# Aryabhatta: The Indian Mathematician

## Life and Works of Aryabhata

**ARYABHATTA**, or, as written by the Arabs, **ARJABAHR**, a celebrated Hindu mathematician, and the earliest known author on Algebra, is now generally believed to have lived about the beginning of our era. Nothing, however, has yet appeared that can give us the slightest information as to the place of his birth, or the time when he lived; nor is there, as far as we know, any tradition or record extent from which we can collect any of the circumstances of his life; even his period is still a matter of dispute. We must, therefore, content ourselves with whatever notices we find of **Aryabhatta **and his system in the various writers on astronomy and other mathematical sciences whose authority is established and cannot be called into doubt.

Aryabhatta is the first writer on astronomy to whom the Hindus do not allow the honour of a divine inspiration. Writers on mathematical science distinctly state that he was the earliest uninspired and a merely human writer on astronomy. This is a notice which sufficiently proves his being an historical character.

## The chief doctrines which Aryabhatta (Aarya-Bhatt) professed were the following:

He affirmed the diurnal revolution of the earth on its axis; an assertion which is fully borne out by a quotation from one of his works, in a commentary on the "Brahmasphut'a-Siddhanta" of Brahmagupta by Prithudakaswami: "The Earth making a revolution produces a daily rising and setting of the stars and planets". At the same time he thought that this revolving of the earth was produced through the agency of a peculiar current of aerial fluid, or spiritus vector ("wind"), to which he assigned a distance of 150 yojanas (114 miles) from the surface of the earth. In opposition to the generally received opinion, he maintained that the moon, the primary planets, and the stars had no light of their own, and were only illumined by the sun; he consequently knew the true cause of solar and lunar eclipses.

Aryabhatta also ascribed to the epicycles, by which the motion of a planet is represented, a form varying from the circle and nearly elliptic. Moreover, he recognized a motion of the nodes and asides of all primary planets, as well as of the moon, and noticed the motion of the equinoctial and solstitial points, which he restricted, however, to an oscillation within the limits of twenty- four degrees, at the rate of one libration in seventy years. The length of AryabhatYa's sidereal year was 356 days 6 hours 12 minutes and 30 seconds. Aryabhatta stated the diameter of the earth at 1050 yojanas and its circumference at 3300 yojanas (25,080 miles). Hence it appears that he held the proportion of the diameter to the periphery of a circle to be seven to twenty-two, which is a nearer approximation than that of Brahmagupta and S'ridhara, who came after him.

The astronomical sects, of which **Arabhatta **is the reputed founder, were distinguished by the name of **Audayakas**, from Udaya, " rising;" implying that they fixed the beginning of the planetary motions on the meridian of Sri Lanka (Ceylon) at sun-rise, in opposition to the **Arddharatrikas**, who began the great astronomical cycle at midnight. Aryabhatta is the author of the " Aryasht'- as'ata" (eight hundred couplets in the Arya metre) and the "Das'agitika" (ten stanzas). The "**Laghwarya-Siddhanta**" is also ascribed to him: but, unfortunately, none of these works have yet been discovered; and we know them only through the numerous quotations from them, with which the works of subsequent writers abound. For an exposition of his numerical system and algebraic doctrine we refer to the article by another renowned scientist called **BHASKARA**.

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Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to believe that there were two different mathematicians called Aryabhata living at the same time. He therefore created a confusion of two different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.

We know the year of Aryabhata's birth since he tells us that he was twenty-three years of age when he wrote Aryabhatiya which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.

We do know that Aryabhata wrote Aryabhatiya in Kusumapura at the time when Pataliputra was the capital of the Gupta empire and a major centre of learning, but there have been numerous other places proposed by historians as his birthplace. Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although the author accepted that he lived most of his life in Kusumapura in the Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the Aryabhatiya.

