Synopsis: In this article we study the references and formulaes from ancient texts that  prove the use of  PI in mathematical calculations during ancient period. The great mathematician Āryabhaṭṭa from approximately 500-600 BCE,  also mentions the exact value Π (PI).

Now a days we all are very familiar with the value of Π (PI). The value of Π is considered to be 3.1416… i.e. Π = 3.1416….. Some 1500 years ago, the great mathematician Āryabhaṭṭa, and approximately several thousands of years ago before that, the composer of Mahābhārata (approximate time : 5 – 6 century BCE) clearly hinted about the exact value (Π=3) of Π (PI).

Even though the word Π (PI) is not mentioned in Mahābhārata, we can still see the calculations done to deduce the value. If we put the examples available in Mahābharata in the exact way Āryabhaṭṭa calculated, we get the same value of Π as a result.

In the same context, a detailed account was presented about the islands and the planets before Dhṛtarāṣṭra by Sañjaya in Bhīṣmaparva of Mahābhārata. Where the paridhi (circumference), vyās (diameter) and thickness of planet Rāhu, Moon and the Sun are mentioned.

Example – The planet Rāhu is round. It’s diameter is 12000 yojana, circumference is 36000 yojana and the thickness is 6000 yojanas. In the same way the diameter of the Moon is 11000 yojana, circumference is 33000 yojana and the thickness is 5900 yojanas. The diameter of the Sun is 10000 yojana, circumference is 30000 yojana and the thickness is 5800 yojanas.

उक्त्वा द्वीपा महाराज ग्रहं वै शृनु तत्वतः ।
स्वर्भानोः कौरवश्रेष्ठ यावदेव प्रमाणतः ॥

अष्टपञ्चाशतं राजन् पिपुलत्वेन चानघ ।
श्रूयते परमोदारः पतगोऽसौ विभावसुः ॥

                                                                         महाभारत भीष्मपर्व १२.३९-४५

uktvā dvīpā mahārāja grahaṃ vai śṛnu tatvataḥ /
svarbhānoḥ kauravaśreṣṭha yāvadeva pramāṇataḥ //

aṣṭapañcāśataṃ rājan pipulatvena cānagha /
śrūyate paramodāraḥ patago’sau vibhāvasuḥ //

                                                                         Mahābhārata Bhīṣmaparva 12.39-45

Here is a great article on   50 Interesting PI Facts

( A yojana is a Sanskrit word. It is a measure of distance that was used in ancient India. A yojana is about 12–15 km. i.e. 60 kosh = 1 yojana and 1 kosh is 200 m )

If this description is reflected in mathematical language the value of  Π (PI) is available as the result. According to Āryabhaṭṭa :

(PI)    Π = paridhi (circumference)

     vyās (diameter)

If the measurement described by Sañjaya is put in above mentioned formula, the exact value comes to 3.  Example :

The paridhi (circumference) of planet Rāhu = 36000 Yojana

The vyās (diameter) of planet Rāhu              = 12000 Yojana

Π = paridhi (circumference)     =    36000  =  3           [ Π = 3 ]

      vyās (diameter)                          12000

The paridhi (circumference) of Candramā = 33000 Yojana

The vyās (diameter) of Candramā              = 11000 Yojana

Π = paridhi (circumference)     =    33000  =  3           [ Π = 3 ]

      vyās (diameter)                        11000

The paridhi (circumference) of Sun = 30000 Yojana

The vyās (diameter) of planet Sun  = 10000 Yojana

Π = paridhi (circumference)     =    30000  =  3           [ Π = 3 ]

      vyās (diameter)                        10000

This is inferable from above description, that the exact value of  Π calculated from the proportion of circumference and diameter of a circle is 3. It can also be concluded that the secret knowledge of circumference and diameter of a circle was very much in use in ancient times. The great mathematician Bhāskarācārya mentioned Π in his composition Līlāvatī :  

व्यासे भनन्दाग्नि (३९२७) हते विभक्तेः

ख वाण सूर्यैः (१२५०) परिधिस्तु सूक्ष्मः ।

द्वाविंशति (२२) घ्न विहतेऽथ शैलेः

स्थूलोऽथवा स्याद् व्यवहारयोग्यः ॥

                                                                    लीलावती (क्षेत्रव्यवहार​​) ४०

vyāse bhanandāgni (3927) hate vibhakteḥ

kha vāṇa sūryaiḥ (1250) paridhistu sūkṣmaḥ /

dvāviṃśati (22) ghna vihate’tha śaileḥ

sthūlo’thavā syād vyavahārayogyaḥ //

                                                                                                       līlāvatī (Kṣetravyavahāra) Verse 40                                                

So, if the diameter of the circle is multiplied by 3927 and the result is divided by 1250, we will get subtle circumference as a result or if the diameter is multiplied by 22 and the result is divided by 7 then the result will be an usable circumference. Thus, Bhāskarācārya mentioned two values of Π (PI), as shown below :

Π = 22 / 7 (Gross value)

Π = 3927 / 1250  = 3.1416  (Subtle value)

The value of Π is also mentioned in Triśatikā  composed by Śrīdharācārya along with citation of the formula of circumference. The rule of calculating area of a circle basing on the circumference and the diameter if the circle is as follows :  Triśatiā (paṭī. gaṇitasāraḥ) vṛttasūtra 45

                                    ___________

Circumference = √ 10 * Diameter

Circumference = Diameter  *  3 ¹⁶/₁₀₀ 

               ___

So, 1 : √­­10   OR  1 : 3.16 …..

So it can be concluded from the description above, that the mathematicians of ancient times reached approximately the same value of Π. As the value of Π is very important in mathematics, the mathematicians from all ancient civilizations tried to calculate the best possible results from their own point of view. Below are few examples from ancient times :

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Different values of Π [1]

Henceforth, it can be concluded from the description above, that the value of Π (PI) which was available and widely used by any civilization, basically indicates the progress in the field of mathematics during that period.

The citation of circumference and diameter in context of Rāhu, Moon and Sun, has been done to calculate the exact (purest) possible value of Π in Mahābhārata where the value of Π is concluded to be 3 (Π = 3). The same is supported in the Old Bible, the Baudhāyanaśulvasūtra , Jain Texts and also by Brahmagupta. Clearly, our ancient texts are full of scientific and practical knowledge.

This is very important to mention that, if anyone draws a circle on a page and divides the circumference with diameter, the value of Π (PI) is always the same, 3. To demonstrate practically, a thin wire is taken and a circle is made by connecting both the edges to make it a circle. Then the diameter is taken into account. The length would be the circumference of the circle. Now, if the circumference is divided by the diameter, the exact value would come to be 3. It is prescribed to use a very thin (fine) wire for the practical.

No matter how many times one tries, the result is always the same : 3. So, it could be possible that the exact value of Π (PI) is actually 3. This is my experience and opinion as a non – mathematician, hence  it is open for debate.

Contributor :  Dr Poornima Singh Rana

English Translator :  Joy Dev

Image Credit :  depositphotos.com