a reservoir of indian theses - value of n and some...
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115
CHAPTER -VI
VALUE OF n AND SOME MATHEMATICAL
CALCULATIONS IN VARIOUS TEXTS
6.1 PREAMBLE
The problem of squaring a circle or circling a square requires
knowledge about the area of the circle, the area of the square, and the
relation between the circumference and the diameter of the circle. The ratio
of the circumference to the diameter of the circle is known as JI.
At the time of Sulbasutras, Sulbakaras were aware of the necessity
to find the value of the ratio of the circumference and the diameter of a
circle which is now known as 71. Aryabatta(499 AD) gave the value of K as
71 =3.1416. At the beginning Indian mathematicians used better
approximations to the value of n.
The Greek letter n indicates the ratio between the circumference of
a circle and its diameter. The exact value of JT is not found in vedic
literature, but the value of K assumed at that time could be found out from
Sulbasutras. The periods of different Sulbasutras are 1) Baudhayana 800
BC 2) Manava 750 BC 3) Apastambha 600BC and Katyayana 200 BC^^^
R.P. Kulkarni, ^"-The value of n known lo Sulhasutrakaras'\ IJHS,Vol-13, 1978, p.32
116
There are evidences to show that the relation between
circumference and the diameter of a circle was known to the people of the
Indus civilization, the Rgvedic period and the people of the Brahmana m
period.
The bullock cart with wheels was known to the people of the Indus
cmWzaiion.^^^ Sphef\ca\ stone weights with f\a\. top and base made of
limestone and quartz were found in the excavation of the city of Mohenjo
Daro and the weights are accurate. This confirms our assumption that
these people might have known the different geometrical properties of a
circle, and the relation between volume and weight.
Chariots of different types were mentioned in the Rgveda^^^. The
number and variety of chariots that were manufactured increased by the
time of the Yajurveda.
At the time of construction, a metallic rim was fitted around the felly.
So it was very necessary to measure the perimeter very accurately
because a very small change in it causes oversize in the rim of the felly or
it is not possible to fit it around the felly. For different types of chariots, the
size of the wheels was also different. The chariot makers might have
known about the relationship between the diameter and perimeter of the
wheel.
" ' Ibid., p.33. ' " Ihid
117
There is a mention of three sacred fires in the Rgveda^^. The shape
of Garhapathya is circle of area one square purusa and the shape of
Ahavaniya is square of the same size^^^. The magnitude of n is necessary
to construct a square of an area equal to the area of a circle. This confirms
the fact that the idea of the value of n might have been developed at the
Brahmana period.
An approximate value of n is given in Baudhayana Sulbasutra as 3.
It is given that the diameter of the yupa is one pada and the circumference
of the pit in which the yupa is to be fixed is three padas^^^
According to Manava Sulbasutra, ^^ the square of two cubit square
is equivalent to a circle of radius one cubit and three angulas.
Area of square = Area of the circle with radius 1 cubit unit
i.e. 2* = 7t [1 + (1/8)] '
Therefore 7t =3.16049
Manava Sulba Sutra also gives the shapes and sizes of the three
sacred fires.^^° The shape of Ahavaniya is square and side length is
1Arafni= 24 Angula. The shape of G'arhapatya is circle and its radius is
13 +7/8 Angula. Daksinagni is semi circular in shape and its radius is IQV?
Angulas. Areas of these three are equal.
' * Rgveda (1.15-12,V. 11.2) (Ref. R.P Kulkami, The value O/K known to Sulbasutrakaras, IJHS,Vol-I3, 1978, p.33)
127
128
Datta B.B, The Sciences of the Sulba.a study in early Hindu Geometry(Ref. Ibid., p.34)
B.Sl.-\-\\3(Ref. Ibid, p.34) ^^^ M.Sl. 1-27, (Ref. . Ibid., p. 37)
''"' Mazumdar N.K,(I922) Mancn'a Sulha Sutram, Journal of dept of letters. Vol. Ill, pp.327-342. University of Culcutta, (Ref Ibid., p.37)
118
Area of Ahavanlya = Area of Garfiapatya => 7c (13 +7/8 y = 576 =>n = 2.99
Area of DaksinSgni = Area of Garfiapatya => n (19.5)^/2 = 576 => TI =3.029 0 m
6.2 APPROXIMATIONS OF K IN THE TRANSFORMATIONS OF CIRCLE
INTO A SQUARE OR SQUARE INTO A CIRCLE
In Katyayana Sulbasutram ^^\ a method is suggested to transform
a square into a circle of the same area. Draw half of the diagonal from the
centre towards a corner. By taking this as the radius draw a circle. By
adding one-third of the radius which lies outside the square with the half
width, draw another circle. This circle will have an area equal to that of the
square.
We can approximate the value of TI from this.
K. SI, 3-13 (Ref. Katyayana Sulbasutra, Ed Khadilkar S.D, p.27)
19
Let ABCD be a square with side x.
OR =1/3 X (X/V2 - x/2 ) + x/2
= X/3V2 + x/3 = x/3 ( 1 + I/V2)
Area of the circle with radius OR = TU [X /3 (1+1/V2)f
=71 Xx^/9 (1+408/577)2
(byB.S.S. 1.61. V2= 577/408)
Area of circle = area of Square. So
71 xx2/9(1+408/577)2 =x2
So 71=3.0852
Another method is transforming a circle into a square. First divide the
diameter of the circle into 15 parts. Then construct a square with side
length 13 parts. ^2 w g can approximate the value of 71 from this.
