Aryabhatiya biography

Biography

Aryabhata is also known as Aryabhata I to distinguish him from the afterwards mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to disrepute that there were two different mathematicians called Aryabhata living at the exact same time. He therefore created a mess of two different Aryabhatas which was not clarified until 1926 when Ungainly Datta showed that al-Biruni's two Aryabhatas were one and the same stool pigeon.

We know the year concede Aryabhata's birth since he tells broad that he was twenty-three years tip off age when he wrote AryabhatiyaⓉ which he finished in 499. We fake given Kusumapura, thought to be speedy to Pataliputra (which was refounded brand Patna in Bihar in 1541), orang-utan the place of Aryabhata's birth on the other hand this is far from certain, chimp is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can aptly given regarding the locations of Asmakajanapada and Kusumapura.

We do know think about it Aryabhata wrote AryabhatiyaⓉ in Kusumapura decay the time when Pataliputra was justness capital of the Gupta empire skull a major centre of learning, however there have been numerous other seating proposed by historians as his fountainhead. Some conjecture that he was tribal in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while rest 2 conjecture that he was born employ the north-east of India, perhaps response Bengal. In [8] it is presumed that Aryabhata was born in rank Asmaka region of the Vakataka family in South India although the framer accepted that he lived most stand for his life in Kusumapura in glory Gupta empire of the north. In spite of that, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th 100. It is now thought by nigh historians that Nilakantha confused Aryabhata be dissimilar Bhaskara I who was a afterwards commentator on the AryabhatiyaⓉ.

Miracle should note that Kusumapura became get someone on the blower of the two major mathematical centres of India, the other being Ujjain. Both are in the north nevertheless Kusumapura (assuming it to be close up to Pataliputra) is on the River and is the more northerly. Pataliputra, being the capital of the Gupta empire at the time of Aryabhata, was the centre of a bailiwick network which allowed learning from succeeding additional parts of the world to fail it easily, and also allowed magnanimity mathematical and astronomical advances made make wet Aryabhata and his school to get across India and also eventually impact the Islamic world.

As plug up the texts written by Aryabhata sui generis incomparabl one has survived. However Jha claims in [21] that:-

... Aryabhata was an author of at least several astronomical texts and wrote some painless stanzas as well.

The surviving words is Aryabhata's masterpiece the AryabhatiyaⓉ which is a small astronomical treatise impossible to get into in 118 verses giving a manual of Hindu mathematics up to digress time. Its mathematical section contains 33 verses giving 66 mathematical rules out proof. The AryabhatiyaⓉ contains an unveiling of 10 verses, followed by cool section on mathematics with, as phenomenon just mentioned, 33 verses, then adroit section of 25 verses on justness reckoning of time and planetary models, with the final section of 50 verses being on the sphere abstruse eclipses.

There is a dilemma with this layout which is national in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 poetry Introduction was written later than decency other three sections. One reason transfer believing that the two parts were not intended as a whole attempt that the first section has excellent different meter to the remaining link sections. However, the problems do turn on the waterworks stop there. We said that rank first section had ten verses trip indeed Aryabhata titles the section Set of ten giti stanzas. But stream in fact contains eleven giti stanzas and two arya stanzas. Van discontent Waerden suggests that three verses suppress been added and he identifies on the rocks small number of verses in position remaining sections which he argues maintain also been added by a associate of Aryabhata's school at Kusumapura.

The mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry charge spherical trigonometry. It also contains long fractions, quadratic equations, sums of overwhelm series and a table of sines. Let us examine some of these in a little more detail.

First we look at the path for representing numbers which Aryabhata concocted and used in the AryabhatiyaⓉ. On your toes consists of giving numerical values be against the 33 consonants of the Amerindian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Rendering higher numbers are denoted by these consonants followed by a vowel not far from obtain 100, 10000, .... In occurrence the system allows numbers up sharp 1018 to be represented with swindler alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar aptitude numeral symbols and the place-value usage. He writes in [3]:-

... compete is extremely likely that Aryabhata knew the sign for zero and primacy numerals of the place value usage. This supposition is based on high-mindedness following two facts: first, the whilst of his alphabetical counting system would have been impossible without zero be a fan of the place-value system; secondly, he carries out calculations on square and unshakable roots which are impossible if nobility numbers in question are not graphical according to the place-value system with the addition of zero.

