CONTRIBUTION OF JAINA MATHEMATICIANS DR. (MRS). PADMAVATHAMMA, M Sc, Ph D Professor of Mathematics (retired) Department of Studies in Mathematics University of Mysore, Manasagangotri Mysore-570 006 E-mail: padma_vathamma@yahoo.com

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Mathematics is one of the important branches of Science from time immemorial. Being an inseparable part of science, it has retained its privileged place as Queen of all Sciences. The contribution of Indian mathematicians towards the development of mathematics is unique and valuable. Zero was first introduced in place value system of notations by Indians. The contributions of ancient Indian mathematicians Āryabhaṭa, Bhāskara, Brahmagupta, Mahāvīrācārya and Bhāskarācārya are world famous even today. Many Jaina mathematicians have saliently contributed. It is perceived among common people that mathematics is difficult to learn. The skill of explaining such difficult material in simple and exact forms is one of the specialties of Jaina mathematicians. 30/01/2010 Dr.Padmavathamma

Tattvārthadhigama Sūtra, Sthānanga Sūtra, Jambūdvīpa Prajñapti, Tiloyapaṇṇatti, Kṣetrasamāsa, Gaṇitasārasangraha and Vyavahāragaṇita are important ancient Jaina mathematical works. Among the important subjects which are available in Jainā philosophy, first priority is given to literature while the second priority is given to mathematics. Hence in Āgamās, it is said lehāiyāvo gaṇiyappahāṇāo, that is, the writing etc. of which the chief (pradhāna) is the counting. From this, it is proved that in educating a child and in the human transactions, mathematics had a very prominent role. 30/01/2010 Dr.Padmavathamma

In Jainā literature there are four anuyogās called prathama, karaṇa, caraṇa and dravya. In karaṇānuyoga, many mathematical operations are used to explain the features of loka and in the explanations of sun, moon, star, island, sea etc.we find the use of mathematics in the following Prakrit works and their commentaries. Sūryaprajñapti, Candraprajñapti, Jambūdvīpaprajñapti, Tiloyapaṇṇatti, Dhavalā commentaries of Ṣaṭkhaṇḍāgama, Gommaṭasāra, Trilokasāra. The above works provide valuable information to know the ancient Indian mathematics. Sūryaprajñapti is called Gaṇitānuyoga. 30/01/2010 Dr.Padmavathamma

The following list shows the names of persons who have contributed to the development of mathematics in Prakrit. Puṣpadanta- Bhūtabali (3rd century A.D) Yativṣabhācārya (5th century A.D) Vīrasenācārya (9th century A.D) Srīdhara (9th century A.D) Srīpati (10th century A.D) Nemicandra Siddhānta Cakravarti (11th century A.D) We do find names of Siddhasena, Bhadrabāhu etc. who have used mathematical formulae in their works, although they were not mathematicians. 30/01/2010 Dr.Padmavathamma

For a Jaina monk, arithmetic and astrology were like adornments For a Jaina monk, arithmetic and astrology were like adornments. Umāsvāti (150 B.C) who was one of the best Jainā philosophers, had mentioned for the first time about the mathematics school at Kusumaoura in Pāṭna. He was living in the ancient Pāṭalīpura which is now the modern Kusumapura in Paṭna. It is quite possible that this mathematics school existed even before the time of the famous Jainā monk Bhadrabāhu (300 B.C) who lived in Kusumapura). His works are – a commentary on Sūryaprajñapti, and Bhadrabāhavi Samhita. 30/01/2010 Dr.Padmavathamma

Initial period (Ādikāla) - 3000 – 500 B.C Ancient India has contributed a lot to the development of mathematics and the part played by the Jainā scholars in this field is significant. The development of mathematics in India may be classified into the following groups. Initial period (Ādikāla) - 3000 – 500 B.C Childhood Period (Śaiśavakāla) – 500 B.C -500 A.D This is also known as Dark Period Middle Period – 500 – 1200 A.D Later Period – 1200 -1800 A.D Modern Period – 1800 A.D onwards 30/01/2010 Dr.Padmavathamma

Mathematical quotations in Ardhamāgadhi and Prakrit are met with in several works. Dhavalā contains a large number of such quotations. A.N.Singh in his article entitled, Mathematics of Dhavalā says “ A study of the Jainā canonical works reveals that mathematics was held in high esteem by the Jainās. In fact, the knowledge of mathematics and astronomy was considered to be one of the principal accomplishments of the Jainā ascetics. ” 30/01/2010 Dr.Padmavathamma

In the present paper we discuss about the following three authors Mathematical material in the Dhavalā may be taken to belong to the period 200 – 600 A.D. Thus Dhavalā becomes a work of first rate importance to the historians of Indian mathematics – the period preceding the fifth century A.D called the dark period. In the present paper we discuss about the following three authors Nemicandra Siddhānta Cakravarti whose works are in Prakrit Mahāvīrācārya whose works are in Sanskrit Rājāditya whose works are in Kannada 30/01/2010 Dr.Padmavathamma

