Friday, 28 November 2014

The legacy of the Kerala mathematicians


The references used in this post are given at the end. General information on the Kerala mathematicians of the Madhava school can be found in Wikipedia and in the "Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures", (ed. Helaine Selin, publ. Springer).

Few people, especially Indians in particular, are aware that mathematicians in India, Kerala to be precise, nearly four centuries before Newton and Leibniz, had developed the fundamentals and key ideas of calculus and used it to obtain several results which are attributed to Western mathematicians today. (Similarly few people know that the so-called Pythagoras theorem was known in the Sulba Sutras centuries before Pythagoras, or that Pascal’s triangle was discovered by Pingala centuries before Pascal, or that the Backus-Naur form of modern computer science was invented by Panini centuries before Backus and Naur, or that Fibonacci numbers were described by Indian mathematicians several decades before Fibonacci. For more information and references look at the earlier posts in this blog.)

While it is quite possible (as I will discuss later) that the work of the Kerala mathematicians was known to Europeans in the 16th-17th centuries, the first time it was described in an academic research journal was in an article in 1834 published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland by Charles M Whish [1], an employee of the East India Company in the Madras region. What Whish described in this article was only the tip of the iceberg, and it is only recently that several aspects and the full depth of the Kerala mathematicians is coming to light. This article by Whish ought to have taken the world by storm, but it lay neglected and ignored for more than a century by mainstream mathematicians and historians.
It was only in the twentieth century that certain Indian mathematicians brought it to the attention of a more general audience. Even then it remained under the esoteric preserve of academics and researchers of mathematical history and has still not become mainstream knowledge.


Fig. The article by Charles Whish on the Kerala mathematicians


The lineage of Madhava and his disciples (Guru-Shishya parampara) and their works


Starting from the 13th century, a lineage of Sanskrit scholars in Kerala specializing in astronomy, mathematics, and linguistics developed the basic ideas of calculus, solutions of transcendental equations by iteration, infinite series expansions for the sine, cosine and inverse tangent functions, infinite series analysis with tests of convergence of infinite series, approximation of transcendental numbers by continued fractions, etc. [2]. These new advances were initiated by Madhava of Sangamagrama and his disciples. He and his disciples formed an unbroken lineage and knowledge was transmitted in the classic Guru-shishya parampara (teacher-disciple tradition; in Sanskrit guru = teacher, shishya = disciple, parampara = tradition). Madhava's lineage is today known as the Kerala school of mathematics in academic circles.



Some historical facts first. Madhava of Sangamagrama was born in the town of Sangamagrama. This is the present day town of Irinjalakuda in Thrissur district of Kerala (shown in red in the map). It is quite close to Kaladi, birth place of Adi Shankara (shown in blue in the map).  Madhava and his disciples were Nambudiri Brahmins and were active in the temple of Guruvayoor, which is a temple dedicated to Lord Sri Krishna. Most of the astronomers and mathematicians of the Madhava tradition were worshippers of Sri Krishna, and even today the temple is one of the holiest spots of pilgrimage for Hindus and is known as the Dwaraka of the south






Fig. A map of Kerala showing the birth places of Madhava (Irinjalakuda - shown in red) and Adi Shankara (Kaladi - shown in blue).

