Ws Lecture1

8/10/2019 Ws Lecture1

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DEVELOPMENT OF MATHEMATICS IN INDIA

Lecture - 1

K. RamasubramanianIIT Bombay

ATM WORKSHOP @IIT BOMBAY

February 2013

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Lecture 1 – OutlineMathematics in the Antiquity: Vedas and ´ Sulbas¯ utras

Introduction

Mathematical references in Vedas

Mathematics in the ´ Sulbas¯ utra texts

Bhuj¯ a-kot . i-karn . a-ny¯ aya (Pythagorean theorem?)Finding the cardinal directions

Tools involved in the construction of altarsNeed for the value of surds in ´ Sulbas¯ utras

Approximations to √ 2 & √ 3Derivation of these expressions

Transforming the shapes of geometrical objects

Circle to square and vice versaApproximate value of π

Citinirm¯ an . am : Construction of Fire-altars

Types of Citis ( ´ Syenaciti )Fabrication of bricks and constructional details

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IntroductionBroad classication of Knowledge – Mun . d . aka-Upanis . ad

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Mathematical references in VedasCitations that unambiguously point to the decimal system being in vogue

We nd a list of powers of 10 in the Kr . s . n . a-Yajur-Veda :

. . .

. . . 1

. . .

2

We also nd a list of odd numbers and multiples of four occuring

in Taittirıya-sam . hit¯ a (4.5.11):

. . .

. . .

We also nd a mantra referring to the notion of transnite ( ∞).

. . .

1¯ Aran . yakam 4.69.2Sam . hit¯ a 7.2.49.

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Mathematics in the ´ Sulbas¯ utra textsWhat are these texts, and where do they fall in the Vedic corpus?

One of the prime occupations of the vedic people seem to havebeen performing sacrices , for which altars of prescribed shapesand sizes were needed.

Recognizing that manuals would be of greatly help inconstructing such altars, the vedic priests have composed aclass of texts called ´ Sulba-s¯ utras .

These texts (composed prior to 800 BCE ), form a part of muchlarger corpus known as Kalpas¯ utras that include:

– Employed in rituals associated with societal welfare.

– Rituals related to household. – Duties 3 and General code of conduct.

– Geometry of the construction of re-altar.

3¯ Adi ´ Sa ˙ nkara in his commentary on Upanis . ads denes the term dharma as

anus . t . hey¯ an¯ am . s¯ am¯ anyavacanam .

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Problems dealt with in the ´ Sulbas¯ utra texts

Besides presenting the geometry related to the design of altars(of different sizes and shapes) the ´ Sulba-s¯ utras also presentvarious interesting approximations for surds .

The motivation for presenting estimates of surds could be tracedto the attempts of vedic priests

to solve the problem of “squaring a circle” and vice versato construct a square whose area is n times the area of agive square, and so on.

The expressions for surds presented in the form

N = N 0 + 1n 1 + 1n 1n 2 + 1n 1 n 2 n 3 + . . . ,

can be understood in different ways, two of which namely,Geometrical construction .

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What does the word ´ Sulbas¯ utra mean? The word sulba stems from the root sulb (to measure).

Since all the measurements were done using ropes or chords, inthe very early times, it seems the word in due course wassynonymously employed to refer to the chords themselves.

The etymological derivation of the word sulba (referring to a rope)can be given as:

4

The exact derivation of the compound word ´ Sulbas¯ utras ,including the grammatical peculiarities is:

=

(

+

)5

Seven ´ Sulbas¯ utras , namely Baudh¯ ayana, ¯ Apastamba, K¯ aty¯ ayana,M¯ anava, Maitr¯ ayan . a, V¯ araha and V¯ adh¯ ula are extant today.

4This derivation is technically referred to as karan . avyutpatti .5This type of derivation is known as bh¯ avavyutpatti and the s¯ utra that

comes into play is ‘ bh¯ ave gha˜ n ’.

