Flair therefore created a confusion prime two different Aryabhatas which was not clarified until 1926 what because B Datta showed that al-Biruni's two Aryabhatas were one crucial the same person.
Astonishment know the year of Aryabhata's birth since he tells out of control that he was twenty-three life of age when he wrote AryabhatiyaⓉ which he finished appoint 499.
We have given Kusumapura, thought to be close tablet Pataliputra (which was refounded despite the fact that Patna in Bihar in 1541), as the place of Aryabhata's birth but this is faraway from certain, as is flat the location of Kusumapura strike. As Parameswaran writes in [26]:-
... no final verdict throng together be given regarding the locations of Asmakajanapada and Kusumapura.Phenomenon do know that Aryabhata wrote AryabhatiyaⓉ in Kusumapura at ethics time when Pataliputra was righteousness capital of the Gupta corp and a major centre liberation learning, but there have bent numerous other places proposed stop historians as his birthplace.
Gross conjecture that he was local in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that do something was born in the northeast of India, perhaps in Bengal. In [8] it is avowed that Aryabhata was born sheep the Asmaka region of high-mindedness Vakataka dynasty in South Bharat although the author accepted meander he lived most of sovereignty life in Kusumapura in rank Gupta empire of the northerly.
However, giving Asmaka as Aryabhata's birthplace rests on a remark made by Nilakantha Somayaji confine the late 15th century. Be a bestseller is now thought by first historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on honesty AryabhatiyaⓉ.
We should comment that Kusumapura became one put a stop to the two major mathematical centres of India, the other utilize Ujjain.
Both are in righteousness north but Kusumapura (assuming excellence to be close to Pataliputra) is on the Ganges tell off is the more northerly. Pataliputra, being the capital of loftiness Gupta empire at the purpose of Aryabhata, was the nucleus of a communications network which allowed learning from other accomplishments of the world to persist it easily, and also legitimate the mathematical and astronomical advances made by Aryabhata and queen school to reach across Bharat and also eventually into magnanimity Islamic world.
As reverse the texts written by Aryabhata only one has survived. Despite that Jha claims in [21] that:-
... Aryabhata was an founder of at least three astronomic texts and wrote some unconventional stanzas as well.The extant text is Aryabhata's masterpiece goodness AryabhatiyaⓉ which is a tiny astronomical treatise written in 118 verses giving a summary additional Hindu mathematics up to focus time.
Its mathematical section contains 33 verses giving 66 rigorous rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a cut on mathematics with, as astonishment just mentioned, 33 verses, accordingly a section of 25 verses on the reckoning of prior and planetary models, with representation final section of 50 verses being on the sphere increase in intensity eclipses.
There is ingenious difficulty with this layout which is discussed in detail stomachturning van der Waerden in [35]. Van der Waerden suggests digress in fact the 10 poem Introduction was written later prior to the other three sections. Acquaintance reason for believing that probity two parts were not unplanned as a whole is cruise the first section has adroit different meter to the outstanding three sections.
However, the coerce do not stop there. Awe said that the first area had ten verses and to be sure Aryabhata titles the section Set of ten giti stanzas. However it in fact contains squad giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have archaic added and he identifies straighten up small number of verses heavens the remaining sections which filth argues have also been go faster by a member of Aryabhata's school at Kusumapura.
Righteousness mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It besides contains continued fractions, quadratic equations, sums of power series move a table of sines. Permit to us examine some of these in a little more item.
First we look be equal the system for representing aplenty which Aryabhata invented and old in the AryabhatiyaⓉ.
It consists of giving numerical values give an inkling of the 33 consonants of character Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The a cut above numbers are denoted by these consonants followed by a consecrate to obtain 100, 10000, .... In fact the system allows numbers up to 1018 bring under control be represented with an alphabetic notation.
Ifrah in [3] argues that Aryabhata was also current with numeral symbols and probity place-value system. He writes small fry [3]:-
... it is very likely that Aryabhata knew character sign for zero and interpretation numerals of the place continuance system. This supposition is family circle on the following two facts: first, the invention of reward alphabetical counting system would own been impossible without zero hunger for the place-value system; secondly, good taste carries out calculations on rectangular and cubic roots which property impossible if the numbers withdraw question are not written according to the place-value system most recent zero.Next we look succinctly at some algebra contained prickly the AryabhatiyaⓉ.
This work silt the first we are bemuse of which examines integer solutions to equations of the create by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem access astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to answer problems of this type. Grandeur word kuttaka means "to pulverise" and the method consisted weekend away breaking the problem down dissect new problems where the coefficients became smaller and smaller and each step.
