A Better Bowstring: The Mādhava Jyā Series

What's better than an improved jyā table and an improved method for interpolation? A full jyā series that lets you compute jyā of any arc, to any degree of precision you wish.

निहत्य चापवर्गेण चापं तत्तत्फलानि च । 
हरेत् समूलयुग्वर्गैस्त्रिज्यावर्गाहतैः क्रमात् ॥
चापं फलानि चाधोऽधो न्यस्योपर्युपरित्यजेत् ।
जीवापत्यै संग्रहोऽस्यैव विद्वान् इत्यदिना कृतः ॥

Multiply the arc by the square of the arc. Multiply successive results by the square of the arc again, and divide by the square of the radius (trjyā) multiplied by sum of successive odd numbers and their squares. Set the arc and the other terms up in a vertical line, and subtract the last from the next, and the result from the next higher, and so on, to get the jīva (jyā). The same can be done by the phrases vidvān and so on.

Or:

\(
jyā(s) = s – s.\frac{s^2}{R^2.(2^2+2)} + s.\frac{s^2}{R^2.(2^2+2)}\frac{s^2}{R^2.(4^2+4)} – s.\frac{s^2}{R^2.(2^2+2)}.\frac{s^2}{R^2.(4^2+4)}.\frac{s^2}{R^2.(6^2+6)} …
\)

निहत्य चापवर्गेण रूपं तत्तत्फलानि च ।
हरेद्विमूलयुग्वर्गैस्त्रिज्यावर्गाहतैः क्रमात् ॥
किन्नु व्यासदलेनैव द्विघ्नेनाद्यं विभज्यताम् ।
फलान्यधोऽधः क्रमशो न्यस्योपर्युपरित्यजेत्
शरापत्यै संग्रहोऽस्यैव स्तेनस्त्रीत्यादिना कृतः ॥

Multiply one by the square of the arc. Multiply successive results by the square of the arc again, and divide by the square of the radius (trjyā) multiplied by difference of successive odd numbers from their squares. Let the divisor of the first term be twice the radius. Set the terms up in a vertical line, and subtract the last from the next, and the result from the next higher, and so on, to get the śara. The same can be done by the phrases stenastrī and so on.

Or:
\(
śara(s) = \frac{s^2}{2R} – \frac{s^2}{2R}.\frac{s^2}{R^2.(4^2-4)} + \frac{s^2}{2R}.\frac{s^2}{R^2.(4^2-4)}.\frac{s^2}{R^2.(6^2-6)} …
\)

Since \( jyā(s) = R.sin(\frac{s}{R})\), and \(n.(n+1) = n^2+n=(n+1)^2-(n+1)\) these two series can be seen to be equivalent to the modern series:
\(
sin(\theta) = {\theta} – \frac{\theta^3}{3!} + \frac{\theta^5}{5!} … \\
vers(\theta) = \frac{\theta^2 }{2!} – \frac{\theta^4}{4!} + \frac{\theta^6}{6!} …
\)

These series yield jyā or śara of any arc, correct to thirds of arc, as we will soon see.

Vidvān – Partly Precomputed Series

Having derived these series, we ask, how many terms do we need to compute? We could go as far as we need to to get the accuracy we need. Mādhava has made things even easier for us by giving us a partially pre-computed series that is accurate to the thirds of arc.

विद्वान्स्तुन्नबलः कवीशनिचयः सर्वार्थशीलस्थिरो ।
निर्विद्धाङ्गनरेन्द्ररुङ् निगदितेष्वेषु क्रमात् पञ्चसु ॥
आधस्त्यात् गुणितादभीष्टधनुषः कृत्या विहृत्यान्तिम-
स्याप्तं शोध्यमुपर्युपर्यथ घनेनैवं धनुष्यन्ततः ॥
The phrases vidvān (44″), tunnabala (33″06”'), kavīśanicaya (16'05”41”'), sarvārthaśīlasthiro (273'57”47”') nirviddhāṅganarendraruṅ (2220'39”40”') are successive multipliers, placed in reverse order, successively multiplied by the square of the arc divided by the square of a quarter-circle, and subtracted from the next, finally multiplied by the cube of the desired arc divided by the cube of a quarter circle, then subtracted from the desired arc.

