08 aryabhatiya ii (kr)

7/23/2019 08 Aryabhatiya II (KR)

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NPTEL COURSE ON

MATHEMATICS IN INDIA:FROM VEDIC PERIOD TO MODERN TIMES

LECTURE 8

Aryabhat.ya of Aryabhat.a - Part 2

K. Ramasubramanian

IIT Bombay
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3/24

Area of a triangle

The formula for the area is presented in halfarya:

The product of the perpendicular[dropped from the vertex

on the base]and half the basegives the measure of the area

of a triangle.

Bhaskara in his commentary observes:

1. the termsamadalakot.imeans the perpendicular droppedfrom the vertex on the base of a triangle, i.e., the altitudeof a triangle.

2. itshould not be interpretedas the upright which bisects thebase of the triangle into two equal parts

3. if we do so, the applicability of the above rule will berestricted to equilateral (sama-tryasra) and isoscelestriangles (dvisama-tryasra).
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Area of a triangleBhaskaras commentary: Classification of triangles

He actually commences the discussion with the classification of triangles.

-

-

. . . are of three types:equilateral, isoceles and scalene.

(a) (b)

(c)
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5/24

Area of a triangleBhaskaras commentary (contd.)

,

-

. . .

? 1

Here Bhaskara explains

1. how the singular usagetribhujasyain the verse has to be understood.

2. how the word sarrahas to be understood, and

3. how the compoundsamadalakot.(= the perpendicular dropped from the vertexon the base) has to be interpreted in this context. In fact, hequotes a maximtojustify his interpretation.

1 (Ex. , )
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6/24

Area of a triangle The formula encoded in thearya

6a

may be expressed as

tribhujaphala = bhujardha samadalakot.

Areaoftriangle = 1

2base height(altitude)

= 1

2

b h
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Area of a triangle

The formula ( 12

base height) as given by Aryabhat.acan be employed

only when the altitude is known.

Ifaltitude is not knownandonly three sides are known, then how do we

find the area?

Bhaskara in his commentary presents a method by which we find

segments of the base(BD&CD)calledabadhasand hence thealtitude (p).

It can be shown that (see next slide)

the diff. in theabadhasis given by

Adiff = BD CD

=

c2 b2

a (b,

c:karn. as)

abadhas = 1

2(bhumi Adiff)

altitude =

karn. a

2 abadh a2

B D C

A

c bp

(a = bhumi = sum of abadhas)

abadha abadha
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Area of a triangle [Bhaskaras commentary (contd.)]

Let the sum and difference of theabadhasbe defined as:Asum=BD+CD and Adiff =BD CD. In the triangle ABC,p2 =c2 BD2 =b2 CD2. Hence,

c2 b2 =(c+ b)(c b)= BD2 CD2 =Asum Adiff

Hence, Adiff = (c+ b)(c b)

Asum

= c2 b2

a

Nowabadhas = 1

2(bhumi Adiff)

From theabadhas, thesamadalakot.i

(altitude, p) has to be obtained.

B D C

A

c bp

(a = bhumi = sum of abadhas)

abadha abadha
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Area of a triangleNumerical example in Bhaskaras commentary

In a scalene triangle[one of] the hypotenuse is thirteen, andthe other is

fifteen.The base is two times seven only. O my friend, [please tell me]

what would be the measure of the area of this [triangle]?

Having posed this problem, Bhaskara presents the solution in prose as

follows:

. . .

, ,

,

Now,samadalakot. =

152 92 =12. Also,

132 52 =12; The area

is 12 14 12= 84.
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Area of a circle

The formula given in the followingarya

7a

may be expressed as

phala = parin. ahardha vis.kambhardha

Area = 1

2circumference semi-diameter= r2.

Commenting on the usage of the word eva Bhaskara observes:

,

,

,

. . . Or, [to be more appropriate] by the use of the wordevathemeans is restricted. . . . is the area of the circlethere is no othermeans. This is not true, since3 r2. . . This alternative method isnot accurate. . .
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Area of a trapezium

The latter half of the following verse in Aryabhat. yagives the

formula for obtaining the area of a trapezium.

2

B C

A D

p

b

c

d

f

In this verse, as per the comm. of

Bhaskara, the wordayamameansbreadth. The termsparsvaandvistararefer to thelength(baseand face) of the trapezium.

The formulae presented here are:

c= fp

f+b , d=

bp

f+b , Area=

1

2(f+b) p.

2Dual usage: ,
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Generalized approach to finding area

In the case of all plane figures,having determined the two

sides,the areais [obtained by finding] their product.

HereBhaskarapoints out that the purpose of the verse is:

to present a certainalternative approachby which we

canverify the areas of all planar figurestrilateral,quadrilateral and circlefor which Aryabhat.ahasalready given formulae.

How is it possible to get the area of any planar figure?

- ,

?

Having obtained the two sides.The word pra expresses some

speciality; Thus it means finding the the two sides specially

(prakars.en. a). What is the speciality of the two sides? It is said
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Generalized approach to finding areaIn the case of triangle, rectangle and trapezium

In the case of a triangle or rectangle or trapezium, its area canbe found using the formula:

B C

A D

b

f

h

phala = vistarardha ayama

Area = 1

2(base+face) height

= 1

2(b+f) p

In the case of a triangle (equilateral, isoceles or scalene) we getthe area by puttingf=0.

In the case of a rectangle, we obtain the area by putting f=b.

For a trapezium, the formula is to be used as such.

