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NPTEL COURSE ONMATHEMATICS IN INDIA:

FROM VEDIC PERIOD TO MODERN TIMES

Lecture 36

Proofs in Indian Mathematics 1

M. D. Srinivas

Centre for Policy Studies, Chennai

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Outline

Upapattisor proofs in Indian mathematical tradition

Early European scholars of Indian Mathematics wereaware ofupapattis

Some important commentaries which presentupapattis

BhaskaracaryaII on the nature and purpose of upapatti

Upapattiofbhuja-kot.i-karn. a-nyaya(Pythagoras theorem) Upapattiofkut.t.akaprocess

Restricted use oftarka(proof by contradiction) in Indian

Mathematics

The Contents ofYuktibhas. a Yuktibhas. ademonstration ofbhuja-kot.i-karn. a-nyaya

Estimating the circumference by successive doubling of

circumscribing polygon

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Upapattisin Indian Mathematics

While there have been several extensive investigations on the

history and achievements of Indian mathematics, there has notbeen much discussion on its methodology, the Indian

mathematicians and philosophers understanding of the nature

and validation of mathematical results and procedures, their

views on the nature of mathematical objects, and so on.

Traditionally, such issues have been dealt with in the detailed

bhas.yasor commentaries, which continued to be written till

recent times, and played a vital role in the traditional scheme of

learning.

It is in such commentaries that we find detailed upapattisor

proofs of the results and procedures, apart from a discussion of

methodological and philosophical issues.
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Early European Scholars Were Aware of Upapattis

In the early stages of modern scholarship on Indian mathe-

matics, we find references to the methods of demonstrationfound in texts of Indian mathematics.

In 1817, H. T. Colebrooke referred to them in his pioneering

and widely circulated translation of Ll avatandBjagan. itaand

the two mathematics chapters ofBrahmasphut.a-siddhanta:

On the subject of demonstrations, it is to be remarked

that the Hindu mathematicians proved propositions

both algebraically and geometrically: as is particularly

noticed byBhaskarahimself, towards the close of his

algebra, where he gives both modes of proof of aremarkable method for the solution of indeterminate

problems, which involve a factum of two unknown

quantities.

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Early European Scholars Were Aware of Upapattis

Similarly, Charles Whish, in his seminal article on Kerala School ofMathematics of 1835, referred to the demonstrations inYuktibhas. a:

A further account of the Yuktibhas. a, the demonstrations ofthe rules for the quadrature of the circle by infinite series,with the series for the sines, cosines, and theirdemonstrations, will be given in a separate paper: I shalltherefore conclude this, by submitting a simple and curiousproof of the 47th proposition of Euclid [the so calledPythagoras theorem], extracted from the Yuktibhas. a.

Whish does not seem to have written any further papers on thedemonstrations of the infinite series as given inYuktibhas. a.

Whishs paper was widely noticed in the scholarly circles of Europe inthe second quarter of nineteenth century.

But it was soon forgotten and there was no study of Yuktibhas. atill1940s, when C. T. Rajagopal and his collaborators wrote pioneering

articles on the proofs outlined in that seminal text.

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The Alleged Absence of Proofs in Indian Mathematics

It has been the scant attention paid, by the modern scholarship of thelast two centuries, to this extensive tradition of commentaries whichhas led to a lack of comprehension of the methodology of Indianmathematics. This is reflected in the often repeated statements onthe absence of logical rigour in Indian mathematics in works onhistory of mathematics such as the following:

As our survey indicates, the Hindus were interested in andcontributed to the arithmetical and computational activitiesof mathematics rather than to the deductive patterns. Theirname for mathematics wasgan. ita, which means thescience of calculation. There is much good procedure andtechnical facility, but no evidence that they considered proofat all. They had rules, but apparently no logical scruples.Moreover, no general methods or new viewpoints werearrived at in any area of mathematics.1

1

Morris Kline: Mathematical Thought from Ancient to Modern Times,Oxford 1972, p.190.

