Mathematics in Ancient and Medieval India c8.s3: Trigonometrical Series

Mathematics in Ancient and Medieval India c8.s3

Language: English

Trigonometrical Series

c8.s3.1

As the ratio of the circumference and diameter can not be expressed in any commensurable quantity, an infinite series for the value of π gradually developed simultaneously with other trigonometrical series namely sine, cosine and tan series. The Sanskrit works, *Tantra-samgraha* and *Karanapaddhati* gave the expansion of the π, sine, cosine and tan functions, the results of which are almost identical with modern values :

c8.s3.2

$1) (π)/4 = 1 - ⅓ + ⅕ - 1/7 +$

c8.s3.3

$2) sin θ = θ - (θ³)/(3!) + (θ⁵)/(5!) -$

c8.s3.4

$3) cos θ = 1 - (θ²)/(2!) + (θ⁴)/(4!) -$

c8.s3.5

$4) θ = ((sin θ)/(cos θ)) - ⅓((sin θ)/(cos θ))³ + ⅕((sin θ)/(cos θ))⁵ -$

c8.s3.6

where θ is small and θ 45°.

c8.s3.7

The original passages and their translations are given. Proofs of these series have been given in the *Yuktibhāṣā*. The (π)/4 series has been discussed in Chapter VI in connection with the development of the concept of integral calculus. Some modern scholars¹ have also made a study of the π series as given in the *Yuktibhāṣā* and expressed the same in modern notation. The tan series known as Gregory's series has been similarly studied also by a number of modern scholars on the basis of *Yuktibhāṣā* in which series was worked out with the previous formulation of number of lemmas¹.

c8.s3.8

¹ Marar, K. and Rajagopal, C. T. ‘On the Hindu Quadrature of the Circle’, *Journal of the Bombay Branch of the Royal Asiatic Society*, **20**, p. 56–77, 1944; Rajagopal, C. T. ‘A neglected chapter of Hindu Mathematics’, *Scripta Mathematica*, **15**, nos. 3–4, p. 201–209, 1949.

c8.s3.9

¹ Marar, K. and Rajagopal, C. T. *Ibid.*, **20**, p. 56–57, 1949; Rajagopal, C. T. and Aiyar, T. V. V. ‘On the Hindu Proof of Gregory’s Series, *Scripta Mathematica*, **17**, nos. 1–2, p. 65–74, 1951.

c8.s3.10

π-*Series* :

c8.s3.11

This series has been discussed in chapter six of the *Karaṇapaddhati* where the value of π has been expressed in a number of different series. The opening verse runs as follows :

c8.s3.12

*vyāsāccaturghnād bahuśaḥ pṛthaksthāt tripañca-saptādyayugāhṛtāni* ।
*vyāse caturghne kramaśastvṛṇaṁ svaṁ kurjāt tadā syāt paridhiḥ susūkṣmaḥ* ॥
( *Karaṇapaddhati*, ch. 6, verse 1 )

c8.s3.13

‘Four times of the diameter is to be divided separately by each of the odd integers 3, 5, 7……; every quotient whose order is even, is taken away from the one preceding it. Combined result of all such small operations, when subtracted from four times the diameter, gives the value of the circumference with progressively greater accuracy’.

c8.s3.14

If C be the circumference and D the diameter, the rule may be expressed as :

c8.s3.15

$C=4D-4D(⅓-⅕)-4 D (1/7-1/9)……...$ $or (π)/4=(1-⅓+⅕-1/7+1/9…………)$

c8.s3.16

The next verse gives the series in a different form as follows :

c8.s3.17

*vyāsād vanasaṅguṇitāt pṛthagāptaṃ tryādyayugvimū-*
*laghanaiḥ* ।
*triguṇavyāse svamṛṇaṃ kramaśaḥ kṛtvāpi pari-*
*dhirāneyaḥ* ॥
( *Karaṇapaddhati*, ch. 6, verse 2 )

c8.s3.18

'Four times the diameter is divided separately by the cubes of the odd integers from 3 onwards, diminished by these integers themselves. The quotients thus obtained are alternately added to and subtracted from thrice the diameter. The result is the circumference'.

