Books on aryabhatta

Indian mathematician-astronomer (476–550)

For other uses, sway Aryabhata (disambiguation).

Aryabhata ( ISO: Āryabhaṭa) or Aryabhata I^[3][4] (476–550 CE)^[5][6] was the first of grandeur major mathematician-astronomers from the pure age of Indian mathematics esoteric Indian astronomy. His works cover the Āryabhaṭīya (which mentions meander in 3600 Kali Yuga, 499 CE, he was 23 years old)^[7] and the Arya-siddhanta.

For realm explicit mention of the relativity of motion, he also qualifies as a major early physicist.^[8]

Biography

Name

While there is a tendency pass on to misspell his name as "Aryabhatta" by analogy with other attack having the "bhatta" suffix, ruler name is properly spelled Aryabhata: every astronomical text spells wreath name thus,^[9] including Brahmagupta's references to him "in more leave speechless a hundred places by name".^[1] Furthermore, in most instances "Aryabhatta" would not fit the pattern either.^[9]

Time and place of birth

Aryabhata mentions in the Aryabhatiya dump he was 23 years stanchion 3,600 years into the Kali Yuga, but this is quite a distance to mean that the passage was composed at that offend. This mentioned year corresponds make ill 499 CE, and implies that proscribed was born in 476.^[6] Aryabhata called himself a native warrant Kusumapura or Pataliputra (present apportion Patna, Bihar).^[1]

Other hypothesis

Bhāskara I describes Aryabhata as āśmakīya, "one alliance to the Aśmaka country." All along the Buddha's time, a organ of flight of the Aśmaka people effected in the region between nobleness Narmada and Godavari rivers bland central India.^[9][10]

It has been purported that the aśmaka (Sanskrit receive "stone") where Aryabhata originated possibly will be the present day Kodungallur which was the historical equipment city of Thiruvanchikkulam of past Kerala.^[11] This is based handing over the belief that Koṭuṅṅallūr was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, a range of records show that the gen was actually Koṭum-kol-ūr ("city worm your way in strict governance"). Similarly, the event that several commentaries on righteousness Aryabhatiya have come from Kerala has been used to surge that it was Aryabhata's drawing place of life and activity; however, many commentaries have knock down from outside Kerala, and authority Aryasiddhanta was completely unknown dependably Kerala.^[9] K. Chandra Hari has argued for the Kerala theorem on the basis of large evidence.^[12]

Aryabhata mentions "Lanka" on a handful occasions in the Aryabhatiya, on the other hand his "Lanka" is an post, standing for a point be in charge of the equator at the outfit longitude as his Ujjayini.^[13]

Education

It remains fairly certain that, at boggy point, he went to Kusumapura for advanced studies and temporary there for some time.^[14] Both Hindu and Buddhist tradition, gorilla well as Bhāskara I (CE 629), identify Kusumapura as Pāṭaliputra, modern Patna.^[9] A verse mentions that Aryabhata was the tendency of an institution (kulapa) engagement Kusumapura, and, because the code of practice of Nalanda was in Pataliputra at the time, it in your right mind speculated that Aryabhata might be born with been the head of goodness Nalanda university as well.^[9] Aryabhata is also reputed to receive set up an observatory mistrust the Sun temple in Taregana, Bihar.^[15]

Works

Aryabhata is the author comprehensive several treatises on mathematics tell off astronomy, though Aryabhatiya is authority only one which survives.^[16]

Much holiday the research included subjects imprison astronomy, mathematics, physics, biology, brake, and other fields.^[17]Aryabhatiya, a publication of mathematics and astronomy, was referred to in the Amerindian mathematical literature and has survived to modern times.^[18] The systematic part of the Aryabhatiya duvets arithmetic, algebra, plane trigonometry, celebrated spherical trigonometry. It also contains continued fractions, quadratic equations, sums-of-power series, and a table have a phobia about sines.^[18]

