Madhava's Sine and Cosine Series

A. K. Bag

Indian National Science Academy 1 Park Street Calcutta 700 016

(Received 29 April 1975)

Indian scholar Madhava (1350-1410) gave a table of almost accurate values of half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. The paper discusses the basis of arriving at such accurate values and has shown that Madhava established the following sine and cosine series before Newton (1642-1727), DeMoivre (1707-38) and Euler (1748), and used these relations to compute his table. These are:
      $sin \theta = \theta-\frac{\theta^3}{3!} + \frac{\theta^5}{5!}-\ldots$
      $cos \theta = 1-\frac{\theta^2}{2!} + \frac{\theta^4}{4!}-\ldots$

The originality of Madhava (1350-1410) in astronomy, particularly in the application of refined mathematics in it in medieval period, is now being slowly recognised. He has given values of 24 half-sine chords for twenty-four arcs in a quarter of a circle drawn at equal intervals of 225', viz. 225', 450', 675', ... 5400'. The corresponding twenty-four sine values given by him are as follows.
      224' 50" 22"'     448' 42" 58"'     670' 40" 16"'
      889' 45" 15"'    1105' 1" 39"'     1315' 34" 7"'
     1520' 28" 35"'    1718' 52" 24"'     1909' 54" 35"'
     2092' 46" 3"'    2266' 39" 50"'     2430' 51" 15"'
     2584' 38" 6"'    2727' 20" 52"'     2858' 22" 55"'
     2977' 10" 34"'    3083' 13" 17"'     3176' 3" 50"
     3255' 18" 22"'    3320' 36" 30"'     3371' 41" 29"
     3408' 20" 11"'    3430' 23" 11"'     3437' 44" 48"

These values are correct to more or less eight to nine places of decimals. How Madhava arrived at such accurate values of sine table, has been discussed in the paper.

The following passage found in the Tantra-samgraha (1501 A.D.) has left distinct hints that the results contained in the lines were of Madhava. The verses run as follows:
      nihatya capa vargena capam tattatphalani ca/
      haret samulayugvargaistrijyavargahataih, kramat//
      capam phalani cddhodhonyasyoparyupari tyajet/
      jivaptyai, sangraho 'syaiva vidvan-ityadinakrtah,//
      nihatya capavargena rupam tattatphaldni' ca/
      hared viniulayugvargaistrijyavargahataih, kramat//
      kintu vydsadalenaiva dvighnenadyam vibhajyatam/
      phalanyadhodhah kramaso nyasyoparyupari tyajet//
      saraptyai, sangraho 'syaiva stenastrityadina krtah/

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 jiva (r sin θ) collected here as found in the expression beginning with vidvan 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 sara (r — r cos θ), collected here as found in the expression beginning with stena.

If tn and t'n be the n-th expression for jiva and sara, then for a small arc s, and radius r,
      $t_n = \frac{s^{2n}.s}{(2^2 + 2)(4^2 + 4)\ldots[(2n)^2 + 2n]r^{2n}}$     $(n = 1, 2, 3,\ldots)$
The successive terms t1, t2, t3 ... are,
      $t_1 = \frac{s^3}{3!r^2}, t_2 = \frac{s^5}{5!r^4}, t_3 = \frac{s^7}{7!r^6}, t_4 = \frac{s^9}{9!r^8},\ldots$
Then according to the rule,
    $jiva = (s - t_1) + (t_2-t_3) + (t_4-t_5) + \ldots$
      $= s-\frac{s^3}{3!r^2} + \frac{s^5}{5!r^4}-\frac{s^7}{7!r^6} + \frac{s^9}{9!r^8}-\frac{s^{11}}{11!r^{10}}+ \ldots$       ... (1)
Again, $t'_n = \frac{s^{2n}.r}{(2^2-2)(4^2-4)\ldots[(2n)^2-2n]r^{2n}}$   $(n = 1, 2, 3,\ldots)$

The successive terms t'1, t'2, t'3 ... are:
      $t'_1 = \frac{s^2}{2!r}, t'_2 = \frac{s^4}{4!r^3},\ldots,t'_6 = \frac{s^{12}}{12!r^{11}} \ldots$
As per rule, $sara = (r-t'_1) + (t'_2-t'_3)+\ldots$
    $= r-\frac{s^2}{2!r}+\frac{s^4}{4!r^3}-\frac{s^6}{6!r^5}+\frac{s^8}{8!r^7}-\frac{s^{10}}{10!r^9}+\frac{s^{12}}{12!r^{11}}-\ldots$       ... (2)
when s = rθ, the eqns (1) and (2) reduce to
        $\begin{array} sin\theta = \theta-\frac{\theta^3}{3!} + \frac{\theta^5}{5!}-\cdots \\ cos \theta = 1-\frac{\theta^2}{2!} + \frac{\theta^4}{4!}-\ldots \end{array}\Bigg\}$       (3)

Fortunately the passages beginning with vidvan and stena referred to in the above verses have been preserved in both Aryabhatiyabhasya of Nilakantha (1443-1545) and Karanapaddhati. 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 t'6, t'5, t'4, t'3, t'2, and t'1 of eqn (2) were given by Madhava (evaha madhavah) when s = 5400' and r = 3437' 44" 48"'. The values are: vidvan (= 44"' = t5), tunna bala ( = 33" 6"' = t4, kaisanicaya ( = 16' 5" 41"' = t3), sarvarthasilasthiro (=273' 57" 47"' = t2), nirvirdhanganarendrarung (= 2220' 39" 40"' = t1) and stena ( = 6"' = t'6), stripisuna (= 5" 12"' = t'5), sugandhinaganud (= 3' 9" 37"' = t'4), bhadrahgabhavyasana ( = 71' 43" 24"'= t'3), minanganarasimha (= 872' 3" 5"' = t'2), unadhanakrtbhureva (= 4241' 9" 0" = t'1).

These values when substituted in eqn (1) containing terms from t1 to t5, jiva comes out to be 3437' 44" 48"', the 24th sine value given in the table of Madhava (here s = 5400'). Similarly if s is replaced gradually by 225', 450', 675' ... Madhava'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 Madhava, followed by the authors of Tantrasamgraha and Karanapaddhati, used the eqns (1) and (2) for the computation of the sine and cosine tables.

How Madhava arrived at the equations (1) and (2) is not yet definitely known. The Tantrasamgraha (ch. 2, verse $12\frac{1}{2}$) of Nilakanbha and Karanapaddhati (ch. 6, verse 19) have both given that for small arc, jiva = $s-\frac{s^3}{3!r^2}$ (approximately). The Yuktibhasa has given the complete rational of the eqns (1) and (2). Its author Jyesthadeva(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 jiva = $s-\frac{s^3}{3!r^2}$ and sara = $\frac{s^2r}{2!r^2}.$.

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 jiva and sara, 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 Madhava before Jyesfehadeva could make use of it.

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)7. It is clear from the discussion that the Indian scholar Madhava(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.
I express my gratitude to Sri S. N. Sen for his kind interest in the paper.


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