Madhava's disciple was Parameshvara (1360-1455). Parameshwara's important works were Goladipika, Bhatadipika, a commentary on the Aryabhatia, Mahabhaskariyabhashya, Siddhantadipika, a supercommentary on Govindaswamin's Mahabhaskariyabhashya, Grahanamandana on eclipses [3]. Damodara was the son and disciple of Parameshvara. No full-fledged work of Damodara remains. Damodara's disciples were the illustrious Nilakantha Somayaji (1445-1545), who was the brilliant author of the Tantra Sangraha and Aryabhatia Bhashya, and Jyeshthadeva (1500-1610), who was also later instructed by Nilakantha. Jyeshthadeva was the author of Yuktibhasha, which was written in Malayalam, and is one of the major works of the Madhava school. Nilakantha's disciple was Shankara Variyar (1500-1560), who was the author of three texts written in Sanskrit: the Kriyakramakari which was a commentary on Bhaskaracharya's Lilavati, Yuktidipika which was an elaborate commentary on the Tantrasangraha written in verse, and Laghuvritti, which was a shorter commentary on the Tantrasangraha in verse. Shankara Variyar was also deeply influenced by Jyeshthadeva. Jyeshthadeva's disciple was Achyuta Pisharati (1550-1621), whose disciple in turn was Melpathur Narayana Bhattathiri (1559-1645). Narayana Bhattathiri was a mathematician and a linguist and wrote scholarly texts on linguistics elaborating on Panini's grammar, but he is most famous as the writer of the Narayaneeyam, a beautiful devotional poem in praise of Lord Krishna which is still sung today in the Guruvayur temple. After Narayana Bhattathiri the other important scholars of the Madhava lineage are Putumana Somayaji (1660-1740), author of Karana Padhati, and Shankara Varman (1800-1838), who wrote the Sadratnamala. Shankara Varman, who was still alive when Charles Whish wrote his article, is described by the latter as "a very intelligent man and acute mathematician. This work (Sadratnamala), which is a complete system of Hindu astronomy, is comprehended in two hundred and eleven verses of different measures, and abounds with fluxional forms and series, to be found in no work of foreign or other Indian countries.''

The picture below shows the timeline of the Kerala mathematicians.


 




Fig. A time line of the Kerala mathematicians.


The works of these scholars are mostly commentaries elaborating on Madhava's research in mathematics and astronomy and contain several new results based on his work. In the next section I will focus the mathematical achievements of these mathematician.

Mathematical highlights of the Madahava school


Now we come to the mathematical work of the Kerala scholars. 

As discussed in the earlier posts, Indian mathematicians and scholars writing in Sanskrit such as Aryabhata, Bhaskara, Brahmagupta, etc. laid the foundations of astronomy, algebra, arithmetic, trigonometry, linguistics in their seminal works (for more information read the earlier posts on this blog). The Kerala mathematicians carried forward the work of these scholars to new levels of sophistication. Madhava's results predate Europe by more than 3 centuries, where these results are today (wrongly) believed to have been originally discovered. Here I shall give a very short summary of some of the contributions of the Kerala mathematicians pertaining to calculus. I will ignore their works in linguistics and astronomy since it is beyond the scope of this blog post.



The towering figure of the Madhava school is of course Madhava. Very few of his works survive in original form. He is known to have written the texts Venvaroha, Sphutachandrapti, Aganitagrahachara, amongst others. His disciples have stated and elaborated several of his results in their own works. One of Madhava's most striking results concerns the series expansions of the sine and cosine functions which has been referred to, for example, by Nilakantha Somayaji (1445-1545) in his Aryabhatia Bhashya of the 13th-14th century (pg. 155 of Ref. [4]) These series are well known:



The importance of these series goes far beyond these series themselves - their importance lies in the fact that it opens a whole new doorway into the analysis of infinitesimals and limiting quantities, which forms the foundation of differential and integral calculus and the branch of mathematical analysis and the study of infinite series and their convergence. In fact Madhava "took the decisive step onward from the finite procedures of ancient mathematics to treat their limit passage to infinity, which is the kernel of modern classical analysis." [5] This doorway was opened in Europe a full four centuries after Madhava composed his works, and as I will discuss later, it was not perhaps an entirely independent development.

