Archive for the ‘Education’ Category

Swatch Bharath Abhyan

October 1, 2015

Swatch Bharath Abhyan

(Clean India Campaign)

It is just one year since the above campaign was launched by our Prime Minister Shri Modi. Many reviews of ‘progress-so-far’ have appeared in print, visual, electronic and social media. Ignoring the politically biased views, generally there is a concern that the campaign has not achieved the desired result so far. Mr. Modi has perhaps anticipated such apathy to the campaign and hence has given himself and the government 5 full years up to 2019 to achieve reasonable cleanliness. Still it is a good idea to review the ‘progress-so-far’ and take some positive actions to improve the progress based on our experience till now.

What makes the public places and surroundings unclean? There are about 10 types of wastes generated by individuals, families and institutions. They are:

Kitchen/Garden waste

Personal and Health care waste

Stationery waste

Plastic waste

Packing waste

Party/event waste

Industrial waste

Construction waste

Electrical/Electronic waste

Metal waste

General public need a lot of guidance and facilities to dispose of these wastes appropriately. I am attempting here to give my own ideas on how to dispose of Plastic Wastes in an environmentally friendly way.

  1. Single plastic bags should never be disposed of on their own. It is more likely to fly off anywhere and block any drain or air passage and block water seepage to the ground and below.
  2. Any thin plastic bag should be disposed off tying its ends together in to a bundle so that it cannot balloon and fly off.
  3. At home a number of such tied up thin plastic bags should all be gathered together in to a larger plastic bag/bundle and disposed of separately. This will enable and encourage the trash pickers to collect them and deposit them for recycling.
  4. Thicker plastic bags should be reused as much as possible. There should be municipal collection facilities where we can deposit them, after packing them neatly.
  5. Waste plastic sheets (thin and thick) should be treated the same was as bags.
  6. Plastic bottles should also be deposited in municipal collection facilities as above.
  7. Disused plastic containers and other thicker materials like boxes, mugs, buckets, furniture and fittings should all be gathered together and handed over to trash dealers personally.
  8. Housing societies and apartment complexes should have a dry waste collection day, once in a month (say, last Saturday of the month). On this day all the residents should deposit their plastic waste material collected as above in to a common bin provided for this purpose. The trash dealers may be requested to collect the same at the end of the day.
  9. Municipal ward offices should announce one day in a month (say, last Sunday of the month) as dry waste collection day and a truck should go around the ward collecting such wastes.
  10. The slums and low cost housing areas should be more actively involved in this Clean India Campaign, for it to succeed.

Such methods of waste disposal as above should be evolved for all types of wastes. They should be publicized periodically in all media, especially the vernacular ones.

Let us all have a Clean India and a Green India. Vande Mataram.

Ways To Overcome Negative Emotions

September 17, 2015

This is an extract from ‘The Speaking Tree’ columns of Times of India dated 31 Aug 2015, by Mr. Sanjay Teotia.

“Emotional balance is vital for personal well-being and the health of our communities. The path to freedom is paved with positive emotions. Negative emotions prevent inner transformation” – We do want to control negative emotions, but HOW is the question. This article suggests some logical methods for the same, which I want to share with my readers. I have listed them below:

  1. For emotional balance, never react suddenly to anything. First think, then, try to find out facts, assess the situation and then give your opinion or take action.
  2. Never be judgmental. Try to find out positive things or reasons in negative and adverse situations also.
  3. Always try to speak positive things. Emotional balance is genetic as well as acquired.
  4. It also depends on your group or friends circle. If you are surrounded by negative-thinking people who are always complaining and criticising others, then you too tend to think negatively. So try to be in the company of positive people.
  5. Emotional balance occurs when we allow ourselves to feel whatever comes up, without stifling it or being overwhelmed by it, and learn to accept our feelings without judgment.
  6. Always avoid confrontation, try to find out the middle path. Negative emotions create negative aura and spread negative vibes from the person who has negative emotions.
  7. Prayer, meditation, faith in God and kind heartedness help you remain positive.
  8. Adverse conditions or tough times come in everyone’s life but nothing is permanent. As the good time passes, in the same way bad times will also pass. Have faith in God, as it protects us from negative emotions, negative psychology, negative thinking and negative perception.
  9. Negative emotions makes the person sadist by nature, he never wants to see others happy. Such persons have complaints about God also. When they are in difficult situations they ask God, `Why me in this situation?’. They forget their fun times. They never said `Why me?’ when they went through happy times.

