Vedas Sulvasutras2

8/21/2019 Vedas Sulvasutras2

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NPTEL COURSE ON

MATHEMATICS IN INDIA:FROM VEDIC PERIOD TO MODERN TIMES

LECTURE 3

Vedas and Sulbasutras - Part 2

K. Ramasubramanian

IIT Bombay

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OutlineMathematics in the Antiquity: Vedas and ´ Sulbas¯ utras – Part 2

Sum of unequal squares & implication of the procedure?

A note on the terminology employed (karanı )

Applications of ´ Sulba theorem

Constructing a square that is difference of two squares Transforming a rectangle into a square To construct a square that is n times a given square

To transform a square into a circle (approx. measure preserving)

Approximation for√

2

Citi – Fire altar (types, shapes, etc)

Fabrication of bricks, Constructional details

General observations

References

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Constructing a square that is sum of unequal squaresAn application of the ´ Sulba -theorem

(BSS I.50)

Desirous of combining different squares, may you mark the rectangularportion of the larger [square] with a side (karan . y¯ a ) of the smaller one(kanıyasah . ). The diagonal of this rectangle (vr . ddhra ) is the side of thesum of the two [squares].

A

C

D

EB

F

G

HI

J

K The term vr . dhra in the above s¯ utra refers to the rectangle ABEF.

Asking us to mark this rectangle, all thatthe text says is the cord AEaks.n . ay¯ arajjuh . gives the side of the sum

of the squares. In other words,

AE 2 = ABCD + CGHI

= AB 2 + CG 2

= AB 2 + BE 2.

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Implication of the above construction ?

Scholars trained in the Euclidean tradition, puzzled by the mere

statement of theorem, without the so called ‘proofs’ always wonderedwhether the Sulbakaras knew the proof of ´ Sulba -theorem, or was it

purely based on empirical guess work?

Though ´ Sulvak¯ aras do not give explicit proofs, it is quite implicit in the

procedures described by them. In fact, the previous description of

construction clearly forms an example of that.

A

C

D

EB

F

G

HI

J

K In the figure, ABCD and CGHI are the

two squares to be combined. E is a pointon BC such that CG = BE .

ABEF is the rectangle that is formed.Now the sum of the two squares may be

expressed as

ABCD + CGHI = ABE + AEF + EHJ + HEG + FDIJ

= ADK + AEF + EHJ + HKI + FDIJ

= AEHK ,

which unambigously proves thetheorem.

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A note on the terminology employed

Before introducing ´ Sulva -theorem, Katyayana has exclusivelydevoted one s¯ utra to clarify the different terminologies that

would be employed to refer to cords in different contexts.

, ,

,

,

karan . ı, tatkaran . ı . . . all refer to cords. (KSS 2.3)

The commentary by Mahıdhara (c. 1589 CE)—explaining the

origin of the five names given in the above s¯ utra —is quiteedifying.

That which limits or produces the length or area is

karan .ı

(producer).

,

,

,

That which produces an area that is twice etc. is called

tatkaran . ı (that-producer); For example, dvikaran . ı,

trikaran . ı, catuh . karan . ı , and so on.

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A note on the terminology employed (contd.)

,

That by which . . . is measured is called tirya ˙ nm¯ anı

(transverse-measurer) . . .

,

That by which the sides are measured is called p¯ arsvam anı

(side measurer); It refers to the cords on either sides that is

stretched along the east-west direction.

,

,

,

That which makes the area look like eyes [i.e., splits into two]

is called aks.n . ay¯ a (diagonal); The mid-cord that connects the

corners. Once it is fixed, the square looks like an eye, and

hence the term aks . n . ay¯ a is used to refer to the diagonal.

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Different connotations of the word karan . ı

1. = side of a square

By the side of the smaller [square] . . . (BSS 2.1)

2. = square root

[In a rectangle] with upright one pada and base three padas, the diagonal-rope is √

10. (KSS 2.4)

3. = a certain unit of measure

,

,

,

(KSS 2.9)

Note: Though karan . ı seems to have ‘different’ connotations, on taking a

closer look, it becomes evident that some of these meanings converge to the

same thing—that which makes a square of area a . Obviously ‘that =√

a .