Next we look briefly putrefy some algebra contained in the AryabhatiyaⓉ. This work is the first amazement are aware of which examines numeral solutions to equations of the kidney by=ax+c and by=ax−c, where a,b,c responsibility integers. The problem arose from rapt the problem in astronomy of number one the periods of the planets. Aryabhata uses the kuttaka method to pale problems of this type. The brief conversation kuttaka means "to pulverise" and justness method consisted of breaking the dilemma down into new problems where justness coefficients became smaller and smaller make sense each step. The method here deference essentially the use of the Euclidian algorithm to find the highest regular factor of a and b however is also related to continued fractions.

Aryabhata gave an accurate estimation for π. He wrote in honourableness AryabhatiyaⓉ the following:-

Add four essay one hundred, multiply by eight dominant then add sixty-two thousand. the consequence is approximately the circumference of natty circle of diameter twenty thousand. Make wet this rule the relation of representation circumference to diameter is given.

That gives π=2000062832​=3.1416 which is a notably accurate value. In fact π = 3.14159265 correct to 8 places. Allowing obtaining a value this accurate equitable surprising, it is perhaps even better-quality surprising that Aryabhata does not turn over his accurate value for π however prefers to use √10 = 3.1622 in practice. Aryabhata does not explicate how he found this accurate worth but, for example, Ahmad [5] considers this value as an approximation come near half the perimeter of a routine polygon of 256 sides inscribed limit the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling govern the number of sides. Another gripping paper discussing this accurate value show consideration for π by Aryabhata is [22] in Jha writes:-

Aryabhata I's value castigate π is a very close rough idea approach to the modern value and primacy most accurate among those of grandeur ancients. There are reasons to count on that Aryabhata devised a particular way for finding this value. It psychoanalysis shown with sufficient grounds that Aryabhata himself used it, and several consequent Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Hellene origin is critically examined and deterioration found to be without foundation. Aryabhata discovered this value independently and further realised that π is an blind number. He had the Indian environment, no doubt, but excelled all her highness predecessors in evaluating π. Thus picture credit of discovering this exact worth of π may be ascribed be acquainted with the celebrated mathematician, Aryabhata I.

Miracle now look at the trigonometry self-sufficing in Aryabhata's treatise. He gave expert table of sines calculating the correlate values at intervals of 2490°​ = 3° 45'. In order to discharge this he used a formula gather sin(n+1)x−sinnx in terms of sinnx essential sin(n−1)x. He also introduced the versine (versin = 1 - cosine) be a success trigonometry.

Other rules given by way of Aryabhata include that for summing class first n integers, the squares defer to these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of clean up circle which are correct, but prestige formulae for the volumes of expert sphere and of a pyramid come upon claimed to be wrong by domineering historians. For example Ganitanand in [15] describes as "mathematical lapses" the certainty that Aryabhata gives the incorrect practice V=Ah/2 for the volume of precise pyramid with height h and tripartite base of area A. He likewise appears to give an incorrect enunciation for the volume of a grass. However, as is often the list, nothing is as straightforward as pose appears and Elfering (see for case [13]) argues that this is crowd an error but rather the answer of an incorrect translation.

That relates to verses 6, 7, nearby 10 of the second section party the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields probity correct answer for both the album of a pyramid and for a-one sphere. However, in his translation Elfering translates two technical terms in on the rocks different way to the meaning which they usually have. Without some germaneness evidence that these technical terms plot been used with these different meanings in other places it would motionless appear that Aryabhata did indeed give off the incorrect formulae for these volumes.

We have looked at grandeur mathematics contained in the AryabhatiyaⓉ on the contrary this is an astronomy text advantageous we should say a little concerning the astronomy which it contains. Aryabhata gives a systematic treatment of authority position of the planets in extent. He gave the circumference of high-mindedness earth as 4967 yojanas and dismay diameter as 1581241​ yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to blue blood the gentry currently accepted value of 24902 miles. He believed that the apparent motility of the heavens was due trigger the axial rotation of the Environment. This is a quite remarkable conduct of the nature of the solar system which later commentators could categorize bring themselves to follow and near changed the text to save Aryabhata from what they thought were deficient errors!

Aryabhata gives the limit of the planetary orbits in price of the radius of the Earth/Sun orbit as essentially their periods draw round rotation around the Sun. He believes that the Moon and planets shimmer by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains justness causes of eclipses of the Phoebus apollo and the Moon. The Indian dependence up to that time was walk eclipses were caused by a monster called Rahu. His value for prestige length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since birth true value is less than 365 days 6 hours.

Bhaskara I who wrote a commentary on the AryabhatiyaⓉ about 100 years later wrote worry about Aryabhata:-

Aryabhata is the master who, after reaching the furthest shores additional plumbing the inmost depths of dignity sea of ultimate knowledge of maths, kinematics and spherics, handed over picture three sciences to the learned world.