If d is the diameter of the circle, then area = TI: x (d/2) ^= TC x (d^/4)
Side length of the square = (13/15) d
Area= (13/15)=^ x d^ =(169/225) x d^
' " K.Sl. 3-14 (Ref. Katyayana Sulbasutra, Ed. Khadilkar S.D, p.27) (MSI 10.3.2,13; Ap.SI 36-8,B.S 1-60) (Ref. R.P. Kulkami, The value of 7t Known to Sulbakaras, IJHS, Vol, 13,1978, p.34)
120
Area of Square =Area of Circle. So 71 c|2/4=169 d^/225.
So 71=3.004
Another method is described in Baudhayana Sulbasutra ^ ^ is given
as follows.
To transform a circle into a square, divide its diameter into 8 parts
and then divide one of these parts into 29 parts and subtract 28 of these
and also the sixth part of the preceding subdivision less the eighth part of
the last gives the side of the square of same area.
Side length x = 7d/8 + d / 8x29 + d/(8 x 29 x 6) - d/(8 x 29 x 6x8 )
= 7d/8 + d/(8 X 29) + 7d/(8 x 29 x 6 x 8)
= d/8x (9799/1392)
= 1.759877r
Area of square = Area of circle
x2=7ir2 => 71 = 3.0971
Dvarkanath Yajva, a great commentator of Sulbasutras tried to
improve the approximate value of n and he obtained the value of % as
71=3.157991.^^ Better values of TT as found by Manava is 71 = 3.1604 and
Dvarakanath Yajva is n=3.157991 and n =3.14159. Sulbasutras give the
value of 71 ranging from 3.09 to 3.16.
' " B.SJ. ]-59(Ref. R.P.KuikamiJhe value of w Known fo SuJbakaras, JJHS, Vol. 13, 1978, p.36).
' ^ Karthyayana Sulbasutra, Ch-3, p.67.
121
(In Baudhayana Sulbasutra^^^ there is a description of converting a
square into a chariot wheel. This starts with a very specific design using
bricks of a certain size, but ultimately the design is described in bricks of
entirely different shapes with circular arcs. First start with an area equal to
225 square bricks. Then continue by 64 more bricks so that one has a new
square equal to 17^ equal to 289 bricks. If we equate the square of side 15
units to a circle of diameter 17 we have 289 714 = 225. Therefore, TT =
900/289)
The construction of a circle of diameter 19 from a square of side 17
gives the value of n as 36l7t/4 = 289. Therefore, TI =1156/361. The above
value is very similar to the one value in Satapatha Brahmana which can be
expressed as 900/288.^^ The value of n as it appears in
Manavasulbasutra where the chariot wheel altar uses 344 bricks Instead of
the 289 of Baudhayana Sulbasulra is 1075/344. It is clear that the
representation of a square side 15 by a circle of diameter 17 is the best of
the three approximations of TI.
' " B.Sl, 16.6-11 (Ref. Subash. C. Kak,"rhree Old Indian Values of TU" UHS, 32(4) ,1997 , p.3IO) '^'' ^.^{Ref. Subash. C. Kak,"rhree Old Indian Values OIK" UHS, 32(4),I997.)
122
There ^ ^ is a method of drawing a semicircle, the area of which is equal to
the area of the given square.
D
B
R S
Draw a circle of radius r such that the area of this circle is equal to
the area of the square. Draw another square PQRS circumscribing this
circle. Then, draw another semicircle with OP as radius. Then the area of
this semicircle is equal to the area of the square ABCD. We can deduce the
value of n used in this transformation.
' " M.Sl. (10.1.1.8) (Ref. R.P Kulkami, "The value of Ji known to Sulbasutrakaras", IJHS, Vol-13, 1978,p.37)
Madhyako?j pramaneha mandalam parilekhayet | Atiriktatribhagena sarvam tu sahaniandlam | Caturasre"aksnay3"rajjurmadhyatah saiinipiatayet | Parilekhya tadardhenardhamandalameva tat |
123
Let X be the side length of ABCD. The measure of the radius r is
r = x/2+(1/3)[(x/V2)-(x/2)] = x/3+(x/3V2) = (1/6)I2x+V2x]^^
The length of side of PQRS = 2r
= (2x+V2x)/3
Radius of the semicircle = V2.r
= V2 [(2x+V2x)/6]
Area of semicircle = Area of ABCD => TT. [V2(2X+V2X)/6]^ (1/2) = x
o 7i/2[x(V2+1)/3]=^ = x
7t = 2.9/(V2+1)=^ = 3.088
That is in the transformation of the given square into a semicircle the
value of K assumed in Mahava Sulbasutra is 3.088.
Another method of drawing a circle of area equal to the area of a
square is given in Mahava Sulbasutra^^^. This says that "One shall divide a
square into 9 parts, the segments of the circumscribed circle into 3 parts
(By lengthening these lines), he shall remove the fifth part from the height
of such a lengthened line measured from its middle, draw a circle with this
as the radius". Then the area of the circle is equal to the area of the square.
