Usually it is customery to regard Usually it is customery to regard Lilawati, Lilawati, Bijaganita,Grahaganita and Bijaganita,Grahaganita and Goladhyaya Goladhyaya as the four parts of the as the four parts of the Siddhanta Siromani Siddhanta Siromani to make it a to make it a comprehensive treatise of comprehensive treatise of Bhaskaracharya’s Bhaskaracharya’s Hindu Hindu mathematical science. His two othermathematical science. His two other work are the work are the Karana-Kutuhala and Karana-Kutuhala and Sisyadhivrddida- tantra.Sisyadhivrddida- tantra.

JyotpattiJyotpattiThe Goladhayaya treats of a topic The Goladhayaya treats of a topic

called Jyotpatti at two places. The first called Jyotpatti at two places. The first occasion is the fifth chapter, called Sphuta-occasion is the fifth chapter, called Sphuta-gati-vgati-vāāsansanāā, where the first six stanzas are , where the first six stanzas are devoted to it. These contain some basic devoted to it. These contain some basic defination and rules of Hindu trignometry. defination and rules of Hindu trignometry. The VB on these ends with the words.The VB on these ends with the words.

atovsistām jyotpattimaagre vaksyāmahatovsistām jyotpattimaagre vaksyāmah‘‘The remaining Jyotpatti we shall deal ahead’The remaining Jyotpatti we shall deal ahead’

True to this promise , the authorTrue to this promise , the author dealt with Jyotpatti again at the end of dealt with Jyotpatti again at the end of SSG. This Jyotpatti work consists of 25SSG. This Jyotpatti work consists of 25 Sanskrit stanzas and may be considered Sanskrit stanzas and may be considered to be the last (i.e.14to be the last (i.e.14th)th) chapter of SSG,but chapter of SSG,but it is better to regard it as an appendix toit is better to regard it as an appendix to SSG which,srictle speaking and SSG which,srictle speaking and according to auther’s own statement, according to auther’s own statement, starts with the word starts with the word siddhi siddhi and end with and end with the word the word vrddhimvrddhim after which only the after which only the jyotpatti jyotpatti starts.starts.

The Jyotpatti tract ( of 25 stanzas) has The Jyotpatti tract ( of 25 stanzas) has manymany

trignometrical riles which make their first trignometrical riles which make their first appearance in India through this work.appearance in India through this work.

These include:These include:

1)The exact values of Sines of 181)The exact values of Sines of 18° and 36°° and 36°

2) Addition and subtraction Theorems for the 2) Addition and subtraction Theorems for the function.function.

Lilavati was translated by Abdul-Fayd Faydi in Lilavati was translated by Abdul-Fayd Faydi in

1587 AD, Bijaganita by Ata Allah Rushdi in 1587 AD, Bijaganita by Ata Allah Rushdi in

1635AD, the Zij-i-Sarumani (1797AD) by Safdar 1635AD, the Zij-i-Sarumani (1797AD) by Safdar

Alikhan is presumably the persian translation ofAlikhan is presumably the persian translation of

Siddhanta SiromaniSiddhanta Siromani..

..JYA: The Indian sine Function..JYA: The Indian sine Function

One of the great contribution of India to One of the great contribution of India to world mathematics is the invention of the world mathematics is the invention of the basic trignometric function called sine(jyā). basic trignometric function called sine(jyā). The motivation for this came from The motivation for this came from discussions on astronomy regarding the discussions on astronomy regarding the position of planet on the celestiaal circle. position of planet on the celestiaal circle. The superior Indian sine not only drove out The superior Indian sine not only drove out the greek chord but also gave rise to a the greek chord but also gave rise to a fully develped science of trignometry.fully develped science of trignometry.

A

B

C

D

O

M P

P'

a

θ O

P

P'

θ

Fig 1

Fig 2

N

A

In fig 1 there is a circle of arbitary radius In fig 1 there is a circle of arbitary radius R. take anyarc AP=a, let PN be the R. take anyarc AP=a, let PN be the perpendicular from P on the radius OA. perpendicular from P on the radius OA. Then PN is called the Indian sine (jyā) of Then PN is called the Indian sine (jyā) of the arc AP. Let PN produced meet the the arc AP. Let PN produced meet the circle at P’. Then the geometrical figure circle at P’. Then the geometrical figure PAP’P resembles the shape (Fig 2) of a PAP’P resembles the shape (Fig 2) of a bow with string. That is why the Indians bow with string. That is why the Indians called the arc (of a circle) as called the arc (of a circle) as capa (a bow) capa (a bow) and the chord as and the chord as jyjyāā or jiv or jivā ā (bow string).(bow string).

. . Since the Indian sine PN of the arc AP is Since the Indian sine PN of the arc AP is half of the chord PP’ it was called half of the chord PP’ it was called ardhyajayardhyajayā ā (half chord) and PP’was often (half chord) and PP’was often called purnajya. called purnajya. SuffixSuffix ardha was dropped ardha was dropped and the Indian sine was commanly called and the Indian sine was commanly called jyā.jyā.