c8.s3.19

Thus, $C=3D+4D[1/(3³-3)-1/(5³-5)+1/(7³-7)-···]$

c8.s3.20

We can express this also in the following form : $π=3+4[1/(3³-3)-1/(5³-5)+1/(7³-7)-········]$ $or,(π-3)/4=1/(3³-3)-1/(5³-5)+1/(7³-7)-1/(9³-9)+········$ $or,(π-3)/4=[(1/2.3.4-1/4.5.6)+(1/6.7.8-1/8.9.10)+······]$ $or,(π-3)/6=⅔[(1/2.3.4-1/4.5.6)+(1/6.7.8-1/8.9.10)+···]$ $=[1/((2.2²-1)²-2²)+1/((2.4²-1)²-4²).$ $.+1/((2.6²-1)²-6²)+········]$

c8.s3.21

The last named form of the second series has in fact been given in separate verse as follows:

c8.s3.22

*vargairyujāṃ vā dviguṇairnirekairvargākṛtairvarjita-*
*yugmavargaiḥ |*
*vyāsaṃ ca ṣaḍghnaṃ vibhajet phalaṃ svaṃ vyāse*
*trinighne paridhistadā syāt ||*
( *Karaṇapaddhati*, ch. 6, verse 4 )

c8.s3.23

‘Six times the diameter is divided separately by the square of twice the squares of even integers ( 2, 4, 6... ) minus one, diminished by the squares of the even integers themselves. The sum of the resulting quotients increased by thrice the diameter is the circumference’.

c8.s3.24

Thus, $C = 3D + 6D [ 1/((2 2² - 1)² - 2²) + 1/((2.4² - 1)² - 4²).$ $. + 1/((2.6² - 1)² - 6²) + ......]$

c8.s3.25

*Convergency of π-Series :*

c8.s3.26

*Tantrasaṃgraha* adopts various approximations to (π)/4. Among these let us discuss the following two which have already been noted by Whish¹. The two approximations to the series (π)/4 are as follows: $i) (π)/4 = 1 - ⅓ + ⅕ - ... ± 1/n$ $((n+1)/2)/((n+1)² + 1) ......[T]$

c8.s3.27

¹ *Transaction of the Royal Asiatic Society of Great Britain and Ireland*, **8**, p. 512, 1935.

c8.s3.28

$and ii) (π)/4 = 1 - ⅓ + ⅕ - ± 1/n (((n + 1)²)/4 + 1)/([(n + 1)² + 4 + 1] ((n + 1)/2)) [T]$

c8.s3.29

In the original series (π)/4 = 1 - ⅓ + ⅕ - 1/7 - + (1) let S ((n + 1)/2) and S ((n - 1)/2) denote sums of the first 2m + 1 and 2m terms respectively, where n = 4m + 1. These provide rational approximations to (π)/4 where n is large. Let us try to improve them by adding corrections—f(n + 1) and f(n - 1) respectively, where f(x) is a function to be suitably chosen. The improved approximations being,

c8.s3.30

$T((n + 1)/2) = S((n + 1)/2) - f(n + 1) 2$

c8.s3.31

$T((n - 1)/2) = S((n - 1)/2) + f(n - 1) 3$

c8.s3.32

T((n + 1)/2) will be the partial sum of a certain series where general term U_n is given by T((n + 1)/2) - T((n - 1)/2).

c8.s3.33

Subtracting ( 3 ) from ( 2 ), we have,

c8.s3.34

$U_n = 1/n - f(n + 1) - f(n - 1) 4$

c8.s3.35

Changing successively n to n-2, n-4, 3, multiplying alternately by 1 and -1, and adding, we obtain, $-- U₃ + U₅ -- + U_n$ $= - ⅓ + ⅕ + 1/n + f (2) - f (n+1)$

c8.s3.36

If f(n) -- → 0 as n -- → ∞, we have from (1)

c8.s3.37

$(π)/4 = 1 - f (2) - U₃ + U₅ + U_n 5$

c8.s3.38

Suppose f (x) is assumed to be of the form

c8.s3.39

$2f(x) = a₁/x + a₂/(x²) + a₃/(x³) 6$

c8.s3.40

Expanding f (n+1) and f (n-1) by Taylor's Theorem, we have form (4), $1/n = U_n + 2[ f (n) + (fii(n))/(2!) + (fiv(n))/(4!) +$ $+ 1/((2p)!) f 2p(n+θ₁) - f 2p(n-θ₂) ] 7$ where 0 < θ₁ < 1, 0 < θ₂ < 1

