CONTRIBUTION OF JAINA MATHEMATICIANS

CONTRIBUTION OF JAINA CONTRIBUTION OF JAINA MATHEMATICIANSMATHEMATICIANS

DR. (MRS). PADMAVATHAMMA, M Sc, Ph DProfessor of Mathematics (retired)Department of Studies in MathematicsUniversity of Mysore, ManasagangotriMysore-570 006E-mail: [email protected]

30/01/2010 Dr.Padmavathamma 2

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 Āryabha a, Bhāskara, Brahmagupta, Mahāvīrācārya and ṭBhāskarācārya 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.

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Tattvārthadhigama Sūtra, Sthānanga Sūtra, Jambūdvīpa Prajñapti, Tiloyapa atti, ṇṇK etrasamāsa, Ga itasārasangraha and ṣ ṇVyavahāraga ita are important ancient Jaina ṇmathematical 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 ga iyappahā āo, ṇ ṇ 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.


In Jainā literature there are four anuyogās called prathama, kara a, cara a and dravya. In ṇ ṇkara ānuyoga, many mathematical operations are ṇused 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, Tiloyapa atti, Dhavalā ṇṇcommentaries of a kha āgama, Gomma asāra, Ṣ ṭ ṇḍ ṭTrilokasāra.

The above works provide valuable information to know the ancient Indian mathematics. Sūryaprajñapti is called Ga itānuyogaṇ .


The following list shows the names of persons who have contributed to the development of mathematics in Prakrit.

i. Pu padanta- Bhūtabali (3ṣ rd century A.D)

ii. Yativabhācārya (5ṣ th century A.D)

iii.Vīrasenācārya (9th century A.D)

iv. Srīdhara (9th century A.D)

v. Srīpati (10th century A.D)

vi. 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.


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 Pā na. He was living in the ancient Pā alīpura which ṭ ṭis now the modern Kusumapura in Pa na.ṭ

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.


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.

i. Initial period (Ādikāla) - 3000 – 500 B.C

ii. Childhood Period (Śaiśavakāla) – 500 B.C -500 A.D

iii. This is also known as Dark Period

iv. Middle Period – 500 – 1200 A.D

v. Later Period – 1200 -1800 A.D

vi. Modern Period – 1800 A.D onwards


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. ”


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

i. Nemicandra Siddhānta Cakravarti whose works are in Prakrit

ii. Mahāvīrācārya whose works are in Sanskrit

iii.Rājāditya whose works are in Kannada


Nemicandra Siddhānta CakravartiNemicandra Siddhānta Cakravarti INTRODUCTIONINTRODUCTION

There are four celebrated ascetic sanghās in the History of Jainās in South India. These sanghās are Nandi, Simha, Sena and Deva.

The Deśīya ga a is a branch of the Nandi sangha and ṇoriginated 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 ga a helped ṇSivamara to found the Ganga dynasty, one of the ancient royal kingdoms of India [1].


Of the twenty scholars in this ga a who were ṇhonoured with the title “Siddhānta Cakravarti” , ācārya Nemicandra is most known for his work in mathematics.

Ascetic Lineage of Nemicandra: It is known from the Gomma asāra (Karmakā a part) that ṭ ṇḍNemicandra was the disciple of Abhayanandi, whose preceptor was Gu anandi. Vīranandi, a ṇcolleague and contemporary of Nemicandra was alsoa disciple of Abhayanandi. Vīranandi, the author of the Candraprabha Mahākāvya, has received homage from Nemicandra in the Gomma asāra (Chapter 6, verse 396).ṭ


Nemicandra as a mathematicianNemicandra as a mathematicianExcept 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.


The works of NemicandraThe works of NemicandraNemicandra Siddhānta Cakravarti is the author of

Dravyasamgraha, Gomma asāra, Labdhisāra, ṭKśapa asāra and Trilokasāra. This paper is mostly ṇconcerned 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.


In the study, he has found numerical symbols (anka samdśṭ i) in both the ṭLabdhisāra and in its commentaries [4]. Furthermore, he has compiled a comprehensive glossary of technical terms relevant to Labdhisāra.


THE MATHEMATICS OF NEMICANDRĀ’S THE MATHEMATICS OF NEMICANDRĀ’S WORKSWORKS

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


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


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.


Arithmetical Arithmetical ProgressionsProgressions 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.


Geometrical Geometrical ProgressionsProgressions 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.


CircleCircleNemicandrā’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 is taken as √10.π

To determine the accurate circumference and area of the Jambūdvīpa the second law is used.