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Some Important Commentaries Which Discuss

Upapattis

1. Bhas. yaofBhaskaraI (c.629) on Aryabhat.yaof Aryabhat.a

(c.499)

2. Bhas. yaofGovindasvamin(c.800) onMahabhaskaryaof

BhaskaraI (c.629)

3. Vasanabhas. yaofCaturveda Pr. thudakasvamin(c.860) on

Brahmasphut.asiddhantaofBrahmagupta(c.628)

4. Vivaran. aofBhaskaracaryaII (c.1150) on Sis. yadhvr. ddhida-

tantraofLalla(c.748)

5. VasanaofBhaskaracaryaII (c.1150) on his own Bjagan. ita

6. Mitaks. araorVasanaofBhaskaracaryaII (c.1150) on his

ownSiddhantasiroman. i

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Upapattis

7. Siddhantadpik aofParamesvara(c.1431) on theBhas. yaof

Govindasvamin(c.800) onMahabhaskaryaofBhaskaraI

(c.629)

8. Aryabhat.yabh as. yaofNlakan. t.ha Somasutvan(c.1501) onAryabhat.yaof Aryabhat.a

9. Yuktibhas. a(in Malayalam) ofJyes.t.hadeva(c.1530)

10. Yuktidpik aof Sankara Variyar(c.1530) onTantrasa ngraha

(c.1500) ofNlakan.

t.ha Somasutvan

11. Kriyakramakarof Sankara Variyar(c.1535) onLl avatof

BhaskaracaryaII (c.1150)

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Upapattis

12. Gan.

itayuktayah.

, Tracts on Rationale in Mathematical

Astronomy by various Kerala Astronomers (c.16th-19th

century)

13. Suryaprakasaof Suryadasa(c.1538) onBhaskaracaryas

Bjagan. ita(c.1150)

14. BuddhivilasinofGan. esa Daivajna(c.1545) onLl avatofBhaskaracaryaII (c.1150)

15. Bjanav a nkuraorBjapallavamof Kr. s.n. a Daivajna(c.1600)

onBjagan. itaofBhaskaracaryaII (c.1150)

16. Vasanavarttika, commentary ofNr. sim.ha Daivajna(c.1621)onVasanabhas. yaofBhaskaracaryaII on his own

Siddhantasiroman. i(c.1150).

17. MarciofMunsvara(c.1630) onSiddhantasiroman. iof

BhaskaracaryaII (c.1150).

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Kr. s.n. a Daivajnaon the Importance of Upapatti

The following passage from Kr. s.n. a Daivajnas commentary onBjagan. itabrings out the general understanding of the Indian

mathematicians that citing any number of favourable instances (evenan infinite number of them) where a result seems to hold, does notamount to establishing it as a valid result in mathematics. Only whenthe result is supported by an upapattior demonstration can the resultbe accepted as valid.

?

,

,

,

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Kr. s.n. a Daivajnaon the Importance of Upapatti

How can we state without proof (upapatti) that twice the product oftwo quantities when added or subtracted from the sum of theirsquares is equal to the square of the sum or difference of thosequantities? That it is seen to be so in a few instances is indeed of noconsequence. Otherwise, even the statement that four times theproduct of two quantities is equal to the square of their sum, wouldhave to be accepted as valid. For, that is also seen to be true in somecases. For instance take the numbers 2, 2. Their product is 4, four

times which will be 16, which is also the square of their sum 4. Ortake the numbers 3, 3. Four times their product is 36, which is alsothe square of their sum 6. Or take the numbers 4, 4. Their product is16, which when multiplied by four gives 64, which is also the squareof their sum 8. Hence the fact that a result is seen to be true in some

cases is of no consequence, as it is possible that one would comeacross contrary instances also. Hence it is necessary that one wouldhave to provide a proof (yukti) for the rule that twice the product oftwo quantities when added or subtracted from the sum of theirsquares results in the square of the sum or difference of thosequantities. We shall provide the proof (upapatti) in the end of thesection onekavarn. a-madhyamaharan. a.

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BhaskaraI on Upapatti(c.629)

In his discussion of Aryabhat.as approximate value of the ratio

of the circumference and diameter of a circle, BhaskaraI notes

that the approximate value is given, as the exact value cannot

be given. He then goes on to argue that other values which

have been proposed are without any justification:

( )

-

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BhaskaraI onUpapatti

It is the view of the Acaryathat there is no means by which the exact

circumference can be obtained. Is it not true that there is thefollowing?

The square root of the ten times the diameter is the circumference ofa circle.

