*Before I begin, a word on the references: Bhāskara's major texts (Līlavātī,*

*B*

*ī*

*jagaṇita*

*, Grahagaṇita and Golādhyāya) in Sanskrit are available at the Anandāshram Sanskrit series at*

*this*

*website in pdf format. A very good translation of the Līlavātī is due to K. S. Patwardhan, S. A. Naimpally, and S. L. Singh (Līlavātī of Bhāskarācārya, publ. Motilal Banarsidass, Delhi). A very good general reference is the "Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures", ed. Heleine Selin, publ. Springer 2008.*

*And now I begin...*

Bhāskarachārya II (1114-1185 AD) is probably the most well known among the Indian mathematicians and astronomers. He came from Vijjadavida near the Sahya mountain,
identified with modern Bijapur in Mysore [1-2].

*Siddhānta Shiroma*

*ṇ*

*i*and

*Kara*

*ṇ*

*a-Kutuhala,*as well as a commentary on the mathematician Lalla’s text

*Shishyadhivrddhida-tantra*. Of these, the

*Siddhānta Shiroma*

*ṇ*

*i*is his most important and well known work. It consists of four parts:

*Līlavātī*,

*B*

*ī*

*jagaṇita*,

*Grahagaṇita*and

*Golādhyāya*. While the last three are more advanced texts with several original results of extraordinary brilliance,

*Līlavātī*is more of a textbook for students learning mathematics. It was extremely popular as a text book in Indian

*Ashramas*and

*Gurukuls*for quite some time. It is written in a light-hearted, sometimes humorous style and neatly describes nearly all the mathematics that one learns till high school, and with lots of practice problems for the student. A reading of the

*Līlavātī*will show that nearly all elementary school mathematics, and to a very large extent high school mathematics as well, is actually Indian mathematics.

*Līlavātī*. All the arithmetic rules taught in elementary school for addition, subtraction, and division, multiplication, finding square roots and cube roots for integers are described in the

*Līlavātī*in a compact, easy to refer manner in the part which deals with arithmetic (this forms the very beginning of the book and serves to recapitulate the basics). The rules taught in school for manipulating fractions are covered as well in the same chapter. This includes addition, subtraction, multiplication, and division of fractions, as well as finding square roots and cube roots of fractions. It contains several results from geometry such as properties of triangles and circles, finding areas of geometric figures, etc. Although trigonometric results are not explicitly presented in the

*Līlavātī*, Bhaskara extensively makes use of trigonometric concepts for astronomical calculations in the rest of the

*Siddhānta Shiroma*

*ṇ*

*i*. It may be remembered that trigonometry originated in India, and that most of the trigonometry taught in high school today was developed by Indian mathematicians, which was transferred to the west via Arabic texts. Even the names 'sine' and 'cosine' are Indian in origin.

*Līlavātī*, and Bhāskara's work in general, really stands apart. The basic method in elementary algebra taught in school is of letting a symbol represent an unknown quantity, and to find the value of that quantity by setting up and solving an algebraic equation. Well, this method is described in the

*Līlavātī*in full detail. In fact the chapter dealing with algebraic equations in the

*Līlavātī*seems to be straight out of a modern school text book, with problems such as

Translation (from [3]):

The

*Līlavātī*gives the standard method of solving a quadratic equation by completing the squares, and gives the formula for the general solution. For a quadratic equation

*ax*

^{2}+

*bx*+

*c*= 0, the general solution is

*Mahābh*

*ā*

*rata*, is

Translation (from [3]):

*x*, the information in the above problem leads to the following equation:

*x*

^{2}- 40

*x*+ 400 = 64

*x*, or

*x*

^{2}- 104

*x*+ 400 = 0

*x*= 4 and 100. Clearly 100 is the answer since Arjuna has already used more than 10 arrows according to the information in the problem.

*One of the problems on quadratic equations in Līlavātī is based on the fight between Arjuna and Karna in the Mah*

*ā*

*bhārata war.*

A topic taught in high school is the topic of arithmetic and geometric progressions, and related results such as the sum of the first

*n*whole numbers, the sum of the squares of the first

*n*whole numbers, and the sum of the cubes of the first

*n*whole numbers. An arithmetic progression is a sequence of numbers such that the difference between any two consecutive numbers is a fixed amount, this fixed amount being called the common difference, while in a geometric progression it the ratio between any two consecutive numbers of the sequence that is fixed, and which is called the common ratio. All the basic results of arithmetic and geometric progressions taught in high school were well known to Indian mathematicians. For example, the

*Aryabhaṭia*of Aryabhaṭa gives the sum of a given number of terms of an arithmetic progression, how to find the number of terms of an arithmetic progression given the sum, etc. Bhāskara revises all these results in the chapter

*Shredhivyavahara*in the

*Līlavātī*. In particular, he gives the following results (also given in

*Aryabhaṭia*):

which expresses the following to identities in verse form:

which expresses the following identities in verse form:

*Aryabhaṭia*. Other results such as finding the first term, or the common difference, or the number terms of an arithmetic progression if the other quantities are given, are also presented, accompanied by several practice problems for the student, which look straight from a modern school text book.

