Contemporary Relevance of Lilavati (of Bhaskaracarya)

Contemporary Relevance of L¢l¡vat¢ of Bh¡skar¡c¡rya N.K. Sundareswaran L¢l¡vat¢ is the most popular text on Indian Mathematics. It forms, actually, a part of Siddh¡nta¿iroma¸i. The greater work Siddh¡nta¿iroma¸i contains four chapters or sections. L¢l¡vat¢, the first section deals mainly with arithmetics. Elementary algebra, mensuration and geometry also find place in this section. The other three sections are B¢jaga¸ita, Grahaga¸ita and Go½¡dhy¡ya. B¢jaga¸ita deals mainly with algebra. Grahaga¸ita deals with problems and theories in connection with planetary motions. Go½¡dhy¡ya, the last section deals with astronomy proper. The author of the work is Bh¡skara. He is often referred to as Bh¡skara II or Bh¡skar¡c¡rya in order to differentiate from his namesake Bh¡skara I. Actually Bh¡skara I was also a very good mathematician. He is the first commentator of the celebrated work Ëryabha¶¢ya of Ëryabha¶a I. Apart from this commentary, he has written another work on Ëryabha¶¢ya. It is named as Mah¡bh¡skar¢ya. Scholars regard this work as complementary of Ëryabha¶¢ya. Now our author Bh¡skar¡c¡rya was more popular. His fame rests mainly on Lil¡vat¢. In his Bijaga¸ita, he gives a solution method for second degree indeterminate equations of the type Nx2 + 1 = y2 and Nx2 + C = y2 . This method is known by the name Cakrav¡½a method. Today these equations are known as Pell‟s equations. Actually the Cakrav¡½a method given by Bh¡skar¡c¡rya is only an improvement of another method given by Brahmagupta (an ancient Indian mathematician of 6th century). Hankel, a famous German mathematician, describes the Cakrav¡½a method of solution given by Bh¡skar¡c¡rya as “the finest thing achieved in the theory of numbers before Lagrange”1 There is an interesting story behind Pell‟s equation. Fermat, a famous modern mathematician (of 17th century) challenged all mathematicians to find an infinity of integer solutions of the equation x 2Ay2 = 1, where A is any non-square integer. Mathematicians were unable to solve the problem until in 1738 Euler published a general solution. The problem belongs to the type of equations which where solved by Brahmagupta and Bh¡skar¡c¡rya. “The Aryabhata group”, a group of researchers in the history of mathematics, having their based at University of Exter, U.K. (which investigates whether solid evidences are there to prove that mathematics, or at least some branches of modern mathematics migrated from India to Europe in the medieval times) argues that Fermat and Euler might have got the idea from the works of Brahmagupta and Bh¡skar¡c¡rya.2 Another striking point in the Siddh¡nta¿iroma¸i is that, here the author implicitly speaks of differential calculus. While discussing about the instantaneous motion of planets in spaÀ¶¡dhik¡ra, he says: 1 2 See A Concise History of Science in India, D.M. Bose et al., p. 168 See the article “Transmission of the Calculus from Kerala to Europe”, Proceedings of the International Seminar and Colloquium on 1500 years of Ëryabhateeyam, Kerala Sastra Sahitya Parishad, Nov. 2002 2 „If y and y1 are the mean anomalies of a planet, at the ends of consecutive intervals then sin y1 – sin y = (y1 – y) cos y. Of course, the language and notations are