What's In A Name?

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?

A Brief Excursion into Indian Trigonometry

Consider the arc CS, centred at O. OC = r is a radius. CB is perpendicular to OS. We take angle COB as \(\theta\). In terms of modern trigonometry,
\( sin(\theta) = \frac{CB}{OC}\),
\(cos(\theta) = \frac{OB}{OC}\), and
\(tan(\theta) = \frac{CB}{OB}\).
Indian “trigonometry”, or more accurately, “circleometry” refers to the half-chord CB as the jyā (“bowstring”) or bhuja (“arm”) of arc CS, and OB as the koti–jyā (“pointed bowstring”), and BS as the utkrama–jyā or śara (“arrow”). OC, the radius is the karṇa. Strictly, it is the arc, not the angle that is the independent variable. However, circles are considered to be 21600′ in circumference by convention, which converts the arc into what we would now call an angle. Most calculations can be done in these units, until we need to convert into units of real length, when we use the “rule of three” (त्रैराशिकम्)‌ – or the principle of proportionality. The familiar relation
\( sin^2(\theta) + cos^2(\theta) = 1 \)
becomes the jyā-koti-karṇa-nyāya (ज्याकोटिकर्णन्याय:)
\( jyā^2 + koti^2 = karṇa^2 \)

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,
\( EB_1 = (\frac{r}{n})\frac{r}{k_1} \)

Illustrating similarity

To describe the same figure we are looking at, Yuktibhāṣā uses a very interesting analogy, which may be accessible to those who have seen traditional Kerala Architecture. Consider the roof-frame of a rectangular building, with frame-rods (കഴുക്കോല്). The connector-rod (വള) enters the first (straight) frame-rod perpendicularly, but enters other frame-rods, which are at an angle to the original straight one, at an angle to the perpendicular. To make the point about similarity of triangles \( A_1OE\) and \(A_1EB_1\) , Yuktibhāṣā points out that the frame-rods are analogous to \(OA_i\) and the connector rod analogous to \(EA_i\). By a parallel arms argument, the angle between the connecting rod and the perpendicular to a frame-rod must be the same as the angle between the frame-rod and the “straight” frame rod.

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.

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. 

This is an interesting Sanskrit verse, relating the circumference and diameter of a circle, found in an equally interesting Mathematical work in Malayalam called Yuktibhāṣā (യുക്തിഭാഷാ – loosely translated as “Proofs in common language”), attributed to the mathematician-astronomer Jyeṣṭhadeva who lived in Kerala, India in the early 1500s CE.

The verse itself is attributed to “Madhava”. The same verse turns up in at least one other work of similar age, and is attributed to Madhava again. It turns out that Madhava was the founder, or early leading light in a school of mathematical astronomers who flourished in Kerala from the 1300s (or possibly earlier) all the way to the 1800s CE.

The same computation – an infinite series for \( \pi \) in modern terms – was rediscovered in Europe in the mid 17th Century – nearly 300 years later than Madhava and 150 years later than Yuktibhāṣā – by Gottfried Leibniz, and was known after him until broader recognition of the contributions of Madhava and his successors kicked in recently.

To understand this verse, the school that created it, and their motivations, we must go back in time a bit, to their inspiration – Aryabhata.

Aryabhata – where it all began

Āryabhaṭa (आर्यभटः)‌ lived in Kusumapura, also called, Pataliputra (modern Patna), around 500CE . His only surviving work, Āryabhaṭīyaṃ (आर्यभटीयम्), is a full statement of the Indian astronomical framework, including all mathematics required for it, in a surprisingly concise set of 121 verses. Because of such brevity, it is mostly understood through commentaries (भाष्यम्), most famously of his student Bhaskara I (the numeral added to distinguish him from the more illustrious Bhaskaracharya half a century later). Other commentaries exist, most relevantly the आर्यभटीयभाष्यम् of Nilakantha Somayaji (whom we will meet later), which refers to many of Madhava's improvements.