' ^ M.Sl. 10.1.1.8, Baudhayana Sulbasutra - 1.58 M.Si. 10.3.2. IO(Ref. R.P Kulkami, "The value of TT known to Sulbasutrakaras", UHS, Vol-13, 1978, p.34)
' " M.Sl. (IO.I.8)(Ref. R.P Kulkami, "The value of TT known to Sulbasutrakaras",IJHS,Vol-l3, 1978, p.38) Caturasram navadha kuryaddhanuh kotyastridhatrfdha i Utsedhat parTcamam lumpet purisench tavasamam |
124
R
p
r
0
Q
M
B
Let ABCD be a square of side x. Let O be its centre. Draw a circumscribed
circle. P and Q are points on the side AB such that AP = PQ = QB = x/3.
Let M be the foot of the J_r from O on MR.
Radius of the circle is r = MR - (1/5) MR
Let the side be x, O be the center.
Therefore, AB = x
OA = X/V2 = OS = OR
OM = PQ/2 = x/6
MR = V(0R2-0M^)
MR = OS- TS = OT = V(OR^ - TR= )
= V[(x/V2) 2 - (x/6)']
=^[{xV2) - (x^/36)]
= V[(x^/2)(1-(1/18))]
=V( 17x736)
=(Vl7/6)x
The area of the circle is equal to the Area of the square
^ Tir = x => 71 [MR - (1/5)MR] ^ = x
^ n [(V17/6) X - (1/5) (V17/6) x] ^ = x=
o Tt[(Vl7/6)x]^ [1-(1/5)]^ = x
125
o 71 (17/36) (4/5)^=1
o 71= 25x9/(17x4) = 3.308
The^'"' period of Baudhayana is 800 BC and the period of Katyayana
is 200 BC. The values of TT obtained by different constructions are given
below 141
Sulbasutra
1) B.SI. 1-113
2) M.SI. 1.27
3) M.SI. (Mazumdar)
4) M.SI. 10.1.1.8
5) M.SI. 10.1.8
6) K.SI. 3-13
7) K.SI. 3-14
8) B.SI. 1-60
9) B.SI. 1-59
10)Dvarkanath Yajva
11)Aryabhatiyam2-10
12)B.SI. 2-10
13)Leelavathy(p-277)
14)K.SI. 3-13
15)B.SI. 16-6-11
Value of 71
3
3.16049
2.99,3.029
3.088
3.308
3.0852
3.004
3.004
3.0971
3.157991
3.1416
3.14159
3.1415926535
3.088
3.114
The concept of n is closely related to the concept of the circle. Early
Indian mathematics was developed in accordance with practical need.
R.P Kulkami, 'The value of K known to Sulhasutrakaras\\mSyo\-n, 1978, p.40) Ibid..
126
6.3 VALUE OF 7C IN LILAVATY AND IN YUKTI DEEPIKA
71 is given as a series in Leelavaty.^^^
The perimeter of a circle is obtained by multiplying the diameter by 4 and
dividing this by 1,3,5,... and giving + and - sign alternatively.
i.e.,2 7rr = 4d/1-4d/3 + 4d/5-4d/7 +
= 4d(1 - 1 / 3 + 1/5-1/7+ )
i.e.,7i = 1-1/3 +1 /5 -1 /7 +
This series is now known as Euler Series. But Madhavan (1320 - 1425)
defined this about 200 years before the time of Euler.
Anothei series for TT is found in YuktiDeepika^*^. This is as follows.
The perimeter is given as a series in terms of the diameter. The first term is
obtained by multiplying the diameter of a circle vi -::.The 2 '^ term is
obtained by dividing the 1®' term by 3 next term is obtained by dividing this
term by 3 and so on. These terms are multiplied respectively by 1,3,5 etc.
and adding the term in the odd places and subtracting the terms in the
even places, we get the perimeter of the circle.
i.e., 27rr =Vl2d/1 - Vl2d/(3x3) + Vl2d/(5x32) - Vl2d/(7x3^) +
Therefore, n=Vl2 x (1 -1/3x3+ 1/5x3^-1/7x3^+ )
''*' C.Krishanan Namboodiri, BharathiyaSas{racintaJ99S, Arshaprikasam Prasidheekarana Samithi, Kozhikode, p.255
'" YuktiDeepika -2. 212-214 (Ref. C.Krishanan Namboodiri, Bharathiya SaslracinlaJ99&, Arshaprikasam Prasidheekarana Samithi, Kozhikode, p.257)
127
More accurate value of TI is given in Leelavaty.^^ This says that the
perimeter of a circle of diameter 9x10^^ is 2827433388233. From this we
can find the value of TT as n= perimeter/diameter= 2827433388233/(9x10)^''
=3.1415926535
The approximate circumference of a circle of diameter 20000 is 62832. '*^
The history of Indian mathematics generally begins with the
geometry of Sulbasutras. Thus geometry is related to the construction of
altars described in Satapatha Brahmana. The transformation of a square
altar into a circular one is a part of the construction of altars. Satapatha
Brahmana 7.1.1.118 - 31 describes the construction of a circular
Garhapathya altar using different types of bricks. Garhapathya altar is
circular in shape and Ahavaniya is square in shape and are of equal area.
According to Sulbasutras^'^, their area is equal to one vyama. One
Purusa= one Vyama = Five Aratnis =120 Arigulas. The construction of
Garhapatya altar is using square and oblong bricks. The square bricks are
of 24x24 square arigulas. The oblong bricks are of 48x24 square ahgulas.