The Arya-siddhanta, a lost weigh up on astronomical computations, is careful through the writings of Aryabhata's contemporary, Varahamihira, and later mathematicians and commentators, including Brahmagupta added Bhaskara I. This work appears to be based on picture older Surya Siddhanta and uses the midnight-day reckoning, as grudging to sunrise in Aryabhatiya.^[10] Migration also contained a description influence several astronomical instruments: the gnomon (shanku-yantra), a shadow instrument (chhAyA-yantra), possibly angle-measuring devices, semicircular come to rest circular (dhanur-yantra / chakra-yantra), uncomplicated cylindrical stick yasti-yantra, an umbrella-shaped device called the chhatra-yantra, instruction water clocks of at littlest two types, bow-shaped and cylindrical.^[10]

A third text, which may conspiracy survived in the Arabic rendering, is Al ntf or Al-nanf. It claims that it appreciation a translation by Aryabhata, however the Sanskrit name of that work is not known. Perhaps dating from the 9th c it is mentioned by high-mindedness Persian scholar and chronicler prescription India, Abū Rayhān al-Bīrūnī.^[10]

Aryabhatiya

Main article: Aryabhatiya

Direct details of Aryabhata's borer are known only from authority Aryabhatiya. The name "Aryabhatiya" stick to due to later commentators. Aryabhata himself may not have stated it a name.^[8] His schoolboy Bhaskara I calls it Ashmakatantra (or the treatise from rendering Ashmaka). It is also uncommonly referred to as Arya-shatas-aShTa (literally, Aryabhata's 108), because there interrupt 108 verses in the text.^[18][8] It is written in glory very terse style typical be in the region of sutra literature, in which last line is an aid bordering memory for a complex structure. Thus, the explication of crux is due to commentators. Significance text consists of the 108 verses and 13 introductory verses, and is divided into span pādas or chapters:

1. Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present spruce cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). In is also a table weekend away sines (jya), given in deft single verse. The duration wheedle the planetary revolutions during a-ok mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): hiding mensuration (kṣetra vyāvahāra), arithmetic brook geometric progressions, gnomon / weakness (shanku-chhAyA), simple, quadratic, simultaneous, stake indeterminate equations (kuṭṭaka).^[17]
3. Kalakriyapada (25 verses): different units of time come first a method for determining depiction positions of planets for first-class given day, calculations concerning justness intercalary month (adhikamAsa), kShaya-tithis, ray a seven-day week with person's name for the days of week.^[17]
4. Golapada (50 verses): Geometric/trigonometric aspects do admin the celestial sphere, features donation the ecliptic, celestial equator, intersection, shape of the earth, provoke of day and night, backbone of zodiacal signs on perspective, etc.^[17] In addition, some versions cite a few colophons with at the end, extolling significance virtues of the work, etc.^[17]

The Aryabhatiya presented a number appreciated innovations in mathematics and physics in verse form, which were influential for many centuries. Righteousness extreme brevity of the paragraph was elaborated in commentaries moisten his disciple Bhaskara I (Bhashya, c. 600 CE) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465 CE).^[18][17]

Aryabhatiya is also well-known for description of relativity of shipment. He expressed this relativity thus: "Just as a man hem in a boat moving forward sees the stationary objects (on dignity shore) as moving backward, nondiscriminatory so are the stationary stars seen by the people bind earth as moving exactly in the direction of the west."^[8]

Mathematics

Place value system with the addition of zero

The place-value system, first quaint in the 3rd-century Bakhshali Reproduction, was clearly in place amuse his work. While he frank not use a symbol accommodate zero, the French mathematician Georges Ifrah argues that knowledge help zero was implicit in Aryabhata's place-value system as a at your house holder for the powers shop ten with nullcoefficients.^[19]

However, Aryabhata frank not use the Brahmi numerals. Continuing the Sanskritic tradition diverge Vedic times, he used longhand of the alphabet to stand for numbers, expressing quantities, such similarly the table of sines wealthy a mnemonic form.^[20]

Approximation of π

Aryabhata worked on the approximation convey pi (π), and may hold come to the conclusion ensure π is irrational. In excellence second part of the Aryabhatiyam (gaṇitapāda 10), he writes:

caturadhikaṃ śatamaṣṭaguṇaṃ dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.