Examples dvikaran . ı, trikaran . ı , dasakaran . ı , and so on.

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Constructing a square that is difference of two squares

[BSS 2.2]

Desirous of subtracting a square from another square, may you mark therectangular portion of the larger [square] with a side (karan . y¯ a ) of thesmaller one that you want to remove. With the [cord corresponding to thelarger] side of the rectangle turned into a diagonal (aks.n . ay¯ a ) touch theother side. Wherever that intersects, chop off that portion. Whatever

remains after chopping, gives the measure of the difference.

000

000

000

000

000

000

000

000

000

000

000

111

111

111

111

111

111

111

111

111

111

111

H

A x E

F

G

B

CD

P

Problem : Find the side of square which is thedifference of the squares ABCD and AEGH.

Solution : Obtain the vr . dhra (rectangle) AEFD,and with radius EF mark a point P on AD. APgives the desired measure.

It is evident from the figure

AP 2 = EP 2 − AE 2

= AD 2 − AE 2. (EP = AD )

= ABCD − AEGH

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Transforming a rectangle into a squareSequel to finding the sum and difference of squares

,

,

[BSS 2.5]

terms in s¯ utra correspondence with figure

rectangle ABCD

east-west cord (AB)

the portion XYCB

square RSNM

by placing

It is evident from the figure

DP 2 = EP 2 − DE 2

= AE 2

−RS 2.

= AENF − RSNM

= HIJK

EA

D

B C

A D

P

R S

M N

JK

F

X Y

H I

E

N

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To construct a square that is n times a given square

Katyayana gives an ingenious method to construct a squarewhose area is n times the area of a given square.

( n + 1 ) a

2 na

A

B CD

(n−1)a

2

( n + 1 ) a

[KSS 6.7]

As much . . . one less than that forms the

base . . . the arrow of that [triangle] makes

that (gives the required number√

n ).

In the figure BD = 1

2 BC = (n −1

2 )a . Considering ABD ,

AD 2 = AB 2 − BD 2 =

n + 1

2

2

a 2 −

n − 1

2

2

a 2

= a 2

4 [(n + 1)2 − (n − 1)2] =

a 2

4 × 4n = na 2

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To transform a square into a circle

A B

CD

P

M

E

O

W

AB = 2a (given)

OP = r (to find)

OD =√

2 a

ME = OE −OM

=√

2 a − a

= a (√

2− 1)

[BSS 2.9]

= semi-diagonal (OD)

= from centre to the east

= whatever [portion] remains

= with one-third of that

As per the prescription given,

Radius OP = r = a + 1

3ME

= a 1 + 1

3

(√

2

−1)

= a

3(2 +

√ 2).

How to find√

2?

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How did Sulvak¯ aras specify the value of√

2?

The following s¯ utra gives an approximation to√

2:

,

,

, [BSS 2.12]

√ 2 ≈ 1+

1

3+

1

3× 4

1− 1

34

(1)

= 577408

= 1.414215686

What is noteworthy here is the use of the word in the

s¯ utra , which literally means ‘that which has some speciality’(speciality ≡ being approximate)

How did the ´ Sulvak¯ aras arrive at (1)?

Several explanations have been offered over the last centuries.Here we will discuss the geometrical construction approach.

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2Rationale for the expression

√ 2 = 1 + 1

3 + 1

3.4 − 1

3.4.34 by Geometrical Construction

Consider two squares ABCD and BEFC (sides of unit length).

The second square BEFC is divided into three strips. The third strip is further divided into many parts, and these parts

are rearranged (as shown) with a void at Q .

Now, each side of the new square APQR = 1 + 13 + 1

3.4.

III

III

III

IIS I

1

2

3

II III 1

A B

CD

P

QR

E

F

void that

remains

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2Rationale for the expression (contd.)

The area of the void at Q is

13.4

2.

Suppose we were to strip off a segment of breadth b from eitherside of this square, such that the area of the stripped off portionis exactly equal to that of the void at Q , then we have,

2b 1 + 1

3

+ 1

3.4−

b 2 = 1

3.4

2

.

