The logic behind व्यासे वारिधिनिहते …

व्यासे वारिधिनिहते रूपहृते व्याससागराभिहते । 
त्रिशरादिविषमसंख्याभक्तमृणं स्वं पृथक्क्रमात् कुर्यात् ॥
from the diameter multiplied by four and divided by one 
reduce and add in turn, the diameter multiplied by four, and 
respectively divided by the odd numbers three, five and so on. 

In the previous article in this series, we made a brief acquaintance with the Madhava series described by this verse.
\( \frac{\pi}{4} = 1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} \)

How though, did Madhava and his successors establish this result, and the others mentioned in the article? Was this a one-off, or did they establish a sufficient body of techniques that deserve to be called Calculus?

Building up the rationale

Consider a circle inscribed in a square. What the figure above shows is the top-right quadrant, which contains a quarter of the circumference. The top point, where the circle touches the square is marked E (for “East”) and the rightmost point as S (for “South”). This follows the convention of Indian Mathematics, where figures are oriented with East towards the top and North to the left. Arcs usually start from the East and grow Northward, as we shall see in later reasoning. Here, we follow Yuktibhāṣā, which we have seen in the previous article, and look at the quadrant with the South-East corner (the अग्निकोणः)‌ as in the above figure, as the problem is symmetric.

Having marked the centre of the circle as O, and the corner of the quadrant as A (for “Agni”, as above), we draw the diagonal OA of the square. If the diameter of the circle is considered to be d, we know that the side of the quadrant would be d/2, or r, the radius. Divide the line segment EA into n equal parts, and mark the intermediate equidistant points as \( A_1, A_2 … A_{n-1} \), each of which we join to the center O. Let OA meet the circle at C, and \( OA_1, OA_2 … OA_{n-1} \) respectively meet the circle at \( C_1, C_2 … C_{n-1} \). We draw \( EB_1, C_1B_2 … C_{n-1}B_n \) perpendicular to \( OA_1, OA_2 … OA_{n-1}, OA \) . respectively. We also draw \( A_1P_2, A_2P_3 … A_{n-1}P_n \), also perpendicular to \( OA_1, OA_2 … OA_{n-1}, OA \) respectively.

Arc ES is a quarter of the circle, and Arc EC is therefore an eighth part of the circumference. \( EC = EC_1 + C_1C_2 + … C_{n-1}C\) . If we can approximate the arcs \( C_{k-1}C_k \), we can approximate the circumference. But how do we do that?

Each of the karṇas \( OA_i = k_i \) are determined by the jyā-koti-karṇa-nyāya as \( k_i^2 = OE^2 + EA_i^2 = r^2 + (\frac{ir}{n})^2\)

We approximate the arcs with their jyās, and say
\(
\begin{align*} EC &= EC_1 + C_1C_2 + … C_{n-1}C \\
&\approx EB_1 + C_1B_2 + … C_{n-1}N_n
\end{align*}
\)

Triangle \( A_1OE\) can be seen to be similar to Triangle \( A_1EB_1\) and hence \( \frac{EB_1}{EA_1} = \frac{OE}{OA_1}\)
and hence,
\( EB_1 = (\frac{r}{n})\frac{r}{k_1} \)

We can compute further jyās \(C_1B_2\) thus, using similarity arguments similar to the above:
\( \frac{OE}{OA_2} = \frac{A_1P_2}{A_1A_2}\)
\( \frac{r}{k_2} = \frac{A_1P_2}{\frac{r}{n}}\)
and
\( \frac{C_1B_2}{OC_1} = \frac{A_1P_2}{OA_1}\)
\( \frac{C_1B_2}{r} = \frac{A_1P_2}{k_1}\)
and hence
\( C_1B_2 = (\frac{r}{n}).(\frac{r^2}{k_1.k_2})\)

Thus,
\( \begin{align*}
EC &\approx EC_1 + C_1C_2 + … C_iC_{i+1} + … + C_{n-1}C \\
&= (\frac{r}{n})\frac{r}{k_1} + (\frac{r}{n}).(\frac{r^2}{k_1.k_2}) + … + (\frac{r}{n}).(\frac{r^2}{k_i.k_{i+1}}) + … (\frac{r}{n}).(\frac{r^2}{k_{n-1}.k_n}) \\
&= (\frac{r}{n})\frac{r^2}{k_0k_1} + (\frac{r}{n}).(\frac{r^2}{k_1.k_2}) + … + (\frac{r}{n}).(\frac{r^2}{k_i.k_{i+1}}) + … (\frac{r}{n}).(\frac{r^2}{k_{n-1}.k_n}) \end{align*}\)
since \(k_0 = r\)
This is a formidable equation, but can be simplified further with an approximation. Before we go there, we note that Yuktibhāṣā points out that the approximate equality above tends to a full equality when the number of segments (ie n) increases, and each segment is very small.

