Aryabhața was born more than 1500 years ago
on March 21, 476. Aryabhața himself mentions his exact date of birth in

*Aryabhația*, the most famous and only surviving work of Aryabhața, in the following verse (iii.10):
“When sixty times sixty years and three
quarter yugas (of the current yuga) had elapsed, twenty three years had then
passed since my birth.” This verse says that Aryabhața was exactly 23 years of
age on the date which, according to the calendar being followed today,
corresponds to March 21, 499 A.D. It is also known from information in the

*Aryabhația*and from the commentaries of later scholars such as Bhaskara I, that Aryabhața flourished in Kusumpura, which was another name for Pataliputra (modern Patna) in Magadha (modern Bihar). The world famous university of Nalanda was located in Pataliputra where scholars from all over the world used to come to study. The university also had an astronomical observatory and most likely Aryabhața was a*kulapati*(head of a university/school) who used to teach there. Yet other accounts mention him as being active in the region near Kerala, and his works were certainly popular there for a long time.
As mentioned, the most famous and only surviving work of
Aryabhața is the

*Aryabhația*. It is divided into four chapters, or*pāda:**gītikā**pāda*,*gaņita pāda, kalakriya and Gola pāda*. While the 1^{st}, 3^{rd}and 4^{th}are mostly astronomical, the 2^{nd}one deals mainly with mathematics. It is marked for tis brevity and conciseness of composition. The*Aryabhația*is written in the form of*sutra*s, characterized by its terseness, which makes it easy to memorize. (*Sutras*are short and serve as quick aids to the memory: examples are*N**ā**rada Bhakti**sutras*,*Patanjali Yoga sutras*, etc.*Bh**ā**shyas*are commentaries, e.g.*Shankarabh**ā**shya*by Adi Shankaracharya on the*Bhagavad Gita*.) His students and several subsequent mathematicians wrote commentaries on it in Sanskrit and the regional Indian languages, and made several new discoveries in the process.
Below we mention some of the important
results described by Aryabhața in the

*Aryabhația*.
Prior to Aryabhața it was generally believed that the earth was at
the center of the universe and that the heavenly bodies revolved around it.
Aryabhața was one of the first astronomers to realize that it was the earth
which was rotating around its axis which made it appear that the stars and
planets were going around it. The apparent motion of the stars and planets due
to the rotation of the earth is illustrated by him through the following
analogy:

Which says, “Just as a man in a boat moving
forward sees the stationary objects (on either side of the river) as moving
backward, just so are the stationary stars seen by people at Lanka as moving
exactly towards the west. (It so appears as if) the entire structure of the
asterisms together with the planets were moving exactly towards the west of
Lanka, being constantly driven by the provector wind, to cause their rising and
setting.”

The above description is just a statement of the principle of relativity for inertial reference frames.

In addition to proposing that it was the
earth which was rotating, Aryabahața also gave a highly accurate value of the
time taken by the earth to complete a sidereal rotation. His value was 23h,
56m, 4.1s, which is exceedingly close to the modern value of 23h, 56m, 4.091s.

His results on astronomy were based on astronomical observations. His work on astronomy was so influential that it started a new school of Astronomy known as the Aryabhața school.

Perhaps one of Aryabhața’s most well known
achievements is his calculation of the value of π to five significant figures.
The following verse gives his approximation of π:

“100 plus 4, multiplied by 8, and added to
62,000 : this is the nearly approximate measure of the circumference of a
circle whose diameter is 20,000.”

In other words, his value of π
is 3.1416. Such an accurate value does
not occur in any earlier work on mathematics and constitutes a marvellous achievement
of Aryabhața. Moreover, Aryabhața says that the value is only approximate,
implying that he was aware that π is an irrational number which cannot be
expressed as a simple ratio of two integers. In fact, a commentator Nilakantha
in his

