Applied Geometry in SulbaSutras

8/12/2019 Applied Geometry in SulbaSutras

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Applied Geometry of the Sulba Sutras

John F. Price*School of MathematicsLiniversitv of New South WalesSydnel', NSW 2052, Australiajohnp Osherlockinvesting.com

Abstract

The Sulba Sutras, part of the Vedic literature of In-dia, describe many geometrical properties and con-structions such as the classical elationship

a2 + b 2 : c 2

between he sides of a right-angle triangle and arith-metical formulas such as calculating the square ootof two a ccurate to five decimal places. Althoughthis article presents some of these constructions,its main purpose is to show how to consider eachof the main Sulba SDtras as a finely cr afted, inte-grated manual for the construction of citis or cere-monial platforms. Certain k ey words, however, sug-gest that the applications go far beyond this.

1 Introduction

The Sulba S[tras are part of the Veclic iterature,an enormous body of work consisting of thousandsof book s covering hundreds of thousands of pages.This section starts with a brief review of the posi-tion of the Sulba S[tras within this literature andthen describes some of its significa nt features. Sec-tion 2 is a discussion of some of the geometry foundin the Sulba Sutras while Section 3 is the same forarithmetical constructions. Section 4 looks at some

of the applications to the building of citis and Sec-tion 5 integrates he different sections and discussesthe significance of the applied geometry of the SulbaSUtras. I am grateful to Ken Hince for assistancewith the diagrams and to Tom Egenes or helpingme avoid some of the pitfalls of Sanskrit transliter-ation and translation.

*This article was written while the author was in the De.partment of Mathematics, Maharishi University of Manage-ment, Fairfield, IA .

1 .1 Vedic Literature and the SulbaSDtras

To understand the geometry of the Sulba Sltrasand their applications, it is helpful first to under-

stand their place in Vedic knowledge as expressedthrough the Vedic iterature.

Nlaharishi Nlahesh Yogi explains that the Vedicliterature is in fortr. parts consisting of the fourVedas plus six sections of six parts each. These sec-tions are the Vedas. he Vedangas. he Uparigas, heUpa-Vedas, he BrahmaTas. and the Pratishakhy as.Each of these parts "expresses specific quality ofconsciousness" 14, . 141]. This m eans hat often wehave to look beyond the surface meanings of manyof the texts to find their deeper significance.

The Sulba Sutras form part of the Kalpa Sltraswhich in turn are a part of the Vedangas. There

are four main Sulba SUtras, the Baudhayana. theApastamba, the N1anava, and the K5tvavana, anda number of smaller ones. One of the meanings of6ulba is "string, cord or rope." The general for-mats of the main Sulba Sltras are the same; eachstarts with sections on geometrical and arithmeti-cal constructions and ends rvith details of how tobuild citis which, for the moment, u'e interpret asceremonial platforms or altars. The measurementsfor the geometrical constructions are performed bydrawing arcs with different radii and centers usinga cord or 6ulba.

There are numerous translations and referencesfor the Sulba Sntras. The two that form the basisof this article are 17] u'hich contains the full textsof the above four Sulba Sutras in Sanskrit) and [9](which is a commentary on and English translationof the Baudhayana Sulba Sutra). Another usefulbook is l8l since it contains both a transliterationof the foui main Sulba SDtras nto the Roman al-phabet and an English translation.

It is timely to be Iooking at some of the mathe-matics contained in the Vedic literature because of

46


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the rene wed understanding brought about by Ma-harishi of the practical benefits to modern Iife ofthis ancient Vedic knowledge. Details and furtherreferences an be found in [5].

L.2 Features of the Sulba SutrasFor me, there are three outstanding features of theSulba Sotras: the wholeness and consistency oftheir geometrical results and constructions, the el-egance and beauty of the citis, and the indicationthat the Sutras have a much deeper purpose.

L.z.I Integrated wholeness of the SulbaSutras

When each of the main Sulba Sutras is viewe d as awhole, instead of a collec tion of parts, then a strik-

ing level of unity and efficiency becomes apparent.There are exactly the right geometrical construc-tions to the precise degree of accuracy necessaryfor the artisans to build the citis. Nothing is redun-dant. This point is nicely made by David Henderson

[2] who argues that the units o f measurement usedeasily ead to the accuracy of the diagonal of one ofthe main bricks of "roughly one-thousandth of thethickness of a sesame seed."

