vedas sulvasutras2
TRANSCRIPT
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 1/35
NPTEL COURSE ON
MATHEMATICS IN INDIA:FROM VEDIC PERIOD TO MODERN TIMES
LECTURE 3
Vedas and Sulbasutras - Part 2
K. Ramasubramanian
IIT Bombay
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 2/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 3/35
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.
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 4/35
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.
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 5/35
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.
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 6/35
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.
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 7/35
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.
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 8/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 9/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 10/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 11/35
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?
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 12/35
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.
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 13/35
Approximation for√
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 14/35
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 +
1
3 +
1
3.4 − 1
3.4.34
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 15/35
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√
( )
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 16/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 17/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 18/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 19/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 20/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 21/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 22/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 23/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 24/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 25/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 26/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 27/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 28/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 29/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 30/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 31/35
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)
General observations
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 32/35
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.
General observations
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 33/35
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
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 34/35
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.
Thanks!
8/21/2019 Vedas Sulvasutras2
http://slidepdf.com/reader/full/vedas-sulvasutras2 35/35
THANK YOU