UNDERSTANDING THE ARYABHATTA
There were also other Jain mathematicians whose work also contributed to
mathematics. But what makes Aryabhatiya so different is the way it was written.
The entire script was written in Sanskrit and hence reads like a poetic
verse rather than a practical manual. There are exactly 123 stanzas in the book
and without a tutor, the book would seem ambiguous.
The book is divided into four sections– Gitikapada, Ganitapada,
Kalakriyapada and the Golapada, each covering various fields.
Gitikapada dealt with time, especially large units of time. Ganitapada
covered mathematics of measurement, arithmetic and geometric progressions.
Kalakriyapada told how one could determine the positions of the planet for any given day and finally, Golapada dealt with the earth’s shape and its celestial presence.
He also said that the moon has no light and shines because it reflects light from the sun. He also proved wrong the false belief that eclipse is caused because of the shadows formed by the shadows cast by the earth and the moon. Aryabhatta used epicycles in a similar manner to the Greek Philosopher Ptolemy to illustrate the inconsistent movement of some planets. This great astronomer wrote the famous treatise Aryabhatiya, which was based on astronomy in 499 AD. This treatise was acknowledged as a masterpiece. In honour of this excellent work Aryabhatta was made head of the Nalanda University by the Gupta ruler Buddhagupta.
UNDERSTANDING THE ARYABHATTA
EARLY LIFE
CHILDHOOD
Aryabhata was born in the region lying between Narmada and Godavari, which was known as Ashmaka and is now identified with Maharashtra, though early Buddhist texts describe Ashmaka as being further south, dakShiNApath or the Deccan, while still other texts describe the Ashmakas as having fought Alexander, which would put them further north. Other traditions in India claim that he was from Kerala and that he traveled to the North, or that he was a Maga Brahmin from Gujarat.
CAREER AND ADULTHOOD
A verse mentions that Aryabhata was the head of an institution (kulapa)
at Kusumapura. Since, the University of Nalanda was in
Pataliputra, and had an astronomical observatory; it is probable that he was
its head too.
Direct details of his work are known only from the Aryabhatiya. His
disciple Bhaskara I calls it Ashmakatantra (or the treatise from the Ashmaka).
The Aryabhatiya is also occasionally referred to as Arya-shatas-aShTa
(literally, Aryabhata’s 108), because there are 108 verses in the text. It also
has 13 introductory verses, and is divided into four pādas or chapters.
Aryabhatiya’s first chapter, Gitikapada, with its large units of time —
kalpa, manvantra, and Yuga — introduces a different cosmology. The duration of
the planetary revolutions during a mahayuga is given as 4.32 million years.
Ganitapada, the second chapter of Aryabhatiya has 33 verses covering
mensuration (kṣetra vyāvahāra), arithmetic and geometric progressions, gnomon
or shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate
equations.
Aryabhatiya’s third chapter Kalakriyapada explains different units of
time, a method for determining the positions of planets for a given day, and a
seven-day week with names for the days of week.
The last chapter of the Aryabhatiya, Golapada describes
Geometric/trigonometric aspects of the celestial sphere, features of the
ecliptic, celestial equator, shape of the earth, cause of day and night, and
zodiacal signs on horizon.
He did not use a symbol for zero; its knowledge was implicit in his
place-value system as a place holder for the powers of ten with null
coefficients.
He did not use the Brahmi numerals, and continued the Sanskritic
tradition from Vedic times of using letters of the alphabet to denote numbers,
expressing quantities in a mnemonic form.
He worked on the approximation for pi thus — add four
to 100, multiply by eight, and then add 62,000, the circumference of a circle
with a diameter of 20,000 can be approached.
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.
In Ganitapada, he gives the area of a triangle as: “for a triangle, the
result of a perpendicular with the half-side is the area”. He discussed ‘sine’
by the name of ardha-jya or half-chord.
Like other ancient Indian mathematicians, he too was interested in
finding integer solutions to Diophantine equations with the form ax + by = c;
he called it the kuṭṭaka (meaning breaking into pieces) method.
His contribution to the study of Algebra is immense. In Aryabhatiya,
Aryabhata provided elegant results for the summation of series of squares and
cubes through well tried formulae.
