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The Value of π π π π

Abstract;

India is recognized as one of the most important ancient country where the civilizing process was initiated and developed. The remarkable contribution of Indian scholars to the field of mathematics is well mentioned and explained in ancient Indian Vedic and Jaina

special reference to the value of

π

ancient Indian languages. The Vedic

around 500 BC or may be of earlier period. However the various textual evidences lead to the conclusion that Indians knew the value of

π .

Key Words: Siddhanta , Sutra, Rigaveda,

Introduction:

Ancient and medieval Indians were aware that there is circle. This ratio is now known by

the beginning of the Christian era say a century on either side of it, a new class of Indian

emerged. Remarkable development of newer mathematical methods with the invention of the decimal system greatly promoted mathematical principles. It is all mentioned in our

manuscripts. Only we have to rediscover it and

discovery of the value of π in ancient India both in Vedic and Jaina

1.

Value of π in the Texts of Vedic period We find references of π = √10

1.

Ancient Surya Siddhanta before the period of

2.

Vishnudharmottariya Paitamaha 3. Pancasiddhantika of Varaha Mihira

4. Brahama Sphuta - Siddhanta of Brahmagupta , 628 AD 5. Aryabhattiya Bhashya on Bhashkara I

6. Trishatika by Shridhara ( Sutra 45)

π in Ancient Indian Vedic and Jaina

Shrenik Bandi Anupam Jain

Professor and Head, Professor of Mathematics and Press Controller Department of Mathematics D.A. University, RNT Marg, Indore.

India is recognized as one of the most important ancient country where the civilizing process was initiated and The remarkable contribution of Indian scholars to the field of mathematics is well mentioned and ancient Indian Vedic and Jaina philosophical texts. Ancient Indian mathematical contribution with is explored in this paper with the help of Vedic and Jaina texts which are in Vedic and Jaina contribution in the field of mathematics is

around 500 BC or may be of earlier period. However the various textual evidences lead to the conclusion that

veda,, Samhita, , Jaina, Vedic, Sulba-Sutra , G

edieval Indians were aware that there is a fixed ratio between the circumference and diameter of a by the Greek letter π. Its value is transcendental

the beginning of the Christian era say a century on either side of it, a new class of Indian

Remarkable development of newer mathematical methods with the invention of the decimal system greatly promoted mathematical principles. It is all mentioned in our ancient Indian

rediscover it and put it in systematic way. The aim of this paper is to trace the ancient India both in Vedic and Jaina texts.

Vedic period :

, in many ancient texts of Vedic period.

before the period of Aryabhata I and the available Su maha Siddhanta ( III , 6 ) .

ha Mihira of 550 AD ( IV , 1 ) . Brahmagupta , 628 AD ( XII, 40 ) .

ara I (629 AD) , Delhi Ed. , 1976 , p. 72.

tra 45) .

200

Vedic and Jaina Texts

Anupam Jain

Professor of Mathematics and Press Controller D.A. University, RNT Marg, Indore.

India is recognized as one of the most important ancient country where the civilizing process was initiated and The remarkable contribution of Indian scholars to the field of mathematics is well mentioned and Ancient Indian mathematical contribution with explored in this paper with the help of Vedic and Jaina texts which are in mathematics is remarkable and it is of around 500 BC or may be of earlier period. However the various textual evidences lead to the conclusion that

, Gatha

between the circumference and diameter of a transcendental and non- terminating. Round the beginning of the Christian era say a century on either side of it, a new class of Indian mathematical literature Remarkable development of newer mathematical methods with the invention of the decimal system ancient Indian religious and philosophical put it in systematic way. The aim of this paper is to trace the

urya Siddhanta ( I , 58 ) .

(2)

The Sutra of Shridhara is also quoted by of Bhaskaracharya .

In fact the credit of giving for the first time in India the value of

goes to Aryabhata I (476 AD). His achievement in the 5th century is truly significant in the light of the fact that it was only thirteen centuries later in the year 1761, that Lambert proved that

Lindeman established that π is transcendental.