So the area of the square in the figure below = 4/5 x 4/5 square purusas. m
Therefore, the diameter of the circle = V [(4/5) * + (4/5) * ]
= V32/5ptymsa
Therefore, radius =2V2 /5
Therefore, area= 7t r = 8 7i/25
'"^ Leelavathy, (Ref. C.Krishanan Naniboodiri, Bharathiya Sastracinta, 1998, Arshaprikasam Prasidheekarana Samithi, Kozhikode, p.259)
"" Aryahhaliya (ganitapada 10) [Scientific Heritage of India, Mathematics, Edited by KG Paulose, published by Govt. Sanskrit College committee Ihripunithura, p. 30]
'•"•Zf.SV 7.4-5 (Ref Subash.C.Kak," Three Old Indian Values of TT" I JHS,32 ( 4 ) , I 9 9 7 )
128
But the area is taken to be 1 square purusa. So 8 7i/25 = 1 or TT
=25/8=3.125
W
6.4 SOME NON INDIAN VALUES FOR 71
6.4.1 Egyptian Values^*^
Some mathematical ideas are obtained from the Egyptian hierographics
(sacred carvings) on tombs and monuments. The main sources of
information are the Egyptian Papyri. One of the problems of Rhind Papyrus
says that the area of the circle with radius 9/2 is equal to the area of a
square with side length 8.
From this,
n (9/2)' = 8'
=> 81 71 = 256
71 = 256/81 =3.16
'• ^ Ramakrishnan, ''The concept of Pi through the ages", M.Phil Dissertation, Dept of Mathematics, CUSAT
129
6.4.2 Chinese Values^**
The Chinese found various values for 71. Chou Pei Suan Ching, the oldest
mathennatical classic, gives the value of 71 as 3. Some other approximations
are VIO, 92/29, 142/45,3.14159 and soon.
6.4.3 Greek Values^^^
The first scientific attempt to compute the value of n is done by Archimedes
of Syracuse. According to him it is 3 + 10/71 < n < 3 + 1/7. Marcus Vitruvius
Pollio, the author of 'De architecture' approximated the value of n used by
Claudius Rolemy of Alexandria and it is 377/120.
6.4.4 Gregory Series and approximations of n^^
The Scotch mathematician James Gregory obtained an infinite series for
tan'^x.
ie, tan-'x = x-(x^/3)+( x^/5)-(x^/7)+
But this was discovered by Madhava about 300 years ago.
When x=1, the series becomes
TT/4 = 1-(1/3)+(1/5H1/7)+ which is known as Leibnitz Series.
From this we can approximate the value of 71.
The result 7id = C = V(12d=^)-[V(12d^)]/3.3 + [V(12d2)]/3 .5 +
""' /hid, ''" Ibid..
"'Ibid.,
130
given in 'Tantrasaiigraha' of Nilakanta Somayaji about 200 years ago is
obtained by putting x = 1/V3 in the Gregory series.
6.5 IRRATIONALITY OF 71 151
Aryabhata I has given the asanna (approximate) value for n as
62832/20,000. From the term asanna, the irrationality of n is implied.
Sangamagrama of Madhava says that "Multiply the square of the diameter
by 12 and extract the square root of the product. This is the first term.
Divide the first term by 3 to obtain the second, the second by 3 to obtain
the third and so on. Divide the terms, in order by the odd numbers 1,3,5.
And so on. Add the odd order term to and subtract the even order terms
*rom the proceeding. The result will give the circumference'.
ie, C = TT D = (Vl2D^) - (Vl2D=^)/3.3 + (Vl2D^)/5.32 - (Vl2D'^)/7.3^ +
Therefore TI =Vl2[1-(1/3.3)+(1/5.32)-<1/7.3=')+ ]
This is an infinite series. The irrationality of TI is recognized from this.
6.6 SOME OTHER INDIAN VALUES
6.6.1 The Value of n in the Mahabharata^^^
The approximate value of n used in the Mahabharata is 3. There are
some verses about TI in the sixth pan/a called Bhisma pan/a of
Mahabharata. This says that the diameter of Rahu is 12000 yojanas and
perimeter is 36000 yojanas.
'" Ibid., ' " Ibid.,
131
Perimeter Diameter
Moon 33000 11000
Sun 30000 10000
6.6.2 The Value of n In the Puranas^^
Brahmanda pumna states that:
nava yojana sahasro vistaro Bhaskarasya tu
vistarat trigunascasya parinahastu mandala".
i.e., 9000 yqjanas is the perimeter of the Sun and three times the
diameter in the circumference in the peripheral circle. Matsya, Aditya and
Vayu Puranas state that the circumference of the circle Is three times its
diameter.
The discovery of Gregory Series is an important factor for calculating
the value of TI.
6.7 SOME IMPORTANT RESULTS GIVEN IN YUKTIBHASA
Mathematics and astronomy flourished in Kerala at a higher level.
Yuktibhcisa based on Tantra Sangraha contains proofs for many
mathematical rules. The mathematicians of Kerala achieved these results
about 300 years before when the Europeans rediscovered them.
The concern for accuracy in the results of computations is reflected in the
calculations.
' " /hid..
132
It is believed that Tantrasamgraham is a revised edition of
"Drugganitapaddhati" oi \/a6assery Parameswaran Namboodiri. The author
of Tantrasamgraham is Kelallur Somayajippad, and was written around AD
1500. Yuktibhasa is based on this text and was written around 1639. Its «
author is Brahmadattan Namboodiri.^^ It is believed that he belongs to the
"Parangode lllam" of Alathur village. The nnethod of illustration in this book
is very appreciable. It describes about the mathematical part of Jyotisastra.