Nemicandra Siddhānta Cakravarti INTRODUCTION There are four celebrated ascetic sanghās in the History of Jainās in South India. These sanghās are Nandi, Simha, Sena and Deva. The Deśīya gaṇa is a branch of the Nandi sangha and originated in the lands called Deśa which extended from river Cauvery in the south to river Godāvari in the north, Sahyadri hills in the east to Palghat in the west (present day Kerala). Jain ācārya Simhanandi belonging to this gaṇa helped Sivamara to found the Ganga dynasty, one of the ancient royal kingdoms of India [1]. 30/01/2010 Dr.Padmavathamma

Of the twenty scholars in this gaṇa who were honoured with the title “Siddhānta Cakravarti” , ācārya Nemicandra is most known for his work in mathematics. Ascetic Lineage of Nemicandra: It is known from the Gommaṭasāra (Karmakāṇḍa part) that Nemicandra was the disciple of Abhayanandi, whose preceptor was Guṇanandi. Vīranandi, a colleague and contemporary of Nemicandra was alsoa disciple of Abhayanandi. Vīranandi, the author of the Candraprabha Mahākāvya, has received homage from Nemicandra in the Gommaṭasāra (Chapter 6, verse 396). 30/01/2010 Dr.Padmavathamma

Nemicandra as a mathematician Except the Trilokasāra which gives cosmological description, all other works of Nemicandra are related to Jain philosophy. His profound knowledge of mathematics i.e rules related to circle and its segments, permutations and combinations are all employed in his works However, the earlier mathematicians in India had also known this science, before him. It is known, however, some of his examples and illustrations on combinations were never seen in the Hindu mathematics. The pioneering research work of Prof.B.Datta [2] and Prof.L.C.Jain [3] would definitely throw more light on the mathematics of Nemicandra. 30/01/2010 Dr.Padmavathamma

The works of Nemicandra Nemicandra Siddhānta Cakravarti is the author of Dravyasamgraha, Gommaṭasāra, Labdhisāra, Kśapaṇasāra and Trilokasāra. This paper is mostly concerned with the first two and the fourth works to explore their mathematical approach. Art present, Prof.L.C.Jain has studied the mathematical and scientific matters contained in the Labdhisāra under the auspicious of Indian National Science Academy. He has worked out mathematical and system theoretical aspects including explanations of algebraic and geometric expressions, based on the verses of the commentaries. 30/01/2010 Dr.Padmavathamma

In the study, he has found numerical symbols (anka samdśṭi) in both the Labdhisāra and in its commentaries [4]. Furthermore, he has compiled a comprehensive glossary of technical terms relevant to Labdhisāra. 30/01/2010 Dr.Padmavathamma

THE MATHEMATICS OF NEMICANDRĀ’S WORKS Before stating the laws of indices of Nemicandra it is better to give his terminology. If N = 2n tyhen n is called the ardhacheda of N. How many times a given number can be halved will be the ardhacheda. Sometimes the word ardha is left out and only cheda is used. In general, if N = xn then n is the cheda of N with respect to the base x. If N = 22n then n is called the ardhacheda of the ardhacheda of N 30/01/2010 Dr.Padmavathamma

Nemicandra gives the following rule: If the ardhacheda of the multiplication is added to the ardhacheda of the multiplier, then the ardhacheda of the product is obtained. This will mean more chedas as shown in the following formula: 2m x 2n = 2m+n If the ardhacheda of the divisor is subtracted from the ardhacheda of the dividend then the ardhacheda of the quotient is obtained. 2m ÷ 2n = 2m-n 30/01/2010 Dr.Padmavathamma

xm x xn = xm+n , xm ÷ xn = xm-n , (xm)n = xmn If the distributed number is multiplied by the substituted number then the ardhacheda of the resulting number is got. This means that if m is distributed into its units and each similar unit is replaced by N then the resulting number is R = Nm . If N = 2n then according to the rule we get the following R = 2nm . If the ardhacheda of the distributed number is added to the ardhacheda of the substituted number then vargaśalāka of the resulting number is obtained. From the rule of Nemicandra it is evident that he knew the following rules of indices: xm x xn = xm+n , xm ÷ xn = xm-n , (xm)n = xmn In Trilokasāra fourteen types of series are used to explain the samkhyamāna and upamamāna. 30/01/2010 Dr.Padmavathamma

Arithmetical Progressions The following rule is given by Nemicandra in relation to arithmetical progressions. Multiply the number obtained by subtracting the number of terms by one and the common difference. If the product is added to the first term then the last term is obtained and if this product is subtracted from the last term then the first term is obtained. If half the sum of the first and the last terms are multiplied by the number of terms, then the sum of the series is obtained. 30/01/2010 Dr.Padmavathamma

Geometrical Progressions To find out the sum of a geometrical progression Nemicandra provides the following rule: Multiply the common ratio as many times as the number of terms. Subtract one from the product and then divide by the number obtained by subtracting one from the common ratio and multiply by the first term. The resulting number will be the sum of the geometrical progression. Algebraically this can be written as S = a (rn - 1) ÷ (r – 1), where a is the first term, r is the common ratio and S is the sum of the series. 30/01/2010 Dr.Padmavathamma

