Aryabhata

Âryabha a (Devanâgarî: ] [ ) (AD 476 – 550) is the first of thegreat mathematician-astronomers of the classical age of Indianmathematics. He was born at Muziris (the modern dayKodungallour village) near Trissur. Available evidence suggest thathe went to Kusumapura for higher studies. He lived inKusumapura, which his commentator Bhâskara 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 workwas the Aryabhatiya and the Aryabhatta-siddhanta. TheAryabhatiya presented a number of innovations in mathematics andastronomy in verse form, which were influential for manycenturies. The extreme brevity of the text was elaborated incommentaries 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[1]Manuscript was clearly in place in his work. He may have been the first mathematician to use letters of the alphabet to denote unknown quantities.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 asecond model (ardha-rAtrikA, midnight), are lost, but can be partly reconstructed from thediscussion in Brahmagupta's khanDakhAdyaka. In some texts he seems to ascribe the apparentmotions of the heavens to the earth's rotation.

The AryabhatiyaPi as Irrational

The number system we use today known as Hindu-Arabic number system was developed by Indian mathematicians and spread around the world by Arabs. In Aryabhatiya, Aryabhatta stated "StanamStanam Dasa Gunam" or in English "Place to Place Ten Times in Value". As per Tobias Denzig,discovery of the place value notation is a world event. Later zero was added to the Aryabhatta'snumber system by Brahmagupta. Aryabhata worked on the approximation for Pi, and may haverealized that ð is irrational. In the second part of the Aryabhatiyam. In other words,, correct to five digits. The commentator Nilakantha Somayaji,(Kerala School, 15th c.) has argued that the word âsanna (approaching), appearing just before thelast 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 wasproved in Europe only in 1761 (Lambert). Aryabhata's 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 issometimes traced to the Indian mathematician Aryabhata who, about 520 A.D., devised a positionaldecimal number system that contained a word, "kha," for the idea of a placeholder. By 876, based onan 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 anactual independent number, not just a place-holder, and wrote rules for adding and subtracting zerofrom other numbers. The Indian writings were passed on to al-Khwarizmi (from whose name wederive the term algorithm) and thence to Leonardo Fibonacci and others who continued to developthe concept and the number.

Mensuration and Trigonometry

In Ganitapada 6, Aryabhata gives the area of triangle astribhujasya phalashariram samadalakoti bhujardhasamvargah (for a triangle, the result of aperpendicular with the half-side is the area.)

But he gave an incorrect rule for the volume of a pyramid. Aryabhata was not concerned with demonstrating his formulas. Aryabhata, in his work Aryabhata-Siddhanta, first defined the sine asthe 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 otkramjya for inverse sine.Aryabhata's tables for the sines (from which the rest can be computed), is presented in a singlerhyming stanza, with each syllable standing for increments at intervals of 225 minutes of arc or 3degrees 45'. Using a compact alphabetic code called varga/avarga, he defines the sines for a circle ofcircumference 21600 (radius 3438). He uses the alphabetic code to define a set ofincrements :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 valuecorresponding to the eighth term (hasjha, 199 (ha=100 + s=90 + jha=9), is the sum of all theincrements before it, totalling 1719. The entire table for 90 degrees is given as follows:225,224,222,219,215,210,205,199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,22,7So 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 beknown as the Aryabhata cipher.

Motions of the Solar System

Aryabhata described a geocentric model of the solar system, in which the Sun and Moon are eachcarried by epicycles which in turn revolve around the Earth. In this model, which is also found in thePaitâmahasiddhânta (ca. AD 425), the motions of the planets are each governed by two epicycles, a[5]smaller manda (slow) epicycle and a larger œ îghra (fast) epicycle. The positions and periods of theplanets were calculated relative to uniformly moving points, which in the case of Mercury andVenus, 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 thezodiac. Most historians of astronomy consider that this two epicycle model reflects elements of pre-Ptolemaic Greek astronomy. Another element in Aryabhata's model, the œ îghrocca, the basicplanetary 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.He states that the Moon and planets shine by reflected sunlight. He also correctly explains eclipses ofthe 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 thecelestial 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 motioncaused by the rotation of the earth:

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

However, in the next verse he describes the motion of the stars and planets as real: “The cause oftheir rising and setting is due to the fact the circle of the asterisms together with the planets driven bythe 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.Aryabhatta make numerous references to Lanka where there is a doubt whether he was originallyfrom Sri Lanka, island nation south of India.Aryabhata's computation of Earth's circumference as 24,835 miles, which was only 0.2% smallerthan the actual value of 24,902 miles. This approximation improved on the computation by theAlexandrinan mathematician Erastosthenes (c.200 BC), whose exact computation is not known inmodern units.

Sidereal periods

Considered in modern English units of time, Aryabhata calculated the sidereal rotation (the rotationof the earth referenced the fixed stars) as 23 hours 56 minutes and 4.1 seconds; the modern value is23:56:4.091. Similarly, his value for the length of the sidereal year at 365 days 6 hours 12 minutes30 seconds is an error of 3 minutes 20 seconds over the length of a year. The notion of sidereal timewas known in most other astronomical systems of the time, but this computation was likely the mostaccurate in the period.

Heliocentrism

Aryabhata's computations are consistent with a heliocentric motion of the planets orbiting the sunand the earth spinning on its own axis. While he is not the first to say this, his authority was certainlymost influential. The earlier Indian astronomical texts Shatapatha Brahmana (c. 9th-7th centuryBC), Aitareya Brahmana (c. 9th-7th century BC) and Vishnu Purana (c. 1st century BC) containearly concepts of a heliocentric model. Heraclides of Pontus (4th c. BC) is sometimes credited with aheliocentric theory. Aristarchus of Samos (3rd century BC) is usually credited with knowing of theheliocentric 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 centuryArabic edition of the Âryabhatîya was translated into Latin in the 13th century, well beforeCopernicus and may have influenced European astronomy, though a direct connection withCopernicus cannot be established. 10th century Arabic scholar Al Baruni states that Aryabhatta'sfolowers believe earth to rotate around the sun. Then he casually adds that this notion does not createany mathematical difficulties. In Indian astronomy sun is always at the center in the "sugrocha"system. It is fair to say earth rotating around the sun was known to Aryabhatta at least 1,000 years before Copernicus.

Diophantine Equations

A problem of great interest to Indian mathematicians since very ancient times concerned diophantineequations. These involve integer solutions to equations such as ax + b = cy. Here is an example fromBhaskara's commentary on Aryabhatiya: :

Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder whendivided 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 inthe 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 ( ) method. Kuttaka meanspulverizing, that is breaking into small pieces, and the method involved a recursive algorithm forwriting the original factors in terms of smaller numbers. Today this algorithm, as elaborated byBhaskara in AD 621, is the standard method for solving first order Diophantine equations, and it isoften referred to as the Aryabhata algorithm. See details of the Kuttaka method in this link http://www.ias.ac.in/resonance/Oct2002/pdf/Oct2002p6-22.pdf%7Carticle

Aryabhata's astronomical calculation methods have been in continuous use for the practical purposesof fixing the Panchanga Hindu calendar.Recently Aryabhata was a theme in the RSA Conference 2006, Indocrypt 2005, which had a sessionon Vedic mathematics.The lunar crater Aryabhata is named in his honour.