Aryabhata

For the Indian satellite, see Aryabhata (satellite).

Aryabhata (आर्यभट) Āryabhaṭa) (476 – 550) is the first of the great mathematician- of the classical age of India. He lived in Kusumapura, which his commentator Bhāskara I (629 AD) identifies with Pataliputra (modern Patna).

Contents

1 Main Contributions
2 Pi as Irrational
3 Mensuration and Trigonometry
4 Motion of the Earth
5 Diophantine Equations
6 Continued relevance

6.1 Confusion of identity

7 References
8 External links

Main Contributions

Aryabhata was the first in the line of brilliant mathematician-astronomers of classical India, whose major work was the Aryabhatiya and the Aryabhatta-sidhanta. Not only did he help revolutionise mathematics, but also provided a heliocentric model of the universe much more advanced that the geocentric model of Ptolemy that had taken hold in Europe and 1000 years prior to Copernicus's heliocentric theory.

Among his main achievements was the development of notation in mathematics, which was and is remains hugely significant. His heliocentric model of the earth and planets was also very advanced, and argued for elliptical orbits of the planets around the sun.

Pi as Irrational

Aryabhata worked on the approximation for Pi, and may have realized that [\pi] is irrational. In the second part of the Aryabhatiya (gaṇitapāda 10), he writes:

''chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vr^ttapariṇahaḥ.''
"Add four to 100, multiply by eight and then add sixty-two thousand. By this rule is the circumference of a circle of diameter 20,000 approximately given"

In other words, [\pi \approx 62832/20000 = 3.1416], correct to four rounded-off decimal places. The commentator Nilakantha Somayaji, (Kerala, 15th c.) has argued that the word āsanna (approaching), appearing just before the last word, here means not only that this is an approximation, but that the value is incommensurable (or irrational). If this is correct, it is quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 (Lambert).

Mensuration and Trigonometry

In Ganitapada 6, Aryabhata gives the area of triangle as

tribhujasya falashariram samadalakoti bhujardhasamvargah (for a triangle, the result of a perpendicular with the half-side is the area.)

Aryabhata, in his work Aryabhata-Siddhanta, first defined the sine as the modern relationship between half an angle and half a chord. He also defined the cosine, versine, and inverse sine. He used the words jya for sine, kojya for cosine, ukramajya for versine, and otkram jya for inverse sine.

Aryabhata's tables for the sines (from which the rest can be computed), is presented in a single rhyming stanza, with each syllable standing for increments at intervals of 225 minutes of arc or 3 degrees 45'. Using a compact alphabetic code called varga/avarga, he defines the sines for a circle of circumference 21600 (radius [\approx] 3438). He uses the alphabetic code to define a set of increments :makhi bhakhi fakhi dhakhi Nakhi N~akhi M~akhi hasjha .... Here "makhi" stands for 25 (ma) + 200 (khi), and the corresponding sine value (for 225 minutes of arc) is 225 / 3438. The value corresponding to the eighth term (hasjha, 199 (ha=100 + s=90 + jha=9), is the sum of all the increments before it, totalling 1719. The entire table for 90 degrees is given as follows:

225,224,222,219.215,210,205,199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,,22,7

So we see that sin(15) (sum of first four terms) = 890/3438 = 0.258871 (correct value = 0.258819, correct to four significant digits). The value of sin(30) (corresponding to hasjha) is 1719/3438 = 0.5; this is of course, exact. His alphabetic code (there are many such codes in Sanskrit) has come to be known as the Aryabhata cipher.

Motion of the Earth

In the fourth book of his Aryabhatiya, Goladhyaya or Golapada, Aryabhata is dealing with the celestial sphere, shape of the earth, cause of day and night etc. In golapAda.6 he says:

bhugolaH sarvato vr.ttaH (The earth is circular everywhere)

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.

Another statement, referring to the island of Sri 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 (ie. on the equator) as moving exactly towards the West. [achalAni bhAni samapashchimagAni - golapAda.9]

In another occasion he says “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”

Arayabhata has number of references to Sri Lanka in Aryabhatiya. His Sri Lankan connection has not been thoroughly investigated.

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.

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.

The 8th century Arabic edition of the Āryabhatīya was translated into Latin in the 13th century, well before Copernicus. Through this translation, European mathematicians may have learned methods for calculating sines and cosines, as well as square and cube roots, and it is likely that some of Aryabhata's results also influenced European astronomy.

Aryabhata clearly stated that the earth is rotating around its own axis. This theory had been proposed by the greek astronomer Heraclides of Pontus in the 4th century B.C. As contacts between hellenistic states and India are well documented, and greek scientists went to India, it cannot be excluded that Aryabbhata was aware of Heraclides' theory.

Aryabhata calculated also the positions of the planets relative to sun (this method is known as "sugrocha"). This is not the same as stating that planets move around the Sun, and cannot be taken as an indication that Aryabhata had in mind the heliocentric system (the first to propose a heliocentric system is recorded as being Aristarchus of Samos in the 3rd century BC).

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 commentry 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 800BC. Aryabhata's method of solving such problems, called the kuttaka method. Kuttaka means pulverizing, 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 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 [link].

Continued relevance

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

Recently Aryabhata was a theme in the RSA Conference 2006. Indocrypt 2005 had an invited talk on Vedic mathmatics. The cryptography community seems to be rediscovering more and more interesting results from ancient Indian mathematics, of which Aryabhata is no doubt the leading luminary.

Confusion of identity

There has been some confusion regarding Aryabhatta's identity. Another notable Indian mathematician, Aryabhata II flourished sometime between 950 and 1100 AD and is one source of confusion. The Persian historian al-Biruni incorrectly believed that there were two famous Indian mathematicians named Aryabhata who lived around 500 AD. The subsequent confusion continued for some time, but in 1926 B Datta showed that al-Biruni's two Aryabhattas were one and the same.However there is a precise mention of the year of birth of Aryabhata in the Aryabhatiya (3-10) which corresponds to 476 AD .

References

External links

John J. O'Connor and Edmund F. Robertson. [] at the MacTutor History of Mathematics archive.
Amartya K Dutta, [Diophantine equations: The Kuttaka], Resonance, October 2002. Also see earlier overview: [Mathematics in Ancient India,]
[An essay on Aryabhata with references]
[RSA Conference 2006]
[Aryabhata and Diophantus' son, Hindustan Times Storytelling Science column, Nov 2004]

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