Circle Nemicandrā’s rule regarding circles is included in the following: The (gross) circumference of a circle will be three times its diameter. The (accurate) circumference is the square-root of ten times the square of the diameter. The accurate area is obtained if we multiply one- fourth of the diameter and the circumference. Here the value of π is taken as √10. To determine the accurate circumference and area of the Jambūdvīpa the second law is used. 30/01/2010 Dr.Padmavathamma

The Prism, Cone and Sphere According to Nemicandra the volume of the prism = (area of ) base x height, the volume of the cone = (1\3) base x height and the volume of the sphere = (9\2) (radius)3 For example, to measure the volume of a heap of (mustard like) seeds which resemble a cone, Nemicandra has given the following formula. Volume = (circumference\6)2 x height In such cases it was supposed that height = (1\11) circumference. And finally we may simply note at this juncture that Nemicandra has also provided mathematical rules regarding the segments of a circle, a trapezium and many other permutations and combinations. 30/01/2010 Dr.Padmavathamma

REFERENCES S.C.Ghośal,Dravyasangraha, Motilal Banarsidas, New Delhi, 1989 B.Datta, Mathematics of Nemicandra, Jainā Antiquary, Vol.1,No.2, Arrah, 1935, p.25-44 L.C.Jain, Divergent Sequences Locating Transfinite Sets in Trilokasāra, Indian Journal of History of Science, Vol.12, No.1, 1977, p.57-75 The Labdhisāra, Vol.1, mSSMK Jain Trust, Katni,India, 1994, p.5 30/01/2010 Dr.Padmavathamma

Introduction Mahāvīrācārya was a famous Jaina mathematician who succeeded after the well-known scholars Āryabhaṭa (C. 5th century A.D.), Varāhamihira (C. 6th century A.D.) and Brahmagupta (C. 7th century A.D.). Not much is known about his life. According to the literature available, he hailed from Karnataka. He enjoyed the patronage of the Rāṣtrakūṭa king Amoghavarṣa Npatuṅga, who ruled in Mānyakheṭa (South India) from 815 A.D. to 877 A.D. The period of Npatuṅgā's rule was well-known for political stability and the development of art and culture. 30/01/2010 Dr.Padmavathamma

Mahāvīrācārya is the author of the Sanskrit work Gaṇitasārasaṅgraha (abbr. GSS), which is on elementary mathematics. It is a compilation work on universal (Laukika) mathematics which is based on non-universal (Lokottara) mathematics contained in Jaina Āgamās. It provides a valuable source of information on ancient Indian mathematics. The Gaṇitasārasaṅgraha was not available for a long time. 30/01/2010 Dr.Padmavathamma

Discovery of Gaṇitasārasaṅgraha Professor M. Rangācārya was appointed as a professor of Sanskrit and comparative philology at the Presidency College, Madras (Chennai). He also took charge of the office of the Government Oriental Manuscripts Library. The Director of Public Instruction, Mr. G. H. Stuart, directed Rangācārya to find out whether there were any manuscripts in the library which could throw new light on the History of Hindu Mathematics. If so, to publish it with an English translation and notes. Rangācārya first found three incomplete manuscripts of Mahāvīrācaryā's GSS. 30/01/2010 Dr.Padmavathamma

Out of the three manuscripts, one is written on paper in Grantha characters with a running commentary in Sanskrit. The other two are palm-leaf manuscripts in Kanarese characters, which contain a brief statement in the Kanarese language of the figures, relating to the various illustrative problems as also of the answers to the same problems. At the instance of Mr. G.H. Stuart, Prof. Rangācārya tried to get more manuscripts from other places and finally succeeded in getting two more manuscripts. 30/01/2010 Dr.Padmavathamma

One was from the Government Oriental Library at Mysore. This is a transcription on paper in Kanarese characters of an original palm-leaf manuscript. It contains the whole of the work with a short commentary in Kannada by Vallabha. The other, that is, the fifth manuscript is also a transcription on paper in Kanarese characters of a palm-leaf manuscript found in the Jain Math at Muḍbidri (South Canara). This manuscript also contains the whole work and gives a brief statement of the problems and their answers. 30/01/2010 Dr.Padmavathamma

The Madras Government published this valuable work in 1912 [18]. After carefully studying and examining these five manuscripts, Prof. Rangācārya was successful in translating Gaṇitasārasaṅgraha into English and in writing mathematical notes wherever necessary. The Madras Government published this valuable work in 1912 [18]. Dr. D.E. Smith, Professor of Mathematics, Teacher's College, Columbia University, New York, has written an introduction to this book. In fact, he had read a paper on GSS at the fourth International congress of Mathematicians held at Rome in April 1908. 30/01/2010 Dr.Padmavathamma

Translators / Commentators of GSS Vallabha (Daivajña-Vallabha) has written commentaries both in Kannada and Telugu for GSS. Pāvalurimallaṇṇa [2] has translated GSS into Telugu. From D. Pingree [17] it is clear that a commentary by Varadarāja and a Rājasthāni translation by Amicandra (in 1842) are also there. L.C. Jain translated GSS into Hindi in 1963 and edited it with collation of additional manuscripts and with detailed historical introduction from beginning of the histrical era upto Mahāvīrācārya. This was published by Jain Samskriti Samrakṣka Sangha, Sholapur [9]. 30/01/2010 Dr.Padmavathamma

