Pothayanar and Pythagoras

The following proposition in italics in going around in the Internet for sometime now. It is another proposition of solving the hypotenuse length problem which is different from what was given by Pythagoras. This has been interpreted from a quatrain of Pothayanar, a Tamil poet. 

The proposition could be true for some values, but I found that it is not true for all values of the sides of a right angled triangle. I worked out one way to contradict the proposition in the image below the text in italics:

It is a simple task to find the square of a number, but finding the square root of a number is not so easy. There is no simple formula to find the square root of a number.

An ancient Tamil mathematician/poet Pothayanar, who lived 800 years before the Pythagoras had given a quatrain of four lines articulating the method of finding the length of the hypotenuse of a right-angle triangle without the need to find the square or the square-root, only using the length of the sides, and simple fractions. 

Here is the English translation of the quatrain:

Divide the horizontal into eight,           

Delete one portion, and add the remaining,

to half of vertical to result you’ve got.

The answer would be hypotenuse of the triangle.  

The Tamil poem by poet Pothayanar is :

ஓடும் நீளம் தனை ஒரேஎட்டுக்

கூறு ஆக்கி கூறிலே ஒன்றைத்

தள்ளி குன்றத்தில் பாதியாய்ச் சேர்த்தால்

வருவது கர்ணம் தானே. – போதையனார்

The advantage of the ancient theorem is that  there is no need to use a square / square root function.

But before we jump to conclusions let us see how this ancient and simple formula works :

Let us take the three sides of the right-angle triangle to be A, B, and C, where C be the hypotenuse. 

Let us take A and B to be the horizontal and perpendicular 

sides respectively. 

If we are to divide A into eight parts and takeaway one eight, it would be 7/8A. 

The half of the vertical side will be 1/2B. 

Thus, the result should be :

C= 7/8A + 1/2B

Let us give some numbers and try : 

 Firstly Say A=8 and B=6

By Pythagoras theorem, C equals √ (8x8+6x6) Which is √ (64+ 36) = √100 =10.

Now, according to the quatrain :

C should be 7/8 A+ ½ B 

 7/8 of A (8) = 7 and ½ of B (6) =3 

Together they add up to give hypotenuse to be 7+3=10  

 Second let us try with taking A=28 and B=21 then 

by Pythagoras theorem C= √ (21x21+28x28)

C = √ (441+784)  

which is =√1225 = 35

According to quatrain : hypotenuse becomes 7/8A + 1/2 B.

7/8 A=7/8 (28) = 24.5 and 1/2B= 1/2 (21) = 10.5

Thus 24.5 + 10.5= 35.

 Third let us try with taking A= 12 and B= 5 then 

By Pythagoras theorem C= √ (12x12) + (5x5) = (144+ 25) √169 =13.

According to the ancient Tamil quatrain :  the hypotenuse becomes 7/8A + 1/2B

7/8(12) = 10.5 1/2 (5) = 2.5

Thus 10.5 +2.5 =13 

Pothayanar must have been a great mathematician, who got lost like fruit hidden in the foliage of the tree. 

The discoveries of the Greek scientists and mathematicians spread far and wide along with their conquests in the world.  

Unfortunately, in ancient India,  many great intellectuals, and their knowledge / findings were  lost to the world owing to various reasons and events. 

Our schools teach the  Pythagoras Theorem to our children. They should also teach Pothayanar's theorem as an alternate and easier method, as explained above.