Everything about Aryabhata totally explained
Āryabhaṭa (
Devanāgarī: आर्यभट) (AD
476 –
550) is the first in the line of great mathematician-astronomers from the classical age of
Indian mathematics and
Indian astronomy. Aryabhatta is the father of the
Hindu-Arabic number system which has become universal today. His most famous works are the
Aryabhatiya(AD
499 at age of 23 years) and
Arya-Siddhanta.
Biography
Aryabhata was born in the region lying between Narmada and Godavari, which was known as
Ashmaka. Ashmaka's exact location isn't defined. Some identify it with central India including
Maharashtra and
Madhya Pradesh, though early Buddhist texts describe Ashmaka as being further south,
dakShiNApath or the
Deccan, while 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 travelled to the North, or that he was a
Maga Brahmin from
Gujarat.
However, it's fairly certain that at some point, he went to
Kusumapura for higher studies, and that he lived here for some time.
Bhāskara I (AD
629) identifies Kusumapura as Pataliputra (modern
Patna). He lived there in the dying years of the
Gupta empire, the time which is known as the golden age of India, when it was already under
Hun attack in the Northeast, during the reign of
Buddhagupta and some of the smaller kings before
Vishnugupta.
Works
Aryabhattah is the author of several treatises on mathematics and astronomy, some of which are lost.
His major work,
Aryabhatiya, a compendium of mathematics and astronomy, was extensively referred to in the Indian mathematical literature, and has survived to modern times. The mathematical part of the Aryabhatiya covers arithmetic, algebra, plane trigonometry and spherical trigonometry. It also contains continued fractions, quadratic equations, sums of power series and a table of sines.
The
Arya-siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata's contemporary
Varahamihira, as well as through 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. This also contained a description of several astronomical instruments, the
gnomon (
shanku-yantra), a shadow instrument (
chhAyA-yantra), possibly angle-measuring devices, semi-circle and circle shaped (
dhanur-yantra /
chakra-yantra), a cylindrical stick
yasti-yantra, an umbrella-shaped device called
chhatra-yantra, and
water clocks of at least two types, bow-shaped and cylindrical. ; he certainly didn't use the symbol, but the French mathematician
Georges Ifrah argues that knowledge of zero was implicit in Aryabhata's place-value system as a place holder for the powers of ten with null coefficients.
However, Aryabhata didn't 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.
Pi as Irrational
Aryabhata worked on the approximation for
Pi (
), and may have realized that
is irrational. In the second part of the Aryabhatiyam (gaṇitapāda 10), he writes:
chaturadhikam śatamaśṭaguṇam dvāśaśṭistathā sahasrāṇām
Ayutadvayaviśkambhasyāsanno vrîttapariṇahaḥ.
"Add four to 100, multiply by eight and then add 62,000. By this rule the circumference of a circle of diameter 20,000 can be approached."
Aryabhata interpreted the word
āsanna (approaching), appearing just before the last word, as saying that not only that's this an approximation, but that the value is incommensurable (or
irrational). If this is correct, it's quite a sophisticated insight, for the irrationality of pi was proved in Europe only in 1761 by
Lambert).
After Aryabhatiya was translated into
Arabic (ca. 820 AD)
this approximation was mentioned in
Al-Khwarizmi's book on algebra
His great contribution to mensuration and trigonometry is used in the current international mathematics.
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 + b = cy, a topic that has come to be known as
diophantine equations. Here is an
example from
Bhaskara'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.
for example find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general,
diophantine equations can be notoriously difficult. Such equations were considered extensively in the ancient Vedic text
Sulba Sutras, the more ancient parts of which may date back to
800 BCE. Aryabhata's method of solving such problems, called the
kuṭṭaka (कूटटक) method. Kuttaka means pulverizing, that's breaking into small pieces, and the method involved a recursive algorithm for writing the original factors in terms of smaller numbers. Today this algorithm,
as elaborated by Bhaskara in AD
621, is the standard method for solving first order Diophantine equations,
and it's often referred to as the
Aryabhata algorithm.
The diophantine equations are of interest in
cryptology, and the
RSA Conference, 2006, focused on the
kuttaka method and earlier work in the
Sulvasutras.
Astronomy
Aryabhata's system of astronomy was called the
audAyaka system (days are reckoned from
uday, dawn at
lanka, equator). Some of his later writings on astronomy, which apparently
proposed a second model (
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.
Motions of the Solar System
Aryabhata appears to have believed that the earth rotates about its axis. This is made clear in the statement, referring to
Lanka , which describes the movement of the stars as a relative motion caused by the rotation of the earth:
» Like a man in a boat moving forward sees the stationary objects as moving backward, just so are the stationary stars seen by the people in lankA (for example on the equator) as moving exactly towards the West. [
achalAnibhAni samapashchimagAni - golapAda.9]
But the next verse describes the motion of the stars and planets as real movements: “The cause of their rising and setting is due to the fact the circle of the asterisms together with the planets driven by the provector wind, constantly moves westwards at Lanka”.
Lanka (lit.
Sri Lanka) is here a reference point on the equator, which was taken as the equivalent to the reference meridian for astronomical calculations.
Aryabhata described 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 order of the planets in terms of distance from earth are taken as: the
Moon,
Mercury,
Venus, the
Sun,
Mars,
Jupiter,
Saturn, and the
asterisms 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.
Eclipses
He states that the
Moon and planets shine by reflected sunlight. Instead of the prevailing cosmogyny where 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-shadow (verse gola.37), and discusses at length the size and extent of this earth-shadow (verses gola.38-48), and then the computation, and the size of the eclipsed part during eclipses. Subsequent Indian astronomers improved on these calculations, but his methods provided the core. This computational paradigm was so accurate that the 18th century scientist
Guillaume le Gentil, during a visit to Pondicherry, found the Indian computations of the duration of the
lunar eclipse of
1765-08-30 to be short by 41 seconds, whereas his charts (by Tobias Mayer,
1752) were long by 68 seconds. A detailed rebuttal to this heliocentric interpretation is in a review which describes
B. L. van der Waerden's book as "show[ing] a complete misunderstanding of Indian planetary theory [that] is flatly contradicted by every word of Āryabhata's description," although some concede that Āryabhata's system stems from an earlier heliocentric model of which he was unaware. It has even been claimed that he considered the planet's paths to be
elliptical, although no primary evidence for this has been cited. Though
Aristarchus of Samos (3rd century BC) and sometimes
Heraclides of Pontus (4th century BC) are usually credited with knowing the heliocentric theory, the version of
Greek astronomy known in ancient India,
Paulisa Siddhanta (possibly by a
Paul of
Alexandria) makes no reference to a Heliocentric theory.
Legacy
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 (ca.
820), was particularly influential. Some of his results are cited by
Al-Khwarizmi, and he's referred to by the 10th 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 (11th c.), were translated into Latin as the
Tables of Toledo (12th c.), 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
Panchangam, or
Hindu calendar, These were also transmitted to the Islamic world, and formed the basis for the
Jalali calendar introduced
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.
India's first satellite
Aryabhata, was named after him. The
lunar crater Aryabhata is named in his honour. An Institute for conducting research in Astronomy, Astrophysics and atmospheric sciences has been named as Aryabhatta Research Institute of observational sciences (ARIES) near Nainital, India.
The interschool
Aryabhatta Maths Competition is named after him.
Further Information
Get more info on 'Aryabhata'.
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