our own and not the author‟s. But his statement ʤɨ¤ÉÉvÉǺªÉ EòÉäÊ]õVªÉÉMÉÖhÉκjÉVªÉɽþ®ú& ¡ò±ÉÆ nùÉäVªÉǪÉÉä®úxiÉ®ú¨É when translated into modern mathematical expression, is the same as above. Scholars have pointed out that he even further goes to state that the derivative vanishes at a maxima.3 Actually this should not be surprising. It is now well known and widely accepted that ancient and medieval India produced brilliant mathematicians who anticipated many of the modern finding on mathematics. And Bh¡skar¡c¡rya is said to be the culminating point of such mathematical and astronomical investigations. His work L¢l¡vat¢, in fact, does not contain any further novelties than what are found in the works of his predecessors. But it was very popular. It ranks with Bhagavadg¢t¡ and K¡vyaprak¡¿a (a comprehensive work on Indian poetics written by Mamma¶a Bha¶¶a, a Kashmiri scholar of 11th century) in the abundance of commentaries. The author himself wrote a commentary. It goes by the name V¡san¡bh¡Àya. The excellence and popularity of the work is caused by two facts. (1) The rules are lucid and have been illustrated by many examples. (2) The language in sweet sounding. Actually the work is based on and modeled after the Ga¸itas¡rasa´graha of Mah¡v¢ra (a Jaina mathematician 3 See Indian Mathematics and Astronomy – Some land marks, S. Balachandra Rao, Janadeep Publications, Bangalore, 2000, p. 146. 3 of 9th century). Many examples are taken from this latter work as well as the Tri¿atik¡ of ár¢dhar¡c¡rya. In order to illustrate the lucidity and sweetness of language, a few examples are cited below. The very first verse, which not only serves as a benediction, but has clear mention of subject matter too, runs as follows: |ÉÒËiÉ ¦ÉHòVÉxɺªÉ ªÉÉä VÉxɪÉiÉä Ê´ÉPxÉÆ Ê´ÉÊxÉPxÉxÉ º¨ÉÞiɺiÉÆ ´ÉÞxnùÉ®úEò´ÉÞxnù´ÉÎxnùiÉ{ÉnÆù xÉi´ÉÉ ¨ÉiÉRÂóMÉÉxÉxɨÉÂ* {ÉÉ]õÓ ºÉnÂMù ÉÊhÉiɺªÉ ´ÉÎS¨É SÉiÉÖ®ú|ÉÒÊiÉ|ÉnùÉÆ |ɺ¡Öò]õÉÆ ºÉÆÊIÉ{iÉÉIÉ®úEòÉä¨É³ýɨɱÉ{Énèù±ÉÉÇʳýiªÉ±ÉÒ±ÉÉ´ÉiÉÒ¨ÉÂ** Here the author himself has stated the factors which make for the excellence of the work. He says; “I shall deal with P¡¶iga¸ita (arithmetics) in a precise and clear language to that it will definitely please and attract the wise people. Sportive nature (ease) caused by the simplicity and sweetness (±ÉÉʳýiªÉ) is the soul of this work”. When we go through the work, we would definitely perceive that this is not a tall claim. Let us see another verse which is given as example for the quadratic equations (MÉÖhÉEò¨ÉÇ, in author‟s language). ¤ÉɱÉä ¨É®úɳýEòÖ ±É¨ÉÚ±Énù³ýÉÊxÉ ºÉ{iÉ iÉÒ®äú ʴɱÉɺɦɮú¨ÉxlÉ®úMÉÉhªÉ{ɶªÉ¨ÉÂ* EÖò´ÉÇSSÉ EäòʳýEò±É½Æþ Eò³ý½ÆþºÉªÉÖM¨ÉÆ ¶Éä¹ÉÆ VɱÉä ´Énù ¨É®úɳýEòÖ ±É|ɨÉÉhɨÉÂ** This poetic expression resounding poetic diction and figure of sound would attract even an uninterested urchin to the bewitching world of mathematics. 