Simplifying

Yuktibhāṣā notes here that the divisor is rather inconvenient (ഹാരകന്നാനുരൂപം – हारकन्नानुरूपम्) and proposes a simplification
\( \frac{1}{k_i.k_{i+1}} \approx (\frac{1}{2}) (\frac{1}{k_i^2}+\frac{1}{k_{i+1}^2})\)
which is justified by the argument that two adjacent \( k_i \) will tend to each other.
Thus, we have:
\( EC \approx \frac{r}{n}. (\frac{1}{2}).(\frac{r^2}{k_0^2}+\frac{r^2}{k_1^2}+\frac{r^2}{k_1^2}+\frac{r^2}{k_2^2}+ … +\frac{r^2}{k_{n-2}^2}+\frac{r^2}{k_{n-1}^2} + \frac{r^2}{k_{n-1}^2}+\frac{r^2}{k_n^2}) \)
One more minor asymmetry remains, which an also be approximated away at large n
\( k_n = \sqrt{2}.k_0\)
We note that both of these appear only once in the expression, unlike others which appear twice. At the cost of an error \(\frac{r}{4.n}\) which vanishes for large n, we replace one instance of \(k_0\) with \(k_n\).
\( EC \approx \frac{r}{n}. (\frac{r^2}{k_1^2}+ … \frac{r^2}{k_{n-1}^2}+\frac{r^2}{k_n^2})\)
Now we have a nice and symmetric expression, and more importantly, one that can be summed with the Sankalita technique known since Aryabhata, and improved by Narayana Pandita, combined with a nice sleight-of-hand, called shodyaphala (शेध्यफलम्).

Noting that \(k_i\) is an inconvenient denominator, and applying shodyaphala and the expression \(k_i^2 = r^2+(\frac{i.r}{n})^2\) for a single term in the series, we get
\( \frac{r}{n}. \frac{r^2}{k_i^2} = \frac{r}{n} – \frac{r}{n}\frac{(\frac{ir}{n})^2}{r^2}+\frac{r}{n}\frac{(\frac{ir}{n})^3}{r^3}- …\)
Writing out the entire series this way (this is elaborate, but spare a thought for those working this out before 1400CE!):
\( \begin{align*}
EC &\approx (\frac{r}{n} – \frac{r}{n}\frac{(\frac{r}{n})^2}{r^2}+\frac{r}{n}\frac{(\frac{r}{n})^4}{r^4}- …) \\
&\quad-(\frac{r}{n} – \frac{r}{n}\frac{(\frac{2r}{n})^2}{r^2}+\frac{r}{n}\frac{(\frac{2r}{n})^4}{r^4}- …) \\
&\quad+(\frac{r}{n} – \frac{r}{n}\frac{(\frac{3r}{n})^2}{r^2}+\frac{r}{n}\frac{(\frac{3r}{n})^4}{r^4}- …) \\
&\quad… \\
&\quad-(\frac{r}{n} – \frac{r}{n}\frac{(\frac{nr}{n})^2}{r^2}+\frac{r}{n}\frac{(\frac{nr}{n})^4}{r^4}- …)
\end{align*}\)
Each line in the above equation corresponds to one term in \( EC \approx \frac{r}{n}. (\frac{r^2}{k_1^2}+ … \frac{r^2}{k_{n-1}^2}+\frac{r^2}{k_n^2})\)

Now we group similar powers, and rewrite as
\(\begin{align*}
EC &\approx \frac{r}{n} (1+1+ …+1) \\
&\quad- \frac{r}{n}\frac{(\frac{r}{n})^2}{r^2} (1^2 + 2^2 + … n^2) \\
&\quad+ \frac{r}{n}\frac{(\frac{r}{n})^4}{r^4} (1^4 + 2^4 + … n^4) \\
&\quad- \frac{r}{n}\frac{(\frac{r}{n})^6}{r^6} (1^6 + 2^6 + … n^6) \\
&\quad …
\end{align*}\)
Whereas we formerly had a finite number of terms (n), but with infinite length, we now have an infinite number of terms with finite length (n) – not that it makes a difference, since we want to make n very large anyway. We are now ready to use sankalita (सङ्कलितम्)‌ compact these terms

Using the sama-ghata-sankalita approximation for small units
\( (1^k+2^k…+n^k) \approx \frac{n^{k+1}}{k+1} \), we have
\(
EC \approx \frac{r}{n} (1+1+ …+1) – \\
\frac{r}{n}\frac{(\frac{r}{n})^2}{r^2} (1^2 + 2^2 + … n^2) + \\
\frac{r}{n}\frac{(\frac{r}{n})^4}{r^4} (1^4 + 2^4 + … n^4) – \\
\frac{r}{n}\frac{(\frac{r}{n})^6}{r^6} (1^6 + 2^6 + … n^6) + \\
…\\
\approx r – \frac{r}{3} + \frac{r}{5} – \frac{r}{7} + …
\)
Now since EC is an eighth of the circumference C
\( \frac{C}{8} = r – \frac{r}{3} + \frac{r}{5} – \frac{r}{7} + … \\
C = 4.d – \frac{4.d}{3} + \frac{4.d}{5} – \frac{4.d}{7} + …\) (using d = 2.r)
and therefore,
\( \frac{\pi}{4} = 1 – \frac{1}{3} + \frac{1}{5} – \frac{1}{7} \)
Which is the verse व्यासे वारिधिनिहते

In Conclusion

Examining Yuktibhāṣā's argument, we can see the Kerala School built their Calculus around the ideas of

• Sankalita of small units
• Shodyaphala to get rid of inconvienient denominators in sums
• Approximations of sums by dropping insignificant terms.

We shall see a lot more of this in further articles as we examine further refinements of this series, and series for trigonometric functions, and dive into some of the mathematics in more detail.