On the computation of the arcsin function in the Kerala school of astronomy and mathematics

The formula is stated in verses 17–19 Chapter VII of Mahābhāskarīya. The verses and their English translation are given below (see [4] p. 45 for the verses and p. 207 for the English translation; see also [3]):

makhyādirahitaṃ karma vakṣyate tatsamāsataḥ $\scriptstyle|$
cakrārdhāṃśakasamūhādviśodhyā ye bhujāṃśakāḥ $\scriptstyle||$
taccheṣuguṇitā dviṣṭhaāḥ śodhyāḥ khābhreṣukhābdhitaṭaḥ $\scriptstyle|$
caturthāṃśena śeṣasya dviṣṭhamantyaphalaṃ hatam $\scriptstyle||$
bāhukoṭyoḥ phalaṃ kṛtsnaṃ kramotkramaguṇasya vā $\scriptstyle|$
labhyate candratīkṣṇāṃśvostārāṇāṃ vāpi tattvataḥ $\scriptstyle||$

“ I briefly state the rule (for finding the bhujaphala and kotiphala etc.) without making use of the Rsine-differences, 225, etc.

Subtract the degrees of the bhuja (or koti) from the degrees of half a circle (i.e., $180^{\circ}$). Then multiply the remainder by the degrees of the bhuja (or koti) and put down the result at two places. At one place subtract the result from 40500. By one-fourth of the remainder (thus obtained) divide the result at the other place as multiplied by the antyaphala (i.e., the epicyclic radius). Thus is obtained the entire bahuphala (or kotiphala) for the Sun, Moon, or the star-planets. So also are obtained the direct and inverse Rsines.”

In the above verses, the reference to “Rsine-differences, 225, etc.” is a reference to Āryabhaṭa’s sine table for a discussion on this table).

Let $x^{\circ}$ be the angle subtended by the arc whose jyā) is sought and let $r$ be the radius of the circle. Bhāskara I’s approximation formula can be expressed in these notations as follows:

Since $\text{jy\={a}}\,(x^{\circ})=r\sin(x^{\circ})$, we have

This is Bhāskara I’s approximation formula to the sine functions in modern notations. Note that this has to be understood in the following form:

If angles are measured in radians, Eq.(3) can be put in the following form:

Some properties of the formula can be easily observed.

1.  (1)

   The formula gives the exact values of $\sin(x^{\circ})$ for $x=0^{\circ},30^{\circ},90^{\circ},150^{\circ}$ and $180^{\circ}$, namely, the values $0,\frac{1}{2},1,\frac{1}{2},0$.

2.  (2)

   The formula is symmetrical about $90^{\circ}$, that is, the formula gives the same value for $\sin(90^{\circ}-x^{\circ})$ and $\sin(90^{\circ}+x^{\circ})$.

Figure 1 shows the percentage error in the value of $\sin(x^{\circ})$ computed using Bhāskara I’s approximation formula for the sine function. It can be seen that the maximum relative error is less than 1.8%.

As already indicated, neither Bhāskara nor his followers have given the rationale of the sine approximation formula. In the literature on the history of Indian mathematics one can see several modern rationales for the formula (see, for example, [3], [9] and [10]). We present below one such rationale which is an adaption of a rationale given in [10]. Our rationale is to look for the simplest function that satisfies the two properties (1) and (2) of Bhāskara I’s approximation formula.

The simplest function which satisfies property (2) and which takes the value $0$ when $x=0,180$ is

Now, $f(30)=\frac{1}{2}$ if $k=\frac{1}{9000}$ and $f(90)=1$ if $k=\frac{1}{8100}$. This suggests that, for $f(x)$ to satisfy property (1), $k$ should be of the form $\frac{1}{g(x)}$ with $g(x)$ satisfying property (2) and with $g(30)=9000$ and $g(90)=8100$.

The simplest general function which satisfies property (2) is

Since $g(30)=9000$ and $g(90)=8100$, we must have

Solving these equations we get $a=\frac{1}{4}\times 40500$ and $b=-\frac{1}{4}$.

Thus the function

satisfies the properties (1) and (2). This is precisely Bhāskara I’s sine approximation formula.

