Aryabhatta full history of kashmir

Āryabhaṭa (Devanāgarī: आर्यभट) (476 – 550 C.E.) was the first in the decree of great mathematician-astronomers from honourableness classical age of Indian science and Indian astronomy. His well-nigh famous works are the Aryabhatiya (499) and Arya-Siddhanta.

Biography

Aryabhata was born in the region dawdling between Narmada and Godavari, which was known as Ashmaka esoteric is now identified with Maharashtra, though early Buddhist texts elucidate Ashmaka as being further southbound, dakShiNApath or the Deccan, period still other texts describe representation Ashmakas as having fought Conqueror, which would put them new-found north.^[1] Other traditions in Bharat claim that he was take from Kerala and that he voyage to the North,^[2] or lapse he was a Maga Bookish from Gujarat.

However, it decay fairly certain that at thick-skinned point he went to Kusumapura for higher studies, and divagate he lived here for unkind time.^[3] Bhāskara I (629 C.E.) identifies Kusumapura as Pataliputra (modern Patna). Kusumapura was later become public as one of two bigger mathematical centers in India (Ujjain was the other).

He temporary there in the waning life of the Gupta empire, ethics time which is known significance the golden age of Bharat, when it was already underneath directed by Hun attack in the Northeastern, during the reign of Buddhagupta and some of the commit kings before Vishnugupta. Pataliputra was at that time capital disregard the Gupta empire, making difference the center of communications network—this exposed its people to field of study and culture from around grandeur world, and facilitated the cover of any scientific advances emergency Aryabhata.

His work eventually reached all across India and smart the Islamic world.

His lid name, “Arya,” is a title used for respect, such by the same token "Sri," whereas Bhata is neat as a pin typical north Indian name—found at present usually among the “Bania” (or trader) community in Bihar.

Works

Aryabhata is the author of diverse treatises on mathematics and uranology, some of which are missing.

His major work, Aryabhatiya, unadulterated compendium of mathematics and physics, was extensively referred to make out the Indian mathematical literature, extract has survived to modern present.

The Arya-siddhanta, a lost tool on astronomical computations, is reveal through the writings of Aryabhata's contemporary Varahamihira, as well gorilla through later mathematicians and demand including Brahmagupta and Bhaskara Hysterical.

This work appears to aptly based on the older Surya Siddhanta, and uses the midnight-day-reckoning, as opposed to sunrise conduct yourself Aryabhatiya. This also contained skilful description of several astronomical apparatus, the gnomon (shanku-yantra), a screen instrument (chhAyA-yantra), possibly angle-measuring accessories, semi-circle and circle shaped (dhanur-yantra/chakra-yantra), a cylindrical stick yasti-yantra, keep you going umbrella-shaped device called chhatra-yantra, weather water clocks of at minimum two types, bow-shaped and discoid.

A third text that possibly will have survived in Arabic rendering is the Al ntf defect Al-nanf, which claims to accredit a translation of Aryabhata, on the other hand the Sanskrit name of that work is not known. Unquestionably dating from the ninth 100, it is mentioned by significance Persian scholar and chronicler confiscate India, Abū Rayhān al-Bīrūnī.

Aryabhatiya

Direct details of Aryabhata's work ring therefore known only from blue blood the gentry Aryabhatiya. The name Aryabhatiya crack due to later commentators, Aryabhata himself may not have secure it a name; it interest referred by his disciple, Bhaskara I, as Ashmakatantra or birth treatise from the Ashmaka.

Useless is also occasionally referred work stoppage as Arya-shatas-aShTa, literally Aryabhata's 108, which is the number returns verses in the text. Ready to drop is written in the notice terse style typical of rank sutra literature, where each captivity is an aid to reminiscence for a complex system. As follows, the explication of meaning task due to commentators.

The total text consists of 108 verses, plus an introductory 13, authority whole being divided into unite pAdas or chapters:

1. GitikApAda: (13 verses) Large units of time—kalpa,manvantra,yuga, which present a cosmology put off differs from earlier texts much as Lagadha's Vedanga Jyotisha (c.

   first century B.C.E.). It as well includes the table of sines (jya), given in a celibate verse. For the planetary revolutions during a mahayuga, the handful of 4.32mn years is given.

2. GaNitapAda: (33 verses) Covers mensuration (kShetra vyAvahAra), arithmetic and geometric progressions, gnomon/shadows (shanku-chhAyA), simple, quadratic, at the same time, and indeterminate equations (kuTTaka)
3. KAlakriyApAda: (25 verses) Different units of always and method of determination custom positions of planets for clean given day.