Jyā AP= (chord PP’)/2Jyā AP= (chord PP’)/2Hence ,”The Indian sine of an arc in a circle Hence ,”The Indian sine of an arc in a circle

was defined as the length of half the chord was defined as the length of half the chord of twice the arc.”of twice the arc.”

Other ancient Trignometrical functions wereOther ancient Trignometrical functions were also defined. The lenghth PM(=ON) was also defined. The lenghth PM(=ON) was called called KotijayaKotijaya (cosine) of arc AP. It (cosine) of arc AP. It corresponds to the sine of the complimentrycorresponds to the sine of the complimentry arc BP. The length NA is called the arc BP. The length NA is called the Utkrama –Jya(versed sine) of the arc Utkrama –Jya(versed sine) of the arc a.a.JyJyā ā a= R Sina= R Sinθθ;; Koti JyKoti Jyāā= R = R

Cos Cos θθ..UtkramaJyUtkramaJyā ā = Versed R Sin= Versed R Sinθθ=R- R Cos =R- R Cos θθ..

When the radius of the circle ofWhen the radius of the circle of reference is unity (i.e. R=1). Thereference is unity (i.e. R=1). The ancient Indian Jyā and Kotijyā ancient Indian Jyā and Kotijyā become equal in value to the modernbecome equal in value to the modern sine and cosine respectivly. But in sine and cosine respectivly. But in one respect the ancient definations one respect the ancient definations as lenghth have an advantage overas lenghth have an advantage over the modern definations. In case of the modern definations. In case of angle angle θθ being equal to 90°, the being equal to 90°, the modern sine and cosine can be modern sine and cosine can be defined only as limiting cases(as no defined only as limiting cases(as no traingle can be formed when traingle can be formed when θθ=90°) =90°) while the ancient Indian definationwhile the ancient Indian definations present no difficulty and directly gives present no difficulty and directly give

JyJyāā (c/4)=R Sin (c/4)=R Sin 90°=BO=R90°=BO=RKotijyKotijyā ā (c/4)=R Cos (c/4)=R Cos 90°=0 where c is 90°=0 where c is the the circumference of the reference circle circumference of the reference circle and and versed R Sin 90°= OA=Rversed R Sin 90°= OA=R

Thus the original indian Mathematical termThus the original indian Mathematical term Jyā Jyā and and jivā ,jivā ,after its long journy ,to west after its long journy ,to west Asia, Spain and England came back to Asia, Spain and England came back to India, after more than thousand years, India, after more than thousand years, during the British rule as during the British rule as sine.sine.

The Jyotpatti (the science of calculation for the construction of Jya of an arc i.e. trigonometry), on acquiring the knowledge of which the rank of acharya is conferred on an astronomer is told for the gratification of various scholars in mathematics by Bhaskara II. In verses 2 and 3 Bhaskara II described the graphical method of obtaining various sine values by actually drawing the circle by taking desired radius in angulas.

Hence the Jya of an arc may be defined as the length of half the chord of twice the arc

Now I describe alternative method based on Now I describe alternative method based on mathematical calculations to know Jyā(R mathematical calculations to know Jyā(R sine) of various arc and the accurate sine) of various arc and the accurate measure of them.measure of them.

The square of the radius is diminished by The square of the radius is diminished by the square of R sine of an arc the square the square of R sine of an arc the square root of the result certainly equal to kotijyā (R root of the result certainly equal to kotijyā (R cosine) of the arc.cosine) of the arc.i.e.√R² - (jyāi.e.√R² - (jyāάά) ²=Kojyā) ²=Kojyāάά……..(1)……..(1)

Subtract from the radius the direct R sine of an arc Subtract from the radius the direct R sine of an arc and of its complement, the result will be the versedand of its complement, the result will be the versedR sine( utkarmajya) of the complement and the R sine( utkarmajya) of the complement and the arc. Subtract from the radius the versed R sine of arc. Subtract from the radius the versed R sine of R and of its complement, the remainder will be R and of its complement, the remainder will be sine of complement and the arc.sine of complement and the arc.i.e. R – jyā i.e. R – jyā αα = utjya(90 - = utjya(90 - αα ) ) R – jyā(90 - R – jyā(90 - αα ) = utjya ) = utjya αα where where αα and 90 are the length of arc of radius R and 90 are the length of arc of radius R subtaining angle of subtaining angle of θθ and 90 respectively at the and 90 respectively at the centre of the circle.centre of the circle.

Now I describe alternative method based on Now I describe alternative method based on mathematical calculations to know Jya(R sine) of mathematical calculations to know Jya(R sine) of various arc and the accurate measure of them.various arc and the accurate measure of them.