*Shredhivyavaharaha*from

*Līlavātī*:

*chhanda sutras*(200 BCE) investigated prosody using mathematics and in the process came up with the binomial theorem and the

*Meru Prastara*(so-called ‘Pascal’s triangle’). Even earlier to Pingala, Sushruta said in

*Sushruta Samhita*that from the six different basic tastes (sweet, sour etc.), different tastes can be created from different combinations of these tastes, and the total number of these tastes is 63 (2

^{6}-1). (The reason is that each taste is either absent or present in a particular combination (i.e. 2 possibilities for each taste), and the number of basic tastes being 6, the total number of possibilities is 2

^{6}i.e. 64. However the case where none of the tastes is absent is not relevant, and hence the total number of tastes possible is one minus this number, i.e. 63.)

*n*objects, in how many ways can we select

*r*objects out of them? If only the number of combinations, and not the respective permutations in each of these combinations, is required, then the answer is denoted by

*(*

^{n}C_{r}*C*for combinations), and if the respective permutations are also required, then the answer is denoted by

*.*

^{n}P_{r}*was Mahāvīra, an important mathematician from the 9*

^{n}C_{r}^{th}century from Mysore, India, who was the author of the famous text

*Ga*

*ṇ*

*itasārasangraha*. (Pg. xix of [5], [6]). Mahāvīra’s formula, invariably taught in high school mathematics today in the context of combinatorics, is given by:

This formula is recapitulated by Bhāskara in
the

*Līlavātī*in the chapter*Mishravyavahara*:
Translation:

*n*, (

*n*-1), (

*n*-2), ... Divide them by 1, 2, 3, ..., to get

*n*/1, (

*n*-1)/2, (

*n*-2)/3, . . . Then the number of combinations of n things taken 1, 2, 3, . . . at a time are

*n*/1,

*n*(

*n*-1)/(1 x 2),

*n*(

*n*-1)(

*n*-2)/(1 x 2 x 3). . . . respectively. Or the number of combinations of

*n*things taken

*r*at a time are [

*n*(

*n*-1)(

*n*-2). . . (

*n*-

*r*+1)] / [1 x 2 x 3. . .

*r!*]. This can be used to solve the problem when

*r*= 1, 2, 3, . . .

*n*objects in the following verse in the

*Līlavāt*

*ī,*chapter

*ankap*

*ā*

*sha*:

*n*objects is 1 x 2 x 3 x ... x

*n*=

*n*! The second part tells how to find the sum of all possible numbers which can be formed using a given set of digits, and the third part is some practice problems based on the first two parts. The translation (from [3]) goes as:

*n*) different digits (or objects), write 1 in the first place, 2, 3, 4, . . . up to the number of objects (

*n*) and multiply them.

*(This is the first part)*. Divide the product of the number of permutations and the sum of the given (

*n*) digits by the number of the given digits (i.e. by

*n*); write the quotient the given number of times (i.e.

*n*times) in a column but leaving one-digit place each time; add them; the result is the sum of the numbers formed (by permuting the given

*n*digits).

*(This is the second part.)*Using (i) 2, 8, (ii) 3, 8, 9, (iii) 2, 3, . . . 9 how many different numbers can be formed? What is the sum of numbers, so formed, in each case?”

*(This is the third part.)*

*of*

^{n}P_{r}*r*objects from n objects is required, it is easy to see, using the formula for

*, along with the above result, that the answer is simply*

^{n}C_{r}*times*

^{n}C_{r}*r*! i.e.

*n*objects are possible when, out of those

*n*objects,

*r*objects are similar? If those

*r*objects had been different, the number of permutations would have been

*n*! But since those

*r*objects are similar and can be permuted amongst themselves

*r*! times, the correct answer is

*n*!/

*r*! This is exactly what Bhāskara states in the following verse:

Translation:

*n*digits or objects, if certain digits are alike, then form the product of the number of permutations of those places at which alike digits occur (assuming that each block of alike digits has different digits), and divide the number of permutations of all the given (n) digits (assuming them different) obtained by the previous method by this product. And the sum of the numbers formed is obtained by the previous method.”

*n*objects,

*r*objects of kind are alike,

*s*objects of another kind are alike,

*t*objects of yet another kind are alike, then the total number of permutations is

*n*!/(

*r*!

*s*!

*t*!...).