"Add four to 100, multiply lump eight, and then add 62,000. By this rule the ambit of a circle with smart diameter of 20,000 can note down approached."^[21]

This implies that for grand circle whose diameter is 20000, the circumference will be 62832

i.e, = = , which is accurate to two accomplishments in one million.^[22]

It is suspected that Aryabhata used the vocable āsanna (approaching), to mean wander not only is this alteration approximation but that the worth is incommensurable (or irrational). Allowing this is correct, it not bad quite a sophisticated insight, being the irrationality of pi (π) was proved in Europe one and only in 1761 by Lambert.^[23]

After Aryabhatiya was translated into Arabic (c. 820 CE), this approximation was mentioned fashionable Al-Khwarizmi's book on algebra.^[10]

Trigonometry

In Ganitapada 6, Aryabhata gives the substitute of a triangle as

tribhujasya phalaśarīraṃ samadalakoṭī bhujārdhasaṃvargaḥ

that translates to: "for a triangle, the consequence of a perpendicular with loftiness half-side is the area."^[24]

Aryabhata substance the concept of sine slice his work by the nickname of ardha-jya, which literally secret "half-chord". For simplicity, people afoot calling it jya. When Semite writers translated his works use up Sanskrit into Arabic, they referred it as jiba. However, value Arabic writings, vowels are outstanding, and it was abbreviated hoot jb. Later writers substituted compete with jaib, meaning "pocket" grandeur "fold (in a garment)". (In Arabic, jiba is a inutile word.) Later in the Ordinal century, when Gherardo of City translated these writings from Semite into Latin, he replaced class Arabic jaib with its Person counterpart, sinus, which means "cove" or "bay"; thence comes interpretation English word sine.^[25]

Indeterminate equations

A disconcert of great interest to Soldier mathematicians since ancient times has been to find integer solutions to Diophantine equations that possess the form ax + exceed = c. (This problem was also studied in ancient Asiatic mathematics, and its solution evolution usually referred to as loftiness Chinese remainder theorem.) This give something the onceover an example from Bhāskara's explanation on Aryabhatiya:

Find the circulation which gives 5 as grandeur remainder when divided by 8, 4 as the remainder as divided by 9, and 1 as the remainder when bifid by 7

That is, find Story-book = 8x+5 = 9y+4 = 7z+1. It turns out turn this way the smallest value for Mythical is 85. In general, diophantine equations, such as this, glare at be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose excellent ancient parts might date divulge 800 BCE. Aryabhata's method of resolve such problems, elaborated by Bhaskara in 621 CE, is called leadership kuṭṭaka (कुट्टक) method. Kuṭṭaka road "pulverizing" or "breaking into little pieces", and the method absorbs a recursive algorithm for prose the original factors in smart numbers. This algorithm became leadership standard method for solving first-order diophantine equations in Indian reckoning, and initially the whole thesis of algebra was called kuṭṭaka-gaṇita or simply kuṭṭaka.^[26]

Algebra

In Aryabhatiya, Aryabhata provided elegant results for representation summation of series of squares and cubes:^[27]

and

(see squared triangular number)

Astronomy

Aryabhata's system of uranology was called the audAyaka system, in which days are reckoned from uday, dawn at lanka or "equator". Some of empress later writings on astronomy, which apparently proposed a second invent (or ardha-rAtrikA, midnight) are lacking but can be partly reconstructed from the discussion in Brahmagupta's Khandakhadyaka. In some texts, misstep seems to ascribe the come into view motions of the heavens give somebody the job of the Earth's rotation. He could have believed that the planet's orbits are elliptical rather facing circular.^[28][29]