4 Feel the clarity and simplicity of expression in the verse. ªÉªÉÉäªÉÉæMÉ& ¶ÉiÉÆ ºÉèEòÆ Ê´ÉªÉÉäMÉ& {É\SÉ˴ɶÉÊiÉ&* iÉÉè ®úɶÉÒ ´Énù ¨Éä ÊIÉ|ÉÆ ´ÉäÎiºÉ ºÉRÂóGò¨ÉhÉÆ ªÉÊnù** A literal translation would be: “If you know the method of transition (ºÉRÂóGò¨ÉhÉ), tell the numbers whose sum is 101 and the difference is 25. Let us have a glance at the concluding verse. It runs as ªÉä¹ÉÉÆ ºÉÖVÉÉÊiÉMÉÖhÉ´ÉMÉÇʴɦÉÚʹÉiÉÉRÂóMÉÒ ¶ÉÖrùÉÊJɱɴªÉ´É¾þÊiÉ& JɱÉÖ Eòh`öºÉHòÉ* ±ÉÒ±ÉÉ´ÉiÉÒ½þ ºÉ®úºÉÉäÊHò¨ÉÖnùɽþ®úxiÉÒ iÉä¹ÉÉÆ ºÉnè´ù É ºÉÖJɺɨ{ÉnÖ{ù ÉèÊiÉ ´ÉÞÊrù¨ÉÂ** It means: „Those who have mastered (by-hearted) this work L¢l¡vati, which speaks clearly of fractions, quadratic equations, square, cubes, square roots and cube roots (etc.) and which abounds in sweet and fine examples; their happiness and prosperity will prosper for ever.” The verse has got another meaning: “Those who are embraced by a sportive damsel - of attractive charms and who is properly adorned by beautiful ornaments, who speaks very clearly and sweetly, who does not deceive – would definitely attain bliss.” In fact these two meanings are to be connected like this: “Just as a righteous and wise damsel of bewitching beauty would unfailingly attract and please a young man, so would this work on arithmetics, dealing with all the topics in an unambiguous and sweet language will never fail to attract the young ones.” 5 This is the kind of expression which the author successfully and masterly employs in this work. Now before deliberating on the present-day relevance of this work, let us get acquainted with the structure of the work as well as the time and place of the author. Bh¡skar¡c¡rya gives his year of birth in the 58th stanza of Go½¡dhy¡ya as ®úºÉMÉÖhÉ{ÉÚhÉǨɽþҺɨɶÉEòxÉÞ{ɺɨɪÉä%¦É´Éx¨É¨ÉÉäi{ÉÊkÉ&* ®úºÉMÉÖhɴɹÉæhÉ ¨ÉªÉÉ ÊºÉrùÉxiÉʶɮúÉä¨ÉhÉÒ ®úÊSÉiÉ&** (I was born in the ¿aka year 1036 and I completed the work at the age of 36). He gives the numbers 1036 and 36 by using a system of verbal notation of numerals. It is called Bh£tasa´khy¡samprad¡ya. The áaka year 1036 corresponds to 1114 AD. Hence we get the year of Bh¡skar¡c¡rya‟s birth as 1114 A.D. and the date of composition of Siddh¡nta¿iromani as 1150 AD. He gives his father‟s name as Mahe¿vara in Bijaga¸ita and Go½¡dhy¡ya. In the former work he says +ɺÉÒx¨É½ä·þ É®ú <ÊiÉ |ÉÊlÉiÉ& {ÉÞÊlÉ´ªÉɨÉÉSÉɪÉǴɪÉÇ{Énù´ÉÓ Ê´ÉnÖù¹ÉÉÆ |É{ÉzÉ&* ±É¤vÉɴɤÉÉävÉEòʱÉEòÉÆ iÉiÉ B´É SÉGäò iÉVVÉäxÉ ¤ÉÒVÉMÉÊhÉiÉÆ ±ÉPÉÖ ¦ÉɺEò®äúhÉ** From this verse we gather the following facts. (1) His name was Bh¡skara. 6 (2) His father was an excellent teacher. He was fortunate to have brilliant students. (3) Bh¡skara learnt Astronomy and Mathematics from his own father. (4) Father‟s name was Mahe¿vara. In the Go½¡dhy¡ya, he gives some more details. The relevant passage runs as follws: +ɺÉÒiºÉÁEÖò±ÉÉSɱÉÉʸÉiÉ{ÉÖ®äú jÉèÊ´ÉtÊ´ÉuùVVÉxÉä xÉÉxÉɺÉVVÉxÉvÉÉΨxÉ Ê´ÉVVɱÉÊ´Ébä÷ ¶ÉÉÎhb÷±ªÉMÉÉäjÉÉä ÊuùVÉ&* ¸ÉÉèiɺ¨ÉÉiÉÇÊ´ÉSÉÉ®úºÉÉ®úSÉiÉÖ®úÉä Êxɶ¶Éä¹ÉÊ´ÉtÉÊxÉÊvÉ& ºÉÉvÉÚxÉɨɴÉÊvɨÉǽ·äþ É®úEÞòiÉÒ nè´ù ÉYÉSÉÚbÉ÷ ¨ÉÊhÉ&** iÉVVɺiÉSSÉ®úhÉÉ®úÊ´ÉxnùªÉÖMɳý|ÉÉ{iÉ|ɺÉÉnù& ºÉÖvÉÒ& ¨ÉÖMvÉÉän¤ù ÉÉävÉEò®Æú Ê´ÉnùMvÉMÉhÉEò|ÉÒÊiÉ|ÉnÆù |ɺ¡Öò]õ¨ÉÂ* BiÉnÂù´ªÉHòºÉnÖÊù HòªÉÖÊHò¤É½Ö³ þ Æý ½ä±þ ÉÉ´ÉM¨ªÉÆ Ê´ÉnùÉÆ ÊºÉrùÉxiÉOÉlÉxÉÆ EÖò¤ÉÖÊrù¨ÉlÉxÉÆ SÉGäò EòʴɦÉÉǺEò®ú&** Here we get the following facts. (1) Bh¡skar¡c¡rya belonged to ᡸ·ilya gotra (2) He hailed from the place Vijjalavi·a (3) His father Mahe¿vara was an astrologer (4) He was well-versed in Vedic knowledge, especially ritualistic scholarship. 