We should note that Kusumapura became one of the two major mathematical centres of India, the other being Ujjain. Both are in the north but Kusumapura (assuming it to be close to Pataliputra) is on the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to reach it easily, and also allowed the mathematical and astronomical advances made by Aryabhata and his school to reach across India and also eventually into the Islamic world.

As to the texts written by Aryabhata only one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least three astronomical texts and wrote some free stanzas as well.

The surviving text is Aryabhata's masterpiece the Aryabhatiya which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to that time. Its mathematical section contains 33 verses giving 66 mathematical rules without proof. The Aryabhatiya contains an introduction of 10 verses, followed by a section on mathematics with, as we just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with the final section of 50 verses being on the sphere and eclipses.

There is a difficulty with this layout which is discussed in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 verse Introduction was written later than the other three sections. One reason for believing that the two parts were not intended as a whole is that the first section has a different meter to the remaining three sections. However, the problems do not stop there. We said that the first section had ten verses and indeed Aryabhata titles the section Set of ten giti stanzas. But it in fact contains eleven giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have been added and he identifies a small number of verses in the remaining sections which he argues have also been added by a member of Aryabhata's school at Kusumapura.

The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines. Let us examine some of these in a little more detail.

First we look at the system for representing numbers which Aryabhata invented and used in the Aryabhatiya. It consists of giving numerical values to the 33 consonants of the Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, .... In fact the system allows numbers up to 1018to be represented with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols and the place-value system. He writes in [3]:-

... it is extremely likely that Aryabhata knew the sign for zero and the numerals of the place value system. This supposition is based on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero or the place-value system; secondly, he carries out calculations on square and cubic roots which are impossible if the numbers in question are not written according to the place-value system and zero.

Next we look briefly at some algebra contained in the Aryabhatiya. This work is the first we are aware of which examines integer solutions to equations of the form by = ax + c and by = ax - c, where a, b, c are integers. The problem arose from studying the problem in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to solve problems of this type. The word kuttaka means "to pulverise" and the method consisted of breaking the problem down into new problems where the coefficients became smaller and smaller with each step. The method here is essentially the use of the Euclidean algorithm to find the highest common factor of a and b but is also related to continued fractions.

Aryabhata gave an accurate approximation for ?. He wrote in the Aryabhatiya the following:-

Add four to one hundred, multiply by eight and then add sixty-two thousand. the result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given.

This gives ? = 62832/20000 = 3.1416 which is a surprisingly accurate value. In fact ? = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, it is perhaps even more surprising that Aryabhata does not use his accurate value for ? but prefers to use ?10 = 3.1622 in practice. Aryabhata does not explain how he found this accurate value but, for example, Ahmad [5] considers this value as an approximation to half the perimeter of a regular polygon of 256 sides inscribed in the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of ? by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of ? is a very close approximation to the modern value and the most accurate among those of the ancients. There are reasons to believe that Aryabhata devised a particular method for finding this value. It is shown with sufficient grounds that Aryabhata himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of ? is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered this value independently and also realised that ? is an irrational number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating ?. Thus the credit of discovering this exact value of ? may be ascribed to the celebrated mathematician, Aryabhata I.

We now look at the trigonometry contained in Aryabhata's treatise. He gave a table of sines calculating the approximate values at intervals of 90°/24 = 3° 45'. In order to do this he used a formula for sin(n+1)x - sin nx in terms of sin nx and sin (n-1)x. He also introduced the versine (versin = 1 - cosine) into trigonometry.

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Name –Aryabhata I

Born -476 , Counrty- Kusumapura (now Patna), India

Aryabhatta (476-550 A.D.) was born in Patliputra in Magadha, modern Patna in Bihar. Many are of the view that he was born in the south of India especially Kerala and lived in Magadha at the time of the Gupta rulers; time which is known as the golden age of India. There is no evidence that he was born outside Patliputra and traveled to Magadha, the centre of education and learning for his studies where he even set up a coaching centre. His first name "Arya" is hardly a south Indian name while "Bhatt" (or Bhatta) is a typical north Indian name even found today specially among the "Bania" (or trader) community.