The square of the radius is diminished by the The square of the radius is diminished by the square of R sine of an arc the square root of the square of R sine of an arc the square root of the result certainly equal to kotijya (R cosine) of the result certainly equal to kotijya (R cosine) of the arc.arc.

i.e. √ R² – (jyāi.e. √ R² – (jyāαα) ²) ² = Kojyā= Kojyāαα ……..(1) ……..(1)

Subtract from the radius the direct R sine of an arc Subtract from the radius the direct R sine of an arc and of its complement, the result will be the versedand of its complement, the result will be the versedR sine( utkarmajya) of the complement and the R sine( utkarmajya) of the complement and the arc. Subtract from the radius the versed R sine of arc. Subtract from the radius the versed R sine of R and of its complement, the remainder will be R and of its complement, the remainder will be sine of complement and the arc.sine of complement and the arc.i.e. R – jyā i.e. R – jyā αα = utjya(90 - = utjya(90 - αα ) ) R – jyā(90 - R – jyā(90 - αα ) = utjya ) = utjya αα where where αα and 90 are the length of arc of radius R and 90 are the length of arc of radius R subtaining angle of subtaining angle of θθ and 90 respectively at the and 90 respectively at the centre of the circle.centre of the circle.

““Half the radius is (equal to) the jya Half the radius is (equal to) the jya (Rsine) of the arc of 30˚ ;and it is (also equal (Rsine) of the arc of 30˚ ;and it is (also equal to) kotijyā (Rsine) of the arc of 60˚.to) kotijyā (Rsine) of the arc of 60˚.

Square root of the half the square of the Square root of the half the square of the radius (or the jya of the arc of the three signs) radius (or the jya of the arc of the three signs) is thw jya (rsine) of the arc of 45˚ (and hence is thw jya (rsine) of the arc of 45˚ (and hence the Kojya or Rcosine of the arc obtained from the Kojya or Rcosine of the arc obtained from that is also equal to that itself)”. That is, jya 30˚ that is also equal to that itself)”. That is, jya 30˚ =(R/2)=Kojya 60˚ or Rsin30˚ = (R/2)=RCos60˚ =(R/2)=Kojya 60˚ or Rsin30˚ = (R/2)=RCos60˚ jya45˚ = √(R²/2) or Rsin45˚ = R/ √2 jya45˚ = √(R²/2) or Rsin45˚ = R/ √2

By inscribing regular polygons in a By inscribing regular polygons in a circle, the Hindu astronomers could get circle, the Hindu astronomers could get the value of jyas of the arcs of 30, 45, 60, the value of jyas of the arcs of 30, 45, 60, 36 and 18. These are called the five 36 and 18. These are called the five fundamental jyas. The rationale of the fundamental jyas. The rationale of the above formulae given by Bhaskaracarya is above formulae given by Bhaskaracarya is as follows. as follows.

““The side of a regular hexagon inscribed in a circle is The side of a regular hexagon inscribed in a circle is equal to its radius; this is well known and has been equal to its radius; this is well known and has been

stated in (my) Arithmetic. Hence follows that the Rsine of stated in (my) Arithmetic. Hence follows that the Rsine of 30˚ is half the radius. 30˚ is half the radius.

““The Rsine of 60˚ is equal to the Rcosine of 30˚, the The Rsine of 60˚ is equal to the Rcosine of 30˚, the Rsine of which is equal to half the semi-diameter”.Rsine of which is equal to half the semi-diameter”.

“The square root of five times the square of the “The square root of five times the square of the square of the radius is subtracted from five times the square of the radius is subtracted from five times the square of the radius and tha remaineder is divided by square of the radius and tha remaineder is divided by eight ; the square-root of the quotient is the R sine of eight ; the square-root of the quotient is the R sine of

36˚. 36˚. “ Or the radius multiplied by 5878 and divided by “ Or the radius multiplied by 5878 and divided by 10000, is the Rsine of 36˚. The RCosine of that is 10000, is the Rsine of 36˚. The RCosine of that is

the Rsine of 54˚”.the Rsine of 54˚”.Jya 36˚ =√⅛(5R²- √ 5R ²) =5878 R/10,000 Jya 36˚ =√⅛(5R²- √ 5R ²) =5878 R/10,000

Where 36 is the arc of radius R subtending an Where 36 is the arc of radius R subtending an angle of 36˚ at its centre. angle of 36˚ at its centre.

That, is R sin 36˚ = √ ⅛(5R²- √ 5R ²) =5878 R/10,000That, is R sin 36˚ = √ ⅛(5R²- √ 5R ²) =5878 R/10,000or sin 36˚ = √ ⅛(5- √5) = 5878 R/10,000or sin 36˚ = √ ⅛(5- √5) = 5878 R/10,000Since √5 = 2.237411 approximatelySince √5 = 2.237411 approximately

5-√5 = 2.762589…… 5-√5 = 2.762589……√⅛ (5 - √5 ²) =√.345323…… = .5878 approximately √⅛ (5 - √5 ²) =√.345323…… = .5878 approximately

“ “ The square root of five times the square of the The square root of five times the square of the radius is diminished by the radius and the remainder is radius is diminished by the radius and the remainder is divided by four; the result is the exact value of the Rsine divided by four; the result is the exact value of the Rsine of 18.”of 18.”

jyā18˚=(1/4)(√ 5R² – R); or Sin 18˚ =1/4(√ 5 -1)jyā18˚=(1/4)(√ 5R² – R); or Sin 18˚ =1/4(√ 5 -1)Where 18 is the arc of radius R subtending an angle of Where 18 is the arc of radius R subtending an angle of 18˚ at its centre.18˚ at its centre.