Pi Approximation Day is celebrated all over the world on July 22nd.

The story of Pi (Π) is interesting and long. It is intertwined with the development of thinking of mankind not only in India but the world over.

Since Pi has a connection with circles, we shall look at the manner in which the circular shape was understood and used by the ancients in India.

#### The most valuable ratio

A circle is one of the perfect and most naturally occurring shapes in the world. It is a figure where every point along the line called the circle or its circumference, is equi-distant from a point called the centre of the circle, lying within the area bound by this line.

This distance was named as *TriJya *by the ancients, after *Jya*, meaning the chord of the bow. The ancients had noticed a relation between this distance *Trijya *and the length of the line that made up the circle. Modern Geometry calls this *TriJya*, the Radius and twice *Trijya *as Diameter.

They had found that if the length of the circular line (the length called circumference) was a little over 3 times the diameter, then the line would go round the centre such that at every point, it would always be equidistant from the centre point.

The ratio of the length of the circular line (circumference) to the width of the circle (diameter) had to be 3.x for the shape to become a circle.

##### A Simple Thread of Reason

It is simple:

Take a rope and a stick such that the rope has to go around the stick.

If the length of the rope is just 1 x stick length, then it would only be able to align along the stick itself.

If the length of the rope is such that it is 2 x stick length, then it would align along the stick twice. But, it would lie along the length itself and not be able to uniformly maintain a distance from the centre of the stick.

If the rope length is 4 times the stick length, then it would align along the stick 4 times. No different from 2 x stick length.

This meant that the length of the rope must be between 2 and 4 times stick length, i.e. 3, for it to be able to go around the stick such that it will stay exactly at the same distance from the centre of the stick at all times.

This ratio 3, came to be referred to as *paridhi-vyasa sambandhana *in Indian thought.

#### Pi – the Paridhi-Vyasa Sambandhana

*Paridhi *means the circumference. It denotes that, which separates space into “this, flat, receptacle, area (*dhi*)” and the space “beyond this area (*pari*)”. It is the line around this area.

*Vyasa *means the diameter. *Vyasa *is that which disjoins, severs, distributes, rearranges. The diameter severs or rearranges the circle into 2 halves or semicircles.

This is why *Vyasa *is also the term for a compiler, since, as an arranger he reorganizes and arranges, distributes data suitably while giving it a coherent shape.

*Sambandhana *denotes relationship. *Paridhi-Vyasa Sambandhana *is the relationship between the diameter and the circumference of a circle which in present times is known to every school child as the constant Π approximated to 3.14, a value, very close to the value 3 for the *Paridhi-Vyasa Sambandhana*.

##### Pi – The Ratio

**But why did the ancients want to be so precise about their circles?**

**What did they use circles for?**

#### Circular shapes for Usage

##### Vedic Altars

Ancient Indians, who were Nature worshippers, are referred to as the Vedic civilization, for their understanding and close association with Nature.

One of their ways of veneration was through Vedic rituals centred around use of fire, hymns and altars. The shapes of altars played a very important role in the lives of these people known as *Ritwik*.

The *Ritwik *stayed in synergy with *Rta*, the order in the cosmos, Nature, using

- natural geometric shapes, patterns (altars),
- patterns of incantation and rhythm (hymns, chants),
- repetitive and ordered acts of offering (rites).

The *Ritwik’s *life was centred around *Rta *– order, pattern, repetition, rhythm, cycle.

Such use of altars, hymns, chants, offerings, rites came to collectively be called a Ritual.

Rituals, sharing etymological root with *Ritwik*, were for the *Ritwik, *the source of energy for mind, body and environs.

Such profound rituals of the *Ritwik*, came to be called erroneously and limitedly as “sacrifices” by some colonial commentators, due to the tangible and visible aspects of offerings.

##### Vedic Texts

It is hard to say when this Vedic period started or when the *Veda *were composed. But what one can confirm with a fair amount of certainty, is that the *Veda *and ancillary texts were last recompiled in 3100 BCE, as against composed for the very first time. This means that, this window of 3100 BCE must have been a prominent era of the Vedic period. Ritual practices would have been taken up vigorously and procedures for setting up and using altars would have been adhered to, then.

The *Sulva Sutra, *the oldest work on geometry gives procedures for construction of geometrical altars (*vedi*) using rope (*rajju*) and gnomon (*sanku*).

The compendium of these *Sutra *is dated to have been worked upon until 800 BCE though one cannot isolate and say which ones and when.

Companions to these *Sutra *were the *Brahmana *texts which outlined the process of Vedic rituals. Popular among these, the *Shatapatha Brahmana *also contains many mathematical instructions to build venues, altars and conduct rituals.

**A circular Vedic Fire Altar excavated at Harappa dating to before 2500 BCE**

When colonial historians had dated the *Veda *to around the 1st Millennium BCE, the *Sulva Sutra *and other texts too came to be bound by this upper limit.