Let $m=\text{jy\={a}}\,\,(s)$, where $s$ is an arc, measured in degrees, of a circle of radius $r$, then:

Brahmagupta has not given any rationale for the expression in Eq.(5). It can be seen that it follows from from Eq.(1). Rrom Eq.(1) we have:

This can be written as

This is a quadratic equation in $s$ and solving it for $s$ we get precisely the formula in Eq.(5). The negative sign of the radical is retained as it is assumed that $s$ is less than $90^{\circ}$.

Somyājī, in Tantrasṅgraha, has given an an approximate expression for jyā  $(s)$ in the following verse (see verse 17 Chapter 2 in [6]):

śiṣṭacāpaghanaṣaṣṭhabhāgato vistarārdhakṙtibhaktavarjitam $\scriptstyle|$
śiṣṭacāpamiha śiñjinī  bhavet spaṣṭatā bhavati cālpatāvaśāt $\scriptstyle||$

“Divide one-sixth of the cube of the remaining arc by the square of the trijyā. This quantity when subtracted from the remaining arc becomes the śiñjinī  (the dorjyā or jyā) corresponding to the remaining arc). The value is accurate because of the smallness [of the arc].”

Somayājī in verse 37 Chapter 2 of Tantrasaṅgraha has indicated that the result in Eq.(6) can be used to compute the arc lengths corresponding to a given jyā provided that the arcs are small. However, Somayājī did not explicitly state how the value of $s$ is to be computed from this equation. But, fortunately, Śankara Vāriyar (c. 1500–1560 CE), in his Laghuvivṛtī commentary on Tantrasaṅgraha, has explained how exactly this can be done (see [6] pp. 92–93). We rewrite Eq.(6) in the following form

and then, in the right side, we replace $s$ by jyā  $(s)$ to get

This is Somayājī’s first method for computing values of arc lengths corresponding to small arcs.

Here we show that Somayājī’s expression to compute the arcsin function given by Eq.(8) is equivalent to the power series expression for $\arcsin(x)$ truncated at the term containing $x^{3}$.

Taking $s=r\theta$ and jyā  $(s)=r\sin\theta$, Eq.(8) can be written as

Let $x=\sin\theta$ so that $\theta=\arcsin(x)$. Then we have:

Thus Eq.(8) is equivalent to the polynomial approximation to $\arcsin(x)$ given by Eq.(9). Incidentally, the approximation to $\arcsin(x)$ given by Eq.(9) is the approximation to $\arcsin(x)$ obtained by neglecting terms containing $x^{5}$ and higher powers of $x$ in the power series expression for $\arcsin(x)$, namely,

The above described procedure can be summarised as an algorithm thus. Let $m$ be the jyā of an arc $s$ of a circle of radius $r$. Given $m$, to compute a sequence of approximations $s_{i}$ to $s$ and to generate an approximate value of $s$:

1.   Step 1.

   Set $s_{0}=m$, $\Delta_{0}=0$.

2.   Step 2.

   For $i=1,2,3,\ldots$, repeat the following until successive values of $s_{i}$-s are nearly equal:

   1. (a)

      $\Delta_{i}=\frac{(m+\Delta_{i-1})^{3}}{6r^{2}}.$

   2. (b)

      $s_{i}=m+\Delta_{i}.$

3.   Step 3.

   When $s_{i}\approx s_{i+1}$, then $s\approx s_{i}$.

The algorithm described in Section 5.1 has completely unexpectedly thrown up an important integer sequence which appears in a large number of combinatorial problems. The Online Encyclopedia of Integer Sequences has characterised this as a “core integer sequence”. To see how the particular integer sequence emerges via Śankara Vāriyar’s algorithm, let us reformulate the algorithm in the following form: we write $m=x$, $t=\frac{1}{6r^{2}}$, $\Delta_{i}=y_{i}$, $s_{0}=x$ and $y_{0}=0$. Then we have, for $i=1,2,\ldots$:

Let us compute a few values of $s_{i}$:

Consider the integers that appear as coefficients in the parenthesised expressions in the $s_{i}$-s:

This is precisely the sequence A001764 in the Online Encyclopedia of Integer Sequences (OIES) (see [5]). This sequence, which appears in a large number of combinatorial problems, has been characterised as a core sequence in the Encyclopedia. As per the Encyclopedia, the $j$-th entry $a_{j}$ in the above integer sequence is given by

Thus we have:

Moreover, there is a closed form expression for the infinite series $\sum_{n=0}^{\infty}a_{n}t^{n}x^{2n+1}$ (for example, see [1]), namely,

To begin with, we assume that the arcs and the corresponding jyā-s are all small. Let $s$ be a small arc and let jyā  $(s)=m$. We are given $m$ and we have to find $s$. From Eq.(7), we have

Somayājī assumes that $m$ and $r$ are in minutes so that $s-m$ given by Eq.(13) is also in minutes. If

then we have

From this, we get

Using this in Eq.(14), we get

This means that, if the difference between arc length and jyā is $k^{\prime\prime}$, then the arc length is given Eq.(17). The lookup table gives the the values of $s$ for $k=1,2,\ldots,24$.