\(
jyā(s) = s – \frac{s^3}{5400^3}.(2220'39”40”' – \frac{s^2}{5400^2}.(273'57”47”' – \frac{s^2}{5400^2}.(16'05”41”' – \frac{s^2}{5400^2}.(33″06”' – \frac{s^2}{5400^2}.44”'))))
\)

We also have the convenient encodings नानाज्ञानतपोधरः for \(5400^2 = 29160000\) and अज्ञाननुन्ने नवतत्त्वसंशयः for \(5400^3 = 157464000000\)

Similarly for śara:
स्त्येनस्त्रीपिशुनस्सुगन्धिनगनुद्भद्राङ्गभव्यासनो
मीनाङ्गो नरसिंह ऊनधनकृद्भूरेव षट्स्वेषु तु ।
आधस्त्यात् गुणितादभीष्टधनुषः कृत्या विहृत्यान्तिम-
स्याप्तं शोध्यमुपर्युपर्यथ फलं स्यादुत्क्रमज्यान्त्यजम् ॥

\(
śara(s) = \frac{s^2}{5400^2}.(4241'09”0”' – \frac{s^2}{5400^2}.(872'03”05”' – \frac{s^2}{5400^2}.(71'43”24”' – \frac{s^2}{5400^2}.(3'09”37”' – \frac{s^2}{5400^2}.(5″12”' – \frac{s^2}{5400^2}.6”')))))
\)

Where do we get these coefficients from? It can be seen that these are derived from computing the Mādhava series for an arc of 5400′ (and hence the division by 5400′ in the formula, by त्रैराशिकम् )

The Mādhava Jyā Table

For those un-inclined to compute the series for arbitrary arcs, worry not – the Mādhava jyā table – understood to have been computed using the vidvān method – exists. Similar to the Āryabhaṭa table, Mādhava jyā table is a verse-encoded table of jyās at the standard 24 points, accurate to the thirds, ie: 3600 times more accurate than Āryabhaṭa. This can be used with the इष्टदोःकोटिधनुषोः method of interpolation to obtain intermediate values.

	Verse	Value	Modern
1	श्रेष्ठं नाम वरिष्ठानां	224,50,22,0	224,50,21,49
2	हिमाद्रिर्वेदभावनः	448,42,58,0	448,42,57,35
3	तपनो भानु सूक्तज्ञो	670,40,16,0	670,40,16,2
4	मध्यमं विद्धि दोहनं	889,45,15,0	889,45,15,36
5	धिगाज्योनाशनं कष्टं	1105,1,39,0	1105,1,38,56
6	छन्नभोगा शयाम्बिका	1315,34,7,0	1315,34,7,26
7	मृगाहारो नरेशोयं	1520,28,35,0	1520,28,35,27
8	वीरो रणजयोत्सुकः	1718,52,24,0	1718,52,24,11
9	मूलं विशुद्धं नाळस्य	1909,54,35,0	1909,54,35,11
10	गानेषु विरळा नराः	2092,46,3,0	2092,46,3,29
11	अशुद्धिगुप्ता चोरश्रीः	2266,39,50,0	2266,39,50,12
12	शङ्कुकर्णो नगेश्वरः	2430,51,15,0	2430,51,14,35
13	तनुजो गर्भजो मित्रं	2584,38,6,0	2584,38,5,31
14	श्रीमानत्र सुखी सखे	2727,20,52,0	2727,20,52,22
15	शशी रात्रौ हिमाहारौ	2858,22,55,0	2858,22,55,6
16	वेगज्ञः पथि सिन्धुरः	2977,10,34,0	2977,10,33,43
17	छायालयो गजो नीलो	3083,13,17,0	3083,13,16,56
18	निर्मलो नास्ति सत्कुले	3176,3,50,0	3176,3,49,57
19	रात्रौ दर्पणमभ्राङ्गं	3255,18,22,0	3255,18,21,34
20	नागस्तुङ्गनखो बली	3320,36,30,0	3320,36,30,12
21	धीरो युवा कथालोलः	3371,41,29,0	3371,41,29,8
22	पूज्यो नारीजनैर्भगः	3408,20,11,0	3408,20,10,56
23	कन्यागारे नागवल्ली	3430,23,11,0	3430,23,10,38
24	देवो विश्वस्थली भृगुः	3437,44,48,0	3437,44,48,22

A minor point to note is the use of ळ encoding to 9 in a couple of lines.

Why Three Methods?

Comparing निहत्य चापवर्गेण with the famous व्यासे वारिधिनिहते series, we see that it is tougher to calculate through, and therefore could not have been expected to be used by someone with basic ability. It must be presumed that Mādhava and his followers left multiple options available for those with varying skills in computation.

Yukti for the Mādhava Jyā Series

As usual, we have detailed derivations of these available from Yuktibhāṣā, but we will leave that for the next article. As with the series itself, these proofs are involved, and require a bit of diligence.