Sunday, 20 July 2014

Bhāskara I

As mentioned earlier, Aryabhața had a huge impact on the development of mathematics and astronomy in the centuries to come. His work was popularized and expanded upon by various commentaries of later scholars. The most well known of these is the Aryabhațiabhashya, written by the great astronomer-mathematician Bhāskara I, who lived about a century after Aryabhața. Apart from being one of the most exhaustive and comprehensive commentaries of the Aryabhața, it contains many original contributions as well. In addition, he wrote two more treatises dealing with  astronomy and mathematics: the Mahabhāskariya and the Laghubhāskariya. His books were very popular as text books in the centres of learning throughout India.
Bhāskara I lived in the period approximately from 600 to 680. The exact place where he was born and brought up is not known with certainty. It is suggested that he probably lived and taught at ValabhI in Surashtra (modern saurashtra or Kathiawar). He was a great devotee of Lord Shiva. In the Aryabhațiabhāshya, for example, one finds the following verse in praise of Lord Shiva:
 
“I bow to Lord Shiva whose Lotus feet are rubbed by the foreheads of the Lotus-born (Brahmā) and Krishna, to Him, a mere recollection of whose name is a source of fortune and end of misfortune for gods, demons, and men.” (Aryabhațiabhashya, Ganitapada, Verse 1)
Also,

 “I bow to Shambhu, who is the cause of creation and destruction of the universe, (the different parts of) whose body are the Sun etc., and who is as effulgent as tens of thousands of suns.” (Ganitapada section, Aryabhațiabhashya).

And also from Aryabhațiabhashya:

 
“I bow to God Shiva, the support of the three worlds, who though inconceivable by nature, being mercy incarnate, assumed the eight forms, viz. The Sun, the Moon, the Ether, Fire, Air, Water, Earth, and the Sacrificer, for the good of the entire world.” (opening verse of the section on Kālakriyāpāda of the Aryabhațiabhashya)

Or from the opening verse of Mahābhāskarīya,

 
“I bow to God Shambhu who bears on His forehead a digit of the moon illumining all directions by its rays, to Him whose feet are adored by the gods and who is a source of all knowledge.” (Opening verse of Mahābhāskarīya)

 
Bhāskara I was a devotee of Lord Shiva

The Āryabhațiabhāshya, which he wrote in 629, establishes Bhāskara as the greatest authority on Aryabhața. Bhāskara had deep reverence for Aryabhața and referred to him as his Guru, though they lived more than a century apart. His work shows deep knowledge of trigonometry, algebra and geometry, which was mainly used to address astronomical problems such as the motion of the heavenly objects.

The technique of symbolic algebra, where an unknown quantity is denoted by a symbol, is introduced in school. This technique has become so widespread over the years that it is taken as ‘trivial’ or ‘obvious’, just as the concept of zero and the number system. However, it is largely forgotten today by the world in general and more strangely by Indians themselves, that the pioneers of symbolic algebra were Indian mathematicians. By Bhāskara’s time, algebra had reached a high level of sophistication in India. An important contribution of Bhāskara in this regard was the further development of the kuttaka algorithm of Aryabhața and his classification of it into the so-called residual and non-residual pulverizer methods. It may be recalled that the kuttaka algorithm is used to find integral solutions in x and y to the equation ax + by = c. Kuttaka or kuttakarah in Sanskrit literally means to pound or pulverize something into smaller and smaller bits, hence the kuttaka is also known as the pulverizer algorithm. Today it is also called the Aryabhața algorithm in honor of Aryabhața.

Such kind of equations occurred in astronomical calculations. A table of the least integral solutions (the integral solutions which are smallest in magnitude) for the equation ax-1=by using kuttaka is given below for the sun (from [1] Appendix II, pg 335. Similar tables are given for the moon and planets as well). As said in [1], pg lxxxii: “Bhāskara I’s tables giving the least integral solutions of the equation ax-1=by corresponding to all sets of values of a and b that may arise in astronomical problems based on the pulveriser are given towards the end of the commentary on the Ganita-pada. They are meant to facilitate the solution of astronomical problems based on the theory of the pulveriser and form a unique feature of Bhāskara I’s mathematics, as tables of the kind are not to be met with in any other known works on Hindu mathematics.” It is clear from the table that the smallest integral solutions to the equations are not so small after all!

