Mathematics in Ancient and Medieval India c8: Chapter V Trigonometry

Mathematics in Ancient and Medieval India c8

Language: English

c8.s1 Spherical Trigonometry — 4 parts
c8.s2 Value of π — 17 parts
c8.s3 Trigonometrical Series — 135 parts

Chapter V Trigonometry

c8.2

The work of Hipparchus ( c. 150 B. C. ), Menelaus of Alexandria ( c. 100 A. D. ) and Ptolemy ( c. 150 A. D. ) in connection with astronomical problems in the early Christian Era led to the growth of trigonometrical knowledge in Greece in a rather fragmentary character. It was however at a later date ( 4th to 12th century A.D. ) the Indian developed the subject in a systematic manner almost resembling its modern form. This was then transmitted to the Arabs by the beginning of 9th century A.D., who introduced some further improvement. From the Arabs, the knowledge went to Europe where a detailed account of trigonometrical knowledge first appeared in the work *De Triangulis* of Regiomontanus ( 1533 A.D. ).

c8.3

In ancient Indian Mathematics, Trigonometry forms an integral part of Astronomy. It is undoubtedly a later development. Earliest reference to trigonometrical concept is found in the *Sūryasidhānta* ( c. 400 A. D. ), *Pañcasiddhāntikā* of Varāhamihira (c. 505 A.D.) *Brāhmasphuṭasiddhānta* of Brahmagupta ( c. 628 A.D. ) *Śiṣyadhivṛddhida* of Lalla ( 768 A.D. ). More or less systematic study on the subject was made by Bhāskara II ( c. 1150 A. D. ) in the section on *Jyotpatti* in his *Siddhāntaśiromaṇi* and by Kamalākara in the *Sphuṭādhyāya* chapter of his *Siddhāntatattvaviveka*.

c8.4

Bhāskara II¹ laid particular emphasis on the study of trigonometry in order to obtain proficiency in astronomy. The idea on values of π, sine, cosine expressed in series was first found in the works of Mādhava and then appeared in the *Karaṇa Paddhati*, *Tantrasaṃgraha* and *Yuktibhāṣā*

c8.5

¹ *ācāryānām padaviṃ jyotpattyā jñātayā yato yāti* ( *Siddhāntaśiromaṇi-jyotpatti*, sl. 1 ) "The *ācārya* ( master ) title ( in astronomy ) is offered on him who has acquired sufficient knowledge of trigonometry".

c8.6

*Preliminary Indian Methods in Plane Trigonometry*

c8.7

( 1 ) *Trigonometical functions* :—In India, like Babylon and Greece, the study of the properties of the right-angled triangle inscribed in a circle in connection with the use of gnomon in astronomical measurement was given much importance. The circle was divided into quadrants and circumference into many equal parts. Three functions² *Jyā*, *Kojyā* and *Utkramajyā* were used in the development of this new geometry, known as plane trigonometry. The modern equivalents of these functions (Fig. 18) are as follows :—

c8.8

² Synonymous terms of *Jyā* are *bhujajyā*, *jīvā*, *maurvi*, *kramajyā*, *dorjyā* etc., and of *Kojyā* are *koṭijyā*, *koṭi*. By definition, it denotes the compliment of the sine arc.

c8.9

Fig. 18

c8.10

Jyā A &= PM = R Sin A / Kojyā A &= OM = R Cos A / Utkramajyā A &= MB = OB - OM / &= R - R Cos A / &= R ( 1 - Cos A ) / &= R Versin A where R is the Radius³ of the circle.

c8.11

³ *śiñjini*, *trijyā* etc. ( synonymous terms ).

c8.12

( 2 ) *Calculation of R Sin A, A⩽90°*

c8.13

Varāhamihira (c. 505 A. D.) in his *Pañcasiddhāntikā*¹ gave the following formulae :

c8.14

¹ ch. 4. v. 19.

c8.15

$R Sin 30⁰ = R/2$

c8.16

$R Sin 60⁰ = (√3)/2 R$

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$R Sin 90⁰ = R = 120′$

c8.18

$( R Sin A )² = R/2( R--R Cos 2A )²$

c8.19

and ( R Sin A )² + ( R Cos A )² = R²

c8.20

With the help of these formulae Varāhamihira perhaps calculated the values of R Sines for arcual measure ranging from 3⁰ 45′ to its 24 multiples i.e ( 3⁰ 45′ × 24 = 90⁰ ).

c8.21

Bhāskara I ( c 600 A. D. ) in his *Mahābhāskarīya* ( ch. 7, v. 17–19 ) has also set the following approximate formula for calculating the R Sine of an acute angle without the use of a table.

c8.22

$R Sin A = (R(180⁰ - A) A)/([40500 - (180⁰ - A)A]R/4)$

c8.23

where A is in degrees. Equivalent forms of the formula have been given by almost all subsequent scholars of mathematics and astronomy. The formula in modern notation may be written as

c8.24

$Sin B = (16B (π - B))/(5π² - 4B(π - B)) where B radians$ correspond to A degrees.

c8.25

Putting B = (π)/3, and (π)/7 we get

c8.26

Sin (π)/3 = .8643..., Sin ((π)/4) = .70058...,

c8.27

and Sin ((π)/7) = .4313... which are correct upto 2 places of decimals. The values of Sin π, Sin (π)/2 and Sin (π)/3 come out to be accurate. Bhāskara I ascribes this formula to Āryabhaṭa I¹. It occurs in the *Brāhma-sphuṭasiddhānta* and in several later works also.²

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¹ See his commentary on *Āryabhaṭīya*, i. 11.

