Let POTHAYANAAR get his due - 457

Alternate Method to Calculate the Length of the  Hypotenuse as done by using the Pythagoras Theorem 

 The Tamilian land, centuries before the dawn of the common era had built dams and dikes, palaces and great cities during the Sangam era. I was wondering how the great turrets in temples and great highways were built without any knowledge about the Pythagoras theorem was made possible. My search did bring some information to the effect that to find the hypotenuse of a right-angle triangle independent of the Pythagoras theorem, which enunciate that sum of the square of both sides of the right angle will be equal to the square of the hypotenuse, of the triangle. It is not a simple task to find the square of a number, but finding the square root of a number is a herculean task. There is no simple formula to find the square root of a number.

 To my own surprise and astonishment, I found that an ancient Tamil Mathematician/Poet by the name of  Pothayanar, who had lived 800 years before the common era, has 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 fraction. Here is the English translation of the quatrain.

Divide the horizontal to eight,           

Delete one portion, and add the remaining,

To half of vertical to result you’ve got’

Available would-be hypotenuse of the triangle.  

The Tamil poem by poet Pothayanar goes as follows; 

ஓடும் நீளம் தனை ஒரேஎட்டுக்

கூறு ஆக்கி கூறிலே ஒன்றைத்

தள்ளி குன்றத்தில் பாதியாய்ச் சேர்த்தால்

வருவது கர்ணம் தானே. – போதையனார்

 The advantage of the ancient theorem is that one need not use a square root function which may be a bit cumbersome. But before we jump to conclusion let us see how our new found 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 B to be the horizontal side. If we are to divide 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: Say A=8 and B=6

 

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

According to the quatrain; C becomes 7/8A + ½ B 

Now 7/8 of A = 7 and ½ of B =3 And they add up to give hypotenuse to be 7=3=10  

 

Now 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.

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

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

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

7/8(12) = 10. 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 great discoveries of the Greeks scientists and Mathematicians helped the humanity because of the valour with which they conquered the  then known world. Unfortunately, not only the great intellectuals but also their findings were lost to the world. Let’s in our thought thank him for his discovery.