Archive for June, 2013

The River – நதி

June 23, 2013

                         நதி

தாயின்று எழுந்து நீராவியாய், பின்

வான்நின்று பொழியும் நல்மேகமாய்,

பூநின்று செல்லும் நீரோட்டமாய், நதி

தாயொன்றி மகிழும் கடல் கூடியே. –  1

தான்நின்று பல்லோர்க்கும் அமுதாகி, நதி

தாள்சென்று அடையாது நஞ்சுற்றே

உயிர்குன்றி ஒசிந்து உணர்வற்று, தன்

உடல்குன்றி வீழ்ந்து ஓய்ந்ததுவே. –  2

ஆஒற்றிக் கரந்த பால் எனினும்

அதன்கன்றிர்க்கும் ஓரளவு ஈவது போல்

உயர்குன்றில் விழுந்த நதி நீரும், சிறிது

தாய்சென்று அடைவதே தருமம் அன்றோ. –  3

வேரின்றி வளராது விருட்சம், தன்

காலின்றி வாழாது கால்நடைகள்

நீரின்று அமையாது உலகு, எனின்

வானின்று அமையாது ஒழுக்கு. –  4

English Translation

Rising from its source as vapours

Falling from the benevolent clouds

Flowing through earth as streams – rivers

Folding joyously into the laps of seas. – 1

Holding its flow to feed thousands, but

Stalling on its way with filth and toxins

Losing its life, form and feeling – river

Falling a victim to greed and neglect. –  2

Tending the cow and drawing the milk, but

Leaving a bit for its calf to drink – like wise

Allowing the waters to reach its source – river

Ending its flow in a holy communion. –  3

No growth for trees without their roots

No life for animals without their feet

No human race without the waters -never

Any peace sans water resources –  4

(The last two lines are adopted from Tirukkural-Tamil)

I am re-publishing this poem after seeing the man-made disaster in Uttarakhand.

L V Nagarajan – 23 June 2013

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Square Root of 2, by Baudayana

June 16, 2013

Square Root of 2, by Baudayana

L V Nagarajan

Baudhayana is a great mathematician of ancient India estimated to have lived during 800BC. He was an expert mathematician, architect, astronomer and a Hindu high priest. He has proposed several mathematical formulas (Sulva Sutras), some of them with proofs. His statement and proof of the so-called Pythagoras theorem is so simple and elegant.

In ancient times, a Square was held as an important geometrical figure. Every area was expressed in so many squares. There was considerable interest in finding an equivalent square for every area, including circle, rectangle, triangle etc.

Baudhāyana, gives the length of the diagonal of a square in terms of its sides, which is equivalent to a formula for the square root of 2:

samasya dvikaraṇī. pramāṇaṃ tṛtīyena vardhayet
tac caturthenātmacatustriṃśonena saviśeṣaḥ

Sama – Square; Dvikarani – Diagonal (dividing the square into two), or Root of Two

Pramanam – Unit measure; tṛtīyena vardhayet – increased by a third

Tat caturtena (vardhayet) – that itself increased by a fourth, atma – itself;

Caturtrimsah savisesah – is in excess by 34th part

In English syntax, it will read as below:

The diagonal of a square of unit measure (is given by) increasing the unit measure by a third and that again by a fourth (of the previous amount). This by itself is in excess by a 34th part (of the previous amount).

That is,

√2 = 1 + 1/3 + ¼ (1/3) = 17/12

But as per the sulba-sutra above, this in excess by a 34th part of the previous amount.

Hence

√2 = 1 + 1/3 + ¼ (1/3) – 1/34[(¼ (1/3)]

= 1 + 1/3 + 1/(3*4) + 1/(3*4*34) = 577/408

The above value is correct to five decimals.

There have been several explanations as to how this formula was evolved. Apparently, initially (even on works of late 6th century AD) an approximate value of 17/12 was used for √2, which is nothing but [1+ 1/3 + 1/(3*4)].

1. One theory is they just used actual measurement by ropes to arrive at these fractions – they first tried unit rope length and then 1/2 of the same length. As it was too long they next tried 1/3. It was just short and hence ¼(1/3) was added to it. It was quite close and hence 17/12 was used initially. However it was found slightly longer (savisesah) and when measured by rope again it was found longer by 1/34[1/4(1/3)]. This was also found minutely longer (savisesah) but accepted as a sufficiently accurate value. – That was a simple explanation.