c8.s3.41

Since f (n) = 0(1/n), fii(n), fiv(n) etc. are 0(1/(n³)), 0(1/(n⁵)) etc. Hence if we propose that U_n = 0(1/(n2p+1)), we obtain from (7) $1/n + 0(1/(n2p+1)) = 2[f (n) + (fii(n))/(2!)$ $+ 1/((2p-2)!) f2p-2(n) ] 8$

c8.s3.42

Substituting (6) in (8), we can calculate a₁, a₂_2p. We verify that a₂ = a₄ = = a_2p = 0. Any function of the type (6) can be taken as f (x), after calculating a₁, a₂_2p in this way, the further coefficients a_2p+1 etc.. being chosen arbitrarily in a convenient manner. We then use ( 4 ) and ( 5 ) to get a series for π in which $U_n = 0 ( 1/(n2p+1) ) .$

c8.s3.43

*Case i* ) Putting p = 2 in (8), we obtain from (6) and (8), $a₁ = 1, a₃ = - 1, a₂ = a₄ = 0.$ Now from (6) $2f (n) & = 1/n - 1/(n³) + 1/(n⁵) - 1/(n⁷) + / & = n/(n² + 1)$ $Hence f (n + 1) = (n + 12)/((n + 1)² + 1)$

c8.s3.44

From (2), we get the approximation

c8.s3.45

$(π)/4 = 1 - ⅓ + ⅕ - ± 1/n (n + 12)/((n + 1)² + 1)$

c8.s3.46

*Case ii* ) Putting p = 3, we obtain as before from ( 6 ) and ( 8 ) $a₁ = 1, a₃ = - 1, a₅ = 5, a₂ = a₄ = a₆ = 0$ From (6), $2f (n) & = 1/n - 1/(n³) + 5/(n⁵) - (5²)/(n⁷) + / & = (n² + 4)/(n(n² + 5))$ $Hence f (n + 1) = ((n + 1)²4 + 1)/((n + 12) [ (n + 1)² + 4 + 1 ])$

c8.s3.47

After this rational approximation, the series becomes, $(π)/4 = 1 - ⅓ + ⅕ - ± 1/n$ $(((n + 1)²)/4 + 1)/(( (n + 1)/2 ) [ (n + 1)² + 4 + 1 ])$

c8.s3.48

This shows that the author of *Tantrasaṃgraha* had definitely the knowledge of slowly converging series of which there is no indication in the *Karaṇapaddhati*.

c8.s3.49

*Sine and Cosine Series* :

c8.s3.50

The following passage containing sine and cosine series is available in the *Tantra-saṃgraha*¹ ( 1501 A. D. ) It has left distinct hints that the results contained in the lines were of Mādhava ( c. 1400 A. D. ). The verses run as follows :

c8.s3.51

¹ Vide *Yuktibhāṣā*, Pt. I, edited with notes by Ramavarma ( Maru ) and Tampuran and A. R. Akhilesvara Iyre, p. 190, Trichure , 1948.

c8.s3.52

*nihatya cāpa vargeṇa cāpam tattatphalāni ca* ।
*haret samūlayugvargaistrijyāvargahataiḥ kramāt* ॥
*cāpam phalāni cādhodhonyasyoparyupari tyajet* ।
*jivāptyai, saṅgraho ‘syaiva vidvān-ityādinakṛtaḥ* ॥
*nihatya cāpavargeṇa rūpam tattatphalāni’ ca* ।
*hared vimūlayugvargaistrijyāvargahataiḥ kramāt* ॥
*kintu vyāsadalenaiva dvighnenādyam vibhājyatām* ।
*phalānyadhodhaḥ kramaśo nyasyoparyupari tyajet* ॥
*śarāptyai, saṅgraho ‘syaiva stenastrītyādinā kṛtaḥ* ।

