Fun with Sankalitas

In the previous article in the series, we saw how the famous Madhava circumference (Pi) series was justified using an argument based on the geometry of Kerala roofs, and the mathematical techniques of shodyaphala and sankalita (सङकलितम्). Since the idea of sankalita is central to the Calculus of the Kerala school, it makes sense to spend some time exploring it.

The mulasankalita (मूलसङ्कलितम्)

\( 1+2+…n=\frac{n.(n+1)}{2} \)

The मूलसङ्कलितम् has been known in Indian Mathematics since Aryabhata. Yuktibhāṣā proves this known result in two ways, and then extends it to the limiting case of very small segments thus:

Graphical yukti (proof)

Consider the मूलसङ्कलितम् arranged as a two-dimensional figure as here, with the shaded squares expressing successive numbers. The shaded area in this particular figure represents the sum of the first 10 numbers \( 1+2+3+…+10\). If we invert the same shape (the unshaded part of the figure), and place it next to the original uninverted sankalita, we get a rectangle of height 10 and width one more than 10 = 11 units, whose area is therefore 10*11 = 110. The area of the shaded part is therefore half of it, which is \(110/2=55\). Since this argument works for any number, we see that \( 1+2+…n=\frac{n.(n+1)}{2} \). This relationship is the basic one used to prove higher order sankalitas and limits.

Algebraic yukti

Yuktibhāṣā considers the case of a very large n, which occurs in the sankalita form that turns up often in Kerala Calculus. Consider the case where a finite length r is divided into a very large number of segments n.

We change our scale such that each segment is of unit length (and therefore, r = n units in our new scale), and the sum of the segments is expressed as:
\( S_n = (1+2+…n) \)
This can be seen to be equal to:
\(
\begin{align*}
S_n &= ((n-(n-1))+(n-(n-2))+ …n-1+n) \\
&= n*n – ((n-1)+(n-2)+…1) \\
&= n^2 – S_{n-1} \\
\end{align*}
\)

Subcase: regular n

For the regular case:
\( \begin{align*}
S_n&= n^2 – (S_n – n) &&(\text{substituting} \quad S_{n-1} = S_n -n) \\
2. S_n &= n^2 + n \\
S_n &= \frac{n(n+1)}{2}
\end{align*}\)
Which is the same result as before, but derived purely algebraically. But the more interesting case for us in the context of Kerala Calculus is that of a very large n

Subcase: Large n

If n is very large – which means the unit is very small, close to atomic (अणुः), we see that
\( \begin{align*}
S_n&= n^2 – S_n &&(\text{substituting} \quad S_{n-1} \approx S_n) \\
2.S_n &\approx n^2 \\
S_n &\approx \frac{n^2}{2}
\end{align*}\)
In this scale, the length r is n units, and therefore the sum of these segments is equivalent to \(r^2/2\) in this scale.

Extending the Sankalitas – power sums for large n

Extending the mulasankalita, we move on to samaghatasankalitas (समघातसङ्कलितानि) or power sums. Yuktibhāṣā derives their expressions for the limit of large n

(If you find too many details not to your taste, you can skip directly to the summary of results)

Vargasankalita (वर्गसङ्कलितम्)- sums of squares for large n

Using some interesting sleight of hand, and the earlier expression for \(S_n\), the मूलसङ्कलितम् for large n, we can calculate the sum of squares, वर्गसङ्कलितम्, thus:
\(
\begin{align*}
S_n^{(2)} &= 1^2+2^2+3^2+…n^2 \\
&= 1.1 + 2.2 + 3.3 + …n.n \\
&= n.1 + n.2 + …n.n \\
&\quad-((n-1).1 + (n-2).2 + ….1.(n-1)) \\
&= n.S_n \\
&\quad- ((n-1)+(n-2)+(n-3)….+1) \\
&\quad- ((n-2)+(n-3)….+1) \\
&\quad- ((n-3)…..+1) \\
&\quad- … \\
&\quad- 1
\end{align*}
\)
(the triangular sum above is interesting, and we will see it again later)
We can simplify this using the approximation for S_n for large n we derived earlier.
\(
\begin{align*}
S_n^{(2)} &= n.S_n – S_{n-1} – S_{n-2} – … S_1 \\
&\approx \frac{n.n^2}{2} – \frac{(n-1)^2}{2} – \frac{(n-2)^2}{2} – \frac{(n-3)^2}{2} …. \\
&= \frac{n.n^2}{2} – \frac{1}{2}.((n-1)^2 + (n-2)^2 + …. 1) \\
&= \frac{n.n^2}{2} – \frac{1}{2}. S_{n-1}^{(2)}
\end{align*}
\)

Since for large n \( S_n^{(2)} \approx S_{n-1}^{(2)} \), we have
\(
S_n^{(2)} \approx \frac{n.n^2}{2} – \frac{1}{2}. S_{n}^{(2)} \\
S_n^{(2)} \approx \frac{n^3}{3}
\)
for large n

