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Aryabhata

Aryabhata is the first one of the important mathematician-astronomers of the classical age of India.

Main Contributions

The Earth was taken to be spinning on its axis and the periods of the planets were given with respect to the sun in other words, it was heliocentric.

Aryabhata calculated the Sidereal day (the rotation of the earth against the fixed stars) as 23 hours 56 minutes and 4.1seconds; 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 30 seconds is only 3 minutes 20 seconds longer than the true value (over 365 days). The very notion of sidereal time was very advanced for the time, so this kind of accurate computation speaks of a very sophisticated understanding of the universe.

Aryabhata states that the Moon and planets shine by reflected sunlight and he believes that the orbits of the planets are ellipses.

He correctly explains the causes of eclipses of the Sun and the Moon.

Aryabhata was the first astronomer to make an attempt at measuring the Earth's circumference since Erastosthenes (circa 200 BC). Aryabhata accurately calculated the Earth's circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation remained the best result until the Industrial age

Introduction of Zero

Aryabhata stated that "Stanam stanam dasa gunam" or place to place in ten times in value, which may be the origin of the modern decimal based place value notation; his positional number system included a zero in his letter code for numerals (which allowed him to express numbers as words) in his mathematical astronomy. Pi as Irrational

Mensuration and Trigonometry

Brahmagupta

Brahmagupta is a person to be assigned the most important discovery in all of mathematics, finding of zero, that would be Brahmagupta. The Brahmasphutasiddhanta, written circa 628 AD, is the earliest known text to treat zero as a number in its own right. It goes well beyond that, however, stating rules for arithmetic on negative numbers and zero which are quite close to the modern understanding. The major divergence is that Brahmagupta attempted to define division by zero, which is left undefined in modern mathematics. His definition is not terribly useful; for instance, he states that 0/0 = 0.

Some of the important contributions made by Brahmagupta in astronomy are: methods for calculations of the motions and places of various planets, their rising and setting, conjunctions, and the calculation of eclipses of the sun and the moon. Brahmagupta criticized the Puranic view that the earth was flat or hollow like a bowl. Instead, he observed that the earth and heaven were round. However he wrongly believed that the earth did not move

Some Other Discovrys of Brahmagupta

Brahmagupta's formula Brahmagupta's identity Brahmagupta's theorem Brahmagupta interpolation formula Brahmagupta matrix Brahmagupta's trapezium Brahmagupta's problem Brahmagupta's polynomial

Bhaskara 2

In many ways, Bhaskara represents the peak of mathematical and astronomical knowledge in the 12th century. He reached an understanding of calculus, astronomy, the number systems, and solving equations, which were not to be achieved anywhere else in the world for several centuries or more. His main works were

A proof of the Pythagorean Theorem by calculating the same area in two different ways and then canceling out terms to get a2 + b2 = c2.

Proved that anything divided by zero is infinity in addition to establishing that infinity divided by anything remains infinity.

Positive and negative numbers

Trigonometry

Calculus#

Astronomy

Mean longitudes of the planets.

True longitudes of the planets.

The three problems of diurnal rotation.

Lunar eclipses.

Solar eclipses.

Latitudes of the planets.

Risings and settings.

The Moon's crescent.

Conjunctions of the planets with each other.

Conjunctions of the planets with the fixed stars.

The patas of the Sun and Moon

References

http://de.wikipedia.org/wiki/Aryabhata

http://de.wikipedia.org/wiki/Brahmagupta

http://de.wikipedia.org/wiki/Bhaskara

Folien

• Classical.ppt: Classical