The Prism, Cone and The Prism, Cone and SphereSphere 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.


REFERENCESREFERENCES

1. S.C.Ghośal,Dravyasangraha, Motilal Banarsidas, New Delhi, 1989

2. B.Datta, Mathematics of Nemicandra, Jainā Antiquary, Vol.1,No.2, Arrah, 1935, p.25-44

3. L.C.Jain, Divergent Sequences Locating Transfinite Sets in Trilokasāra, Indian Journal of History of Science, Vol.12, No.1, 1977, p.57-75

4. The Labdhisāra, Vol.1, mSSMK Jain Trust, Katni,India, 1994, p.5


IntroductionIntroductionMahāvīrācārya was a famous Jaina mathematician

who succeeded after the well-known scholars Āryabha a (C. 5th century A.D.), Varāhamihira (C. 6th ṭcentury 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 Rā trakū a king ṣ ṭAmoghavar a Npatu ga, who ruled in Mānyakheṭ a ṣ ṅ ṭ(South India) from 815 A.D. to 877 A.D.

The period of Npatuṭ gā's rule was well-known for ṅpolitical stability and the development of art and culture.


Mahāvīrācārya is the author of the Sanskrit work Ga itasārasa grahaṇ ṅ (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 Ga itasārasa grahaṇ ṅ was not available for a long time.


Discovery of Discovery of GaṇitasārasaṅgrahaGaṇitasārasaṅgraha 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.


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.


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 Mu bidri (South Canara). ḍ

This manuscript also contains the whole work and gives a brief statement of the problems and their answers.


After carefully studying and examining these five manuscripts, Prof. Rangācārya was successful in translating Ga itasārasa grahaṇ ṅ 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.


Translators / Commentators Translators / Commentators of GSSof GSSVallabha (Daivajña-Vallabha) has written

commentaries both in Kannada and Telugu for GSS.

Pāvalurimalla a [2] has translated GSS into ṇṇTelugu.

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 Samrak ka ṣSangha, Sholapur [9].


New Edition of New Edition of GaṇitasārasaṅgrahaGaṇitasārasaṅgrahaMahāvīrācārya 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.


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 Ga itabhārati, vol. 25, Nos. 1-4, 2003, p. ṇ197-199.


TELUGU EDITION OF TELUGU EDITION OF GSSGSSGSS was translated into Telugu by

Vidwan T. Subbarao and edited by P.V. Aru āchalam. This was published [21] ṇby the Telugu Academy, Hyderabad in the year 2003.


Other works of Other works of MahāvīrācāryaMahāvīrācāryaDifferent research scholars have agreed

that Mahāvīrācārya is also the author of the following four works:

1. attrimśika ( attrim Ṣ Ṣ Ṣatika)2. Jyoti apa alaṣ ṭ3. K etra Ga itaṣ ṇ4. Chattīsapūrva Uttara Pratisaha


Mathematics of GSSMathematics of GSSThe 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 Kara ānuyoga and the Dravyānuyoga Groups.ṇ

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:


Translation – Sanskrit Translation – Sanskrit shlokashloka 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 Sārasa grahaṅ , a small work on arithmetic, which is (however) not small in value.


The List of Chapters Detailed The List of Chapters Detailed in GSS in GSS

1. Samjñadhikārah (On Terminology)2. Parikarma Vyavahārah (Treatment on Algebraic

operations)3. Kalā Savar a Vyavahārah ṇ (Treatment on Fractions)4. Prakīr aka Vyavahārahṇ (Miscellaneous Problems on

Fractions)5. Trairāśika Vyavahārah (The Rule of Three)6. Miśraka Vyavahārah (Mixed Problems)7. K etraga ita Vyavahārahṣ ṇ (Calculations Relating to

the Measurement of Areas)8. Khāta Vyavahārah (Calculations Regarding

Excavations)9. Chāyā Vyavahārah (Calculations Relating to

Shadows)


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


Translation – Sanskrit Translation – Sanskrit shlokashloka 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.


This rule will be clear from the This rule will be clear from the following examples:following examples:1. To find the square of 12

12 = 12 × 1 × 2 = 422 = 4

Therefore the square of 12 = 144

2. To find the square of 13112 = 12 × 1 × 3 = 62 × 1 × 1 = 232 = 92 × 3 × 1 = 612 = 1

Therefore the square of 131 = 17 1 6 1


Mahāvīrācārya has discussed various algebraic operations involving zero. He writes (GSS, chapter 1, Verse No. 49) :


Translation – Sanskrit Translation – Sanskrit shlokashloka 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 = AActually 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.


