Indian Mathematicians and the Value of Pi

India has been a cra­dle not only to refined civ­i­liza­tion­al best-prac­tices but also to mul­ti­tudi­nous sci­en­tif­ic devel­op­ments. Evi­dence and knowl­edge of volu­mi­nous lit­er­a­ture pro­duced in Indi­an sci­en­tif­ic pur­suits has been well estab­lished by seri­ous researchers. In con­trast, there is preva­lent igno­rance about facts and feats that we have inher­it­ed.

Through a series of short arti­cles, we have set forth to cov­er a few high­lights in the devel­op­ment of Jyotiṣa (Astron­o­my) and Gaṇitā (Math­e­mat­ics). The devel­op­ment of both these fields can broad­ly be put in three eras:

1. Vedic or Pre-Sid­dhan­tic Era,

2. Sid­dhan­tic or Clas­si­cal Era and

3. Post-Sid­dhan­tic or Medieval Era.

Hav­ing delved into the Vedic peri­od, we have moved ahead in the time­line and pre­sent­ed an intro­duc­to­ry glimpse into the phe­nom­e­nal break­throughs in Chan­das-śās­tra com­posed by Piṅ­gala-nāga around 3rd cen­tu­ry BCE. In this arti­cle, we shall cut across the time­lines of the three eras of devel­op­ment of math­e­mat­ics in India in order to appre­ci­ate how India’s con­tri­bu­tion to approx­i­mat­ing π is immense­ly sig­nif­i­cant. The recent date that passed — 22nd of July is cel­e­brat­ed as world π day. We will throw an inter­est­ing light on for how long Indi­ans have rumi­nat­ed on π and have sig­nif­i­cant­ly con­tributed to bet­ter approx­i­ma­tions in the glob­al stage.

What is real­ly π?

Since the Vedic era Indi­an math­e­mati­cians have been deal­ing exten­sive­ly with the geom­e­try of the cir­cle and have always tried to esti­mate this val­ue as the ratio between the cir­cum­fer­ence (paridhi/parinaha) to the diam­e­ter (vyāsa/viṣkambha) of the cir­cle (vṛt­ta). Bharatiya Gani­ta Parampara knew that this num­ber can­not be expressed as a frac­tion and hence it is a surd. Nonethe­less for prac­ti­cal pur­pos­es such as con­struc­tion, area trans­for­ma­tion and in var­i­ous places of math and astron­o­my, Indi­an math­e­mati­cians approx­i­mat­ed the val­ue of π and have strived immense­ly to improve the effi­cien­cy.

Sig­nif­i­cant leaps relat­ed to π

The Śul­vasū­tras (c. 800 BCE) give the val­ue of π close to 3.088. Śata­pathabrāh­maṇa (<600 BCE) gave 25/8 = 3.125 cor­rect to 1 dec­i­mal place (Geo­met­ric method).

In Āryabhaṭı̄ya (499 CE), the mag­num opus of Āryab­haṭa, an approx­i­ma­tion which is cor­rect to four dec­i­mal places is present.चतुरधिकं शतमष्टगुणं द्वाषष्टिस्तथा सहस्राणाम्।

अयुतद्वयविष्कम्भस्य ‘आसन्नो’ वृत्तपरिणाहः ॥

One hun­dred plus four mul­ti­plied by eight and added to six­ty-two thou­sand.

This is the approx­i­mate mea­sure of the cir­cum­fer­ence of a cir­cle whose diam­e­ter is twen­ty-thou­sand.

Thus as per the above verse, the approx­i­mat­ed val­ue for is 62832/20000 = 3.1416. And this is cor­rect to four dec­i­mal places and high­ly valu­able for pre­ci­sion in prac­ti­cal aspects.

The same val­ue has been giv­en by Bhāskarācārya II (1114 CE) in Lı̄lāvatı̄ by remov­ing a fac­tor of 8 from the denom­i­na­tor and the numer­a­tor. The num­bers employed by Bhāskara are in Bhū­ta-saṅkhya sys­tem.व्यासे भनन्दाग्निहतेे विभक्ते खबाणसूर्यैः परिधिः स सूक्ष्मः।

द्वाविंशतिघ्ने विहृतेऽथ शैलैः स्थूलोऽथवा स्याद्व्यवहारयाेग्यः~॥१९९॥

When the diam­e­ter is mul­ti­plied by 3927 and divid­ed by 1250, that [result-

ing] cir­cum­fer­ence is near pre­cise. Else, if [the diam­e­ter is] mul­ti­plied by

22 and divid­ed by 7, then [the result­ing cir­cum­fer­ence] would be a gross

[val­ue] good for prac­ti­cal use.

The fol­low­ing accu­rate val­ue of π (cor­rect to 11 dec­i­mal places) has been giv­en by Mād­ha­va (c. 1340 CE), who is the pio­neer of the Ker­ala school of astron­o­my and math­e­mat­ics. विबुधनेत्रगजाहिहुताशनत्रिगुणवेदभवारणबाहवः।

नवनिखर्वमिते वृतिविस्तरे परिधिमानमिदं जगदुर्बुधाः॥

The π val­ue giv­en above is: π = 2827433388233 / 9 x 1011 = 3.141592653592…

The 13 dig­it num­ber appear­ing in the numer­a­tor has been spec­i­fied using the Bhū­ta-saṅkhya sys­tem, where­as the denom­i­na­tor is spec­i­fied by word numer­als. In the Bhū­ta-saṅkhyā sys­tem, vibud­ha =33, netra =2, gaja =8, ahi =8, hutāśana =3, triguṇa =3, veda =4, bha =27, vāraṇa =8, bāhu =2. In word numer­als, nikhar­va rep­re­sents 10 11. Hence, nava-nikhar­va = 9 × 10 11.

Only in 1706 CE William Jones made Greek π as this ratio’s nota­tion. In 1914 CE Srini­vasa Ramanu­jan derived a set of infi­nite series that seemed to be the fastest way to approx­i­mate π. How­ev­er, these series were nev­er employed for this pur­pose until 1985, when it was used to com­pute 17 mil­lion terms of the con­tin­ued frac­tion of π (Mod­u­lar Equa­tion method).

As of Jan 2020 we have 50 tril­lion terms of π by algorithms(Modular Equa­tion method). It is impor­tant to also cel­e­brate the lin­eage of Indi­an math­e­mati­cians who achieved break­throughs that con­tin­u­ous­ly advanced π, impact­ing our leaps in allied areas of trigonom­e­try, and cal­cu­lus.

It is indeed enthralling to know in depth about the rich sci­en­tif­ic her­itage of the Indi­an civ­i­liza­tion. We shall con­tin­ue to see oth­er inge­nious achieve­ments in Indi­an math­e­mat­ics in the fol­low­ing edi­tions of Parni­ka.

Aum Tat Sat!