FAMOUS FOR - Whatever this origin, it cannot be argued that he lived in Patliputra where he wrote his famous treatise the "Aryabhatta-siddhanta" but more famously the "Aryabhatiya", the only work to have survived. It contains mathematical and astronomical theories that have been revealed to be quite accurate in modern mathematics. For instance he wrote that if 4 is added to 100 and then multiplied by 8 then added to 62,000 then divided by 20,000 the answer will be equal to the circumference of a circle of diameter twenty thousand. This calculates to 3.1416 close to the actual value Pi (3.14159). But his greatest contribution has to be zero. His other works include algebra, arithmetic, trigonometry, quadratic equations and the sine table.

Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost. His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.[3]

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Ab? Rayh?n al-B?r?n?.[3]

Education -However, it is fairly certain that at some point, he went to Kusumapura for higher studies, and that he lived here for some time.[2] Bh?skara I (AD 629) identifies Kusumapura as Pataliputra (modern Patna). He lived there in the dying years of the Gupta empire, the time which is known as the golden age of India, when it was already under Hun attack in the Northeast, during the reign of Buddhagupta and some of the smaller kings before Vishnugupta.

FAMOUS FOR ----

Aryabhatta’s Representation of the Numerals and Zero

Aryabhatta developed a representation of numbers using the letters of the Indian alphabet as his symbols. The ways in which he carries out his calculations suggest that he knew and understood the place value system of representing numbers, and that he also knew about zero.

He also calculated the value of pi with an accuracy unknown before him. He estimated it to be 3.1416, which is actually correct to four decimal places. However, he does not explain how he arrived at this value and, surprisingly enough, uses the value of the square root of ten for his calculations requiring pi.

He also gave formulae for the sums of the first n integers. He did the same thing for the squares and cubes of the integers.

Aryabhatta Constructed Sine Tables

Aryabhatta was able to construct sine tables, giving values for the sine function at intervals of 3 45’ (three degrees and forty five minutes). He studied trigonometry on plane surfaces as well as spherical surfaces. He estimated the circumference and the diameter of the earth with great accuracy.

His text gives many mathematical rules without proofs. He gives formulae for the areas of circles and triangles, and also for the volume of spheres and pyramids. His formulae for the volume of spheres and pyramids are believed to be incorrect, though some argue that this is due to errors in translation

Aryabhatta’s Insights in Astronomy

Aryabhatta studied the orbits of the planets around the sun and said that they were elliptical. He worked with the equations that predict the positions of the planets. These studies enabled him to predict eclipses.

He suggested that it is the earth that rotates around its axis, and not the stars that move in the sky, as was believed until then. He also said that the moon shines due to reflected light

Remarks … Aryabhatta is the first writer on astronomy to whom the Hindus do not allow the honour of a divine inspiration. Writers on mathematical science distinctly state that he was the earliest uninspired and a merely human writer on astronomy. This is a notice which sufficiently proves his being an historical character.

ARYABHATTA, or, as written by the Arabs, ARJABAHR, a celebrated Hindu mathematician, and the earliest known author on Algebra, is now generally believed to have lived about the beginning of our era. Nothing, however, has yet appeared that can give us the slightest information as to the place of his birth, or the time when he lived; nor is there, as far as we know, any tradition or record extent from which we can collect any of the circumstances of his life; even his period is still a matter of dispute. We must, therefore, content ourselves with whatever notices we find of Aryabhatta and his system in the various writers on astronomy and other mathematical sciences whose authority is established and cannot be called into doubt.

Aryabhatta is the first writer on astronomy to whom the Hindus do not allow the honour of a divine inspiration. Writers on mathematical science distinctly state that he was the earliest uninspired and a merely human writer on astronomy. This is a notice which sufficiently proves his being an historical character.