My Thanks to Mr. Sanjay Teotia.

Soul tied to physical body

May 28, 2014

Soul tied to physical body

L V Nagarajan

1.0 Soul, in a spiritual context

Last spring, I attended a spiritual workshop conducted in our neighborhood. In one of the sessions the Guru involved me in a demo conversation to bring out the concept of Soul or Atma.

He asked me : “Who are you?”

I replied, “I am Nagarajan”

“No that is your name. I know it. But who are You?”

After some thought I replied: “I am an Electrical Engineer.”

“No that is your profession. But who are YOU?”

After some more thought I pointed to my body somewhere near the heart and said – “This is Me.”

“No that is your body. But who are Y-O-U? Can you tell me who are you without referring to your extensions like name, profession, your body and such things. They are all temporary and subject to change.”

“How? Body can change ….?”

Guru did not reply. He went on to describe Soul or Atma in a spiritual context and how it will finally merge with the superior SOUL or PARAMATMA. But I was intrigued by the unanswered question – like name and profession, can you change your body also?

2.0 Soul, in Yogic context

I was surprised to find the answer to this question recently through the columns of Times of India. Writing in ‘The Speaking Trees’ of 30th April 2014, Sadhguru Jaggi Vasudev says: “Breath is not just the exchange of oxygen and carbon dioxide. For different levels of thought and emotion that you go through, your breath takes on different types of patterns. When you are angry, peaceful, happy or sad, your breath goes through subtle changes. Whichever way you breathe, that is the way you think. Whichever way you think, that is the way you breathe. Breath can be used as a tool to do many things with body and mind. Pranayama is the science whereby consciously breathing in a particular way, the very way you think, feel, you understand and experience life can be changed”. He further adds: “Breath is like the hand of the Divine. You don’t feel it. It is not the sensations caused by the air. This breath that you do not experience is referred to as Koorma Nadi. It is a string which ties you with this body, an unbroken string. If i take away your breath, you and your body will fall apart because the being and the body are bound by the Koorma Nadi. This is a big deception. There are two, but they are pretending to be one. There are two people here, the body and being, two diametrically opposite ones, but they pretend that they are one. If you travel through breath, deep into yourself, to the deepest core of breath, it will take you to that point where you are actually tied to the body. Once you know where and how you are tied, you can untie it at will. Consciously, you can shed the body as effortlessly as you would shed your clothes. When you know where your clothes are tied, it is easy to drop them. When you don’t know where it is tied, whichever way you pull, it does not come off. You have to tear them apart. Similarly, if you do not know where your body is tied to you, if you want to drop it, you have to damage or break it in some way. But if you know where it is tied, you can very clearly hold it at a distance. When you want to drop it, you can just drop it consciously. Life becomes very different. When somebody willfully sheds the body completely, we say this is mahasamadhi. This is mukti or ultimate liberation. It is a great sense of equanimity where there is no difference between what is inside the body and what is outside the body. The game is up. This is something every yogi longs for. Consciously or not, every human being is working towards this.”

Yes, here we have the answer. You are different from your body. I find this as a Yogic or elemental way looking at your soul. A yogic practice to realize oneself separated from one’s own body. This is perhaps the way Sri Ramana Maharshi found the answer for his monumental question WHO AM I? Subsequently he even achieves out-of-body experience and preaches these concepts to thousands of his followers and devotees. All this said and done, this is still not a complete answer to my question – ‘Like name and profession, can you change your body also?’

