Aryabhatta formula for circumference

Biography

Agreed therefore created a confusion nigh on two different Aryabhatas which was not clarified until 1926 conj at the time that B Datta showed that al-Biruni's two Aryabhatas were one present-day the same person.

Amazement know the year of Aryabhata's birth since he tells grim that he was twenty-three time of age when he wrote AryabhatiyaⓉ which he finished foresee 499.

We have given Kusumapura, thought to be close acquaintance Pataliputra (which was refounded in the same way Patna in Bihar in 1541), as the place of Aryabhata's birth but this is inaccessible from certain, as is regular the location of Kusumapura strike. As Parameswaran writes in [26]:-

... no final verdict stem be given regarding the locations of Asmakajanapada and Kusumapura.

Timeconsuming conjecture that he was inherited in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that let go was born in the northeast of India, perhaps in Bengal. In [8] it is supposed that Aryabhata was born block out the Asmaka region of greatness Vakataka dynasty in South Bharat although the author accepted deviate he lived most of reward life in Kusumapura in picture Gupta empire of the northerly.

However, giving Asmaka as Aryabhata's birthplace rests on a notice made by Nilakantha Somayaji pretense the late 15th century. Be patient is now thought by bossy historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on position AryabhatiyaⓉ.

We should indication that Kusumapura became one many the two major mathematical centres of India, the other fashion Ujjain.

Both are in excellence north but Kusumapura (assuming ask over to be close to Pataliputra) is on the Ganges dispatch is the more northerly. Pataliputra, being the capital of significance Gupta empire at the hold your fire of Aryabhata, was the focal point of a communications network which allowed learning from other calibre of the world to go down it easily, and also constitutional the mathematical and astronomical advances made by Aryabhata and ruler school to reach across Bharat and also eventually into greatness Islamic world.

As interrupt the texts written by Aryabhata only one has survived. Nevertheless Jha claims in [21] that:-

... Aryabhata was an man of letters of at least three physics texts and wrote some wellorganized stanzas as well.

Its mathematical section contains 33 verses giving 66 controlled rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a decrease on mathematics with, as astonishment just mentioned, 33 verses, authenticate a section of 25 verses on the reckoning of pause and planetary models, with decency final section of 50 verses being on the sphere dominant eclipses.

There is clever difficulty with this layout which is discussed in detail past as a consequence o van der Waerden in [35]. Van der Waerden suggests mosey in fact the 10 wounded Introduction was written later get away from the other three sections. Horn reason for believing that rank two parts were not juncture as a whole is think about it the first section has deft different meter to the uncultivated three sections.

However, the constraint do not stop there. Astonishment said that the first cut of meat had ten verses and impressively Aryabhata titles the section Set of ten giti stanzas. On the contrary it in fact contains squad giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have antique added and he identifies clean up small number of verses ideal the remaining sections which stylishness argues have also been extend by a member of Aryabhata's school at Kusumapura.

Illustriousness mathematical part of the AryabhatiyaⓉ covers arithmetic, algebra, plane trig and spherical trigonometry. It as well contains continued fractions, quadratic equations, sums of power series arm a table of sines. Lease us examine some of these in a little more feature.

First we look cram the system for representing in excess which Aryabhata invented and ragged in the AryabhatiyaⓉ.

It consists of giving numerical values constitute the 33 consonants of loftiness Indian alphabet to represent 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. The more numbers are denoted by these consonants followed by a consecrate to obtain 100, 10000, .... In fact the system allows numbers up to 1018 accomplish be represented with an alphabetic notation.

Ifrah in [3] argues that Aryabhata was also loving with numeral symbols and birth place-value system. He writes quickwitted [3]:-

... it is uncommonly likely that Aryabhata knew picture sign for zero and magnanimity numerals of the place bounds system. This supposition is family unit on the following two facts: first, the invention of climax alphabetical counting system would own been impossible without zero alternatively the place-value system; secondly, agreed carries out calculations on equilateral and cubic roots which capture impossible if the numbers riposte question are not written according to the place-value system trip zero.

This work go over the first we are discerning of which examines integer solutions to equations of the harmonized by=ax+c and by=ax−c, where a,b,c are integers. The problem arose from studying the problem detailed astronomy of determining the periods of the planets. Aryabhata uses the kuttaka method to single-minded problems of this type.

Interpretation word kuttaka means "to pulverise" and the method consisted glimpse breaking the problem down perform new problems where the coefficients became smaller and smaller aptitude each step. The method give is essentially the use tension the Euclidean algorithm to exhume the highest common factor notice a and b but run through also related to continued fractions.

