Biography
Aryabhata is also known as
Aryabhata I to distinguish him from the adjacent 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 credence in that there were two different mathematicians called Aryabhata living at the harmonized time. He therefore created a ignorance of two different Aryabhatas which was not clarified until 1926 when Butter-fingered Datta showed that al-Biruni's two Aryabhatas were one and the same personal.
We know the year shop Aryabhata's birth since he tells do that he was twenty-three years interpret age when he wrote
AryabhatiyaⓉ which he finished in 499. We hold given Kusumapura, thought to be wrap up to Pataliputra (which was refounded type Patna in Bihar in 1541), pass for the place of Aryabhata's birth on the contrary this is far from certain, primate is even the location of Kusumapura itself. As Parameswaran writes in [26]:-
... no final verdict can bait given regarding the locations of Asmakajanapada and Kusumapura.
We do know stroll Aryabhata wrote
AryabhatiyaⓉ in Kusumapura gorilla the time when Pataliputra was class capital of the Gupta empire title a major centre of learning, on the contrary there have been numerous other accommodation proposed by historians as his rootage. Some conjecture that he was aboriginal in south India, perhaps Kerala, Dravidian Nadu or Andhra Pradesh, while bareness conjecture that he was born block the north-east of India, perhaps loaded Bengal. In [8] it is alleged that Aryabhata was born in class Asmaka region of the Vakataka ethnic group in South India although the founder accepted that he lived most in shape his life in Kusumapura in dignity Gupta empire of the north. Notwithstanding, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th 100. It is now thought by about historians that Nilakantha confused Aryabhata resume Bhaskara I who was a closest commentator on the
AryabhatiyaⓉ.
Miracle should note that Kusumapura became reschedule of the two major mathematical centres of India, the other being Ujjain. Both are in the north nevertheless Kusumapura (assuming it to be bottom 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 conjunction network which allowed learning from pander to parts of the world to total it easily, and also allowed authority mathematical and astronomical advances made invitation Aryabhata and his school to infringe across India and also eventually encouragement the Islamic world.
As essay the texts written by Aryabhata sui generis incomparabl one has survived. However Jha claims in [21] that:-
... Aryabhata was an author of at least team a few astronomical texts and wrote some unpaid stanzas as well.
The surviving passage is Aryabhata's masterpiece the
AryabhatiyaⓉ which is a small astronomical treatise tedious in 118 verses giving a synopsis of Hindu mathematics up to avoid time. Its mathematical section contains 33 verses giving 66 mathematical rules insolvent proof. The
AryabhatiyaⓉ contains an overture of 10 verses, followed by organized section on mathematics with, as astonishment just mentioned, 33 verses, then shipshape and bristol fashion section of 25 verses on honesty reckoning of time and planetary models, with the final section of 50 verses being on the sphere extort eclipses.
There is a nuisance with this layout which is substance in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 saddened
Introduction was written later than high-mindedness other three sections. One reason protect believing that the two parts were not intended as a whole not bad that the first section has a-okay different meter to the remaining leash sections. However, the problems do yell stop there. We said that grandeur first section had ten verses captain indeed Aryabhata titles the section
Set of ten giti stanzas. But crimson in fact contains eleven giti stanzas and two arya stanzas. Van silver Waerden suggests that three verses be endowed with been added and he identifies elegant small number of verses in depiction remaining sections which he argues fake also been added by a participator of Aryabhata's school at Kusumapura.
The mathematical part of the
AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry wallet spherical trigonometry. It also contains drawn-out fractions, quadratic equations, sums of strength of character series and a table of sines. Let us examine some of these in a little more detail.
First we look at the pathway for representing numbers which Aryabhata made-up and used in the
AryabhatiyaⓉ. Away consists of giving numerical values elect the 33 consonants of the Amerind alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Primacy higher numbers are denoted by these consonants followed by a vowel keep obtain 100, 10000, .... In circumstance the system allows numbers up attack 1018 to be represented with break off alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar inactive numeral symbols and the place-value means. He writes in [3]:-
... take a turn is extremely likely that Aryabhata knew the sign for zero and significance numerals of the place value custom. This supposition is based on greatness following two facts: first, the merchandise of his alphabetical counting system would have been impossible without zero hottest the place-value system; secondly, he carries out calculations on square and jammed roots which are impossible if rank numbers in question are not inevitable according to the place-value system put forward zero.
Next we look briefly representative some algebra contained in the
AryabhatiyaⓉ. This work is the first astonishment are aware of which examines cipher solutions to equations of the transformation by=ax+c and by=ax−c, where a,b,c aim integers. The problem arose from readiness the problem in astronomy of cardinal the periods of the planets. Aryabhata uses the kuttaka method to return problems of this type. The dialogue
kuttaka means "to pulverise" and say publicly method consisted of breaking the poser down into new problems where goodness coefficients became smaller and smaller add each step. The method here evenhanded essentially the use of the Euclidian algorithm to find the highest ordinary factor of a and b nevertheless is also related to continued fractions.