There is also a remarkable degree of internal con-sistency such as the way that the 'square to circle,''circle to square' and 'square root of two' construc-tions fit togeth er with an accuracy of 0.0003%. (De-tails are given in Section 5.) A related discussionin the literature is whether or not the authors ofthe Sutras knew their construction of the root oftwo was an approximation.l Viewing the Sttrasas utilitarian construction manuals suggested hatthey knew that they wer e describing an approxima-tion but that they achieved what they set out to do,namely to provide the first terms of an expansion ofthe root of two sufficient to en sure the reciprocityof the 'square to circle' and 'circle to square' con-structions.

If mathematicians were asked to write such a

manual,it is likely that they would do two things,

firstly give the construction procedures to level ofaccuracy appropriate for the actual constructions,and secondly, for their own enjoyment, show toother mathematical re aders that they really un-derstood that they were dealing with approxima-tions. Both these eatures are observed n the SulbaSutras.

lThis discussion hinges on the meaning of oi.fesa, which

[1] and others take to mean, in this context , a small excessquantity or difference. See also [8, p. t68].

1.2.2 Beauty of the citis

Each of the citis are low platforms consisting of lar''-ers of carefully shaped and arr anged bricks. Someare quite simple shapes such as a square or a rhom-bus while others are much more involved such as afalcon in flight with curved wings, a chariot wheelcomplete with spokes, or a tortoise with extendedhead and legs. These latter designs are particu-Iarly beautiful and elegant depictions of powerfuland archetypal symbols, the falcon as the great birdthat can soar to heav en. he wheel as the 'wheel oflife,' and the tortoise as the representativ e of sta-bility and perseverance.

L.2.3 Deeper significance

Sanskrit is a rich language full of subtle nuances.\\brds can have quite different meanings becauseof their context and, in any case, frequently thereis no reasonable English equivalent.

There are a number of key terms in the SulbaSutras which, because of their etymology and pho-netics, suggest hat there is a much deeper signifi-cance o the Sftras. One is the word cili introducedabove. In the context of the Sulba Sltras, the usualtranslation is a type of ceremonial platform but itis close o the word cil which means consciousness.Another is ued'i which is usually translated as theplace or area of ground on which the citi is con-structed. But since the word uedo means "pureknowledge, compiete knowledge"

[],p 3], vedi also

means an enlightened person, a person "who pos-sesses eda."

A third is purusa which is usually translated asa rrnit of metuslrrement btained by the,height of aman with upstretched arms (Nfanava Sulba SltraIV, 5) or as 120 aiguLas (Baudheyana Sulba SutraI, I 2I), a measuremen t based on sizes of certaingrains.

However, in l3l puruso is defined as "the unin-volved witnessing quality of inteliigence, he unified. . . self-referral state of intelligence at the bas is ofall creativity" (p. 109). Thus we could easily infer

that a more expanded role of the Sulba Sutras is asa description of consciousness. Further discussionof this point is given in Section 5.

When these and other examples are combinedwith the general direction of ail the Vedic liter-ature towards describing "qualities of conscious-ness," we are led to the conclu sion that the SulbaS[tras are describing something much beyond pro-cedures or building brick platforms, no matter howfar-reaching their purpose. This theme is referredto again in the concluding section. In this article


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48 Geometry at Work

E

,lWFigure 1: Steps or the construction of a square.

the focus is on the geometrical content of the text,but because of the range of meanings of the keyterrns, they u'ill generally be left in their original(but transliterated) form. In a later article, I hopeto develop some of these deeper hemes of the SulbaSttras.

2 Geometry

Nlost of the geometric procedures described n theSulba S[tras start rvith the laying out of a prac[which is a line in the east-west direction. This lineis then incorporated into the final geometric objectsor constructions, generally as a center ine or line ofs)'mmetry. This section describes some of the maingeometric constructions given in the Sttras.

2.L Construction of a square with aside of given length

From verses I, 22-28 of the Baudhayana SulbaSutra (BSS)2, he procedure s to start with a pracrand a center point (line EPW in Figure 1) and,by describing circles with certain centers and radii,construct a square ABC D in which ,E and lV arethe midpoints of AB and C D. The steps of the

construction are displayed n Figure 1.

2.2 Theorem on the squareof the diagonal

Verse . 48 of BSS states:

'There are tn'o main numberings of the verses of the BSS,one used by Thibaut [9] and one attributed to A. Biirk in [8 ]and used there. In this article we follow the numberine usedbv Thibaut.