Neglecting b 2 (as it is too small), we get

b = 1

3.4

2

×3.4

34

= 1

3.4.34

.

Hence the side of the resulting square

√ 2 = 1 +

1

3 +

1

3.4 − 1

3.4.34

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Approximate value of πAn estimate of the value of π used by ´ Sulvak¯ aras

If 2a is the side of the square, then we saw that the prescription

given in the text amounts to taking the radius of the circle to be

r = a

1 +

1

3(√

2− 1)

(2)

If we were to impose the constraint that the transformation has

to be measure preserving, then it translates to the conditionπr 2 = 4a 2.

From the relation given above we have,

π 1

3

(2 +√

2)2

≈4. (3)

Using the value of√

2 given in the text we get

π ≈ 3.0883, (4)

which is correct only to one decimal place.

V l f√

( )

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Value of√

3 (trikaran . ı )Geometrical construction described by Datta

000111

I II III IV VIVS

III’

IV’

1V

V

VI

VI

2

3

1

2

3

V’

V’

VI’

VI’

1

2

1

2

BA

CD

P

QR

E F

H G

void that remains

to the filled

Each side of the new largersquare APQR = 1 + 2

3 + 1

3.5

So for closer approximation, letthe side of the new square bediminished by an amount y ,

such that

2y

1 +

2

3 +

1

3.5

−y 2 =

1

3.5

2

.

Neglecting y 2 as too small, weget y = 1

3.5.52, nearly.

Thus we get√ 3 = 1 + 2

3 + 1

3.5 − 1

3.5.52

P bl f i i l

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Problem of squaring a circle

1

,

2

3 [BSS 2.10]

With the desire of turning a circle into a square [with the same area]dividing the diameter into 8 parts . . .

2a = 7d

8 +

d

8 −

28d

8.29 +

d

8.29.6 − d

8.29.6.8

(5)

This may be rewritten as

2a = d

1− 1

8 +

1

8.29 − 1

8.29.6 +

1

8.29.6.8

(6)

1 ,

2 ,

3( ) –

Ci i Fi lt

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Citi : Fire altar

– Platform constructed of burnt bircks and mud mortar.

: [the locus] unto which things are broughtinto [and arranged].

(

)= assembling or fetching together

Fire altars are of two types. The ones used for

—daily ritual.

—intended for specific wish fulfilment.

The fire altars are of different shapes. They include (isosceles

triangle), (rhombus), (chariot wheel),

(a particular type of vessel/water jar), (tortoise),

(bird, falcon type), etc.

Number of bricks used is 1000 (

...), 2000, and 3000.

Altar has multiples of five layers, with 200 bricks in each layer.

T f Fi lt

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Types of Fire altars (representative list)

Different types of wish-fulfilling fire-altars are described in Vedas.

,

. . .

, . . . . . .

The table below presents a list some of them, along with theshapes and the purpose as stated in the text.

Name of the citi Its shape Who has to perfom

Form of a bird Desirous of cattle

,

Form of bird Desirous of heaven

Isoceles triangle Annihilation of rivals

Chariot wheel Desirous of region

Form of a trough Abundance in food

Table: Different citis , their shapes and purpose.

On the height and the shape of iti

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On the height and the shape of citis Measurements were case-based (based on the performer) and not ‘standardized’

Taittirıya-sam . hit¯ a , prescribing the height of the citi observes:4

, ,

,

,

Knee-deep should he pile when he piles for the first time,

and indeed he mounts this world with g¯ ayatrı , naval-deep

should he pile when he piles the second time, . . . neck-deep

should he pile when he piles the third time . . .

Elsewhere (5.5.3) observing on the shape of the fire-altar it is

said that it should be akin to the shadow cast by the bird.

5

4Taittirıya-sam .

hit¯ a 5.6.8.5

(BSS.8.5)

S iti Falcon shaped fire altars

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Syenaciti —Falcon-shaped fire-altars

The origin of ´ Syenaciti can be traced back to vedas .

For instance in s . ad . vim . sa br ahman . a belonging to s¯ amaveda ,

. . .