c8.s3.53

*English Translation* : Multiply the arc by the square of itself ( multiplication being repeated any number of times ) and divide the result by the product of the square of even numbers increased by that number and square of the radius (the multiplication being repeated same number of times). The arc and the results obtained from above are placed one below the other and are subtracted systematically one from its above. These together give the *jīvā* ( r sin θ ) collected here as found in the expression beginning with *vidvān* etc. Multiply the unit (i.e. radius) by the square of the arc (multiplication being repeated any number of times) and divide the result by the product of square of even number decreased by that number and square of the radius (multiplication being repeated same number of times). Place the results one below the other and subtract one from its above. These together give the *śara* ( r-r cos θ ) collected here as found in the expression beginning with *stena*.

c8.s3.54

If t_n and t'_n be the n-th expression for *jīvā* and *śara*, then for a small are s and radius r.

c8.s3.55

$t_n = (s2n. s)/((2²+2)(4²+4)...[(2n)²+2n]r2n) (n = 1,2,3...).$ The successive terms t₁, t₂, t₃...are, $t₁ = (s³)/(3!r²), t₂ = (s⁵)/(5!r⁴), t₃ = (s⁷)/(7!r⁶), t₄ = (s⁹)/(9!r⁸), ......$ Then according to the rule, $jīvā = (s - t₁) + (t₂ - t₃) + (t₄ - t₅) + ......$ $= s - (s³)/(3!r²) + (s⁵)/(5!r⁴) - (s⁷)/(7!r⁶) + (s⁹)/(9!r⁸) - (s¹¹)/(11!r¹⁰) + ...... (1)$ Again, $t'_n = (s2n. r)/((2²-2)(4²-4)...[(2n²)-2n]r2n) (n = 1,2,3,...)$ The successive tetms t'₁, t'₂, t'₃, ... are : $t'₁ = (s²)/(2!r), t₂ = (s⁴)/(4!r³), ... t'₆ = (s¹²)/(12!r¹¹)...$

c8.s3.56

As per rule, *śara* = (r-t₁')+(t₂'-t₃')+ ... ...

c8.s3.57

$= r - (s²)/(2!r) + (s⁴)/(4!r³) - (s⁶)/(6!r⁵) + (s⁸)/(8!r⁷) - (s¹⁰)/(10!r⁹) + (s¹²)/(12!r¹¹) - ... (2)$

c8.s3.58

when s = r θ, the eqns (1) and (2) reduce to

c8.s3.59

.\sin\ \theta = \theta\ - (\theta³)/(3!) + (\theta⁵)/(5!) - (\theta⁷)/(7!) + (\theta⁹)/(9!) - \ ... / \cos\ \theta = 1\ - (\theta²)/(2!) + (\theta⁴)/(4!) - (\theta⁶)/(6!) + (\theta⁸)/(8!) - \ ... } ...\ (3)

c8.s3.60

Fortunately the passages beginning with *vidvān* and *stena* referred to in the above verses have been preserved in both *Āryabhaṭīyabhāṣya* (TSS 101, p. 113; *yukti-bhāṣā*, p. 145) of Nīlakaṇṭha (1443-1545) and *Karaṇa-paddhati* (ch. 6. vss. 14-15). In the former it has been clearly stated that the values of the first five terms t₅, t₄, t₃, t₂, t₁ of the eqn (1) and of t₆', t₅', t₄', t₃', t'₂ and t₁' of eqn (2) were given by Mādhava (*evāha mādhavah*) when s = 5400' and r = 3437' 44″ 43″'. The values are : *vidvā* (= 44″ = t₅), *tunna bala* (= 33″ 6″' = t₄), *kaviśanicaya* (= 16' 5″ 41″' = t₃), *sarvārthaśīlasthiro* (= 273' 57″ 47″' = t₂), *nirvirdhāṅga - narendrarung* (= 2220' 39″ 40″' = t₁) and *stena* (= 6″' = t₆'), *stripiśuna* (= 5″ 12″' = t₅') *sugandhinaganud* (= 3' 9″ 37″' = t₄'), *bhadrāṅgabhavyāsana* (= 71' 43″ 24″' = t₃'), *mīnāṅganara-simha* (= 872' 3″ 5″' = t₂'), *unadhanakṛtbhūreva* (= 4241' 9″ 0″' = t₁).