His system of astronomy was called the audayaka system, in which days
are reckoned from uday, dawn at lanka or “equator”. His later writings, which
apparently proposed the ardha-rAtrikA, or midnight model, are lost.
He correctly believed 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, challenging the prevailing view.
In Aryabhatiya, he writes that ‘setting and rising of planets’ is a
perception similar to that of someone in a boat going forward sees an unmoving
(object) going backward.
He correctly asserted that the planets shine due to the reflection of
sunlight, and that the eclipses occur due to the shadows of moon and earth, and
not caused by a demon called “Rahu”!
He correctly deduced that the orbits of the planets are ellipses; this
is another great discovery not credited to him but to Johannes Kepler (a German
astronomer, born AD 1571).
UNDERSTANDING THE ARYABHATTA
WORKS AND INVENTIONS
1. Aryabhata (476–550
CE) was the first of the major mathematician- astronomers from the classical
age of Indian mathematics and Indian astronomy
2. : Works . Aryabhatiya :mathematics Place value system and zero Pi as
irrational Mensuration and trigonometry Indeterminate equations Algebra
:astronomy Motions of the solar system Eclipses Sidereal periods Heliocentrism
3. His most famous work, Aryabhattiya is a detailed text on mathematics
and astronomy. The mathematical part of the Aryabhatiya covers arithmetic,
algebra and trigonometry. It also contains continued fractions, quadratic
equations, sums of power series and a table of Aryabhattiya
4. Place value system and zero The place-value system, first seen in the
3rd-century Bakhshali Manuscript, was clearly in place in his work. He used
letters of the alphabet to denote numbers, expressing quantities, such as the
table of sines
5. Pi as irrational Aryabhata worked on the approximation for pi and may
have come to the conclusion that pi is irrational. How he arrived it?? He wrote
that if 4 is added to 100 and then multiplied by 8 then added to 62,000 then
divided by 20,000 the answer will be equal to the circumference of a circle of
diameter twenty thousand. This calculates to 3.1416 close to the actual value
Pi (3.14159).
actual value Pi (3.14159)
6. 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. Aryabhata discussed the concept of sine in his work by the name of
ardha-jya, which literally means "half-chord".
7. find N = 8x+5 = 9y+4 = 7z+1??? Aryabhata's method of solving such
problems, elaborated by Bhaskara in 621 CE, is called the kuṭṭaka (कु ट्टक) method. Kuttaka means
"pulverizing" or "breaking into small pieces", and the
method involves a recursive algorithm for writing the original factors in
smaller Indeterminate equations
8. Algebra In Aryabhatiya, Aryabhata provided elegant results for the
summation of series of squares and cubes.
9. Motions of the solar system Aryabhatta was aware that the earth
rotates on its axis. The earth rotates round the sun and the moon moves round
the earth. He discovered the positions of the nine planets and related them to
their rotation round the sun.
10. 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 Rahu
andKetu (identified as the pseudo- planetary lunar nodes), he explains eclipses
in terms of shadows cast by and falling on Eclipses
11. 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; the
modern
12. 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.
UNDERSTANDING THE ARYABHATTA
ARYABHATIYA
Aryabhatiya
is the book written by aryabhatta.Direct details
of Aryabhata's work are therefore known only from the Aryabhatiya. The
name Aryabhatiya is due to later commentators, Aryabhata himself may not have
given it a name; it is referred by his disciple, Bhaskara I,
as Ashmakatantra or the treatise from the Ashmaka. It is also
occasionally referred to as Arya-shatas-aShTa, literally Aryabhata's 108,
which is the number of verses in the text. It is written in the very terse
style typical of the sutra literature, where each line is an aid to memory for
a complex system. Thus, the explication of meaning is due to commentators. The
entire text consists of 108 verses, plus an introductory 13, the whole being
divided into four pAdas or chapters:
1. GitikApAda: (13 verses) Large units of time—kalpa, manvantra, yuga, which present a cosmology that differs from earlier texts such as Lagadha's Vedanga Jyotisha (c. first century B.C.E.). It also includes the table of sines (jya), given in a single verse. For the planetary revolutions during a mahayuga, the number of 4.32mn years is given.