6

Another version of the same statement perhaps on another

Br¯ ahman . a which is more popular goes as

These sentences are cited in the Mım am . s¯ a text in connectionwith the discussion on deciding the meaning of the word syena

that appears in the vidhi (injunction)

6S . ad . vim . sa br¯ ahman . a 4.2.3.

Measurement units used in construction

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Measurement units used in construction

7

a ˙ ngula = 14 an . u or 34 tila

ks . udrapada = 10a ˙ ngula

pr¯ adesa = 12a ˙ ngula

pada = 15a ˙ ngula

prakrama = 30a ˙ ngula

aratni = 2pr¯ adesa = 24a ˙ ngula

vy¯ ay¯ ama = 4aratni purus . a = 5aratni

7Baudh¯ ayana-sulbas utra ,1.3

Construction of Syenaciti: I

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Construction of Syenaciti : ITypes of bricks: 1, 2 and 3

Bricks of geometrical shapes other than rectilinear are needed.

The five types of bricks used:

1. B 1 —one-fourth brick (caturthı )—30× 30 a ˙ ngulas ; i.e., asquare whose side is 1

4 pu .

2. B 2 —half brick (ardh¯ a )—obtained by cutting the one-fourthsquare brick diagonally; each of 2 sides equals a ˙ ngulas and

the hypotenuse 30 30√ 2 a ˙ ngulas 3. B 3 —quarter brick (p¯ ady¯ a )—obtained by cutting B 1

diagonally; each of 2 sides equals 15√

2 a ˙ ngulas andhypotenuse 30 a ˙ ng .

A

B C

D A D

B C

A D

CB

B 32B1

B

30

30 30

3 0

30

1 5

2 2

Types of bricks: 4 and 5

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Types of bricks: 4 and 5

B 4 —four-sided quarterbrick (caturasra-p ady¯ a )—of

sides equal to 22 12 , 15, 7 1

2

and 15√

2 a ˙ ng . The area is

15× 15 a ˙ ng , the same asthat of B 3.

B 5 —(ham . samukhı )- half

brick obtained by joining

two B 4s, along their

common longest side.

E

C

B 4

A

F

✑

215

15

B 5

B

C

E

B

A

D

D

2

30

15

7

12

7

15 2

Outline of body and head of Syena I

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Outline of body and head of Syena I

A

B C

D

G

H

IJ

K

L

240

150

FE

45

45

12

82

60

A

B C

D

30

30

´ Syenaciti: Falcon-shape

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Syenaciti : Falcon-shape

0000000000000

0

0

0

0

0

0

0

0000000000

1111111111111

1

1

1

1

1

1

1

11111111110 00 00 00 00 01 11 11 11 11 10 0 00 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 11 1 10 0 00 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 11 1 1 0 0 00 0 00 0 0

0 0 0

0 0 0

1 1 11 1 11 1 1

1 1 1

1 1 1

0 0 00 0 00 0 01 1 11 1 11 1 1 0 00 00 00 01 11 11 11 1

0 00 00 00 00 01 11 11 11 11 1

0 0 00 0 00 0 0

0 0 0

0 0 0

0 0 0

1 1 11 1 11 1 1

1 1 1

1 1 1

1 1 1

0 0 00 0 00 0 00 0 00 0 01 1 11 1 11 1 11 1 11 1 1

Number of bricks used

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Number of bricks used

Parts of the citi B 1 B 2 B 3 B 4 B 5 Total

Head 1 6 6 1 14Body 30 6 10 46

Wings 30 62 16 108Tail 8 4 20 32

Total 69 72 52 6 1 200

´ Syenaciti: second layer

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Syenaciti : second layerNumber of bricks used in the second layer

Parts B 1 B 2 B 3 B 4 B 5 Total

Head 10 10Body 12 28 4 4 48Wings 48 28 34 110

Tail 8 4 18 2 32Total 68 70 56 6 200

Fabrication of bricks

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Fabrication of bricksIngredients to be added to the mixture of clay employed in manufacting the bricks

. . .

Extracts of gum from certain trees (pal¯ asa )

. . .

Hair of the goat, of a bullock, horse, etc.