But with the last recompilation of the Veda being traceable to 3100 BCE, the *Sulva Sutra *and other Vedic texts such as the *Shatapatha Brahmana *can also be dated to atleast 3100 BCE if not earlier.

##### Vedic Altars – Only One of the Needs

But Vedic altars were only one of the needs for circles. It was a need of only a particular set of people who were entrusted with safeguarding the psychological and physiological wellbeing of the society by keeping them in tune with Nature as well as sustaining Nature itself through daily rituals.

There were others who were in charge of the development of the township and in development of trade. They needed more precise circles as even a small error could make their living very shaky.

#### Circular Structures that Towered

##### Mehergarh Tower

**Circular tower at Mehergarh dating to before 5000 BCE, implying existence of such mathematical and architectural skills atleast from 6000 BCE.**

The use of such precise mathematics and geometry can be seen in one of the ancient structures of the world – a circular tower which is not only a perfect circle but also stands perfectly straight even after 8000 years.

A perfect circular tower has been excavated at the ancient town of Mehergarh in the Sindhu Sarasvati belt, dating back to beyond 5000 BCE. Such a perfect structure which has not only stood tall but has also withstood the test of time, indicates the perfection in ratios and proportions applied during design as well as construction.

**What is making us go around in circles today is the question, “why did they need to build such tall circular towers?”?**

**Were they water wells? Were they storage wells? Were they observation towers? Were they astronomical tools? Were they signalling beacons?**

**Or were they so advanced that just like us today, they chose to build circular columns and towers out of aesthetics?**

Whatever be the motive, precise mathematics to plan and build circular structures was the need of those who practiced professions such as engineering and architecture.

#### Circular sky to map

##### Living Under The Skies

The ancients did not stop with Nature worship alone. They were also in tune with Nature, to the extent that the motions of objects in the sky determined how they would lead their lives. For, it is these motions that gave rise to day, night, seasons and years.

##### Predicting Seasons

All the shapes, right from that of the earth, the moon, the sun are all circular.

All the motions, right from the rotation of the earth, to the moon around the earth, to the earth around the sun, to the planets around the sun, are all based on circular orbits.

The need to calculate these motions in “space” and predict the arrival of “time” meant the calculation of distances and speed of these astronomical objects along circular paths.

Thus precise mathematics to draw up circular paths and precisely calculate these motions was the need of those who practiced professions such as agriculture.

#### Circular seas to navigate

Even though India is classified as an agricultural civilization, trade has been one of India’s vibrant professions right from the times the *Veda *were composed, atleast more than 8000 years ago, since *Ramayana *is dateable to 7100 years ago.

Many *sloka *in the *Veda *mention about trade and there are also *slokas *that speak about navigation across oceans for this trade.

India has thus been navigating the seas from over 8000 to 10000 years ago.

##### Travelling Around the Globe

Ancient Indians had travelled the world over for trade as well as making contact with people and lands as far away as Central and South Americas. There are many evidences to show an Indo-American connect right from over 5000 years ago.

They had travelled eastwards to arrive at lands to their west for, they knew that the Earth was round and waters encircled the earth. So sailing the seas, one could go from one end of the earth to the other – i.e. go around the earth.

##### Plotting Maps and Distances

This need for navigation, gave rise to precise mapping of stars in the spherical sky with locations and distances across flat seas.

These needs translated into the development of many branches of study, differentiated today as Geometry, Trigonometry, Calculus and so on, all of which were lapped up by the entire world community as these needs are common to all mankind.

The ancient Indians were also adept in dividing the earth into zones of time using meridians to locate and navigate to remote places.

Thus precise mathematics to work with a spherical globe and circular lines called meridians was the need of those who practiced professions such as trade and navigation.

#### The formulation of Pi

It would therefore not be amiss to say that the ancients who drew these perfect circular shapes and built structures were in the know of the ratio of the length of the periphery of a circle to its width.

To be able to work with chords, arc and sines of a circular path in the sky and map it to distances on earth, the ancient Indians would have had to first know the ratio of length between the circumference and the diameter.

**So, when was Pi born?**

#### Birth of Pi

The concept of Pi in India is pegged to the *Sulva Sutra, *which go back to the date of the accompanying *Brahmana*, which again go back as far as the *Veda *atleast 8000 years ago. These *Sulva Sutra *are dated to have been last edited in 800 BCE.

The precisely dateable, physical proof for the usage of the concept of Pi though lies in Egyptian papyrus scrolls, dating to around 2000 BCE.

The list of mathematicians who have subsequently worked on this ratio Pi and have achieved converging results is quite long. It is almost as along as the number of digits after decimal for Pi.

#### Birth of the notion of Pi

The list of people who have worked on the value of Pi and those who will continue to do so in future may also be endless just as the infinite series of Pi. In this never ending evolution of the mathematical expression of Pi, what can perhaps be fixed though, is its birth.