3.0 Soul, in a scientific context

Idly I turned my eyes away from ‘The Speaking Trees’ to the next page of the same issue of Times of India. To my surprise I found the missing part of the answer in another news item on science pages. It talks of ‘A device to let you ‘virtually’ swap your body with another’. Here goes the report: 

 “A group of artists based in Barcelona has created an unusual virtual reality device that can allow you to experience what it might be like to step into the skin of another person. The device, called ‘The Machine to be Another’ lets people experience life in another person’s body. Participants in a body swapping experiment at the ‘Be Another’ lab, don an ‘Oculus Rift’ virtual reality headset with a camera rigged to the top of it. The video from each camera is piped to the other person, so what you see is the exact view of your partner. If she moves her arm, you see it. If you move your arm, she sees it. To get used to seeing another person’s body without actually having control of it, participants start by moving their arms and legs very slowly, so that the other can follow along. Eventually, this slow movement becomes comfortable, and participants start to feel as though they are living in another person’s body, BBC News reported.”

Is the above an attempt to look at your soul from scientific aspect? Now we can look up to Jagat Guru Adi Sankara. He had achieved this feat of entering another body but without the aid of such devices as above. Even some lesser mortals have achieved this feat and this is known as one of ashta-ma-siddis, the eight great feats. In Tamil it is known as Koodu-vittu-koodu-paaydal, or ‘from one shell to another’.

We may meditate on this.

Ref:

  1. THE SPEAKING TREE, Life Breath & The Ultimate Expansion, by Sadhguru Jaggi Vasudev, Time of India 30th April 2014, page 20.
  2. A device to let you virtually swap bodies with another, BBC News report,  Time of India 30th April 2014, page 21

LVN/28 May 2014

Baudhayana’s Circles

September 21, 2013

Western biographers credit Archimedes of Syracuse (287-212 B.C) with the analytical evaluation of the factor Pi associated with circle, within a close range of 31/7 to 310/71. However in the process of establishing this, they also recognize him as the first person to realize that the same factor is associated with both perimeter and area of the circle. He is also said to be the first person to propose and prove that “Area of the circle = ½ x Perimeter x Radius”. Is it really so? Let us look at another person, one Baudhayana, from ancient India (800BC) who also worked on the circles earlier to Archimedes, by a few centuries.

It was known to Baudhayana, or even to people earlier to him, that the perimeter of the circle depends only on its radius or diameter and that it is actually proportional to the radius or diameter. Though it has not been stated explicitly, it is clear from various sutras that they were well aware that for similar figures, the ratio of the areas equals the square of the ratio of the lengths of the corresponding sides. It was also known that the area of the circle depends only on its radius or diameter and that it is actually proportional to the square of the radius. That is, for a circle, it was known that: – Area = Ka x r2, and Perimeter P = Kp x (2r). But do they know that, Ka = Kp = Pi ?.

During 800 BC, this Indian high priest, Baudhayana, has formulated, in his Sulvasutra (I-48), the so-called Pythagoras theorem, centuries before Pythagoras (572 BC). In another sutra (I-51) he has given a general rule for finding the square root of any number, both geometrically and arithmetically. In his Sutra (I-61) he found the value of √2 to a great accuracy and has given the procedure for the same. This Indian mathematician could construct a circle almost equal in area to a square and vice versa. He has described such procedures in his sutras (I-58 and I-59). All these were achieved in 800BC!

As Baudhayana was designing a religious altar for performing the Hindu rites, he constructed a square within a square as below:


Normal+

He observed the inner square is exactly half of the bigger square in area.  This led him to formulate, in his Sutra I-48, the so called Pythagoras theorem, which was reinvented by Pythagoras a few centuries later. Please refer to my earlier blog on “Baudhayana’s (Pythagoras) Theorem”.

Subsequently Baudhayana wanted to evolve procedures for constructing circular altars. He constructed two circles circumscribing the two squares shown above. Now, just as the areas of the squares, he realized that the inner circle should be exactly half of the bigger circle in area. Yes, he knows that the area of the circle is proportional to the square of its radius and the above construction proves the same. By the same logic, just as the perimeters of the two squares, the perimeter of outer circle should also be √2 times the perimeter of the inner circle. This proves the known fact, that the perimeter of the circle is proportional to its radius. Now it is known beyond any doubt that for a circle,

Area = Ka x r2, and Perimeter P = Kp x (2r).