Chapters 6 and 7 describe the theorem of hypotenuse, properties of
triangles, construction of a circle from a square and so on. All these are
applicable in Vastusastra especially in the construction of the roof and in
the construction of Yajnakundas. Yuktibhasa describes many properties
which we have seen now in modern mathematics like the theorem of
hypotenuse (Pythagoras theorem). From these we can assume that there
were many scholars among the ancient Indians.
The fact that "The product of the height of a segment of a circle and
its complementary segment is equal to the square of the half of the
chord^^^" is given in Yuktibhasa.
\ ^ c
w
^
B
' Yuktibhasa, Ed. A.R Akhileswara Iyer, Mangalodayam Ltd. Publishers, Fhrissur
IbuL, ch-7 p. 273.
133
We can prove this using modern theory.
Consider two complementary arcs QAP and QBP. Let O be the centre of
the circle and C be the midpoint of PQ.
Clearly AC is perpendicular to the chord PQ and bisects it, Since AC
passes through O.
Consider AAPB, a triangle inscribed within a semicircle. Therefore
<APB=90°
Therefore AB2=AP^ + PB^ (1)
AAPC is a right angled triangle. Therefore
AP^ =AC^ + CP^ (2)
Substituting in 8quation(1), we have AB^ =AC^ + CP^ + PB^
(AC+CB) 2 = AC ' + CP= +( PC^+CB^)
(From APCB, PB^ = PC^+CB^)
Therefore AC= +BC= +2 AC.BC=AC^+2PC2+BC=^
Therefore AC.BC=PC^
That is the product of the height of a segment of a circle and its
complementary segment is equal to the square of half of the chord. Some
important results in Yuktibhasa are given below.
The common chord of two intersecting circle is perpendicular to the line of
centres. ^^
"" Ibid..
134
The surface area of a sphere is given by the product of circumference and
Its diameter ^ ^ and its volume is given by 1/6 ^ of the product of its
Diameter ^^ These are evident since
Surface area = 4 jir^ = (2 TT r)x 2r
=Circumference x Diameter
Volume = 4/37ur3 = 1/6 x 2 TC r x (2r) =
=1/6 X Circumference x (Diameter)
In modern mathematics, we can prove this using integral calculus.
The area of a circle is given by the product of semi circumference
and the semi diameter. Some formulae and their proofs are given in
Yuktibhasa. Yuktibhasa is based on Tantrasamgraham by Kelallur
Nilakantha Somayajipad. This book gives illustration and proof of the
*
operations and subjects described in Tantrasamgraham. The author unifies
the proofs of subjects of Yuktibhasa which were scattered in different
families and places and transferred heretically. The style of description is
very appreciable. Starting from the basic point and illustrating different
branches based on this point and re-examining these after illustration is a
very good method.
We can see the trigonometrical results,
Sin^A - Sin^ B = Sin(A+B) x Sin(A-B)
' " Ibid.. " " /hid.
135
This also states that the area of a cyclic quadrilateral is the product of
the three diagonals divided by twice the circum diameter^^®.
SinA. Sin B = Sin^ (A+B / 2) - Sin^ (A-B / 2) with the help of the cyclic
quadrilateral. ^^ Let ABCD be the quadrilateral. DE || to BC AE is the third
diagonal.
AM is x'to BC .-.I'to DE. DN x'to BC.
.-.Area of the quadrilateral = V^ BC x AL + V^ BC(DN)
= Va BC (AL + DN) = 72 BC (AL + LM)
= VaBCxAM
from triangle ADE, AM = (AD x AE)/2R, where 2R is the diameter of the
circum circle.
[For AABC a/Sin A = b / SinB = c / Sin C = 2R]
.-.Area of quadrilateral = V BC x (AD x AE)/2R
= 72 BC X AD X AE/Diameter
or Diameter = 14 BC(AD) AE/ Area of the quadrilateral.
Ibid.xh-l pp.228-238. Ibid.., ch-7 pp.224-227.
136
Yuktibhasa also says that the product of the sides of a triangle
divided by the circum diameter is the altitude to the base.^^V
6.7.1 Statement and proof of the formula (a+b)*= a* + 2ab + b*
We can see the statement and proof of the formula (a+b)^ = a^ + 2ab + b
in Yuktibhasa. It is said that to square a number, split this into two and
add twice the product of these numbers to the sum of the squares of
them.^^^
Proof:
There is given a geometric proof for this.
Consider a square of side x+y say D APQS. Let ABND be a square
of side 'x and MNRQ a square of side 'y'.
Area of QAPQS = Area of DABND + Area of D BPMN + Area of D MNRQ
+ Area of D RNDS.
Therefore, (x+y) * = x* + xy +y* + xy
=x' + y* + 2xy^^^
/hid.,ch-l pp. 231
"'" /hid, ch-],p.\9. "'' /bid,ch-],p.2\
I.U
D N
M
0
S R
There is a reference of another formula (a+b) * = (a-b) * + 4ab. i.e., If
four times the product of two numbers is added to the square of the
difference of these two, we will get the square of the sum of these
numbers. We can see a geometric proof for this in Yuktibhasa.
/ ^
s
A 8
5>' c
L Let OP=x and OQ=y.