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² Gupta, R. C. ‘Bhāskara I's Approximation to Sine’ *Indian Journal of History of Science*, 2, No. 1, p. 121-36, 1967.

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( 3 ) *Calculation of Sin A*, A < 90º

c8.31

Bhāskara I ( c. 600 A D. ), Brahmagupta and others have made use of the formulae :

c8.32

R Sin (90º + A) = R Sin (90º – A) = R Cos A R Sin (180º + A) = – R Sin A R Sin (270º + A) = – R Sin (90º – A) = – R Cos A where A90º

c8.33

The basis of the above formulae may be explained as follows :

c8.34

Fig. 19

c8.35

Let AOB, BOC, COD and DOA be four quadrants of a mean circular orbit of a planet, A being the Planet's apogee from which the mean anomaly is measured anticlockwise. If P₁, P₂, P₃ and P₄, be the Planet's positions at four quadrants respectively, then

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*In Quadrant I* arc traversed = A P₁ = bhūja; arc to be traversed = P₁ B = koṭi

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*In Quadrant II* arc traversed = BP₂ = koṭi; arc to be traversed = P₂C = bhūja

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*In Quadrant III* arc traversed = CP₃ = bhūja; arc to be traversed = P₃D = koṭi

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*In Quadrant IV* arc traversed = DP₄ = koṭi; arc to be traversed = P₄A = bhūja

c8.40

The R Sine of the *bhūja* is defined to be the R Sine of the corresponding anomaly and the R Sine of the *koṭi* is defined to be the R Cosine of the corresponding anomaly. The R Sine is positive in the first and second quadrants and negative in the third and fourth quadrants. The R cosine is positive in the first and fourth quadrants and negative in the second and third quadrants. Then we have, $R Sin AP₁ = P₁E R Cos AP₁ = P₁F$ $R Sin AP₂ = P₂H R Cos AP₂ = -P₂G$ $R Sin AP₃ = -P₃M R Cos AP₃ = -P₃N$ $R Sin AP₄ = -P₄S R Cos AP₄ = PT₄$ This is equivalent to $R Sin (90⁰ + A) = R Sin (90⁰ - A) = R Cos A$ $R Sin (180⁰ + A) = -R Sin A$ $R Sin (270⁰ + A) = -R Sin (90′ - A) = -R Cos A$ $Where A < 90⁰$ Similarly R Cosine formulae can also be deduced.

c8.41

Bhāskara I¹ has given several other formulae which may be given without proof as follows : $R Sin (90⁰ + A) = R Sin 90⁰ - R versin A$ $R Sin (180⁰ + A) = 90⁰ - R versin 90⁰ - R Sin A$ $R Sin (270⁰ + A) = R Sin 90⁰ - R versin 90⁰$ $- R Sin 90⁰ + R versin A where A < 90⁰$

c8.42

¹ *Mahābhāskarīya*, iv. 2.

c8.43

These formulae, though not explicitly stated, were given earlier by Āryabhaṭa I in his *Āryabhaṭīya* (iii, 22, 1st part see commentaries also) and used by Brahmagupta (*Brāhmasphuṭasiddhānta*, ii, 15–16).

c8.44

*Deduction of Formulae*

c8.45

In Varāhmihira’s work, we find the following formulae :

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I. ‘The *Jyā* of an arc diminished from a quadrant is the square-root of the difference of the square of the radius and the square of the *Jyā* of the arc’.²

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² *Pañcasiddhāntikā*, ch. 4. v. 23 (1st part of the rule)

c8.48

That is, $Jya((π)/2-A)=√(R²-Jya²A)$ $or, R Sin((π)/2-A)=√(R²-(R Sin A)²)$

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Expressed in modern form, it becomes, $Sin((π)/2-A)=√(1-Sin²A)$

c8.50

The rule gives the correct result.

