The works of Aryabhata dealt with mainly
mathematics and
astronomy. He also worked on the approximation for
pi.
Biography
Name
While there is a tendency to misspell his name as "Aryabhatta" by analogy with other names having the "
bhatta" suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus,
[6] including
Brahmagupta's references to him "in more than a hundred places by name".
[7] Furthermore, in most instances "Aryabhatta" does not fit the metre either.
[6]Time and place of birth
Aryabhata mentions in the
Aryabhatiya that it was composed 3,630 years into the
Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476.
[4]Education
It is fairly certain that, at some point, he went to Kusumapura for advanced studies and lived there for some time.
[8] Both Hindu and Buddhist tradition, as well as
Bhāskara I (CE 629), identify Kusumapura as
Pāṭaliputra, modern
Patna.
[6] A verse mentions that Aryabhata was the head of an institution (
kulapati) at Kusumapura, and, because the university of
Nalanda was in Pataliputra at the time and had an astronomical observatory, it is speculated that Aryabhata might have been the head of the Nalanda university as well.
[6] Aryabhata is also reputed to have set up an observatory at the Sun temple in
Taregana, Bihar.
[9]Other hypotheses
Some archeological evidence suggests that Aryabhata could have originated from the present day
Kodungallur which was the historical capital city of
Thiruvanchikkulam of ancient Kerala.
[10] For instance, one hypothesis was that
aśmaka (Sanskrit for "stone") may be the region in Kerala that is now known as Koṭuṅṅallūr, based on the belief that it was earlier known as Koṭum-Kal-l-ūr ("city of hard stones"); however, old records show that the city was actually Koṭum-kol-ūr ("city of strict governance"). Similarly, the fact that several commentaries on the Aryabhatiya have come from Kerala were used to suggest that it was Aryabhata's main place of life and activity; however, many commentaries have come from outside Kerala.
Aryabhata mentions "Lanka" on several occasions in the
Aryabhatiya, but his "Lanka" is an abstraction, standing for a point on the equator at the same longitude as his
Ujjayini.
[11]Works
The
Arya-siddhanta, a lot work on astronomical computations, is known through the writings of Aryabhata's contemporary,
Varahamihira, and later mathematicians and commentators, including
Brahmagupta and
Bhaskara I. This work appears to be based on the older
Surya Siddhanta and uses the midnight-day reckoning, as opposed to sunrise in
Aryabhatiya. It also contained a description of several astronomical instruments: the
gnomon (
shanku-yantra), a shadow instrument (
chhAyA-yantra), possibly angle-measuring devices, semicircular and circular (
dhanur-yantra /
chakra-yantra), a cylindrical stick
yasti-yantra, an umbrella-shaped device called the
chhatra-yantra, and
water clocks of at least two types, bow-shaped and cylindrical.
[12]A third text, which may have survived in the
Arabic translation, is
Al ntf or
Al-nanf. It claims that it is a translation by Aryabhata, but the Sanskrit name of this work is not known.
Aryabhatiya
Main article:
AryabhatiyaDirect details of Aryabhata's work are known only from the
Aryabhatiya. The name "Aryabhatiya" is due to later commentators. Aryabhata himself may not have given it a name. His disciple
Bhaskara I calls it
Ashmakatantra (or the treatise from the Ashmaka). It is also occasionally referred to as
Arya-shatas-aShTa (literally, Aryabhata's 108), because there are 108 verses in the text. It is written in the very terse style typical of
sutra literature, in which each line is an aid to memory for a complex system. Thus, the explication of meaning is due to commentators. The text consists of the 108 verses and 13 introductory verses, and is divided into four
pādas or chapters:
- Gitikapada: (13 verses): large units of time—kalpa, manvantra, and yuga—which present a cosmology different from earlier texts such as Lagadha's Vedanga Jyotisha (c. 1st century BCE). There is also a table of sines (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
- Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations
- Kalakriyapada (25 verses): different units of time and a method for determining the positions of planets for a given day, calculations concerning the intercalary month (adhikamAsa), kShaya-tithis, and a seven-day week with names for the days of week.
- Golapada (50 verses): Geometric/trigonometric aspects of the celestial sphere, features of the ecliptic, celestial equator, node, shape of the earth, cause of day and night, rising of zodiacal signs on horizon, etc. In addition, some versions cite a few colophons added at the end, extolling the virtues of the work, etc.
The Aryabhatiya presented a number of innovations in mathematics and astronomy in verse form, which were influential for many centuries. The extreme brevity of the text was elaborated in commentaries by his disciple Bhaskara I (
Bhashya, c. 600 CE) and by
Nilakantha Somayaji in his
Aryabhatiya Bhasya, (1465 CE). He was not only the first to find the radius of the earth but was the only one in ancient time including the Greeks and the Romans to find the volume of the earth.