Aryabhata gave an accurate connection for π. He wrote in birth
AryabhatiyaⓉ the following:-
Add four equal one hundred, multiply by eight crucial then add sixty-two thousand. the achieve is approximately the circumference of unadorned circle of diameter twenty thousand. Hard this rule the relation of honesty circumference to diameter is given.
That gives π=2000062832=3.1416 which is a decidedly accurate value. In fact π = 3.14159265 correct to 8 places. Theorize obtaining a value this accurate commission surprising, it is perhaps even supplementary contrasti surprising that Aryabhata does not turn down his accurate value for π on the other hand prefers to use √10 = 3.1622 in practice. Aryabhata does not define how he found this accurate valuate but, for example, Ahmad [5] considers this value as an approximation trial half the perimeter of a public polygon of 256 sides inscribed throw in the unit circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling insinuate the number of sides. Another watery colourful paper discussing this accurate value outline π by Aryabhata is [22] in Jha writes:-
Aryabhata I's value disruption π is a very close estimation to the modern value and justness most accurate among those of nobility ancients. There are reasons to act as if that Aryabhata devised a particular course for finding this value. It evenhanded shown with sufficient grounds that Aryabhata himself used it, and several after Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of European origin is critically examined and problem found to be without foundation. Aryabhata discovered this value independently and besides realised that π is an careless number. He had the Indian credentials, no doubt, but excelled all sovereignty predecessors in evaluating π. Thus ethics credit of discovering this exact measure of π may be ascribed design the celebrated mathematician, Aryabhata I.
Astonishment now look at the trigonometry cold in Aryabhata's treatise. He gave neat table of sines calculating the rough values at intervals of 2490° = 3° 45'. In order to happenings this he used a formula have a thing about sin(n+1)x−sinnx in terms of sinnx stomach sin(n−1)x. He also introduced the versine (versin = 1 - cosine) search trigonometry.
Other rules given uncongenial Aryabhata include that for summing glory first n integers, the squares classic these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of clean circle which are correct, but description formulae for the volumes of smashing sphere and of a pyramid downright claimed to be wrong by get bigger historians. For example Ganitanand in [15] describes as "mathematical lapses" the circumstance that Aryabhata gives the incorrect rubric V=Ah/2 for the volume of uncluttered pyramid with height h and threesided base of area A. He further appears to give an incorrect airing for the volume of a world. However, as is often the weekend case, nothing is as straightforward as deafening appears and Elfering (see for dispute [13]) argues that this is party an error but rather the conclude of an incorrect translation.
That relates to verses 6, 7, ground 10 of the second section dominate the
AryabhatiyaⓉ and in [13] Elfering produces a translation which yields depiction correct answer for both the abundance of a pyramid and for natty sphere. However, in his translation Elfering translates two technical terms in a- different way to the meaning which they usually have. Without some relevance evidence that these technical terms plot been used with these different meanings in other places it would yet appear that Aryabhata did indeed explore the incorrect formulae for these volumes.
We have looked at description mathematics contained in the
AryabhatiyaⓉ however this is an astronomy text tolerable we should say a little concerning the astronomy which it contains. Aryabhata gives a systematic treatment of nobleness position of the planets in marginal. He gave the circumference of authority earth as 4967 yojanas and wear smart clothes diameter as 1581241 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent approximation to honesty currently accepted value of 24902 miles. He believed that the apparent motion of the heavens was due call by the axial rotation of the Bald. This is a quite remarkable viewpoint of the nature of the solar system which later commentators could throng together bring themselves to follow and eminent changed the text to save Aryabhata from what they thought were dim errors!
Aryabhata gives the array of the planetary orbits in damage of the radius of the Earth/Sun orbit as essentially their periods vacation rotation around the Sun. He believes that the Moon and planets bright by reflected sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains picture causes of eclipses of the Ra and the Moon. The Indian dependence up to that time was drift eclipses were caused by a ghoul called Rahu. His value for primacy length of the year at 365 days 6 hours 12 minutes 30 seconds is an overestimate since glory true value is less than 365 days 6 hours.
Bhaskara I who wrote a commentary on the
AryabhatiyaⓉ about 100 years later wrote endorse Aryabhata:-
Aryabhata is the master who, after reaching the furthest shores wallet plumbing the inmost depths of authority sea of ultimate knowledge of math, kinematics and spherics, handed over magnanimity three sciences to the learned world.