7 (5) Bh¡skar¡c¡rya was trained by his father in Astronomy and mathematics. (6) Bh¡skar¡c¡rya was a poet. (7) He was very much confident that by the perusal of his work, a wise man can easily understand the science and thus attain pleasure. Bh¡skar¡c¡rya clearly states that his birth place is Vijjalavi·a, a city situated near Sahya mountains. Scholars hold different opinions regarding the identification of the place. Some take it to be Bijapur of Karnataka. Some take it to be the town Bedara near Hyderabad. Bedara is 80 k.m. away from Sholapur. This identification is first made by Faizi, a court poet of Akbar, who translated L¢l¡vat¢ into Persian language as suggested by the latter N¤simhadaivajµa in his commentary V¡san¡v¡rttika on L¢l¡vat¢ states that Bh¡sk¡r¡c¡rya hailed from Maharashtra.4 Another commentator Mun¢¿vara describes Vijjalavi·a (in his commentary named ¨É½þÉ®úɹ]Ånõ äù¶ÉÉxiÉMÉÇiÉÊ´Énù¦ÉÉÇ{É®ú{ɪÉÉÇªÉ Mar¢cik¡) Ê´É®úÉ]õnäù¶ÉÉnùÊ{É as ÊxÉEò]äõ ºÉÁEÖò±ÉÉxiÉMÉÇiɦÉÚ|Énä¶ù Éä MÉÉänùɴɪÉÉÇ xÉÉÊiÉnÚ®ù äú {É\SÉGòÉä¶ÉÉxiÉ®äú Ê´ÉVVɱÉÊ´Éb÷¨É (Vijjalavi·a is a place in Maharashtra, situated five kro¿as away from Godavari in the Vidarbha region near Sahya mountains). Based on this some scholars identify Vijjalavi·a as a place near Malegaon of Maharashtra. The text L¢l¡vat¢ contains 266 verses which are not divided into sections or chapters. It is the commentators and editors of the text who 4 Ê´ÉVVɱÉÊ´Éb÷ÊxÉ´ÉɺÉÒ {ÉÊ´ÉÊjÉiÉnùhb÷EòÉ®úhªÉ& ¨É½þÉ®úɹ]ÅÉõ xÉɨÉɸɪÉÉä ¨É½äþ·É®úxÉxnùxÉ& ¸ÉÒ¦ÉɺEò®úÉSÉɪÉÇ& - Quoted in L¢l¡vat¢ of Bh¡skar¡c¡rya, MLBD, New Delhi, 2006. 8 have divided these verses into several small sections. After giving a benediction the works starts with a table of measures and weights. Then the work proceeds to explain simple arithmetics starting with counting numbers, place value of digits, and fundamental operations of counting numbers. Then different methods for finding square, cube and square root and cube root of numbers are given. Fundamental operations of tractions and operations with zero are explained. Then follows <¹]õEò¨ÉÇ, where a method to find an unknown quantity when the result of some fundamental operations made on it is given. MÉÖhÉEò¨ÉÇ is another section where quadratic equations are dealt with. Then follows rule of three, rule of five etc., and reverse proportion. Problems on simple interest, permutations and combinations and progression are dealt with. Problems to find out the volume of different shapes of solids, formulae to find out the area of different geometrical figures, evaluation of , wood cutting and the shadow problems are of special interest. And the last section deals with pulverization or the solution of indeterminate equations of the type ax + by = c. The style of presentation employed is uniform. In the beginning of every section or each set of problems, the formula or the general mode of solution is given first in the Kara¸as£tra. Then it is illustrated by several examples. Subscribing to the general trend of ancient and medieval Indian Mathematicians, the rationale behind (or the ways and means by which the results or the formulae are arrived at) are not explained. But from the illustration of examples, a keen researcher could (or rather should) make out the approach and methodology employed by the medieval Indian 9 mathematician in solving the problems. This exactly is the point where a modern student or researcher should focus his attention. All the types of problems and findings discussed in L¢l¡vat¢ are well known and solved out by modern mathematics using western methodology and