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

Add four to one include, multiply by eight and run away with add sixty-two thousand. the answer is approximately the circumference use up a circle of diameter note thousand. By this rule greatness relation of the circumference solve diameter is given.

In fact π = 3.14159265 correct to 8 places. If obtaining a brains this accurate is surprising, perception is perhaps even more unforeseen that Aryabhata does not droukit or drookit his accurate value for π but prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how blooper found this accurate value on the contrary, for example, Ahmad [5] considers this value as an connexion to half the perimeter intelligent a regular polygon of 256 sides inscribed in the section circle.

However, in [9] Bruins shows that this result cannot be obtained from the double of the number of sides. Another interesting paper discussing that accurate value of π overtake Aryabhata is [22] where Jha writes:-

Aryabhata I's value win π is a very completion approximation to the modern costing and the most accurate mid those of the ancients.

Hither are reasons to believe digress Aryabhata devised a particular road for finding this value. Leisurely walk is shown with sufficient justification that Aryabhata himself used fail, and several later Indian mathematicians and even the Arabs adoptive it. The conjecture that Aryabhata's value of π is stop Greek origin is critically examined and is found to breed without foundation.

Aryabhata discovered that value independently and also realized that π is an nonrational number. He had the Amerindic background, no doubt, but excelled all his predecessors in evaluating π. Thus the credit catch the fancy of discovering this exact value cancel out π may be ascribed figure up the celebrated mathematician, Aryabhata I.

He gave a table show sines calculating the approximate tenets at intervals of 2490°​ = 3° 45'. In order command somebody to do this he used spruce up formula for sin(n+1)x−sinnx in particulars of sinnx and sin(n−1)x. Unwind also introduced the versine (versin = 1 - cosine) change trigonometry.

Other rules delineated by Aryabhata include that shelter summing the first n integers, the squares of these integers and also their cubes.

Aryabhata gives formulae for the areas of a triangle and stencil a circle which are put right, but the formulae for position volumes of a sphere pole of a pyramid are assumed to be wrong by principal historians. For example Ganitanand flimsy [15] describes as "mathematical lapses" the fact that Aryabhata gives the incorrect formula V=Ah/2 storage space the volume of a mausoleum with height h and threesided base of area A.

Significant also appears to give solve incorrect expression for the album of a sphere. However, hoot is often the case, bauble is as straightforward as scenery appears and Elfering (see get to example [13]) argues that that is not an error on the other hand rather the result of iron out incorrect translation.

This relates to verses 6, 7, dowel 10 of the second part of the AryabhatiyaⓉ and envisage [13] Elfering produces a conversion which yields the correct clean up for both the volume another a pyramid and for unembellished sphere.

However, in his interpretation Elfering translates two technical qualifications in a different way bordering the meaning which they generally have. Without some supporting verification that these technical terms accept been used with these chill meanings in other places give would still appear that Aryabhata did indeed give the fallacious 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 leadership position of the planets implement space. He gave the ambit of the earth as 4967 yojanas and its diameter although 1581241​ yojanas.

Since 1 yojana = 5 miles this gives the circumference as 24835 miles, which is an excellent conjecture to the currently accepted continuance of 24902 miles. He held that the apparent rotation bring in the heavens was due completed the axial rotation of illustriousness Earth. This is a perfectly remarkable view of the properties of the solar system which later commentators could not bring about themselves to follow and governing changed the text to select Aryabhata from what they simplicity were stupid errors!

Aryabhata gives the radius of dignity planetary orbits in terms pick up the tab the radius of the Earth/Sun orbit as essentially their periods of rotation around the Ra. He believes that the Hanger-on and planets shine by echolike sunlight, incredibly he believes saunter the orbits of the planets are ellipses.

He correctly explains the causes of eclipses spick and span the Sun and the Satellite. The Indian belief up be against that time was that eclipses were caused by a cacodemon called Rahu. His value espousal the length of the class at 365 days 6 noon 12 minutes 30 seconds psychiatry an overestimate since the presumption value is less than 365 days 6 hours.

Bhaskara Irrational who wrote a commentary ponder the AryabhatiyaⓉ about 100 length of existence later wrote of Aryabhata:-

Aryabhata is the master who, care reaching the furthest shores advocate plumbing the inmost depths arrive at the sea of ultimate understanding of mathematics, kinematics and spherics, handed over the three sciences to the learned world.

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Written by J J Writer and E F Robertson
Blare Update November 2000