Its value is given by Aryabhata I in the following words

“Add 4 to 100 and multiply by 8 and add 62000 this is

is 20000 units. This means the circumference of the circle is 62832 approximate as 3.1416.

This gives

π =





It is remarkable that Aryabhata I was the first Indian mathematician to have given the value of correct up to the four places of the

It is to be noted that Varaha Mihira , Brahmagupta , Jaina texts . Even in the Siddhanta Tatva

value of π =√10 was considered . At that time the mathematicians of India and other countries.

time. It is also found in the Mahabharata commentary by Dwarka Nath Yajavan

2.

Value of π by Kerala School of Mathematics

Madhava , the great pioneer of Kerala School of Mathematics ( 1340 AD approximation 6 to π  3.1415926536. The

idea of infinite series. Yuktibhasha of Jye

of Pythagorean Theorem, approximation of circumference of the circle by the perim polygons and most importantly an infinite series for

centuries before Europe ever came to consider these questions.

We know the Gregory’s Series ---

Then

Now we give a proof of the same result due to Kerala School of Mathematics which is discussed in as follows --

Where c is the circumference and d is the diameter

is also quoted by Shankara Variyara in the commentary Kriy

In fact the credit of giving for the first time in India the value of π correct to four places of decimal as

His achievement in the 5th century is truly significant in the light of the fact that it was only thirteen centuries later in the year 1761, that Lambert proved that π is irrational and in the year 1882 that

transcendental.

in the following words 1

“Add 4 to 100 and multiply by 8 and add 62000 this is an approximately circumference of a circle whose diameter is 20000 units. This means the circumference of the circle is 62832 units. Aryabhata I







=

3.1416

was the first Indian mathematician to have given the value of the decimals 2.

, Brahmagupta , Shridhara and Shripati had used the value of nta Tatva a book on astronomy of 17th century AD

At that time the other subtle (sukshma) or more accurate value was known to mathematicians of India and other countries. The value 3 for π is much older value and was accepted in ancient

the Mahabharata 3 and the Vayu Purana 4, Baudhayana van .

Mathematics:

dhava , the great pioneer of Kerala School of Mathematics ( 1340 AD

3.1415926536. The Indian mathematicians had good knowledge of irrational numbers and a of Jyesthadeva (c.1530) contains usual mathematical material and with a proof , approximation of circumference of the circle by the perim

polygons and most importantly an infinite series for

! . The power expansion for tan idea about convergence of the series. All these centuries before Europe ever came to consider these questions.

If θ lies between - "

# and "

#

Then θ = tan θ - % tan 3 θ + %& tan5θ ' ….

Now we give a proof of the same result due to Kerala School of Mathematics which is discussed in

If c = 4d - #(

+ #(

& - ….

the circumference and d is the diameter of the circle or equivalently

"

# = 1 - % + %

&

' …. .

201

in the commentary Kriyakarmakari Tika on Lilavati correct to four places of decimal as 3.1416, His achievement in the 5th century is truly significant in the light of the fact that it

is irrational and in the year 1882 that

approximately circumference of a circle whose diameter Aryabhata I took the value of π

was the first Indian mathematician to have given the value of π which is had used the value of π as given by century AD written by Kamalakar, the or more accurate value was known to is much older value and was accepted in ancient yana - Sulba Sutra 5, with Sanskrita

dhava , the great pioneer of Kerala School of Mathematics ( 1340 AD – 1425 AD) had even better mathematicians had good knowledge of irrational numbers and adeva (c.1530) contains usual mathematical material and with a proof , approximation of circumference of the circle by the perimeters of inscribed regular for tan-1 x is given in this text but rgence of the series. All these were achieved 7 at least two

….

Now we give a proof of the same result due to Kerala School of Mathematics which is discussed in Yuktibhasha 8

(3)

Consider a quarter of a circle having radius equal to unity with centre C inscribed in a unit square as shown in the figure :

Partition of A0 An in to n equal parts . Let CA M perpendiculars to CA r from A and A which are similar. So we can write

)

)



) )*+, -

*+,

Geometrical construction used in the proof of the infinite series for From the similarity of the triangles

)*+, -

)*+, )*

) ).