c8.s3.61

These values when substituted in eqn (1) containing terms from t₁ to t₅, *jīvā* comes out to be 3437' 44″ 48″', the 24th sine value given in the table of Mādhava (here s = 5400'). Similarly if s is replaced gradually by 225ʹ, 450ʹ, 675ʹ… Mādhava’s sine table is obtained. Proceeding in a similar way and substituting values in eqn(2), the cosine table is obtained. This evidently shows that Mādhava, followed by the authors of *Tantra-saṃgraha* and *Karaṇapaddhati*, used the eqns (1) and (2) for the computation of the sine and cosine tables.

c8.s3.62

How Mādhava arrived at the equations (1) and (2) is not yet definitely known. The *Tantrasaṃgraha* (ch. 2, verse 12½) of Nīlakaṇṭha and *Karaṇapaddhati* (ch. 6, verse 19) have both given that for small arc, jīvā=s-(s³)/(3!r²) (approximate'y).

c8.s3.63

The *Yuktibhāṣā* has given the complete rational of the eqns (1) and (2). Its author Jyeṣṭhadeva (c. 1500-1600) in an effort to find an expression for the difference between any arc and its sine chord, divided the circumference of the quarter of a circle into n equal divisions and considered the first and second sine differences. He then found the sum of the first n sine differences and cosine differences by considering all sine chords to be equal to corresponding arc and the small unit of the circumference to be equal to one unit, which evidently gives jīvā=s-(s³)/(3!r²) and śara=(s²r)/(2!r²).

c8.s3.64

Since sine values are not actually equal to its arc length, further correction was applied *ad-infinitum* to each of the terms of the values obtained for *jīvā* and *śara*, which ultimately gives rise to the eqns (1) and (2) It would not be quite unlikely to presume that the rational was first established by Mādhava before Jyeṣṭhadeva could make use of it.

c8.s3.65

In western mathematics Newton ( 1642-1727 ) is often given credit for the expansion of sine and cosine series No. ( 3 ). The result was established later algebraically on a solid foundation by De Moivre ( 1707-38 ) and Euler ( 1748 ). It is clear from the discussion that the Indian scholar Mādhava ( 1350-1410 ) used and possibly established the series (1), (2) and (3) of course in finite form before Newton, De Moivre and Euler, and laid the foundation of his sine table.

c8.s3.66

*Tan Series* :

c8.s3.67

The verse from the *Karanapaddhati* runs as follows :

c8.s3.68

*vyāsārdhena hatādabhiṣṭaguṇataḥ koṭyāptamādyam*
*phalaṃ jyāvargeṇa vinighnamādimaphalaṃ*
*tattatphalaṃ cāharet* ।
*kṛtyā koṭiguṇasya tatra tu phaleṣvekatripañcādi-*
*bhirbhakteṣvojayutaistajet samanjutim*
*jīvādhanuśiśasyate* ॥

c8.s3.69

‘*Jyā* of the arc ( *dhanuṣ* ) is to be multiplied by the semi-diameter ( *vyāsārdha* ) and is divided by the *koṭi*. This is the first term ( of the tan series ). The value ( of the first term ) when multiplied by the square of the *jyā* and divided by the square of the *koṭi* gives the second term. This process is repeated. The successive terms are divided by the odd integers 1, 3, 5… . Now when the consecutive terms in the series starting from the first term are alternately subtracted and added gives the circumference’.

c8.s3.70

In modern notation ( Fig. 20 ), it becomes

c8.s3.71

$arc BP = OP [ PM/OM - ⅓ (PM³)/(OM³) + ⅕ (PM⁵)/(OM⁵) - ]$

c8.s3.72

$or s = r [ (r sin θ)/(r cos θ) - ⅓ ( (r sin θ)/(r cos θ) )³ + ⅕ ( (r sin θ)/(r cos θ) )⁵ - ]$ Hence, θ = tan θ - ⅓ tan ³ θ + ⅕ an⁵ θ - when s = r θ

c8.s3.73

³ Vide also 'The Sine and Cosine Power Series in Hindu Mathematics' by Rajagopal, C. T, and Venkaṭaraman, A. *Journal of the Asiatic Society of Bengal*, 3rd Series, 15, p. 1-13, 1949.