The tenth verse of the second chapter (Ganitapada) of Āryabhaṭīyaṃ is thus:

 चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम्
अयुतद्वयविष्कम्भकस्यासन्नो वृत्तपरिणाहः ।।
Of a circle with diameter 2000, the circumference is approximately
 four plus hundred multiplied by eight, and added to 62,000 (=62,832)

The word आसन्न (approximately) has always been considered to be relevant in the commentarial tradition. Bhaskara I says “आसन्नपरिधिः कस्मादुच्यते, न पुनः स्पुटपरिधिरेवोच्यते?‌ एवं‌ मन्यन्ते – स उपाय एव नास्ति येन सूक्ष्मपरिधिरानीयते” (Why is the approximate circumference explained, why not the exact one? It is understood that there exists no means of calculating a precise circumference). Thus, the Aryabhatan tradition held that

1. The value of the circumference of a circle of diameter 10000 is approximately 31416
2. There is no way to calculate a precise value

This does not, of course, preclude finding better approximations, and likely spurred the search for improvements.

Bhaskaracharya – Rough approximation for common use

Bhāskara (c. 1114–1185), commonly called Bhāskarācārya (“Bhāskara, the teacher”) in Indian texts and Bhaskara II in English texts to distinguish himself from his namesake, is most famous as the author of Līlāvatī, probably the most famous arithmetic text in India. He also composed Bījagaṇita (a text on algebra), and Grahagaṇita and Golādhyāya on astronomy.

व्यासे भनन्दाग्निहते विभक्ते खबाणसूर्यैः परिधिः स सूक्ष्मः । 
द्वाविंशतिघ्ने विहृतेऽथ शैलैः स्थूलोऽथवा स्याद् व्यवहारयोग्यः ।।
The diameter multiplied by 3927, and divided by 1250 
yields the circumference. Multiplied by 22 and divided by 7 
yields the rough circumference for common use

Bhāskarācārya does not seem to have been a fan of greater precision in this regard. 3927/1250 = 3.1416, which is the same as Aryabhata's estimate. 22/7 is less accurate, but is good enough for regular use, as schoolchildren know.

The Kerala School

The Kerala school flourished barely three centuries after Bhāskarācārya, compared to the seven that separate him with Āryabhaṭa. There were prior notable figures in this lineage, such as Vararuci, who was a contemporary of Āryabhaṭa and originator of the Katapayadi (कटपयादि) system, and Sankaranarayana , who lived in the 800s CE, and was a reputed astronomer and commentator or the work of Bhaskara 1. However, nothing much is known of other figures in between them and Madhava of Sangamagrama. That they are undoubtedly connected to Vararuci and Sankaranarayana is evidenced by their common use of the Katapayadi system to encode numbers in Sanskrit, unlike Aryabhata's unique syllabic encoding or the bhutasankhya method favoured by other Indian mathematicians.

Iriññāttappiḷḷi Mādhavan Nampūtiri (ഇരിഞ്ഞാറ്റപ്പിള്ളി മാധവൻ നമ്പൂതിരി), commonly referred to in English as Madhava of Sangamagrama can be legitimately considered the founder of the school. The innovations that his student lineage attribute to him are truly exceptional, and evidently sparked a period of innovation that spanned several generations of his student lineage.

Perhaps the most illustrious member of this lineage after Madhava himself, and definitely one whose work we have the most knowledge about is Nilakantha Somayaji, author of the Sanskrit work तन्त्रसङ्ग्रहः (Tantrasangraha – “Concise Theory”) that describes the high point of Indian Mathematical Astronomy, and contained very significant improvements in Aryabhata's astronomical model, just as Madhava had improved the mathematics underlying the model.

The leading lights, Madhava, as well as Nilakantha Somayaji are marked by a deep understanding of the work of Āryabhaṭa, plus the curiosity and inventiveness to substantially improve on it.

Beyond these two, the most important is Jyeṣṭhadeva, the author of Yuktibhāṣā, which is valuable as a very detailed expostion of the logic behind the innovations of Madhava and Nilakantha, which allow us to demonstrate exactly how the mathematics behind the verse at the beginning of this article was arrived at.

Indian texts on mathematical topics are divided into tantra and upapatti/yukti categories. tantra texts present the theory in a terse and memorizable form. The commentaries on them (bhāṣya) often contained explanations and rationale. The various commentaries on Āryabhaṭīyaṃ and Līlāvatī are studied along with the texts themselves, without which the texts are understandably not easy to grasp.

