Wednesday, January 25, 2006

The Tamilian Statement of the Pythagoras Theorem

Pythagoras Theorem, as every one would have studied in their high school mathematics, is defined as follows

"In any right triangle, the area of the square whose side is the hypotenuse (the side of the triangle opposite the right angle) is equal to the sum of the areas of the squares of the other two sides."

so the hypotenuse can be calculated as h = (a² + b²)½, that is square root of (a² + b²). But there is an amazing theorem in Tamil, rather a statement to find out the hypotenuse without the use of square-roots. This was stated, by Bodhayanar or Buddhayana, even before the time of Pythagoras. The statement goes like this

"Odum Neelam Thanai Ore Ettu
Kooru thaaki Koorilae Ondrai
Thalli Kundrathil Paadhiyai Saerthal

Varuvathu Karnam Thane"

which means split the base (Neelam) into 8 equal parts (Kooru) and subtract 1 part from it and add half the height (Kundru) which gives the hypotenuse (Karnam). So lets work this out with an example. Consider a right triangle, of dimensions base 8 units and height 6 units so according to the pythagoras theorem the hypotenuse is square root of (64 + 36) which is 10. Now lets try the same with the above theorem

Divide the base into 8 equal parts so in this case each part is 1 unit

Subtract 1 part from the 8 parts so the remaining is 7 units ----- (1)

Add half the height which is 1/2 of 6 = 3 units -----(2)

Now adding (1) and (2) gives the total 10 units which is the hypotenuse of the triangle.
Amazing isn't it? Also if you notice there is no use of any square roots, but simple basic arithmetic addition, subtraction, multiplication and division. The only condition for this to work is that the base has to be greater than the height. So we can summarize the above as follows. If x, y are the base and height of a right triangle respectively then the hypotenuse

z = 7/8*x + 1/2*y where x > y

This works 100% accurately for all integral pythagorean triplets (i.e, the right triangles with dimensions that are round numbers like [3,4,5], [8,6,10], [5,12,13] etc) for the others it works with 95% to 99% accuracy

I don't have any rigorous mathematical proof to it, but it works!

More to come, until then...

Tuesday, January 10, 2006

The Power of Yoga

Yoga, the catch-phrase of today. For anything metaphysical, the first resort is to yoga. If you are stressed, mentally pressurized, wanting some peace of mind, the first suggestion that everybody would give is to practise yoga. What is yoga actually? Well, it is an organised way to keep your mind and physique in good shape and condition. Apart from that, practising yoga is said to be a cure for some diseases and a prevention from many. Well, as the old quote goes- Prevention is better than Cure. But in what ways!. Thirumoolar describes in his masterpiece, Thirumandhiram, many facets of yoga. In the following one he describes about what are the ailments that yoga cures when practised during different times of the day

"Anjanam Pondrudal Iyy-arrum Andhiyil
Vanjaga Vaadham Arum Madhiyaanathil
Senchiru Kaalaiyil Seydhidil Pitharum
Nanjara Sonnom Narai Thirai Naasame"

before explaining this song, let me digress a bit to tell something about the human body. Three main components drive the human body, called the Vaadham (Air), Piththam (Fire), Silaethumam or Saythumam (Water). The components in the appropriate proportions will have the body acting normally. If one of the them decreases or increases in proportion, it results in some ailment. Getting back to the song, Our body is like the anjanam (kajal), soft, in which, the ailments due to disproportionate Silaethumam can be cured by practising yoga in the evening, and of Vaadham(Air) during the afternoon and of Piththam (Heat/Fire) during the early dawn. And as a closing note of this verse, Thirumoolar says that what he has told is the truth and practising yoga will ensure that your hair colour will remain black making you look young and you would not have any eye related diseases.

"Pullinum Mikka Puraviyai Merkondal
Kall Unna Vendadhavanai KaliTharum
Thulli Nadapikkum Sombu Thavirpikkum
Ulladhu Sonnom Unarvudayorkkay"

This song refers to breathing techniques practised in yoga, it means those who adopt the practise of taking in air that is faster compared to a bird, then it gives the pleasure and briskness as you would drink the Kall (a drink prepared from the sap of toddy palm). The breathing practice will make you agile, will prevent fatigue. And Thirumoolar ends this with a quote, that "I have told to those who can realise"

"Thirigindra Vaayuvai Neripada Ullay
Nin malam aakidil Uruppu
Sivakkum Romam Karukkum Purappattu
Pogaan Puri Sadaiyonae"

meaning, "If channelised properly, within you, the air that goes around you, Your skins and organs will get rejuvenated, the hair will get darker and the Lord Shiva (Puri Sadaiyone) will always reside in your body"

More to come about Yoga, Thirumoolar and Thirumandhiram, until then...