Table 1 lists the values of the arc lengths for various values of the difference between the arc lengths and the corresponding jyā-s as given in the Laghuvivṛtī commentary of Tantrasaṅgraha (see [6] pp. 94–95). The Laghuvivṛtī commentary lists the values of $m=\text{jy\={a}}\,\,(s)$ given by Eq.(16) for $k=1,2,\ldots,24$ whereas Table 1 lists the values of $\text{jy\={a}}\,\,(s)$ as well as the values of $s$ given by Eq.(17). In the table, we have also given the kaṭapayādi encodings of the values of jyā  $(s)$-s. These encodings are taken from the following verses given in the Laghuvivṛtī commentary.

lavaṇaṃ nindyaṃ kapilā gopī cararāśayastavārthitayā $\scriptstyle|$
laghunoddiṣṭo rājñaḥ praḷayo dhāmnāṃ trinetra narakapuram $\scriptstyle||$
savadhūṭīndro jalasūradrīhimavān gurustriśaṅkuvaraḥ $\scriptstyle|$
varado vajrī tilabhūrmeruḥ kālena tatra nṛpaticaraḥ $\scriptstyle||$
tilakaṃ sāndraṃ dhāvatisarit na me kuñjaro nivṛttajaraḥ $\scriptstyle|$
śreṣṭhakaḷatramamāśādhātrī dhūpo’gnīnāmbutilavanagaḥ $\scriptstyle||$

The values given by the Laghuvivṛtī commentary are remarkably accurate. For example, for $k=24$, the commentary gives $s=305^{\prime}\,\,0^{\prime\prime}$ whereas a modern computation using Maxima software, with $r=\frac{21600}{2\pi}$, yielded $s=304^{\prime}\,\,58.03^{\prime\prime}$, a difference of nearly $2^{\prime\prime}$ only.

The lookup table gives the values of the arc lengths corresponding to a predefined set of jyā values. If the jyā whose arc length is to be determined happens to be very close to one of the values listed in the table, then the corresponding arc length can be read off from the table. For example, if $\text{jy\={a}}\,\,(s)=200^{\prime}$ which is very close to the value $202^{\prime}\,\,08^{\prime\prime}$ listed in the table, then we may take $s$ as the corresponding arc length given in the table, namely, $202^{\prime}\,\,15^{\prime\prime}$.

It is interesting to speculate why the table contains only values up to $k=24^{\prime\prime}$. One important application of the table might have been the computation of the true longitudes of the Moon which involved the computation of an expression of the form (see [6] p. 90)

The maximum value of $\tfrac{7}{80}\sin(\theta_{0}-\theta_{m})$ is

Hence in the computation of the true longitudes of the Moon, one would be required to compute arc lengths corresponding to jyā-s less than this maximum value only. Note that the value of jyā corresponding to $k=23^{\prime\prime}$ is $300^{\prime}\,\,18^{\prime\prime}$ and that corresponding to $k=24^{\prime\prime}$ is $304^{\prime}\,\,36^{\prime\prime}$, and that the maximum value lies between these two values.

Nīlakaṇṭha Somayājī’s formula appears in verse 14 Chapter 2 of tantrasaṅgraha (see [6] p. 68).

jyayorāsannayorbhedabhaktastatkoṭiyogataḥ $\scriptstyle|$
chedastena hṛtādvighnā trijyā taddhanurantaram $\scriptstyle||$

“The sum of the kojyā-s divided by the difference of those two jyā-s, which are close to each other, forms the cheda (divisor). Twice the trijyā divided by this is the difference between the corresponding arcs.”

Tantrasaṅgraha does not give any rationale for Eq.(18). The editors of a modern edition of Tantrasaṅgraha have suggested that a geometrical argument could be the method by which Indian astronomers arrived at the result (see [6] pp. 69–70). We present such an argument below which is a slight variant of the argument given in [6].