But that is not enough. Baudhayana wants to construct circular altars of specific areas. He needs to know the values of Ka and Kp. Baudhayana and his ilk were more interested in the area of the circle than its perimeter.

At this point, Baudhayana would make an important observation. Areas and perimeters of many regular polygons, including the squares above, can be related to each other just as the case of circles. The perimeters and areas of some simple regular polygons are listed below (‘r’ is the distance from the centre of the polygon to its sides):

Equi. Triangle–  Perimeter = K3(2r) & Area = K3(r2); with K3 =  3√3

Square-  Perimeter = K4(2r) & Area = K4(r2); with K4 = 4

Hexagon-  Perimeter = K6(2r) & Area = K6(r2); with K6 =  2√3

Octagon-  Perimeter = K8(2r) & Area = K8 (r2); with K8 =  8(√2-1)

It may also be noticed that the values of the constants, Ki’s, are gradually reducing from about 5 to about 3.3, by the time we reach the octagon. Another fascinating feature with these polygons is: all their areas are ‘½ r’ times their perimeters. Anybody would have been tempted to conclude from above that for circles also, Kp= Ka= K0. This will automatically make Area of the circle = ½ r x (Perimeter). However Baudhayana wanted to prove this.

Let us now consider an N-gon, a regular polygon of N-sides. Let ‘r’ be the distance of the sides from the centre. Let each side be equal to ‘s’. The area of the triangle one side makes along with the centre is (½)sr. Hence the total area of this N-gon is ½Nsr.

So, for N-gon -> Perimeter = Ns & Area = ½Nsr; with Kn = Ns/2r

Here again, Area = (r/2) x Perimeter. In all the above regular polygons, a circle of radius r can be inscribed. Now let us assume that, for this circle, constants Kp and Ka are different. The areas and perimeters of the above polygons are steadily reducing but still remaining more than those of this circle. (i.e.) Kn(2r) > Kp(2r) and Kn(r2) > Ka(r2). Hence any Kn will have to be greater than both K1 and K2. In the N-gon last considered, Ns being the perimeter, it reduces constantly as number of sides N increases. Kn = Ns/2r will also reduce gradually and finally will converge to a finite value, now known as Pi. Baudhayana realized that there is an N-gon with such a perimeter, whose Kn is just more than Kp by any arbitrarily small amount ‘∂’. Similarly Baudhayana concluded that there is an N-gon of such an area, whose Kn is just more than Ka by any small amount ‘ε’.  However both these Kn’s are greater than Kp and Ka. (i.e.) Kp + ∂ > Ka, and,  Ka + ε  > Kp, for any small amounts of ∂’s and ε’s. The above is possible only when Kp=Ka.

Thus Baudhayana concluded, Kp = Ka = Ko, which is now known as Pi. This automatically makes, Area of the circle = ½ r x (Perimeter).

So we may conclude that, the above facts about the circles were already known to Baudhayana and other ancient Indian mathematicians even before Archimedes. Credit is surely due to Archimedes for narrowing down the value of Pi between 31/7  and 310/71. However, Baudhayana on his own has narrowed down the value of Pi to be between 40/12 and 40/13 (i.e. 31/3 and 31/13). It is the value 40/13 he has used in his Sulva Sutra I-58, (for Circling the Square or to find a circle equal in area to a square), as will be demonstrated later. Before going to I-58, let us see how he derived the above values for Pi.

Ancient mathematicians could have already found the value of Pi with limited accuracy, by actual measurements of diameter and perimeter of the circle by using ropes as per the practices existing in those days. It could have given them, at best, a value between 3.11 and 3.17 (i.e. Pi +/- 1%). So, the attempts continued to analytically find the value of Pi.

Even before Baudhayana, the upper limit for the value of Pi was fixed as 4 by considering a circle inscribed in a square. The lower limit was fixed as 3 by considering a regular hexagon along with its circum-circle. But after Baudhayana’s (Pythagoras) theorem, these limits could be narrowed down to be between 3 and 2√3 by considering both circum-circle and in-circle of a hexagon as below.