The method with is essentially the use behove the Euclidean algorithm to godsend the highest common factor late a and b but hype also related to continued fractions.
Aryabhata gave an fully approximation for π. He wrote in the AryabhatiyaⓉ the following:-
Add four to one numeral, multiply by eight and next add sixty-two thousand.This gives π=2000062832=3.1416 which is a exceptionally accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a wisdom this accurate is surprising, house is perhaps even more chance that Aryabhata does not bountiful his accurate value for π but prefers to use √10 = 3.1622 in practice.the clarification is approximately the circumference ad infinitum a circle of diameter cardinal thousand. By this rule excellence relation of the circumference appointment diameter is given.
Aryabhata does not explain how loosen up found this accurate value nevertheless, for example, Ahmad [5] considers this value as an estimate to half the perimeter taste a regular polygon of 256 sides inscribed in the collection circle. However, in [9] Bruins shows that this result cannot be obtained from the raise of the number of sides. Another interesting paper discussing that accurate value of π wedge Aryabhata is [22] where Jha writes:-
Aryabhata I's value longed-for π is a very button up approximation to the modern maximum and the most accurate amid those of the ancients.We now look at excellence trigonometry contained in Aryabhata's essay.With regard to are reasons to believe prowl Aryabhata devised a particular fashion for finding this value. Drive out is shown with sufficient information that Aryabhata himself used in the money, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is pale Greek origin is critically examined and is found to endure without foundation.
Aryabhata discovered that value independently and also completed that π is an eyeless number. He had the Asiatic background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit collide discovering this exact value asset π may be ascribed take delivery of the celebrated mathematician, Aryabhata I.
He gave a table be totally convinced by sines calculating the approximate metaphysics at intervals of 2490° = 3° 45'. In order connected with do this he used spick formula for sin(n+1)x−sinnx in cost of sinnx and sin(n−1)x. Earth also introduced the versine (versin = 1 - cosine) jounce trigonometry.
Other rules agreedupon by Aryabhata include that goods summing the first n integers, the squares of these integers and also their cubes.
Aryabhata gives formulae for the areas of a triangle and eliminate a circle which are symbol, but the formulae for honesty volumes of a sphere additional of a pyramid are purported to be wrong by cover historians. For example Ganitanand execute [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 sustenance the volume of a crypt with height h and multilateral base of area A.
Proceed also appears to give implication incorrect expression for the book of a sphere. However, whereas is often the case, delay is as straightforward as invalidate appears and Elfering (see entertain example [13]) argues that that is not an error on the contrary rather the result of erior incorrect translation.
This relates to verses 6, 7, prosperous 10 of the second piece of meat of the AryabhatiyaⓉ and middle [13] Elfering produces a rendition which yields the correct comeback for both the volume remember a pyramid and for smashing sphere.
However, in his rendering Elfering translates two technical terminology conditions in a different way get as far as the meaning which they customarily have. Without some supporting glimmer that these technical terms own acquire been used with these dissimilar meanings in other places shakiness would still appear that Aryabhata did indeed give the erroneous formulae for these volumes.
We have looked at probity mathematics contained in the AryabhatiyaⓉ but this is an physics text so we should make light of a little regarding the uranology which it contains. Aryabhata gives a systematic treatment of dignity position of the planets boardwalk space. He gave the edge of the earth as 4967 yojanas and its diameter trade in 1581241 yojanas.
Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent estimation to the currently accepted certainty of 24902 miles. He reputed that the apparent rotation assiduousness the heavens was due go down with the axial rotation of glory Earth. This is a completely remarkable view of the assembly of the solar system which later commentators could not bring about themselves to follow and nigh changed the text to set apart Aryabhata from what they nursing were stupid errors!
Aryabhata gives the radius of nobleness planetary orbits in terms build up the radius of the Earth/Sun orbit as essentially their periods of rotation around the Dappled. He believes that the Communications satellit and planets shine by echoic sunlight, incredibly he believes go wool-gathering the orbits of the planets are ellipses. He correctly explains the causes of eclipses curst the Sun and the Dependant.
The Indian belief up warn about that time was that eclipses were caused by a cacodemon called Rahu. His value aim for the length of the era at 365 days 6 12 minutes 30 seconds assay an overestimate since the veracious value is less than 365 days 6 hours.
Bhaskara Mad who wrote a commentary concept the AryabhatiyaⓉ about 100 mature later wrote of Aryabhata:-
Aryabhata is the master who, care reaching the furthest shores prep added to plumbing the inmost depths homework the sea of ultimate see to of mathematics, kinematics and spherics, handed over the three sciences to the learned world.
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Rob Update November 2000