[citation needed]Mathematics
Place value system and zero
However, Aryabhata did not use the Brahmi numerals. Continuing the
Sanskritic tradition from
Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a
mnemonic form.
[14]Approximation of π
Aryabhata worked on the approximation for
pi (
), and may have come to the conclusion that
is irrational. In the second part of the
Aryabhatiyam (
gaṇitapāda 10), he writes:
caturadhikam śatamaṣṭaguṇam dvāṣaṣṭistathā sahasrāṇām
ayutadvayaviṣkambhasyāsanno vṛttapariṇāhaḥ.
"Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached."
[15]
This implies that the ratio of the circumference to the diameter is ((4 + 100) × 8 + 62000)/20000 = 62832/20000 = 3.1416, which is accurate to five
significant figures.
It is speculated that Aryabhata used the word
āsanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (or
irrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by
Lambert.
[16]After Aryabhatiya was translated into
Arabic (c. 820 CE) this approximation was mentioned in
Al-Khwarizmi's book on algebra.
[12]Trigonometry
In Ganitapada 6, Aryabhata gives the area of a triangle as
- tribhujasya phalashariram samadalakoti bhujardhasamvargah
that translates to: "for a triangle, the result of a perpendicular with the half-side is the area."
[17]Aryabhata discussed the concept of
sine in his work by the name of
ardha-jya, which literally means "half-chord". For simplicity, people started calling it
jya. When Arabic writers translated his works from
Sanskrit into Arabic, they referred it as
jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as
jb. Later writers substituted it with
jaib, meaning "pocket" or "fold (in a garment)". (In Arabic,
jiba is a meaningless word.) Later in the 12th century, when
Gherardo of Cremona translated these writings from Arabic into Latin, he replaced the Arabic
jaib with its Latin counterpart,
sinus, which means "cove" or "bay"; thence comes the English
sine. Alphabetic code has been used by him to define a set of increments. If we use Aryabhata's table and calculate the value of sin(30) (corresponding to hasjha) which is 1719/3438 = 0.5; the value is correct. His alphabetic code is commonly known as the Aryabhata cipher.
[18]Indeterminate equations
A problem of great interest to
Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + by = c, a topic that has come to be known as
diophantine equations. This is an example from
Bhāskara's commentary on Aryabhatiya:
- Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7
That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text
Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata's method of solving such problems is called the
kuṭṭaka (कुट्टक) method.
Kuttaka means "pulverizing" or "breaking into small pieces", and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the
Aryabhata algorithm.
[19] The diophantine equations are of interest in
cryptology, and the
RSA Conference, 2006, focused on the
kuttaka method and earlier work in the
Sulbasutras.
Algebra
In
Aryabhatiya Aryabhata provided elegant results for the summation of
series of squares and cubes:
[20] 
and
Astronomy
Aryabhata's system of astronomy was called the
audAyaka system, in which days are reckoned from
uday, dawn at
lanka or "equator". Some of his later writings on astronomy, which apparently proposed a second model (or
ardha-rAtrikA, midnight) are lost but can be partly reconstructed from the discussion in
Brahmagupta's
khanDakhAdyaka. In some texts, he seems to ascribe the apparent motions of the heavens to the
Earth's rotation. He may have believed that the planet's orbits as
elliptical rather than circular.
[21][22]Motions of the solar system
Aryabhata correctly insisted that the earth rotates about its axis daily, and that the apparent movement of the stars is a relative motion caused by the rotation of the earth, contrary to the then-prevailing view in other parts of the world, that the sky rotated. This is indicated in the first chapter of the
Aryabhatiya, where he gives the number of rotations of the earth in a
yuga,
[23] and made more explicit in his
gola chapter:
[24] In the same way that someone in a boat going forward sees an unmoving [object] going backward, so [someone] on the equator sees the unmoving stars going uniformly westward. The cause of rising and setting [is that] the sphere of the stars together with the planets [apparently?] turns due west at the equator, constantly pushed by the cosmic wind.
Aryabhata described a
geocentric model of the solar system, in which the Sun and Moon are each carried by
epicycles. They in turn revolve around the Earth. In this model, which is also found in the
Paitāmahasiddhānta (c. CE 425), the motions of the planets are each governed by two epicycles, a smaller
manda (slow) and a larger
śīghra (fast).
[25] The order of the planets in terms of distance from earth is taken as: the
Moon,
Mercury,
Venus, the
Sun,
Mars,
Jupiter,
Saturn, and the
asterisms."
[12]The positions and periods of the planets was calculated relative to uniformly moving points. In the case of Mercury and Venus, they move around the Earth at the same mean speed as the Sun. In the case of Mars, Jupiter, and Saturn, they move around the Earth at specific speeds, representing each planet's motion through the zodiac. Most historians of astronomy consider that this two-epicycle model reflects elements of pre-Ptolemaic
Greek astronomy.
