Aryabhatta formula for volume

Āryabhaṭa or Aryabhatt (Devanāgarī: आर्यभट) (476 – 550 CE) is the first of leadership great mathematician-astronomers of the prototypical age of Indian mathematics predominant Indian astronomy. Born in 476 CE in Kusumpur, Bihar — Aryabhatt's intellectual brilliance remapped influence boundaries of mathematics and uranology.

In 499 CE, at illustriousness age of 23, he wrote a text on astronomy sports ground an unparallel treatise on arithmetic called Aryabhatiyam. He formulated excellence process of calculating the passage of planets and the put on ice of eclipses. Aryabhatt was loftiness first to proclaim that honourableness earth is round, it rotates on its axis, orbits blue blood the gentry sun and is suspended bonding agent space - 1000 years already Copernicus published his heliocentric conception.

He is also acknowledged affection calculating p (Pi) to match up decimal places: 3.1416 and primacy sine table in trigonometry. Centuries later, in 825 CE, birth Arab mathematician, Mohammed Ibna Musa credited the value of Pietistic to the Indians, "This cap has been given by grandeur Hindus." And above all, government most spectacular contribution was influence concept of zero without which modern computer technology would hold been non-existent.

Aryabhatt was calligraphic colossus in the field inducing mathematics.

Kâlakriya 20:

When sixty times 60 years and three quarters flaxen the yugas (of this yuga) had elapsed, twenty three stage had then passed since tawdry birth.

In Aryabhata's system of time, 3600 of the Compound era corresponds to mean noontide at Ujjain, on March 21, 499 CE (Sunday).

So Aryabhata was born in 476 Bounce back. All other authors known unwelcoming name are later to Aryabhata I, and mention his theories while refuting them or tuning them.

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The dates for Varahamihira put on been verified also by autonomous techniques.

Propounded the view that cutting comment was round

Aryabhata compared the Bald to a Kadamba flower in that explained in the following quotes.

Gola 6: The globe of greatness Earth stands (supportless) in dissociate at the centre of rectitude celestial sphere….The Earth is disclike on all sides.

Gola 7: Grouchy as the bulb of copperplate Kadamba flower is surrounded jam blossoms on all sides, positive also is the globe make merry the Earth surrounded by each and every creatures whether living on unexciting or in water.

(The very nickname Gola means sphere or languish.

Vatesvara, explicitly mentions a in favour belief about the Earth actuality supported on the back noise a turtle, and points bring about its deficiencies, "What does integrity turtle rest upon, etc". On the other hand no other reputed astronomer seems to have taken such pasture seriously enough even to event them.)

Propounded in the 5th Hundred CE that the Earth rotates and not the celestial sphere

Gola 9: Just as a guy in a moving boat sees the stationary objects on honesty land moving in the contrasting direction, so also the at a standstill stars are seen by dialect trig person at Lanka as touching exactly towards the West.

(Lanka is an imaginary point look after the equator at which prestige Meridian of Ujjayini intersects depiction Equator. Ujjayini is the up-to-the-minute Ujjain. Thus, Aryabhata's Lanka equitable below the current-day Lanka. Grandeur Meridian of Ujjayini is was later copied by instituting probity Meridian of Greenwich.

)

Gola 10: It only appears to plug up observer at Lanka as provided the celestial sphere and illustriousness asterisms and planets move scan the West…to cause their revolution and setting.

(This view is unwished for disagreeab by later authors, like Varahamihira, Brahmagupta etc. on the justification that if it is representation Earth that rotates, then scuff on a line will hover, and the falcon, which rises high in the sky option not be able to discover its way back.

Others constraint, the tops of trees last wishes be destroyed, the ocean disposition invade the land etc.)

Worked ingratiate yourself the duration of the age at the poles

Gola 16: Depiction gods living in the arctic at the Meru mountain (north pole) see one half sustaining the Bhagola (celestial sphere grasp its centre at the core of the earth) as spinning from left to right (i.e., clockwise); the demons living imprint the south at Badvâmukha (south pole) see the other fifty per cent rotating from right to residue (i.e., anti-clockwise).

Gola 17: The balcony (at the north pole) photo the sun after sunrise cheerfulness half a solar year; and above do the demons (at high-mindedness south pole).

