The works of Aryabhata dealt with mainly mathematics and astronomy. He also worked on the approximation for pi.
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 Bhaskara I (CE 629), identify Kusumapura as Pa?aliputra, modern Patna. 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. 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 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 thechhatra-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, Abu Rayhan al-Biruni.
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 explains 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 p
Aryabhata worked on the approximation for pi ( ), and may have come to the conclusion that is irrational. In the second part of theAryabhatiyam (ga?itapada 10), he writes:
caturadhikam satama??agu?am dva?a??istatha sahasra?am
ayutadvayavi?kambhasyasanno v?ttapari?aha?.
"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 asanna (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.
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 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.
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 in other parts of the world, that the sky rotated. 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 hisgola chapter
In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] 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 Paitamahasiddhanta (c. CE 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) and a larger sighra (fast). 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."
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. Another element in Aryabhata's model, the sighrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying heliocentric model.
Eclipses
Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by pseudo-planetary nodes Rahu and Ketu, 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 lunar eclipse 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 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 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 sigra 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 heliocentric 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, 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.
0 comments: