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=y^{2}, 7y^{2}+1=z^{2}. 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 Nx^{2}+1=y^{2}. 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 180

^{o}. 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 180

^{o}. (x is expressed in radians in the above formula, i.e. 180^{o}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 180^{o}, 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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