c8.s3.74

The formula holds good if θ 45°.

c8.s3.75

The *Karaṇapaddhati* gives no proof of these series. These series also appear in the *Tantrasaṃgraha* (1465-1545 A.D.) without proof to which Whish already drew attention.¹ The proof of all these can be found in the *Yuktibhāṣā*, an exposition of Nīlakaṇṭha's *Tantrasaṃgraha*.

c8.s3.76

¹ *Transaction of the Royal Asiatic Society of Great Britain and Ireland*, 3, pt. III, p. 513, 516, 519.

c8.s3.77

In the general history of mathematics, the π and tan series are associated with the name of Gregory (1638-1675 A. D.).²

c8.s3.78

² *Chamber's Encyclopaedia*, 10, p. 227, 1935, Cajori, F. *A History of Mathematics*, 2nd revised and enlarged edition, p. 206.

c8.s3.79

*Sine and Cosine Series (Proof)*

c8.s3.80

In the chapter on *Calculus* (*vide* chapter VI) under differentiation, we have shown how to obtain the changes in the sine chord (*bhūjajyā*) and the cosine chord (*koṭijyā*) for small differences in the arc. It has been shown there :—

c8.s3.81

i) *bhūjakhaṇḍa* or sine difference of 1st order $= (R cos x dx)/R$ ii ) *koṭikhaṇḍa* or cosine difference of 1st order $= (R sin x dx)/R$ Now the difference between any two *bhūjakhaṇḍas* = Sine difference of the second order = koṭikhaṇḍa × dx/R = R sin x ((dx)²)/(R²)

c8.s3.82

This is used in the *Yuktibhāṣā*¹ to find an expression for the difference between any arc and its sine chord. Let x ( = AP) be any part of the arc (Fig. 24) in the

c8.s3.83

¹ Saraswati, T. A. ‘Series Mathematics after Bhāskara II’, Bulletin of the National Institute of Sciences of India, 21, p. 320-43, 1963.

c8.s3.84

Fig. 24

c8.s3.85

first quadrant be divided into n equal divisions each being dx. Let p₁, p₂, p₃_n be the sine chords for parts of the arc x corresponding to parts of the arc represented by dx, 2dx, 3dx... . Let p'₁, p'₂, p'₃... be the corresponding sine differences. Then p'₁ = p₁ = R sin x $p'₁ - p'₂ = R sin x. ((dx)²)/(R²) = p₁ ((dx)²)/(R²)$ $or p'₂ = p'₁ - p₁ ((dx)²)/(R²)$ $= p₁ - p₁ ((dx)²)/(R²)$ $p'₃ = p'₂ - p₂ ((dx)²)/(R²) = p₁ - p₁ ((dx)²)/(R²) - p₂ ((dx)²)/(R²)$ $$ $$ Similarly, $p'_n = p₁ - p₁ ((dx)²)/(R²) - p₂ ((dx)²)/(R²) - - p_n-1 ((dx)²)/(R²)$ $p'₁ + p'₂ + p'_n &= np₁ - [(n-1)p₁ + (n-2)p₂ + / & + 2p_n-2 + p_n-1] ((dx)²)/(R²)$ or p_n = sum of all these sine differences $&= np₁ - [(n-1)p₁ + (n-2)p₂ + ... / & + 2p_n-2 + p_n-1] ((dx)²)/(R²)$ $&= whole arc - [(n-1)p₁ + (n-2)p₂ + / & + p_n-1] ((dx)²)/(R²)$

c8.s3.86

If n be sufficiently large, the arc corresponding to p₁ may be approximately equal to p₁, then the approximate length of the whole arc is given by np₁ = S ( say ).

c8.s3.87

or S-p_n = [(n-1)p₁ + (n-2)p₂ + + p_n-1] ((dx)²)/(R²) = [p₁ + (p₁ + p₂) + (p₁ + p₂ + p₃) + (n-1)times] ((dx)²)/(R²)

c8.s3.88

or, difference between the arc and sine chord = 2nd order *saṁkalita* of the sines (1).