c8.51

II. 'Twice any desired arc is substracted from 90°; the *Jyā* of the remainder is subtracted from the radius. The square-root of the result multiplied by sixty (i.e. half, of the radius) is the *Jyā* of that arc. By deducting that square (i.e. Jya²A) from the square of the radius, the square of *Kojyā* is obtained.¹

c8.52

¹ *ichāṁśadviguṇona tribhojyayonā trayusya cāpajyā* । *ṣaṣṭiguṇā sā karaṇi tayā dhruvonāvaśeṣasya* ॥ (*Pañcasiddhāntikā*, ch. 4, v. 5)

c8.53

That is, $(Jya A)²=R/2[R-Jya((π)/2-2A)]$ $or, (Jya A)²=R/2[R-Kojya 2 A]$ $and R²-(Jya A)²=(Kojya A)²$ $or, Jya² A+Kojya² A=R²$

c8.54

These two rules are equivalent respectively to our modern formulae : $Sin²A=½(1-Cos 2A) and Sin²A+Cos²A=1$

c8.55

III. 'To find the *Jyā* of any desired arc, subtract from the radius, the *Jyā* of quarter of a circle reduced by double the arc and then add the square of half the result to the square of half the *Jyā* of the double arc followed by the square root of this summation result’.¹

c8.56

¹ *śeṣeṣviṣṭeṣu dhanurdviguṇapadāyojya śeṣaguṇahinā* । *trijyā tadarddhāvargodviguṇajyārddhasya saṁyojyaḥ* ॥ ( *Pañcasiddhāntikā*, ch. 4, v. 3 )

c8.57

This gives, $(Jya A)² = [(Jya 2A)/2]² + [(R - Jya((π)/2-2A))/2]²$

c8.58

This corresponds to our modern formula, $Sin²A = (Sin² 2A)/4 + (Versin² 2A)/4$

c8.59

Lalla (c. 768 A. D.) gave the following rules about the relationship of trigonometrical functions :

c8.60

1. ‘The square of the *Jyā* is subtracted from the square of the radius, the square root of the remainder is the *Kojyā*, or *Kojyā* is the *Jyā* of a quadrant minus *bhūja* arc’²

c8.61

Which means,
i) √(R²-(Jya A)²) = Kojya A and ii) Kojya A = Jya((π)/2-A)

c8.62

These are identical with modern formulae respectively as follows : $√(1-Sin² A) = Cos A and Cos A = Sin((π)/2-A)$

c8.63

² *Śiṣyadhivṛddhida*, ch. 2, v. 30.

c8.64

In Brahmagupta’s *Brāhmasphuṭasiddhānta*, we find the following relations :

c8.65

1. Half the diameter diminished by the *Utkramajyā* of an arc or its complement is the *Jīva* or *Jyā* of the respective complementary arc. The square-root of the difference of the square of the radius and that of the *Jyā* of an arc or its complements is the *Jyā* of the respective complementary arc’.¹

c8.66

¹ vyāsadalamita,a jīvābhūjakoṭyaṃsotkramajyāyāhīnaṃ । koṭibhujayāvyāsārddhakṛtiviśeṣātpadamcānyā ॥ ( *Brahmaspuṭasiddhānta*, ch. 14, v. 7 )

c8.67

This can be written as :

c8.68

i) *Jyā* ( (π)/2 - A ) = R - Utkramajyā A

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ii) *Jyā* A = R - Utkramajyā ( (π)/2 - A )

c8.70

and iii) √(R² - (Jyā A)²) = Jīvā ( (π)/2 - A )

c8.71

The modern equivalents are respectively,

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Sin ( (π)/2 - A ) = Cos A

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Sin A = Cos ( (π)/2 - A )

c8.74

and, √(1 - Sin² A) = Sin ( (π)/2 - A )

c8.75

In the *Mahāsiddhānta* of Āryabhaṭṭa II, we find the following formulae :

c8.76

1. ‘The square-root of half the result obtained by adding to or subtracted from the maximum value of the *Jyā*, the product of the radius and the *Jyā* of an arc gives the *Jyā* of an arc equal to twelve times the first arc (i.e., 3⁰45' × 12 = 45⁰) increased or decreased by half of the arc (under consideration )'.¹

c8.77

¹ *iṣṭajyāgajyāhatihīnāḍhyauparamasiñjinīvargo* । *taddalamūle te staḥ saṃkhyārdhonāḍhyakarabhavau piṇḍau* ॥ ( *Mahāsiddhānta*, ch. 3, v. 2 ) *gajyā* = *trijyā*, *paramasiñjinī* = *trijyā*, *hīnāḍhyau* = *rahita sahitau*, *saṃkhyārdhonāḍhyakarabhavau* = *abhiṣṭardhena hīnā yuktāśca dvādaśa śeṣasaṃkhyāsamau* ( 3⁰45' × 12 = 45⁰ ).

c8.78

That is,

c8.79

$Jya((π)/4 ± A/2) = √((R² ± R Jya A)/2)$

c8.80

Expressed in modern form, it becomes,

c8.81

$Sin((π)/4 ± A/2) = √((1 ± Sin A)/2)$

c8.82

Bhāskara II² ( 1150 A. D ) made a comprehensive list of the trigonometrical formulae ( *vide Jyotpatti* section ) including those of the previous workers, as given below. These formulae were found arranged serially and shows a genetic relation among them.