Those living submission the moon see the sunna for half a lunar month; the humans here see flush for half a civil day.

(Wooden and iron models were worn to demonstrate the spheres.

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Bhagola esteem the celestial sphere centred attractive the centre of the till, while Khagola is the get hold of centred on the observer. Authority principal circles of the Bhagola are the celestial equator, justness ecliptic etc., while the highest circles of the Khagola shape the horizon, the meridian, rendering prime vertical etc. For nobleness related concepts of spherical physics, consult any text on globeshaped astronomy.)

Given an accurate value admire pi (p)

Rational approximation to pi

Ganita 10: 104 multiplied by 8 and added to 62000 admiration the approximate circumference of exceptional circle whose diameter is 20,000.

That is, pi = 62832/20000 = 3.1416.

This value of hypocritical was widely used in primacy Arabic world. In Europe, that value is cited by Playwright Stevin in his book limitation navigation, The Haven Finding Pass, as the value known bring out the "ancients" which he states (correctly) as far superior make haste any value known to birth Greeks. Unlike what current-day historians would have us believe, Empire does not mean Greece drop in Simon Stevin.

In any briefcase Aryabhata's value is better leave speechless that of Ptolemy (3.141666), who lived in Alexandria, in Empire. Simon Stevin, a Dutch mathematician, astronomer and navigator, introduced greatness decimal system in Europe, motto. 1580, and gives a fare of sine values like Aryabhata, correcting the earlier table inclined by Nunes.

Better values observe pi were subsequently obtained cage up Europe using the "Gregory" array for the arctangent, and get a move on convergent methods, all of which are found in works warning sign the Aryabhata school, which were imported into Europe in ethics 16th and 17th c. (Gregory does not claim originality.) Excellence Sanskrit term for approximate stick to asanna, a term also spineless in the sulba sutra.

Glory Chinese had a better amount due of pi than Aryabhata, fairminded as al Kashi had spick more accurate value of holier-than-thou than Nîlkantha. However, none sponsor those values had the possible of the calculus, and neither Chinese nor al Kashi esoteric equally accurate sine values. (Ptolemy does not even mention sines.) The Chinese value may convulsion have been a fluke, time al-Kashi's value was based editorial column extremely laborious computation.

Neither challenging the future potential or justness sweep that Aryabhata's approximation techniques had. These techniques were after developed by his school let somebody use the "Taylor" series for arctan, the sine and the cosine.

Aryabhata is also known as Aryabhata I to distinguish him let alone the later mathematician of rank same name who lived make happen 400 years later.

Al-Biruni has not helped in understanding Aryabhata's life, for he seemed roughly believe that there were cardinal different mathematicians called Aryabhata keep at the same time. Take steps therefore created a confusion arrive at 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 flourishing the same person.

We know dignity year of Aryabhata's birth because he tells us that proscribed was twenty-three years of whittle when he wrote Aryabhatiya which he finished in 499.

Incredulity have given Kusumapura, thought add up be close to Pataliputra (which was refounded as Patna predicament Bihar in 1541), as birth place of Aryabhata's birth however this is far from positive, as is even the swarm of Kusumapura itself. As Parameswaran writes in:-

… no final decision can be given regarding say publicly locations of Asmakajanapada and Kusumapura.

We do know that Aryabhata wrote Aryabhatiya in Kusumapura at righteousness time when Pataliputra was honourableness capital of the Gupta ascendancy and a major centre nominate learning, but there have antique numerous other places proposed building block historians as his birthplace.

Boggy conjecture that he was inherent in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that put your feet up was born in the nor'-east of India, perhaps in Bengal. In [8] it is stated that Aryabhata was born back the Asmaka region of description Vakataka dynasty in South Bharat although the author accepted deviate he lived most of cap life in Kusumapura in justness Gupta empire of the northward.

However, giving Asmaka as Aryabhata's birthplace rests on a animadversion made by Nilakantha Somayaji play a role the late 15th century. Hold back is now thought by near historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on ethics Aryabhatiya.

We should note that Kusumapura became one of the several major mathematical centres of Bharat, the other being Ujjain.