Tlie diagonal of a rectangle produces both(areas) which its length and breadth pro-duce separately.

There appears to be no direct mention of areas nthis verse. When, however, t is combined with thesubseqrrent ne.

This is seen n rectangles with sides hreeand four, twelve and five, fifteen and eight,seven and t'nventl'-four, lvelve and thirty-five. fifteen and thirty-six ,

it is clear that it is an equivalent statement tothe theorem named in the west after Pythagoras,namely that in a right-angle triangle with sides o,b and c ( c the h1 'potenuse) . 2 +b2: c2 . Note a lsothat all possible pairs (a, b) are given (except one)rvhich (i) allolv an integer solution of a2 + b2 : c2with 1 I o, 12 and a I b, and (ii) are coprirne.(The interested reader might like to check this.)This accounts for five of the listed pairs, the lastpair (15.36) being derivable rom the earlier pair( 5 , 1 2 ) .

Further evidence that Baudhayana had a clearunderstanding of this result and its usefulness sprovided by the next two constructions.

2.3 A square equal to the sum of twounequal squares

Verse I. 50 of BSS describes he construction of asquare with area equal to the sum of the areas oftwo unequal squares. Suppose that the two givensquares are ABCD ard EFGH with AB > EF.Mark off points J.K on AB and DC wllh AJ :DK : ,EF as shor,,,'n n Figure 2. Then the lineAK 1s he side of a square with area equal to thesum of the areas of ABCD and EFGH.


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Applied Geometry of the Sulba Sttras .19

Figure 2: Sum of two squares.

llE>ll

E F

Figure 3: Difference of t"'r'o squares.

2.4 A square equal to the differenceof two squares

The subsequent verse (I, 51) describes he construc-tion of a squ are with area equal to the difference oftwo unequal squares. With the ncltation the sameas the preceding example, form an arc D L with cen-ter A as shown in Figure 3. Then J-L is the side ofthe required square.

This follows from the facts that (AJ)z + (J L)2 :(AL)2 , AL: AB, and AJ : EF, so tha t (JL) ' :( A B ) ' - ( E F ) '

2.5 Converting a rectangle into asquare

The method of converting a rectangle nto a squarewith the same area described n BSS I. 54 makes useof the previous construction. Start with a rectangleABCD with AB > CD as shown in Figure 4 andform a square AEFD. The excess portion is cutinto equal halves and one half is piaced on the sideof the square. This gives two squares, a larger oneAGJC' and a smaller one FHJB'; the requiredsquare is the difference of these two squares. InFigure 4 the side of this square is GI, where .L is

determined by EL: EB'.To see his, denote the sides of the rectangle by

AB : a and AD:b.

Then the sideGI

satisfies

(GL)2 : (EL) ' - (EG) ' : (EB' )2 - @G) '. , , 2 , , , 2

L d - o \ l 0 - 0 \: ( b -" I

- (" 1 : u b\ j , / \ z /

and so the ar ea of the shaded square equals he areaof the initial rectangle. as required.

Datta l1] suggests hat the steps n this construc-tion of marking off a square, dividing the excess,and rearranpging he parts could be the basis of themethod descr ibed ater n the BSS ,61 (see .1) offinding the square root of trvo. By repeating these

steps, he shos.s hat the successive pproximationsto the square root of tlvo described n I, 61 are ob-tained. If this is the procedure hey used, hen thisis another example of the tightly knit methods andlogic of the Sttras.

2.6 Converting a square into a circle

Verse , 58 describes he procedure for constructinga c ircle with area approxlmately equal to that ofa given square. Start with a square ABCD as in


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bt J Geometry at Work

c'

B

Figure 4: Steps in converting a rectangle to a square.

and hence

Figure 5 with center O. Draw an arc DG withcenter O so that OG is parallel to AD. Supposethat OG intersects DC at the point .F. Let 1/ be apoint 1/3 of the distance from F to G. Then OHis the radius of the required circle.

To see what is going on here, let 2a be the lengthof the side of the squa.re ABC D and r the radius ofthe constructed circle. Then

r : O H : O F + F HI: oF + 4(oG oF)

1 -a + 7 @ t / 2 - a )

' : i( '*t')If we substitute the values r :3.I4I593 and yE :1.414214 we get that the area of the constructedcircle is

A r e a : r r 2 :4 .0690 IL . . x a 2

which is within abofi 1.770 of the correct value ofA

In the next verse Baudhayana describes how togo in the opposite direction, namely from a circleto a square.

