Monday, 14 July 2014

Aryabhața

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 1st, 3rd and 4th are mostly astronomical, the 2nd one deals mainly with mathematics. It is marked for tis brevity and conciseness of composition. The Aryabhația is written in the form of sutras, characterized by its terseness, which makes it easy to memorize. (Sutras are short and serve as quick aids to the memory: examples are Nārada Bhakti sutras, Patanjali Yoga sutras, etc. Bhāshyas are commentaries, e.g. Shankarabhāshya 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-bhāshya says why asanna is used: “Why has an approximate value been mentioned here (in Aryabhația) 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 rth term of the progression be denoted by tr, so that tr = 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+1th 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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