Āryabhaṭīya or
Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the
magnum opusMasterpiece in modern usage refers to a creation that has been given much critical praise, especially one that is considered the greatest work of a person's career or to a work of outstanding creativity, skill or workmanship....
and only extant work of the 5th century
Indian mathematicianIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
,
ĀryabhaṭaAryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
.
Structure and style
The text is written in
SanskritSanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
and divided into four sections, covering a total of 121 verses that describe different results using a mnemonic style typical for such works in India.
1. 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(ca. 1st century BCE). There is also a table of
sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
s (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
3. 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.
4. 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.
It is highly likely that the study of the
Aryabhatiya was meant to be accompanied by the teachings of a well-versed tutor. While some of the verses have a logical flow, some don't and its lack of coherence makes it extremely difficult for a casual reader to follow.
Indian mathematical works often used word numerals before Aryabhata, but the
Aryabhatiya is the oldest extant Indian work with alphabet numerals. That is, he used letters of the alphabet to form words with consonants giving digits and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without it. At the same time, this system of numeration allows for poetic license even in the author's choice of numbers.
Cf.
{{DISPLAYTITLE:Āryabhaṭīya}}
Āryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opusMasterpiece in modern usage refers to a creation that has been given much critical praise, especially one that is considered the greatest work of a person's career or to a work of outstanding creativity, skill or workmanship....
and only extant work of the 5th century Indian mathematicianIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
, ĀryabhaṭaAryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
.
Structure and style
The text is written in SanskritSanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
and divided into four sections, covering a total of 121 verses that describe different results using a mnemonic style typical for such works in India.
1. 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(ca. 1st century BCE). There is also a table of sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
s (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
3. 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.
4. 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.
It is highly likely that the study of the Aryabhatiya was meant to be accompanied by the teachings of a well-versed tutor. While some of the verses have a logical flow, some don't and its lack of coherence makes it extremely difficult for a casual reader to follow.
Indian mathematical works often used word numerals before Aryabhata, but the Aryabhatiya is the oldest extant Indian work with alphabet numerals. That is, he used letters of the alphabet to form words with consonants giving digits and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without it. At the same time, this system of numeration allows for poetic license even in the author's choice of numbers. Cf.
{{DISPLAYTITLE:Āryabhaṭīya}}
Āryabhaṭīya or Āryabhaṭīyaṃ, a Sanskrit astronomical treatise, is the magnum opusMasterpiece in modern usage refers to a creation that has been given much critical praise, especially one that is considered the greatest work of a person's career or to a work of outstanding creativity, skill or workmanship....
and only extant work of the 5th century Indian mathematicianIndian mathematics emerged in the Indian subcontinent from 1200 BCE until the end of the 18th century. In the classical period of Indian mathematics , important contributions were made by scholars like Aryabhata, Brahmagupta, and Bhaskara II. The decimal number system in use today was first...
, ĀryabhaṭaAryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
.
Structure and style
The text is written in SanskritSanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
and divided into four sections, covering a total of 121 verses that describe different results using a mnemonic style typical for such works in India.
1. 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(ca. 1st century BCE). There is also a table of sineIn mathematics, the sine function is a function of an angle. In a right triangle, sine gives the ratio of the length of the side opposite to an angle to the length of the hypotenuse.Sine is usually listed first amongst the trigonometric functions....
s (jya), given in a single verse. The duration of the planetary revolutions during a mahayuga is given as 4.32 million years.
2. Ganitapada (33 verses): covering mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon / shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
3. 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.
4. 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.
It is highly likely that the study of the Aryabhatiya was meant to be accompanied by the teachings of a well-versed tutor. While some of the verses have a logical flow, some don't and its lack of coherence makes it extremely difficult for a casual reader to follow.
Indian mathematical works often used word numerals before Aryabhata, but the Aryabhatiya is the oldest extant Indian work with alphabet numerals. That is, he used letters of the alphabet to form words with consonants giving digits and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without it. At the same time, this system of numeration allows for poetic license even in the author's choice of numbers. Cf. {{unicode, the Sanskrit numerals.
Contents
Crowning glory of Aryabhatiya is the decimal place value notation without which modern, Western-based, mathematics, science and commerce would be impossible. Prior to AryabhataAryabhata was the first in the line of great mathematician-astronomers from the classical age of Indian mathematics and Indian astronomy...