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.]


139 × 109 = 15151152207 × 73 = 1111111114287143 × 7 = 10001000112345679 × 9 = 111111111142857143 × 7 = 100000000111011011 × 91 = 100200200127994681 × 411 = 12345654321333333666667 × 33 =

11000011000011

Chapters III and IV deal respectively with kalāsavar a and Prakīr aka Vyavahāra and are ṇ ṇcompletely devoted to fractions. Types of fractions and operations on fractions have been discussed in detail. Some points worthy to be noted are given below.


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:


32

1

3

1....

3

1

3

1

2

11

222 nn

21

.2

1

21

.212

1....

21

.4.3

1

21

.3.2

11

nnn

....1

211

2

1

1

aanan

a

ann

a

n

rr

r

rr

r

aana

a

aanaan

a

........... 11121

1


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 Ga itasārasa grahaṇ ṅ 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] :-


Translation – Sanskrit Translation – Sanskrit shloka 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.


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.

!!

!

....2.1

1....1

rnr

n

r

rnnnnCr


SeriesSeries 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, Āryabha a and ṭBrahmagupta, in Mahāvīrācārya's GSS great detail and systematic analysis of series are available.

In words of Lal [14]:"No doubt, his (Mahāvīrācāryā's) predecessors, Āryabha a (476 A.D.) and Brahmagupta (599 A.D. ), ṭhad 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."


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]

,

2

21n

anS

20 No. Verse ,2,16 Chapter

,2

.852

d

an

ddda

22] No. Verse 2,Chapter [16,

,2)1(

n

dnnS

a

24 No. Verse 2,Chapter 16,

, 225

1

1

a

nnd 198 No. Verse 7,Chapter 16,


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.

1

r

aarS

n

n


6

)12)(1(21 222 .....

nnnn

2333

2

)1(21 .....

nn

n

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. ]


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 ]


Translation – Sanskrit Translation – Sanskrit shlokashloka 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]:


Translation – Sanskrit Translation – Sanskrit shlokashloka 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.

52

10

2

10210


CONCLUDING REMARKSCONCLUDING REMARKSMahāvīrācārya 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 Ga itasārasa grahaṇ ṅ will definitely become interested in mathematics.


Through Ga itasārasa graha mathematics acquired ṇ ṅan 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.


REFERENCESREFERENCES1. Agrawal, M. B. - Mahāvīrācārya ki Jain Ga ita ko ṇ

Dena, Jaina Siddhāntha Bhāskara, Arrah, 24-1, 42-47.1967.

2. Agrawal, M. B. - Ga ita evam Jyoti a ke Virasamem ṇ ṣJainācāryon 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 - Ga ita ka itihās, U. P. Hindi Granth ṇAcademy, 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.


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 - Ga itasārasa graha, Hindi Edition, Sholapur, ṇ ṅ1963.

10.Jain, L. C - Bhārtīya Ga itasāstr evam Jain Lokottara Ga it, ṇ ṇJain 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 - Rāśtrakū a yug kā Jain Sāhity ṭsamvardhana 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.


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 - Tīrtha kara Mahāvīra Aur unkī ṅĀcārya Parampara.

16. Padmavathamma, Ga itasārasa graha, Kannada Edition, ṇ ṅSri 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 - Ga itasārasa graha, English Edition, ṇ ṅMadras Government, 1912.

19. Shāstri, N.C. - Bharatīya Jyoti a kā pośak Jain Jyoti ṣ ṣSāhity Varnī Abhinandan Granth Sagar, 1950, 470-484.

20. Subbarao, T. - Ga itasārasa graha, Telugu edition, ṇ ṅ(edited by P.V. Arunachalam) Telugu Academy, Hyderabad, 2003.


RājādityaRājāditya

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.

i. Vyavahāraga itaṇii. K etraga itaṣ ṇiii.Vyavahāraratna

iv. Līlāvati

v. Chitrahasuge

vi. Jainaga ita Sūtrodāhara aṇ ṇ


Out of the above six works of Rājāditya, only Vyavahāraga ita ṇhas 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 Vyavahāraga ita is based on three palm-leaf ṇmanuscripts 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, Uttamabhavabhūsa a, Jinapadakamalamadhukara were ṇconferred on him. It is evident from his compositions of verses that he is not only a mathematician but also a good poet.


In the Introduction to Vyavahāraga ita he has ṇbeautifully described his native place Poovinabāge.