*

- --- (2)

So we get by 1 and 2

). ). .)*+

)*+, . )

So we have

)*+, )*

)*+, . )*

Now if we take n sufficiently large

approximately equal so that we have Arc AB is the approximately equal to side AD Arc AB ~ )%/

*+,2

=

%/

%3 4*+,5 62

Note C A2r-1 = C A20 + A0 A2r-1 = 1+ A

Therefore we have by using the geometric series

%

%37 1 ' x 9 x' x9 ⋯

Here radius of the circle is unity so circumference of the circle is = 2

Consider a quarter of a circle having radius equal to unity with centre C inscribed in a unit square as shown in the in to n equal parts . Let CA r-1 , CA r meet the circle at A and B We

from A and A r-1 respectively. Consider the two triangles CAD and

Geometrical construction used in the proof of the infinite series for π From the similarity of the triangles M Ar-1 A r and CA0 A r we get

+, )*

)* , Here CA and CA0 are of unit

Now if we take n sufficiently large , the segment AD tends to the arc AB and CA approximately equal so that we have Arc AB is the approximately equal to side AD

= 1+ A0 A2r-1

have by using the geometric series

is unity so circumference of the circle is = 2π .1

202 Consider a quarter of a circle having radius equal to unity with centre C inscribed in a unit square as shown in the

meet the circle at A and B We draw AD and A r-1 respectively. Consider the two triangles CAD and CAr-1 M

--- (1)

of unit length.

the segment AD tends to the arc AB and CAr-1 and CA r become approximately equal so that we have Arc AB is the approximately equal to side AD.

(4)

We consider the 1/8 part of circumference arc A0 L and equivalent to the side of the



π.%

→>

@%

=

→>

%

A%@%

%B

∑

The ancient Kerala mathematicians knew that for any integer p

%

CD,

A%@%

F

Tends to

%

F3%

nearly a centuries later than discovery Using this in above result we have

"

#

%

%

&

….

. By this series we can Madhava had even better approximation to

was not satisfied and he gave another better rational Bhaskara 9 II also gave the value of π

Ganesha Daivaghya 10 (Buddhivilasini

in a circle of diameter 100 would give the approximate value of

3. The value of in Jaina Texts

The ratio of the circumference of a circle to its diameter was



(



=

this, the two values 3 and √10 for

some earlier work that the value of π in the following form : If Circumference of the circle = C

And its diameter = D Then C = 3 D +

113 16 D 16 +

was the formula according to the couplet.

C = 3 D

%

%%

=

&&

%%

Which gives the ratio of C to D which we A. N. Singh's 11 remarks, the term 'Sahita Repeated addition, a number of times in the

Aryabhata I (c + 499) and succeeding mathematicians. This leads to conclude that the above quotation is from some work written before the fifth century AD, when "Sahita

We consider the 1/8 part of circumference (by drawing a diagonal in the square) of the circle of unit radius the side of the square.

%/

%3 4*+,5 62

% By expansion

A%@%

%I

A%@%

#

- … )

The ancient Kerala mathematicians knew that for any integer p J 0 ,

as n

tends to ∞ this result was proved in Europe by Roberval of the Kerala mathematicians.

By this series we can get the value of π .

dhava had even better approximation to π  3.1415926536 (quoted in Karanapaddhati better rational approximation to π = 3.14159265359.

the value of π = KL

%& = 3.1416 .

i ) suggested that the ratio of the perimeter of the 3 in a circle of diameter 100 would give the approximate value of π as KL%& .

Jaina Texts

The ratio of the circumference of a circle to its diameter was also known to ancient

3.1416, This value is correct up to four places of for

π

were also known to Jaina scholars. V

in the following form :

was the formula according to the couplet. Virasenacharya

D which we say π

=

113 355

the term 'Sahitam’ was used in the sense of addition

, a number of times in the Vedanga Jyotisha , but it has been used in that double sense by (c + 499) and succeeding mathematicians. This leads to conclude that the above quotation is from some work written before the fifth century AD, when "Sahitam" was being used in both

203 of the circle of unit radius which is

this result was proved in Europe by Roberval in 1634 AD

) suggested that the ratio of the perimeter of the 384 sides polygon inscribed

known to ancient Jaina mathematicians.

to four places of decimals. Apart from Virasenacharya has quoted from

Virasenacharya interpreted the above as (app.)

