Aryabhatta
Born
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476 CE
prob. Ashmaka |
Died
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550 CE
|
Era
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Gupta era
|
Region
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India
|
Main interests
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Mathematics, Astronomy
|
Notable ideas
|
Explanation of Lunar eclipse and Solar eclipse, Rotation of earth on its
axis, Reflection of light by moon, Sinusoidal functions, Solution of single
variable quadratic equation,
Value of π correct to 4
decimal places, Circumference of Earth to 99.8% accuracy, Calculation of the
length of Sidereal year
|
Major works
|
Āryabhaṭīya, Arya-siddhanta
|
Biography
Name
While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus, including Brahmagupta's references to him "in more than a hundred places by name" Furthermore, in most instances "Aryabhatta" does not fit the metre either.Time and place of birth
Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476.Aryabhata provides no information about his place of birth. The only information comes from Bhāskara I, who describes Aryabhata as āśmakīya, "one belonging to the aśmaka country." During the Buddha's time, a branch of the Aśmaka people settled in the region between the Narmada and Godavari rivers in central India; Aryabhata is believed to have been born there.
Education
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time. Both Hindu and Buddhist tradition, as well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna. A verse mentions that Aryabhata was the head of an institution (kulapa) at Kusumapura, and, because the university of Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well. Aryabhata is also reputed to have set up an observatory at the Sun temple in Taregana, Bihar.Works
Aryabhata is the author of several treatises on mathematics and astronomy, some of which are lost.His major work, Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry, and spherical trigonometry. It also contains continued fractions, quadratic equations, 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. 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, Abū Rayhān al-Bīrūnī.
Aryabhatiya
Main article: 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). Mathematics
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 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 coefficients.However, Aryabhata did not use the Brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form.
Approximation of π
Aryabhata worked on the approximation for pi , and may have come to the conclusion that is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."
This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five significant figures.
It is speculated that Aryabhata used the word āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert. After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi's book on algebra.
Trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as
tribhujasya phalashariram
samadalakoti bhujardhasamvargah
that
translates to: "for a triangle, the result of a perpendicular with the
half-side is the area." Aryabhata discussed the concept of sine 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 Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb. Later writers substituted it with jaib, meaning "pocket" or "fold (in a garment)". (In Arabic, jiba is a meaningless word.) Later in the 12th century, when Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic jaib with its Latin counterpart, sinus, which means "cove" or "bay"; thence comes the English sine. Alphabetic code has been used by him to define a set of increments. If we use Aryabhata's table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.
Indeterminate equations
A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as diophantine equations. This is an example from Bhāskara's commentary on Aryabhatiya:
Find the number which gives
5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and
1 as the remainder when divided by 7
That is,
find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85.
In general, diophantine equations, such as this, can be notoriously difficult.
They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient
parts might date to 800 BCE. Aryabhata's method of solving such problems is
called the kuṭṭaka (कुट्टक) method. Kuttaka means "pulverizing" or "breaking
into small pieces", and the method involves a recursive algorithm for
writing the original factors in smaller numbers. Today this algorithm,
elaborated by Bhaskara in 621 CE, is the standard method for solving
first-order diophantine equations and is often referred to as the Aryabhata algorithm.The diophantine
equations are of interest in cryptology, and the RSA Conference, 2006, focused on
the kuttaka method and earlier work in the Sulbasutras.
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