Aryabhata

Āryabhaṭa (Devanāgarī: आर्यभट) (AD 476 – 550) is the first of the great mathematician-astronomers of the classical age of Indian mathematics and Indian astronomy. He was born at Muziris (the modern day Kodungallour village) near Thrissur, Kerala. K. Chandra Hari, a senior geoscientist at the Institute of Reservoir Studies of Oil and Natural Gas Commission, Ahmedabad has refuted this popular opinion and claims that based on his interpretation of Aryabhatta's system of measurements and writings, it is highly likely that he belonged to the modern Ponnani-Chamravattom area (latitude 10N51 and longitude 75E45) in Kerala in the 6th Century AD.The HinduAvailable evidence suggest that he went to Kusumapura for higher studies. He lived in Kusumapura, which his commentator Bhāskara I (AD 629) identifies as Pataliputra (modern Patna). Aryabhata was the first in the line of brilliant mathematician-astronomers of classical Indian mathematics, whose major work was the Aryabhatiya and the Aryabhatta-siddhanta. 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, ca. 600) and by Nilakantha Somayaji in his Aryabhatiya Bhasya, (1465). The number place-value system, first seen in the 3rd century Bakhshali Manuscript was clearly in place in his work.^[1] He may have been the first mathematician to use letters of the alphabet to denote unknown quantities.^[2]

Aryabhata's system of astronomy was called the audAyaka system (days are reckoned from uday, dawn at lanka, equator). Some of his later writings on astronomy, which apparently proposed a second model (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.

The Aryabhatiya

Pi as Irrational

Aryabhata worked on the approximation for Pi, and may have realized that $\pi$ is irrational. In the second part of the Aryabhatiyam. In other words, $\pi \approx 62832/20000=3.1416$ , correct to five digits.

Mensuration and Trigonometry

In Ganitapada 6, Aryabhata gives the area of triangle as

tribhujasya phalashariram samadalakoti bhujardhasamvargah (for a triangle, the result of a perpendicular with the half-side is the area.)^[3]

Motions of the Solar System

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle.^[4] The positions and periods of the planets were calculated relative to uniformly moving points, which in the case of Mercury and Venus, move around the Earth at the same speed as the mean Sun and in the case of Mars, Jupiter, and Saturn 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.^[5] 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.^[6]

He states that the Moon and planets shine by reflected sunlight. He also correctly explains eclipses of the Sun and the Moon, and presents methods for their calculation and prediction.

Another statement, referring to Lanka , describes the movement of the stars as a relative motion caused by the rotation of the earth:

Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in lankA (i.e. on the equator) as moving exactly towards the West. [achalAni bhAni samapashchimagAni - golapAda.9]

However, in the next verse he describes the motion of the stars and planets as real: “The cause of their rising and setting is due to the fact the circle of the asterisms together with the planets driven by the provector wind, constantly moves westwards at Lanka”.

Lanka here is a reference point on the equator, which was taken as the equivalent to the reference meridian for astronomical calculations.

Aryabhata's computation of Earth's circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation improved on the computation by the Alexandrinan mathematician Erastosthenes (c.200 BC), whose exact computation is not known in modern units.

Sidereal periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotation of the earth referenced 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 30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal time was known in most other astronomical systems of the time, but this computation was likely the most accurate in the period.

Heliocentrism

Āryabhata claims that the Earth turns on its own axis and some elements of his planetary epicyclic models rotate at the same speed as the motion of the planet around the Sun. This has suggested to some interpreters that Āryabhata's calculations were based on an underlying heliocentric model in which the planets orbit the Sun.^[7]^[8] A detailed rebuttal to this heliocentric interpretation is in a review which describes van der Waerden's book as "show[ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Āryabhata's description,"^[9] although some concede that Āryabhata's system stems from an earlier heliocentric model of which he was unaware.^[10] It has even been claimed that he considered the planet's paths to be elliptical, although no primary evidence for this has been cited.^[11] Though Aristarchus of Samos (3rd century BC) and sometimes Heraclides of Pontus (4th century BC) are usually credited with knowing the heliocentric theory, the version of Greek astronomy known in ancient India, Paulisa Siddhanta (possibly by a Paul of Alexandria) makes no reference to a Heliocentric theory. The Āryabhatīya influenced many early Arabic astronomical tables (zijes) that were translated into Latin in the 12th century, thereby influencing European astronomy. The 10th century Arabic scholar Al-Biruni states that Āryabhata's followers believed the Earth to rotates on its axis. Then he casually adds that this notion does not create any mathematical difficulties.^{[citation needed]}

Diophantine Equations

A problem of great interest to Indian mathematicians since very ancient times concerned diophantine equations. These involve integer solutions to equations such as ax + b = cy. Here is an example from Bhaskara'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.

i.e. find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text Sulba Sutras, the more ancient parts of which may date back to 800 BCE. Aryabhata's method of solving such problems, called the kuttaka (कूटाक) method. Kuttaka means pulverizing, that is breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm, as elaborated by Bhaskara in AD 621, is the standard method for solving first order Diophantine equations, and it is often referred to as the Aryabhata algorithm. See details of the Kuttaka method in this [2]. RSA Conference 2006, Indocrypt 2005, which had a session on Vedic mathematics.

The lunar crater Aryabhata is named in his honour.

Continued Relevance

Aryabhata's astronomical calculation methods have been in continuous use for the practical purposes of fixing the Panchanga, or Hindu calendar.