Here also, there is only a tradition, and not a proof (upapatti), that thecircumference when the diameter is one is square-root of ten. Then itis contended that the circumference when diameter is unity, whenmeasured by means of direct perception (pratyaks. a), is thesquare-root of ten (dasa-karan. ). That is incorrect, because themagnitude of the square-roots (karan. ) cannot be stated exactly. It

may be further contended that when the circumference associatedwith that diameter (one) is enclosed with the diagonal of a rectanglewhose breadth and length are one and three respectively, i.e. onewhose length is square root of ten only, it (the circumference) has thatlength. But that too has to be established.

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BhaskaraII onUpapatti(c.1150)InSiddhantasiroman. i,BhaskaracaryaII (1150) presents theraison detreofupapattiin the Indian mathematical tradition:

Without the knowledge of upapattis, by merely masteringthe calculations (gan. ita) described here, from themadhyamadhikara(the first chapter of Siddhantasiroman. i)onwards, of the [motion of the] heavenly bodies, amathematician will not be respected in the scholarlyassemblies; without theupapattishe himself will not be freeof doubt (nih. sa msaya). Sinceupapattiis clearly perceivablein the (armillary) sphere like a berry in the hand, I thereforebegin theGoladhyaya(section on spherics) to explain the

upapattis.

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BhaskaraII onUpapatti

The same has been stated byGan. esa Daivajnain the

introduction to his commentaryBuddhivilasin(c. 1540) on

Ll avatof Bhaskaracarya

II

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Thus, according to the Indian mathematical texts, the purposeofupapattiis mainly:

(i) To remove confusion and doubts regarding the validity and

interpretation of mathematical results and procedures;

and,

(ii) To obtain assent in the community of mathematicians.

This is very different from the ideal of proof in the

Greco-European tradition which is to irrefutably establish theabsolute truth of a mathematical proposition.

Bh k II U i

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In hisBjagan. ita-vasana,BhaskaracaryaII (c.1150) refers to the longtradition ofupapattisin Indian mathematics.

. . .

The demonstration follows. It is twofold in each case: Onegeometrical and the other algebraic. There, the geometricalone is stated... Then the algebraic demonstration is stated,that is also geometry-based. This procedure [of upapatti]has been earlier presented in a concise instructional formby ancient teachers. For those who cannot comprehend thegeometric demonstration, to them, this algebraicdemonstration is to be presented.

Here, Bhaskaraalso refers to the ks. etragata(geometric) andrasigata(algebraic) demonstrations. To understand them, we shall consider

the two proofs given byBhaskaraof thebhuja-kot.i-karn. a-nyaya.

U tti f Bh j K ti K N

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UpapattiofBhuja-Kot.i-Karn. a-Nyaya

In themadhyamaharan. asectionBhaskaraposes the following problem

In a right angled triangle with sides 15 and 20 what is the

hypotenuse? Also give the demonstration for this traditional

method of calculation.

HereBhaskaragives two proofs. First the geometrical:

/

/

,

U tti f Bh j K ti K N

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UpapattiofBhuja-Kot.i-Karn. a-Nyaya

ya =

225

ya

+

400

ya

ya2 = 625

ya = 25

Let the hypotenuse (karn. a) beya. Take thehypotenuse as the base. Then the perpendicular

to the hypotenuse from the opposite vertexdivides the triangle into two triangles [which aresimilar to the original]. Now by the rule ofproportion (trairasika), ifyais the hypotenusethebhujais 15, then when thisbhuja15 is thehypotenuse, thebhuja, which is now the abadha

(segment of the base) to the side of the originalbhujawill be (225/ya). Again ifyais thehypotenuse, thekot.iis 20, then when thiskot.i20 is the hypotenuse, thekot.i, which is now thesegment of base to the side of the (original)kot.i

will be (400/ya). Adding the two segments(abadhas) ofyathe hypotenuse and equating thesum to (the hypotenuse)ya, we getya= 25, thesquare root of the sum of the squares of bhuja

andkot.i. The base segements are 9, 16 and the

perpendicular is 12. See the figure.

U tti of Bh j K ti K N

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UpapattiofBhuj a-Kot.i-Karn. a-Ny aya

Now the algebraic demonstration:

AsBhaskarahas noted, this algebraic demonstration is also

geometrical in nature.