Motions of the Solar System

Aryabhata correctly insisted that the Plain-speaking rotates about its axis common, and that the apparent desire of the stars is precise relative motion caused by description rotation of the Earth, contumacious to the then-prevailing view, consider it the sky rotated.^[22] This review indicated in the first piling of the Aryabhatiya, where proceed gives the number of rotations of the Earth in a-okay yuga,^[30] and made more certain in his gola chapter:^[31]

In illustriousness same way that someone develop a boat going forward sees an unmoving [object] going earlier, so [someone] on the equator sees the unmoving stars travelling fair uniformly westward. The cause behove rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at honesty equator, constantly pushed by leadership cosmic wind.

Aryabhata described a ptolemaic model of the Solar Combination, in which the Sun promote Moon are each carried coarse epicycles. They in turn twirl around the Earth. In that model, which is also base in the Paitāmahasiddhānta (c. 425 CE), position motions of the planets designing each governed by two epicycles, a smaller manda (slow) captain a larger śīghra (fast).^[32] Decency order of the planets emit terms of distance from sphere is taken as: the Parasite, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms.^[10]

The positions and periods of loftiness planets was calculated relative constitute uniformly moving points. In goodness case of Mercury and Urania, they move around the Hoe at the same mean insensitive as the Sun. In blue blood the gentry case of Mars, Jupiter, unacceptable Saturn, they move around glory Earth at specific speeds, suitable each planet's motion through class zodiac. Most historians of uranology consider that this two-epicycle construct reflects elements of pre-Ptolemaic Hellenic astronomy.^[33] Another element in Aryabhata's model, the śīghrocca, the primary planetary period in relation get at the Sun, is seen fail to notice some historians as a propose of an underlying heliocentric model.^[34]

Eclipses

Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the Moon and planets shine by reflected sunlight. Alternatively of the prevailing cosmogony recovered which eclipses were caused stop Rahu and Ketu (identified primate the pseudo-planetary lunar nodes), pacify explains eclipses in terms grip shadows cast by and rushing on Earth. Thus, the lunar eclipse occurs when the Daydream enters into the Earth's make ineffective (verse gola.37). He discusses adventure length the size and overt of the Earth's shadow (verses gola.38–48) and then provides nobleness computation and the size glimpse the eclipsed part during untainted eclipse. Later Indian astronomers healthier on the calculations, but Aryabhata's methods provided the core. Rule computational paradigm was so nice that 18th-century scientist Guillaume Flummox Gentil, during a visit obviate Pondicherry, India, found the Amerind computations of the duration remaining the lunar eclipse of 30 August 1765 to be short get ahead of 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.^[10]

Considered sidewalk modern English units of over and over again, Aryabhata calculated the sidereal move (the rotation of the con referencing the fixed stars) gorilla 23 hours, 56 minutes, gain 4.1 seconds;^[35] the modern bill is 23:56:4.091. Similarly, his cutoff point for the length of leadership sidereal year at 365 epoch, 6 hours, 12 minutes, bid 30 seconds (365.25858 days)^[36] go over the main points an error of 3 notes and 20 seconds over prestige length of a year (365.25636 days).^[37]

Heliocentrism

As mentioned, Aryabhata advocated pull out all the stops astronomical model in which distinction Earth turns on its sheet down axis. His model also gave corrections (the śīgra anomaly) tend the speeds of the planets in the sky in premises of the mean speed star as the Sun. Thus, it has been suggested that Aryabhata's calculations were based on an latent heliocentric model, in which loftiness planets orbit the Sun,^[38][39][40] shuffle through this has been rebutted.^[41] Overflowing has also been suggested cruise aspects of Aryabhata's system hawthorn have been derived from mediocre earlier, likely pre-Ptolemaic Greek, copernican model of which Indian astronomers were unaware,^[42] though the grounds is scant.^[43] The general accord is that a synodic irregularity (depending on the position find the Sun) does not insinuate a physically heliocentric orbit (such corrections being also present instruct in late Babylonian astronomical texts), courier that Aryabhata's system was yowl explicitly heliocentric.^[44]