2. GaNitapAda: (33 verses) Covers mensuration (kShetra vyAvahAra), arithmetic and geometric progressions, gnomon/shadows (shanku-chhAyA), simple, quadratic, simultaneous, and indeterminate equations (kuTTaka)
3. KAlakriyApAda: (25 verses) Different units of time and method of determination of positions of planets for a given day. Calculations concerning the intercalary month (adhikamAsa), kShaya-tithis. Presents a seven-day week, with names for 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.
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) and by Nilakantha Somayaji in his Aryabhatiya Bhasya (1465).
UNDERSTANDING THE ARYABHATTA
LEGACY
Aryabhata's work was of great influence in the
Indian astronomical tradition, and influenced several neighboring cultures
through translations. The Arabic translation during the Islamic Golden Age (c.
820), was particularly influential. Some of his results are cited by Al-Khwarizmi, and he is
referred to by the tenth century Arabic scholar Al-Biruni, who states that
Āryabhata's followers believed the Earth to rotate on its axis.
His definitions of sine, as well as
cosine (kojya), versine (ukramajya), and inverse sine (otkram
jya), influenced the birth of trigonometry. He was also the
first to specify sine and versine (1-cosx) tables, in 3.75° intervals from 0°
to 90° to an accuracy of 4 decimal places.
In fact, the modern names "sine" and
"cosine," are a mis-transcription of the
words jya and kojya as introduced by Aryabhata. They were
transcribed as jiba and kojiba in Arabic. They were then misinterpreted
by Gerard of Cremona while translating an Arabic geometry text to Latin; he
took jiba to be the Arabic word jaib, which means "fold in a
garment," L. sinus (c. 1150).
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 were used to compute many Arabic
astronomical tables (zijes). In particular, the astronomical tables in the work
of the Arabic Spain scientist Al-Zarqali (eleventh century), were translated
into Latin as the Tables of Toledo (twelfth century), and remained the most
accurate Ephemeris used in Europe for centuries.
Calendric calculations worked out by Aryabhata and
followers have been in continuous use in India for the practical purposes of
fixing the Panchanga, or Hindu calendar, These were also transmitted to the
Islamic world, and formed the basis for the Jalali calendar introduced in 1073,
by a group of astronomers including Omar Khayyam, versions of
which (modified in 1925) are the national calendars in use in iran and afghanistan today. The
Jalali calendar determines its dates 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 lower in the Jalali calendar than in the Gregorian
calendar.
UNDERSTANDING THE ARYABHATTA
NAMED IN HIS HONOUR
- India's
first satellite Aryabhata, was named after him.
- The lunar
crater Aryabhata is named in his honor.
- The
interschool Aryabhata Maths Competition is named after him.
UNDERSTANDING THE ARYABHATTA
LAST DAYS
Aryabhata is believed to have
died around 550 A.D. He has left an amazing legacy to be sure. A great many
modern mathematicians and astronomers look towards his early work for
inspiration.
UNDERSTANDING THE ARYABHATTA
SOME FACTS ABOUT ARYABHATTA
Aryabhata
is credited to have set up an observatory at the Sun temple in Taregana, Bihar.
Some
sources suggest that Kerala was Aryabhata's main place of life and activity but
others refute this statement.
He
served as the head of an institution (kulapa) at Kusumapura and might have also
been the head of the Nalanda university.
Some
scholars claim that the Arabic text ‘Al ntf’ or ‘Al-nanf’ is a translation of
one of his works.
His most
famous text, ‘Aryabhatiya’, consists of 108 verses and 13 introductory verses.
Aryabhata
did not use the Brahmi numerals; he used letters of the alphabet to denote
numbers.
It is
probable that he might have come to the conclusion that 'pi' is irrational.
He
discussed the concept of ‘sine’ in his work by the name of “ardha-jya”, which
literally means "half-chord".
Calendric
calculations devised by Aryabhata are used for fixing the ‘Panchangam’ (the
Hindu Calendar).
He
correctly stated that the earth rotates about its axis daily.
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