c8.83

1) R² - (R Sin A)² = (R Cos A)² (v.4).

c8.84

² Vide *Jyotpatti* section of *Siddhāntaśiromaṇi* of Bhāskara II.

c8.85

2) R Sin A = R Cos((π)/2 - A) (v. 5).

c8.86

3) 'Half the root of the sum of the squares of *Kramajyā* and *Utkramajyā* of any arc is the *Jyā* of half the arc; or, the square-root of half the product of the radius and the *Utkramajyā* is the *Jyā* of half that arc.'³

c8.87

³ *kramotkramajyākṛtiyogamūlād dalam tadardhāṃśaka śiñjinī syāt* । *trijyotkramajyā nihatirdalasya mūlaṃ tadardhāṃśaka śiñjinī vā* ॥ ( v. 10. )

c8.88

These are :

c8.89

i) ½√(Fya² A + Utkramajya² A) = Jya A/2

c8.90

and ii) √((R × Utkramajya A)/2) = Jya A/2

c8.91

In Modern notation, the formulae are :

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½√(Sin² A + Versin² A) = Sin A/2

c8.93

and √((Versine A)/2) = Sin A/2

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4) Jya((90⁰)/2) = √((R² ± R × Jya A)/2) ... ... (v.12)

c8.95

In modern notation, these are equivalent to,

c8.96

Sin((90⁰ ± A)/2) = √((1 ± Sin A)/2)

c8.97

5) ‘Square of the difference of the *Jyās* of the two desired arcs is to be added with the square of the difference of their *Kojyās*, the square-root of this (result) when halved gives the *Jyā* of (an arc equal to) half the difference of those two arcs.’¹

c8.98

¹ *yaddorjayorantaramiṣṭayoryat kṛtijyāyontat kṛtiyoga mūlaṃ* । *dalikṛtaṃ syād bhūjayorviyoga-khaṇḍasyajivaivamanekadhā vā* ॥ (v. 13)

c8.99

That is,

c8.100

½√((Jya A - Jya B)² + (Kojya A - Kojya B)²)
= Jya (A - B)/2

c8.101

or, in modern notation, this comes out as, $½ √((Sin A - Sin B)² + (Cos A - Cos B)²)$ $= Sin (A - B)/2$

c8.102

6) 'The square-root of half the square of the difference of *Jyā* and *Kojyā* of an arc is equal to the *Jyā* of another arc equal to half the difference of the arc and its complement'.¹

c8.103

¹ *doḥkoṭijīvāvavarasya vargo dalikṛtastasya padena tulya*। *syātkoṭibāhorvivarārdhajivā vakṣyetha mūlagraṇam vināpi*॥ ( v. 14 ).

c8.104

That is, $√(((Jya A - Kojya A)²)/2) = Jya [ ((90⁰ - A) - A)/2 ]$

c8.105

It may be written in modern form as follows : $√(((Sin A - Cos A)²)/2) = Sin [ ((90⁰ - A) - A)/2 ]$

c8.106

7) R Sin (90⁰ - 2 A) = R - ((R Sin A)²)/(R/2) ...... (v.15)

c8.107

8) '*Jyās* of two desired arcs are multiplied by the *Koṭijyās* of the other, each product is divided by the radius; the sum of the quotients is the *Jyā* of the sum of the angles and their difference is the *Jyā* of their difference.'²

c8.108

² *cāpayoriṣṭyordorjyemithaḥkoṭijyākahate*। *trijyābhakte tayoraikyam taccāpaikasya dorjakāḥ*॥ ( v. 21 and 22½ ).

c8.109

That is, $Jya (A ± B) = (Jya A × Kojya B)/R ± (Jya B × Kojya A)/R$

c8.110

In modern equivalents, these reduce to :

c8.111

Sin (A)=Sin A Cos B A Sin B

c8.112

These two relations were called respectively *samāsa bhāvanā* and *antara bhāvanā* by Bhāskara II (*1150 A. D.*). Almost the same list of trigonometrical formulae of Bhāskara II appears in the work *Siddhāntatattvaviveka* of Kamalākara besides many other important formulae.¹ The *Karaṇapaddhati* (ch vi, vs. 9–11) gives also some correct formulae as follows :

c8.113

¹ Sin 2 A = 2 Sin A Cos A (ch. 3, v. 73); Sin 3 A = 3 Sin A – 4 Sin³A (ch. 3, v. 74); Sin (A±B) = Sin A Cos B±Cos A Sin B (ch. 3, v. 68); Cos (A±B) = Cos A Cos BSin A Sin B (ch. 3, v. 69); etc. [in modern equivalents.]

