31415926536
A famous verse in a later work from the Kerala school, with a कटपयादि mnemonic of pi to ten decimal places goes:
आनूननून्नाननुनुन्ननित्यैस् समाहताश्चक्रकलाविभक्ताः ।
करणपद्धतिः – पुतुमनसोमयाजी
चण्डांशुचन्द्राधमकुम्भिपालैर्व्यासस्तदर्धं त्रिभमौर्विका स्यात् ।।
The circumference of a circle in minutes of arc is multiplied by आनूननून्नाननुनुन्ननित्यम् (=10000000000) and divided by चण्डांशुचन्द्राधमकुम्भिपालः = (31415926536) to yield the diameter ….
This is accurate to 11 digits including the leading 3.
कटपयादि encoding
नञावचश्च शून्यानि संख्या: कटपयादय:।
मिश्रे तूपान्त्यहल् संख्या न च चिन्त्यो हलस्वर:॥
न, ञ and vowels are zero, numerals 1-9 are encoded starting with क, ट, प, य. Only the last consonant of a conjunct is considered, and consonants without attached vowels are ignored.
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 0 |
| क | ख | ग | घ | ङ | च | छ | ज | झ | ञ |
| ट | ठ | ड | ढ | ण | त | थ | द | ध | न |
| प | फ | ब | भ | म | |||||
| य | र | ल | व | श | ष | स | ह | ळ | अ-औ |
Numbers are encoded least significant digit first, according to the maxim संख्यानां वामतो गतिः
By this, we get आनूननून्नाननुनुन्ननित्यम् = 10000000000 and चण्डांशुचन्द्राधमकुम्भिपालः = 31415926536,
How was चण्डांशुचन्द्राधमकुम्भिपालः calculated?
In the previous article in the series, we saw how the व्यासे वारिधिनिहते series converged too slowly to be of practical use, and how it could be made useful by adding a संस्कार, or correction factor. We saw two correction factors, the first order \(\frac{4d}{2(p+1)}\) and the second order \(\frac{\frac{p+1}{2}}{(p+1)^2+1}\), and how they made the series converge well. Neither, however, is good enough to calculate this result in less than 30 terms, which is what would be manually feasible.
Third order संस्कारः (correction)
अन्ते समसङ्ख्यादलवर्गः सैको गुणः सैव पुनः
युगगुणितो रूपयुतः समसङ्ख्यादलहतः भवेद्धारः
| Terms | Result with third order correction | Accuracy (places after decimal point) |
| 10 | 3.141592705349 | 6 |
| 20 | 3.141592654020 | 9 |
| 28 | 3.141592653631 | 10 |
| 30 | 3.141592653615 | 10 |
| 40 | 3.141592653593 | 11 |
| 50 | 3.141592653591 | 11 |
| 100 | 3.141592653590 | 11 |
By calculating to 28 terms, we achieve our desired result of चण्डांशुचन्द्राधमकुम्भिपालः = 3145926536.
Madhava Series Summary
The basic series is
व्यासे वारिधिनिहते रूपहृते व्याससागराभिहते ।
त्रिशरादिविषमसंख्याभक्तमृणं स्वं पृथक्क्रमात् कुर्यात् ॥
To make this converge faster, we add the संस्कारः (correction):
यत्सङ्ख्ययात्र हरणे कृते निवृत्ता हृतिस्तु जामितया
तस्या ऊर्ध्वगता या समसङ्ख्या तद्दलं गुणोऽन्ते स्यात् ||
तद्वर्गो रूपयुतो हारो व्यासाब्धिघातात् प्राग्वत्
ताभ्यामाप्तं स्वमृणे कृते धने क्षेप एव करणीयः ||
Or the more accurate संस्कारः (correction):
अन्ते समसङ्ख्यादलवर्गः सैको गुणः सैव पुनः
युगगुणितो रूपयुतः समसङ्ख्यादलहतः भवेद्धारः
We can calculate the mnemonic चण्डांशुचन्द्राधमकुम्भिपालः = (31415926536) by summing व्यासे… to 28 terms and adding the latter correction.
Alternate series
By incorporating the correction term into the series itself, Yuktibhāṣā derives quicker-converging series as alternatives to व्यासे वारिधिनिहते. There are equivalent to using the correction terms, so are more of a curiosity than providing any further benefit.
We have seen the स्थौल्य (error term) before: \( E_p = \frac{1}{a_{p-1}} + \frac{1}{a_{p}} – \frac{1}{p} \) . We fold it into the series thus
\(\begin{align*}C &= 4.d – \frac{4.d}{3} + \frac{4.d}{5} – \frac{4.d}{7} + … \pm \frac{4.d}{p} \mp \frac{4d}{a_p} \\
&= 4.d [1 – \frac{1}{a_1} + (\frac{1}{a_1} + \frac{1}{a_3} – \frac{1}{3}) – (\frac{1}{a_3} + \frac{1}{a_5} – \frac{1}{5}) … \mp (\frac{1}{a_{p-1}}+\frac{1}{a_p}-\frac{1}{p}) \\
&= 4.d [ 1 – \frac{1}{a_1}] + 4d [E_3-E_5+…\mp E_p]
\end{align*}\)
By simple rearrangement, we have transformed the original series with संस्कारः into alternate series using the respective स्थौल्य. We have calculated the स्थौल्य for first and second order corrections previously. Unfortunately, the स्थौल्य for the most accurate correction is hard to use. Thus, we get the two alternate series incorporating संस्कारः
First alternate series
We get this by incorporating the first order correction
व्यासाद् वारिधिनिहतात् पृथगाप्तं त्र्याद्ययुग्विमूलघनैः
त्रिघ्नव्यासे स्वमृणं क्रमशः कृत्वा परिधिरानेयः ||
Second alternate series
Incorporating the second order error term instead, we get:
समपञ्चाहतयो या रुपाद्ययुजां चतुर्घ्नमूलयुताः
ताभिः षोडशगुणितात् पृथगाहृतेषु विषमयुतेः
समफलयुतिमपहाय स्यादिष्टव्याससंभवः परिधिः
\(C = 16d.\frac{1}{1^5+4.1} – 16d.\frac{1}{3^5+4.3} + 16d.\frac{1}{5^5+4.5} ….\)