New Edition of Gaṇitasārasaṅgraha Mahāvīrācārya happens to be from Karnataka, the motherland of Kannada. There was a need for the Kannada version so that Kannadigas could appreciate the beauty of the mathematics contained in GSS. This deficiency was met by the author. She has translated the original Sanskrit verses into Kannada and also translated the Sanskrit verses into English. Since both the English and Hindi editions of GSS were out of print for a long time, the English translation was also included in the new work. This new edition of GSS [16] was published by Sri Siddhānta Keerthi, Granthamala of Sri Hombuja Jain Math, Shimoga District, Karnataka in the year 2000. 30/01/2010 Dr.Padmavathamma

This new edition of GSS is a rare and unique text This new edition of GSS is a rare and unique text. It appears as a combination of three fine fragrant flowers blooming from the same creeper. The new style and the format followed in this book are of high standard and very attractive. The Sanskrit, English and Kannada versions which are like the gemtrios are accommodated in the same volume. Similar to the mingling of three sacred rivers, this single book embodies the presentation of the text in three different languages- Sanskrit, English and Kannada. The review of this book by S. Balachandra Rao has appeared in Gaṇitabhārati, vol. 25, Nos. 1-4, 2003, p. 197-199. 30/01/2010 Dr.Padmavathamma

TELUGU EDITION OF GSS GSS was translated into Telugu by Vidwan T. Subbarao and edited by P.V. Aruṇāchalam. This was published [21] by the Telugu Academy, Hyderabad in the year 2003. 30/01/2010 Dr.Padmavathamma

Other works of Mahāvīrācārya Different research scholars have agreed that Mahāvīrācārya is also the author of the following four works: 1. Ṣattrimśika (Ṣattrim Ṣatika) 2. Jyotiṣapaṭala 3. Kṣetra Gaṇita 4. Chattīsapūrva Uttara Pratisaha 30/01/2010 Dr.Padmavathamma

Mathematics of GSS The style of GSS is in the form of a text book. This is a collection of the South Indian mathematics which was embedded in Digambara Jaina texts of the Karaṇānuyoga and the Dravyānuyoga Groups. Keeping in view the Jaina Karma Theory in its Pūrvā's tradition of the Digambara Jaina School, it can be said that the mathematics of GSS is the one that has come through mathematico philosophical texts of the Digambara Jainācāryās and definitely not the sole contribution of Mahāvīrācārya alone. This is made clear by Mahāvīrācārya himself in the following stanzas 17, 18, and 19 of the first chapter of GSS. However the rules and formulae in the existing literature which appear first in GSS can be certainly credited to him: 30/01/2010 Dr.Padmavathamma

Translation – Sanskrit shloka With the help of the holy sages, who are worthy to be worshipped by the lords of the world and of their disciples and disciples, who constitute the well-known jointed series of preceptors, I glean from the great ocean of the knowledge of numbers a little of its essence, in the manner in which gems are (picked up) from the sea, gold is from the stony rock and the pearl from the oyster shell and give out, according to the power of my intelligence, the Sārasaṅgraha, a small work on arithmetic, which is (however) not small in value. 30/01/2010 Dr.Padmavathamma

The List of Chapters Detailed in GSS 1. Samjñadhikārah (On Terminology) 2. Parikarma Vyavahārah (Treatment on Algebraic operations) 3. Kalā Savarṇa Vyavahārah (Treatment on Fractions) 4. Prakīrṇaka Vyavahārah (Miscellaneous Problems on Fractions) 5. Trairāśika Vyavahārah (The Rule of Three) 6. Miśraka Vyavahārah (Mixed Problems) 7. Kṣetragaṇita Vyavahārah (Calculations Relating to the Measurement of Areas) 8. Khāta Vyavahārah (Calculations Regarding Excavations) 9. Chāyā Vyavahārah (Calculations Relating to Shadows) 30/01/2010 Dr.Padmavathamma

Chapters I and II deal with the six algebraic operations multiplications, division, squaring, cubing, extraction of square roots and cube roots. Arithmetic and geometric series have also been discussed. In case of multiplication, four rules are given with examples. Along with the rule of division, the modern rule is also explained. There is a special rule for squaring which is as follows: SANSKRIT SHLOKA 30/01/2010 Dr.Padmavathamma

Translation – Sanskrit shloka Get the square of the last figure (in the number, the order of counting the figures being from the right to the left) and then multiply this last (figure), after it is doubled and pushed on (to the right by one notational place), by (the figures found in) the remaining places. Each of the remaining figures (in the number) is to be pushed on (by one place ) and then dealt with similarly. This is the method of squaring. 30/01/2010 Dr.Padmavathamma

This rule will be clear from the following examples: 1. To find the square of 12 12 = 1 2 × 1 × 2 = 4 22 = 4 Therefore the square of 12 = 144 2. To find the square of 131 2 × 1 × 3 = 6 2 × 1 × 1 = 2 32 = 9 2 × 3 × 1 = 6 12 = 1 Therefore the square of 131 = 17 1 6 1 30/01/2010 Dr.Padmavathamma