 
Apart from writing the Āryabhațiabhāshya, he is also famous as the creator of two other important astronomical works: Mahābhāskariya and Laghubhāskariya. Mahābhāskariya  was in fact the first work written by him, followed by Āryabhațiabhāshya and then Laghubhāskariya. The Laghubhāskariya involves involves the knowledge of the integral solutions of the  following multiple equations: x+y=a square number, x-y=asquare number, xy+1=a square number and the knowledge of the integral solutions of these simultaneous equations: 8x+1=y2, 7y2+1=z2. This is also found in Brahmasphutasiddhanta by Brahmagupta. Brahmagupta stated the general solution of the above multiple equations and also a method for solving the equation Nx2+1=y2. The occurrence of the above mentioned problems in the two works written independently about the same time gives us an idea of the development of Hindu algebra in the first half of the seventh century AD. (pg xlvii of [1])

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 in a unit circle. 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 Aryabhața’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.

Fundamental contributions to trigonometry were made by Bhāskara in his works. He is responsible for the development of the modern trigonometric circle, shown in Fig.

 
 Among other discoveries in trigonometry, he is famous for devising an approximate formula for finding  the sine of an angle between 0 and 180o. This is given in Mahābhāskariya, vii. 17-19, and also in Brahmasphuta Siddhanta by Brahmagupta.

 
The approximation for the sine expressed in the verses above is the following:

 
which is valid for x between 0 and 180o. (x is expressed in radians in the above formula, i.e. 180o equals π radians.) The accuracy of the above approximation is remarkable. If we call Δ(x) the difference between sin(x) and Bhaskara’s approximation, and plot Δ(x) as a function of x, with x varying from 0 to 180o, we get the graph shown in the figure below.

 
The maximum error is not more than 0.002 - very small indeed considering the simple form of Bhaskara’s approximation, and is especially useful when one wants a quick estimate of the sine without recourse to a sine table. Later on, astronomers such as Madhava of Sangamagrama around 1300 pioneered the techniques of infinite series with which they could calculate the trigonometric functions to extraordinary degrees of accuracy.

Some of the features of the Mahābhāskariya include (Pg  xlii of [1]) the application of indeterminate equations of the first degree to problems in astronomy (Mahābhāskariya, i. 41-52), discussion of planetary motion, calculation of eclipses, and finding distances of sun, moon and the planets. It also describes the correct method of calculating the illuminated part of the moon, and is an improvement over the method in the Surya Siddhanta (Mahābhāskariya, vi, 5-7). In contrast to stereotypical accounts of Indian civilization by present day historians, this shows the open and dynamic nature of Indian science , which was steeped in the scientific attitude and was not blindly stuck to a single text or authority. The Indian scientists were willing to criticize and improve upon earlier works as  and when the need arose. (Present day Sanskrit scholars should do the same. Criticizing and improvement does not imply disrespect). Mahābhāskariya, vi, 5-7.

His other notable contribution, the Laghubhāskariya, “proved to be a excellent textbook for beginners in astronomy on account of its conciseness, clear and simple expression and judicious arrangement.” (pg xlviii) It was meant to be an abridged version of the Mahābhāskariya, written for younger students. The content of this book was mainly astronomical, with algebra, trigonometry and geometry being applied to calculate the motion of planets, and to phenomena such as eclipses and conjunctions.

Bhaskara I’s contributions had a deep impact on the development of mathematics in India and the rest of the world. In his honor, India’s second satellite, launched on June 7 1979, was named Bhaskara I. (http://www.isro.org/satellites/Bhaskara-I.aspx)

 
References
 
[1] Aryabhatia of Aryabhata with the commentary of Bhaskara I and Someshvara, critically edited by Kripa Shankar Shukla, Indian National Science Academy, 1976
 
[2] Laghubhāskariya, edited and translated into English by Kripa Shankar Shukla, Department of mathematics and astronomy, Lucknow University, 1963
 
[3] Mahābhāskariya, edited and translated into English by Ram Ballabh, Department of mathematics and astronomy, Lucknow University, 1960
 
 

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