Here we have a page from Nilakantha's Aryabhatia Bhashya (see [6]):





A closely related discovery by Madhava is the series for the inverse tangent function, which in modern terminology is expressed as:




 Madhava has been credited for this discovery, for example, by Nilakantha Somayaji (1445-1545) in his Tantrasangraha ([7] and [1]) and by Shankara Varman (1800-1838) in Sadratnamala. A proper proof of this result is also given in Jyeshthadeva's (1500-1610) Yuktibhasha. The verse in Sanskrit from the Sadratnamala is shown in the following figure



The tranlsation is: "The radius into the sine divided by the cosine is the first quote: this multiplied by the square of the sine, and divided by the square of the cosine, is the second quote; this second, and those obtained continually in the same way, multiply and divide by the square of the sine and the square of the cosine respectively: divide the quotes in order by 1, 3, 5, 7, 11, etc. respectively, and the difference of the sum of the 1st, 3rd,5th, etc. and of the 2nd, 4th, 6th, etc., will be the arc whose sine was taken.'' Verse taken from Ref. [8], translation from [1]. The Sanskrit words and their meanings as far as I know (thanks to [9]). phala - result / area;  guna - multiple; bhuja/bahu - side (of a triangle/swaure etc.); dhanus - arc of a circle; varga - square; koti - height; yugma - even.





A quick proof goes as follows:




Since arctan(1) = π/4, if we put x = 1 in the above series for arctan, we get another one of Madhava's result, namely, a series expansion for π:
Shankara Variyar (1500-1560) in his Kriyakramakari quotes the following verse of Madhava stating the above result ([10] Glimpses of the history of mathematics in India, by K. Ramasubramanian, IIT Mumbai):

which translates to: "The diameter multiplied by four and divided by unity (is found and saved). Again the products of the diameter and four are divided by the odd numbers like three, five, etc., and the results are subtracted and added in order." vyaasa - diameter; trisharadi - 3, 5, etc. (?); vishamasankhya - odd number; kramat - in order / sequence; kuryat - should be done


This result has been stated and attributed to Madhava by Nilakantha and other scholars of the Kerala school have also stated this result and attribute it to Madhava. This is the first time in the history of mathematics that an infinite series expression for π was developed anywhere in the world, and hence the series should be called the Madhava series. Instead, what we have for its name is the Gregory series, or the Gregory-Leibniz series, even though it was developed in 1668 by James Gregory three centuries after Madhava.

Such was their dexterity in handling infinite series, that the Kerala mathematicians obtained a plethora of series expansions for π. A few examples mentioned in [4] are given below:




A verse from Sadratnamala describing one of these results is shown in the first verse of the following figure - this seems to be second equation in the above equations. The second verse shows the value of π calculated to 10 decimal places:

Translation of first verse: “Square the diameter and multiply the product by 12, and extract the root of this product; the root obtained will be the modulus of odd quotes, which if you divide by 3, the quotient will be the modulus of even quotes. Divide each modulus continually by 9, and the quotient thus obtained from the former, divide by double the numbers 1, 3, 5, 7, 9, etc. minus 1 respectively, and the quotient obtained by the latter, by double the number 2, 4, 6, 8, 10, etc. minus 1 respectively, add up the new obtained quotes, and subtract the sum of those gotten from the even from the sum of those gotten from the odd modulus, the remainder is the circumference of the circle.” (I think this refers to the second equation in Eqn. (22).) vyaasasya vargaad - from the square of the diameter (vyaasa = diameter, varga = square) Translation of the second verse: “Square the diameter and multiply the product by 12, and extract the root of this product; this root divide continually by 3, and the quotients thus obtained by 1, 3, 5, 7, 9, 11, etc., and subtract the sum of the 2nd, 4th, 6th, 8th of the last obtained quotes from the sum of the 1st, 3rd, 5th, 7th, 9th, etc. If you do thus, and measure the diameter of a great circle by 100000000000000000 equal parts, the circumference will be equal to 314159265358979324 of such parts.” Sanskrit verses taken from [8] and English translations from [1].

