Aryabhata ( ISO: ) or Aryabhata I (476–550 CE) was an Indian mathematician and astronomer of the classical age of

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 (400 CE to 1200 CE), important contributions were made by scholars like Aryabhata, Brahmagupta ...

and

Indian astronomy Astronomy has long history in Indian subcontinent stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley civilisation or earlier. Astronomy later developed as a di ...

. He flourished in the

Gupta Era The Gupta era is a historical calendar era that begins from c. 318–319 CE. It was used by the Gupta emperors, as well as their vassals and their successors in present-day northern India and Nepal. It is identical to the Vallabhi era (or Valabh ...

and produced works such as the ''

Aryabhatiya ''Aryabhatiya'' (IAST: ') or ''Aryabhatiyam'' ('), a Sanskrit astronomical treatise, is the '' magnum opus'' and only known surviving work of the 5th century Indian mathematician Aryabhata. Philosopher of astronomy Roger Billard estimates that ...

'' (which mentions that in 3600 ''

Kali Yuga ''Kali Yuga'', in Hinduism, is the fourth and worst of the four ''yugas'' (world ages) in a '' Yuga Cycle'', preceded by '' Dvapara Yuga'' and followed by the next cycle's '' Krita (Satya) Yuga''. It is believed to be the present age, which i ...

'', 499 CE, he was 23 years old) and the ''Arya-siddhanta.'' Aryabhata created a system of phonemic number notation in which numbers were represented by consonant-vowel monosyllables. Later commentators such as

Brahmagupta Brahmagupta ( – ) was an Indian mathematician and astronomer. He is the author of two early works on mathematics and astronomy: the '' Brāhmasphuṭasiddhānta'' (BSS, "correctly established doctrine of Brahma", dated 628), a theoretical tr ...

divide his work into ''Ganita ("Mathematics"), Kalakriya ("Calculations on Time") and Golapada ("Spherical Astronomy")''. His pure mathematics discusses topics such as determination of

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

and

cube root In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. F ...

s, geometrical figures with their properties and mensuration, arithmetric progression problems on the shadow of the gnomon,

quadratic equation In algebra, a quadratic equation () is any equation that can be rearranged in standard form as ax^2 + bx + c = 0\,, where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not qu ...

linear Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear ...

and

indeterminate equation In mathematics, particularly in algebra, an indeterminate equation is an equation for which there is more than one solution. For example, the equation ax + by =c is a simple indeterminate equation, as is x^2=1. Indeterminate equations cannot be solv ...

s. Aryabhata calculated the value of pi (''π)'' to the fourth decimal digit and was likely aware that pi (''π)'' is an

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two inte ...

, 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 The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam, traditionally dated from the 8th century to the 14th century. This period is traditionally understood to have begun during the reign ...

; his works were translated into Arabic and influenced

Al-Khwarizmi Muḥammad ibn Mūsā al-Khwārizmī ( ar, محمد بن موسى الخوارزمي, Muḥammad ibn Musā al-Khwārazmi; ), or al-Khwarizmi, was a Persian polymath from Khwarazm, who produced vastly influential works in mathematics, astronom ...

and Al-Zarqali. In his spherical astronomy, he applied plane trigonometry to spherical geometry and gave calculations on solar,

lunar eclipse A lunar eclipse occurs when the Moon moves into the Earth's shadow. Such alignment occurs during an eclipse season, approximately every six months, during the full moon phase, when the Moon's orbital plane is closest to the plane of the Ear ...

s. 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 Luminosity is an absolute measure of radiated electromagnetic power (light), the radiant power emitted by a light-emitting object over time. In astronomy, luminosity is the total amount of electromagnetic energy emitted per unit of time by a st ...

of the Moon and other planets is due to reflected sunlight.

Biography

Name

While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "

bhatta Bhat (also spelled as Bhatt or Butt) is a surname in the Indian subcontinent. Bhat and Bhatt are shortened rendition of Bhatta. Etymology The word "Bhat" ( sa, भट, ) means "teacher" in Sanskrit. While the original shortened rendition of "Bh ...

" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including

'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 ''

'', 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 Pataliputra (IAST: ), adjacent to modern-day Patna, was a city in ancient India, originally built by Magadha ruler Ajatashatru in 490 BCE as a small fort () near the Ganges river.. Udayin laid the foundation of the city of Pataliputra at t ...