2.7 Converting a circle into a squareThibaut's translation of Verse , 59 is :

If you wish to turn a circle nto a square,divide the dianreter into eight parts andone of these eight parts into twenty-nineparts; of these twentl.nine parts removetrventr'-eight ancl moreover the sixth part(of the one part left) less the eighth part(of the sixth part).

In modern mathematicai notation, starting with acircle of diameter d, the length of the side of thecorresponding quare s:

I e n g t h a _ l + ^ j -8 8 ' < 2 9

----.......

.-__-,-r--

Figure 5: Converting a square to a circle

( 1 )

tc

H

I


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d / r 1 \l - - - l

8 x 2 9 \ 6 6 x 8 /9785

: - x d .111 3 6

Taking n:

3.141593 s before, weget

and then sequentially estimate and reduce the s izeof the errors. Let 11 be the first error so that

The correct value is 1 and so, just as in the previ-ous section, the result is accurate to approximately1.7%. The precision of these accuracies will belooked at in more detail in Section 5 belorv.

3 Arithmetic

We shall first consider the example in theBaudhayana Sulba Sltra of the calcuiation of thesquare root of two and then bok at t'hat couid bethe underlying principle.

3.1 The square root of two

Verse , 61 of BSS writes:

Increase the measure by a third and this(third) again by its own fourth less itsthirty-fourth part; this is the (length of)

the diagonal of a square (whose side s tnemeasure).

Using modern notation, the assertion s that if thelength of the side of the original square is a, thenthe length of its diagonal is

( 9785 2 ,z / r , z:\ t " * /

o - I 4 o: 0 . 9 8 3 0 4 5 . ..

r+-) "

11 + : - | r r

: J '

Squaring both sides and neglecting erurs n z givesrt : #. Now define the next approximation 12by

1 1t / z : t + )

J - 3 ^ 4 + " 'Repeating the preceding step gives

1& z -

3 x 4 x 3 ' 1

which yields the approximation given in the BSSjust described.

Another repetition of this step gives the approx-

imation:

J '= I3 " 4 -1

3 x 4 x 3 4

3 x 4 x 3 4 x 2 x 5 7 7665857 : 1.4142r3562:t74. .

u'hich is accurate o 13 decimal places.Anotlier approach has been suggested bV [t]

which rnaintains the geometric flavor of the Sutras.The idea is to use the first steps of the method de-scribed in 2.5 to convert a rectangle of size 2 x 1into a squarc of sicle O. lt these steps are repeatedthe above sequence f approximations is obtained.

Henderson {2]shou-s ust horv natural the proce-dure is and hon' it can be gcneralizecl o a largeclass of real numbers. [Ie also cibserr,es ]rat samesequential pproxirnatiorr r, yI can be obtained byapplying Newt,ou's retllo(l or finding the roots ofan algebraic equation. Takr' /(.r1 : .r2 - 2 n'ithan initial approximation of .r'e Jl3 for its posi-tive root. Nev,.ton's rethod is that the successireapproximations are given br'

"_ f ( r , ' ) _ r ' r , _4 n + r - . i n

f , ( r . l , 2' l \ r n /

It is easily checked hat this gives the approxirna-t ions descr ibed rr he Sr r lba Si r t ras .

4 Applications to theconstruction of citis

In this section we look at the applications of thegeometrical and arithrnetical constructions of the

( 2 )

Area of squareArea of circle

1l + - +.)

40832

diagonal :

3.2 Discussion

Given that JZ : t.l,t+213. . . , it is not surprisingthat many commentators have proposed ways thatexplain the underlying method of achieving suchan accurate result. The common thread pa^ssingthrough most of the explanations is to start withthe initial approximation

l t ll 1 + - - l - - -\ ^ 3 3 x 457 740 81 . 4 1 4 2 1 5 . . . x a

1/ n

' / i=r+


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52 Geometry at Work

Figure 6: Possible configuration of the garhaptya citi for the odd and even layers.

Sulba Sutras to the constr uction of citis. As statedin Section 1, this is ust the first level of applicationsof the S[tras.3

4.L Generalintroduction

Each of the citi s is constr ucted from five layer s ofbricks, the first, third and fifth layers being of thesame design, as are the seco nd and fourth. Quite alengthy sequence of units is used; the two that arereferred to the mos t are the angula and the purusawhich were discussed n 1.2.3 above. They are ap-proximately 0.75 nches and 7 feet 6 inches with 120aigulas equal to a purusa.