In Figure 2, we assume that $s_{2}-s_{1}=\mathchoice{\vbox{ \hbox{\leavevmode\resizebox{15.71458pt}{0.0pt}{{\char 62\relax}}} \nointerlineskip\hbox{$\displaystyle PQ$}}}{\vbox{ \hbox{\leavevmode\resizebox{15.71458pt}{0.0pt}{{\char 62\relax}}} \nointerlineskip\hbox{$\textstyle PQ$}}}{\vbox{ \hbox{\leavevmode\resizebox{11.0002pt}{0.0pt}{{\char 62\relax}}} \nointerlineskip\hbox{$\scriptstyle PQ$}}}{\vbox{ \hbox{\leavevmode\resizebox{7.85728pt}{0.0pt}{{\char 62\relax}}} \nointerlineskip\hbox{$\scriptscriptstyle PQ$}}}$ is small and so we take

We shall now compute the length of the chord $PQ$.

Let $J$ be the midpoint of the chord $PQ$. We drop perpendiculars $PL,QM,JK$ to $OA$ and $PD$ to $QM$. $K$ is the midpoint of $ML$. From the figure we have:

Since $J$ is the midpoint of $PQ$, $OJ$ is perpendicular to $PQ$. It follows that the triangles $PQD$ and $JOK$ are similar. hence:

Using the expressions for $QD$ and $OK$, we have:

Now, $JO\approx r$ and so we have

Combining Eq.(22) and Eq.(23) we get

Let $s$ be an unknown arc whose jyā is known, say, jyā  $(s)=m$. Using the value of jyā  $(s)$, the value of kojyā  $(s)$ can be calculated using the the following result:

Now, let $m$ be between two consecutive jyā values, say jyā  $(s_{1})$ and jyā  $(s_{2})$, listed in a table of jyā-s, say Mādhava’s sine table, and let $s_{1}<s_{2}$. Let $m$ be closer to jyā  $(s_{1})$. Using Eq.(18), we have:

from which we get $s=s_{1}+p$. If $m$ is closer to jyā  $(s_{2})$, we have to calculate $s_{2}-s$ and subtract it from $s_{2}$ to get $s$.

To illustrate the method with a numerical example, let us calculate arcsin  $(3000^{\prime})=s$. So, here $m=\text{jy\={a}}\,\,(s)=3000^{\prime}$. We perform all computations in the units of arc-thirds and hence the radius $r$ will be taken as

and $m$ as

We first calculate kojā  $(s)$:

In Mādhava’s sine table, the jyā value $3000^{\prime}$ lies between $\text{jy\={a}}\,\,(3600^{\prime})=2977^{\prime}\,\,10^{\prime\prime}\,\,34^{\prime\prime\prime}$ and $\text{jy\={a}}\,\,(2825^{\prime})=3083^{\prime}\,\,13^{\prime\prime}\,\,17^{\prime\prime\prime}$ and is closer to $\text{jy\={a}}\,\,(3600^{\prime})$. Hence we take $s_{1}=3600^{\prime}$ and

Further, we have:

Therefore, we have

and hence

This is amazingly accurate because by a modern computation we see that

and the difference of this from the value of $m$ is only $1^{\prime\prime\prime}$.

Let $C$ be the circumference of a circle of diameter $D$. Let $C$ be unknown but an approximation $C^{\ast}$ to $C$ be known. The procedure described in Yuktibhāṣā yields a better approximation to $C$.

We consider a circle of radius $D$ and an arc of length $\frac{C^{\ast}}{4}$ on this circle (see Figure 3). We calculate $a^{\ast}=\text{jy\={a}}\,(\frac{C^{\ast}}{4})$ using the Mādhava-Newton series for jyā:

We next calculate $b^{\ast}=\text{koj\={a}}\,(\frac{a^{\ast}}{4})$ as follows:

The arc $\frac{C}{4}$ is one-eighth of the circle of radius $D$. Hence we have:¹¹1Yuktibhāṣā has given the rationale of these results in an earlier section (see Gaṇita-Yukti-Bhāṣā Section 7.2.2; [7] p. 211).

We assume that $b^{\ast}>a^{\ast}$ so that $C>C^{\ast}$. By jīve-parspara nyāya, we have