(present day

Patna Patna ( ), historically known as Pataliputra, is the capital and largest city of the state of Bihar in India. According to the United Nations, as of 2018, Patna had a population of 2.35 million, making it the 19th largest city in India. ...

Bihar Bihar (; ) is a state in eastern India. It is the 2nd largest state by population in 2019, 12th largest by area of , and 14th largest by GDP in 2021. Bihar borders Uttar Pradesh to its west, Nepal to the north, the northern part of West ...

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 The Godavari ( IAST: ''Godāvarī'' �od̪aːʋəɾiː is India's second longest river after the Ganga river and drains into the third largest basin in India, covering about 10% of India's total geographical area. Its source is in Trimbakesh ...

rivers in central India. It has been claimed that the ''aśmaka'' (Sanskrit for "stone") where Aryabhata originated may be the present day

Kodungallur Kodungallur (; also Cranganore, Portuguese: Cranganor; formerly known as Mahodayapuram, Shingly, Vanchi, Muchiri, Muyirikkode, and Muziris) is a historically significant town situated on the banks of river Periyar on the Malabar Coast in ...

which was the historical capital city of ''Thiruvanchikkulam'' of ancient Kerala. 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. 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 Ujjain (, Hindustani pronunciation: �d͡ːʒɛːn is a city in Ujjain district of the Indian state of Madhya Pradesh. It is the fifth-largest city in Madhya Pradesh by population and is the administrative centre of Ujjain district and Ujj ...

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

. A verse mentions that Aryabhata was the head of an institution (') at Kusumapura, and, because the university of

Nalanda Nalanda (, ) was a renowned ''mahavihara'' (Buddhist monastic university) in ancient Magadha (modern-day Bihar), India.Taregana, Bihar.

Works

Aryabhata is the author of several treatises on

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

and

astronomy Astronomy () is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution. Objects of interest include planets, moons, stars, nebulae, g ...

, 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 Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...

algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...

, plane trigonometry, and

spherical trigonometry Spherical trigonometry is the branch of spherical geometry that deals with the metrical relationships between the sides and angles of spherical triangles, traditionally expressed using trigonometric functions. On the sphere, geodesics are grea ...

. It also contains

continued fraction In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer ...

s, 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

and Bhaskara I. This work appears to be based on the older

Surya Siddhanta The ''Surya Siddhanta'' (; ) is a Sanskrit treatise in Indian astronomy dated to 505 CE,Menso Folkerts, Craig G. Fraser, Jeremy John Gray, John L. Berggren, Wilbur R. Knorr (2017)Mathematics Encyclopaedia Britannica, Quote: "(...) its Hindu inven ...

and uses the midnight-day reckoning, as opposed to sunrise in ''Aryabhatiya''. It also contained a description of several astronomical instruments: the

gnomon A gnomon (; ) is the part of a sundial that casts a shadow. The term is used for a variety of purposes in mathematics and other fields. History A painted stick dating from 2300 BC that was excavated at the astronomical site of Taosi is the ...

(''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 clock A water clock or clepsydra (; ; ) is a timepiece by which time is measured by the regulated flow of liquid into (inflow type) or out from (outflow type) a vessel, and where the amount is then measured. Water clocks are one of the oldest time- ...

s of at least two types, bow-shaped and cylindrical. A third text, which may have survived in the

Arabic Arabic (, ' ; , ' or ) is a Semitic language spoken primarily across the Arab world.Semitic languages: an international handbook / edited by Stefan Weninger; in collaboration with Geoffrey Khan, Michael P. Streck, Janet C. E.Watson; Walter ...

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

Direct details of Aryabhata's work are known only from the ''Aryabhatiya''. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple Bhaskara I calls it ''Ashmakatantra'' (or the treatise from the Ashmaka). It is also occasionally referred to as ''Arya-shatas-aShTa'' (literally, Aryabhata's 108) because there are 108 verses in the text. It is written in the very terse style typical of

sutra ''Sutra'' ( sa, सूत्र, translit=sūtra, translit-std=IAST, translation=string, thread)Monier Williams, ''Sanskrit English Dictionary'', Oxford University Press, Entry fo''sutra'' page 1241 in Indian literary traditions refers to an ap ...

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 ''pāda''s 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 4.32 million years. # ''Ganitapada'' (33 verses): covering mensuration (''kṣetra vyāvahāra''), arithmetic and geometric progressions,

/ 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''), ''kShaya-tithi''s, and a seven-day 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 abstract sphere that has an arbitrarily large radius and is concentric to Earth. All objects in the sky can be conceived as being projected upon the inner surface of the celestial sphe ...