The heights of each layer are 6.1 aigulas whichis abou t 4.8 inches and the successive ayers arebuilt so that no joins iie along each other. ThisIast requirement is sometimes difficult to achieveand adds to the aesthetics of the finished productas much as to its streng th. Generally each ayer has200 bricks with the exception of the garhaptya citiwhich has 21 bricks in each ayer.

For each design (with the exception of thegarhaptya citi), the citi is first constructed with anarea of 7.5 square purusas, then with 8.5 square

purusas. and so on up to 101.5 quare purusas BSSII, 1-6). Verse I, 12 explains how these ncreases nsize are to be brought about. If the origi nal citi of7.5 sq. purusas is to be increased by q sq. purusas,a square with area one sq. purusa has substitutedfor it a square with area | + (2qll5) sq. purusas.The sides of this square form the unit of the newconstruction replacing the original purusa. Hence

JPhotographs from Kerala, India of citis and associatedceremonies are contained in 16l.

bhe area of the enlarged citi is

: 7.5 + q sq. purusas,

as requrreo.NIany of the designs described in the BSS have

a number of variations. For simplicity not ali thevariations will be described and we shall choose oneof them usually without commenting that there areother possibilities.

4.2 The gdrhaptya citiThis citi, described in BSS II, 66 69, is on evyayama square and consis ts of 21 bricks on eachlevel. (A vyayama is 96 angulas or about 6 feet;BSS I, 2l). Three tvpes of bricks are used: one-sixth. one-fourth and one-third of a vyayama. Theodd layers consist of 9 bricks of the first type and12 of the second, while the even layers consist of6 bricks of the third type and 16 of the first. Onepossible arrangement is shown in Figure 6.

4.3 The Syena citi

This is a particularly beautiful depiction of a fal-con (6yena) in flight, its construction steps beingdescribed n BSS III, 62-104. See Figures 7 and 8.

4.4 The rathacakra citi

This citi is in the shape of a chariot wheel (ratha-cakra) wtth nave (or center), spokes and felly (orrim), its c onstruction being described n BSS III,187-214. It requires seven types of bricks for the

z . s x


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Applied Geometry of the Sulba SDtras 53

Figure 7: The Syena iti: lavers 1. 3, and 5.

Figure 9: Design

odd la1'ers nd nine types for the even ayers. Thereseems o be some flexibility about the final design,Figure 9 being one possibility.

The initial calculations for determining the differ-ent parts of the '*'heel are in terms of square brickseach of area 1/30 square purusas. Since the finalarea is require d to be 7.5 square purusas, the num-ber of bricks is 7.5 x 30 : 225. The nave of thewheel consists of 16 of these bricks. the spokes 64and the rim 145, making 225 in all.

The spaces between the spokes are equal in areato the spokes and so, if these spaces are included,the overall area is 225 + 64 : 289 bricks. (No-tice that these numbers satisfy 152 + 82 : 172and are one right-angle-triangle triples describedabove.) Hence the radius of the outer rim of the

Figure 8: The Syena citi: lavers 2 and 4

of the rathacakra citi.

wheel is equal to the radius of a circle equal in areato a square of side 17 bricks so the methods de-scribed n subsection .6 could be used o constructthis circle. Similarly. the inner radius of the rim isthe radius of a circle equal in area to a square withside the square root of 16 + 6.1 64 : 14.1, ramely,12. Finaliy. the radius of the nave comes from asquare of side 4.

4.5 The kurma citi

Another fascinating series of constructions are inthe form of a tortoise (kArma). In the BSS there aretwo types of these constructions, one s described ashaving twisted linibs (o,akraigaica) and the other ashaving rounded Iinbs (parimandalaica). Figures 10


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Geometry at Work

Figure 10: The korma citi: layers 1, 3, and 5.

and 11 depict the first type.The construction for the odd layers starts with a

square of sid e 300 angulas and then the four cornersare removed by isosceles riangles with equal sidesof 30 angulas. Head, iegs, sides and tail are nowadded with the result shown in Figure 10.

For the even layers, the starti ng step is a squareof side 270 arigul as which is offset from the basicsquare for the odd laye rs by 15 angulas. The plan

of the final construction is shown in Fisure 11 .