Statue of Aryabhata The chief doctrines which Aryabhatta (Aarya-Bhatt) professed were the following:

He affirmed the diurnal revolution of the earth on its axis; an assertion which is fully borne out by a quotation from one of his works, in a commentary on the "Brahmasphut'a-Siddhanta" of Brahmagupta by Prithudakaswami: "The Earth making a revolution produces a daily rising and setting of the stars and planets". At the same time he thought that this revolving of the earth was produced through the agency of a peculiar current of aerial fluid, or spiritus vector ("wind"), to which he assigned a distance of 150 yojanas (114 miles) from the surface of the earth. In opposition to the generally received opinion, he maintained that the moon, the primary planets, and the stars had no light of their own, and were only illumined by the sun; he consequently knew the true cause of solar and lunar eclipses.

Aryabhatta also ascribed to the epicycles, by which the motion of a planet is represented, a form varying from the circle and nearly elliptic. Moreover, he recognized a motion of the nodes and asides of all primary planets, as well as of the moon, and noticed the motion of the equinoctial and solstitial points, which he restricted, however, to an oscillation within the limits of twenty- four degrees, at the rate of one libration in seventy years. The length of AryabhatYa's sidereal year was 356 days 6 hours 12 minutes and 30 seconds. Aryabhatta stated the diameter of the earth at 1050 yojanas and its circumference at 3300 yojanas (25,080 miles). Hence it appears that he held the proportion of the diameter to the periphery of a circle to be seven to twenty-two, which is a nearer approximation than that of Brahmagupta and S'ridhara, who came after him.

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INDIAN MATHEMATICS - BRAHMAGUPTA

Brahmagupta

Brahmagupta (598–668 AD)

The great 7th Century Indian mathematician and astronomer Brahmagupta wrote some important works on both mathematics and astronomy. He was from the state of Rajasthan of northwest India (he is often referred to as Bhillamalacarya, the teacher from Bhillamala), and later became the head of the astronomical observatory at Ujjain in central India. Most of his works are composed in elliptic verse, a common practice in Indian mathematics at the time, and consequently have something of a poetic ring to them.

It seems likely that Brahmagupta's works, especially his most famous text, the “Brahmasphutasiddhanta”, were brought by the 8th Century Abbasid caliph Al-Mansur to his newly founded centre of learning at Baghdad on the banks of the Tigris, providing an important link between Indian mathematics and astronomy and the nascent upsurge in science and mathematics in the Islamic world.

In his work on arithmetic, Brahmagupta explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots. He also gave rules for dealing with five types of combinations of fractions. He gave the sum of the squares of the first n natural numbers as n(n + 1)(2n + 1)? 6 and the sum of the cubes of the first n natural numbers as (n(n + 1)?2)².

Brahmagupta’s rules for dealing with zero and negative numbers

Brahmagupta’s rules for dealing with zero and negative numbers

Brahmagupta’s genius, though, came in his treatment of the concept of (then relatively new) the number zero. Although often also attributed to the 7th Century Indian mathematician Bhaskara I, his “Brahmasphutasiddhanta” is probably the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit as was done by the Babylonians, or as a symbol for a lack of quantity as was done by the Greeks and Romans.

Brahmagupta established the basic mathematical rules for dealing with zero (1 + 0 = 1; 1 - 0 = 1; and 1 x 0 = 0), although his understanding of division by zero was incomplete (he thought that 1 ÷ 0 = 0). Almost 500 years later, in the 12th Century, another Indian mathematician, Bhaskara II, showed that the answer should be infinity, not zero (on the grounds that 1 can be divided into an infinite number of pieces of size zero), an answer that was considered correct for centuries. However, this logic does not explain why 2 ÷ 0, 7 ÷ 0, etc, should also be zero - the modern view is that a number divided by zero is actually "undefined" (i.e. it doesn't make sense).