U tti of Bh j K ti K N

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UpapattiofBhuj a-Kot.i-Karn. a-Ny aya

Let the hypotenuse beya. The area of the triangle which is half theproduct of thebhuja(15) andkot.i(20) is 150. Consider a squarewhose sides are formed out of the hypotenuse of these triangles.

In the centre is formed a square whose side is 5, the difference ofbhujaand kot.i, and whose area is 25. The area of the four triangles is600. Thus, adding these, the area of the big square is 625. Takingya2 =25,we get the hypotenuse to be 25.

The aboveupapattiis based on the algebraic identity

(a b)2 +4.1

2ab=a2 +b2

Krsna Daivajnas Upapatti of Kuttaka Process

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Kr. s.n. a Daivajna sUpapattiofKut.t.akaProcess

As an example of anupapattiwhich proceeds in a sequence of steps,we may briefly consider the detailed upapattifor thekut.t.aka

procedure given byKr. s.n. a Daivajna(c.1600) in his commentaryBjapallavaon Bjagan. itaof Bhaskara.

Thekut.t.akaprocedure is for solving first order indeterminateequations of the form

(ax+c)

b

=y

Here,a, b, care given integers (calledbhajya, bhajakaand ks. epa) andx, yare to be solved for in integers.

Kr. s.n. afirst shows that the solutions for x, ydo not vary if we factor allthree numbersa, b, cby the same common factor.

He then shows that if a and b have a common factor then the aboveequation will not have a solution unlesscis also divisible by the same.

He then gives theupapattifor the process of finding the apavarta nka

(greatest common divisor) ofa and b by mutual division (the so-called

Euclidean algorithm).


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Kr. s.n. athen provides a detailed justification for the kut.t.akamethod offinding the solution by making a vall(table) of the quotients obtainedin the above mutual division, based on a detailed analysis of thevarious operations in reverse (vyasta-vidhi).

In doing the reverse computation on the vall(vallyupasam.hara) thenumbers obtained, at each stage, are shown to be the solutions tothekut

.t.akaproblem for the successive pairs of remainders (taken in

reverse order from the end) which arise in the mutual division ofaandb.

After analysing the reverse process of computation with thevall,Kr. s.n. ashows how the solutions thus obtained are for positive and

negative ks. epa, depending upon whether there are odd or evennumber of coefficients generated in the above mutual division.

And this indeed leads to the different procedures to be adopted forsolving the equation depending on whether there are odd or evennumber of quotients in the mutual division.


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As an illustration,Kr. s.n. aconsiders the equation (173x+3)

71 =y

withbhajya173,bhajaka71 andks. epa3.In the mutual division of 173 and 71 we get the quotients 2, 2, 3

and 2 and remainders 31, 9, 4 and 1.

If we do the reverse computation on thevallformed by 2, 2, 3,

2, 1, 3 and 0, we first get 6, 3 as the labdhiandgun. a, whichsatisfy the equation

(9.33)4 =6, with the remainders 9, 4

serving asbhajyaandbhajaka.

In the reverse computation on thevall, we then get 21, 6 as

labdhiandgun. a, which satisfy the equation (31.6+3)

9 =21, withthe remainders 31, 9 serving as bhajyaandbhajaka.

And so on, till we get 117 and 48 as labdhiandgun. a, which

satisfy the equation (173.48+3)

71 =117

Use of Tarka in Upapatti

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Use ofTarkainUpapatti

The method of proof by contradiction is referred to as tarkain Indianlogic. We see that this method is employed in order to show thenon-existence of an entity.

For instance,Kr. s.n. a Daivajnaessentially employstarkato show thenon-existence of the square-root of a negative number whilecommenting on the statement of Bhaskarathat a negative number

has no root.

. . .

-

-

-


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Thus according toKr. s.n. a

The square-root can be obtained only for a square. Anegative number is not a square. Hence how can we

consider its square-root? It might however be argued:

Why will a negative number not be a square? Surely it

is not a royal fiat... Agreed. Let it be stated by youwho claim that a negative number is a square as to

whose square it is; surely not of a positive number, for

the square of a positive number is always positive by

the rule... not also of a negative number. Because

then also the square will be positive by the rule... Thisbeing the case, we do not see any such number

whose square becomes negative...