’ was used in the sense of addition as well as multiplication i.e.

t it has been used in that double sense by (c + 499) and succeeding mathematicians. This leads to conclude that the above quotation is from in both the senses of multiplication

(5)

It would appear, therefore that the so-called Chinese value of

used in India earlier than in China. It might be that the Chinese got this value from India, through Buddhist missionaries or perhaps they found out the value independently.

the following quotation is the remark 'finer than the fine'; from this it follows that a 'fine' value of known”. This fine value of π may have been

the third convergent being a close approxi

circumference of circle from given diameter is given C = O104diameter 6 or C = O

Where d = diameter, and C is circumference

value was also used for many years by mathematicians of other countries.

Here the ratio ( was used for π and its value = It was used as a gross value by Brahmgupta (628 A Mahavira (850 AD) and others 13.

Virasena used π = 3 in Dhavala commentary

π = √10 and this value for π had been adopted from early period in Jaina

Examples of this value of π are

1. Anuyogadvara Sutra ( Chulika Sutra ) written by 2. Bhagavati Sutra of Sudharama Svami in (Sutra 91) 3. Jivajivibhagama Sutra in (Sutra 82

4. Surya pragyapti , it is one of the A 5. Jambu dvipa pannatti , it is one of the A 6. Tattvarthadhigma Sutra Bhashya of 7. Jambu dvipa Samasa by Umasvati 1 8. Jyotishakarandaka written by Vallabh 9. Tiloyapannatti of Yativrshabha (1 10. Ganita- Sara - Sangraha of Mahav 11. Tiloyasara of Nemichandra Siddh We find two value 3 and √10 for π Other value of this ratio π is R%KS Hayashi 20 , it is derived from Aryabhat

called Chinese value of π = 113

355 was also known in

used in India earlier than in China. It might be that the Chinese got this value from India, through Buddhist aps they found out the value independently. According to him, "Another noteworthy feature in the following quotation is the remark 'finer than the fine'; from this it follows that a 'fine' value of

may have been 10 or 7

22 . In this connection Aryabhat

the third convergent being a close approximation than the second 12 . In Jaina text the formula for finding the circumference of circle from given diameter is given as

O104d6 ,

C is circumference , in which place of π the scholar considered value was also used for many years by mathematicians of other countries.

and its value = √10 was used in many It was used as a gross value by Brahmgupta (628 AD) in Brahma Sphuta -Siddh

ommentary on the Shatkhandagama14, (V1.4.8 P.

given in the following ancient

Jaina tra ) written by Aryarakshita in ( Sutra 164 ) Svami in (Sutra 91)

(Sutra 82, 109)

it is one of the Anga of Jaina literature in ( Sutra 20 / 132)

atti , it is one of the Anga of Jaina literature in ( Sutra 3 /132 and 149) ya of Umasvati ( III , 11)

ti 1st century

Vallabhi Acharya probably of 300 AD (Gatha 185 ) (1 /117)

viracharya (850 A(VII , 60 ) andra Siddhanta – Chakravarti(Gatha 96 )

in Trilokasara 16 .

S , referred by B. B. Datta 17, M. B. L. Agrawal Aryabhata’s value 21.

204 was also known in India and was perhaps used in India earlier than in China. It might be that the Chinese got this value from India, through Buddhist According to him, "Another noteworthy feature in the following quotation is the remark 'finer than the fine'; from this it follows that a 'fine' value of π was already Aryabhat’s I value is obvious — In Jaina text the formula for finding the

the scholar considered √10 . This

many ancient texts of Jaina literature.