The Kerala mathematicians invested a lot of effort in making accurate astronomical observations, based on which they developed their astronomical models (see [3]). Says Nilakantha: "One has to realize that the five siddhantas had been correct at a particular time. Therefore, one should search for a siddhanta that does not show discord with actual observations (at the present time). Such accordance with observation has to be ascertained by (astronomical) observers during times of eclipses etc. When siddhantas show discord, that is, when an earlier siddhanta is in discord, observations should be made of revolutions etc. (which would give results in accord with actual observations) and a new siddhanta enunciated."

Irrationality of  π

Already in the 5th century, i.e. 800 years before the Kerala mathematicians, Aryabhata was well aware of the irrationality of the ratio of circumference to diameter of a circle (π). In the following verse in the Aryabhatia,


Aryabhata says that his value for π (3.1416) is an approximation and not the actual value. Nilakantha explains in the Aryabhatiabhashya, his commentary on the Aryabhatia, why Aryabhata called this value an approximation:






which translates to: “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 aproached).” [11]

The possible transmission of the Kerala mathematics to  Europe

Although the development of calculus is conventionally attributed to Newton and Leibniz, it is quite possible that its development by these two mathematicians was not unassisted and independent of the Kerala mathematics.  In a separate I will discuss this more detail. Here I will just give an overview of the circumstantial evidence which supports this possibility. The information given below is from a slide show I created earlier. (Here is a link to the slide show.)

During the middle ages, Europe was centuries behind India in mathematical knowledge. It was only in 1202 that the Indian number system was popularized in Europe by Fibonacci through the Arabs (hence the number system is mistakenly called the Arabic number system). This was the time when Europe was engulfed in the Dark Ages and abject poverty, hence trade and conquest of wealthy nations such as India assumed paramount importance. But trade implies navigating the seas, which needs knowledge of astronomy, for e.g. calculating latitude and longitude, which in turn requires knowledge of trigonometry, tables of sines, cosines, etc. A reliable calendar is also a must for these purposes. Now neither did Europeans have this knowledge nor was their calendar reliable enough for navigation, which resulted in the loss of several ships as well as severe economic and human losses. Since the church dominated the scene in Europe, navigation and calendar reform became priority programs by the church. Lucrative prizes were offered by the church to anyone who could provide accurate navigational techniques and calendrical reforms.

At the same time, the Hindus in India had all this information in the form of the Sanskrit texts of Bhaskara, Aryabhata, etc. Indian navigators, assisted by reliable navigation techniques and an accurate calendar, also used to trade with several countries. It became important for the church to obtain this information from the Hindus. But Hindus were 'pagans', 'heathens' and 'idol worshippers' who had to be 'civilized' (christianized). It would never do for the church to admit that the 'pagan' Hindus had superior knowledge. Hence although it privately sought 'pagan' learning, publicly the church continued to deny that there was any learning among the 'pagans'. There was great pressure among the people to deny the possibility of the pagans possessing useful knowledge, since death in the form of being burnt at the stake for being a 'heretic' would be a certainty for such a person.

The opportunity and means for obtaining the knowledge appeared from 1499 onwards. In this year, Vasco da Gama arrived at the Malabar coast in Kerala and established a direct link to Europe via Lisbon. Missionary activity began in earnest in 1540 when Francis Xavier arrived in Goa and subsequently made Kerala a hub of missionary activity. (The missionary activity was in no way a peaceful or benign one and hides several hideous facts, but I'll save this for a separate post.) Some time later, Christoph Clavius (1538-1612), a Jesuit mathematician and astronomer, included mathematics in the curriculum of Jesuit priests at Collegio Romano. Christoph Clavius was later to head the calendar reform committee of the church which resulted in the present Gregorian calendar. The first batch of Jesuit priests mathematically trained by Clavius reached Malabar in Kerala (including the city of Cochin, the epicenter of the Kerala mathematics) 1578 onwards. These included Matteo Ricci, Johann Schreck, and Antonio Rubino. These priests learnt the local language and were in close touch with local scholars and royal personages. Rubino and Ricci have also been recorded in correspondence as answering requests for astronomical information from the Kerala mathematical and astronomical sources. It is thus clear that the purpose of these mathematically trained priests is to acquire Indian knowledge navigation, astronomy, and the calendar. The Gregorian calendar reform took place soon after in 1582. In 1607, Christoph Clavius published tables of sines and cosines for different angles to a high degree of accuracy (seven to eight decimal places) without explaining how he carried out these calculations. (As mentioned earlier, these tables were important in navigation.)