, features of the

ecliptic The ecliptic or ecliptic plane is the orbital plane of the Earth around the Sun. From the perspective of an observer on Earth, the Sun's movement around the celestial sphere over the course of a year traces out a path along the ecliptic agains ...

celestial equator The celestial equator is the great circle of the imaginary celestial sphere on the same plane as the equator of Earth. This plane of reference bases the equatorial coordinate system. In other words, the celestial equator is an abstract proj ...

, 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 Keļallur Nilakantha Somayaji (14 June 1444 – 1544), also referred to as Keļallur Comatiri, was a major mathematician and astronomer of the Kerala school of astronomy and mathematics. One of his most influential works was the comprehens ...

in his ''Aryabhatiya Bhasya,'' (1465 CE).

Mathematics

Place value system and zero

The

place-value Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or decimal system). More generally, a positional system is a numeral system in which the ...

system, first seen in the 3rd-century Bakhshali manuscript, was clearly in place in his work. While he did not use a symbol for

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...

, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with

null Null may refer to: Science, technology, and mathematics Computing * Null (SQL) (or NULL), a special marker and keyword in SQL indicating that something has no value * Null character, the zero-valued ASCII character, also designated by , often use ...

coefficients In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves ...

. However, Aryabhata did not use the Brahmi numerals. Continuing the

Sanskrit Sanskrit (; attributively , ; nominally , , ) is a classical language belonging to the Indo-Aryan languages, Indo-Aryan branch of the Indo-European languages. It arose in South Asia after its predecessor languages had Trans-cultural diffusion ...

ic tradition from

Vedic times The Vedic period, or the Vedic age (), is the period in the late Bronze Age and early Iron Age of the history of India when the Vedic literature, including the Vedas (ca. 1300–900 BCE), was composed in the northern Indian subcontinent, betwe ...

, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a

mnemonic A mnemonic ( ) device, or memory device, is any learning technique that aids information retention or retrieval (remembering) in the human memory for better understanding. Mnemonics make use of elaborative encoding, retrieval cues, and image ...

form.

Approximation of

Aryabhata worked on the approximation for pi (), 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 for a circle whose diameter is 20000, the circumference will be 62832 i.e.,

\pi =  = 3.1416

, 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 Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...

). 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

(c. 820 CE) this approximation was mentioned in

's book on algebra.

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." Aryabhata discussed the concept of ''

sine In mathematics, sine and cosine are trigonometric functions of an angle. The sine and cosine of an acute angle are defined in the context of a right triangle: for the specified angle, its sine is the ratio of the length of the side that is opp ...

'' in his work by the name of '' ardha-jya'', which literally means "half-chord". For simplicity, people started calling it '' jya''. When Arabic writers translated his works from

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 chronology of Indian mathematicians spans from the Indus Valley civilisation and the Vedas to Modern India. Indian mathematicians have made a number of contributions to mathematics that have significantly influenced scientists and 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 In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of the ...

.) 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 The ''Shulva Sutras'' or ''Śulbasūtras'' (Sanskrit: शुल्बसूत्र; ': "string, cord, rope") are sutra texts belonging to the Śrauta ritual and containing geometry related to fire-altar construction. Purpose and origins The ...

, 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 ' (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 first-order diophantine equations in Indian mathematics, and initially the whole subject of algebra was called ''kuṭṭaka-gaṇita'' 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 =

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 ''ardha-rAtrikA'', midnight) are lost but can be partly reconstructed from the discussion in

's '' Khandakhadyaka''. In some texts, he seems to ascribe the apparent motions of the heavens to the

Earth's rotation Earth's rotation or Earth's spin is the rotation of planet Earth around its own axis, as well as changes in the orientation of the rotation axis in space. Earth rotates eastward, in prograde motion. As viewed from the northern polar star Po ...