c8.114

i) √(R². R Sin A) = R(Sin A/2 A/2)

c8.115

ii) RSin A/2 = (R(Sin A/2+CosA/2)+R(Sin A/2-CosA/2))/2

c8.116

iii) R CosA/2 = (R(Sin A/2+CosA/2)-R(SinA/2-CosA/2))/2

c8.117

iv) √(((R Sin A)²+(R-R Cos A)²)/2) = R Sin A/2

c8.118

v) √(((R Cos A)²+(R+R Cos A)²)/2) = R Cos A/2

c8.119

vi) √((R².R Cos A)/2) = R Sin A/2 , R Cos A/2

c8.120

Bhāskara II gave no proof to these theorems. The commentator of Kamalākara (1658 A. D.) mentions that Kamalākara gave two proofs of these relations, none of which are available.¹ It is also believed that even earlier workers gave similar proofs, but these could not be traced. As corollaries to the general theorem of *samāsa bhāvanā* and *antara bhāvnā*, Bhāskara II indicated how to derive the functions of multiple angles. That Bhāskara II was aware of the cosine formulae has also been, attested by Muniśvara and Kamalākara².

c8.121

¹ *Siddhāntatattvaviveka* of Kamalākara, ch. 3, v. 70 (Commentary.)

c8.122

Proofs of the above formulae had been derived perhaps as follows :

c8.123

(1) The proof follows directly if the Radius is taken as the hypotenuse, the *Jyā* as the perpendicular and *Kojyā* as the base of the right angled triangle.

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(2) The results follow directly from the definition of *Jyā*, *Kojyā* and *Utkramajyā* of an arc.

c8.125

Fig. 20

c8.126

(3) i) In the right-angled triangle (Fig. 20) PMB, PM² + MB² = PB² or ½√(PM² + MB²) = ½ PB or ½√*Jyā*² A + *Utkramajyā*² A = *Jyā* A/2 (arc BP = A)

c8.127

ii) Let K = *Koṭijyā*, U = *Utkramajyā*, R = Radius and J = *Jyā*. Now K = R − U K² = R² − 2RU + U² or R² − K² = 2RU − U² or J² = 2RU - U² or (J² + U²)/4 = RU/2 √((J + U²)/4) = √(RU/2) or √((Jyā² A + Utkramajyā² A)/4) = √((R × Utkramajyā A)/2) or Jyā A/2 = √((R × Utkramajyā A)/2) ... ... by 3 (i)

c8.128

² *Golādhyāya*, Anandasrama Sanskrit Series No. 122, p. 152; *Siddhāntatattva-viveka*, ed by Sudhakara, p. 113.

c8.129

In modern notation, it becomes, Sin A/2 = √((Versin A)/2)

c8.130

(4) Let the two diameters BOB' and COC' of the circle of centre O cut each other at right angle (Fig. 21) Let P be any point on the quadrant COB' such that = A. Hence = 90° + A. Let S be the middle point of PB then SB is the Jyā of (90° + A)/2

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Fig. 21

c8.133

Now PB² = PM² + MB² = K² + (R + J)² where R = Radius, J = Jyā and K = Koṭijyā of the arc PC.

c8.134

or PB² = K² + R² + 2RJ + J² = R² - J² + R² + 2RJ + J²

c8.135

= 2R² + 2RJ = 2(R² + RJ)

c8.136

Again SB = ½ PB = √((2(R² + RJ))/2) = √((R² + RJ)/2)

c8.137

i.e., Jya((90⁰ + A))/2 = √((R² + R × Jya A)/2)

c8.138

Similarly the other relation my be deduced.

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(5) In the Fig. 22, let arc BQ = A and arc BP = B. OP = OQ = R = Radius.

c8.140

Then QK = QN - KN = QN - PM = Jya A - Jya B PK = PS - KS = OM - ON = Kojya B - Kojya A. Now QP = Jya of the arc PQ i.e., A--B or QP/2 = Jya of the arc ((A-B))/2

c8.141

Fig. 22

c8.142

Since, ½√(QK² + PK²) = QP/2

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we can write,

c8.144

½√((Jya A - Jya B)² + (Kojya B - Kojya A)²) = Jya((A-B))/2

c8.145

or ½√((Sin A - Sin B)² + (Cos A - Cos B)²) = Sin (A-B)/2

c8.146

(6) Substituting 90⁰ - A for A, and A for B in the relation (5), the result follows

c8.147

(7) From (3), we have √((R × Utkramajya A)/2) = JyaA/2 or R\=a A=2Jy\=a² A/2

c8.148

or Utkramajy\=a 2 A=(2Jy\=a² A)/R (Substituting 2A for A)

c8.149

or R-Utkramajy\=a 2A=R-(2Jy\=a² A)/R

c8.150

or Kramajy\=a 2A=R-(2Jy\=a² A)/R

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or Jy\=a (90°-2A)=R-(2Jy\=a² A)/R

c8.152

or, R Sin (90°-2 A)=R-((R Sin A)²)/(R/2)

c8.153

or Sin (90°-2A)=1-2 Sin² A.

c8.154

(8) In the quarter of the circle (Fig. 23), let the arcs GX=GE=EC. The perpendiculars are drawn from the points, G, E, C on OX and OY.