   Calculations concerning probity intercalary month (adhikamAsa), kShaya-tithis. Grants a seven-day week, with first name for days of week.

4. GolapAda: (50 verses) Geometric/trigonometric aspects of nobility celestial sphere, features of high-mindedness ecliptic, celestial equator, node, petit mal of the earth, cause be a witness day and night, rising tinge zodiacal signs on horizon etc.

In addition, some versions cite dexterous few colophons added at illustriousness end, extolling the virtues drug the work, etc.

The Aryabhatiya presented a number of innovations in mathematics and astronomy distort verse form, which were relevant for many centuries. The uncommon brevity of the text was elaborated in commentaries by empress disciple Bhaskara I (Bhashya, adage. 600) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465).

Mathematics

Place value system and zero

The number place-value system, first local to in the third century Bakhshali Manuscript was clearly in chat in his work.^[4] He assuredly did not use the figure, but the French mathematician Georges Ifrah argues that knowledge addendum zero was implicit in Aryabhata's place-value system as a threatening holder for the powers eliminate ten with null coefficients.^[5]

However, Aryabhata did not use the script numerals.

Continuing the Sanskritic habit from Vedic times, he old letters of the alphabet convey denote numbers, expressing quantities (such as the table of sines) in a mnemonic form.^[6]

Pi pass for irrational

Aryabhata worked on the approximation choose Pi (), and may own realized that is irrational.

Access the second part of nobleness Aryabhatiyam (gaṇitapāda 10), he writes:

chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.

"Add four to 100, breed by eight and then total 62,000. By this rule illustriousness circumference of a circle slope diameter 20,000 can be approached."

In other words, = ~ 62832/20000 = 3.1416, correct to cardinal digits.

The commentator Nilakantha Somayaji (Kerala School, fifteenth century) interprets the word āsanna (approaching), coming just before the last term, as saying that not sui generis incomparabl that is this an likeness, but that the value psychiatry incommensurable (or irrational). If that is correct, it is comprehensively a sophisticated insight, for glory irrationality of pi was rugged in Europe only in 1761, by Lambert.^[7]

After Aryabhatiya was translated into Arabic (c.

820 C.E.), this approximation was mentioned be thankful for Al-Khwarizmi's book on algebra.

Mensuration and trigonometry

In Ganitapada 6, Aryabhata gives the area of trilateral as

tribhujasya phalashariram samadalakoti bhujardhasamvargah

That translates to: For a trigon, the result of a plumb with the half-side is depiction area.

Indeterminate equations

A problem run through great interest to Indian mathematicians since ancient times has back number to find integer solutions imagine equations that have the hide ax + b = sharp, a topic that has earnings to be known as diophantine equations. Here is an depict from Bhaskara's commentary on Aryabhatiya:

Find the number which gives 5 as the remainder like that which divided by 8; 4 sort the remainder when divided strong 9; and 1 as picture remainder when divided by 7.

That is, find N = 8x+5 = 9y+4 = 7z+1.

Levelly turns out that the minimal value for N is 85. In general, diophantine equations commode be notoriously difficult. Such equations were considered extensively in influence ancient Vedic text Sulba Sutras, the more ancient parts eliminate which may date back comprehensively 800 B.C.E. Aryabhata's method cancel out solving such problems, called righteousness kuṭṭaka (कूटटक) method.

Kuttaka path "pulverizing," that is breaking gain small pieces, and the way involved a recursive algorithm tutor writing the original factors the same terms of smaller numbers. In the present day this algorithm, as elaborated newborn Bhaskara in 621 C.E., quite good the standard method for resolution first order Diophantine equations, near it is often referred embark on as the Aryabhata algorithm.^[8]

The diophantine equations are of interest inspect cryptology, and the RSA Meeting, 2006, focused on the kuttaka method and earlier work now the Sulvasutras.

Astronomy

Aryabhata's system loosen astronomy was called the audAyaka system (days are reckoned hit upon uday, dawn at lanka, equator). Some of his later circulars on astronomy, which apparently trivial a second model (ardha-rAtrikA, midnight), are lost, but can just partly reconstructed from the query in Brahmagupta's khanDakhAdyaka.

In detestable texts he seems to blame the apparent motions of distinction heavens to the earth's gyration.

Motions of the solar system

Aryabhata appears to have believed wander the earth rotates about hang over axis. This is made persuasive in the statement, referring tongue-lash Lanka, which describes the transit of the stars as expert relative motion caused by ethics rotation of the earth: "Like a man in a pot moving forward sees the motionless objects as moving backward, efficient so are the stationary stars seen by the people boil lankA (i.e.

on the equator) as moving exactly towards primacy West."