, Babylonians used 60 based place value notation which never gained momentum. Mathematics of Aryabhatta went to Europe through Arabs and was known as "Modus Indorum" or the method of the Indians. This method is none other than our arithmetic today.
The Aryabhatiya begins with an introduction called the "Dasagitika" or "Ten Giti Stanzas." This begins by paying tribute to BrahmanIn Hinduism, Brahman is the one supreme, universal Spirit that is the origin and support of the phenomenal universe. Brahman is sometimes referred to as the Absolute or Godhead which is the Divine Ground of all being...
, the "Cosmic spirit" in Hinduism. Next, Aryabhata lays out the numeration system used in the work. It includes a listing of astronomical constants and the sine table. The book then goes on to give an overview of Aryabhata's astronomical findings.
Most of the mathematics is contained in the next part, the "Ganitapada" or "Mathematics."
The next section is the "Kalakriya" or "The Reckoning of Time." In it, he divides up days, months, and years according to the movement of celestial bodies. He divides up history astrologically - it is from this exposition that historians deduced that the Aryabhatiya was written in c. 499 C.E. It also contains rules for computing the longitudes of planets using eccentricsIn mathematics, the eccentricity, denoted e or \varepsilon, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.In particular,...
and epicycles.
In the final section, the "Gola" or "The Sphere," Aryabhata goes into great detail describing the celestial relationship between the Earth and the cosmos. This section is noted for describing the rotation of the earth on its axis. It further uses the armillary sphereAn armillary sphere is a model of objects in the sky , consisting of a spherical framework of rings, centred on Earth, that represent lines of celestial longitude and latitude and other astronomically important features such as the ecliptic...
and details rules relating to problems of trigonometry and the computation of eclipses.
Significance
The treatise uses a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth. In this model, which is also found in the Paitāmahasiddhānta (ca. AD 425), the motions of the planets are each governed by two epicycles, a smaller manda (slow) epicycle and a larger śīghra (fast) epicycle.
The Aryabhatiya has also been interpreted as advocating heliocentrismHeliocentrism, or heliocentricism, is the astronomical model in which the Earth and planets revolve around a stationary Sun at the center of the universe. The word comes from the Greek . Historically, heliocentrism was opposed to geocentrism, which placed the Earth at the center...
, in that the Earth was taken to be spinning on its axis, the periods of the planets were given with respect to the sun, and it states: "Whoever knows this Dasagitika Sutra which describes the movements of the Earth and the planets in the sphere of the asterismsIn astronomy, an asterism is a pattern of stars recognized on Earth's night sky. It may form part of an official constellation, or be composed of stars from more than one. Like constellations, asterisms are in most cases composed of stars which, while they are visible in the same general direction,...
passes through the paths of the planets and asterisms and goes to the higher Brahman." According to this view, it was heliocentric.
Aryabhata asserted that the Moon, planets, and asterismsIn astronomy, an asterism is a pattern of stars recognized on Earth's night sky. It may form part of an official constellation, or be composed of stars from more than one. Like constellations, asterisms are in most cases composed of stars which, while they are visible in the same general direction,...
shine by reflected sunlight. He also correctly explained the causes of eclipses of the Sun and the Moon. His value for the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds is only 3 minutes 20 seconds longer than the true value of 365 days 6 hours 9 minutes 10 seconds. In this book, the day was reckoned from one sunrise to the next, whereas in his "Āryabhata-siddhānta" he took the day from one midnight to another. There was also difference in some astronomical parameters.
A close approximation to π is given as : "Add four to one hundred, multiply by eight and then add sixty-two thousand. The result is approximately the circumference of a circle of diameter twenty thousand. By this rule the relation of the circumference to diameter is given." In other words, π ≈ 62832/20000 = 3.1416, correct to four rounded-off decimal places.
Aryabhata was the first astronomer to make an attempt at measuring the Earth's circumference since EratosthenesEratosthenes of Cyrene was a Greek mathematician, poet, athlete, geographer, astronomer, and music theorist.He was the first person to use the word "geography" and invented the discipline of geography as we understand it...
(circa 200 BC). Aryabhata accurately calculated the Earth's circumference as 24,835 miles, which was only 0.2% smaller than the actual value of 24,902 miles. This approximation remained the most accurate for over a thousand years.
Significant verses
chaturadhikaM shatamaShTaguNaM dvAShaShTistathA sahasrANAm
AyutadvayaviShkambhasyAsanno vr^ttapariNahaH.