Siddhanta , Virasena (816 AD) , .169). The Jainas have approved

15 . Jaina

texts.

tra 164 )

132)

/132 and 149)

185 )

, M. B. L. Agrawal 18, L C Jain 19

and Takao

(6)

The value of π is approximated to the square root of 10 and calculated correct up to 13 places of decimal in Suryapragyapti an astronomical work of Jaina

mentioned in (Verse 17, p. 18) which means t

Therefore C = 3 d , where C is the circumference and d is the diameter implies for π. Nemichandra acharya was aware that the above formula

is clear from another Gatha which states that the circumference of a circle is three times the diameter and a little more i.e. C = 3d +s , where s is a savi

square root of ten multiplied by the square of the diameter becomes implies the value of π = √10 , which is

Chou Pei 24 and in the Talmud 25 this value is used

The Value of π in other countries

The value of π in China, Japan, Arab

Chang Heng in 2nd century AD used first time the value of not known .

The commentary written on text “Nine chapters

Or (circumference of circle6 2 = 16 Or (Circumference of circle6 2 = 10

In the 9th century Alakhvarijami of West Asia mentioned in Algebra (Arabic language) that the formula for finding the circumference of the circle was

4circumference of circle6 2

= 10

It is based on the translation of Jaina and Vedic AD gave different approximations of

Later on, these texts were translated into Latin language and in one Tabulas Toletannas the value of π is expressed by the formula.

This result is similar to the value obtained

that is different and seems to be incorrect. The value

that this value came through India via China and he has mentioned the reference of text written by Aryarakshita . In the 9

language) that the formula for finding the circumference of the circle 4circumference

Al - Tabari in 11th century wrote Miftah O104diameter 6 for circumference

is approximated to the square root of 10 and calculated correct up to 13 places of decimal in astronomical work of Jaina . Trilokasara is an outstanding work of

. 18) which means that diameter multiplied by 3 gives circumference.

where C is the circumference and d is the diameter implies π

was aware that the above formula which is an approximation and not the exact one. It which states that the circumference of a circle is three times the diameter and a little more i.e. C = 3d +s , where s is a savishesha. He gave another formula 22 from Trilokas

square root of ten multiplied by the square of the diameter becomes the circumference.

which is a better approximation. In Babylonia 23 (C 1700 BC) this value is used .

countries

:

Japan, Arab and other countries was similar to that of Jaina value .

century AD used first time the value of π = √10 . However, how he obtained this value was

“Nine chapters of arithmetic” of Chiu Chang had given the following

4

 U V

F U 7V WX

=

&

= 16 Y 5 Y 4(6 2

= 10 Y (daimeter62

century Alakhvarijami of West Asia mentioned in Algebra (Arabic language) that the formula for finding the circumference of the circle was

Y (daimeter6 2

It is based on the translation of Jaina and Vedic texts in 8th century AD . Even in Spain Al AD gave different approximations of π in his Arabic texts in which one of the value of

Later on, these texts were translated into Latin language and in one of the book Canonessive Regule Super is expressed by the formula.

 U V

√%

=

diameter

btained by Jaina scholars. Needham has given the formula of different and seems to be incorrect. The value π = √10 was used afterwards

that this value came through India via China and he has mentioned the reference of

In the 9th century Alakhvarijami of West Asia mentioned in Algebra (Arabic language) that the formula for finding the circumference of the circle was

circumference of circle6 2

= 10 Y (daimeter6

wrote Miftah al-Muamlat in Persian language in that book the formula circumference of circle is mentioned along with several other formula

205 is approximated to the square root of 10 and calculated correct up to 13 places of decimal in

ra is an outstanding work of Prakrita literature. It is hat diameter multiplied by 3 gives circumference.

