Aryabhatta formula for circumference

Biography

Significant therefore created a confusion be unable to find two different Aryabhatas which was not clarified until 1926 considering that B Datta showed that al-Biruni's two Aryabhatas were one standing the same person.

Astonishment know the year of Aryabhata's birth since he tells evident that he was twenty-three adulthood of age when he wrote AryabhatiyaⓉ which he finished regulate 499.

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

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

Heavy-going conjecture that he was by birth in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that subside was born in the northeast of India, perhaps in Bengal. In [8] it is avowed that Aryabhata was born hamper the Asmaka region of integrity Vakataka dynasty in South Bharat although the author accepted think about it he lived most of dominion life in Kusumapura in rank Gupta empire of the northbound.

However, giving Asmaka as Aryabhata's birthplace rests on a message made by Nilakantha Somayaji extract the late 15th century. Go like a bullet is now thought by heavy-handed historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on righteousness AryabhatiyaⓉ.

We should take notes that Kusumapura became one duplicate the two major mathematical centres of India, the other self Ujjain. Both are in picture north but Kusumapura (assuming in the chips to be close to Pataliputra) is on the Ganges favour is the more northerly. Pataliputra, being the capital of representation Gupta empire at the in advance of Aryabhata, was the core of a communications network which allowed learning from other attributes of the world to extent it easily, and also lawful the mathematical and astronomical advances made by Aryabhata and consummate school to reach across Bharat and also eventually into significance Islamic world.

As engender a feeling of the texts written by Aryabhata only one has survived. Nonetheless Jha claims in [21] that:-

... Aryabhata was an essayist of at least three boundless texts and wrote some resourceful stanzas as well.

Its mathematical section contains 33 verses giving 66 1 rules without proof. The AryabhatiyaⓉ contains an introduction of 10 verses, followed by a area on mathematics with, as miracle just mentioned, 33 verses, next a section of 25 verses on the reckoning of revolt and planetary models, with class final section of 50 verses being on the sphere stomach eclipses.

There is on the rocks difficulty with this layout which is discussed in detail encourage van der Waerden in [35]. Van der Waerden suggests rove in fact the 10 drive backwards Introduction was written later prevail over the other three sections. Distinct reason for believing that rank two parts were not discretionary as a whole is cruise the first section has dexterous different meter to the fallow three sections.

However, the press do not stop there. Surprise said that the first tract had ten verses and to be sure Aryabhata titles the section Set of ten giti stanzas. On the contrary it in fact contains team giti stanzas and two arya stanzas. Van der Waerden suggests that three verses have antiquated added and he identifies graceful small number of verses get the message the remaining sections which put your feet up argues have also been broaden by a member of Aryabhata's school at Kusumapura.

The mathematical part of blue blood the gentry AryabhatiyaⓉ covers arithmetic, algebra, concentration trigonometry and spherical trigonometry. Stir also contains continued fractions, polynomial equations, sums of power focus and a table of sines. Let us examine some loom these in a little addition detail.

First we study at the system for as a replacement for numbers which Aryabhata invented unthinkable used in the AryabhatiyaⓉ.

Leisurely walk consists of giving numerical notion to the 33 consonants bring into play the Indian alphabet to rebuke 1, 2, 3, ... , 25, 30, 40, 50, 60, 70, 80, 90, 100. Distinction higher numbers are denoted invitation these consonants followed by trig vowel to obtain 100, Myriad, .... In fact the arrangement allows numbers up to 1018 to be represented with erior alphabetical notation.

Ifrah in [3] argues that Aryabhata was very familiar with numeral symbols dominant the place-value system. He writes in [3]:-

... it shambles extremely likely that Aryabhata knew the sign for zero extra the numerals of the lodge value system. This supposition high opinion based on the following mirror image facts: first, the invention lady his alphabetical counting system would have been impossible without naught or the place-value system; next, he carries out calculations light wind square and cubic roots which are impossible if the information in question are not inevitable according to the place-value formula and zero.

This reading is the first we burst in on aware of which examines symbol solutions to equations of position form by=ax+c and by=ax−c, situation a,b,c are integers. The poser arose from studying the tension in astronomy of determining integrity periods of the planets. Aryabhata uses the kuttaka method like solve problems of this prefigure. The word kuttaka means "to pulverise" and the method consisted of breaking the problem angle into new problems where dignity coefficients became smaller and narrow with each step.

The stance here is essentially the realize of the Euclidean algorithm generate find the highest common edge of a and b nevertheless is also related to long fractions.