Brahmagupta’s view of numbers as abstract entities, rather than just for counting and measuring, allowed him to make yet another huge conceptual leap which would have profound consequence for future mathematics. Previously, the sum 3 - 4, for example, was considered to be either meaningless or, at best, just zero. Brahmagupta, however, realized that there could be such a thing as a negative number, which he referred to as “debt” as a opposed to “property”. He expounded on the rules for dealing with negative numbers (e.g. a negative times a negative is a positive, a negative times a positive is a negative, etc).

Furthermore, he pointed out, quadratic equations (of the type x2 + 2 = 11, for example) could in theory have two possible solutions, one of which could be negative, because 32 = 9 and -32 = 9. In addition to his work on solutions to general linear equations and quadratic equations, Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables), and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657.

Brahmagupta’s Theorem on cyclic quadrilaterals

Brahmagupta’s Theorem on cyclic quadrilaterals

Brahmagupta even attempted to write down these rather abstract concepts, using the initials of the names of colours to represent unknowns in his equations, one of the earliest intimations of what we now know as algebra.

Brahmagupta dedicated a substantial portion of his work to geometry and trigonometry. He established ?10 (3.162277) as a good practical approximation for ? (3.141593), and gave a formula, now known as Brahmagupta's Formula, for the area of a cyclic quadrilateral, as well as a celebrated theorem on the diagonals of a cyclic quadrilateral, usually referred to as Brahmagupta's Theorem.

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Aryabhatta came to this world on the 476 A.D at Patliputra in Magadha which is known as the modern Patna in Bihar. Some people were saying that he was born in the South of India mostly Kerala. But it cannot be disproved that he was not born in Patlipura and then travelled to Magadha where he was educated and established a coaching centre. His first name is “Arya” which is a South Indian name and “Bhatt” or “Bhatta” a normal north Indian name which could be seen among the trader people in India.

No matter where he could be originated from, people cannot dispute that he resided in Patliputra because he wrote one of his popular “Aryabhatta-siddhanta” but “Aryabhatiya” was much more popular than the former. This is the only work that Aryabhatta do for his survival. His writing consists of mathematical theory and astronomical theory which was viewed to be perfect in modern mathematics. For example, it was written in his theory that when you add 4 to 100 and multiply the result with 8, then add the answer to 62,000 and divide it by 20000, the result will be the same thing as the circumference with diameter twenty thousand. The calculation of 3.1416 is nearly the same with the true value of Pi which is 3.14159. Aryabhatta’s strongest contribution was zero. Another aspect of mathematics that he worked upon is arithemetic, algebra, quadratic equations, trigonometry and sine table.

Aryabhatta was aware that the earth rotates on its axis. The earth rotates round the sun and the moon moves round the earth. He discovered the 9 planets position and related them to their rotation round the sun. Aryabhatta said the light received from planets and the moon is gotten from sun. He also made mention on the eclipse of the sun, moon, day and night, earth contours and the 365 days of the year as the exact length of the year. Aryabhatta also revealed that the earth circumference is 24835 miles when compared to the modern day calculation which is 24900 miles.

Aryabhatta have unusually great intelligence and well skilled in the sense that all his theories has became wonders to some mathematicians of the present age. The Greeks and the Arabs developed some of his works to suit their present demands. Aryabhatta was the first inventor of the earth sphericity and also discovered that earth rotates round the sun. He was the one that created the formula (a + b)2 = a2 + b2 + 2ab. He also created a solution formula of solving the following equations:

1 + 2 + 3 + 4 + 5 + ……………… + n = n (n + 1)/2

12 + 22 + 32 + 42 + 52 + ……………….. + n2 = n (n + 1) (2n + 1)/6

13 + 23 + 33 + 43 + 53 + ………………….. n3 = (n (n + 1)/2)2

14 + 24 + 34 + 44 + 54 + ………………….. + n4 = (n (n + 1) (2n + 1) (3n2 + 3n – 1))/30

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aryabhatta

Aryabhata

From Wikipedia, the free encyclopedia

For other uses, see Aryabhata (disambiguation).