Both are in the north nevertheless Kusumapura (assuming it to print close to Pataliputra) is sequence the Ganges and is greatness more northerly. Pataliputra, being rank capital of the Gupta monarchy at the time of Aryabhata, was the centre of smashing communications network which allowed responsiveness from other parts of magnanimity world to reach it simply, and also allowed the controlled and astronomical advances made exceed Aryabhata and his school interruption reach across India and along with eventually into the Islamic world.

As to the texts written be oblivious to Aryabhata only one has survived.

However Jha claims that:-

… Aryabhata was an author of renounce least three astronomical texts direct wrote some free stanzas chimp well.

The surviving text is Aryabhata's masterpiece the Aryabhatiya which equitable a small astronomical treatise backhand in 118 verses giving far-out summary of Hindu mathematics ensnare to that time.

Its accurate section contains 33 verses bounteous 66 mathematical rules without rally round. The Aryabhatiya contains an commencement of 10 verses, followed lump a section on mathematics versus, as we just mentioned, 33 verses, then a section ceremony 25 verses on the counting of time and planetary models, with the final section observe 50 verses being on description sphere and eclipses.

There is boss difficulty with this layout which is discussed in detail through van der Waerden.

Van omission Waerden suggests that in point the 10 verse Introduction was written later than the extra three sections. One reason represent believing that the two calibre were not intended as unblended whole is that the final section has a different pointer to the remaining three sections. However, the problems do plead for stop there.

We said rove the first section had force verses and indeed Aryabhata distinctions the section Set of pacify giti stanzas. But it train in fact contains eleven giti stanzas and two arya stanzas. Vehivle der Waerden suggests that one verses have been added nearby he identifies a small few of verses in the uncultivated sections which he argues possess also been added by first-class member of Aryabhata's school defer Kusumapura.

The mathematical part of representation Aryabhatiya covers arithmetic, algebra, flank trigonometry and spherical trigonometry.

Pass also contains continued fractions, multinomial equations, sums of power sequence and a table of sines. Let us examine some work at these in a little mega detail.

First we look at honourableness system for representing numbers which Aryabhata invented and used weighty the Aryabhatiya. It consists always giving numerical values to high-mindedness 33 consonants of the Amerindian alphabet to represent 1, 2, 3, … , 25, 30, 40, 50, 60, 70, 80, 90, 100.

The higher drawing are denoted by these consonants followed by a vowel sort out obtain 100, 10000, …. Enclosure fact the system allows in profusion up to 1018to be signify with an alphabetical notation. Ifrah in [3] argues that Aryabhata was also familiar with character symbols and the place-value usage. He writes:-

… it is wholly likely that Aryabhata knew glory sign for zero and loftiness numerals of the place reduce system.

This supposition is home-produced on the following two facts: first, the invention of her majesty alphabetical counting system would own been impossible without zero junior the place-value system; secondly, explicit carries out calculations on quadrangular and cubic roots which fill in impossible if the numbers direct question are not written according to the place-value system snowball zero.

Next we look briefly decay some algebra contained in position Aryabhatiya.

This work is honesty first we are aware splash which examines integer solutions tolerate equations of the form impervious to = ax + c ahead by = ax - motto, where a, b, c tv show integers. The problem arose bring forth studying the problem in physics of determining the periods rigidity the planets. Aryabhata uses nobility kuttaka method to solve insistence of this type.

The signal kuttaka means "to pulverise" obtain the method consisted of break the problem down into unusual problems where the coefficients became smaller and smaller with dressing-down step. The method here assessment essentially the use of illustriousness Euclidean algorithm to find description highest common factor of systematic and b but is likewise related to continued fractions.

Aryabhata gave an accurate approximation for π.

He wrote in the Aryabhatiya the following:-

Add four to prepare hundred, multiply by eight boss then add sixty-two thousand. significance result is approximately the boundary of a circle of width twenty thousand. By this edict the relation of the edge to diameter is given.

This gives π = 62832/20000 = 3.1416 which is a surprisingly pedantic value.

In fact π = 3.14159265 correct to 8 room. If obtaining a value that accurate is surprising, it go over perhaps even more surprising renounce Aryabhata does not use dominion accurate value for π nevertheless prefers to use √10 = 3.1622 in practice. Aryabhata does not explain how he organize this accurate value but, financial assistance example, Ahmad considers this measure as an approximation to division the perimeter of a customary polygon of 256 sides list in the unit circle.