c8.156

$CB=CM+MB$ $=CM+IL$

c8.157

Now CM:CI=EF:EO

c8.158

$∴ CM=(CI)/EO=(GP)/R$

c8.159

Again, IL:EK=IO:EO

c8.160

$∴ IL=(EK)/EO=(EK)/R$

c8.161

$( IO=GH) . 23$

c8.162

$Now CB=CM+IL=(GP)/R+(EK)/R (1)$

c8.163

If the successive *bhujajy\={a}s* are denoted by J₁, J₂, J₃ and successive *Ko\d{t}ijy\={a}s* by C₁, C₂, C₃, then J₃=(J₂C₁+J₁C₂)/R from (1)

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By generalising the result, $J_m+n = (J_mC_n + J_nC_m)/R$ or Jyā (A + B) = (Jyā A × Kojyā B + Jya B × Kojya A)/R

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Mādhava (*Āryabhaṭīyabhāṣya*, TSS 101, p. 58) used the following result to obtain the formula : R Sin (A ± B) = √((R Sin A)² - (lamba)²) ± √((R Sin B)² - (lamba)²) where lamba = (R Sin A.R Sin B)/R

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*Sine Table and its Origin*

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The construction of the *Jyā* table (or Indian sine table i.e. the values of R Sin θ) or the calculation of the values of the *Jyās* of different arcs in a quadrant was given a great importance by Indian astronomers as it was used to calculate the planetary positions as accurately as possible. For this, the circumference of each quarter of a circle was divided into 24 equal parts, each part covering 225 units (= 3°45') of the circumference. The twenty-four *Jyā* values as given by Varāhamihira,¹ Āryabhaṭa², Govindasvāmī³ and others for every increase of 3°45' are arranged in Table I.

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¹ Bag, A. K. 'A note on the Sine Table I,' *Indian Journal of History of Science*, **7**, No. 1, p. 71–74, 1972; note that in the 5th jyā a term *adhikā* is added.

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² Bag, A. K. 'Sine Table in Ancient India,' *Indian Journal of History of Science*, **4**, Vos. 1 & 2, p. 79–85, 1969; note that 16th jyā according to Āryabhaṭa I and Bhāskara is 2977.

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³ Gupta, R. C. 'Fractional Parts of Āryabhaṭa's Sines and Certain Rules found in Govindasvāmī's Bhāṣya on the Mahābhāskariya', *Indian Journal of History of Science*, **6**, No. 1, p. 51–59, 1971.

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The values of R Sin nh (where h = 3⁰ 45', n = 1, 2,...24 ) as found in different Indian works are tabulated below :

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| S. No. | *Pañcasiddhāntikā* (505 A. D.) of Varāhamihira | *Sūryasiddhānta* Āryabhaṭa I Āryabhaṭa II Bhāskara II | Brahmagupta ( 628 A. D. ) | Govindasvāmi ( 850 A. D. ) | Mādhava (1400 A.D.) ( *Tantrasaṅgraha* ) TSS–188 p. 19. ch. 1, V. 15–22 |
| | (1) | (2) | (3) | (4) | (5) |
| 1. | 7′51′ | 225′ | 214′ | 224′50′23″ | 224′50′22″ |
| 2. | 15′40″ | 449′ | 427′ | 448′42′53″ | 448′42′58″ |
| 3. | 23′25″ | 671′ | 638′ | 670′40′11″ | 670′40′16″ |
| 4. | 31′ 4″ | 890′ | 846′ | 889′45″ 8″ | 889′45″15″ |
| 5. | 38′34.5″ | 1105′ | 1051′ | 1105′ 1″30″ | 1105′ 1″39″ |
| 6. | 45′56″ | 1315′ | 1251′ | 1315′33′56″ | 1315′34″ 7″ |
| 7. | 53′ 5″ | 1520′ | 1446′ | 1520′28′22″ | 1520′28″35″ |
| 8. | 60′ 0′ | 1719′ | 1635′ | 1718′52′10″ | 1718′52″24″ |
| 9. | 66′40″ | 1910′ | 1817′ | 1909′54′19″ | 1909′54′35″ |
| 10. | 73′ 3″ | 2093′ | 1991′ | 2092′45″46″ | 2092′46″ 3″ |
| 11. | 79′ 7″ | 2267′ | 2156′ | 2266′38″44″ | 2266′39″50″ |

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TABLE I (Continued)

