Sunday, 27 July 2014

Bhāskara II

Before I begin, a word on the references: Bhāskara's major texts (Līlavātī, Bījagaita, Grahagaita 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]. 
He is the author of the mathematical and astronomical treatises Siddhānta Shiromai and Karaa-Kutuhala, as well as a commentary on the mathematician Lalla’s text Shishyadhivrddhida-tantra. Of these, the Siddhānta Shiromai is his most important and well known work. It consists of four parts: Līlavātī, Bījagaita,Grahagaita 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.  
 Arithmetic, geometry, algebra, and trigonometry are the major subjects taught in elementary and high-school mathematics. All these topics are covered in much detail in the 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 Shiromai. 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.
However, it is in the field of algebra that the 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]):
“From a bunch of lotuses, one third is offered to Lord Shiva, one fifth to Lord Vishnu, one sixth to the sun, one fourth to the goddess. The remaining six are offered to the Guru. Find quickly the number of lotuses in the bunch.” (The answer turns out to be 120.)

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 ax2+bx+c = 0, the general solution is 
After describing this method and giving the above formula, Bhāskara has set out several exercises for the student to practice. One such exercise, based on the Sanskrit epic Mahābhārata, is
Translation (from [3]):
“Arjuna became furious in the war and in order to kill Karna, picked up some arrows. With half the arrows, he destroyed all of Karna’s arrows. He killed all of Karna’s horses with four times the square root of the arrows. He destroyed the spear with six arrows. He used one arrow each to destroy the top of the chariot, the flag, and the bow of Karna. Finally he cut off Karna’s head with another arrow. How many arrows did Arjuna discharge?”
 Calling the total number of arrows picked up by Arjuna as x, the information in the above problem leads to the following equation:

Taking x/2 and 10 to the left and multiplying throughout with 2 we get

Squaring the equation then gives
x2 - 40x + 400 = 64x, or
x2 - 104x + 400 = 0
This equation can then be solved using the formula given above, or directly by factoring since the factorization is easy in this case. This gives 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 Mahābhārata 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 Aryabhaia 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 Aryabhaia): 

which expresses the following to identities in verse form:
The second result above is not exactly for an arithmetic progression but for sum of arithmetic progression's with increasing numbers of terms. Bhāskara then gives the following verses
which expresses the following identities in verse form:
Again, the above two results were given earlier by Aryabhaa in the Aryabhaia. 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.

Regarding geometric progressions. Bhāskara gives a method to find the sum of a given number of terms of a geometric progression in the chapter Shredhivyavaharaha from Līlavātī:


 Which states the following result for the sum S of the first n terms of a geometric progression with first term a and common ration r :
 Another topic taught in high school mathematics is the topic of combinatorics. This essentially involves counting and listing the number of ways an event can occur. (A typical example could be: if ten people come to a party and each person shakes hands with all the other persons present, how many handshakes have there been in total?) In India, combinatorics was important in the context of prosody. As mentioned earlier, Pingala’s 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 (26-1). (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 26 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.)

 Typical problems in combinatorics at the high school level involve permutations and combinations, i.e. given 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 nCr (C for combinations), and if the respective permutations are also required, then the answer is denoted by nPr.

The first mathematician in the history of the world to give the formula for nCr was Mahāvīra, an important mathematician from the 9th century from Mysore, India, who was the author of the famous text Gaitasārasangraha. (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:

Starting with the number n write down 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, . . .

 In addition, he also gives the result for the number of permutations of n objects in the following verse in the Līlavātī, chapter ankapāsha:

The above verse actually consists of three parts. The first part states that the number of permutations of 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:

 “To find the number of permutations of given (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.)

 If the permutations nPr of r objects from n objects is required, it is easy to see, using the formula for nCr, along with the above result, that the answer is simply nCr times r! i.e.
How many permutations of 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:
“To find the total number of permutations of given 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.”
In other words, if out of 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."

 The fifth case is of interest here. Looking at the earlier verses in the chapter, it is clear that Bhāskara is intuitively referring to what is called a limit. Using the rules from his earlier verses, the conditions in (v) can be expressed in the more familiar terminology of limits as:
which gives x = 14. This approach towards handling such tricky quantities forms the very foundation of differential calculus. Indeed, at another point in Siddhānta Shiromai, he explicitly demonstrates and makes use of the relation
to find the 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.
Closely connected to the concept of zero is the concept of infinity and  Bhāskara discusses these concepts in the same chapter. In the section dealing with rules for handling zero, he says that any (finite) number divided by zero is such that it “remains immutable in form and concept both … and that any finite number added to it or subtracted from it will not alter its value.” Calling this ‘immutable’ as ‘khahara’, Bhāskara gives a beautiful description of it in the Bījagaita in the chapter dealing with zero:

“There is no change in infinite (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:
“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.)
A very often quoted verse with regard to the nature of infinity and Bhāskara’s above description of khahara is the 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 Dx2+1=y2. 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!

Unaware of the work of the Indian mathematicians, this problem caught the fancy of European mathematicians in the 17th 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 Dx2+1=y2 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:
 “…his correspondence with Digby, and, through Digby, with the English mathematicians WALLIS and BROUCKNER occupies the next year and a half, from January 1657 to June 1658. It begins with a challenge to Wallis and Brouckner, but at the same time also to Frenicle, Schooten “and all others in Europe” to solve a few problems, with special emphasis upon what later became known (through a mistake of Euler’s) as “Pell’s equation”.  What would have been Fermat’s astonishment if some missionary, just back from India, had told him that his problem had been successfully tackled there by native mathematicians almost six centuries earlier!”
-André Weil, “Number Theory, an approach through history from Hammurapi to Legendre” (pp. 81-82)
It has been noted by the historian of mathematics C. O. Selenius that “…the 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.
Also, Florian Cajori, the noted historian, summed up the matter in an extraordinarily suggestive manner: ‘The perversity of fate has willed it that the equation y2 = nx2 +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.)
Apart from the topics mentioned above, other elementary topics which Bhāskara covers in 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:
“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.”

“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.”

There is yet another instance where this humor is displayed. In the section which deals with manipulating fractions, there is an exercise for the student which involves the computation of a rather elaborate kind of a fraction to find out how much money a miser gave to a beggar, and after doing the computation, it turns out that the money the miser gave to the beggar was the smallest possible currency of those days!

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:

“The earth attracts inert bodies in space towards itself. The attracted body appears to fall down on the earth. Since the space is homogeneous, where will the earth fall?”
As mentioned, several other advanced material from Siddhānta Shiromai 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.
[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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