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# 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**.

## Comments

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aryabhatta

Aryabhata

From Wikipedia, the free encyclopedia

For other uses, see Aryabhata (disambiguation).

tandii baba

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

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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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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, 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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