It is well known that the concept of zero and
the number system originated in India. What is less well known is that Indian
mathematicians also knew how to handle tricky quantities of the 0/0 type – for example, in the chapter on
zero in

*Līlavātī*, Bhāskara asks of the reader to find, using the rules described earlier in the text, a number which, when multiplied by zero and added to half the result, and the result, upon being multiplied by three and then divided by zero, give 63. The original Sanskrit verse from*Līlavātī*is shown below:
The above verse is a series of exercises for the student regarding zero.
It says:

"Find (i) 5+0 (ii) the square of 0, the cube of 0, the square root of 0, the cube root of 0, (iii) 0x5 (iv) 10/0, (v) A certain number is multiplied by zero and added to half the result. If the sum so obtained is first multiplied by three and then divided by zero, the result is 63. Find the original number."

*x*= 14. This approach towards handling such tricky quantities forms the very foundation of differential calculus. Indeed, at another point in

*Siddhānta Shiroma*

*ṇ*

*i*, he explicitly demonstrates and makes use of the relation

*instantaneous velocity*of a planet, which he calls

*tātkalika gati*(the Sanskrit phrase for instantaneous velocity). Such a relation was in fact used earlier by the astronomer mathematician Manjula in 932 AD. It may be recalled that Brahmagupta had proposed an interpolation method using second order differences, which is equivalent to the Newton-Stirling formula. Bhāskara has also described this and commented on it in his text. This formula implicitly makes use of the notion of derivatives, which is of course the key ingredient of differential calculus. Bhāskara also knew that the derivative vanishes at points of extrema, and has also stated in his work what is today known as the Rolle’s theorem of differential calculus (ref). Thus, Bhāskara can be credited with laying the seeds of differential calculus, and implicitly used ideas of Integral calculus ideas in finding the volume and area of a sphere. The principles of integral and differential calculus were developed by the Kerala astronomers fourteenth century onwards.

*khahara*’, Bhāskara gives a beautiful description of it in the

*B*

*ī*

*jagaṇita*in the chapter dealing with zero:

Translation:

*khahara*) figure if something is added to or subtracted from it. It is like: there is no change in the infinite Lord Vishnu (Almighty) due to the dissolution or creation of abounding living beings.”

*Bhāskara illustrated the concept of infinity, or khahara, by comparing it to the infinitude of Lord Vishnu.*

A better understanding Bhāskara’s
description of khahara can be obtained from Sri Krishna’s
words in the

*Bhagavad Gita,*chapter 9, presented below:
Translation:

“The
whole of this universe is permeated by Me as unmanifest Divinity, and all
beings dwell within Me. But I am not present in them. || All those beings
abide not in Me, but behold the wonderful power of my Divine Yoga; though the
sustainer and creator of beings, I Myself in reality dwell not in those beings. || Just as the extensive air, which is moving everywhere, (being born of
ether) ever remains in ether, likewise know that all beings, who have
originated from my

*Sankalpa*(intention), abide in Me. || Arjuna, during the final dissolution all beings enter enter My*Prakriti*(the prime cause), and at the beginning of creation, I send them forth again. || Wielding my nature I procreate, again and again (according to their respective Karmas) all this multitude of beings subject to the influence of their own nature. (9.4 - 9.8)”*Explanatory note:-*

(So, although the number of beings is changing during universal creation and dissolution, the one independent Reality, which is what Lord Sri Krishna is an Avatar of, is unchanging. This explains the seeming contradiction in verses 4 and 5 since, if Sri Krishna was in those beings, which are subject to birth and death, then He would also be subject to birth and death. On the other hand, He, as the supreme independent Reality, has to be in them, since those beings, subject to birth and death, cannot have an independent existence. So, although the number of beings is changing, Lord Krishna (or Lord Vishnu) is not changing, which is what Bhāskara is trying to explain by comparing khahara with Lord Vishnu.)

*shanti paath*from the

*Ishavasyopanishad*:

which means:

*Om*. That (

*Brahman*) is complete. This (universe) is complete. When you remove completeness from completeness, what remains is also complete.

*Om Shanti Shanti Shanti*.”

