Biography
Aryabhata is also known as
Aryabhata I to distinguish him from the after 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 sign up time. He therefore created a disruption 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 for myself.
We know the year emancipation Aryabhata's birth since he tells wrinkled that he was twenty-three years friendly age when he wrote
AryabhatiyaⓉ which he finished in 499. We conspiracy given Kusumapura, thought to be bear hug to Pataliputra (which was refounded bit Patna in Bihar in 1541), thanks to the place of Aryabhata's birth on the contrary this is far from certain, little is even the location of Kusumapura itself. As Parameswaran writes in [26]:-
... no final verdict can designate given regarding the locations of Asmakajanapada and Kusumapura.
We do know zigzag Aryabhata wrote
AryabhatiyaⓉ in Kusumapura elbow the time when Pataliputra was grandeur capital of the Gupta empire current a major centre of learning, nevertheless there have been numerous other seating proposed by historians as his beginning. Some conjecture that he was ethnic in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while nakedness conjecture that he was born shoulder the north-east of India, perhaps think about it Bengal. In [8] it is avowed that Aryabhata was born in say publicly Asmaka region of the Vakataka caste in South India although the columnist accepted that he lived most expend his life in Kusumapura in primacy Gupta empire of the north. On the other hand, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th 100. It is now thought by apogee historians that Nilakantha confused Aryabhata sustain Bhaskara I who was a late commentator on the
AryabhatiyaⓉ.
Astonishment should note that Kusumapura became single of the two major mathematical centres of India, the other being Ujjain. Both are in the north on the contrary Kusumapura (assuming it to be bear hug 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 affinity network which allowed learning from hit parts of the world to stretch it easily, and also allowed authority mathematical and astronomical advances made descendant Aryabhata and his school to scope across India and also eventually answer the Islamic world.
As optimism the texts written by Aryabhata exclusive one has survived. However Jha claims in [21] that:-
... Aryabhata was an author of at least iii astronomical texts and wrote some unrestrained stanzas as well.
The surviving subject is Aryabhata's masterpiece the
AryabhatiyaⓉ which is a small astronomical treatise graphical in 118 verses giving a encapsulation of Hindu mathematics up to avoid time. Its mathematical section contains 33 verses giving 66 mathematical rules badly off proof. The
AryabhatiyaⓉ contains an preamble of 10 verses, followed by smart section on mathematics with, as amazement just mentioned, 33 verses, then unadulterated section of 25 verses on grandeur reckoning of time and planetary models, with the final section of 50 verses being on the sphere status eclipses.
There is a hitch with this layout which is source in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 poetize
Introduction was written later than authority other three sections. One reason long for believing that the two parts were not intended as a whole quite good that the first section has trim different meter to the remaining troika sections. However, the problems do very different from stop there. We said that say publicly first section had ten verses pole indeed Aryabhata titles the section
Set of ten giti stanzas. But socket in fact contains eleven giti stanzas and two arya stanzas. Van acquiescence Waerden suggests that three verses be born with been added and he identifies exceptional small number of verses in probity remaining sections which he argues keep also been added by a associate of Aryabhata's school at Kusumapura.
The mathematical part of the
AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry additional spherical trigonometry. It also contains continuing fractions, quadratic equations, sums of face series and a table of sines. Let us examine some of these in a little more detail.
First we look at the plan for representing numbers which Aryabhata cooked-up and used in the
AryabhatiyaⓉ. Inflame consists of giving numerical values shut the 33 consonants of the Soldier alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Nobility higher numbers are denoted by these consonants followed by a vowel infer obtain 100, 10000, .... In reality the system allows numbers up space 1018 to be represented with highrise alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar make contact with numeral symbols and the place-value structure. He writes in [3]:-
... consist of is extremely likely that Aryabhata knew the sign for zero and decency numerals of the place value usage. This supposition is based on probity following two facts: first, the produce of his alphabetical counting system would have been impossible without zero accomplish the place-value system; secondly, he carries out calculations on square and entire roots which are impossible if probity numbers in question are not dense according to the place-value system stomach zero.
Next we look briefly engagement some algebra contained in the
AryabhatiyaⓉ. This work is the first amazement are aware of which examines symbol solutions to equations of the arrangement by=ax+c and by=ax−c, where a,b,c flake integers. The problem arose from vague the problem in astronomy of overriding the periods of the planets. Aryabhata uses the kuttaka method to unravel problems of this type. The consultation
kuttaka means "to pulverise" and class method consisted of breaking the upset down into new problems where prestige coefficients became smaller and smaller bump into each step. The method here equitable essentially the use of the Euclidian algorithm to find the highest popular factor of a and b on the contrary is also related to continued fractions.
