Information AboutAryabhata |
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Aryabhata (आर्यभट) Āryabhaṭa) ( of the classical age of India . He lived in Kusumapura, which his commentator Bhāskara I ( 629 AD) identifies with Pataliputra (modern Patna ). MAIN CONTRIBUTIONS : see also separate article on Aryabhatiya Aryabhata's Magnum Opus, the ''Āryabhatīya'', is a concise text of 123 stanzas in Sanskrit that describe different results using a mnemonic style typical of the Indian tradition. The Earth was taken to be spinning on its axis and the periods of the Planet s were given with respect to the Sun (in other words, it was Heliocentric , see below). This book is divided into four chapters: (i) the astronomical constants and the sine table (ii) mathematics required for computations (gaNitapāda) (iii) division of time and rules for computing the longitudes of planets using eccentrics and epicycles (iv) the armillary sphere, rules relating to problems of trigonometry and the computation of eclipses (golādhyaya). In the book, the day was reckoned from one sunrise to the next, whereas in his "Āryabhata-siddhānta" he took the day from one midnight to another. There was also difference in some astronomical parameters. PI AS IRRATIONAL Aryabhata worked on the approximation for Pi , and may have realized that is irrational. In the second part of the Aryabhatiya (gaNitapAda 10), he writes:
In other words, , correct to four rounded-off decimal places. The commentator nIlakaNTha, ( Kerala , 15th c.) has argued that the word ''Asanna'' (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 Europe discovered that pi was irrational 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 3438). He uses the alpbabetic 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 . HELIOCENTRICISM 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 laMkA (ie. on the equator) as moving exactly towards the West. bhAni samapashchimagAni'' - golapAda.9 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 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. Aryabbhata's Heliocentricism predates Copernicus by almost a thousand years. 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. 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 621AD , 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 | ||||||||||||||||||||||||
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