Mahāvīrācārya has discussed various algebraic operations involving zero. He writes (GSS, chapter 1, Verse No. 49) : 30/01/2010 Dr.Padmavathamma

Translation – Sanskrit shloka A number multiplied by zero is zero and that (number) remains unchanged when it is divided by ; combined with or diminished by zero. Multiplication and other operations in relation to zero (give rise to) zero and then in the operation of addition, the zero becomes the same as what is added to it. Algebraically, the above can be expressed as follows [16, p.10]: A × 0 = 0, A + 0 = A, A – 0 = A, A 0 = A Actually Mahāvīrācāryā's rule related to multiplication, addition and subtraction is correct, but his rule related to division has another interpretation, implying that the division by zero means the non-existence of divisor. Sridhara (c. 10th century A.D) who was, perhaps, not earlier to Mahāvīrācārya has not considered division by zero. 30/01/2010 Dr.Padmavathamma

It is a point to be noted that according to concept of improper or mathematical infinity, the correct answer as a limiting value was known to Brahmagupta 300 years earlier. Bhāskarācārya (1150 A.D.) has given the symbol Khahara for the result of division by zero and rightly assigns to it the value of mathematical infinity. The concept of proper infinities in the Jaina Āgamās was to come with George Cantor in the sixties of the nineteenth century, Mahāvīrācārya obviously thinks that a division by zero is not division at all. Multiplications of those numbers which lead to numbers of a necklace (that is numbers which are same when read either from right or left) are very interesting : [16, Chapter II, Examples 3 et seq.] 30/01/2010 Dr.Padmavathamma

139 × 109 = 15151 152207 × 73 = 11111111 14287143 × 7 = 100010001 12345679 × 9 = 111111111 142857143 × 7 = 1000000001 11011011 × 91 = 1002002001 27994681 × 411 = 12345654321 333333666667 × 33 = 11000011000011 Chapters III and IV deal respectively with kalāsavarṇa and Prakīrṇaka Vyavahāra and are completely devoted to fractions. Types of fractions and operations on fractions have been discussed in detail. Some points worthy to be noted are given below. 30/01/2010 Dr.Padmavathamma

Credit goes to Mahāvīrācārya for expressions of unit fractions as the sum of unit fractions. In words of Brijmohan [5], "No other Indian mathematician has even touched upon this". This problem has created much interest to Ahmes (Papyrus, 1050 B.C.). In modern notation the relevant problems [16, Chapter III, 75-78] can be expressed as follows: 30/01/2010 Dr.Padmavathamma

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In Jaina Āgamās, permutations and combinations play an important role. In Chapter V, Mahāvīrācārya has given utmost importance to Rule of Three and most part of it is devoted to Rule of Three and its generalised forms. In Jaina Āgamās, permutations and combinations play an important role. As a result in Gaṇitasārasaṅgraha also it has been explained in Chapter VI in great detail. Actually the following rule gives the number which can be chosen (out of n given things) r at a time. The concerned verse is [16, Chapter VI, verse No. 218] :- 30/01/2010 Dr.Padmavathamma

Translation – Sanskrit shloka Beginning with one and increasing by one, let the numbers going up to the given number of things be written down in regular order and in the inverse order (respectively) in an upper and a lower (horizontal) row, (If) the product (of one, two, three or more of the numbers in the upper row) taken from right to left be divided by the (corresponding) product (of one, two, three or more of the numbers in the lower row) also taken from right to left, (the quantity required in each such case of combination) is (obtained as) the result. 30/01/2010 Dr.Padmavathamma

Algebraically, The credit goes to Mahāvīrācārya for the above formula since he appears to be the first to collect it in the world as above. It is also interesting to note that the same formula became prevalent again through Herign in 17th century A.D. 30/01/2010 Dr.Padmavathamma

Series Excluding Vedic mathematics, in Jaina literature, the use of series has appeared in the Kalpasūtra (c. 2nd century A.D.). Although information can be obtained from the works of the earlier mathematicians, Āryabhaṭa and Brahmagupta, in Mahāvīrācārya's GSS great detail and systematic analysis of series are available. In words of Lal [14]: "No doubt, his (Mahāvīrācāryā's) predecessors, Āryabhaṭa (476 A.D.) and Brahmagupta (599 A.D. ), had contribution to the subject, yet Mahāvīrācārya can be named to be the first amongst them, who put the subject elaborately using lucid method and charming language." 30/01/2010 Dr.Padmavathamma

If a, d and s respectively denote the first term, the common difference and the sum of n terms of an arithmetical progression, Mahāvīrācāryā's rules are as follows: In relation to geometrical progressions the following rule is given to find out the sum of first n terms [16, Chapter 2, Verse No. 97] 30/01/2010 Dr.Padmavathamma

Where a is the first term and r is the common ratio. Mahāvīrācārya has not only considered arithmetical and geometrical progressions, he has also dealt with many other series. To find out the sum of the squares of the first n terms of an arithmetical progression the formula is and the sum of the cubes of the first n terms is Actually we find many such formulae in this chapter. 30/01/2010 Dr.Padmavathamma