AM=y, PM=y
Therefore, AB = x-y,
SM = x = QN
a
A/
M
138
Therefore, Area of D PQRS = Area of u ABCD + Area of D POBM +
Area of D OQNC + Area of U NRLD + Area of D LSMA
Therefore, (x+y)' = (x-y) * + xy + xy + xy + xy
= (x-y) ' + 4xy
6.7.2 Statement and proof of the Formula x*+y' - 2xy = (x-y)^.
Another property given in Yuktibhasa is that if we subtract twice the product
of two numbers from the sum of their squares, we obtain the square of their
difference.
x*+y 2 - 2xy = (x-y)^ ^^
The geometrical proof given in Yuktibhasa is given below.
p s •
Let PQ = x+y
PB = x = CQ
BQ = y = AP = CR = DS
Area of rABCD = Area of nPQRS - Area [AAPB - ABQC - ACDR - AASD]
=Area of nPQRS-Area of [ 72 r]PBNA+ Yz nBQCL + 72; iCTDR + 72 i AMDS]
/hid.
139
= (x+y) * - V2 [xy + xy + xy + xy]
= (x+y) * - 2xy
= X* + y*
from the above formula.
Area of D MNLT = Area of D ABCD - Area of AABN - Area of ABCL - Area
of ACTD - Area of ADAM
= Area of OABCD -Area of[ Ya D PBNA + 72 DBQCL + V^ nCTDR + VzU MDS]
i.e. (x-y) * = (x* + y*)-[ !/ xy + V xy + Vaxy +Vixy ]
= X* + y* - 2xy
6.7.3 Some trignometrical results in Yuktibhasa
Another property given in Yuktibhasa is that, the difference of the
squares of the two sides of a triangle is equal to the difference of the
squares of their projections on the base.^^
We can prove this using trigonometry. B
A
Let ABC be a triangle and let AB = x, BC = y. Let AP be the projection of
AB on the base AC and PC be the projection of BC on the base AC.
Clearly ABP and BCP are right angled triangles. Then
/hid, ch-\,pJ5.
140
BP2 = AB2-AP^ fromAABP
Also, BP^ = BC^ - PC * from BPC
Therefore, AB^ - AP^ = BC^ - PC^
Therefore, AB' - BC^ = AP= - PC^
i.e., the difference of squares of these two sides is equal to the difference
of squares of their projections on the base. Condition of similarity of right
angled triangles given in Yuktibhasa is given below:'^
(1)Two right angled triangles are similar if the hypotenuse {kamam)
and one side {bhuja) of one triangle is parallel to the hypotenuse and
one side of the other respectively or one side (koti) and hypotenuse *
of one triangle is perpendicular to one side (koti) and hypotenuse of
the other triangle respectively.
(2) The three sides of one triangle are perpendicular to the three sides
of the other respectively.
(3) Two triangles are similar if the three sides of one triangle are parallel
to the three sides of the other triangle respectively.
From this it is clear that ancient people in India were awarw of the
basic axioms of trigonometry. They used these properties for the
measurement of the rafters etc. in the construction of roofs.
Another property seen in Yuktibhasa is that if two sides of a
triangle are equal, the perpendicular from the vertex bisects the base. If
they are unequal, the foot of the perpendicular is nearer the shorter side. ®''
"^ Ibid., ch-6, p.89
//7/J.,ch-7,p.l44.
141
This property was indicated in the process of inscribing a hexagon
of sides equal to the radius within a circle. We can prove this using the
simple properties of trigonometry.
A
B M C
Let ABC be a triangle in which AB = AC. Let M be the foot of the
perpendicular from A on BC. ClearlyWBM and AACM are right angled
triangles. By the theorem of hypotenuse
BM2 = AB»-AM2
MC* = A C - AM*
If AB= AC, then clearly BM* = MC^
Therefore, BM = MC.
If AB < AC, then BM < MC. ie, the foot of the perpendicular M is nearer to
the shorter side AB.
Another property given in the process of inscribing a hexagon
within a circle is that six chords of length equal to the radius can be placed
in order inside a circle, i.e., the side length of the hexagon inscribed within
a circle is equal to the radius of the circle.
We can give a simple proof for this.
142
D
Consider AOBC, where OB = OC = radius of the circle.
Clearly <BOC = 360/6 = 60.
Since AOBC is an isosceles triangle, we have the opposite angles of the
equal sides equal.
Therefore, <B + <C = 180 - <BOC
= 180-60=120
Therefore, <B = <C = 60.
Therefore, AOBC is an equilateral triangle and therefore BC = OB = OC.
Therefore, BC is the radius of the circle.
Yuktibhasa gives the area of the triangle as half of the product of the
base and altitude. ®® There is also given a proof for this.
"•" JhiJ., ch-7, p.222.
143
B C S O R
Let ABC be a triangle. Let M be the mid point of AB, and N be the
mid point of AC. Draw a line through M perpendicular to BC and a line
through N perpendicular to BC. Choose points P and Q on a line parallel to
BC through A, such that PQRS is a rectangle. Let S and R be the feet of
the perpendiculars on BC respectively. Clearly triangles PAM and SBM are
similar, (three sides parallel). Also BM = AM Therefore, corresponding
sides are equal. So these two triangles are equal triangles. Similarly, AAQN
= A CRN. Cut AMBS and paste it at the place of APAM with the vertices B
at the position of A and S at the position of P. Similarly, cut A RCN and
paste it at the place ofA QAN with C at the position of A and R at the
position of Q. We obtain a rectangle PQRS.