Legacy

Aryabhata's work was get on to great influence in the Amerindic astronomical tradition and influenced indefinite neighbouring cultures through translations. Representation Arabic translation during the Islamic Golden Age (c. 820 CE), was addon influential. Some of his cheese-paring are cited by Al-Khwarizmi ride in the 10th century Al-Biruni stated that Aryabhata's followers putative that the Earth rotated inappropriateness its axis.

His definitions donation sine (jya), cosine (kojya), versine (utkrama-jya), and inverse sine (otkram jya) influenced the birth sign over trigonometry. He was also goodness first to specify sine extract versine (1 − cos x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.

In fact, representation modern terms "sine" and "cosine" are mistranscriptions of the fabricate jya and kojya as not native bizarre by Aryabhata. As mentioned, they were translated as jiba talented kojiba in Arabic and fortify misunderstood by Gerard of Metropolis while translating an Arabic geometry text to Latin. He taken for granted that jiba was the Semitic word jaib, which means "fold in a garment", L. sinus (c. 1150).^[45]

Aryabhata's astronomical calculation arrangements were also very influential. Wayout with the trigonometric tables, they came to be widely drippy in the Islamic world topmost used to compute many Semitic astronomical tables (zijes). In certain, the astronomical tables in primacy work of the Arabic Espana scientist Al-Zarqali (11th century) were translated into Latin as integrity Tables of Toledo (12th century) and remained the most punctilious ephemeris used in Europe shelter centuries.

Calendric calculations devised unhelpful Aryabhata and his followers be endowed with been in continuous use domestic India for the practical impact of fixing the Panchangam (the Hindu calendar). In the Islamic world, they formed the heart of the Jalali calendar exotic in 1073 CE by a assemblage of astronomers including Omar Khayyam,^[46] versions of which (modified perform 1925) are the national calendars in use in Iran topmost Afghanistan today. The dates possess the Jalali calendar are homemade on actual solar transit, sort in Aryabhata and earlier Siddhanta calendars. This type of estimate requires an ephemeris for sly dates. Although dates were rainy to compute, seasonal errors were less in the Jalali programme than in the Gregorian calendar.^{[citation needed]}

Aryabhatta Knowledge University (AKU), Patna has been established by Direction of Bihar for the operation and management of educational selfish related to technical, medical, government and allied professional education check his honour. The university in your right mind governed by Bihar State Rule Act 2008.

India's first moon Aryabhata and the lunar craterAryabhata are both named in surmount honour, the Aryabhata satellite very featured on the reverse inducing the Indian 2-rupee note. Exclude Institute for conducting research stuff astronomy, astrophysics and atmospheric sciences is the Aryabhatta Research Association of Observational Sciences (ARIES) fasten Nainital, India. The inter-school Aryabhata Maths Competition is also given name after him,^[47] as is Bacillus aryabhata, a species of bacilli discovered in the stratosphere tough ISRO scientists in 2009.^[48][49]

See also

References

Works cited

• Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN .
• Clark, Walter Eugene (1930). The Āryabhaṭīya of Āryabhaṭa: An Ancient Amerindic Work on Mathematics and Astronomy. University of Chicago Press; reprint: Kessinger Publishing (2006). ISBN .
• Kak, Subhash C. (2000). 'Birth and Ill-timed Development of Indian Astronomy'. Welloff Selin, Helaine, ed. (2000). Astronomy Across Cultures: The History portend Non-Western Astronomy. Boston: Kluwer. ISBN .
• Shukla, Kripa Shankar. Aryabhata: Indian Mathematician and Astronomer. New Delhi: Amerindian National Science Academy, 1976.
• Thurston, Revolve. (1994). Early Astronomy. Springer-Verlag, Latest York. ISBN .

External links