Chapter VII deals with problems on mensuration. Since this is used extensively in the creation of the universe, lot of information is available from Jaina works. Here, Mahāvīrācārya not only discusses figures which have been already considered by his predecessors but also deals with many new figures and has given formulae to find their areas. Chapter VIII is about calculations regarding excavations. ] 30/01/2010 Dr.Padmavathamma

Chapter IX deals with shadow problems Chapter IX deals with shadow problems. Here many formulae and examples are given to find out the length of the shadow in day time from which time can be calculated. Let us look at the following example, [ 16, Chapter 9, verse no. 38 ½ - 39 ½ ] 30/01/2010 Dr.Padmavathamma

Translation – Sanskrit shloka At the time when, in the course of a forenoon the human shadow is twice the human height what in relation to a (vertical excavation of) square (section) measuring 10 hastas in each dimension, will be the height of the shadow on the western wall caused by the eastern wall (there of) ? O mathematician, give out if you know, how you may arrive at the value of the shadow that has ascended up (a perpendicular wall). To work out this problem, Mahāvīrācārya gives the following rule [16, Chapter IX, Verse No.21]: 30/01/2010 Dr.Padmavathamma

Translation – Sanskrit shloka The height of the style is multiplied by the measure of the human shadow (in terms of the man's height). The (resulting) product is diminished by the measure of the interval between the wall and the style. The difference (so obtained) is divided by the very measure of the human shadow (referred to above). The quotient so obtained happens to be the measure of (that position of) the style's shadow which is on the wall. Using the above rule, it is clear that the height of the shadow on the western wall caused by the eastern wall is in Hastas. 30/01/2010 Dr.Padmavathamma

CONCLUDING REMARKS Mahāvīrācārya occupies unique place in the History of Mathematics in India. His contributions towards imaginary numberrs, least common multiples, number of combinations, solution of algebraic equations and application of algebra to janyavyavahāra and in determining the areas of many strange or unfamiliar figures are of immense importance. Each one who reads Gaṇitasārasaṅgraha will definitely become interested in mathematics. 30/01/2010 Dr.Padmavathamma

He emphasized theoretical and practical implications. Through Gaṇitasārasaṅgraha mathematics acquired an identity of its own. Actually before Mahāvīrācārya, mathematics was in the garb of Jyotisha or it was a handmaid of religious rituals. Mahāvīrācārya gave the subject a form, an identity and an independent existence. He emphasized theoretical and practical implications. For higher education, when people looked at Varanasi, Ujjaian, Pataliputra, Nalanda, Takshashila etc. Mahāvīrācārya established a great center for learning in Karnataka. As such he earned an esteemed place in the galaxy of Indian mathematics. 30/01/2010 Dr.Padmavathamma

REFERENCES 1. Agrawal, M. B. - Mahāvīrācārya ki Jain Gaṇita ko Dena, Jaina Siddhāntha Bhāskara, Arrah, 24-1, 42-47.1967. 2. Agrawal, M. B. - Gaṇita evam Jyotiṣa ke Virasamem Jainācāryon kā yogadān, Agra University, Doctoral Thesis 1972, 377. 3. Ambalal Sha - Jaina Sāhitya brhad itihās, parts, parśvanāth vidyāśram Śodha Samstha, Varanasi, 1965, 160-61. 4. Bell, E. T - Development of Mathematics, Macraw Hill, New York, 1940. 5. Brijmohan - Gaṇita ka itihās, U. P. Hindi Granth Academy, Lucknow, 1965. 6. Gupta, R. C - Mahāvīrācāryā's Rule for the volume of Frustum - like Solids, Aligarh Journal of Oriental Studies, Vol III, 1986, No.1, 31-38. 30/01/2010 Dr.Padmavathamma

7. Jain, A. –Śattrimśika ya Śattrim Śatika, Jain Siddhānth Bhāskar, Āra, 34-2, 1982, 39-40. 8.Jain, A. L - Mahāvīrācārya, Vyaktitva evam krititva - Jaina Śodhanka 47, Mathura, 1981, 258-260. 9. Jain, L.C - Gaṇitasārasaṅgraha, Hindi Edition, Sholapur, 1963. 10.Jain, L. C - Bhārtīya Gaṇitasāstr evam Jain Lokottara Gaṇit, Jain Viśva Bhārati, Ladnu, 1973, 33-41. 11.Jain, P. - Jain Dharm ka prācīn itihās, part 2, P. S. Jain, Motor company, Delhi, 1974. 12.Jyotiprasad Jain - Rāśtrakūṭa yug kā Jain Sāhity samvardhana me yogadān, Siddānthācārya Kailāschand Abhinandan Granth, Rewa 1980, 273-280. 13.Kāsalīvāla, K.- Rājasthān ke Jaina Śastr Bhandarom ki Granth Sūci Shri Mahāvīrji Atishay Kshetra, Shri Mahāvīrji, 1957. 30/01/2010 Dr.Padmavathamma