5 Discussion

Having listed some of the constru ctions of the SulbaSDtras, we are now in a position to look at how theyfit together. A key example is the duality of the'circle-square' constructions. As explained in 3.1,verse , 61 of BSS gives a value of the square root oftwo as ffi. Uring this value, there is a remarkableduality between the 'square to circle' and 'circle tosquare'results.

Suppose hat we start with a square of side 2. Byequation (1), the diameter d of the correspondingcircle is

,1 -

:

Now use equation 2)

Figure 11: The klrrna citi: Iavers 2 and .t .

side :

2 *J 1 : 2 . 0 0 0 0 0 6 .

6 , 8 1 5 . 2 3 2

an accuracy of 0.0003%.A second example is the use made of the 'dif-

ference of two squares' construction to construct asquare with area equal to a given rectangle. Also, asexplained in Section 2, construction may well formthe basis of the method of finding the root of two.

Other examples are the conversion of squa,res ocircles or the construction ofcitis in circular shapessuch as the rath acakra citi, or the methods used toincrease he size of the citis by scaling up all the di-mensions. These and other similal results show thelevel of integration and completeness f the body ofresults n the Sulba Sttras.

In the opening section. several examples weregiven of key words in the Sulba Sttras that eitherhad alternative deeper meanings or were related tosuch words. The study of the meaning of wordsin Sanskrit is a large and technical field foundedon the work of Palini. Because of the technicalnature of the area, drawing the specific conclusionthat the Sttras were also intended to be dealingwith the field of consciousness "r'ould equire con-siderably more work. But there is an example that

2 - ( r - 5 7 7 \3 \ - l o 8 /1 3 , 6 3 0 . 0 56 ,8 1 5 , 3 2

9785111 3 6

9 r - \

5lz + ttz)2 ( ) _ 1 5 7 7 \3 \ - 4 0 8 / '

to get the side of the square


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Applied Geornetry of the Sulba Sfitras

explicitly connects he construction of the citis withconsciousness, amely Verse II, 81 of BSS. It readsthat after having constructed a citi for the thirdtime, tlren a. chandaicil is to be constructed. Theword chandas means mantra o r mantras which are

the 'structures of pure knowledge, he sounds of theVeda' [4, p. 3]. Commentators nterpret this verseas ndicating that the fourth and later constnrctionsare to be calried out on the level of consciousnesswith mantras replacing the actual bricks. (See, orexanrple . 8 . 9] )

As with all the Vedic iterature. he Sulbar Dtrascan be read and interpreted on many leveis. At thevery least, they provide a fascinating chapter in thegrowtli of geometrical and arithmetical knowledgeand its application to the design aud constrttctionof complex brick platforms. But there are manyindications, some of which have been pointed outabove, that they are also a descriptiort. or perhapsa map, of the structure and qualities of the field ofconsciousness.

References

[1] B. Datta The Science f the iuLba. CalcuttaUniversitl ' Press, Calcutta. 1932.

[2] David W. Henderson. Square roots in the Sulbas[tras. In C. Gorini, editor. Geonetrg at Work,pages 39 45. N{AA, Washingtoli. DC. 2000.

f3] Maharishi \,Iahesh Yogi. L'Iaharisht's AbsoluteTheory of Gouernmenf. N{VLI Press. Vlodrop.the Netherlands, 1993.

[4] N,Iaharishi N{ahesh Yogi. Vedic Knowledge orEueryone. N'IVU Press, Vlodrop, the Nether-lands, 1994.

[5] N'IERU. Scienti.fi,c Research on Maharish'i'sTranscendental Meditation and TIv[ Sid.hi Pro-gramme: Collected Papers, Volumes 1-5, 1977.

f6] Ajit N{ookerjee. Ritual Art oJIndia. Thames

and Hudson, London, 1985.

[7] Satya Prakash and Usha Jyotishmati. The SutbaSutras: Terts on the Vedic Geometry. RatnaKumari Svadhyava Sansthana, Allahabad, In-dia. 1979.

[8] S. N. Sen and A. K. Bag. The Sulbashtrasof Baudhayana, Apastamba, Katyayana and,Manaua. Indian National Science Academy.New Delhi, India, 1983.

George F. W. Thibaut. Mathematzcs ,n theMaking in Ancient India. K. P. Bagchi & Co.Calcutta, India, 1984. This is a reprint of twoarticles by Thibaut: On the Sulvasittras, J. Asi-atic Soc. Bengal, 1875 and Baudhayana Su)va

Sitram, Paldit, 187 5-L877.

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