Ghanasankalita (घनसङ्कलितम्) – sums of cubes for large n

Using a very similar argument, we can derive an approximation for the sum of cubes (घनसङ्कलितम्) and all higher power sums (समघातसङ्कलितानि).
\( \begin{align*}
S_n^{(3)} &= 1^3+2^3+3^3+…n^3 \\
&= 1.1^2 + 2.1^2 + … n.n^2 \\
&= n.(1^2+2^2 + ….+n^2) \\
&\quad- ((n-1).1^2 + (n-2).2^2 + ….1.(n-1)^2)
\end{align*}\)
Which simplifies, analogously to the case of the वर्गसङ्कलितम्, to
\(
S_n^{(3)} \approx n . S_n^{(2)} – \frac{1}{3}.S_n^{(3)} \\
S_n^{(3)} \approx \frac{n^4}{4}
\)
for large n

Samaghatasankalita (समघातसङ्कलितम्) – sums of higher powers for large n

Using a very similar argument, we can derive an approximation for the sum of higher powers (समघातसङ्कलितम्)
\( \begin{align*}
S_n^{(k)} &= 1^k+2^k+3^k+…n^k \\
&= 1.1^{k-1} + 2.1^{k-1} + … n.n^{k-1} \\
&= n.(1^{k-1}+2^{k-1} + ….+n^{k-1}) \\
&\quad- ((n-1).1^{k-1}+ (n-2).2^{k-1} + ….1.(n-1)^{k-1})
\end{align*}\)
Which simplifies to
\(
S_n^{(k)} \approx n . S_n^{(k-1)} – \frac{1}{k}.S_n^{(k)} \\
S_n^{(k)} \approx \frac{n^{k+1}}{k+1}
\)
for large n
The astute reader would have noted the analogy with the modern \( \int x^k.dx = \frac{x^{k+1}}{k+1} \). As we would expect from this analogy, we can calculate areas, surface areas and volumes using these sankalitas, as we shall see soon enough. Before that, though, we take a small detour to the concept of the sankalita-sankalita, also called the sum of sums, or second order sum and its higher level analogs.

Extending the Sankalita – the Sankalita-sankalita (सङ्कलितसङ्कलितम्)

We can extend the mulasankalitas in another way too. Instead of adding higher powers of numbers, we can add mulasankalitas themselves! The सङ्कलितसङ्कलितम् is defined as the sum of mulasankalitas up to a given number
\(
\begin{align*}
SS^{(2)}_n &= S_n + S_{n-1} + …. + 1 \\
&= n + (n-1) + (n-2) + … + 1 \\
&\quad+ (n-1) + (n-2) + … + 1 \\
&\quad+ (n-2) + … + 1 \\
&\quad … \\
&\quad + 1
\end{align*}
\)
We saw this triangular sum in the expression for vargasankalita, but did not explicitly name it then.
We can also define higher order sankalita-sankalitas. For example the sankalita-sankalita-sankalita
\(
SS^{(3)}_n = SS^{(2)}_n + SS^{(2)}_{n-1} + SS^{(2)}_{n-2} + …+SS^{(2)}_1
\)
and the general k-th order sankalita-sankalita as
\(
SS^{(k)}_n = SS^{(k-1)}_n + S^{(k-1)}_{n-1} + S^{(k-1)}_{n-1} + … + S^{(k-1)}_{1}
\)

Narayana Pandita (c. 1350 CE) has derived the general expression for the sankalita-sankalitas as
\(
\begin{align*}
SS^{(2)}_n &= \frac{n.(n+1)(n+2)}{1.2.3}\\
SS^{(3)}_n &= \frac{n.(n+1)(n+2)(n+3)}{1.2.3.4}\\
… \\
SS^{(k)}_n &= \frac{n.(n+1)….(n+k)}{1.2.3..(k+1)}
\end{align*}
\)

Sankalita-sankalitas for large n

By directly considering the case of large n in Narayana Pandita's formula, we can see that \(SS^{(k)}_n \approx \frac{n^{k+1}}{(k+1)!} \), where ! denotes the factorial \(k! = 1.2.3..k\)

Yuktibhāṣā calculates this explicitly thus:
\(
\begin{align*}
SS^{(2)}_n &= S_n + S_{n-1} + …. + 1 \\
&\approx \frac{n^2}{2}+ \frac{(n-1)^2}{2} + … 1 &&\text{(for large n)}\\
&= \frac{1}{2}.S^{(2)}_n \\
&= \frac{n^3}{2.3}
\end{align*}
\)
and works up to: \(SS^{(k)}_n \approx \frac{n^{k+1}}{(k+1)!} \) for large n.

In the previous article, we saw how वर्गसङ्कलितम् was used to derive the Madhava Pi series. In the next article, we will see how the area of a circle, and the volume of a sphere are calculated using these.