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.