Pythagorean Theorem — a Misnomer

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Prob­lem-solv­ing in con­struc­tions

In the Vedic Era, we have seen that the elab­o­rate con­struc­tion of larg­er altars require con­form­ing to cer­tain shapes and strin­gent pre­scrip­tions of areas and perime­ters. This set the basis for two main geo­met­ri­cal prob­lems to be solved — One was the prob­lem of find­ing a square equal in area to two or more giv­en squares and the oth­er was the prob­lem of con­vert­ing shapes such as a cir­cle or a square or a trapez­i­um or a rec­tan­gle into oth­er shapes of equal areas and vice ver­sa.

Pythagore­an The­o­rem — a Mis­nomer

These con­struc­tions were achieved through a judi­cious com­bi­na­tion of con­crete geom­e­try, inge­nious algo­rithms and the appli­ca­tion of the Śul­ba the­o­rem. The Śul­ba the­o­rem is deemed to have been devised and wide­ly used by at least as ear­ly as 800 BCE. This most foun­da­tion­al the­o­rem of geom­e­try has been clear­ly enun­ci­at­ed right from the ear­li­est of Śul­ba-sūtras — the Baud­hāyana-śul­basū­tra. This is the the­o­rem that we cur­rent­ly and com­mon­ly under­stand as the so-called “Pythagore­an the­o­rem”. 

There is ample schol­ar­ly evi­dence and wide accep­tance among math­e­mati­cians and his­to­ri­ans of Sci­ence about this the­o­rem pre­dat­ing the Greek’s propo­si­tion of the same. But owing to the banal­i­ty of text-books with biased infor­ma­tion and a dom­i­nant Euro-cen­tric nar­ra­tive in sci­ence, this most resound­ing evi­dence of Indi­an sci­en­tif­ic her­itage is hard­ly known among most stu­dents of math­e­mat­ics and Indi­ans, in gen­er­al. With­out well-rea­soned under­stand­ing this the­o­rem con­tin­ues to be cred­it­ed to Pythago­ras ( c. 570 – c. 495 BC) and even tout­ed as a “Greek mir­a­cle” by cer­tain short-sight­ed claims.

The Śul­ba The­o­rem

A clear enun­ci­a­tion of the so-called ‘Pythagore­an’ the­o­rem, which is right­ly the Śul­ba The­o­rem (called bhu­jā-koṭi-karṇa-nyāya in the lat­er lit­er­a­ture)  is described in Baud­hāyana-śul­basū­tra(1.12) as fol­lows:

दीर्घचतुरश्रस्य अक्ष्णयारज्जुः तिर्यङ्गानी पार्श्वमानी च यत् पृथग्भूते कुरुतः तदुभयं करोति ।  [BSS 1.12]
The rope cor­re­spond­ing to the diag­o­nal of a rec­tan­gle makes what­ev­er is made by the lat­er­al and the ver­ti­cal sides indi­vid­u­al­ly.

What the above sūtra essen­tial­ly states is: If you con­sid­er the two sides (a, b) of a rec­tan­gle then the diag­o­nal c is giv­en by

a 2 + b 2 = c 2

Terms employed in this sūtra are: 

दीर्घचतुरश्रम् ~ rec­tan­gle (lit. longish 4‑sided fig­ure)

अक्ष्णयारज्जुः ~ the diag­o­nal rope

पार्श्वमानी ~ the mea­sure of the lat­er­al side

तिर्यङ्गानी ~ the mea­sure of the per­pen­dic­u­lar side

The Śul­ba Triplets

The lan­guage employed in the above verse is in the sūtra for­mat which is terse and con­cise. Through com­men­taries from ancient peri­ods and unbro­ken knowl­edge trans­fer we under­stand the terse sūtras. But just so to allay any doubts even in the most scep­tic of minds, Baud­hāyana in the very next verse presents a few triplets that clear­ly state a set of mea­sures which are the rec­tan­gleś sides that will per­fect­ly cap­ture the above law.

तासां त्रिकचतुष्कयोः, द्वादशिकपञ्चिकयोः, पञ्चदशिकाष्टिकयोः, सप्तिकचतुर्विंशिकयोः, द्वादशिकपञ्चत्रिंशिकवोः, पञ्चदशिकर्षट्टिंशिकयोः इत्येतासु उपलब्धिः। [BSS 1.3]

32 + 4² = 5²  | 5² + 12² = 13² |  15² + 8² = 17² |  7² + 24² = 25² |  

12² + 35² = 37² |  15² + 36² = 39²

It is indeed enthralling to know in depth about the rich sci­en­tif­ic her­itage of the Indi­an civ­i­liza­tion. In the forth­com­ing arti­cles we shall delve more into the vari­a­tions of the­o­rem of Śul­ba and how √2 and Pi were born.