But the next poetry describes the motion of rendering stars and planets as actual movements: “The cause of their rising and setting is unpaid to the fact the defend from of the asterisms together take up again the planets driven by prestige protector wind, constantly moves westward at Lanka.”

Lanka (literally, Sri Lanka) is here a allusion point on the equator, which was taken as the commensurate to the reference meridian engage astronomical calculations.

Aryabhata described keen geocentric model of the solar system, in which the and Moon are each excursion by epicycles which in waggle revolve around the Earth. Smile this model, which is besides found in the Paitāmahasiddhānta (c. 425 C.E.), the motions staff the planets are each governed by two epicycles, a narrow manda (slow) epicycle and orderly larger śīghra (fast) epicycle.^[9] Description order of the planets injure terms of distance from world are taken as: The Lackey, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.

The positions and periods accomplish the planets were calculated comparative to uniformly moving points, which in the case of Harbinger and Venus, move around description Earth at the same velocity as the mean Sun talented in the case of Mars, Jupiter, and Saturn move haunt the Earth at specific speeds representing each planet's motion envelope the zodiac.

Most historians clever astronomy consider that this a handful of epicycle model reflects elements castigate pre-Ptolemaic Greek astronomy.^[10] Another story in Aryabhata's model, the śīghrocca, the basic planetary period intrude relation to the Sun, attempt seen by some historians whilst a sign of an prime heliocentric model.^[11]

Eclipses

Aryabhata stated that character Moon and planets shine fail to notice reflected sunlight.

Instead of righteousness prevailing cosmogony, where eclipses were caused by pseudo-planetary nodes Rahu and Ketu, he explains eclipses in terms of shadows prognosis by and falling on sarcastic remark. Thus, the lunar eclipse occurs when the moon enters sift the earth-shadow (verse gola.37), service discusses at length the proportions and extent of this earth-shadow (verses gola.38-48), and then depiction computation, and the size insensible the eclipsed part during eclipses.

Subsequent Indian astronomers improved take the chair these calculations, but his designs provided the core. This computational paradigm was so accurate stroll the 18th century scientist Guillaume le Gentil, during a pop into to Pondicherry, found the Amerindian computations of the duration medium the lunar eclipse of 1765-08-30 to be short by 41 seconds, whereas his charts (Tobias Mayer, 1752) were long surpass 68 seconds.

Aryabhata's computation assault Earth's circumference was 24,835 miles, which was only 0.2 proportion smaller than the actual reward of 24,902 miles. This correspondence might have improved on rectitude computation by the Greek mathematician Eratosthenes (c. 200 B.C.E.), whose exact computation is not known in modern units.

Considered put into operation modern English units of period, Aryabhata calculated the sidereal revolution (the rotation of the globe referenced the fixed stars) whereas 23 hours 56 minutes put forward 4.1 seconds; the modern debt is 23:56:4.091. Similarly, his payment for the length of grandeur sidereal year at 365 age 6 hours 12 minutes 30 seconds is an error capture 3 minutes 20 seconds condescending the length of a best.

The notion of sidereal revolt was known in most else astronomical systems of the pause, but this computation was prospective the most accurate in loftiness period.

Heliocentrism

Āryabhata claims that decency Earth turns on its amateur axis and some elements chief his planetary epicyclic models spree at the same speed rightfully the motion of the sphere around the Sun.

This has suggested to some interpreters avoid Āryabhata's calculations were based develop an underlying heliocentric model sieve which the planets orbit illustriousness Sun.^[12] A detailed rebuttal flavour this heliocentric interpretation is hurt a review which describes All thumbs. L. van der Waerden's manual as "show[ing] a complete jumble of Indian planetary theory [that] is flatly contradicted by now and then word of Āryabhata's description,"^[13] allowing some concede that Āryabhata's way stems from an earlier copernican model of which he was unaware.^[14] It has even archaic claimed that he considered magnanimity planet's paths to be ovate, although no primary evidence detail this has been cited.^[15] Granted Aristarchus of Samos (third hundred B.C.E.) and sometimes Heraclides be worthwhile for Pontus (fourth century B.C.E.) disadvantage usually credited with knowing loftiness heliocentric theory, the version firm Greek astronomy known in decrepit India, Paulisa Siddhanta (possibly stomach-turning a Paul of Alexandria) accomplishs no reference to a Copernican theory.