How do we get the area of a circle in this generalized approach?
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Generalized approach to finding areaThe case of a circle

In the case of a circle it is said:

r (vishkambhardha)

,

In the case of a circle,the

semi-diameter is the length, [and]the

semi-circumference is the breadth orheight, that constitutes the rectangle

( ayatacaturasraks.etram)

Area = vis.kambhardha paridhyardha

= semi-diameter semi-circumference= r r

In his commentaryBhaskaraalso takes up morecomplex caseslikedrum,tusk of an elephant, etc. and demonstrates how to

find the area of those shapes as well.
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Purpose of the verse: Computation or verification?

?

, ?

, ?

, 3

But how does the computation of the area and the verification happen in onestroke?If this rule is meant for verification, then how does it help in

computating the area? And [conversely], if this rule is meant for computation

of area, then how can it be used for verification? There is nothing wrong with

that.What has been designed for one purpose, is found to be serving

another purpose. That is as in the following case Canals are constructed

for the sake of [watering] paddy [field]. And from these [canals] water is drunk

and also used for cleansing. It is similar in this case too.

3 , - - ,

Ch d f i h f h i f f i l
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Chord of one-sixth of the circumference of a circle

O A

B

R

GR/2

C

60

The chord of one-sixthof thecircumference[of a circle] isequal tothe semi-diameter.

It is evident from the figure

BC=chord 60 =R,

sinceOBC is an equilateral triangle.This is what is given in the verse.

It is pointed out byBhaskarathat the purpose of the verse willbecome clear from the later verse samavr. tta-paridhipadam. :

-

A i l f
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Approximate value of

The Sulba-sutra-s, give the value of close to 3.088.

Aryabhat.a(499 AD) gives an approximation which is correct to

four decimal places.

=(100

+4) 8

+62000

20000 = 62832

20000 =3.1416

The same value has been given in Ll avat4 by removing a factorof 8 from the denominator and the numerator.

= 3927

1250=3.1416 thats same as Aryabhat.as value.

4Llavatof Bhaskaracarya, verse 199.

A i t l f
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Approximate value of Bhaskaras edifying discussion while analysing the meaning of the term asanna

?

,

,

5 ,

(

) -

[The word]asannameansclose to or nearby.It is close to what? To

the accurate circumference.How does one understand that it is

close to accurate valueand not the one that is practically used

(vyavaharikasya), since thesupposition of the unstatedis equally valid

in both the cases. There is nothing wrong [in the interpretation]. This

is only a [valid] doubt. [But it must be remembered] that in allinstances of doubts, the following principle gets invoked Clear

understanding arises out of [traditional] explanation (just because

there is a doubt . . . )Therefore we would interpret as . . .

5In the printed edition, the text reads as and it seems to be

an editorial error while splitting

A i t l f
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Approximate value of Bhaskaras edifying discussion while analysing the meaning of the term asanna

6

,

,

, ?

Or else,by the word , that which is close to [the accurate value]

is not conveyed. . . . If an approximation to practical value [is what is

conveyed],then the circumference obtained from that would be even

worse(more gross).No one would make efforts to get something

worse!. Therefore it is established . . . Why is . . . exact circumferenceis not stated?It is considered so [by scholars] there is absolutely

no means by which one can get the exact circumference.

6In the printed edition, the text reads as The

absense of splittingan editorial errorseems to have created a confusion,and hence in Agathe Kellers translation . . .

Constr ction of the sine table
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Construction of the sine-table

A quadrant is divided into24 equal parts, so that each arc bit= 90

24=345 =225.

Aryabhat.apresentstwo different methods

for findingRsin i, (PiNi)i=1, 2, . . . 24.

The Rsines of theintermediate angles

are determined byinterpolation (I orderor II order).

Finding tabular sines: Geometrical approach
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Finding tabular sines: Geometrical approach

May the quadrant of the circumference of a circle be divided [into as manyparts as desired]. Then,from (right) triangles and rectangles, one can findas many Rsines of equal arcs as one likes,for any given radius.

O A

B

R

E

F

G

C

R/2

60

D We know, chord 60 =R=3438

From the triangle OBC,

Rsin 30= BC=R/2= 1719

Now consider the rectangle OABC. Here,AB=Rsin 60 =OC.

OC=

OB2 BC2 =

R2 R2

2 =2978

Thus it may be noted that from 8th Rsine(30) we immediately get 16th Rsine (60)

In other words,Rsin Rsin(90 ).

Finding tabular sines: Geometrical approach (contd )
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Finding tabular sines: Geometrical approach (contd.)

O A

B

R

E

F

G

C

R/2

60

D

In the triangle CBD,BC=Rsin 30 andCD=OD OC=Rvers30 are known.Hence,BD=chord 30

BD =

(Rsin 30)2 + (Rvers30)2,

is known. Rsin 15 = 12

BD=890.

At this stage, we need to note that

Rsin Rcos Rvers

Rsin &Rvers Rsin

2

Now considering the triangle ODE,

OE =

OD2 DE2

=

R2 (Rsin 15)2

givesRsin 75.

Finding tabular sines: Geometrical approach (contd )
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Finding tabular sines: Geometrical approach (contd.)

Most of the Indian astronomers have presented their sine tables by

dividingthe quadrant (90) into 24 parts.

By the principle outlined above, it can be easily shown that all the 24Rsines can be obtained providedthe 24th,12thand8thRsines are

known.

8th 12th

6th

3rd

16th

22nd

23rd

1st

2nd

4th

20th

10th

11th

21st

18th

9th

5th

14th

7th

17th13th

15th

19th

The circumference of the circlewas taken by Aryabhat.ato be

21600 units. From that using the

approximation for given by him,we getR=24thRsine 3438.

Once this it known,it is

noteworthythat in the proposedscheme of constructing the table,

all that is required isextraction of

square root, for which Aryabhat.a

had clearly evolved an efficient

algorithm.

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Thanks!

THANK YOU

More of Aryabhat. yain the next lecture!
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08 aryabhatiya ii (kr)

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