14. Lal, R. S. and Sinha, S. R - Contribution of Mahāvīrācārya in the Development of Theory of Series - M. E. (Shinan) 15-B, 1987, 83-92. 15. Nemichandra Shāstri - Tīrthaṅkara Mahāvīra Aur unkī Ācārya Parampara. 16. Padmavathamma, Gaṇitasārasaṅgraha, Kannada Edition, Sri Siddhāntakeerthi Granthamāla, Humbuja, Shimoga District, Karnataka, 2000. 17. Pingree, D. - Census of Exact Sciences in Sanskrit, Series A. vol. 4, Philadelphia, 1981, 388 18. Rangācārya, M - Gaṇitasārasaṅgraha, English Edition, Madras Government, 1912. 19. Shāstri, N.C. - Bharatīya Jyotiṣa kā pośak Jain Jyotiṣ Sāhity Varnī Abhinandan Granth Sagar, 1950, 470-484. 20. Subbarao, T. - Gaṇitasārasaṅgraha, Telugu edition, (edited by P.V. Arunachalam) Telugu Academy, Hyderabad, 2003. 30/01/2010 Dr.Padmavathamma

Rājāditya Rājāditya was a Jainā poet. Probably he was the first person who wrote mathematical works in Kannada. He is the author of the following works on mathematics. Vyavahāragaṇita Kṣetragaṇita Vyavahāraratna Līlāvati Chitrahasuge Jainagaṇita Sūtrodāharaṇa 30/01/2010 Dr.Padmavathamma

Out of the above six works of Rājāditya, only Vyavahāragaṇita has been published by the Madras (now Chennai) Government in the year 1955. This was critically edited by Prof.Mariappa Bhat who was the Head of the Kannada Department in the University of Madras. This edition of Vyavahāragaṇita is based on three palm-leaf manuscripts and one paper manuscript which were preserved in the Madras Government. This is written in Kannada Vrtta metres. Rājāditya was known by many other names such as Rājā, Rājavarma, Bhāskara, Bāchayya. Many titles like Vojevedanga, Padyavidyādhara, Uttamabhavabhūsaṇa, Jinapadakamalamadhukara were conferred on him. It is evident from his compositions of verses that he is not only a mathematician but also a good poet. 30/01/2010 Dr.Padmavathamma

In the Introduction to Vyavahāragaṇita he has beautifully described his native place Poovinabāge. This is mostly located in North-Karnātaka and resembles very much Hoovinahaḍagali and Bāgevāḍi. In Vyavahāragaṇita, Rājāditya states that his master was Shubhachandra, father was Shripathi, mother was Vasantha and patrons were Bāhubali-Bharatha. Being handsome, honest, helpful to others and a great scholar, Rājāditya was flourishing very well in some royal court. He was also a devotee of Nemitīrthamkara. Rājādityā’s mathematical works are mainly in the form of verses which is a rare combination of poetic genius and scientific knowledge. 30/01/2010 Dr.Padmavathamma

The published work Vyavahāragaṇita is concerned with commercial arithmetic of olden times. It consists of the following eight sections – Technical terms, Proposition, Proportionate Division, Mixture, Interest, Profit and Loss, Discount and Miscellaneous. In almost every topic, the principle is first stated which is followed by illustration of examples. At the end of every problem we find labdha which is a brief analysis with the answer to that problem. This labdha is followed by ṭīkā which gives a clear explanation. Hundreds of problems in this book are taken from real life which enable us to have an idea of the socio-economic conditions which prevailed in Karnataka during the 12th century A.D. 30/01/2010 Dr.Padmavathamma

The Introductory Chapter of Vyavahāragaṇita ends beautifully thus – It is surprising to note that in many parts of Karnataka, copies of Vyavahāragaṇita were made and were used to teach mathematics for children from 12th century A.D. From the manuscript (No.D.1445) available at Madras Oriental Research Library, it is clear that Vyavahāragaṇita was used not only by Jains but also by Brahmins and others. The Introductory Chapter of Vyavahāragaṇita ends beautifully thus – “ Idu Shubha Chandradeva yogīndra pādāravinda Madhukarāyamaṇam Manasānamdita Sakalagaṇita tattvavilāsa vineyajana vinuta Śri Rājāditya viracitamappa Vyavahāragaṇitado pīṭīkā prakaraṇam samāptam.” 30/01/2010 Dr.Padmavathamma

Regarding the time of Rājāditya there is a controversy between historians and Dr.Venkatasubbiah. In the published work of Vyavahāragaṇita, the name of the king Vishṇunpāla appears in some of the examples. The historians opine that this Vishṇunpāla must be the king Vishṇuvardhana who ruled during 1111-1141 A.D. Besides this, in one of the scriptures of 117 at Shravaṇbelgoa there is a mention that a teacher by name Shubhachandra who expired in the year 1123 A.D. Based on these facts the historians decided that Rājāditya was a poet around 1120 A.D. in the royal court of king Vishṇuvardhana. 30/01/2010 Dr.Padmavathamma