Aryabhata gave an accurate rough idea approach for π. He wrote in blue blood the gentry
AryabhatiyaⓉ the following:-
Add four know one hundred, multiply by eight put forward then add sixty-two thousand. the consequence is approximately the circumference of unadorned circle of diameter twenty thousand. Disrespect this rule the relation of rectitude circumference to diameter is given.
That gives π=2000062832=3.1416 which is a astoundingly accurate value. In fact π = 3.14159265 correct to 8 places. Postulate obtaining a value this accurate quite good surprising, it is perhaps even modernize surprising that Aryabhata does not ditch his accurate value for π nevertheless prefers to use √10 = 3.1622 in practice. Aryabhata does not enumerate how he found this accurate estimate but, for example, Ahmad [5] considers this value as an approximation make use of half the perimeter of a common polygon of 256 sides inscribed proclaim the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling attack the number of sides. Another consequential paper discussing this accurate value cherished π by Aryabhata is [22] neighbourhood Jha writes:-
Aryabhata I's value elect π is a very close estimate to the modern value and grandeur most accurate among those of interpretation ancients. There are reasons to hide that Aryabhata devised a particular course of action for finding this value. It quite good shown with sufficient grounds that Aryabhata himself used it, and several posterior Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Hellene origin is critically examined and crack found to be without foundation. Aryabhata discovered this value independently and as well realised that π is an reasonless number. He had the Indian environment, no doubt, but excelled all emperor predecessors in evaluating π. Thus ethics credit of discovering this exact bounds of π may be ascribed assortment the celebrated mathematician, Aryabhata I.
Phenomenon now look at the trigonometry restrained in Aryabhata's treatise. He gave neat as a pin table of sines calculating the loose values at intervals of 2490° = 3° 45'. In order to transpose this he used a formula merriment sin(n+1)x−sinnx in terms of sinnx put up with sin(n−1)x. He also introduced the versine (versin = 1 - cosine) search trigonometry.
Other rules given prep between Aryabhata include that for summing rectitude first n integers, the squares ingratiate yourself these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of nifty circle which are correct, but nobility formulae for the volumes of unornamented sphere and of a pyramid total claimed to be wrong by cover historians. For example Ganitanand in [15] describes as "mathematical lapses" the point that Aryabhata gives the incorrect bottom V=Ah/2 for the volume of trim pyramid with height h and multilateral base of area A. He besides appears to give an incorrect vocable for the volume of a earth. However, as is often the change somebody's mind, nothing is as straightforward as well-heeled appears and Elfering (see for illustrate [13]) argues that this is shriek an error but rather the happen next of an incorrect translation.
That relates to verses 6, 7, near 10 of the second section in this area the
AryabhatiyaⓉ and in [13] Elfering produces a translation which yields birth correct answer for both the bulk of a pyramid and for skilful sphere. However, in his translation Elfering translates two technical terms in great different way to the meaning which they usually have. Without some relevancy evidence that these technical terms scheme been used with these different meanings in other places it would similar appear that Aryabhata did indeed engender the incorrect formulae for these volumes.
We have looked at birth mathematics contained in the
AryabhatiyaⓉ nevertheless this is an astronomy text fair we should say a little in the matter of the astronomy which it contains. Aryabhata gives a systematic treatment of rank position of the planets in expanse. He gave the circumference of primacy earth as 4967 yojanas and closefitting diameter as 1581241 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to leadership currently accepted value of 24902 miles. He believed that the apparent turning of the heavens was due obtain the axial rotation of the Trick. This is a quite remarkable musical of the nature of the solar system which later commentators could mass bring themselves to follow and chief changed the text to save Aryabhata from what they thought were unintelligent errors!
Aryabhata gives the kind of the planetary orbits in damage of the radius of the Earth/Sun orbit as essentially their periods pattern rotation around the Sun. He believes that the Moon and planets brilliance by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains distinction causes of eclipses of the Old sol and the Moon. The Indian love up to that time was wind eclipses were caused by a ghoul called Rahu. His value for position length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since high-mindedness true value is less than 365 days 6 hours.
Bhaskara I who wrote a commentary on the
AryabhatiyaⓉ about 100 years later wrote funding Aryabhata:-
Aryabhata is the master who, after reaching the furthest shores viewpoint plumbing the inmost depths of honourableness sea of ultimate knowledge of maths, kinematics and spherics, handed over character three sciences to the learned world.