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Indian methods of finding the value and relation between the Circumference and diameter, starting with Vedic period to the methods given by Srinivasa Ramanujan is given in this article.This article discusses the necessity and advantages of imparting these Indian Contributions to students. This paper, share the experience from Pi day celebrations on 22/7 22nd July
pi in sulbasutras – BC 800
The transformation of a square altar into a circular shape was a common practise in Vedic rituals.
Satapatha Brahmana-SB discussed construction of Circular garhapatya altar using bricks of different sizes. The diameter of a circular garhapatya altar is (square root of 32) /5 purusa. The area is taken as that of one square purusa ahavaniya altar,
(π x 32/25) / 4 =1
So we can observe that π = 25/8
1 purusa = 1 vyama = 5 aratnis = 120 angulas.
In Boudhayana Sulbasutra 16.6 describes the construction for converting a square into a circular – chariot wheel.It starts with an area of 225 square bricks and is then augmented with 64 more bricks so that one has a new square equal to17 x 17 = 289 bricks. This leads to an equality of a square of side 15 units into a circle of diameter of 17 units ie: ( π x 289 )/ 4 =225
So we get an approximation π = 900/289.
Manava Sulbasutra gives a value of π as1075 / 344
The values of the ratio of the circumference to the diameter implied in those methods were slightly better than ancient presumed value of 3.
Pi by Jain Mathematicians – Before CE 850
In the ancient Jain texts like Jyotiskarandaka and Jambudvipasamasa of Umasvati ,the ratio -Pi was considered as Square root of 10.
Jambudvipaprajnapati explains dhanuhprstha-the arc and Jiva-the chord and gives approximate formulas like Area of the circle = ( Circumference x d ) / 4
Circumference of Circle = Square root of (10 d x d) = Pi d etc.
Virasena (8th CE ) in his DhavalaTika gives value of pi in a better way
“when the diameter multiplied by 16,combined with 16 and divided by 113 is again combined with thrice the diameter will be most exact.
The equation can be ( 16d+ 16) / 113 + 3d and if we remove 16 the approximate value of Pi will be355 /113
Pi by Aryabhata CE 499
“Chathuradhikam satamastagunam,dwashashtistanatham sahasranam
Aythadwaya vishkambasyasanno vritthaparinamaha”
( The approximate value of the circumference for a diameter of 20000 is 62832.)
And we can find Pi as 62832 / 20 000 =3.1416
The most significant thing here is the word asanna- a near approximation.
Pi by Kerala mathematicians Between CE 1350- 1750
A remarkably close approximation given by Sangamagramma Madhavan (1340-1425 CE) in word numerals-Bhuta Samkhya system
“ Vibhudha netra gajahi hutasana trigunavedabhavaranabhahava
Navanigharvamihe vrithivistare paridhimanamitham jagadbudha”
Viuta= God (33), Netra = Eyes (2), = Elephants (8),= Snakes (8), = fires(3), = Three (3), =
Qualities (3), Vedas (4) Nakshatras= (27) =Elephants(8) and = arms(2)
2 827433388233.
The wise say that this is the measure of circumference of a circle having diameter 9 x10 11 units.The measure of the circumference in a circle of diameter 900 000 000 000 is
2827 433 388 233 From this Pi π = 3.14159265359
In his commentary Nilakanda Somayaji ( 15th CE ) - gave the explanation – by the measure with which the diameter can be measured without a remainder, the circumference measured by the same will certainly leave a remainder when used for measuring the diameter. Though we carry it very far we can achieve smallness of the remainder only, but never without a remainder.
Kriya kramakari of Sankara warrior gives still refined value :
“Vritha vyase hathe nagavedavaha:yathimkhendhubhi
Tithyasivavibhudhyrbhakthe susukhmam Paridhi”
The value of Pi is 104348 /33215 = 3.1415926539211..
Karanapadhati of Puthumana Somayaji (15th CE) gives 31415 926536 as the circumference for a diameter of 10000000000.
Kerala School of mathematics used inscribing polygon method and also used an infinite series to find Pi.
The infinite series in its basic form can be found in the following sloka of Jyeshtadeva in Yuktibhasha quoting Tantrasangraha of Nilakanda somayaji ( 15 th CE)
“Vyase Varidhinihathe roopathe vyasa sagarabhihathe
Trisaradi vishamasankhya bhakth rina swa yadhakramat kuryath”
Means:
In the diameter multiplied by 4 and divided by one, decrease and increase should be made in turn of the diameter multiplied by four and divided one by one by the odd numbers beginning with 3 and 5
So Circumference = 4d – 4d/3 + 4d/5 – 4d/7 + …….
Also we can find Pi/ 4 = 1 – 1/3 + 1/5 – 1/7 + ….
In Mahajyānayana prakāra -sine table methods of Madhava as quoted by his disciples. These infinite series is the beginning of CalcuSankara Varma(18th CElus by Kerala mathematicians since14th Century.
Sadratnamala of ) in ch:4.2
Bhadram budhisiddha janma ganitha sraddhasma-yed bhupatih:
This gives the circumference of the circle with diameter of 1017 as 314159265358979324, and the value of pi is correct up to 17 decimal places.
Facts being this about the evaluation of Pi by Indian Mathematicians, when we talk about the history of Pi, people miss out the contributions of Indian mathematicians from 6th Century to 18th Century.
For eg: the method of Infinite series generally attribute to Leibniz (1646 -1716) or James Gregory ( 1638 – 1675),where as these methods were explained by Sangamagramma Madhavan and his disciples well in advance to these two European mathematicians.
Srinivasa Ramanujan also developed an infinite series for Pi in 1910,and based on this algorithm number theorists developed methods to find out more and more decimal places using computers.
Pi Day celebration
We organised Pi day celebration for students on 22/7 ;22nd July during the past 3 years. This facilitated students to explore the applications of pi in various fields. Students creatively prepared posters, cartoon etc focusing Pi. Organised Pi assembly with formation of Pi shape Diameter and Circumference etc. Pi digit recitation, is also an interesting activity. The Pi day celebration is an ideal event to involve students creatively with deeper connection with the concept. This helped students in intuiting the concept of Pi and its relations to create confidence and interest in geometry and thereby mathematics.
Find your Birth day in Pi
https://www.piday.org/find-birthday-in-pi/
References
[1] KV Sharma etc, Ganita Yuktibasha of Jyeshtadeva, Hindustan Book Ag (2008)
[2] BB Datta,The Jaina School of Mathematics (1929)
[3] Dr T A Sarasvati Amma Geometry in Ancient and Medieval India, MLBD (1999)
[4] MD Srinivas, Kerala School of Astronomy and Mathematics,Mathematics, IIT GN News letter Jun 2012
[5] Dr GG Joseph, Passage to Infinity, Sage Publications 2009
[6] Prof. VPN Nampoori, Sangamagramma Madhavan- a brief, SSM -Kerala,2010
[7] Dr. V Madhukar mallayya, Various articles on Kerala mathematics.
This article published by NCERT :9th NCME 2020
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