Recap
In the previous article in this series, we saw how Āryabhaṭa's jyā table was calculated. It was a remarkably simple recurrence relation, with accuracy to one minute of arc (using the Indian standard circle of circumference 21600′) at 24 points 225′ apart.
प्रथमाच्चापज्यार्धाद्यैरूनं खण्डितं द्वितीयार्धम् ।
तत्प्रथमज्यार्धांशैस्तैस्तैरूनानि शेषाणि ।।
We set both first jyā B1 and the first difference K1 as equal to 225. Then, further differences (खण्डज्या) and actual jyās (पिण्डज्या) are calculated using the recurrence relations
K_2 = K_1 – \frac{B_1}{B_1} \\
K_n = K_{n-1} – \frac{B_{n-1}}{B_1} \\
B_n = K_n + B_{n-1}
\)
The resulting खण्डज्या are encoded in the verse
मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कलार्धज्याः ||
For intermediate points in between the 24 standard tabular points, we use linear interpolation.
Making the tabular jyā more accurate
The Kerala School came up with two approaches to make tabular sines, or पठितज्या as they called it, more accurate. The first, which we shall see in this article, is a modification of Āryabhaṭa's recurrence, plus a better method for interpolation. The second, which we shall see later, resulted in a full infinite series for jyā allowing any degree of accuracy to be achieved.
Improved recurrence
This verse is from Tantrasaṅgraha of Nīlakaṇṭha Somayāji and is also seen in Yuktibhāṣā
विलिप्तदशकोना ज्या राश्यष्टांशधनुः कलाः ।
आद्यज्यार्धात्ततो भक्ते सार्धदेवाश्विभिस्ततः ।।
त्यक्ते द्वितीयखण्डज्या द्वितीयज्या च तद्युतिः ।
ततस्तेनैव हारेण लब्धं शोद्यं द्वितीयतः ।।
खण्डात्तृतीयखण्डस्स्यात् द्वितीयस्तद्युतो गुणः ।
तृतीयस्स्यात्ततश्चैवं चतुर्थाद्याः क्रमात्गुणाः ।।
The jyā of an eighth of a rāśi is its length minus ten seconds
From the first jyā is subtracted itself divided by 233′ 30″ to get the second jya difference and the second jyā is obtained by adding that to the first jyā
Further subtrahends are found by dividing successive jyās with the same divisor, and are subtracted from jyā differences to find successive differences.
Further jyās are obtained by adding successive differences to prior jyās
The Kerala school replaced the Āryabhaṭa recurrence with the precise relation
\(K_n = K_{n-1} – {(\frac{a}{R})}^2.B_{n-1} \\
B_n = K_n + B_{n-1}
\)
where R is the radius, and a is the full-chord of the quarter circumference divided by 24, (ie 225′). So far, this is perfectly precise, as we shall see in the next article where we will look at the saṅkalita for this. To make this computable, they use the approximation \(a = 225′\). This is more accurate than the Āryabhaṭa approximation which sets the corresponding jyā to 225′, since the full-chord is between the jyā and the corresponding arc in length. This leads to the approximations
\({(\frac{a}{R})}^2 \approx 233′ 30″\) (नीलोबालारिः or सार्धदेवाश्वि in bhūtasaṅkhyā).
This was later improved by Śankaravāriyar to \({(\frac{a}{R})}^2 \approx 233′ 32″\) (रङ्गेबालास्त्री)
Also,
\(K_1 \approx 224′ 50″\)
This again was improved by Śankaravāriyar as,
\(K_1 \approx 224′ 50″22‴\)
What we gain here is better accuracy, at the cost of
- A more complex divisor – 233′ 32″ in the place of 225′
- A more complex first sine – 224′ 50″ 22”' in the place of 225′
Comparsion of tabular jyā accuracy
Comparing the accuracy of tabular jyā by these methods, with the modern values, we see that Śankaravāriyar is accurate to a few seconds of arc, while Nīlakaṇṭha's table is off by upto about 20 seconds. Āryabhaṭa, as we have noted is accurate to about a minute of arc.