Yuktibhāṣā is unique in that it is a yukti text that is only loosely associated with a tantra text. It is intended to explain the details of the yukti behind Tantrasangraha, but is organized independently, and can be studied as an independent text. This is of help when we wish to treat the mathematics and the Astronomy of the Kerala school separately, which is complicated to do in the context of a work like Tantrasangraha, which is organized as an Astronomical text which dives into mathematics as and when required from context. Also unusually, Yuktibhāṣā is written in Malayalam, not Sanskrit. The language is recognizable to a modern speaker, but is not easy to follow without the benefit of tradition.

Luckily for us, an exposition in more modern Malayalam, with explanations in modern mathematical notation was published in 1948, by Rama Varma Thampuran and A.R. Akhileshwara Iyer. Also available, is an English translation by K.V. Sarma, with explanatory notes by Professors K Ramasubramanian, M D Srinivas and M S Sriram, published in 2008.

Why did they need more accurate values of Pi?

The answer was, as the reader may have guessed, Astronomy. The Kerala school specialized in increasingly precise astronomical calculations, and while Aryabhata's approximation was good enough for all earthly purposes, there were heavenly ones it was becoming increasingly clear that it wasn't sufficiently accurate for. A thousand years had passed, and small errors were now seen in greater magnification.

Madhava and his successors refined both Aryabhata's mathematics and his astronomical model in the search for better accuracy over a longer period. Nilakantha Somayaji even created a semi-heliocentric model of the universe, similar to the later model of Tycho Brahe, to improve on the accuracy of Aryabhata's geocentric-epicyclic-model. Small wonder then, that they sought to squeeze out accuracy wherever they could, including with better computations of circumference and what we now call trigonometric functions.

Back to the verse …

व्यासे वारिधिनिहते रूपहृते व्याससागराभिहते । 
त्रिशरादिविषमसंख्याभक्तमृणं स्वं पृथक्क्रमात् कुर्यात् ॥

In modern mathematical notation: for a circle with diameter d and circumference C,
\( C = \frac{4.d}{1} – \frac{4.d}{3} + \frac{4.d}{5} – \frac{4.d}{7}… \)
or equivalently,

Presto, we now have the value of the circumference, known to be impossible to calculate exactly, to any degree of accuracy we desire. We would never reach the far shores of \( \pi \) , but get as close as we need to, with sufficient effort.

A reader with some knowledge of Sanskrit would naturally ask – why does वारिधिः / सागरम् (“ocean”) mean four in this verse? Why does शरः (“arrow”) mean five?
There are many encodings of numerals in Sanskrit verse, and this is one of them, called Bhūtasaṃkhyā, which encodes numbers by the names of common objects associated with them. The Kerala school used it only for short numbers, and preferred to use Katapayadi for longer ones, which we will see later.
Also, how was this idea of a non-terminating series and this one, in particular, derived? Surely, it wasn't pulled out of thin air, or transmitted by an alien civilization?
We will see in later posts that

• There was a sound mathematical rationale for this, which built upon work by earlier mathematicians in India, particularly Aryabhata and Narayana Pandita and their work on series summation.
• The way the idea of a limit and the fuzzy idea of an “infinitesmal” were approached was quite different.
• There was much more to the mathematics of the Kerala school than this one series.

For example, we will see other series that converge more rapidly, that were derived systematically from the series we have already seen:

 व्यासाद्वारिधिनिहतात्पृथगाप्तं त्र्याद्ययुग्विमूलघनैः
 त्रिघ्नव्यासे स्वमृणं क्रमशः कृत्वा परिधिरानेयः ||   

or

समपञ्चाहतयो या रुपाद्ययुजां चतुर्घ्नमूलयुताः 
ताभिः षोडशगुणितात्पृथगाहृतेषु विषमयुते 
समफलयुतिमपहाय स्यादिष्टव्याससंभवः परिधिः 

or

We will see similar series for what we now call trigonometric functions, but were understood in India since Aryabhata in terms of half-chords of arcs (ज्या). We will see surface area and volume integrals similar to what high-school textbooks show today, and even a line integral, which turns up earlier than we'd expect, but all using terminology and mathematical techniques that are unfamiliar to a reader educated in modern mathematical language.

And therefore, it would only be fair to start the next post in this series with a backgrounder on mathematics – arithmetic, algebra, and trigonometry – as Madhava and successors saw it.