But Dr.Venkatasubbiah asserts that many kings had the same name Vishṇuvardhana in Hoysaa family. He also proves that many Jainā teachers at different times had the same name Shubhachandra. Hence he considers the topic related to Bharatabāhubali stated in the above verses to determine the time of Rājāditya. On the basis of scriptures, he establishes the family-tree of Dākarasa Damḍanāyaka. Then he shows that these Bāhubalibharatās being the sons of the second Mariyāne Damnḍanāyaka were respectively Damḍanāyaka and Mahāpradhān Sarvādhikāri Māṇikyabhamḍāri in the reign of second Ballāa. After paying the dues to their king second Ballāa, Bāhubalibharata got the possession of the cities Sindhagere etc. in the year 1183 A.D. 30/01/2010 Dr.Padmavathamma

Dr.Venkatasubbiah opines that Rājāditya must have referred to this Ballāa as Vishṇuvardhana. Besides this, there are many evidences to prove that the guru Shubhachandra praised by Rājāditya also lived during this period. For example, from one of the verses in Vyavahāragaṇita, it follows that in 1191 A.D, the king Billama was defeated by Hoysaa Varaballāa in Soratur. Taking this great incident as the subject, Rājāditya who composed mathematical formulae must have lived definitely around 1190 A.D – thus concludes Dr.Venkatasubbiah. Later historians Shri Narasimhacharya and others have also accepted this period (1190 A.D) of Rājāditya. 30/01/2010 Dr.Padmavathamma

From the chapter on terminology in Vyavahāragaṇita we find names upto 40 unit places. They are – ekkam, daham, śatam, dasavīra, lakṣa, dālakṣa, koṭi, dākoṭi, śatakoṭi, arbuda, nyarbuda, kharva, mahākharva, padma, mahāpadma, kṣoṇi, mahākṣoṇi, samkha, mahāśamkha, kṣiti, mahākṣiti, kṣobha, mahākṣobha, nadi, mahānadi, naga, mahānaga, ratha, mahāratha, hari, mahāhari, phaṇi, mahāphaṇi, kratu, mahākratu, sāgara, mahāsāgara, parimita, mahāparimita. There are many synonyms for the numbers 0 upto 9 in Vyavahāragaṇita. Example for “0” : 0 – divi, kaika, śūnya, agra, ambara, gagana, meghamārga, ākāśa, jaladharmārga, bamdhayogaa, viyat, ayana, patha, abhrakham, vattam, agasa, jaladharapatha, vyadvite. 30/01/2010 Dr.Padmavathamma

Regarding Vyavahāragaṇita, Rājāditya says thus – Many names are also there for addition, subtraction, multiplication and division. Regarding Vyavahāragaṇita, Rājāditya says thus – Dhāruṇiyo sakala budhā Dhāramenal svalpa māgiyam vyavahāra Dhāramenal sāramenal Śri Rājādityanaltiyam viracisidam || From the above it follows that his work is brief and is useful for many practical purposes. He says that to know mathematics, ancient mathematical works are sufficient. But for easy transactions and quick references, Vyavahāragaṇita is written. 30/01/2010 Dr.Padmavathamma

Rājāditya must have written Vyavahāragaṇita after carefully examining the previously available mathematical works. Let us look at this example – One rich person appointed a mahout at the rate of one gadyāṇa per day as wage to look after 108 elephants. But he sells one elephant per day. After 25 days mahout left that job. Calculate the amount to be paid to the mahout for these 25 days of work by the rich person. 30/01/2010 Dr.Padmavathamma

(2400) \ (108) = 200 \ 9 = 22 gadyāṇa 2 paṇa 4 vīsa 1 kāṇi. According to a formula given in Vyavahāragaṇita, add 1 to 108 to get 109. Multiply 109 by 25 (the last day). Instead of taking 108, 107, … since 109 X 25 is taken , the excess taken, that is, 1 + 2 + … + 25 has to be subtracted from 109 X 25 = 2725. Now to get 1 + 2 + … + 25, square 25 and add 25 which gives 625 + 25 = 650. Halve this to get 325. Subtract 325 from 2725 to get 2725 – 325 = 2400. Thus looking after 108 elephants for 1 day, 107 for 1 day, 106 for 1 dat etc. is equivalent to looking after 2400 elephants for 1 day. The rest is easy Rule of Three and the answer is – (2400) \ (108) = 200 \ 9 = 22 gadyāṇa 2 paṇa 4 vīsa 1 kāṇi. 30/01/2010 Dr.Padmavathamma

The following table of coins and fractions is found in Vyavahāragaṇita. 4 Kāṇi = 1 Vīsa, 5 Visa = 1 Haga, = 4 Haga = 1 Paṇa, 10 Paṇa = 1 Gadyāṇa. Here the formula used to sum 1 + 2 + … + 25 is now the well-known formula 1 + 2 + … + n = n(n + 1) \ 2 in Algebra. From Vyavahāragaṇita it appears that instead of using money, exchange system was in practice. Problems related to arecanut indicate that arecanuts were sold not by weighing but by actual counting. Rājāditya occupies a unique place in the History of Kannada literature as he was the first person to write mathematics in Kannada. 30/01/2010 Dr.Padmavathamma

30/01/2010 Dr.Padmavathamma