This birth can be traced to the version of the *Sulva Sutra *going way back to the time of the composition of the *Veda*, based on the dexterity of the ancient Vedic Indians.

- who used to construct perfect geometrical shapes for their Vedic altars and potters’ wheel
- who could precisely predict circular motions of stars and planets in the sky
- who could precisely navigate to locations based on mapping the stars in the spherical sky to the flat seas and land.

All of which, point to their mastery over circles and spheres and the mathematics concerning them.

With the Vedic texts dateable to having been last recompiled in 3100 BCE, the birth of the concept of Pi and a value close to 3 for this ratio can be attributed to times going beyond 2000 BCE into the mists of time and mysteries of ancient Indians.

#### Birth of the name Pi

The name Pi and symbol Π for this ratio though, can be traced to William Jones, the British mathematician in 1707 CE who identified this name and symbol for this ratio.

Incidentally, William Jones, the mathematician, was the father of Sir.William Jones, the Philologist and Orientalist who became a Jurist in India during the British rule, founded the Asiatic Society of Bengal to popularize Oriental studies and was one of the first to popularize the idea of Indo-European group of languages, which culminated later in the incorrect and ill- famed Aryan Theory of migration into India.

**William Jones, the Mathematician (The Father) Source – Painting by Willaim Hogarth, National Portrait Gallery**

**Sir. William Jones, the Philologist (The Son)**

*Source – Pictures in the Hall of University, Oxford*

This name Pi and the Greek letter Π as symbol for it, was later popularized by the Swiss mathematician Leonhard Euler, in 1737.

**Leonhard Euler**

*A portrait by Emanuel Handmann in 1753*

##### Pi and the Art of Navigation

William Jones, the father, was a Welsh born mathematician who between 1695 and 1702, was commissioned to teach Maths onboard naval ships. While aboard ships in the sea, William Jones learnt the finer aspects of navigation and produced the work *“A New Compendium of the Whole Art of Navigation” *in 1702.

William Jones was also a close friend of Sir. Isaac Newton and Sir. Edmund Halley. While Newton is credited with the discovery of Calculus, Halley is credited with predicting the path and appearance of the comet which came to be named after him as Halley’s comet.

##### Navigation and Indian Maths

We see that the development of sciences in Europe, especially Maths, Astronomy and their primary use for Navigation, took place over the foundation provided by Indian mathematics which had reached Europe through Arabs and later through Jesuits.

It would therefore not be out of place here to conclude that even though William Jones is credited with having named this ratio of circumference to diameter as Pi, his study and findings could have been based on Indian works available and followed by Europe then.

##### Pi from Pie?

The name *Pi *and the alphabet symbol, is said to come from the Greek word “*Perimetros*” for “perimeter” and hence the attribution of the Greek alphabet ‘Pi’ and its symbol Π.

Another way to ascribe Pi to this ratio could have been the fact that this ratio is used along with arc, angles and chords of circles which divide the circle into small “pies”.

Also the more prevalent use of Pi today as an infinite series comes from the notion of adding up the infinite and minute arcs of a quadrant and comparing this integrated value with the radius or *Trijya *of a circle. A method that William Jones as a mathematician of the 1700s and a friend of Newton, would have been fairly conversant with.

For, this method of arriving at the value of Pi, as worked out by Madhava from Kerala, had reached Europe by then via the Jesuits.

#### Journey of Pi

Pi, as a concept thus was revealed and came into relevance for mankind from many millennia ago.

Pi, as a ratio was known to the Indian civilization as evidenced by the Circular towers, Vedic altars and procedures to create them from 6000 BCE and before.

Pi, as a mathematical concept travelled from India along with other aspects of Mathematics including *Jya *(Sine) in Trigonometry and Calculus needed for navigation along the many millennia since.

Pi, as a number has gained various levels of accuracy with further independent work by various civilizations and their mathematicians especially from the first millennium of the Common Era (CE).

Pi, as an infinite series of expression using Calculus, as in use in the modern world today, was deciphered by the Kerala school of Mathematics in 1350 CE.

Pi, as a name and symbol was coined by William Jones in 1707 CE.

Pi, as a constant was popularized in modern maths today by Leonhard Euler in 1737 CE.

Pi, as Π found its way into basic school books, all over the world in 1900s.

#### Pi – a perimeter around the world

In Pi, Π, we thus see a blend of,

- Ancient’s understanding
- Nature’s marvel
- Beauty of symmetry
- Precision in Maths
- Meticulousness of Mathematicians
- Common Need of Mankind
- Solutions from Maths
- Integrational capability of Maths
- Binding power of flowing knowledge.

The world has been literally and metaphorically bound in a circle of knowledge defined by the Perimeter called Pi, Π.

And, India’s slice, rather pie, of contribution to Pi and to the world of Maths is worthy enough to be branded and celebrated.

More on Pi in our book, **Brand Bharat – Vol-2 – Roots In India.**