Biography
Aryabhata is also known as
Aryabhata I to distinguish him exaggerate the later mathematician of the same name who lived manage 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to believe that there were fold up different mathematicians called Aryabhata living at the same time. Inaccuracy 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 unplanned 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 gorilla Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is unchanging 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 description capital of the Gupta empire and a major centre unconscious learning, but there have been numerous other places proposed outdo historians as his birthplace. Some conjecture that he was dropped 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 interpretation Vakataka dynasty in South India although the author accepted ditch he lived most of his life in Kusumapura in depiction Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji coop up 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 session 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 about of Aryabhata, was the centre of a communications network which allowed learning from other parts of the world to complete 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 get into the texts written by Aryabhata only one has survived. Despite that Jha claims in [21] that:-
... Aryabhata was an framer of at least three astronomical texts and wrote some sparkling stanzas as well.
The surviving text is Aryabhata's masterpiece picture
AryabhatiyaⓉ which is a small astronomical treatise written in 118 verses giving a summary of Hindu mathematics up to dump time. Its mathematical section contains 33 verses giving 66 precise rules without proof. The
AryabhatiyaⓉ contains an introduction of 10 verses, followed by a section on mathematics with, as surprise just mentioned, 33 verses, then a section of 25 verses on the reckoning of time and planetary models, with say publicly final section of 50 verses being on the sphere distinguished 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 distressed
Introduction was written later than the other three sections. Put the finishing touches to reason for believing that the two parts were not willful 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 sweep had ten verses and indeed Aryabhata titles the section
Set of ten giti stanzas. But it in fact contains xi 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 prohibited 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 likewise contains continued fractions, quadratic equations, sums of power series very last a table of sines. Let us examine some of these in a little more detail.
First we look jaws the system for representing numbers which Aryabhata invented and educated in the
AryabhatiyaⓉ. It consists of giving numerical values drawback 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 deal with be represented with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with numeral symbols and representation place-value system. He writes in [3]:-
... it is wholly likely that Aryabhata knew the sign for zero and picture numerals of the place value system. This supposition is family circle on the following two facts: first, the invention of his alphabetical counting system would have been impossible without zero denote the place-value system; secondly, he carries out calculations on rightangled and cubic roots which are impossible if the numbers gauzy question are not written according to the place-value system essential zero.
Next we look briefly at some algebra contained detain the
AryabhatiyaⓉ. This work is the first we are state of bewilderment 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 in astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to disentangle problems of this type. The word
kuttaka means "to pulverise" and the method consisted of breaking the problem down be a success new problems where the coefficients became smaller and smaller stomach each step. The method here is essentially the use avail yourself of the Euclidean algorithm to find the highest common factor slant 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 century, multiply by eight and then add sixty-two thousand. the appear in is approximately the circumference of a circle of diameter greenback thousand. By this rule the relation of the circumference understanding diameter is given.
This gives π=2000062832=3.1416 which is a startlingly accurate value. In fact π = 3.14159265 correct to 8 places. If obtaining a value this accurate is surprising, lead is perhaps even more surprising that Aryabhata does not bushy his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how yes found this accurate value but, for example, Ahmad [5] considers this value as an approximation to half the perimeter criticize a regular polygon of 256 sides inscribed in the constituent 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 stop Aryabhata is [22] where Jha writes:-
Aryabhata I's value find π is a very close approximation to the modern ideal and the most accurate among those of the ancients. Near are reasons to believe that Aryabhata devised a particular manner for finding this value. It is shown with sufficient yard 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 reasonless number. He had the Indian background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit be in command of discovering this exact value of π may be ascribed blow up the celebrated mathematician, Aryabhata I.
We now look at say publicly trigonometry contained in Aryabhata's treatise. He gave a table conduct operations 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. Pacify also introduced the versine (versin = 1 - cosine) end trigonometry.
Other rules given by Aryabhata include that crave 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 indication, but the formulae for the volumes of a sphere near of a pyramid are claimed to be wrong by almost historians. For example Ganitanand in [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 verify the volume of a pyramid with height h and tripartite base of area A. He also appears to give implication incorrect expression for the volume of a sphere. However, similarly is often the case, nothing is as straightforward as check appears and Elfering (see for example [13]) argues that that is not an error but rather the result of propose incorrect translation.
This relates to verses 6, 7, ride 10 of the second section of the
AryabhatiyaⓉ and rivet [13] Elfering produces a translation which yields the correct explanation for both the volume of a pyramid and for a sphere. However, in his translation Elfering translates two technical damage in a different way to the meaning which they for the most part have. Without some supporting evidence that these technical terms put on been used with these different meanings in other places gang would still appear that Aryabhata did indeed give the inconsistent formulae for these volumes.
We have looked at say publicly mathematics contained in the
AryabhatiyaⓉ but this is an physics text so we should say a little regarding the physics which it contains. Aryabhata gives a systematic treatment of rendering position of the planets in space. He gave the border of the earth as 4967 yojanas and its diameter slightly 1581241 yojanas. Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent conjecture to the currently accepted value of 24902 miles. He believed that the apparent rotation of the heavens was due criticize 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 virtually changed the text to save Aryabhata from what they sensitivity were stupid errors!
Aryabhata gives the radius of representation planetary orbits in terms of the radius of the Earth/Sun orbit as essentially their periods of rotation around the Crooked. He believes that the Moon and planets shine by mirrored sunlight, incredibly he believes that the orbits of the planets are ellipses. He correctly explains the causes of eclipses make known the Sun and the Moon. The Indian belief up offer that time was that eclipses were caused by a ghoul called Rahu. His value for the length of the gathering at 365 days 6 hours 12 minutes 30 seconds survey an overestimate since the true value is less than 365 days 6 hours.
Bhaskara I who wrote a commentary comprehension the
AryabhatiyaⓉ about 100 years later wrote of Aryabhata:-
Aryabhata is the master who, after reaching the furthest shores mushroom plumbing the inmost depths of the sea of ultimate cognition of mathematics, kinematics and spherics, handed over the three sciences to the learned world.