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While the method of proof by contradiction or reductio ad

absurdumhas been used to show the non-existence of entities,

the Indian mathematicians do not use this method to show the

existence of entities, whose existence cannot be demonstrated

by other direct means. They have a constructive approach tothe issue of mathematical existence.

It is a general principle of Indian logic that tarkais not accepted

as an independentpraman. a, but only as an aid to other

praman. as.

Indian Logic Excludes Aprasiddha Entities from

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Indian Logic ExcludesAprasiddhaEntities from

Logical Discourse

Naiyayikasor Indian logicians do not grant any scheme of

inference, where a premise which is known to be false is used

to arrive at a conclusion, the status of an independent praman.

a

or means of gaining valid knowledge.

In fact, they go much further in exorcising the logical discourse

of allaprasiddhaterms or terms such as rabbits horn

(sasasr.nga) which are empty, non-denoting or unsubstantiated.

Indian Logic Excludes Aprasiddha Entities from

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Indian Logic ExcludesAprasiddhaEntities from

Logical Discourse

Nyaya...(excludes) from logical discourses anysentence which will ascribe some property (positive or

negative) to a fictitious entity. Vacaspatiremarks that

we can neither affirm nor deny anything of a fictitious

entity, the rabbits horn. Thus nyayaapparently agrees

to settle for a superficial self-contradiction because, informulating the principle that nothing can be affirmed

or denied of a fictitious entity like rabbits horn, nyaya,

in fact violates the same principle. Nyayafeels that

this superficial self-contradiction is less objectionable

(than admitting fictitious entities in logical discourse)...

(This can be seen from the discussion in ) UdayanasAtmatattvaviveka...2

2B. K. Matilal,Logic, Language and Reality, Delhi 1985, p-9.

Yuktibhasa of Jyesthadeva

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Yuktibh as. aofJyes.t.hadeva

The most detailed exposition of upapattisin Indian mathematics isfound in the Malayalam textYuktibhas. a(1530) ofJyes.t.hadeva, a

student ofDamodara, and a junior colleague of Nlakan. t.ha.

At the beginning ofYuktibhas. a,Jyes.t.hadevastates that his purpose isto present the rationale of the results and procedures as expoundedin theTantrasa ngraha.

Many of these rationales have also been presented (mostly in theform of Sanskrit verses) by Sankara Variyar(c.1500-1556) in hiscommentariesKriyakramakar(on Ll avat) andYuktidpik a(onTantrasa ngraha).

Yuktibhas. ahas 15 chapters and is naturally divided into two parts,

Mathematics and Astronomy.

In the Mathematics part, the first five chapters deal with the notion of

numbers, logistics, arithmetic of fractions, the rule of three and the

solution of linear indeterminate equations.

Yuktibhasa of Jyesthadeva

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Yuktibh as. aofJyes.t.hadeva

Chapter VI ofYuktibhas. adeals with the paridhi-vyasa-sambandhaorthe relation between the circumference and diameter of a circle. It

presents a detailed derivation of the Madhavaseries for , includingthe derivation of the binomial series, and the estimate of the sums ofpowers of natural numbers 1k +2k + ...nk for largen. This is followedby a detailed account ofMadhavas method of end correction termsand their use in obtaining rapidly convergent transformed series.

Chapter VII ofYuktibhas. ais concerned withjyanayanaor thecomputation of Rsines. It presents a derivation of the second orderinterpolation formula ofMadhava. This is followed by a detailedderivation of theMadhavaseries for Rsine and Rversine.

In the end of the Mathematics section, Yuktibhas. aalso presentsproofs of various results on cyclic quadrilaterals, as also the formulaefor the surface area and volume of a sphere.

The Astronomy part of Yuktibhas. ahas seven chapters which givedetailed demonstrations of all the results of spherical astronomy.