Therefore, Area of AABC = Area of n PQRS = PS x RS = AG x RS.
The triangles ABMS and ABAC are similar. (Since three sides are parallel.)
Also,
BM = V2 BA.
Therefore, BS = Vz BO
Similarly, CR = V^ CO
Therefore, ^S - CR = 'A (BO + CO)
144
(BC-SR) = 72 BC
.•.SR = y2BC
So the Area of the AABC = AO x RS = AO x 72 BC = Vi BC AO
= V2 base X altitude
Another reference of the triangle inscribed in a semi-circle is that the
chords of any two arcs of a semi-circle are mutually perpendicular.^®^ In a
triangle, the product of the two sides divided by the diameter of the
circumcircle gives the altitude.^^°
We can prove this using some formulae of trigonometry.
The altitude of the AABC is CP, where P is the foot of the perpendicular
from C on AB. From the right angled triangle ACAP.Sin A=CP/AC.
Therefore, CP=AC SinA = b SinA
But from the trigonometric formula a/SinA = b/SinB = c/SinC = 2R where
a=BC, b= AC, c= AB and R is the radius of the circumcircle.
Therefore, a/2R = SinA
'•'' Ihid., ch-7, p.229.
™//)/J., ch-7, p.23l
145
Therefore, CP = b SinA = b x a/2R = ab/2R
ie, altitude = product of the two sides / diameter of the circumcircle.
Another property given in Yuktibhasa about the cyclic quadrilateral is
that the sum of the product of the opposite sides of a cyclic quadrilateral is
equal to the product of the diagonals.^^^
6.8 SOME MATHEMATICAL CALCULATIONS IN B A K S H A L T
MANUSCRIPT
The Bakshali Manuscript is a mathematical work found at Bakshaii,
a village of Peshawar district of the North West Frontier Province of India in
1881. It was written on brich-bark, and each layer of the bark Is white or
pinky white on the outer side, but is a reddish or yellowish buff on the inner
side. The language used in this text is an irregular Sanskrit. The beginning
and end of the manuscript was lost. The author and name of the work were
unknown. It contains problems involving systems of linear equations,
indeterminate equations of the second degree, arithmetical progressions,
quadratic equations, approximate values of square roots and so on. It
contains a large variety of problems relating to our daily life.
' " Ibid., ch-7,p.233
146
This consists of Sutras (rules) and examples. There is given a
method of extracting square roots in this text. The method gives the first
approximation for the square root of
Q=A» + b is A+(b/2A)=qi
The first error qi*-Q=(b/2A) '' = e^.
The second approximation for the square root of Q is
VQ = V (qi» - ei) = qi-(ei/2qi)
=[A+(b/2A)] - (b'/4A^)/(2[A+(b/2A)])
=[A+(b/2A)] -(b'/4A2)[A/(2A^+b)]
= A+ (b/2) [(1 /A)-(b/4A^+2Ab)]
=A +[(4A^b+b2)/(8A='+4Ab)]
The second en-or 62 = (ei /2qi)* ^
There are given some examples.
V41 =V(36+5), qi = 6+5/12. 61=45/144, q2=6 + 745/1848
V481= V(21» + 40)(, qi=21 40/42, ei/8 = 1600/14112, q2=21+ 9020/9681
We can compare these with modern values
Q qi q2 q (approximately)
41 6.41667 6.40313 6.403124
481 21.9524 21.9307 21.9317122
The first one is correct upto four decimal places and the second one
is correct upto two decimal places.
' ^ KAYE G.R, ""The Bakshali Manuscript, A Study in mediaeval mathematics", Cosmo Publications, New Delhi,l98l, p.30.
147
6.9 GENERAL TRIGNOMETRICAL RESULTS
There is a reference of the expression for the circumradius in
Yuktibhasa^^^ which says that the product of two chords divided by the
diameter is equal to the altitude from the point of intersection of the chords
to the line joining their ends^ '*. The theorem "Perpendicular bisectors of the
sides of a triangle are concurrent " is proved for the equilateral triangle
under Aryabhatiya Ganitapada 6^^^
From the fourth Century A.D Indian astronomers gave very large
importance to the construction of Jya^table (Sine - table). It was necessary
to calculate the position of planets accurately.
C
The trignometrical functions Jya^ (PM), Kojya (OM) for a small arc PB is
defined as
Jya A =PM = R SinA, Kojya = OM=R CosA
There is a reference of the value of Jya"30°(Sin 30°) and Jya~60°(Sin 60°)
in Pancasiddhantika^^^ without giving their derivation. They are as follows:
' " Ibid, pp.244-246. ''" /bid, p.23\.
'^'Aryabhatiya-Ganitapada, TrivanJrumSanskritSeries,]930,versus 14-16 '/2
(Ref. T.A Saraswathy, "Development ofMathamatical ideas in India", IJHS, Volume 4, Nov-1969)
''"' Pancasiddhantika,Chapter 4 V-2 (Ref. AmulyaKumar Bag, "SINE TABLE IN ANCIEN I INDIA", National Institute of Science of lndia,IJHS.Vol-4,l969.)