8

Fine powder of burnt bricks ..

9

The above process of strengthening is in practice till date.10

8 ´ Satapatha br¯ ahman . a , 6.5.1.1–6.9Baudh¯ ayana ´ Sulbas¯ utra , 2.78–79

10The addition of fly ash as well as pozzuolana is well known in the

manufacture of cement.

Fabrication of bricks

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Fabrication of bricksHandling the contraction in size of the brick (Sun’s heat + Burning in the kiln)

There will be reduction in the size of the moulded bricks:

11

Different ´ Sulbas¯ utra texts suggest different measures to

handle this problem of contraction

12

Appropriately increase the size of the mould.

,

13

Compensate the loss with the mortar.

11M¯ anava ´ Sulbas¯ utra , 10.3.4.1712M¯ anava ´ Sulbas¯ utra , 10.2.5.213Baudh¯ ayana ´ Sulbas¯ utra , 2.60

Constructional Details

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Co st uct o a eta sSpecifications regarding the arrangement of bricks in different layers

Here the word “bheda ” does not simply meandifference/distinction (in fact, this has to be maintained).

What is meant is a clear segregation between two rowsacross all the layers. This is to be avoided.

Joints should be disjoint! (not continuous)

14

The etymology could be:

(Arbitrary) foreign material should not be employed to fill

the gaps.

The above-mentioned are very important principles from the

view point of civil engineering.

14BSS. 2.22–23. (RPK’s Book)


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The purpose for which the geometry got developed in the Indian context

is construction and transformation of planar figures.

We saw that Bodhayana (prior to 800 BC E) not merely listed theso-called ‘Pythagorean’ triplets, but also gave the theorem in the form of

an explicit statement.

Extensive applications of the theorem in the context of scaling and

transformation of geometrical figures was also discussed.

Though ´ Sulbak¯ aras did not explicitly give proofs —which anyway wasNOT a part of their “oral” tradition (of the antiquity)—it is evident from

several applications discussed, that the proof is implicit.

From the view point of history it may also be worth recalling:

Antiquity? Though the Babylonians of 2nd millenium BC E had

listed triplets in cuneiform tablets, there is no generalstatement in the form of a theorem.

Pythagorean? Since there is hardly any evidence to show Pythagoras

himself was the discoverer of the theorem, some of the

careful historian call it Pythagorean theorem.


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It may be reiterated that ´ Sulbas¯ utra texts were primarily meant for

assisting the Vedic priest in the construction of altars designed for the

performance of a variety of sacrifices. However, these texts shed a lot of light from the view point of

development of mathematics in the antiquity, particularly the use of

arithmetic and albegra, besides geometry.

Use of fractions The expressions used by the ´ Sulvak¯ aras for expressing

surds—in terms of sums of fractions, leading to aremarkable accuracy15 —is quite interesting.

Use of algebra The rules and operations described by them in the

context of scaling geometrical figures unambiguously

demonstrate the use of algebraic equation.

The different citis not only speak of the aesthetic sense, but also of the

creativity and ingenuity of ´ Sulvak¯ aras to work with several constraintsimposed—both in terms of area and volume.

The archaeological excavations at Kausambi (UP) seems to have

revealed a ´ Syenaciti constructed around 200 BC E.

15Five decimal places in the case of √ 2.

References

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K¯ aty¯ ayana-sulbas utram , with the comm. of Karka and

Mahıdhara, Kashi Sanskrit Series, No. 120. (1900?)

[digital version from Univ. of Toronto].

B. B. Datta and A. N. Singh, ‘Hindu Geometry’, Indian

Journal of History of Science , Vol. 15, No. 2, 1979,

pp. 121–188. S. N. Sen and A. K. Bag, The ´ Sulbas¯ utras , INSA, New

Delhi 1983.

Bibhutibhushan Datta, Ancient Hindu Geometry, The

Science of ´ Sulba , Cosmo Publications, New Delhi 1993.

T. A. Sarasvati Amma, Geometry in Ancient and Medieval

India , Motilal Banarsidass, New Delhi 1979; Rep. 2007.

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Vedas Sulvasutras2