2. Another explanation for the evolution of this formula was based on geometrical construction. – Two equal squares, each with side of one unit were taken. One of the squares was vertically divided into three rectangles. Two pieces of the above was placed along the two adjacent sides of the square to form an approximate square of side 1+1/3, but for missing a small square 1/3×1/3. This was also made up by using a piece from remaining rectangle. Hence we get the first approximate value of (1+1/3) for √2.

Square Root 2

But a small piece is still remaining of size (1/3 x 2/3). This was made into 4 equal strips of size [1/4(1/3) x 2/3]. Two pieces, end to end, were kept along one side of the above augmented square and the other two pieces on the other side. Now we get a total area which is the sum of the two squares. The above augmented square is of side (1 + 1/3 + 1/(3×4)) = 17/12, which was found good enough initially. But, we still miss a small portion of (1/12 x 1/12), to complete the square. Hence the size has to be reduced by this extra area (savisesah). – To find this extra measure, (1/12 x 1/12) should be divided by (17/12 +17/12) and that gives 1/(3x4x34). Of course still we have some minute extra area (savisesah). The above construction is explained in the above figures.

Yes, this is really an interesting explanation. But this amazing formula (sutra) evolved in 800 BC deserves a better explanation which I will offer now.

3. Way back in 1967, I was in a class room of IIT/Kanpur. The teacher was Professor Dr. V. Rajaraman, the pioneer of computer education in India. He was teaching us the basic algorithms for programming in Fortran, a (then) popular programming language. One of the very early recursive algorithms he taught us was, to find the square root of a number. It goes thus:

Let N be the number for which square root is required

Make first guess of the square root as r0 – Later, we will know, the guess may be as bad as 10 times N; still the method works as smoothly as ever.

The next guess can be made as r1 = ½ (r0 + N/r0)

Keep improving this value using the recursion: r(n+1) = ½ [r(n) + N/r(n)]

Surprisingly the value converges very fast to √N, to the required level accuracy.

As you have seen in my earlier blog of Evolution of Sine Table by Hindu Maths, our ancient mathematicians have always preferred recursive steps to solve any problems. Hence, in this case also, Baudayana preferred to use recursive steps, exactly as above. However like others, he preferred to calculate individual step sizes as below:

r(n+1) – r(n) = ½[N/r(n) – r(n)], which is same as the above recursive statement.

To find square root of 2, Boudayana used (1+1/3) = 4/3, as the first guess, r(0).

Hence next step, r(1) – r(0) = ½(3/2 – 4/3) = (3/4 – 2/3) = 1/(3×4)

And hence, r(1) = 1 + 1/3 + 1/(3×4) = 17/12 = 1.4166

Next step, r(2) – r(1) = ½(24/17 – 17/12) = [12/17 – 17/(3x4x2)] =  -1/(3x4x34)

And hence,  r(2) = 1 + 1/3 + 1/(3×4) – 1/(3x4x34) = 577/408 = 1.414216 (correct up to 5 decimals)

Baudayana would have gone to the next minute step also (as his savisesah, indicates), as below

Next step, r(3) – r(2) = ½(816/577 – 577/408) = 408/577 – 577/(3x4x34x2) = -1/(3x4x34x1154)

Hence, r(4) =  1 + 1/3 + 1/(3×4) – 1/(3x4x34) -1/(3x4x34x1157) = 1.41421356237469

The above value is correct up to 13 places.

The recursive algorithm is always the first approach of ancient Indian mathematicians.

Just to satisfy myself that it is not just an isolated case, I tried this logic for finding square root of three also. The step sizes came out to be as below:

Start = 1

First Step = ½             , Root = 1.5

2nd Step = 1/(2×2), Root = 1.75

3rd Step = – 1/(2x2x14), Root = 1.7321429

4th Step = – 1/(2x2x14x194), Root = 1.7320508

The above value is correct up to 7 decimal places.

Again hats-off to Baudayana!

Baudhayana’s (Pythagoras) Theorem

June 12, 2013

                              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:

Normal

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:

Offset

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) – 4 x (ab/2)

(i.e) Area of the inner Square = a2  +  b2

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:

 “dīrghasyākṣṇayā rajjuH pārśvamānī, tiryaDaM mānī, 

cha  yatpthagbhUte kurutastadubhayākaroti.”

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

“dīrghasya  akṣṇayā  rajjuH – pārśvamānī, tiryaDaM mānī, 

Cha  yat  pthah  bhUte  kurutah – tat ubhayākaroti.”

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