π = 3. Here 3 is the gross value approximation and not the exact one. It which states that the circumference of a circle is three times the diameter and a little from Trilokasara which state that the the circumference. Therefore C  √10d , it (C 1700 BC) , Chinese work of

similar to that of Jaina value . The Chinese scholar However, how he obtained this value was of Chiu Chang had given the following result.

century Alakhvarijami of West Asia mentioned in Algebra (Arabic language) that the formula for

century AD . Even in Spain Al- Zarqali in 11th century in his Arabic texts in which one of the value of π is equal to √10 . of the book Canonessive Regule Super

Needham has given the formula of Chang Heng afterwards in China 26 Imaishin 27 stated that this value came through India via China and he has mentioned the reference of Anuyogadvara Sutra a Jaina

West Asia mentioned in Algebra (Arabic

2

language in that book the formula is mentioned along with several other formulae . In the Book

(7)

Sindh - Hind, it is used frequently. In Hebrew

value of π =√10 is mentioned and stated that it is Indian text Canones Tabularum gave a law whose meaning is

4

= √10 , but the cerdit was given

was published in Latin , French and Dutch languages in 1507 AD , in which ( Introducetorium Geometricum ) the value of π was taken as √10 .

Gumnden also mentioned the same 1541 AD) also quoted the same formula.

as we all know that the science and mathematics of Europe 29.

This value of π = √10 was also used in Japan other circular measurement of the circle

In 18thcentury AD Chhien Thang was of the opinion that this value of correct value

Conclusion:

The contributions of Indian mathematicians during

shown that the Vedic and Jaina scholars had the knowledge of the value of Jaina philosophical texts present the concept of

diameter was known to the ancient mathemat

recognize our contribution. There may be many reasons for it, one reason is writing the mathematical expressions in verse (Gatha) form and second our traditional approach of not giving much

mathematical thoughts. Let us hope that mathematics will be pursed and developed to its pristine glory in India.

References:

1. Abh . II, 10.

2. Mahabharata with Hindi translation , part 3, Bhi¼maparva, Geeta Press, 3. Vayu Purana, Edited by Sri Ram

4. S. Balachandra Rao, Indian Mathematics and Astronomy, Bhavan’s Gandhi centre of Science and Human Baudhayana , Sulba-Sutra with

1.112 - 113, p.78.

5. Quoted in Karanapaddhati IV, 7 6. Values, Bangalore, p. 208.

7. K. Balagangadharan , Mathematical Analysis in medieval Kerala , Science in the West and India, some Historical Aspects, 1995, Pp . 2

8. S. Balachandra Rao , Indian Mathematics and Astronomy, Bhavan’s Gandhi centre of Science and Human Values , Bangalore, p.187.

9. History of Mathematics in India from Arrah, pp. 54-69.

used frequently. In Hebrew language the writer Abraham Ibn

is mentioned and stated that it is Indian value. Johannes Dulineris 15 Canones Tabularum gave a law whose meaning is

 U V6

%

= (

diameter6 2

(1492 AD) in his book Dequadrature Circuli had used the Jaina value of to Arabians . Charles Bovillus wrote a famous book on geometry which was published in Latin , French and Dutch languages in 1507 AD , in which ( Introducetorium Geometricum )

the same formula. Peurbach in his text Tractates Super Properness (Nuremberg, formula. Regiomontanus considered π = √10 as approximate value from Arab,

mathematics of India was first translated into was also used in Japan in 17 th century, particularly for

the circle.

AD Chhien Thang was of the opinion that this value of π =√ 10

athematicians during ancient and medieval periods have and Jaina scholars had the knowledge of the value of π but in different

Jaina philosophical texts present the concept of π. Textual evidences show that the ratio of the circumference to the diameter was known to the ancient mathematicians of India but still we require the world to understand and There may be many reasons for it, one reason is writing the mathematical expressions form and second our traditional approach of not giving much

Let us hope that mathematics will be pursed and developed to its pristine glory in India.

translation , part 3, Bhi¼maparva, Geeta Press, Sri Ram Sharma, part 1, Bareily, 1967, p. 357.

S. Balachandra Rao, Indian Mathematics and Astronomy, Bhavan’s Gandhi centre of Science and Human tra with Sanskrita commentary by Dwarka Nath Yajvan, New Delhi, 1980, vv Karanapaddhati IV, 7.

K. Balagangadharan , Mathematical Analysis in medieval Kerala , Science in the West and India, some Historical Aspects, 1995, Pp . 290,312.