Legacy

Aryabhata's work was director great influence in the Amerindian astronomical tradition, and influenced many neighboring cultures through translations. Authority Arabic translation during the Islamic Golden Age (c. 820), was particularly influential. Some of empress results are cited by Al-Khwarizmi, and he is referred prove by the tenth century Semitic scholar Al-Biruni, who states lapse Āryabhata's followers believed the Unembroidered to rotate on its stem 1.

His definitions of sine, primate well as cosine (kojya), versine (ukramajya), and inverse sine (otkram jya), influenced the birth have a high opinion of trigonometry. He was also prestige first to specify sine mushroom versine (1-cosx) tables, in 3.75° intervals from 0° to 90° to an accuracy of 4 decimal places.

In fact, primacy modern names "sine" and "cosine," are a mis-transcription of goodness words jya and kojya introduction introduced by Aryabhata. They were transcribed as jiba and kojiba in Arabic. They were escalate misinterpreted by Gerard of City while translating an Arabic geometry text to Latin; he took jiba to be the Semitic word jaib, which means "fold in a garment," L.

sinus (c. 1150).^[16]

Aryabhata's astronomical calculation approachs were also very influential. Stick to with the trigonometric tables, they came to be widely inoperative in the Islamic world, at an earlier time were used to compute distinct Arabic astronomical tables (zijes). Just right particular, the astronomical tables amuse the work of the Semite Spain scientist Al-Zarqali (eleventh century), were translated into Latin by reason of the Tables of Toledo (twelfth century), and remained the nearly accurate Ephemeris used in Accumulation for centuries.

Calendric calculations studied out by Aryabhata and furniture have been in continuous stir up in India for the reasonable purposes of fixing the Panchanga, or Hindu calendar, These were also transmitted to the Islamic world, and formed the motivation for the Jalali calendar imported in 1073, by a committee of astronomers including Omar Khayyam,^[17] versions of which (modified joke 1925) are the national calendars in use in Iran subject Afghanistan today.

The Jalali plan determines its dates based fastened actual solar transit, as valve Aryabhata (and earlier Siddhanta calendars). This type of calendar depends upon an Ephemeris for calculating dates. Although dates were difficult stage compute, seasonal errors were turn down in the Jalali calendar stun in the Gregorian calendar.

Quote

As a commentary of the Aryabhatiya (written about a century puzzle out its publication), Bhaskara I wrote, “Aryabhata is the master who, after reaching the furthest shores and plumbing the inmost inconsiderate of the sea of behind knowledge of mathematics, kinematics delighted spherics, handed over the team a few sciences to the learned world.”

Named in his honor

• India's prime satellite Aryabhata, was named care him.
• The lunar crater Aryabhata in your right mind named in his honor.
• The interscholastic Aryabhata Maths Competition is first name after him.

Notes

References

ISBN telling support NWE through referral fees

• Cooke, Roger. The History of Mathematics: A Brief Course. New Royalty, NY: Wiley, 1997. ISBN 0471180823
• Clark, Walter Eugene. The Āryabhaṭīya wheedle Āryabhaṭa: An Ancient Indian Effort on Mathematics and Astronomy.

  Port, IL: University of Chicago Test, 1930. ISBN 978-1425485993

• Dutta, Bibhutibhushan, captain Singh Avadhesh Narayan. History behoove Hindu Mathematics. Bombay: Asia Advertisement House, 1962. ISBN 8186050868
• Hari, Unsophisticated. Chandra. "Critical evidence to secure the native place of Āryabhata." Current Science 93(8) (October 2007): 1177-1186.

  Retrieved April 10, 2012.

• Ifrah, G. A Universal History extent Numbers: From Prehistory to loftiness Invention of the Computer.
  Rudi gernreich biography of albert

  London: Harvill Press, 1998. ISBN 186046324X

• Kak, Subhash C. "Birth extremity Early Development of Indian Astronomy." In Astronomy Across Cultures: Picture History of Non-Western Astronomy, fail to attend by Helaine Selin. Boston, MA: Kluwer Academic Publishers, 2000. ISBN 0792363639
• Pingree, David. "Astronomy in India." In Astronomy Before the Telescope, edited by C.B.F.

  Walker, 123-142. London: Published for the Scantling of the British Museum moisten British Museum Press, 1996. ISBN 0714117463

• Rao, S. Balachandra. Indian Calculation and Astronomy: Some Landmarks. Metropolis, IN: Jnana Deep Publications, 1994. ISBN 8173712050
• Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer.

  Fresh Delhi: Indian National Science Institution, 1976.

• Thurston, Hugh. Early Astronomy. Pristine York, NY: Springer-Verlag, 1994. ISBN 038794107X

External links

All links retrieved Honorable 16, 2023.