He states that the moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by rahu 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.3848) 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 lunar eclipse of to be short by 41 seconds, whereas his charts (by tobias mayer, 1752) were long. 8 Sidereal periods Considered in modern English units of time, aryabhata calculated the sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and.1 seconds; 30 the modern value is 23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days) 31 is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).
Aryabhatta, biography, adolescence mathematics
Turns due west at the equator, constantly pushed by the cosmic wind. Aryabhata described a geocentric model of the solar system, in which the sun and moon are each carried by epicycles. They in turn revolve around the earth. In this model, which is also found in the paitāmahasiddhānta (c. Ce 425 the motions of the planets are each governed by two epicycles, a smaller manda net (slow) and a larger śīghra (fast). 27 The order of the planets in terms of distance from earth is taken as: the moon, mercury, venus, the sun, mars, jupiter, saturn, and the asterisms." 8 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. 28 Another element in Aryabhata's model, the śīghrocca, the basic planetary period in relation to the sun, is seen by some historians as a sign of an underlying heliocentric model. 29 Eclipses Solar and lunar eclipses were scientifically explained by Aryabhata.
This algorithm became the standard method for solving first-order diophantine equations in Indian mathematics, and initially the whole subject of algebra was called kuṭaka-gaṇita or simply kuṭaka. 20 Algebra In Aryabhatiya, aryabhata provided elegant results for the summation of series of squares and cubes: 21 1222n2n(n1 2n1)6displaystyle 1222cdots n2n(n1 2n1) over n3(12n)2displaystyle 1323cdots n3(12cdots n)2 (see squared triangular number ) Astronomy Aryabhata's system of astronomy was called the audayaka system, in which. 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 Brahmagupta 's Khandakhadyaka. In some texts, he seems to ascribe the apparent motions of the heavens to the earth's rotation. He may have believed that the planet's orbits as elliptical rather than circular. 22 23 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 supermarket caused by the rotation of the earth, contrary to the then-prevailing view, that the sky. 24 This is indicated in the first chapter of the Aryabhatiya, where he gives the number of rotations of the earth in a yuga, 25 and made more explicit in his gola chapter: 26 In the same way that someone in a boat going forward. The cause of rising and setting is that the sphere of the stars together with the planets apparently?
19 Indeterminate equations A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to diophantine equations that have the form ax. (This problem was also studied in ancient Chinese mathematics, and its solution is usually referred to as the Chinese remainder theorem.) This is an example from Bhāskara 's commentary on Aryabhatiya: Find the number which gives 5 as the remainder when divided by 8,. It turns out that the smallest value for n. 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, elaborated by Bhaskara in 621 ce, is called the kuṭaka method. Kuṭaka means "pulverizing" or "breaking into small pieces and the method involves a recursive algorithm for writing the original factors in smaller numbers.
Biography, aryabhatta, essay - 1476 Words
In the writers second part of the Aryabhatiyam ( gaṇitapāda 10 he writes: caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām ayutadvayaviṣkambhasyāsanno vṛttapariṇāha. "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." 16 This implies that the ratio of the circumference to the diameter is (4 100) /20000.1416, 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 (π) was proved in Europe only in 1761 by lambert. 17 After Aryabhatiya was translated into Arabic (c. 820 CE) this approximation was mentioned in Al-Khwarizmi 's book on algebra.
8 Trigonometry In Ganitapada 6, Aryabhata gives the area of a triangle as tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvarga that translates to: "for a triangle, the result of a perpendicular with the half-side is the area." 18 Aryabhata discussed the concept of sine in his work. 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. 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.
Ganitapada (33 verses covering mensuration ( kṣetra vyāvahāra arithmetic and geometric progressions, gnomon / shadows ( shanku - chhaya simple, quadratic, simultaneous, and indeterminate equations ( kuṭaka ). 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, 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,. 600 CE) and by nilakantha somayaji in his Aryabhatiya bhasya, (1465 CE). 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. 14 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. 15 Approximation of π aryabhata worked on the approximation for pi (π and may have come to the conclusion that π is irrational.
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Aryabhata himself may not have given it a name. His disciple Bhaskara i calls it Ashmakatantra (or the treatise from proposal the Ashmaka). It is dates 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 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. There is also a table of sines ( jya given in a single verse. The duration of the planetary revolutions during a mahayuga is given.32 million years.
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 ( 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 clocks of at least two. 8 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. 8 Aryabhatiya main article: Aryabhatiya direct details of Aryabhata's work are known only from the Aryabhatiya. The name "Aryabhatiya" is due to later commentators.
he went to kusumapura for advanced studies and lived there for some time. 12 Both Hindu and Buddhist tradition, as well as Bhāskara i (CE 629 identify kusumapura as Pāṭaliputra, modern Patna. 7 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. 7 Aryabhata is also reputed to have set up an observatory at the sun temple in Taregana, bihar. 13 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.
Kali yuga, when he was 23 years old. This corresponds to 499 ce, and implies that he was born in 476. 5 Aryabhata called himself a native of Kusumapura or Pataliputra (present day patna, bihar ). 1 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 rivers in central India. 7 8 It roles has been claimed that the aśmaka (Sanskrit for "stone where Aryabhata originated may be the present day kodungallur which was the historical capital city of Thiruvanchikkulam of ancient Kerala. 9 This is based on the belief that Koṭuallūr 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. Chandra hari has argued for the kerala hypothesis on the basis of astronomical evidence.
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For other uses, see, aryabhata (disambiguation). Aryabhata iast : Āryabhaṭa ) or, aryabhata i 2 3 (476550. Ce ) 4 5 was the first of the major mathematician - astronomers from the classical age of, indian mathematics and, indian astronomy. His works include the Āryabhaṭīya (499 ce, when he was 23 years old) 6 and the. Contents, 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, 7 including, brahmagupta 's references to him. 1, furthermore, in most instances "Aryabhatta" would not fit the meter either. 7, time and place of birth. Aryabhata mentions in the, aryabhatiya that it was composed 3,600 years into the.