c8.s3.89

Now, *śara* ( or *utkramajyā* ) of the given arc = the sum of the *koṭikhaṇḍas* = p₁dx/R + p₂dx/R + + p_ndx/R = (p₁ + p₂ + + p_n)dx/R

c8.s3.90

Assuming all the sine chords to be equal to the corresponding arc and dx to be equal to one unit, then the approximate value of *utkramajyā* = [ 1 + 2 + 3 + + (x-1) + x ]1/R = 1/R[ x + (x-1) + + 2 + 1 ] = (x²)/2 · 1/R = (x²)/2R (2)

c8.s3.91

Expressing sines p′₁, p′₂, p′₃ in terms of the corresponding arcs, we get,

c8.s3.92

p′₁ = C_1/ 2 · dx/R (where C_1/ 2 is the cosine chord at the middle point of the small arc dx)

c8.s3.93

$= (R-h_½)· dx/R ( h_½ represents the height of.$ $.the half chord )$

c8.s3.94

$Similarly, p₂' = C_1½·dx/R = (R-h_1½)dx/R$

c8.s3.95

$p'_n = C_n-½dx/R = (R-h_n-½)dx/R$

c8.s3.96

$p'₁-p'_n = dx/R(R-h_½- R+h_n-½)$

c8.s3.97

$= dx/R· h_n$

c8.s3.98

$( dx is very small, h_½ can be taken as h₀=0 & h_n-½=h_n)$

c8.s3.99

$= dx/R· śara of the given arc.$

c8.s3.100

$= dx/R·(x²)/2R = (x²dx)/(2R²) = (x²)/(2R²) (since dx = 1 unit)$

c8.s3.101

$Similarly, p'₁-p'_n-1 = ((x-1)²)/(2R²)$

c8.s3.102

$p'₁-p'_n-2 = ((x-2)²)/(2R²)$

c8.s3.103

The sum of the left hand side $= (n-1)p'₁ - (p'₂ + p'₃ + +p'_n-1)$ $= np'₁ - (p'₁ + p'₂ + p'₃ + +p'_n-1)$ $= np₁ - p_n = S-p_n$

c8.s3.104

$Hence, S-p_n = (x²)/(2R²) + ((x-1)²)/(2R²) + ((x-2)²)/(2R²) + from (1)$ or, differences between arc and the sine chord

c8.s3.105

$= 1/(2R²)[x² + (x-1)² + (x-2)² + ...]$

c8.s3.106

$= 1/(2R²) · (x³)/3 = (x³)/(3!R²) ... ... ... (3)$

c8.s3.107

This gives a closer approximation of the value of the difference between arc x and its sine chord. The difference thus obtained has to be applied as correction to each term of left hand side expression of equation ( 2 ) where length of the arc was assumed to be equal to that of its sine chord which was not correct. Hence applying these corrections in equation ( 2 ), the expression of the righthand side becomes—

c8.s3.108

$1/R[(x-(x³)/(3!R²)) + (x-1) - ((x-1)³)/(3!R²) + ...]$

c8.s3.109

$= 1/R[x + (x-1) + ...] - 1/(3!R³)[x³ + (x-1)³ + ...]$

c8.s3.110

$= 1/R[x + (x-1) + ...] - 1/(3!R³) · (x⁴)/4$

c8.s3.111

$= (x²)/(2!R) - (x⁴)/(4!R³)$

c8.s3.112

Hence the correction for the *śara* = (x⁴)/(4!R³)

c8.s3.113

When this correction is applied also to the values of sine differences in the equation ( 3 ) the expression of the righthand side becomes—

c8.s3.114

$1/R[((x²)/2R-(x⁴)/(4!R³))+((x-1)²)/2R-((x-1)⁴)/(4!R³)+]$ $=1/R[(x²)/2R+((x-1)²)/2R+((x-2)²)/2R+]$ $-1/R[(x⁴)/(4!R³)+((x-1)⁴)/(4!R³)+]$ $=1/(2R²)·(x³)/3-1/(4!R⁴)·(x⁵)/5$ $=(x³)/(3!R²)-(x⁵)/(5!R⁴)$

c8.s3.115

Hence to obtain *jya* ( or the sine chord ) the above two corrections are to be applied *ad infinitum* to the independent expressions for the arc sine difference and the *śara*.

