Aryabhata

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For the Indian satellite, see Aryabhata (satellite).

For the numeral system created by Āryabhaṭa, see Āryabhaṭa numeration.

Āryabhaṭa (Devanāgarī: आर्यभट) (AD 476 – 550) is the first of the great mathematician-astronomers of the classical age of Indian mathematics. He was born at 'Ashmak' village near Patliputra^{[citation needed]} (modern Patna). Available 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 Aryabhatiyam and the Aryabhatta-siddhanta. Aryabhatiyam 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.

Contents 1 Pi as Irrational 2 Mensuration and Trigonometry 3 Motions of the Solar System 3.1 The Oldest Accurate Astronomical Constant 3.2 Heliocentrism 4 Diophantine Equations 5 Continued Relevance 6 Confusion of Identity 7 Notes 8 References 9 External links

[edit] Pi as Irrational

Aryabhata worked on the approximation for Pi, and may have realized that π is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.

"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, , correct to five digits. The commentator Nilakantha Somayaji, (Kerala School, 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). Aryabhata greatest contribution is signified by 0 (Zero). Notation for placeholders in positional numbers is found on stone tablets from ancient (3,000 B.C.) Sumeria. Yet, the Greeks had no concept of a number like zero. In terms of modern use, zero is sometimes traced to the Indian mathematician Aryabhata who, about 520 A.D., devised a positional decimal number system that contained a word, "kha," for the idea of a placeholder. By 876, based on an existing tablet inscription with that date, the kha had become the symbol "0". Meanwhile, somewhat after Aryabhata, another Indian, Brahmagupta, developed the concept of the zero as an actual independent number, not just a place-holder, and wrote rules for adding and subtracting zero from other numbers. The Indian writings were passed on to al-Khwarizmi (from whose name we derive the term algorithm) and thence to Leonardo Fibonacci and others who continued to develop the concept and the number.

[edit] 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.)

But he gave an incorrect rule for the volume of a pyramid.^[3] Aryabhata was not concerned with demonstrating his formulas.^[4] 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 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.

[edit] 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.^[5] 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. 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. Aryabhata defines the sizes of the planets' orbits in terms of these periods.^[6][7]

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.

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)

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 to mean the equator, which was known to pass through Sri Lanka.

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.

[edit] The Oldest Accurate Astronomical Constant

In the work The Àryabhatiya of Àryabhata, An Ancient Indian Work on Mathematics and Astronomy, translated by William Eugene Clark, Professor of Sanskrit at Harvard University (The University of Chicago Press, Chicago, Illinois. 1930), Àryabhata writes that 1,582,237,500 rotations of the Earth equal 57,753,336 lunar orbits. This extremely accurate ratio for two fundamental cosmic motions was noticed in the 1990s and published online in 'The Àryabhatiya of Àryabhata: The oldest exact astronomical constant?' in 1998 by James Q. Jacobs. Jacobs determined that the ratio was precise for about 1600 BCE, and that it represents the earliest written astronomic ratio of such accuracy.

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.

[edit] Heliocentrism

Aryabhata's computations are consistent with a heliocentric motion of the planets orbiting the sun and the earth spinning on its own axis. While he is not the first to say this, his authority was certainly most influential. The earlier Indian astronomical texts Shatapatha Brahmana (c. 9th-7th century BC), Aitareya Brahmana (c. 9th-7th century BC) and Vishnu Purana (c. 1st century BC) contain early concepts of a heliocentric model.^{[citation needed]} Heraclides of Pontus (4th c. BC) is sometimes credited with a heliocentric theory. Aristarchus of Samos (3rd century BC) is usually credited with knowing of 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 8th century Arabic edition of the Āryabhatīya was translated into Latin in the 13th century, well before Copernicus and may have influenced European astronomy, though a direct connection with Copernicus cannot be established.

[edit] 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 800BC. Aryabhata's method of solving such problems, called the kuttaka 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 [1].

[edit] Continued Relevance

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

Recently Aryabhata was a theme in the RSA Conference 2006, Indocrypt 2005, which had a session on Vedic mathematics. The cryptography community seems to be rediscovering more and more interesting results from ancient Indian mathematics, and its leading luminary, Aryabhata.

The lunar crater Aryabhata is named in his honour.

[edit] Confusion of Identity

There was confusion regarding Aryabhatta's identity. Another Indian mathematician, Aryabhata II flourished sometime between AD 950 and 1100. Two famous Indian mathematicians named Aryabhata were thought to have lived around AD 500, until, in 1926, B. Datta showed that these two Aryabhattas were one and the same. A precise mention of the year of birth of Aryabhata in the Aryabhatiya (3-10) corresponds to 476.

[edit] Notes

1. ^ P. Z. Ingerman, 'Panini-Backus form', Communications of the ACM 10 (3)(1967), p.137
2. ^ History of Hindu Mathematics/Bibhutibhushan Dutta and Avadhesh Narayan Singh, Asia Publishing House, 1962. (reprint ISBN 81-86050-86-8).
3. ^ Cooke, Roger (1997). "The Mathematics of the Hindus", The History of Mathematics: A Brief Course. Wiley-Interscience, 205. ISBN 0471180823. “Aryabhata gave the correct rule for the are of a triangle and an incorrect rule for the volume of a pyramid. (He claimed that the volume was half the height times the area of the base.)” 
4. ^ Cooke, Roger (1997). "The Mathematics of the Hindus", The History of Mathematics: A Brief Course. Wiley-Interscience, 205. ISBN 0471180823. “It is clear therefore that Aryabhata was not concerned with demonstration;” 
5. ^ David Pingree, "Astronomy in India", pp. 127-9.
6. ^ Otto Neugebauer, "The Transmission of Planetary Theories in Ancient and Medieval Astronomy," Scripta Mathematica, 22(1956): 165-192; reprinted in Otto Neugebauer, Astronomy and History: Selected Essays, New York: Springer-Verlag, 1983, pp. 129-156. ISBN 0-387-90844-7
7. ^ Hugh Thurston, Early Astronomy, New York: Springer-Verlag, 1996, pp. 178-189. ISBN 0-387-94822-8

[edit] References

• Dutta, B. & A.N. Singh (1962), History of Hindu Mathematics, Asia Publishing House, Bombay
• Kak, Subhash C. (2000). 'Birth and Early Development of Indian Astronomy'. In Selin, Helaine (2000), Astronomy Across Cultures: The History of Non-Western Astronomy, Kluwer, Boston, ISBN 0-7923-6363-9
• Pingree, David (1996), "Astronomy in India", in Walker, Christopher, Astronomy before the Telescope, London: British Museum Press, ISBN 0-7141-1746-3

• Rao, S. Balachandra (1994/1998), Indian Mathematics and Astronomy: Some Landmarks, Jnana Deep Publications, Bangalore, ISBN 81-7371-205-0

• Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Indian National Science Academy, 1976.

• Thurston, H. (1994), Early Astronomy, Springer-Verlag, New York, ISBN 0-387-94107-X

[edit] External links

• The Àryabhatiya of Àryabhata: The oldest exact astronomical constant?
• O'Connor, John J., and Edmund F. Robertson. "Aryabhata". 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 - dead link, Feb. 2007
• RSA Conference 2006
• Aryabhata and Diophantus' son, Hindustan Times Storytelling Science column, Nov 2004