148
"Square of the radius R is to be defined constant. The one-fourth part of
that is the square of the Aries. The square root of the two quantities- the
square of the Aries and the Aries lessened from the constant are the Jyas
of 30° and 60° respectively"
ievyya30°=A/(R2/4)=R/2
Jya 60°=V[RMR^/4)] =(V3/2)R.
Another result is^^ "Twice of any desired arc is subtracted from 90°,
the Jyaoi the remainder is subtracted from the radius. The square root of
the result multiplied by half the radius is the Jya of that arc. By deducing
that square from the constant R , the square of the Kojya is obtained,
i.e. 1) (Jya A) ^ = R/2 [R-Jya(Tr/2 - 2A)]
(Jya A) ^ = R/2 [R-Kojya 2A] and
2) R2-(JyaA)=^ = [Jya{TT/2 -A) ]^
i.e. R^-(Jya A) 2 = {Kojya A) ^
or (Jya A) ^ + {Kojya A) ^ = R^
These two results arc equivalent to the identities SinA^ =(1/2) [1-Cos2A]
and Sin='A+Cos='A=1
Mahavira^^^ in the 9* century refers to the circle inscribed in
triangles and in quadrilaterals giving a formula for radius as radius = area /
half the perimeter.
The point of intersection of the Xr bisectors of the sides is the incentre.
' " Pancasiddhantika, Chapter 4 V-2 (Ref. AmulyaKumar Bag, "SINE TABLE IN ANCIENT INDIA", National Institute of Science of India,IJHS.Vol-4,1969.)
' '* T.A.Saraswathy, Develupmeni of Mathematical ideas in India, IJHS, May-Nov. 1969, p.70.
149
Total area of A ABC = area of AQOR + Area of BPOR + Area of CPOQ
.-. Area of AQOR = Area of A A R O + Area of A AOQ
= 72 R O X AR + 72 OQ x AQ
= 72 r (AR + AQ)
B P C
.-.Area of AABC = Ya r (AR+ AQ + BR+ BP + CQ +CP) = 72 r (a +b+ c)
Sulbakaras were aware that the volume of the regular solid with opposite
faces parallel, is the area of the base multiplied by the height.
Brahmagupta^'^^ knows the volume of a pyramid is 1/3' ' of the volume of a
prism with the same height and base. It was the astronomical need of
knowing the circumference, chords and arc of the circle accurately. The
main reason for this achievement was their great interest in series.
6.10 KNOWLEDGE OF GEOMETRY IN INDUS CIVILIZATION
It is clear from the archaeological investigations that the geometry of
Mohenjo Daro and Harappa was well developed. They^^ used a slip of
shell marked with division of 0.67 cm. The error of the scaling is only
0.00762 cm. From the measurement of staircases of buildings of Mohenjo
Daro, it is evident that the fundamental operations like sum subtraction,
" ' T A Saraswathy, Mahaviras trealmcnt of series. Journal of Kanchi University 1,1962, p.43 (Ref: Development of Mathematical ideas in India , T. A Saraswathy, Indian Journal of History Of Science, May-Nov. 1969 )
"'° K.P Kulkami, "Geometry as known to the people of Indus civilization", IJHS Vol-13,1978
150
multiplication and division were known to them. The bricks used for
construction of wells are of trapezium shape. This is because the thickness
of the joints between bricks should not be increasing with the increase in
the radius of the circular wall of the well. This is an example of a good
engineering practice.
6.10.1 Knowledge of Pythagoras^®^
The dimensions of the great bath are given as 11.99 meters long on
the west side and 11.96 meters, long on the east side 6.98 meters broad
on the south side and 6.87 meters broad on the north side. The difference
between the lengths of the opposite sides is 2.54 cm and 10.9 cm
respectively. The difference is very small compared to the other layouts of
rec^ngle brick works. This is an example of their knowledge of drawing a
right angled triangle on the ground. From the well laid out street plans and
from the arrangement of buildings, it is evident that drawing of similar
figures was known to the people of the Indus civilization.
The experiments conducted in Sindh, Baloochistan and Panjab
give knowledge about a civilization 4000 or 5000 years ago. More than
500 seals were obtained. These seals are made of sand and clay and ot
small in shape. Beautiful pictures were sculptured on this.
"" Ibid.,
151
6.10.2 Knowledge about the Circle
The most popular design on pots is composed of a series of
intersecting circles^^^. We can see a drawing of a copper vessel and a
circle containing a flower with four petals such that all circular parts of the
copper vessel are concentric. Some circular stone rings with concentric
rings marked on them were found at Mohenjo Daro of height 24.892 cms to
28.448 cms.^^^
From these we can assume that they knew about the construction of
a circle, how to draw a perpendicular to a given line at a given point,
bisecting a given angle and so on.
The scales^®^ and measures used in Sindhu river ci\'ilization
(culture) are very accurate. They used burnt bricks for construction; and
their measures are 11x5.5x2.5 inches and 9.2x4.5x2.2 inches. Modern
science agrees that these measures are very suitable for the construction
of walls.
'"^ Mackey, B. (1948) '"Early Indus Civilisation", Luzac and Co Limited,London,2™' Edition
(Ref. K.P Kulkami, "Geometry as known to the people of Indus civilization", IJHS Vol-13,1978)
'*' K.P Kulkami, ""Geometry as known to the people of Indus civilization'", IJHS Vol-13,1978
"'' Dr. P.V. Ouseph, "Vaslusastram oru sumagrapadanam" Published by D.C Books, Kottayam.