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Tools involved in the construction of altarsConstructing a perpendicual bisector: Tool that nds application in many contexts

Two commonly employed methods for drawing the perpendicular

bisector of a given straight line were:

(drawing sh-gure)

(folding the cord)

As the former is too well known, and is taught in schools eventoday, we explain the latter as described in ´ Sulbas¯ utras .E

S

W

N

,

,

,

,

– distance between the two pins(along the east-west direction) – by repeating with the string

– hits a nail

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Dertermining the east-west line

Determining the exact east-west line at a given location, is a

pre-requisite for all constructions, be it a residence, a temple, asacricial altar or a re-place.

The procedure for its determination is described thus:

‘

’

[Kt. Su. I 2]

O

X

E

W

N S

A

B

OB − afternoon shadowOA − forenoon shadow

Fixing a pin (or gnomon) on levelledground and drawing a circle with acord measured by the gnomon ,a hexes pins at points on the line (of thecircumference) where the shadow ofthe tip of the gnomon falls. That is thepr¯ acı .

a Astr. Signicance: r > 2OX .

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Time from shadow measurement (not in ´ Sulbas¯ utras )

Figure: Zenith distance and the length of the shadow.

t = ( R sin )− 1 R cos z cos φ cos δ ±R sin ∆ α

∓∆ α.

If φ and δ are known ( ∆ α = f (φ, δ )), then t is known.

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The ´ Sulva (Pythagorean?) theorem

A clear enunciation of the so-called ‘Pythagorean’ theorem – known as bhuj¯ a-kot . i-karn . a-ny¯ aya – is found in ´ Sulva-s¯ utras :6

7

The rope corresponding to the the diagonal of a rectangle makes whatever is made by the lateral and the vertical sides individually.

Terms their meaning – Rectangle (lit. longish 4-sided gure)

– the diagonal rope

– the measure of the lateral side

– the measure of the perpendicular side

6Bodh¯ ayana ´ Sulvas¯ utra 1.48.7 It is an indeclinable that occurs in the Vedic literature:

...

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To construct a square that is n times a given square

Katy ayana 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

[Kt.Sl VI 7]

As much . . . one less than that forms thebase . . . the arrow of that [triangle] makesthat (gives the required number √ n ).

In the Figure BD = 12 BC = (

n − 12 )a . Consider ABD ,

AD 2 = AB 2 −BD 2 =n + 1

2a

2

−n −1

2a

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 circleNeed for nding the value of √ 2 accurately ?

A B

CD

P

M

E

O

W

AB = 2a

OP = r OD = a √ 2ME = a (√ 2 −1)

According to Baudh¯ayana :

= semi-diagonal

= from centre to the east = whatever [portion] remains = with one-third of that

Radius OP = r = a 1 + 1

3(√ 2

−1)

= a

3(2 + √ 2) .

How to nd √ 2?

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Approximate value of π

What would have been the value of π used by Sulvak¯ aras ?

If 2a is the side of the square, we saw that the radius of the circleis given by

r = a 1 + 13

(√ 2 −1) (1)

The vedic priests imposed the constraint that the transformationhas to be measure preserving .

This translates to the condition πr 2 = 4a 2 . Hence we have therelation

π 13 (2 + √ 2)2

= 4 or π ≈3 .0883 , (2)

which is correct only to one decimal place.

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To transform a square into a circleHow did the Sulvak¯ aras specify the value of √ 2?

The following s¯ utra gives an approximation to √ 2:

, , ,

√ 2 ≈ 1+13

+ 1

3

×4 −

13

×4

×34

(3)

= 577

408= 1.414215686

What is noteworthy here is the use of the word in thes¯ 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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Approximation for √ 2Rationale for the expression √ 2 = 1 + 1

3 + 13. 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 partsare 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 thatremains

√

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Approximation for √ 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 + 13

+ 13 .4 −

13 .4 .34

√

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Value of √ 3

000111I 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 remainsto the filled

Each side of the new largersquare APQR = 1 + 23 + 13.5

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

2y 1 + 23

+ 13 .5 −y 2 =

13 .5

2

.