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| 12. | 84′51″ | 2431′ | 2312′ | 2430′50″54″ | 2430′51″15″ |
| 13. | 90′13″ | 2585′ | 2459′ | 2584′37′43″ | 2584′38′ 6″ |
| 14. | 95′12″ | 2728′ | 2594′ | 2727′20″29″ | 2727′20″52″ |
| 15. | 99′46″ | 2859′ | 2719′ | 2858′22′31″ | 2858′22″55″ |
| 16. | 103′55″ | 2978′ | 2832′ | 2977′10″ 9″ | 2977′10″34″ |
| 17. | 107′37′ | 3084′ | 2933′ | 3083′12″51″ | 3083′13″17″ |
| 18. | 110′52″ | 3177′ | 3021′ | 3176′ 3′23″ | 3176′ 3′50″ |
| 19. | 113′37″ | 3256′ | 3096′ | 3255′17″54″ | 3255′18″22″ |
| 20. | 115′55″ | 3321′ | 3159′ | 3320′36″ 2″ | 3320′36″30″ |
| 21. | 117′42′ | 3372′ | 3207′ | 3371′41″ 1″ | 3371′41′29″ |
| 22. | 118′59″ | 3409′ | 3242′ | 3408′19′42″ | 3408′20″11″ |
| 23. | 119′44″ | 3431′ | 3263′ | 3430′22′42″ | 3430′23″11″ |
| 24. | 120′ | 3438′ | 3270′ | 3437′44″19″ | 3437′44′48″ |

c8.175

In working out the values of *Jyā* table, the circles of different radii (24th *jya* = *jya* 90º = *Sinus totus*) viz. 120′, 3438′, 3415′, 3437′, 44″ 19″ etc were used by different workers. Since the radius was expressed in minute, the respective *Jyā* length was expressed in minute. The relative accuracy of the values may be judged from Table II in which the valucs of Sin nh as found from above are compared with their corresponding accurate values.

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TABLE II

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The values of Sin nh (in decimal)

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| S. No. | (1) | (2) | (3) | (4) | (5) | Correct values |
| 1. | .06543 | .06543 | .06544 | .06540 | .06540 | .06540 |
| 2. | .13055 | .13059 | .13058 | .13053 | .13053 | .13053 |
| 3. | .19514 | .19517 | .19511 | .19509 | .19509 | .19509 |
| 4. | .25888 | .25887 | .25872 | .25882 | .25882 | .25882 |
| 5. | .32145 | .32141 | .32141 | .32144 | .32144 | .32144 |
| 6. | .38277 | .38249 | .38257 | .38268 | .38268 | .38268 |
| 7. | .44236 | .44212 | .44220 | .44229 | .44229 | .44228 |
| 8. | .50000 | .50000 | .50000 | .50000 | .50000 | .50000 |
| 9. | .55555 | .55555 | .55565 | .55557 | .55557 | .55558 |
| 10. | .60875 | .60878 | .60887 | .60876 | .60876 | .60876 |
| 11. | .65930 | .65939 | .65929 | .65934 | .65934 | .65934 |
| 12. | .70708 | .70709 | .70703 | .70711 | .70711 | .70711 |

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TABLE II ( Continued )

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| 13. | .75181 | .75189 | .75198 | .75184 | .75184 | .75184 |
| 14. | .79333 | .79348 | .79327 | .79335 | .79335 | .79335 |
| 15. | .83139 | .83159 | .83149 | .83146 | .83147 | .83146 |
| 16. | .86597 | .86620 | .86605 | .86603 | .86603 | .86603 |
| 17. | .89681 | .89703 | .89694 | .89687 | .89687 | .89688 |
| 18. | .92388 | .92408 | .92385 | .92388 | .92388 | .92388 |
| 19. | .94681 | .94706 | .94679 | .94693 | .94693 | .94692 |
| 20. | .96597 | .96597 | .96605 | .96593 | .96593 | .96593 |
| 21. | .98083 | .98080 | .98073 | .98079 | .98079 | .98079 |
| 22. | .99153 | .99156 | .99143 | .99144 | .99144 | .99144 |
| 23. | .99778 | .99799 | .99786 | .99785 | .99785 | .99785 |
| 24. | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 | 1.00000 |

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Varāhamihira gave a fairly accurate (roughly correct upto 4 places of decimals) table of sines though no hint was left how he computed the table. The *Sūryasiddhānta*¹ used the following formula to compute the table :

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¹ *Sūryasiddhānta*, ch. 2, v. 15-16.

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R Sin (n + 1)h = R Sin n h + R Sin nh – R Sin (n – 1)h – (R Sin nh)/(R Sin h)

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where R=3438′

c8.185

and n=0, 1, 2,23 and R Sin h=h=3°45′ $=225′$

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The formulae was perhaps taken up by Āryabhaṭa I. Govindasvami in his commentary on the *Mahābhāskarīya*¹ of Bhāskara I) and Nīlakaṇṭha in his *Tantrasaṃgraha* (1500 A. D.) added some corrections on the *Jyā* values given by Āryabhaṭa I, which present astonishingly accurate table of Sines (correct upto five places of decimals, vide table II). The Sine table given by other scholars namely, Brahmagupta, Śrīpati, Āryabhaṭa II, Bhāskara II, etc. is no improvement over that of Govindasvamī (c. 800–850 A. D.) How the author of the *Sūryasiddhānta* arrived at the rule is not very clear. The result of *Sūryasiddhānta* is correct if Sin h of the denominator of the last expression is replaced by 4 Sin² h/2.