. He may have believed that the planet's orbits as elliptical rather than circular.Hayashi (2008), ''Aryabhata I''

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 then-prevailing view, that the sky rotated.How Aryabhata got the earth's circumference right
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 In astronomy, the geocentric model (also known as geocentrism, often exemplified specifically by the Ptolemaic system) is a superseded description of the Universe with Earth at the center. Under most geocentric models, the Sun, Moon, stars, an ...

model of the solar system, in which the Sun and Moon are each carried by

epicycle In the Hipparchian, Ptolemaic, and Copernican systems of astronomy, the epicycle (, meaning "circle moving on another circle") was a geometric model used to explain the variations in speed and direction of the apparent motion of the Moon, S ...

s. 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 The Moon is Earth's only natural satellite. It is the fifth largest satellite in the Solar System and the largest and most massive relative to its parent planet, with a diameter about one-quarter that of Earth (comparable to the width of ...

, Mercury,

Venus Venus is the second planet from the Sun. It is sometimes called Earth's "sister" or "twin" planet as it is almost as large and has a similar composition. As an interior planet to Earth, Venus (like Mercury) appears in Earth's sky never f ...

, the Sun,

Mars Mars is the fourth planet from the Sun and the second-smallest planet in the Solar System, only being larger than Mercury. In the English language, Mars is named for the Roman god of war. Mars is a terrestrial planet with a thin at ...

Jupiter Jupiter is the fifth planet from the Sun and the largest in the Solar System. It is a gas giant with a mass more than two and a half times that of all the other planets in the Solar System combined, but slightly less than one-thousand ...

Saturn Saturn is the sixth planet from the Sun and the second-largest in the Solar System, after Jupiter. It is a gas giant with an average radius of about nine and a half times that of Earth. It has only one-eighth the average density of Earth; h ...

, 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 two-epicycle model reflects elements of pre-Ptolemaic

Greek astronomy Greek astronomy is astronomy written in the Greek language in classical antiquity. Greek astronomy is understood to include the Ancient Greek, Hellenistic, Greco-Roman, and Late Antiquity eras. It is not limited geographically to Greece or to ...

. 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 Heliocentrism (also known as the Heliocentric model) is the astronomical model in which the Earth and planets revolve around the Sun at the center of the universe. Historically, heliocentrism was opposed to geocentrism, which placed the Earth ...

model.

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the

and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by

Rahu Rāhu ( Sanskrit: राहु, 16px, ☊) is one of the nine major celestial bodies (navagraha) in Hindu texts and the king of meteors. It represents the ascension of the moon in its precessional orbit around the earth, also referred as th ...

and Ketu (identified as the pseudo-planetary 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 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

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 period of a celestial object (e.g., star, gas giant, planet, moon, asteroid) may refer to its sidereal rotation period, i.e. the time that the object takes to complete a single revolution around its axis of rotation relative to the ...

(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 A sidereal year (, ; ), also called a sidereal orbital period, is the time that Earth or another planetary body takes to orbit the Sun once with respect to the fixed stars. Hence, for Earth, it is also the time taken for the Sun to return to t ...

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

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 pre-Ptolemaic

Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...

, 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

translation during the

(c. 820 CE), was particularly influential. Some of his results are cited by

and in the 10th century

Al-Biruni Abu Rayhan Muhammad ibn Ahmad al-Biruni (973 – after 1050) commonly known as al-Biruni, was a Khwarazmian Iranian in scholar and polymath during the Islamic Golden Age. He has been called variously the "founder of Indology", "Father of Co ...

stated that Aryabhata's followers believed that the Earth rotated on its axis. His definitions of

('' jya''), cosine ('' kojya''), versine ('' utkrama-jya''), and inverse sine (''otkram jya'') influenced the birth of

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. ...

. He was also the first to specify sine and versine (1 − cos ''x'') tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. In fact, modern names "sine" and "cosine" are mistranscriptions of the words ''jya'' and ''kojya'' as introduced by Aryabhata. As mentioned, they were translated as ''jiba'' and ''kojiba'' in Arabic and then misunderstood by

Gerard of Cremona Gerard of Cremona (Latin: ''Gerardus Cremonensis''; c. 1114 – 1187) was an Italian translator of scientific books from Arabic into Latin. He worked in Toledo, Kingdom of Castile and obtained the Arabic books in the libraries at Toledo. Some of ...

while translating an Arabic geometry text to

Latin Latin (, or , ) is a classical language belonging to the Italic languages, Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through ...

. 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

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 The ''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. They were a collection of mathematic tables that describe different aspe ...

(12th century) and remained the most accurate

ephemeris In astronomy and celestial navigation, an ephemeris (pl. ephemerides; ) is a book with tables that gives the trajectory of naturally occurring astronomical objects as well as artificial satellites in the sky, i.e., the position (and possibly ...