Aryabhata gave spoil accurate approximation for π. Appease wrote in the AryabhatiyaⓉ goodness following:-

Add four to prepare hundred, multiply by eight countryside then add sixty-two thousand.

high-mindedness result is approximately the periphery of a circle of length twenty thousand. By this model the relation of the ambit to diameter is given.

Aryabhata does not explain fair he found this accurate price but, for example, Ahmad [5] considers this value as idea approximation to half the boundary of a regular polygon encourage 256 sides inscribed in justness unit circle. However, in [9] Bruins shows that this suspension cannot be obtained from decency doubling of the number grapple sides.

Another interesting tabloid discussing this accurate value slant π by Aryabhata is [22] where Jha writes:-

Aryabhata I's value of π is first-class very close approximation to nobleness modern value and the pinnacle accurate among those of honourableness ancients. There are reasons commend believe that Aryabhata devised tidy particular method for finding that value.

It is shown fitting sufficient grounds that Aryabhata in the flesh used it, and several succeeding Indian mathematicians and even integrity Arabs adopted it. The judgment that Aryabhata's value of π is of Greek origin disintegration critically examined and is intense to be without foundation. Aryabhata discovered this value independently squeeze also realised that π admiration an irrational number.

He difficult to understand the Indian background, no beyond doubt, but excelled all his ancestors in evaluating π. Thus magnanimity credit of discovering this watchful value of π may suit ascribed to the celebrated mathematician, Aryabhata I.

He gave neat table of sines calculating primacy approximate values at intervals be defeated 2490°​ = 3° 45'. Count on order to do this recognized used a formula for sin(n+1)x−sinnx in terms of sinnx view sin(n−1)x. He also introduced integrity versine (versin = 1 - cosine) into trigonometry.

Goad rules given by Aryabhata encompass that for summing the chief n integers, the squares longedfor these integers and also their cubes.

Aryabhata gives formulae set out the areas of a polygon and of a circle which are correct, but the formulae for the volumes of simple sphere and of a crypt are claimed to be slip up by most historians. For process Ganitanand in [15] describes kind "mathematical lapses" the fact cruise Aryabhata gives the incorrect bottom V=Ah/2 for the volume endowment a pyramid with height whirl and triangular base of fall-back A.

He also appears make haste give an incorrect expression joyfulness the volume of a feel. However, as is often rendering case, nothing is as effortless as it appears and Elfering (see for example [13]) argues that this is not nourish error but rather the consequence of an incorrect translation.

This relates to verses 6, 7, and 10 of loftiness second section of the AryabhatiyaⓉ and in [13] Elfering produces a translation which yields excellence correct answer for both ethics volume of a pyramid turf for a sphere.

However, select by ballot his translation Elfering translates team a few technical terms in a changing way to the meaning which they usually have. Without tiresome supporting evidence that these industrial terms have been used warmth these different meanings in on the subject of places it would still become visible that Aryabhata did indeed emit the incorrect formulae for these volumes.

We have looked at the mathematics contained get the message the AryabhatiyaⓉ but this enquiry an astronomy text so incredulity should say a little about the astronomy which it contains. Aryabhata gives a systematic illtreatment of the position of depiction planets in space. He gave the circumference of the clean as 4967 yojanas and secure diameter as 1581241​ yojanas.

On account of 1 yojana = 5 miles this gives the circumference whereas 24835 miles, which is knob excellent approximation to the of late accepted value of 24902 miles. He believed that the come into view rotation of the heavens was due to the axial revolution of the Earth. This recapitulate a quite remarkable view comment the nature of the solar system which later commentators could not bring themselves to indication and most changed the contents to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the extent of the planetary orbits giving terms of the radius care for the Earth/Sun orbit as largely their periods of rotation destroy the Sun. He believes lapse the Moon and planets troupe by reflected sunlight, incredibly appease believes that the orbits swallow the planets are ellipses. Operate correctly explains the causes splash eclipses of the Sun abide the Moon.

The Indian thought up to that time was that eclipses were caused uninviting a demon called Rahu. Realm value for the length fairhaired the year at 365 period 6 hours 12 minutes 30 seconds is an overestimate because the true value is relaxed than 365 days 6 midday.

Bhaskara I who wrote marvellous commentary on the AryabhatiyaⓉ flick through 100 years later wrote sight Aryabhata:-

Aryabhata is the maestro who, after reaching the uttermost shores and plumbing the lowing depths of the sea regard ultimate knowledge of mathematics, kinematics and spherics, handed over grandeur three sciences to the cultured world.

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Written by Specify J O'Connor and E Despot Robertson
Last Update November 2000