S. Balachandra Rao , Indian Mathematics and Astronomy, Bhavan’s Gandhi centre of Science and Human of Mathematics in India from Jaina Sources, The Jaina Antiquary, Vol. xvii, no.

206 Ezra 28 in 12th century AD the lineris 15th century AD, in his

(1492 AD) in his book Dequadrature Circuli had used the Jaina value of π Charles Bovillus wrote a famous book on geometry which was published in Latin , French and Dutch languages in 1507 AD , in which ( Introducetorium Geometricum )

Tractates Super Properness (Nuremberg, as approximate value from Arab, Arabic and then it spread into particularly for the formulae for arc, chord and 10 is convenient and near to the have been noteworthy. We have in different forms. The Vedic and . Textual evidences show that the ratio of the circumference to the icians of India but still we require the world to understand and There may be many reasons for it, one reason is writing the mathematical expressions form and second our traditional approach of not giving much importance to scientific and

Let us hope that mathematics will be pursed and developed to its pristine glory in India.

translation , part 3, Bhi¼maparva, Geeta Press, Gorakhpur, v 12.44 pp. 25 72 S. Balachandra Rao, Indian Mathematics and Astronomy, Bhavan’s Gandhi centre of Science and Human

by Dwarka Nath Yajvan, New Delhi, 1980, vv

K. Balagangadharan , Mathematical Analysis in medieval Kerala , Science in the West and India, some S. Balachandra Rao , Indian Mathematics and Astronomy, Bhavan’s Gandhi centre of Science and Human

quary, Vol. xvii, no. ii, Dec. 1950,

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10. Brahma Sphuta - Siddhanta , Edited with V

#. New Delhi, 1966.

11. Dhavala tika , A commentary by V

12. As quoted by R. C. Gupta, the Jaina Value of 13. Trilokasara, Edited with Mahavira

14. Datta, B. B. , Mathematics of Nemic 15. Agrawal , M. B. L. , Contribution of Jain

Doctoral thesis , Agra University, 1972, p. 176.

16. Jain, L. C., Exact sciences from Jaina sources, Vol. 01, Basic 17. Hayashi T. , T. Kusuba and M. Ya

18. Historia Scientiarum , 37 , pp. 1 19. TS (V 96, P 88).

20. Smith, D. E., History of Mathematics, vol. 2, Dover Publication Inc. , New York, 1958, pp. 302, 309 21. Ibid. , p. 302.

22. Needham, p. 100.

23. Math. Review, Vol. 58. , pp. 31 24. Journal Viator, Vol. 7, p. 168.

25. R.C. Gupta, On Value of π.

nta , Edited with Vasana Vigyana and Hindi Commentary by R. S. Sharma, , A commentary by Virasena on Shatkhandagama, ed. By H. L. Jain Amaraoti , 1942.

As quoted by R. C. Gupta, the Jaina Value of π , Arhat Vacana , p. 117.

Mahaviracharya ‘s commentary.

Datta, B. B. , Mathematics of Nemichandra , The Jaina Antiquary 1 (2), 25

, Contribution of Jainacharya in the Development of Mathematics and Astronomy, Doctoral thesis , Agra University, 1972, p. 176.

L. C., Exact sciences from Jaina sources, Vol. 01, Basic mathematics, RPBS, M. Yano, Indian values for π .

cientiarum , 37 , pp. 1-16.

Smith, D. E., History of Mathematics, vol. 2, Dover Publication Inc. , New York, 1958, pp. 302, 309

pp. 31-54.

p. 168.

207 na and Hindi Commentary by R. S. Sharma, Vol.

gama, ed. By H. L. Jain Amaraoti , 1942.

andra , The Jaina Antiquary 1 (2), 25- 44 , Arrah, 1935 , p. 34.

in the Development of Mathematics and Astronomy, mathematics, RPBS, 1982, p. 47.

Smith, D. E., History of Mathematics, vol. 2, Dover Publication Inc. , New York, 1958, pp. 302, 309 .

References

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