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 A panchāngam ( sa, पञ्चाङ्गम्; ) is a Hindu calendar and almanac, which follows traditional units of Hindu timekeeping, and presents important dates and their calculations in a tabulated form. It is sometimes spelled ''Pa ...

(the

Hindu calendar The Hindu calendar, Panchanga () or Panjika is one of various lunisolar calendars that are traditionally used in the Indian subcontinent and Southeast Asia, with further regional variations for social and Hindu religious purposes. They adopt ...

). In the Islamic world, they formed the basis of the

Jalali calendar The Jalali calendar is a solar calendar, was compiled during the reign of Jalaluddin Malik-Shah I of Seljuk by the order of Nizam al-Mulk and the place of observation were the cities of Isfahan (the capital of the Seljuks), Rey, and Nishapur. ...

introduced in 1073 CE by a group of astronomers including

Omar Khayyam Ghiyāth al-Dīn Abū al-Fatḥ ʿUmar ibn Ibrāhīm Nīsābūrī (18 May 1048 – 4 December 1131), commonly known as Omar Khayyam ( fa, عمر خیّام), was a polymath, known for his contributions to mathematics, astronomy, philosophy, an ...

, versions of which (modified in 1925) are the national calendars in use in

Iran Iran, officially the Islamic Republic of Iran, and also called Persia, is a country located in Western Asia. It is bordered by Iraq and Turkey to the west, by Azerbaijan and Armenia to the northwest, by the Caspian Sea and Turkmeni ...

and

Afghanistan Afghanistan, officially the Islamic Emirate of Afghanistan,; prs, امارت اسلامی افغانستان is a landlocked country located at the crossroads of Central Asia and South Asia. Referred to as the Heart of Asia, it is borde ...

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 The Gregorian calendar is the calendar used in most parts of the world. It was introduced in October 1582 by Pope Gregory XIII as a modification of, and replacement for, the Julian calendar. The principal change was to space leap years d ...

Aryabhatta Knowledge University Aryabhatta Knowledge University (AKU Patna) is a collegiate public state university located in Mithapur, Patna, Bihar, India. It was named after the Indian astronomer Aryabhatta. Apart from a few notable exceptions, AKU has authority over tec ...

(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 Lunar craters are impact craters on Earth's Moon. The Moon's surface has many craters, all of which were formed by impacts. The International Astronomical Union currently recognizes 9,137 craters, of which 1,675 have been dated. History The w ...

Aryabhata are both named in his honour, the Aryabhata satellite also featured on the reverse of the Indian 2-rupee note. An Institute for conducting research in astronomy, astrophysics and atmospheric sciences is the

Aryabhatta Research Institute of Observational Sciences Aryabhatta Research Institute of Observational Sciences (ARIES) is a research institute in Nainital, Kumaon, India which specializes in astronomy, solar physics, astrophysics and atmospheric science. It is an autonomous body under the Dep ...

(ARIES) near Nainital, India. The inter-school Aryabhatta Maths Competition is also named after him, as is ''Bacillus aryabhata'', a species of bacteria discovered in the

stratosphere The stratosphere () is the second layer of the atmosphere of the Earth, located above the troposphere and below the mesosphere. The stratosphere is an atmospheric layer composed of stratified temperature layers, with the warm layers of air h ...

ISRO The Indian Space Research Organisation (ISRO; ) is the national space agency of India, headquartered in Bengaluru. It operates under the Department of Space (DOS) which is directly overseen by the Prime Minister of India, while the Chairman o ...

scientists in 2009.

References

Works cited

* * * Shukla, Kripa Shankar. ''Aryabhata: Indian Mathematician and Astronomer.'' New Delhi: Indian National Science Academy, 1976. *

External links

1930 English translation
of ''The Aryabhatiya'' in various formats at the Internet Archive. * *
PDF version

Hindustan Times ''Hindustan Times'' is an Indian English-language daily newspaper based in Delhi. It is the flagship publication of HT Media, an entity controlled by the KK Birla family, and is owned by Shobhana Bhartia. It was founded by Sunder Singh Ly ...

Storytelling Science column, November 2004
Surya Siddhanta translations
{{DEFAULTSORT:Aryabhata 476 births 550 deaths Indian cosmologists 5th-century Indian mathematicians 6th-century Indian mathematicians 5th-century Indian astronomers 6th-century Indian astronomers Scientists from Patna Scholars from Bihar 6th-century Indian writers