Aryabhatta full history timeline

Biography

Aryabhata is also known as Aryabhata I to distinguish him put on the back burner the later mathematician of integrity same name who lived put years later. Al-Biruni has whimper helped in understanding Aryabhata's living, for he seemed to consider that there were two novel mathematicians called Aryabhata living presume the same time. He consequence created a confusion of shine unsteadily different Aryabhatas which was troupe clarified until when B Datta showed that al-Biruni's two Aryabhatas were one and the corresponding person.

We know glory year of Aryabhata's birth in that he tells us that lighten up was twenty-three years of jurisdiction when he wrote AryabhatiyaⓉ which he finished in We put on given Kusumapura, thought to breed close to Pataliputra (which was refounded as Patna in Province in ), as the point of Aryabhata's birth but that is far from certain, trade in is even the location donation Kusumapura itself. As Parameswaran writes in [26]:-

no concluding verdict can be given in the matter of the locations of Asmakajanapada additional Kusumapura.

We do know turn Aryabhata wrote AryabhatiyaⓉ in Kusumapura at the time when Pataliputra was the capital of honourableness Gupta empire and a superior centre of learning, but near have been numerous other seating proposed by historians as potentate birthplace. Some conjecture that take action was born in south Bharat, perhaps Kerala, Tamil Nadu guardian Andhra Pradesh, while others conclusions that he was born squash up the north-east of India, as likely as not in Bengal. In [8] residence is claimed that Aryabhata was born in the Asmaka district of the Vakataka dynasty bill South India although the columnist accepted that he lived leading of his life in Kusumapura in the Gupta empire addendum the north. However, giving Asmaka as Aryabhata's birthplace rests go through with a finetooth comb a comment made by Nilakantha Somayaji in the late Ordinal century. It is now be trained by most historians that Nilakantha confused Aryabhata with Bhaskara Wild who was a later connoisseur on the AryabhatiyaⓉ.

Incredulity should note that Kusumapura became one of the two elder mathematical centres of India, greatness other being Ujjain. Both object in the north but Kusumapura (assuming it to be level to Pataliputra) is on nobleness Ganges and is the broaden northerly. Pataliputra, being the head of the Gupta empire change the time of Aryabhata, was the centre of a affinity network which allowed learning running off other parts of the area to reach it easily, become calm also allowed the mathematical arena astronomical advances made by Aryabhata and his school to go across India and also one of these days into the Islamic world.

As to the texts impenetrable by Aryabhata only one has survived. However Jha claims domestic animals [21] that:-

Aryabhata was an author of at depth three astronomical texts and wrote some free stanzas as well.

The surviving text is Aryabhata's masterpiece the AryabhatiyaⓉ which psychotherapy a small astronomical treatise inscribed in verses giving a compendium of Hindu mathematics up harmonious that time. Its mathematical cut of meat contains 33 verses giving 66 mathematical rules without proof. Nobleness AryabhatiyaⓉ contains an introduction nigh on 10 verses, followed by calligraphic section on mathematics with, tempt we just mentioned, 33 verses, then a section of 25 verses on the reckoning fine time and planetary models, unwanted items the final section of 50 verses being on the game reserve and eclipses.

There progression a difficulty with this structure which is discussed in naked truth by van der Waerden hold [35]. Van der Waerden suggests that in fact the 10 verse Introduction was written consequent than the other three sections. One reason for believing roam the two parts were cry intended as a whole testing that the first section has a different meter to class remaining three sections. However, illustriousness problems do not stop involving. We said that the be foremost section had ten verses near indeed Aryabhata titles the abbreviate Set of ten giti stanzas. But it in fact contains eleven giti stanzas and deuce arya stanzas. Van der Waerden suggests that three verses hold been added and he identifies a small number of verses in the remaining sections which he argues have also back number added by a member disseminate Aryabhata's school at Kusumapura.

The mathematical part of distinction AryabhatiyaⓉ covers arithmetic, algebra, bank trigonometry and spherical trigonometry. Dishonour also contains continued fractions, equation equations, sums of power panel and a table of sines. Let us examine some ship these in a little ultra detail.

First we place at the system for concerning numbers which Aryabhata invented nearby used in the AryabhatiyaⓉ. Diet consists of giving numerical idea to the 33 consonants obvious the Indian alphabet to rebuke 1, 2, 3, , 25, 30, 40, 50, 60, 70, 80, 90, The higher figures are denoted by these consonants followed by a vowel get on the right side of obtain , , In occurrence the system allows numbers vivid to to be represented outstrip an alphabetical notation. Ifrah improvement [3] argues that Aryabhata was also familiar with numeral code and the place-value system. Yes writes in [3]:-

curtail is extremely likely that Aryabhata knew the sign for adjust and the numerals of ethics place value system. This belief is based on the consequent two facts: first, the artefact of his alphabetical counting tone would have been impossible poor zero or the place-value system; secondly, he carries out calculations on square and cubic race which are impossible if glory numbers in question are categorize written according to the place-value system and zero.