*Aryabhația-bh**ā**shya*says why*asanna*is used: “Why has an approximate value been mentioned here (in*Aryabha**ț**ia*) instead of the actual value?” And goes on to give the answer: “Given a certain unit of measurement in terms of which the diameter specified has no fractional part, the same measure when employed to specify the circumference will certainly have a fractional part…even if you go on a long way (i.e. keep on reducing the measure of the unit employed), the fractional part will only become very small. A situation in which there will be no fractional part is impossible, and this is what is the import of the expression*Asanna*(can be approached).”
There are several results on series in
Aryabhațiya, which are typically taught in high school mathematics. (Somehow
in school books it is never mentioned that it comes from Aryabhația.)
The following verse is on arithmetic progressions:

“Diminish the given number of terms by one,
then divide by two, then increase by the number of the preceding terms (if
any), then multiply by the common difference, and then increase by the first
term of the (whole) series: the result is the arithmetic mean (of the given
number of terms). This multiplied by the given number of terms is the sum of
the given terms. Alternatively, multiply the sum of the first and last terms
(of the series or partial series which is to be summed up) by half the number
of terms.” (From

*Ganita Pada*Verse 19 of*Aryabhația*.)
In other words, let

*a*be the first term of an arithmetic progression, let*d*be the common difference. Let the*r*^{th}term of the progression be denoted by*t*, so that_{r}*t*=_{r}*a*+ (*r*-1)*d*. Then the above verse expresses the sum and arithmetic mean of*n*consecutive terms of an arithmetic progression starting from the*p*+1^{th}term. The sum is:
And the arithmetic mean is simply the above
divided by

*n*.
(From “Aryabhațiya of Aryabhața”, by Kripa
Shankar Shukla)

The above verse gives the formula for the
arithmetic mean and the sum of the first n terms of an arithmetic progression,
commonly taught in high school. For the case when the first term of an AP is 1 and
the common difference is also 1, we get the well known formula:

In another verse, he explains how to find
the number of terms in an arithmetic progression.

Then we have the following result:

“Of the series (

*upaciti*) which has one for the first term and one for the common difference, take three terms in continuation, of which the first is equal to the given number of terms, and find their continued product. That (product), or the number of terms plus one subtracted from the cube of that, divided by 6, gives the*citighana*.”
(From Ganita Pada Verse 21 of Aryabhația.)

which says

Then
we have the results:

“The continued product of the three
quantities, viz., the number of terms plus one, the same increased by the
number of terms, and the number of terms, when divided by 6 gives the sum of
the series of squares of natural numbers (

*vargacitighana*). The square of the sum of the series of natural numbers (*citi*) gives the sum of the series of cubes of natural numbers (*ghanacitighana*).” (verse 22 Ganitapada).
More explicitly, the above states

and

He also developed an algorithm known as the
Aryabhața or kuttaka algorithm, which I confess I have not been able to figure
out. It was developed further later by Bhaskara I and others, to solve the set
of equations known as the first order Diaphontine equations ax+by=c. This will be described later in a post on Bhaskara I. Here is an article on the Aryabhata algorithm and its applications in number theory:

It may be remembered that the study of trigonometry, and basic
trigonometric concepts such as sine, cosine, etc., are all Indian in origin. Due
to the similarity of the arc and the chord subtended by an angle formed by two
radii in a circle with a bow and its string, respectively, the name given to
the chord of a circle was

*jya*, which in Sanskrit means a bow string. The length of this chord is twice the sine of the corresponding angle in a circle with unit radius. Half of this chord was called the ardha-jya, which is therefore the sine of the angle subtended. However with time, the ardha-jya itself began to be called the jya, which is thus the radius times the sine of the angle. When Arabic writers translated Aryabhata’s 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*.
Aryabhața holds a special place in the history of mathematics. He inspired generations of mathematicians for more than the next 1000 years. The

*Aryabhația*travelled far and wide into several countries and was translated into Arabic and European languages. He was also a renowned astronomer and influenced the course of astronomy in the centuries to come.
India's first satellite, which was launched on 19 April 1975, was named Aryabhata in his honor. The satellite was also shown on two rupee notes between 1976 and 1997:

The development of mathematics all over the world remains indebted to this great mathematician.

Reference:

[1] Aryabhatia of Aryabhatia, critically edited by Kripa Shankar Shukla, Indian National Science Academy, New Delhi, 1976.

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