This is mostly located in North-Karnātaka and resembles very much Hoovinaha agali and ḍBāgevā i. ḍ

In Vyavahāraga ita, Rājāditya states that his master ṇwas 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.


The published work Vyavahāraga ita is concerned ṇwith 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 īkā which gives a clear ṭexplanation. 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.


It is surprising to note that in many parts of Karnataka, copies of Vyavahāraga ita were made ṇand 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 Vyavahāraga ita was used not only by Jains but also ṇby Brahmins and others.

The Introductory Chapter of Vyavahāraga ita ends ṇbeautifully thus –

“ Idu Shubha Chandradeva yogīndra pādāravinda Madhukarāyama am Manasānamdita ṇSakalaga ita tattvavilāsa vineyajana vinuta Śri ṇRājāditya viracitamappa Vyavahāraga itado ṇ pī īkā prakara am samāptam.”ṭ ṇ


Regarding the time of Rājāditya there is a controversy between historians and Dr.Venkatasubbiah. In the published work of Vyavahāraga ita, the name of the king Vish unṭpāla ṇ ṇappears in some of the examples. The historians opine that this Vish unṭpāla must be the king ṇVish uvardhana who ruled during 1111-1141 A.D. ṇ

Besides this, in one of the scriptures of 117 at Shrava belgoa there is a mention that a teacher by ṇ 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 Vish uvardhana.ṇ


But Dr.Venkatasubbiah asserts that many kings had the same name Vish uvardhana in Hoysaa family. He also ṇ 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 Dam anāyaka. ḍ

Then he shows that these Bāhubalibharatās being the sons of the second Mariyāne Damn anāyaka were ḍrespectively Dam anāyaka and Mahāpradhān ḍSarvādhikāri Mā ikyabham āri in the reign of second ṇ ḍBallāa.

After paying the dues to their king second Ballāa, Bāhubalibharata got the possession of the cities Sindhagere etc. in the year 1183 A.D.


Dr.Venkatasubbiah opines that Rājāditya must have referred to this Ballāa as Vish uvardhana. Besides ṇthis, 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 Vyavahāraga ita, it follows that in 1191 A.D, the king ṇBillama was defeated by Hoysaa Varaballāa 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.


From the chapter on terminology in Vyavahāraga ita ṇwe find names upto 40 unit places.

They are – ekkam, daham, śatam, dasavīra, lak a, ṣdālak a, ko i, dāko i, śatako i, arbuda, nyarbuda, ṣ ṭ ṭ ṭkharva, mahākharva, padma, mahāpadma, k o i, ṣ ṇmahāk o i, samkha, mahāśamkha, k iti, mahāk iti, ṣ ṇ ṣ ṣk obha, mahāk obha, nadi, mahānadi, naga, ṣ ṣmahānaga, ratha, mahāratha, hari, mahāhari, pha i, ṇmahāpha i, kratu, mahākratu, sāgara, mahāsāgara, ṇparimita, mahāparimita.

There are many synonyms for the numbers 0 upto 9 in Vyavahāraga ita. Example for “0” :ṇ0 – divi, kaika, śūnya, agra, ambara, gagana, meghamārga, ākāśa, jaladharmārga, bamdhayogaa, viyat, ayana, patha, abhrakham, vṭattam, agasa, jaladharapatha, vyadvite.


Many names are also there for addition, subtraction, multiplication and division.

Regarding Vyavahāraga ita, Rājāditya says thus – ṇ

Dhāru iyo sakala budhāṇ

Dhāramenal svalpa māgiyam vyavahāraDhāramenal 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, Vyavahāraga ita ṇis written.


Rājāditya must have written Vyavahāraga ita after ṇcarefully examining the previously available mathematical works. Let us look at this example –

One rich person appointed a mahout at the rate of one gadyā a per day as wage to look after 108 ṇelephants. 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.


According to a formula given in Vyavahāraga ita, add 1 to 108 to get 109. ṇMultiply 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 gadyā a 2 pa a 4 ṇ ṇvīsa 1 kā i.ṇ


The following table of coins and fractions is found in Vyavahāraga ita.ṇ

4 Kā i = 1 Vīsa, 5 Visa = 1 Haga, = 4 Haga = 1 Pa a, 10 ṇ ṇPa a = 1 Gadyā a.ṇ ṇ

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 Vyavahāraga ita it appears that instead of using ṇmoney, 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.


CONTRIBUTION OF JAINA MATHEMATICIANS

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