Neglecting y 2

as too small, weget y = 13.5 .52 , nearly.

Thus we get√ 3 = 1 + 23 + 1

3 .5 − 1

3 .5.52

l

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Citi : Fire altar

– Platform constructed of burnt bircks and mud mortar.

: [the locus] unto which things arebrought into [and arranged].

(

)= assembling or fetching together

Shapes of the citis :

(isosceles triangle) (rhombus) (chariot wheel)

(water jar)

(tortoise) (bird, falcon type), etc.

Number of bricks used is 1000 (

...).

Altar has ve layers, hence 200 bricks in each layer

Ci i Fi l

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Citi : Fire altar

Fire altars are of two types. The ones used for

—daily ritual. —intended for wish fullment.

One of the forms of wish-fullling re-altars is constructed in theform of a bird. They are of a few types:

The shape of there-altar is close to that of the shadow cast by the bird while ying.

(BSS.8.5) 8

8

(Taittirıya sam . hit¯ a 5.5.3.2)

S i i F l h d l

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Syenaciti —Falcon-shaped re-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 , . . .

9

Another version of the same statement perhaps on anotherBr¯ ahman . a which is more popular goes as

These sentences are cited in the Mım am . s¯ a text in connection with

the discussion on deciding the meaning of the word syena appears inthe injunction

9s . ad . vim . sa br¯ ahman . a ,4.2.3.

M t it d i t ti

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Measurement units used in construction

10

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 = 24 a ˙ ngula

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

10 Baudh¯ ayana-sulbas utra ,1.3

Constr ction of S iti : 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 ve types of bricks used:1. B 1 —one-fourth brick ( caturthı )—30 ×30 a ˙ ngulas ; i.e., a

square whose side is 14 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 )—ofsides equal to 22 1

2 , 15, 7 12

and 15 √ 2 a ˙ ng . The area is

15 ×15 a ˙ ng , the same asthat of B 3 . B 5 —(ham . samukhı )- halfbrick obtained by joiningtwo B 4s, along theircommon 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

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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 TotalHead 1 6 6 1 14Body 30 6 10 46

Wings 30 62 16 108Tail 8 4 20 32Total 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 TotalHead 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.

11

Fine powder of burnt bricks ..

12

The above process of strengthening is in practice till date. 1311 ´ Satapatha br¯ ahman . a , 6.5.1.1–6.12 Baudh¯ ayana ´ Sulbas¯ utra , 2.78–7913 The addition of y 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:

14

Different ´ Sulbas¯ utra texts suggest different measures tohandle this problem of contraction

15

Appropriately increase the size of the mould.

,

16

Compensate the loss with the mortar.

14 M¯ anava ´ Sulbas¯ utra , 10.3.4.1715 M¯ anava ´ Sulbas¯ utra , 10.2.5.216 Baudh¯ ayana ´ Sulbas¯ utra , 2.60

Constructional Details

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Constructional DetailsSpecications 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)

17

The etymology could be:

(Arbitrary) foreign material should not be employed to llthe gaps.

The above-mentioned are very important principles from theview point of civil engineering.

17 BSS. 2.22–23. (RPK’s Book)

General observations

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General observations

The purpose for which the geometry got developed in the Indiancontext is far different from the context in which it got developedin other civilizations.

The expressions used by the ´ Sulvakaras for expressing surdsseems to be quite interesting . What algorithm ?

The layout and construction of citis are discussed in the texts´ Sulbas¯ utras (Authors are architects of the Vedic age!)

Citis are platforms—having artistic looks—constructed out ofburnt bricks (+ mud mortar).

The archaeological excavations at Kausambi (UP) seems to

have revealed a ´ Syenaciti constructed around 200 BC. The different citis not only speak of the aesthetic sense , but alsoof the creativity and ingenuity of the different ´ Sulvak¯ aras to workwith several constraints imposed.

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