approach. It is the approach and methodology employed by the medieval mathematician which should grab our attention because medieval Indian mathematician inherits a striking legacy of mathematical tradition. In every scientific finding the final result is not that important as the various ways and means or the methodologies employed to arrive at them. In the Indian context it is all the more important. In the ancient and medieval India, a great tradition of mathematics was transmitted orally. When we lost the links of the chain of this oral tradition, the knowledge system itself was lost to us. And we had to sit at the feet of Europeans to learn our own science which, in fact, got transmitted to Europe through Arab and Persian translations in many a case. Hence the methodology and approach is the most important factor which we have to discover or reconstruct, which when blended or contrasted with the modern approach would definitely open up new vistas of knowledge. Unfortunately it is this unique methodology and approach of Indian tradition which is very often neglected in modern studies. The point may be illustrated with one case. The following is an illustrative example given by Bh¡skar¡c¡rya. ¤ÉɱÉä ¨É®úɳýEòÖ ±É¨ÉÚ±Énù³ýÉÊxÉ ºÉ{iÉ iÉÒ®äú ʴɱÉɺɦɮú¨ÉxlÉ®úMÉÉhªÉ{ɶªÉ¨ÉÂ* EÖò´ÉÇSSÉ EäòʳýEò±É½Æþ Eò³ý½ÆþºÉªÉÖM¨ÉÆ ¶Éä¹ÉÆ VɱÉä ´Énù ¨É®úɳýEòÖ ±É|ɨÉÉhɨÉÂ** 10 (There was a flock of swans on a lakeside. Seven times half the square root of the number of swans were moving about near the lake. One amorous pair of swans was playing in water. How many swans were there, in total, in the flock?) When we get such a problem, our approach with modern training would be like this: Let x be the number of swans. Then the number of swans in the shore = ” ” 7 x 2 2 i.e. or i.e. 7 x 2 water = 2 would be the total number of swans. 7 x 2 2 7 x 2 x x 2 49x 4 2 x 2 (squaring both the sides) = x 2 4x 4 i.e. 49x 4x2 16x 16 i.e. 4x2 65x 16 0 11 x 652 4 x 4 x16 2x4 65 ( Solution set of the equation ax2+bx+c = 0 is = 65 = 65 = 65 63 8 = 128 2 or 8 8 b b 2 4ac 2a ) 4225 256 8 3969 8 i.e. x = 16 or 1/4 Now x being the number of swans, it cannot be 1/4. So the total number of swans is 16. Now let us see how this problem is tackled by Bh¡skar¡c¡rya. By using modern notation, his procedure may be explained as follows: Let the total number of swans be x2. Then the number of swans in the shore would be 7x 2 The remaining two birds are in water. That means 12 x2 7x 2 2 In order to solve problems of this kind Bh¡skar¡c¡rya gives a general method. Suppose x2 + bx , say c , is given. We are to find x2. Then the practical procedure to be followed is: Add b2 4 to c and find the square root of the result. Then subtract or add (as the case may be) (b/2) to it. Squaring the arrived at figure we will get the required number i.e. x2. The rationale behind is as follows: x2 bx is given as c (b is also given). We are adding x 2 bx b2 4 is a perfect square, because x root of this would be {x b }. 2 b2 bx 4 2 b2 4 . The sum, that x Thus if we subtract or add b 2 2 b 2 . Square we get x. Squaring it, we get x2, the required number. Here, in the above problem x 2 49 7/2 i.e. 16 2 2 to 2 we get 2 49 81 i.e. . 16 16 7 7/2 9 7 16 i.e. (b / 2) we get i.e. 4. 