** Baudhayana’s Pythagoras Theorem **

“Long long ago, so long ago, nobody knows how long ago” – that is how we used to start our stories in our younger days. But this story starts exactly like this. ** **Long long ago, so long ago, nobody knows how long ago, there lived one Baudhayana, who was an ancient Hindu master. He is dated to have lived during 800BC. He was an expert mathematician, architect, astronomer and a Hindu high priest. Once he was designing a sacrificial alter in the shape of a square. He inscribed another smaller square inside this square as below:

Baudhayana contemplated on this shape and realized the area of the inner square is exactly half the area of the outer square. With the cross-wires drawn as above, it is easy for us also to see this fact.

But the genius of Baudhayana went further. He thought of inscribing an off-set square with in the bigger square as below:

Now he calculated the area of the inner square as:

Area of the inner Square

= Area of the outer square – area of the 4 bordering triangles

= (a + b)^{2 } – 4 x (ab/2)

(i.e) Area of the inner Square = a^{2} + b^{2}

Aaha..! This sounds very familiar. Is this not called Pythagoras Theorem? But how come, it exists in 800 BC, almost 300 years before Pythagoras (570 -495 BC)? That too found by an ancient Indian? Should we call this then, as Baudhayana’s Theorem. But Baudhayana proposed many more such theorems in his Sulva Sutras. His statement of the so called Pythagoras theorem is as below:

* “dīrghasyāk**ṣṇ**ayā rajjuH pārśvamānī, tiryaDaM mānī,*

*cha yatp**ṛ**thagbhUte kurutastadubhayā**ṅ**karoti.”*

*The above verse can be written again, by separating the combined words and syllables, as below:*

*“dīrghasya ak**ṣṇ**ayā rajjuH – pārśvamānī, tiryaDaM mānī,*

*Cha yat p**ṛ**thah bhUte kurutah – tat ubhayā**ṅ**karoti.”*

*Below are the meanings of all the words:*

*Dirgha – Oblong tank or pond*

*Akshnaya – Diagonally or transversely*

*Rajjuh – rope*

*Pārśvamānī = The longer side of the oblong or the side of a square*

*Tiryak –across, oblique, sideways*

*Yat (… tat) – Which ( … the same)*

*Prthah – ( particular) measure*

*bhūta – become, produce*

*kurutaha – they (two) do, both do* (typical Sanskrit dual verb)

*(Yat …) tat – (Which …) the same *

*ubhayā – In two ways, two together*

*Ubhayangkarothi – Produces or effects the two together*

Putting the verse in the English language syntax, it reads as below:

**In an oblong tank – (what) longer side and (the other) oblique side, the measures (or areas) they produce – (the same) (sum of) both, is effected or produced – by a diagonally held rope. **

The natural evolution of this Baudhayana Sutra (Or this Baudhayana Theorem) speaks volumes of its originality. Our salutations to Baudhayana.

In trying to translate this verse into English I was handicapped by two deficiencies – (i) my highly limited knowledge of Sanskrit and, (ii) Non availability of a English-Sanskrit-English technical dictionary. Such a dictionary is very much a need of the hour, as lot more technical people are now trying to understand and interpret the immense contribution of ancient Indians to Science and Technology. For example in the case of this verse, Deergha, Parsva and Triya may mathematically mean the three sides of a right angled triangle. Experts in this field should take initiative in developing such a technical dictionary for Sanskrit.

Ref **:** S.G. Dani, On the Pythagorean triples in the ´ Sulvas¯utras, Current Sci. 85(2003), 219-224;

(available at: http://www.ias.ac.in/currsci/jul252003/contents.htm/)

** **

**L V Nagarajan**

**12 June 2013**

June 13, 2013 at 11:15 am |

Wonderful Indeed. Look forward to more on mathematics