[26] Another element in Aryabhata's model, the
śīghrocca, the basic planetary period in relation to the Sun, is seen by some historians as a sign of an underlying
heliocentric model.
[27]Eclipses
Solar and lunar eclipses were scientifically explained by Aryabhata. He states that the
Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogony in which eclipses were caused by pseudo-planetary nodes
Rahu and
Ketu, he explains eclipses in terms of shadows cast by and falling on Earth. Thus, the lunar eclipse occurs when the moon enters into the Earth's shadow (verse gola.37). He discusses at length the size and extent of the Earth's shadow (verses gola.38–48) and then provides the computation and the size of the eclipsed part during an eclipse. Later Indian astronomers improved on the calculations, but Aryabhata's methods provided the core. His computational paradigm was so accurate that 18th-century scientist
Guillaume Le Gentil, during a visit to Pondicherry, India, found the Indian computations of the duration of the
lunar eclipse of 30 August 1765 to be short by 41 seconds, whereas his charts (by Tobias Mayer, 1752) were long by 68 seconds.
[12]Sidereal periods
Considered in modern English units of time, Aryabhata calculated the
sidereal rotation (the rotation of the earth referencing the fixed stars) as 23 hours, 56 minutes, and 4.1 seconds;
[28] the modern value is 23:56:4.091. Similarly, his value for the length of the
sidereal year at 365 days, 6 hours, 12 minutes, and 30 seconds (365.25858 days)
[29] is an error of 3 minutes and 20 seconds over the length of a year (365.25636 days).
[30]Heliocentrism
As mentioned, Aryabhata advocated an astronomical model in which the Earth turns on its own axis. His model also gave corrections (the
śīgra anomaly) for the speeds of the planets in the sky in terms of the mean speed of the sun. Thus, it has been suggested that Aryabhata's calculations were based on an underlying
heliocentric model, in which the planets orbit the Sun,
[31][32][33] though this has been rebutted.
[34] It has also been suggested that aspects of Aryabhata's system may have been derived from an earlier, likely pre-Ptolemaic
Greek, heliocentric model of which Indian astronomers were unaware,
[35] though the evidence is scant.
[36] The general consensus is that a synodic anomaly (depending on the position of the sun) does not imply a physically heliocentric orbit (such corrections being also present in late
Babylonian astronomical texts), and that Aryabhata's system was not explicitly heliocentric.
[37]Legacy
India's first satellite named after Aryabhata
Aryabhata's work was of great influence in the Indian astronomical tradition and influenced several neighbouring cultures through translations. The
Arabic translation during the
Islamic Golden Age (c. 820 CE), was particularly influenced. Some of his results are cited by
Al-Khwarizmi and in the 10th century
Al-Biruni stated that Aryabhata's followers believed that the Earth rotated on its axis.
His definitions of
sine (
jya), cosine (
kojya), versine (
utkrama-jya), and inverse sine (
otkram jya) influenced the birth of
trigonometry. He was also the first to specify sine and
versine (1 − cos
x) tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places.
In fact, modern names "sine" and "cosine" are mistranscriptions of the words
jya and
kojya as introduced by Aryabhata. As mentioned, they were translated as
jiba and
kojiba in Arabic and then misunderstood by
Gerard of Cremona while translating an Arabic geometry text to
Latin. He assumed that
jiba was the Arabic word
jaib, which means "fold in a garment", L.
sinus (c. 1150).
[38]Aryabhata's astronomical calculation methods were also very influential. Along with the trigonometric tables, they came to be widely used in the Islamic world and used to compute many
Arabic astronomical tables (
zijes). In particular, the astronomical tables in the work of the
Arabic Spain scientist
Al-Zarqali (11th century) were translated into Latin as the
Tables of Toledo (12th century) and remained the most accurate
ephemeris used in Europe for centuries.
Calendric calculations devised by Aryabhata and his followers have been in continuous use in India for the practical purposes of fixing the
Panchangam (the
Hindu calendar). In the Islamic world, they formed the basis of the
Jalali calendar introduced in 1073 CE by a group of astronomers including
Omar Khayyam,
[39] versions of which (modified in 1925) are the national calendars in use in
Iran and
Afghanistan today. The dates of the Jalali calendar are based on actual solar transit, as in Aryabhata and earlier
Siddhanta calendars. This type of calendar requires an ephemeris for calculating dates. Although dates were difficult to compute, seasonal errors were less in the Jalali calendar than in the
Gregorian calendar.
Aryabhatta Knowledge University (AKU), Patna has been established by Government of Bihar for the development and management of educational infrastructure related to technical, medical, management and allied professional education in his honour. The university is governed by Bihar State University Act 2008.
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