YuktibhasaProof ofBhuja-Koti-Karna-Nyaya

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. j . . y y

ABCD, a square with its side equal to thebhuja, is placed on thenorth. The square BPQR, with its side equal to thekot.i, is placed onthe South. It is assumed that thebhujais smaller than thekot.i. MarkM on AP such that AM = BP =kot.i. Hence MP = AB = bhujaand MD

= MQ =karn. a. Cut along MD and MQ, such that the triangles AMDand PMQ just cling at D, Q respectively. Turn them around to coincidewith DCT and QRT. Thus is formed the square DTQM, with its sideequal to thekarn. a. It is thus seen that

karn. a-square MDTQ =bhuja-square ABCD +kot.i-square BPQR

Successive Doubling of Circumscribing Polygon

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g g yg

It appears that the Indian mathematicians (at least in the Aryabhat.an

tradition) employed the method of successive doubling of the circumscribing

square (leading to an octagon, etc.) to find successive approximations to the

circumference of a circle. This method has been described in the Yuktibhas.aand also in theKriyakramakar.

The latter cites the verses ofMadhavain this connection


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g g yg

In the figure,EOSA1 is the first quadrant of the square circumscribingthe given circle. EA1 is half the side of the circumscribing square. Let

OA1 meet the circle atC1. DrawA2C1B2 parallel toES. EA2 is halfthe side of the circumscribing octagon.

Similarly, letOA2 meet the circle at C2. DrawA3C2B3 parallel toEC1.

EA3 is now half the side of a circumscribing regular polygon of 16

sides. And so on.


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g g ygLet half the sides of the circumscribing square, octagon etc., be denoted

l1 =EA1 , l2 =EA2 , l3 =EA3 , . . .

The correspondingkarn. as(diagonals) are

k1 =OA1 , k2 =OA2 , k3 =OA3 , . . .

and theabadhas(intercepts) are

a1 =D1A1 , a2 =D2A2 , a3 =D3A3 , . . .

Now

l1 =r, k1 =

2r, anda1 = r

2.

Using thebhuja-kot.i-karn. a-nyaya(Pythagoras theorem) andtrai-

rasika-ny aya(rule of three for similar triangles), it can be shown that

l2 = l1 (k1 r)

l1a1

k22 = r2 +l22

a2 = k22

r2 l22

2k2


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In the same wayln+1, kn+1 andan+1 can be obtained from

ln, knandan.

These can be shown equivalent to the recursion relation for ln+1which is the half side of the circumscribing polygon of 2n+1

sides:

ln+1=

r

ln

r

2 +l2n12 r

We of course have the initial value l1 =r.

This leads tol2 = (

2 1)rand so on.

Kriyakramakarnotes that

Thus, one can obtain an approximation (to the

circumference of the circle) to any desired level of

accuracy.

References

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1. Ll avatofBhaskarawith comm. BuddhivilasinofGan. esa

Daivajna, Ed Apte, 2 Volumes, Pune 1938.

2. Bjagan. itaofBhaskarawith comm. BjapallavaofKr. s.n. a

Daivajna, Ed. Radhakrishna Sastri, Saraswati Mahal,

Thanjavur 1958.

3. Gan. itayuktibhas. aof Jyes.t.hadeva(in Malayalam),

Gan. itadhyaya, Ed., with Notes in Malayalam, byRamavarma Thampuran and A. R. Akhileswara

Aiyer,Trichur 1948.

4. Gan. itayuktibhas. aof Jyes.t.hadeva(in Malayalam), Ed. with

Tr. by K. V. Sarma with Explanatory Notes by

K. Ramasubramanian, M. D. Srinivas and M. S. Sriram, 2

Volumes, Hindustan Book Agency, Delhi 2008.

5. Kriyakramakarof Sankara VariyaronLl avatof

BhaskaracaryaII: Ed. by K. V. Sarma, Hoshiarpur 1975.

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6. Tantrasa ngrahaofNlakan. t.hawithYuktidpik aof Sankara

Variyar, Ed. K. V. Sarma, Hoshiarpur 1977.

7. S. Parameswaran,The Golden Age of Indian Mathematics,Swadeshi Science Movement, Kochi 1998.

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G. G. Emch, R. Sridharan and M. D. Srinivas, Eds.,

Contributions to the History of Indian Mathematics,

Hindustan Book Agency, Delhi 2005, pp.209-248.

9. C. K. Raju,Cultural Foundations of Mathematics: The

Nature of Mathematical Proof and the Transmission of the

Calculus from India to Europe in the 16th c.CE,Pearson

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Thanks!

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