| Āryabhaṭa | Nīlakaṇṭha | Śankaravāriyar | Modern | |
| 1 | 225 | 224,50,0,0 | 224,50,22,0 | 224,50,21,49 |
|---|---|---|---|---|
| 2 | 449 | 448,42,13,37 | 448,42,58,1 | 448,42,57,35 |
| 3 | 671 | 670,39,9,19 | 670,40,16,55 | 670,40,16,2 |
| 4 | 890 | 889,43,45,11 | 889,45,17,10 | 889,45,15,36 |
| 5 | 1105 | 1104,59,43,37 | 1105,1,41,33 | 1105,1,38,56 |
| 6 | 1315 | 1315,31,45,42 | 1315,34,11,31 | 1315,34,7,26 |
| 7 | 1520 | 1520,25,45,33 | 1520,28,41,30 | 1520,28,35,27 |
| 8 | 1719 | 1718,49,4,4 | 1718,52,32,46 | 1718,52,24,11 |
| 9 | 1910 | 1909,50,42,38 | 1909,54,46,57 | 1909,54,35,11 |
| 10 | 2093 | 2092,41,36,2 | 2092,46,19,8 | 2092,46,3,29 |
| 11 | 2267 | 2266,34,45,13 | 2266,40,10,30 | 2266,39,50,12 |
| 12 | 2431 | 2430,45,29,17 | 2430,51,40,20 | 2430,51,14,35 |
| 13 | 2585 | 2584,31,36,59 | 2584,38,37,34 | 2584,38,5,31 |
| 14 | 2728 | 2727,13,37,36 | 2727,21,31,36 | 2727,20,52,22 |
| 15 | 2859 | 2858,14,51,2 | 2858,23,42,25 | 2858,22,55,6 |
| 16 | 2978 | 2977,1,37,15 | 2977,11,30,3 | 2977,10,33,43 |
| 17 | 3084 | 3083,3,25,0 | 3083,14,23,12 | 3083,13,16,56 |
| 18 | 3177 | 3175,52,59,31 | 3176,5,7,7 | 3176,3,49,57 |
| 19 | 3256 | 3255,6,29,41 | 3255,19,50,32 | 3255,18,21,34 |
| 20 | 3321 | 3320,23,34,1 | 3320,38,11,52 | 3320,36,30,12 |
| 21 | 3372 | 3371,27,26,0 | 3371,43,24,23 | 3371,41,29,8 |
| 22 | 3409 | 3408,4,58,21 | 3408,22,20,35 | 3408,20,10,56 |
| 23 | 3431 | 3430,6,46,22 | 3430,25,35,30 | 3430,23,10,38 |
| 24 | 3438 | 3437,27,10,26 | 3437,47,29,10 | 3437,44,48,22 |
Improved Interpolation
Rather than basic linear interpolation, the Kerala School introduced a more complex interpolation equation to improve accuracy:
इष्टदोःकोटिधनुषोः स्वसमीपसमीरिते
ज्ये द्वे सावयवे न्यस्य कुर्यादूनाधिकं धनुः
द्विघ्नतल्लिप्तिकाप्तैकशरशैलशिखीन्दवः
न्यस्याच्छोधाय च मिथः तत्संस्कारविधित्सया
छित्वैकां प्राक् क्षिपेज्जह्यात् तद्धनुष्याधिकोनके
अन्यस्यामथ तां द्विघ्नां तथा स्यामिति संस्कृतिः
इति ते कृतसंस्कारे स्वगुणौ धनुषोस्तयोः
तत्राल्पीयः कृतिं त्यक्त्वा पदं त्रिज्याकृते परः
For a desired arc, find the tabular jyā and kojyā below and above it
compute the differences of arc length to the tabular points above and below,
Divide 13751 by these and keep aside as divisors for mutual correction
Divide each one (ie jyā below and kojyā above) by this, and subtract from other (jyā above and kojyā below). Mutliply the results by two, and add to the original (ie jyā below and kojyā above) to get the interpolated jyā and kojyā.
Of the two, take the smaller, square it, subtract from the square of the radius and take the root to get the other
Again, this verse is from Tantrasaṅgraha and is also seen in Yuktibhāṣā. The idea is better explained mathematically thus:
\(
D = \frac{13751}{2.d} \\
\begin{align*}
jyā(B+d) &= jyā(B) + \frac{2}{D}(kojyā(B) – \frac{jyā(B)}{D}) \\
jyā(B-d) &= jyā(B) – \frac{2}{D}(kojyā(B) + \frac{jyā(B)}{D}) \\
kojyā(B-d) &= kojyā(B) + \frac{2}{D}(jyā(B) – \frac{kojyā(B)}{D})\\
kojyā(B+d) &= kojyā(B) – \frac{2}{D}(jyā(B) + \frac{kojyā(B)}{D})\\
\end{align*}
\)
How does this work? Where does the mysterious 13751 come from? It turns out that it is a decent approximation for four times the radius of the standard circle. Using the जीवे परस्परन्यायः (which we will see in a later article), which modern readers will know as the equation
\( sin(A+B) = sin(A).cos(B) + cos(A).sin(B)\),
and the approximation for small t
\(
sin(t) = t \\
cos(t) = \sqrt{1-t^2}
\)
and recasting to the jyā framework allows us to derive the equations above.
We aren't done yet
Granted, this gets us better accuracy than Āryabhaṭa, but knowing the Kerala school, the gains from this are clearly not sufficient to satisfy their rather exacting standards. So, this improved method is extended further to generate an infinite series for jyā, just as we saw for the circumference. We will see that the jyā series produces much improved accuracy than the recurrence relation, and will lead us to the famous Madhava jyā table and its interesting mnemonics.