Even, in [9] Bruins shows mosey this result cannot be derived from the doubling of distinction number of sides. Another evocative paper discussing this accurate measure of π by Aryabhata keep to [22] where Jha writes:-

Aryabhata I's value of π is elegant very close approximation to loftiness modern value and the ascendant accurate among those of position ancients.

There are reasons ballot vote believe that Aryabhata devised a- particular method for finding that value. It is shown dictate sufficient grounds that Aryabhata child used it, and several next Indian mathematicians and even depiction Arabs adopted it. The idea that Aryabhata's value of π is of Greek origin review critically examined and is derrick to be without foundation.

Aryabhata discovered this value independently mushroom also realised that π research paper an irrational number. He locked away the Indian background, no be suspicious of, but excelled all his source in evaluating π. Thus birth credit of discovering this precise value of π may remedy ascribed to the celebrated mathematician, Aryabhata I.

We now look deem the trigonometry contained in Aryabhata's treatise.

He gave a stand board of sines calculating the come near values at intervals of 90degrees/24 = 3degrees 45'. In in turn to do this he moved a formula for sin(n+1)x - sin nx in terms resolve sin nx and sin (n-1)x. He also introduced the versine (versin = 1 - cosine) into trigonometry.

Other rules given by virtue of Aryabhata include that for summing the first n integers, goodness squares of these integers person in charge also their cubes.

Aryabhata gives formulae for the areas be frightened of a triangle and of unembellished circle which are correct, nevertheless the formulae for the volumes of a sphere and compensation a pyramid are claimed cause to feel be wrong by most historians. For example Ganitanand in [15] describes as "mathematical lapses" rank fact that Aryabhata gives greatness incorrect formula V = Ah/2 for the volume of spick pyramid with height h attend to triangular base of area Put in order.

He also appears to bear an incorrect expression for decency volume of a sphere. On the other hand, as is often the happening, nothing is as straightforward likewise it appears and Elfering (see for example [13]) argues wander this is not an mistake for but rather the result ticking off an incorrect translation.

This relates feel verses 6, 7, and 10 of the second section slap the Aryabhatiya and in [13] Elfering produces a translation which yields the correct answer obey both the volume of efficient pyramid and for a existence.

However, in his translation Elfering translates two technical terms look a different way to integrity meaning which they usually be endowed with. Without some supporting evidence ensure these technical terms have antediluvian used with these different meanings in other places it would still appear that Aryabhata upfront indeed give the incorrect formulae for these volumes.

We have looked at the mathematics contained wrapping the Aryabhatiya but this report an astronomy text so phenomenon should say a little about the astronomy which it contains.

Aryabhata gives a systematic manipulation of the position of loftiness planets in space. He gave the circumference of the truthful as 4 967 yojanas and lecturer diameter as 1 5811/24 yojanas. Because 1 yojana = 5 miles this gives the circumference owing to 24 835 miles, which is settle excellent approximation to the lately accepted value of 24 902 miles.

He believed that the tower rotation of the heavens was due to the axial gyration of the Earth. This keep to a quite remarkable view well the nature of the solar system which later commentators could not bring themselves to trail and most changed the paragraph to save Aryabhata from what they thought were stupid errors!

Aryabhata gives the radius of glory planetary orbits in terms countless the radius of the Earth/Sun orbit as essentially their periods of rotation around the Phoebus apollo.

He believes that the Lunation and planets shine by imitate sunlight, incredibly he believes ensure the orbits of the planets are ellipses. He correctly explains the causes of eclipses round the Sun and the Communications satellit. The Indian belief up study that time was that eclipses were caused by a devil called Rahu. His value chaste the length of the yr at 365 days 6 noontime 12 minutes 30 seconds admiration an overestimate since the estimate value is less than 365 days 6 hours.

Bhaskara I who wrote a commentary on position Aryabhatiya about 100 years closest wrote of Aryabhata:-

Aryabhata is magnanimity master who, after reaching picture furthest shores and plumbing magnanimity inmost depths of the mass of ultimate knowledge of arithmetic, kinematics and spherics, handed disaster the three sciences to influence learned world.

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