[gaNita pAda, 10]
Add 4 to 100, multiply by 8 and add to 62,000. This is approximately
the circumference of a circle whose diamenter is 20,000.
i.e. 
correct to four places. Even more important however is the word
"Asanna" - approximate, indicating an awareness that even this is an
approximation.
tribhujasya falasharIraM samadalakoTI bhujArdhasaMvargaH
It depicts the area of a triangle.
jyA = sine, koTijyA = cosine
jyA tables :
Circle circumference = minutes of arc = 360x60 = 21600.
Gives radius R = radius of 3438; (exactly 21601.591)
[ with 
, gives 21601.64]
The R sine-differences (at intervals of 225 minutes of arc = 3:45deg),
are given in an alphabetic code as
225,224,222,219.215,210,205,
199,191,183,174,164,154,143,131,119,106,93,79,65,51,37,,22,7
which gives sines for 15 deg as sum of first four = 890 →
sin(15) = 890/3438 = 0.258871 vs. the correct value at 0.258819.
sin(30) = 1719/3438 = 0.5
Expressed as the stanza, using the varga/avarga code:
ka-M 1-5, ca-n~a: 6-10, Ta-Na 11-15, ta-na 16-20, pa-ma 21-25
the avargiya vyanjanas are:
y = 30, r = 40, l=50, v=60, sh=70, Sh=80, s =90 and h=100
makhi (ma=25 + khi=2x100) bhakhi (24+200) fakhi (22+200) dhakhi (219)
Nakhi 215, N~akhi 210, M~akhi 205, hasjha (h=100 + s=90+ jha=9)
skaki (90+ ki=1x00 + ka=1) kiShga (1x100+80+3), shghaki, 70+4+100
kighva (100+4+60) ghlaki (4+50+100) kigra (100+3+40) hakya (100+1+30)
dhaki (19+100) kicha (106) sga (93) shjha (79) Mva (5+60) kla (51)
pta (21+16, could also have been chhya) fa (22) chha (7).
makhi bhakhi dhakhi Nakhi N~akhi M~akhi hasjha
225 224 222 219 215 210 205
skaki kiShga shghaki kighva ghlaki kigra hakya
199 191 183 174 164 154 143
dhaki kicha sga shjha Mva kla pta fa chha
119 106 93 79 65 51 37 22 7
given a carefully chosen radius of 3,438 these values are successive differences of 
to within one digit;
for example,
modern value = 889.820
Both the choice of the radius based on the angle, and the 225 minutes
of arc interpolation
interval, are ideal for the table, better suited than the modern
tables.
Influence
The Aryabhatiya was an extremely influential work as is exhibited by the fact that most notable Indian mathematicians after Aryabhata wrote commentaries on it. At least twelve notable commentaries were written for the Aryabhatiya ranging from the time he was still alive (c. 525) through 1900 ("Aryabhata I" 150-2). The commentators include Bhāskara IBhāskara was a 7th century Indian mathematician, who was apparently the first to write numbers in the Hindu-Arabic decimal system with a circle for the zero, and who gave a unique and remarkable rational approximation of the sine function in his commentary on Aryabhata's work...
and BrahmaguptaBrahmagupta was an Indian mathematician and astronomer who wrote many important works on mathematics and astronomy. His best known work is the Brāhmasphuṭasiddhānta , written in 628 in Bhinmal...
among other notables.
The estimate of the diameter of the Earth in the Tarkīb al‐aflāk of Yaqūb ibn TāriqYaʿqūb ibn Ṭāriq was an 8th-century Persian astronomer and mathematician who lived in Baghdad.- Works :Works ascribed to Yaʿqūb ibn Ṭāriq include:...
, of 2,100 farsakhs, appears to be derived from the estimate of the diameter of the Earth in the Aryabhatiya of 1,050 yojanas.
The work was translated into Arabic around 820 by Al-Khwarizmi, whose On the Calculation with Hindu Numerals was in turn influential in the adoption of the Hindu-Arabic numerals in Europe from the 12th century.
Although the work was influential, there is no definitive English translation.
Aryabhata's methods of astronomical calculations have been in continuous use for practical purposes of fixing the PanchangamA panchāngam is a Hindu astrological almanac, which follows traditional Indian cosmology, and presents important astronomical data in tabulated form. It is sometimes spelled Pancanga, Panchanga, Panchaanga, or Panchānga, and is pronounced Panchānga...
(Hindu calendar)