Hexagon

The circum-radius is ‘a’ and the in-radius is √3/2(a). The perimeter of the hexagon is larger than that of in-circle and less than that of circum-circle. i.e. π(2xa) > 6a and π(2ax√3/2) < 6a. Hence, 2√3 > π > 3.  There is an indirect reference to the value of 3 in an earlier sutra of Baudhayana for constructing altars: “The pits for the sacrificial posts are 1 pada in diameter, 3 padas in circumference.” This gives an approximate value of 3 for Pi. But they knew it is more than 3, as 3 pada is already known as the perimeter for a hexagon inscribed in the above circle. Hence 3 is just the lower limit for value of Pi.

Baudhayana went one step further by considering a regular octagon along with its circum-circle and in-circle. Just as Archimedes found the bounds for value of Pi, by considering 96-gon circum-scribing and in-scribing a given circle, Baudhayana found the bounds, by considering circum-circle and in-circle of an octagon. (yes, a complimentary procedure to what was done by Archimedes, five centuries later). Baudhayana used for √2, the value of 17/12 (normally used in those days) to arrive at these simple fractions, as below. He first considered a square with 12 units as half side. Hence half diagonal will be 17 units considering 17/12 as √2. Referring to the diagram below, the octagon inscribed within this square will have its side as 10.

octagon

The in-radius of this octagon will be 12 units and circum-radius will be 13 units. (‘5,12,13’ , the Pythagoras triple, as known even in Baudhayana’s times defines this octagon!). The perimeter of the octagon is 80 units. As this octagon is sandwiched between the circum-circles and in-circle, (ref Figure above), we may compare their perimeters as below:

Pi(2×13) > 80 > Pi(2×12)

Hence,

Baudhayana’s values of Pi are given by: 40/12 > Pi > 40/13.

Accuracy of Pi was always sought to be improved throughout the history, even after much closer estimates by Archimedes. A later mathematician Manava (650 ~ 300 BC) has stated in his sutra:

Viskambhah pancabhaagasca  viskambhastrigunasca yah.

sa mandalapariksepo na vaalamatiricyate ||      

(Manava  Sulvasutra 10.3.2.13)

(a fifth of the diameter plus three times the diameter, is

the circumference of the circle, not a hair-breadth remains.)

It gives a value of 31/5 as the upper limit for value of Pi, a better value than Baudhayana’s 31/3, and approaching 31/7 of a later day Archimedes: (as per Manava’s statement “not a bit remains” after 3.2 times the diameter).

Baudhayana cleverly used his values of Pi for finding the area of a circle and for drawing a circle of a given area. He found the value of 40/13 to be closer to the eventual value of Pi and has used the same to derive his Sulvasutra I-58 for “Circling the Square”. Baudhayana would have derived this formula as below:

Let ‘a’ be the distance of the sides of the square from its centre. i.e. each side of the square is ‘2a’. To find a circle of radius ‘r’ with an area equivalent to this square, one may write

πr2 = 4a2;        (i.e.), r = (2/√π)a

Square Circle

Baudhayana used 40/13 as value of Pi, and 17/12 as value of √2.

From the picture above we see, a < r < √2a.

Hence he assumed, r = a + (1/x)(√2a-a) = a [1 + (1/x)(√2 – 1)]

For the above square, πr2 = 4a2

r2 = 4a2/π = 4a2 *(13/40) = a2 * 130/100

Hence, r = a * (√130)/10

By Baudhayana’s method (I-51) for finding the square root of any number,

√130 = √(122 – 14)  = 12 – (14/24) = 10 + 17/12

So, r = a(1 + 17/120) = a [1 + (1/3)(51/120)]

≈ a[1 + (1/3)(5/12)] = a[1 + (1/3)(√2 – 1)]

Thus, r = [a + (1/3)(√2a – a)]

