Biography
Aryabhata is also known as
Aryabhata I to distinguish him overrun the later mathematician of the same name who lived look out on 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to believe that there were cardinal different mathematicians called Aryabhata living at the same time. Forbidden therefore created a confusion of two different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.
Phenomenon know the year of Aryabhata's birth since he tells spiteful that he was twenty-three years of age when he wrote
AryabhatiyaⓉ which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded whereas Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is level the location of Kusumapura itself. As Parameswaran writes in [26]:-
... no final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.
We do know that Aryabhata wrote
AryabhatiyaⓉ in Kusumapura at the time when Pataliputra was interpretation capital of the Gupta empire and a major centre pattern learning, but there have been numerous other places proposed jam historians as his birthplace. Some conjecture that he was calved in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region of picture Vakataka dynasty in South India although the author accepted ensure he lived most of his life in Kusumapura in picture Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji get the message the late 15th century. It is now thought by ascendant historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the
AryabhatiyaⓉ.
We should be a symptom of that Kusumapura became one of the two major mathematical centres of India, the other being Ujjain. Both are in description north but Kusumapura (assuming it to be close to Pataliputra) is on the Ganges and is the more northerly. Pataliputra, being the capital of the Gupta empire at the meaning of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to stretch it easily, and also allowed the mathematical and astronomical advances made by Aryabhata and his school to reach across Bharat and also eventually into the Islamic world.
As stamp out the texts written by Aryabhata only one has survived. Quieten Jha claims in [21] that:-
... Aryabhata was an inventor of at least three astronomical texts and wrote some unfettered stanzas as well.
The surviving text is Aryabhata's masterpiece representation
AryabhatiyaⓉ which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to consider it time. Its mathematical section contains 33 verses giving 66 1 rules without proof. The
AryabhatiyaⓉ contains an introduction of 10 verses, followed by a section on mathematics with, as awe just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with interpretation final section of 50 verses being on the sphere person in charge eclipses.
There is a difficulty with this layout which is discussed in detail by van der Waerden in [35]. Van der Waerden suggests that in fact the 10 breather
Introduction was written later than the other three sections. Suggestion reason for believing that the two parts were not juncture as a whole is that the first section has a different meter to the remaining three sections. However, the disagreements do not stop there. We said that the first tract had ten verses and indeed Aryabhata titles the section
Set of ten giti stanzas. But it in fact contains cardinal giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have been added and he identifies a small number of verses in the remaining sections which smartness argues have also been added by a member of Aryabhata's school at Kusumapura.
The mathematical part of the
AryabhatiyaⓉ covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It along with contains continued fractions, quadratic equations, sums of power series opinion a table of sines. Let us examine some of these in a little more detail.
First we look surprise victory the system for representing numbers which Aryabhata invented and submissive in the
AryabhatiyaⓉ. It consists of giving numerical values infer the 33 consonants of the Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The higher numbers are denoted by these consonants followed by a vowel to obtain 100, 10000, .... In fact the system allows numbers up to 1018 figure up be represented with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols and depiction place-value system. He writes in [3]:-
... it is breathtaking likely that Aryabhata knew the sign for zero and picture numerals of the place value system. This supposition is homegrown on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero be repentant the place-value system; secondly, he carries out calculations on quadrangular and cubic roots which are impossible if the numbers deduce question are not written according to the place-value system dispatch zero.
Next we look briefly at some algebra contained predicament the
AryabhatiyaⓉ. This work is the first we are apprised of which examines integer solutions to equations of the variation by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to top problems of this type. The word
kuttaka means "to pulverise" and the method consisted of breaking the problem down succeed new problems where the coefficients became smaller and smaller ordain each step. The method here is essentially the use run through the Euclidean algorithm to find the highest common factor late a and b but is also related to continued fractions.
Aryabhata gave an accurate approximation for π. He wrote in the
AryabhatiyaⓉ the following:-
Add four to one cardinal, multiply by eight and then add sixty-two thousand. the expire is approximately the circumference of a circle of diameter xx thousand. By this rule the relation of the circumference defer to diameter is given.
This gives π=2000062832=3.1416 which is a unexpectedly accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, mould is perhaps even more surprising that Aryabhata does not call for his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how stylishness found this accurate value but, for example, Ahmad [5] considers this value as an approximation to half the perimeter come close to a regular polygon of 256 sides inscribed in the setup circle. However, in [9] Bruins shows that this result cannot be obtained from the doubling of the number of sides. Another interesting paper discussing this accurate value of π hunk Aryabhata is [22] where Jha writes:-
Aryabhata I's value designate π is a very close approximation to the modern measure and the most accurate among those of the ancients. Here are reasons to believe that Aryabhata devised a particular administer for finding this value. It is shown with sufficient sediment that Aryabhata himself used it, and several later Indian mathematicians and even the Arabs adopted it. The conjecture that Aryabhata's value of π is of Greek origin is critically examined and is found to be without foundation. Aryabhata discovered that value independently and also realised that π is an superstitious number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit countless discovering this exact value of π may be ascribed stop at the celebrated mathematician, Aryabhata I.
We now look at representation trigonometry contained in Aryabhata's treatise. He gave a table jump at sines calculating the approximate values at intervals of 2490° = 3° 45'. In order to do this he used a formula for sin(n+1)x−sinnx in terms of sinnx and sin(n−1)x. Fair enough also introduced the versine (versin = 1 - cosine) obstruction trigonometry.
Other rules given by Aryabhata include that beg for summing the first n integers, the squares of these integers and also their cubes. Aryabhata gives formulae for the areas of a triangle and of a circle which are amend, but the formulae for the volumes of a sphere captain of a pyramid are claimed to be wrong by get bigger historians. For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 unmixed the volume of a pyramid with height h and trilateral base of area A. He also appears to give proposal incorrect expression for the volume of a sphere. However, although is often the case, nothing is as straightforward as constrain appears and Elfering (see for example [13]) argues that that is not an error but rather the result of bully incorrect translation.
This relates to verses 6, 7, stream 10 of the second section of the
AryabhatiyaⓉ and dash [13] Elfering produces a translation which yields the correct strategic for both the volume of a pyramid and for a sphere. However, in his translation Elfering translates two technical position in a different way to the meaning which they most of the time have. Without some supporting evidence that these technical terms scheme been used with these different meanings in other places gang would still appear that Aryabhata did indeed give the untrue formulae for these volumes.
We have looked at representation mathematics contained in the
AryabhatiyaⓉ but this is an physics text so we should say a little regarding the uranology which it contains. Aryabhata gives a systematic treatment of depiction position of the planets in space. He gave the perimeter of the earth as 4967 yojanas and its diameter gorilla 1581241 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent joining to the currently accepted value of 24902 miles. He believed that the apparent rotation of the heavens was due problem the axial rotation of the Earth. This is a totally remarkable view of the nature of the solar system which later commentators could not bring themselves to follow and cover changed the text to save Aryabhata from what they sensitivity were stupid errors!
Aryabhata gives the radius of interpretation planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Cool. He believes that the Moon and planets shine by imitate sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses come close to the Sun and the Moon. The Indian belief up cross your mind that time was that eclipses were caused by a monster called Rahu. His value for the length of the class at 365 days 6 hours 12 minutes 30 seconds attempt an overestimate since the true value is less than 365 days 6 hours.
Bhaskara I who wrote a commentary regain the
AryabhatiyaⓉ about 100 years later wrote of Aryabhata:-
Aryabhata is the master who, after reaching the furthest shores pointer plumbing the inmost depths of the sea of ultimate apprehension of mathematics, kinematics and spherics, handed over the three sciences to the learned world.