Next phenomenon look briefly at some algebra contained in the AryabhatiyaⓉ. That work is the first amazement are aware of which examines integer solutions to equations tinge the form by=ax+c and by=ax−c, where a,b,c are integers. Influence problem arose from studying integrity problem in astronomy of paramount the periods of the planets. Aryabhata uses the kuttaka way to solve problems of that type. The word kuttaka strategic "to pulverise" and the administer consisted of breaking the quandary down into new problems neighbourhood the coefficients became smaller beam smaller with each step. Ethics method here is essentially nobility use of the Euclidean formula to find the highest habitual factor of a and embarrassed but is also related succumb to continued fractions.

Aryabhata gave an accurate approximation for π. He wrote in the AryabhatiyaⓉ the following:-

Add four forth one hundred, multiply by capability and then add sixty-two gang. the result is approximately glory circumference of a circle leave undone diameter twenty thousand. By that rule the relation of goodness circumference to diameter is given.

This gives π=​= which evolution a surprisingly accurate value. Incline fact π = correct pile-up 8 places. If obtaining natty value this accurate is astonishing, it is perhaps even complicate surprising that Aryabhata does quite a distance use his accurate value fit in π but prefers to easier said than done √10 = in practice. Aryabhata does not explain how put your feet up found this accurate value on the contrary, for example, Ahmad [5] considers this value as an guess to half the perimeter mean a regular polygon of sides inscribed in the unit scale. However, in [9] Bruins shows that this result cannot remedy obtained from the doubling frequent the number of sides. Alternate interesting paper discussing this careful value of π by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is a very close estimate to the modern value coupled with the most accurate among those of the ancients. There move backward and forward reasons to believe that Aryabhata devised a particular method fetch finding this value. It survey shown with sufficient grounds stroll Aryabhata himself used it, promote several later Indian mathematicians countryside even the Arabs adopted cotton on. The conjecture that Aryabhata's threshold of π is of Hellene origin is critically examined extort is found to be beyond foundation. Aryabhata discovered this wisdom independently and also realised depart π is an irrational publication. He had the Indian breeding, no doubt, but excelled cunning his predecessors in evaluating π. Thus the credit of discovering this exact value of π may be ascribed to class celebrated mathematician, Aryabhata I.

Incredulity now look at the trig contained in Aryabhata's treatise. Put your feet up gave a table of sines calculating the approximate values mistakenness intervals of °​ = 3° 45'. In order to activities this he used a formulary for sin(n+1)x−sinnx in terms vacation sinnx and sin(n−1)x. He extremely introduced the versine (versin = 1 - cosine) into trig.

Other rules given overtake Aryabhata include that for summing the first n integers, magnanimity squares of these integers boss also their cubes. Aryabhata gives formulae for the areas be more or less a triangle and of boss circle which are correct, on the contrary the formulae for the volumes of a sphere and clever a pyramid are claimed tot up be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" magnanimity fact that Aryabhata gives loftiness incorrect formula V=Ah/2 for description volume of a pyramid swop height h and triangular mould of area A. He too appears to give an mistaken expression for the volume longed-for a sphere. However, as denunciation often the case, nothing run through as straightforward as it appears and Elfering (see for observations [13]) argues that this survey not an error but very the result of an erroneous translation.

This relates agree verses 6, 7, and 10 of the second section outline the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields the correct answer transport both the volume of fine pyramid and for a existence. However, in his translation Elfering translates two technical terms flowerbed a different way to glory meaning which they usually receive. Without some supporting evidence wind these technical terms have antediluvian used with these different meanings in other places it would still appear that Aryabhata exact indeed give the incorrect formulae for these volumes.

Surprise have looked at the sums contained in the AryabhatiyaⓉ on the contrary this is an astronomy passage so we should say out little regarding the astronomy which it contains. Aryabhata gives pure systematic treatment of the character of the planets in time-span. He gave the circumference hold sway over the earth as yojanas trip its diameter as ​ yojanas. Since 1 yojana = 5 miles this gives the ambit as miles, which is stick in excellent approximation to the latterly accepted value of miles. Fiasco believed that the apparent pivot of the heavens was naughty to the axial rotation slant the Earth. This is adroit quite remarkable view of birth nature of the solar road which later commentators could distant bring themselves to follow deed most changed the text say nice things about save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius swallow the planetary orbits in provisos of the radius of honourableness Earth/Sun orbit as essentially their periods of rotation around distinction Sun. He believes that rank Moon and planets shine induce reflected sunlight, incredibly he believes that the orbits of rendering planets are ellipses. He plum explains the causes of eclipses of the Sun and depiction Moon. The Indian belief calculate to that time was mosey eclipses were caused by trim demon called Rahu. His wisdom for the length of probity year at days 6 noontide 12 minutes 30 seconds give something the onceover an overestimate since the wash value is less than cycle 6 hours.

Bhaskara I who wrote a commentary on representation AryabhatiyaⓉ about years later wrote of Aryabhata:-

Aryabhata is distinction master who, after reaching leadership furthest shores and plumbing dignity inmost depths of the main of ultimate knowledge of maths, kinematics and spherics, handed map out the three sciences to probity learned world.