Now, is it a mere coincidence that these tables had been already worked out by Madhava to the same degree of accuracy, and is it possible that Clavius's tables had nothing to do with Madhava's tables? Unlikely, considering the circumstantial evidence. Nilakantha quoted these tables of Madhava more than a century earlier in his Aryabhatia Bhashya which was an important text of the Kerala mathematics. (http://en.wikipedia.org/wiki/Madhava's_sine_table also shows below table. Also see [12] Madhava's Sine and Cosine Series by A. K. Bag.)







There is further circumstantial evidence that the knowledge of the Kerala mathematicians travelled to Europe. Tycho Brahe (1546-1601) was the most well known astronomer of the time who in 1597 became the Royal astronomer of the Holy Roman empire upon the invitation of emperor Rudolph II to Prague. He would thus be a natural recipient of any Hindu astronomical texts which might be obtained by the Jesuit priests in Kerala. Tycho Brahe is also well known for his model of planetary motion, in which Mercury, Venus, Mars, Jupiter and Saturn revolve around the sun, but the sun along with these planets revolve around the earth. Is it just a coincidence that exactly the same model was proposed by Nilakantha in his Tantra Sangraha?

Narayana Bhattathiri and the Narayaneeyam

As mentioned earlier, the Kerala scholars of Madhava's lineage were Nambudiri Brahmins and devotees of Lord Vishnu.  The most important spiritual text for Lord Vishnu devotees is the Srimadbhagavad Puran, which consists of 18,000 verses and describes the life and deeds of Sri Krishna, an avatar of the Supreme. It seeks to increasing inner peace and develop dispassion towards the vicissitudes of life by reminding one of its impermance and putting one more and more in touch with the immortal, unchanging reality underlying the whole of creation. Sri Krishna also spoke the Bhagavad Gita to Arjuna on the eve of the Mahabharata war, in which three main paths are outline to realize this unchanging reality - Karma Yoga, Bhakti and Gyana Yoga. Srimadbhagavat Puran is especially valuable, indeed indispensable, for Bhakti Yogis, but is useful for Karma Yogis and Gyana Yogis as well.

One of the most important non-mathematical works of the Kerala scholars is the Narayaneeyam. This was written by Melpathur Narayana Bhattathiri (1559-1645) and is a condensed / abridged version of the Srimadbhagavat Puran. Melpathur was a linguist and was a student of Achyuta Pisharati.The legend associated with how he came to write the Narayaniyam is very beautiful and is well worth mentioning here. I am giving the legend below, which I have taken almost verbatim from the following website: www.narayaneeyam-firststep.org/introduction.html


Achyuta Pisharati, Melpathur Bhattathiri's Guru, suffered from acute rheumatism. Bhattathiri prayed to relieve his Guru of the disease and transfer it to him. His prayer was heard. His Guru regained health and instead the disease came to Narayana Bhattathiri. He was convinced that if he surrendered at the feet of Lord Krishna in the temple of Guruvayur he would be cured of disease. He sought the guidance of Thunchaththu Ramanujan Ezhuthachan in this matter. (Some information on Ezhuthachan: Thunchaththu Ramanujan Ezhuthachan was a renowned Malayalam devotional poet and linguist from that time. He was born in the Shudra caste in south Malabar and later on became a monk. He is considered the father of the Malayalam language and translated the Ramayana and the Mahabharata into Malayalam.) He instructed Bhattathiri to 'start with fish'. The suggestion would have seemed offensive to a Nambudiri Brahmin, who are strict vegetarians, but  Bhattathiri understood the cryptic meaning. The fish is one of the avatars of Lord Vishnu described in the Bhagavat Puran (Matsya avatar - the fish avatar). Bhattathiri was quick to take the indication of glorifying the Lord with the various avatars starting with the Matsyaavatara. This prompted him to compose the abridged version of the Purana in he form of the Sriman Narayaneeyam Stotram.

