamA panchāngam is a Hindu astrological almanac, which follows traditional Indian cosmology, and presents important astronomical data in tabulated form. It is sometimes spelled Pancanga, Panchanga, Panchaanga, or Panchānga, and is pronounced Panchānga...
(the
Hindu calendarThe hindu calendar used in ancient times has undergone many changes in the process of regionalization, and today there are several regional Indian calendars, as well as an Indian national calendar. Nepali calendar, Bengali calendar, Malayalam calendar, Tamil calendar, Telugu calendar, Kannada...
). In the Islamic world, they formed the basis of the Jalali calendar introduced in 1073 CE by a group of astronomers including
Omar KhayyamOmar Khayyám was aPersian polymath: philosopher, mathematician, astronomer and poet. He also wrote treatises on mechanics, geography, mineralogy, music, climatology and theology....
, versions of which (modified in 1925) are the national calendars in use in
IranIran , officially the Islamic Republic of Iran , is a country in Southern and Western Asia. The name "Iran" has been in use natively since the Sassanian era and came into use internationally in 1935, before which the country was known to the Western world as Persia...
and
AfghanistanAfghanistan , officially the Islamic Republic of Afghanistan, is a landlocked country located in the centre of Asia, forming South Asia, Central Asia and the Middle East. With a population of about 29 million, it has an area of , making it the 42nd most populous and 41st largest nation in the world...
today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlier
SiddhantaSiddhanta, a Sanskrit term, roughly translates as the Doctrine or the Tradition. It denotes the established and accepted view of a particular school within Indian philosophy.Hindu philosophy:...
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 calendarThe Gregorian calendar, also known as the Western calendar, or Christian calendar, is the internationally accepted civil calendar. It was introduced by Pope Gregory XIII, after whom the calendar was named, by a decree signed on 24 February 1582, a papal bull known by its opening words Inter...
.
India's first satellite
AryabhataAryabhatta was India's first satellite, named after the great Indian astronomer of the same name. It was launched by the Soviet Union on 19 April 1975 from Kapustin Yar using a Cosmos3M launch vehicle. It was built by the Indian Space Research Organization to gain experience in building and...
and the lunar crater
AryabhataAryabhata, named after Indian astronomer Aryabhata , is the remnant of a lunar impact crater located in the eastern Mare Tranquillitatis. The crater has been almost submerged by lavaflow, and now only an arcshaped ridge formed from the eastern half of the rim remains above the lunar mare. This...
are named in his honour. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the
Aryabhatta Research Institute of Observational SciencesAryabhatta Research Institute of Observational Sciences is one of the leading research Institutes which specializes in Astronomy, Astrophysics and Atmospheric Sciences...
(ARIES) near Nainital, India. The interschool Aryabhata Maths Competition is also named after him, as is
Bacillus aryabhata, a species of bacteria discovered by ISRO scientists in 2009.
See also
 Aryabhatiya
Āryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opus and only extant work of the 5th century Indian mathematician, Āryabhaṭa. Structure and style:...
 Aryabhata's sine table
Āryabhaṭa's sine table is a set of twentyfour of numbers given in the astronomical treatise Āryabhaṭiya composed by the fifth century Indian mathematician and astronomer Āryabhaṭa , for the computation of the halfchords of certain set of arcs of a circle...
 Indian mathematics
Indian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
 List of Indian mathematicians
Other references
of : An Ancient Indian Work on Mathematics and Astronomy
 last=Clark  first=Walter Eugene
 year=1930
 publisher=University of Chicago Press; reprint: Kessinger Publishing (2006)
 isbn=9781425485993
 url=http://www.archive.org/details/The_Aryabhatiya_of_Aryabhata_Clark_1930
 ref = harv
 postscript =
}}
 Kak, Subhash C.
Subhash Kak is an Indian American computer scientist, most notable for his controversial Indological publications on history, the philosophy of science, ancient astronomy, and the history of mathematics...
(2000). 'Birth and Early Development of Indian Astronomy'. In
 Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.
External links