c8.s3.116

Hence

c8.s3.117

$jya=x-(x³)/(3!R²)+(x⁵)/(5!R⁴)- (4)$

c8.s3.118

$And kojya=R-sara of the arc$ $=R-(x²)/(2!R)+(x⁴)/(4!R³)- (5)$

c8.s3.119

Now putting x=R θ, jya=R sin θ and kojya=R cos θ in (4) and (5), we get respectively,

c8.s3.120

$sin θ=θ-(θ³)/(3!)+(θ⁵)/(5!)-$ $and cos θ=1-(θ²)/(2!)+(θ⁴)/(4!)-$

c8.s3.121

*Tan Series (Proof)* : $S = R ( tan θ - ⅓ tan ³ θ + ⅕ tan ⁵ θ - ... )$

c8.s3.122

*Yuktibhāṣā*¹ gives a proof of the series for tan θ = t = 1. This with a slight and obvious alteration in the figure covering all cases. This alteration is almost the only point of departure from its original proof given below. Proof is based on two lemmas.

c8.s3.123

¹ For actual proof vide Mukunda Marar’s article, *Teacher’s Magazine*, 15, p. 28-34, 1940, vide also Marar, K. M. and Rajagopal, C. T. “On the Hindu Quadrature of the Circle”, *Journal of the Bombay Branch of the Royal Asiatic Society*, 20, n. s. p. 66–68, 1944.

c8.s3.124

Fig. 25

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*Lemma 1.* Let P, Q be points on the tangent at A to a circle of unit radius, whose centre is 0; let OP meet the circle in p. If m is the foot of the perpendicular from p to OQ, (Fig. 25) pm is given by, $pm = PQ/OP.OQ$

c8.s3.126

*Lemma 2* In Lemma 1, PQ is small, the arc pq of the circle intercepted between OP and OQ is given by, $S = arc pq = PQ/(OP²)$

c8.s3.127

$or, PQ/(1 + AP²) (OA = R = 1)$

c8.s3.128

Fig. 26

c8.s3.129

*Proof* : Let AOB = θ 45° $or tan θ = AB = t 1$

c8.s3.130

Divide AB into n equal parts denoting the points of division by P₀ (= A), P₁, P₂, ...... P_n-1, P_n (= B). (vide Fig. 26)

c8.s3.131

Let OB cuts the circle in b.

c8.s3.132

Then from *Lemma* 2. $S = _n → ∞ _r = 0n-1 (P_r P_r+1)/(1 + AP_r²)$ $= _n → ∞ _r = 0n-1 (t/n)/(1 + (rt/n)²)$ $= _n → ∞ _r = 0n-1 t/n 1 - (rt/n)² + .$ $. + (-1)ν-1 (rt/n)2ν-2 + ((-1)ν (rt/n)2ν)/(1 + (rt/n)²)$

c8.s3.133

Now making use of the result, $_n → ∞ 1/(np+1) _r=0n-1 rp = 1/(p+1)$ We obtain, $S = t - (t³)/3 + + (-1)ν-1 (t2ν-1)/(2ν-1) + (-1)ν R_ν$ where 0 < R_ν < 1/(2ν+1)

c8.s3.134

When n → ∞, we get Gregory's series 0 < t 1. The validity of the series for -1 t < 0 becomes obvious when we change the sign of t.

c8.s3.135

This testifies to the high degree of development that series mathematics had achieved in India. The development of series mathematics is found in Europe in the works of Newton, Leibniz and Gregory ( 17th century A. D. ) more than a century later. The priority of Indian concept in this connection is therefore beyond question. Further research is necessary to ascertain whether there was any useful contact between India and Europe between fourteenth to seventeenth century A. D. It is not unlikely that such contacts might have occurred during this period when traderoute between the two countries was made through the efforts of navigator like Vasco-da-Gama and others.

Linked texts (3)

Tantrasaṅgraha — Nīlakaṇṭha Somayājī 1500, Sanskrit
Yuktibhāṣā (Gaṇitayuktibhāṣā) — Jyeṣṭhadeva 1520, Malayalam
Karaṇapaddhati — Putumana Somayājī 1660, Sanskrit