In his acute rheumatism Bhattathiri somehow reached the temple of Guruvayoor and fully surrendered himself at the feet of the Lord. He prostrated in deep devotion and started to pray, singing His glory and worship Him everyday. It is said that he composed and rendered one dashakam (10 verses) each day to the Lord. Thus, in 100 days of sincere worship, the Narayaneeyam consisting of 100 dashakams was composed.

These verses written in pain and agony of the author invoked the sympathy and grace of the Lord. At the end of each dashakam, each day, he would pray for the mercy and kindness of the Lord to cure his disease. Soon the Lord's grace showered on him and on the 100th day the Lord blessed him with His vision. Bhattathiri was overwhelmed with ecstay and in the 100th Dashakam he cries out - 'Agre Pashyami' - which in Sanskrit means 'here I see Him in front of me' - and he gives a vivid description of the most enchanting form of the Lord he saw - from head to foot (Keshadi paadam). From that day his ailments vanished and he got totally cured.

see also:
http://en.wikipedia.org/wiki/Melpathur_Narayana_Bhattathiri
http://en.wikipedia.org/wiki/Narayaniyam



References:


[1] "On the Hindu quadrature of the circle, and the infinite series of the proportion  of the circumference to the diameter exhibited in the four Sastras, the Tantra Sangraham, Yukti Bhasha, Carana Padhati, and Sadratnamala" by Charles M Whish, Transactions of the Royal Asiatic Society of Great Britain and Ireland, Vol. 3, No. 3 (1834).

[2]http://www.pas.rochester.edu/~rajeev/papers/canisiustalks.pdf

[3] "Kerala school of astronomy and mathematics" by M. S. Sriram, Pg. 1160, "Encyclopedia of the History of Science, Technology, and Medicine in Non-Western Cultures", (ed. Helaine Selin, publ. Springer).

[4] "Ancient Indian leaps into Mathematics", B. S. Yadav and Manmohan (ed.) Birkhauser, Springer Science (2010).


[5]C. T. Rajagopal and M. S. Rangachari, On an untapped source of medieval Keralese mathematics, Archive for history of exact sciences, Baltimore, Maryland, Vol 18 1978 pp. 89-102.

[6] Aryabhatiam with Bhashya of Nilakantha, edited by K. Sambasiva Sastri, Thiruvanathapuram Sanskrit Series, No. 101. Thiruvananthapuram (1930). 

[7] "The discovery of the series formula for pi by Leibniz, Gregory, and Nilakantha", Ranjan Roy, Mathematics Magazine, Vol. 63, No. 5 (Dec. 1990), pp. 291-306, publ. Mathematical Association of America.

[8] Sadratnamala of Shankara Varman, critically edited by K. V. Sarma, Indian National Science Academy, New Delhi (2001).


[9] "The Golden Age of Indian Mathematics" by S. Parameswaran

[10] Glimpses of the history of mathematics in India, by K. Ramasubramanian, IIT Mumbai


[11] Development of calculus in India: contribution of kerala school (1350-1550 CE), K. Ramasubramanian, IIT Bombay


[12] Madhava's Sine and Cosine Series by A. K. Bag











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:

Translation:
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:
 
 
Translation:
“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:

Translation:
 
“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:
 
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.)
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:
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.”

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:
 

 
Translation:
 
“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.
 
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.