4 2 4 4 4 7 x 2 is given as 2. Adding Its square root is 9/ 4 . Adding Squaring this, we get the required number, i.e. 16. This general method or the simple procedure is given by Bh¡skar¡c¡rya as 13 MÉÖhÉPxɨÉÚ±ÉÉäxɪÉÖiɺªÉ ®úÉú¶ÉänÖùǹ]õºªÉ ªÉÖHòºªÉ MÉÖhÉÉvÉÇEÞòiªÉÉ* ¨ÉÚ±ÉÆ MÉÖhÉÉvÉæxÉ ªÉÖiÉÆ Ê´É½þÒxÉÆ ´ÉMÉÔEÞòiÉÆ |ɹ]Öõ®ú¦ÉÒ¹]õ®úÉʶÉ&** Now if, instead of x2 + bx, x 2 procedure is: multiply it by 1 1 1 n 1 2 x n bx is given, then the and proceed as before. An example for this is the following problem {ÉÉlÉÇ& EòhÉÇ´ÉvÉÉªÉ ¨ÉÉMÉÇhÉMÉhÉÉxÉ GÖòrùÉä ®úhÉä ºÉxnùvÉä iɺªÉÉvÉæxÉ ÊxÉ´ÉɪÉÇ iÉSUô®úMÉhÉÆ ¨ÉÚ±Éè¶SÉiÉÖ̦ɽǪþ ÉÉxÉÂ* ¶É±ªÉÆ ¹ÉÎb¦÷ É®úlÉä¹ÉÖʦÉκjÉʦɮúÊ{É UôjÉÆ v´ÉVÉÆ EòɨÉÖÇEÆò ÊSÉSUäônùɺªÉ ʶɮú& ¶É®äúhÉ EòÊiÉ iÉä ªÉÉxÉVÉÖxÇ É& ºÉxnùvÉä** Furious Arjuna pelted some arrows (Say x2) to kill Kar¸a. With half the arrows, i.e. x2 2 , he prevented the arrows of Kar¸a. And killed Kar¸a‟s horses with four times the square root of the total number of arrows (i.e. 4x). He destroyed áalya with six arrows. He used one arrow each to destroy the UôjÉ (umbrella placed at the top of the chariot), flag and bow of Kar¸a . Finally, he choped off Kar¸a‟s head with an arrow. Tell me how many arrows did Arjuna pelt, in total? Solution is as follows: 14 Total number of arrows = x2 to be found out. No. of arrows used to prevent Kar¸a ‟s arrows = x2 2 . No. of arrows used to kill horses ” =4x to destroy Śalya ” =6 ” ” UôjÉ, v´ÉVÉ & EòɨÉÖÇEò =3 ” ” kill Kar¸a =1 x2 2 i.e. x 2 x 2 4x 10 0 or x2 2 4x 10. This equation is in the form x 2 multiplying this by 1 2 x n bx c where n = 2. 1 1 1 i.e. i.e. (or2) 1 1/ n 1 1/ 2 1/ 2 Then we get x2 - 8x = 20. This is in the form of x2 – bx = c where b = 8 and c = 20. Adding b 2 2 to c we get 20 + 16 = 36. Taking square root we get 6 Adding b/2 (i.e. 4) to it we get 10. Squaring we get 100, the required answer. 15 Conclusion Hence while reading classical texts on Indian Mathematics like L¢l¡vat¢ our approach should be to unravel the methodology and approach, which actually was employed by the authors. In the case of L¢l¡vat¢ , it is especially so. L¢l¡vat¢ is the culminating point of a saga of mathematical activity in India. The mathematical tradition of ancient and medieval India does not reveal the rationales and methodologies behind any of the findings. The unique methodology and approach employed by the greatest of mathematical tradition of the world would naturally and definitely lead to new vistas of knowledge, when unraveled (in many of the modern studies, this methodology is often compromised, unfortunately)5. And this is the relevance of L¢l¡vat¢ today. 5 See for Instance S. Balachandra Rao, op.cit., pp. 147-151. 16 Books Referred to. L¢l¡vat¢ (with the Malayalam comm. of P.K. Koru), Mathrubhumi Press, Kozhikkode, 1953. L¢l¡vat¢ (with a Hindi Comm.), Chowkhamtha Vidyabhavan, Varanasi, 1961. L¢l¡vat¢ of Bh¡skar¡c¡rya Ed. (with Kriy¡kramakar¢ comm.) K.V. Sarma, VVRI, Hoshiarpur, 1975. A Concise history of Science in India, D.M. Bose et. al., INSA, New Delhi, 1989. Indian Mathematics and Astronomy – Some landmarks, S. Balachandra Rao, Jnanadeep Publications, Bangalore, 2000. Proceedings of the International Seminar and Colloquium on 1500 years of Ëryabhateeyam, Kerala Sastra Sahitya Parishad, Kochi, 2002. L¢l¡vat¢ of Bh¡skar¡c¡rya, MLBD, New Delhi, 2006.  17