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¹ *saptāgnirandhrāṇi viyadguṇāgaṃ netrābdhinetraṃ munipañca-vedāḥ ।* *dvyakṣyaṣṭasya saṇṇayanadvirāmā vedāgnibhūtam raviṣaṭkṛ-sānuḥ ।* *randhrābhrapakṣaṃ guṇapāvakāṣṭau cakṣurviyatsapta khachan-drasūryāḥ ।* *rudrāgnicandrā manusaptasoma dasrābhranetraṃ nayanāṃdvisūryam ।* *akṣyabdhipakṣaṃ vasunetrarandhraṃ candrāgnividyā vasukhāṣṭa-chandram ।* *randhreṣuvedaṃ navarūpamidhmaṃ khābhrāgnayas saptaguṇe-dhmasaṃkhyam ॥* *ityuktāstatparādyāḥ syurete hīnādhikāṃśakāḥ ।* *guṇānāṃ te tataḥ śodhyā makhyādau yojitā api ॥* *tritridvirūpa netrai kadvicandrai kendu saṃkhyayāḥ ।* *eka trirūpa netraiśca, jyāvidbhi gaṇakaiḥ kramāt ॥* (*Mahābhāskarīya* of Bhāskara I, ed. by T. S. Kupanna Sastri with the *Bhāṣya* of Govindasvāmī, p. 200–201, Govt. Oriental MS. Library, Madras, 1957).

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*Trijyā* ( *Sinus Totus* )

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How the Indian values of R or *Sinus Totus* have been fixed up is not known. Āryabhaṭa I in connection with the verses on instruments made a reference to a circle of radius 57 degrees besides others. Pṛthūdakasvāmī advised to draw a circle of radius 3270 *aṅgulas* with a pair of compass. This appears to be physically impossible if we consider 1 *aṅgula* = 3/4 inches roughly. It may be presumed that the Indian values of R were perhaps obtained from the relation R=C/(2π), where C= circumference = 360 degrees = 21600 minutes.

c8.190

If we use π=3.1416 given by Āryabhaṭa I, $R&=360/(2× 3.1416)=57+387/1309 degrees / &=57 degrees ( omitting the fraction )$

c8.191

Though Āryabhaṭa I refers to 57 degrees but no sine table is available where he had used R=57 degrees.

c8.192

$Again, R=(360× 60)/(2× 3.1416)=(3437+967/1309) minutes=$ 3438 minutes roughly.

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This has been used by Indian scholars for the computation of Sine table.

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$Like wise R&=(360× 60× 60)/(2× 3.1416)=(206264+424/1309) seconds / &=206264 seconds approx.=3437'44″$

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This has been used by Vaṭeśvara.

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$Again, R = (360 × 60 × 60 × 60)/(2 × 3.1416) = (12375859 + 569/1309)$ $third = 12375859 = 3437'44″19″'$

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This has been used by Govindasvāmī (c. 800–850).

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Mādhava used a far smaller accurate value of π to obtain R = 3437'44″48″'. According to *Karaṇa-paddhati* (TSS. 126, ch. 6, v. 7), radius was calculated as R = (21600 × 10¹⁰)/(31,41,59,26,536) = 3437'44″48″'

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*Computation of Sine Table :*

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Several methods for deductions of the rule given in the *Sūryasiddhānta* and *Āryabhaṭīya* have been worked out by Delambre¹, Burgess², Singh³, Naraharayya⁴ and Ayyanger⁵. The methods given by last two scholars are reproduced by Srinivasiengar⁶. In a few cases, values given in the *Jyā* (Indian sine) table derived from the rule of the *Sūryasiddhānta* do not strictly agree with the modern values. To remove the deviation, Srinivasiengar formulated certain subsidiary rules¹.

Linked texts (13)

Tantrasaṅgraha — Nīlakaṇṭha Somayājī 1500, Sanskrit
Sūryasiddhānta — Maya, Sanskrit
Brāhmasphuṭasiddhānta — Brahmagupta, Sanskrit
Āryabhaṭīya — Āryabhaṭa, Sanskrit
Siddhāntatattvaviveka — Kamalākarabhaṭṭa 1633, Sanskrit
Mahābhāskarīya — Bhāskara I, Sanskrit
Pañcasiddhāntikā — Varāhamihira, Sanskrit
Yuktibhāṣā (Gaṇitayuktibhāṣā) — Jyeṣṭhadeva 1520, Malayalam
Mahāsiddhānta (Āryasiddhānta) — Āryabhaṭa II, Sanskrit
Khaṇḍakhādyaka — Brahmagupta, Sanskrit
Karaṇapaddhati — Putumana Somayājī 1660, Sanskrit
Siddhāntaśiromaṇi — Bhāskara II, Sanskrit
Grahajñāna — Āśādhara, Sanskrit

Mathematics in Ancient and Medieval India c8: Chapter V Trigonometry

Contents

Chapter V Trigonometry