An achievement of
Bhāskara for which he is most well known in the mathematics hall of fame is the
development of the

*chakravāla*algorithm to obtain general integral solutions in*x*and*y*of equations of the type*Dx*^{2}+1=*y*^{2}. Although this method is attributed to Bhāskara, it was also independently developed by an earlier Indian mathematician Jayadeva (950-1000). Brahmagupta had already achieved headway to solve this equation in the form of the lemma which bears his name, and had solved it for*D*= 83 and 92. However, it was the*chakravāla*method which gave the solution for general*D*. In particular, the cases*D*=61 and 109 were especially difficult, but the*chakravāla*algorithm gives the solution in a few lines! For*D*=61, the solutions*x*=226,153,980 and*y*=1,766,319,049 are the*smallest integral*solutions! Likewise, for*D*=109, the*smallest integral*solutions are*x*=15,140,424,455,100 and*y*=158,070,671,986,249!^{th}century, and the equation later came to be known as Pell’s equation. Pierre de Fermat (of the “Fermat’s last theorem” fame) proposed exactly the same problem which Bhaskara had solved, namely, to find integral solutions in

*x*and

*y*for

*Dx*

^{2}+1=

*y*

^{2}with

*D*=61, to fellow mathematician Bernard Frénicle de Bessy in 1657. It is interesting to read what mathematician André Weil has to say in his book

*Number Theory, an approach through history from Hammurapi to Legendre*in this regard:

-André Weil,

*“Number Theory, an approach through history from Hammurapi to Legendre”*(pp. 81-82)*chakravāla*method anticipated the european methods by more than a thousand years. But, as we have seen,

*no European performances*in the whole field of algebra at a time much later than Bhāskara’s, nay

*nearly up to our times, equalled the marvellous complexity and ingenuity of chakravāla*. (emphasis added)” (C.O.Selenius,

*Rationale of the Chakravāla process of Jayadeva and Bhāskara II*, Historia Mathematica 2 (1975), 167-184.) Considering the priority of the Indian mathematicians in considering and solving this equation, Selenius suggests in the same article that this equation should be renamed as the Jayadeva-Bhāskara equation instead of Pell’s equation.

*y*2 =

*nx*2 +1 should now be called Pell’s Problem, while in recognition of Brahmin scholarship it ought to be called the “Hindu Problem.” It is a problem that has exercised the highest faculties of some ofour greatest modern analysts.’ Indian mathematical historians would like to call it the Brahmagupta–Bhāskara problem, keeping in mind that Bhaskara perfected Brahmagupta’s method of solution in the twelfth century; Bhaskara used ‘Chakra-vala,’ or a cyclic process, to improve Brahmagupta’s method by doing away with the necessity of finding a trial solution.” (Ref [9], Pg. 187.)

*Līlavātī*are from geometry, mensuration, and commercial mathematics. These include the properties of various geometrical figures, finding areas and volumes of various geometrical figures, and stating several rules in commercial mathematics which were very useful to traders and merchants. Bhāskara also has a certain humor which is impossible to miss. Consider for example the following verses from

*Līlavātī*:

Translation:

“In a triangle or a polygon, it is impossible for one side to be greater
than the sum of the other sides. It is daring for anyone to say that such a
thing is possible.”

Translation:

“If an idiot says that there is a quadrilateral of sides 2, 6, 3, 12 or
a triangle with sides 3, 6, 9, explain to him that they don’t exist.”

The
material in

*Līlavātī*is just the tip of the iceberg. We have not been able to cover his work in astronomy since such a discussion would be too involved. However, a glimpse into the other parts of*Siddhanta Shiromani*should be enough to convince us of the depth and profundity of Bhāskara's genius. For example, the following verse from the*Golādhyāya*(chapter*Bhuvanakosha*, Stanza 6) asserts, more than 500 years before gravitation was proposed in Europe, that it is because of the earth’s attraction that objects stay on it:
Translation:

*Siddhānta Shiroma*

*ṇ*

*i*has not been discussed here because of personal limitations on part of the author and also because such a discussion would be too involved for the purposes of this article. But this brief overview of Bhāskara should still be an eye-opener for most school students in India in revealing where most of the mathematics they study in school comes from.

To conclude in a poetic strain, Bhāskara II, true to his name, shines like a
sun in the world of mathematics.

References:

[1]

*Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures*, ed. Heleine Selin, publ. Springer 2008.
[3] K. S. Patwardhan, S. A. Naimpally, and S. L. Singh

*,**Līlavātī of Bhāskarācārya,*publ. Motilal Banarsidass, Delhi
[4] N. L. Biggs,

*The Roots of Combinatorics*, Historia Mathematica 6 (1979), 109-136.
[5] T. K. Puttaswamy,

*Mathematical Achievements of Pre-modern Indian Mathematicians*, publ. by Elsevier, 2012.
[6] R. Wilson, J. J. Watkins, R. Graham,

*Combinatorics: Ancient and Modern*, Oxford University Press, 2013.[7] André Weil,

*Number Theory, an approach through history from Hammurapi to Legendre,*Birkhäuser 2007

*.*

[8] C.O.Selenius,

*Rationale of the Chakravāla process of Jayadeva and Bhāskara II*, Historia Mathematica 2 (1975), 167-184.
[9]

*Ancient Indian Leaps into Mathematics*, Yadav and Man Mohan (eds.), Birkhäuser 2011.
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