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Statue of Aryabhata on the grounds of IUCAA, Pune. As there is no known information regarding his appearance, any image of Aryabhata originates from an artist's conception.

Born 476

Died 550

Era Gupta era

Region India

Main interests Maths, Astronomy

Major works ?ryabha??ya, Arya-siddhanta

Aryabhata (IAST: ?ryabha?a, Sanskrit: ??????) (476–550 CE) was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the ?ryabha??ya (499 CE, when he was 23 years old) and the Arya-siddhanta.

Contents [hide]

1 Biography

1.1 Name

1.2 Time and Place of birth

1.3 Education

1.4 Other hypotheses

2 Works

2.1 Aryabhatiya

3 Mathematics

3.1 Place value system and zero

3.2 Approximation of ?

3.3 Trigonometry

3.4 Indeterminate equations

3.5 Algebra

4 Astronomy

4.1 Motions of the solar system

4.2 Eclipses

4.3 Sidereal periods

4.4 Heliocentrism

5 Legacy

6 See also

7 References

7.1 Other references

8 External links

[edit]Biography

[edit]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,[1] including Brahmagupta's references to him "in more than a hundred places by name".[2] Furthermore, in most instances "Aryabhatta" does not fit the metre either.[1]

[edit]Time and Place of birth

Aryabhata mentions in the Aryabhatiya that it was composed 3,630 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476 CE.[citation needed]

Aryabhata provides no information about his place of birth. The only information comes from Bh?skara I, who describes Aryabhata as ??mak?ya, "one belonging to the a?maka country." The Asmaka were one of the 16 of Ancient Indian Mahajanapadas, and the only one situated south of the Vindhyas. Since Aryabhata lived in and around 480, he resided in The Gupta's, near the end of the Golden Age

It is widely attested that during the mid-first millennium BCE, a branch of the A?maka people settled in the region between the Narmada and Godavari rivers in central India, and it is possible Aryabhata was born there.[1][3] 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 Kerala and lived in Magadha at the time of the Gupta rulers.

[edit]Education

It is fairly certain that, at some point, he went to Kusumapura for advanced studies and that he lived there for some time.[4] Both Hindu and Buddhist tradition, as well as Bh?skara I (CE 629), identify Kusumapura as P??aliputra, modern Patna.[1] A verse mentions that Aryabhata was the head of an institution (kulapati) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well.[1] Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.[5]

[edit]Other hypotheses

Some archeological evidences suggests that Aryabhata could have originated from the present day Kodungallur in Tamilakam, modern Kerala. For instance, one hypothesis was that a?maka (Sanskrit for "stone") may be the region in Kerala that is now known as Ko?u??all?r, based on the belief that it was earlier known as Ko?um-Kal-l-?r ("city of hard stones"); however, old records show that the city was actually Ko?um-kol-?r ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Tamilakam were used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Tamilakam, and the Aryasiddhanta was completely unknown in Kerala. [6]

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.[7]

[edit]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, sums-of-power series, and a table of sines.

The Arya-siddhanta, a lot work on astronomical computations, is known through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta and Bhaskara I. This work appears to be based on the older Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in Aryabhatiya. It also contained a description of several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (dhanur-yantra / chakra-yantra), a cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, and water clocks of at least two types, bow-shaped and cylindrical.[3]

A third text, which may have survived in the Arabic translation, is Al ntf or Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known. Probably dating from the 9th century, it is mentioned by the Persian scholar and chronicler of India, Ab? Rayh?n al-B?r?n?.[3]

[edit]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 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 p?das 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 (k?etra vy?vah?ra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations

Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.

Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465 CE).

[edit]Mathematics

[edit]Place value system and zero

The place-value 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, the Fren

aryabhatta

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