So Baudhayana formulates his sulvasūtra I-58 as below:

caturaśraṃ maṇḍalaṃ cikīrṣann

akṣṇayārdhaṃ madhyātprācīmabhyāpātayet |

yadatiśiṣyate tasya saha

tṛtīyena maṇḍalaṃ parilikhet

 

caturaśraṃ = Square, Mandalam = Circle,

akṣṇayārdhaṃ = Half Diagonal,  

madhyātprācīm = From centre towards east,

abhyāpātayet = laiddown, yadatiśiṣyate = portion in excess,

tasya saha tṛtīyena = using only a third of this, 

parilikhet – Draw around

To make a square into a circle, draw half its diagonal from the centre towards the East; then describe a circle using only a third of the portion which is in excess.

i.e. Using the above formula your are able to draw a circle of given area (=4a2), where a is the measure of half the side of the above square. The radius of this circle is given as:

r = [a+1/3(√2a – a)] = [1+1/3(√2 – 1)] a

Let us see with value of Pi as 40/13, how the areas of square and circle compare.

Area of Square = 4a2

Area of circle = 40/13 x [1 + (1/3)(√2-1)]a2

= (40/13) x (41/36)2 a2  = (67240/16848) a2 = 3.9909 a2

Remarkably close.

As per this construction the value of Pi we obtain as per today’s value of √2 is, Pi = 3.088312

However in the reverse process of squaring the circle, he has gone for corrective fractions as will be demonstrated by Baudhayana’s sutra I-59, as given below:

maṇḍalaṃ caturaśraṃ cikīrṣanviṣkambhamaṣṭau

bhāgānkṛtvā bhāgamekonatriṃśadhā

vibhajyāṣṭāviṃśatibhāgānuddharet |

bhāgasya ca ṣaṣṭhamaṣṭamabhāgonam

viṣkambham = Diameter; aṣṭau bhāgānkṛtvā = making eight parts; 

bhāgaekona = take out one part,

triṃśadhā vibhajya = 29 parts of this part;

āṣṭāviṃśatibhāgānuddharet = of these remove 28 parts; 

bhāgasya ca = from this part also,

ṣaṣṭhamaṣṭamabhāgonam = remove (1/6 minus 1/8 of 1/6).

If you wish to turn a circle into a square, divide the diameter into eight parts and one of these parts into twenty-nine parts: of these twenty-nine parts remove twenty-eight and moreover the sixth part (of the one part left) less the eighth part (of the sixth part). 

The above formula is to make a square of area, equal to a given circle. Baudhayana could have really inverted the earlier formula I-58 to obtain this. But this sutra appears quite complicated. It is so only because Baudhayana in this case used a better value for √2. As per his sutra I-61,

√2 = 1 + (1/3) + (1/3*4) – (1/3*4*34) = 577/408

We may be wondering how handicapped the ancient mathematicians were without the present day decimal point system. However ancient Indians were so facile with fractions they never needed the decimal point system. Even as late as 19th century, Indians were using fractional multiplications tables of ½, ¼, 1/8 , 1/16 , 1/32 and even 3/16  in their every day arithmetic calculations. Baudhayana uses this amazing fractional arithmetic to arrive at the above formula.

We know from I.58, r = a + (1/3)(√2a-a) = [1+1/3(√2 – 1)]a

So we may write, a = r/[1+(1/3)(√2-1)]

With √2 = 577/408,

we get, a = r/[1+(1/3)(169/408)]= (1224/1393)r

Now for fractional magic:

1224/1393 = (1224/1392)*(1392/1393) = (51/58)/(1393/1392)

= [1-(7/56)(56/58)]/[1 + (1/1392)]

= [1-(1/8)(28/29)]*[1- (1/1392)]

= [1-(28/8*29)]*[1 – (1/8*29*6)]

(assuming 28/29 ≈ 1, in the last term),

= 1 – [28/(8*29)] – [1/(8*29*6)] + [1/(8*29*8*6)], .

Thus Baudhayana obtains the final formula of I-59 as:

(With Side of square as ‘s’ and the diameter of the circle as ‘d’)

 s = [1- 28/(8*29) – 1/(8*29*6) + 1/(8*29*6*8)] x d

i.e., s = 0.878682 * d

This increases the value of Pi marginally from 3.088312 to 3.088326.

Conclusion:

It was after studying the book “Journey Through Genius” by William Dunham, I got interested in the History of Mathematics. I read several books to know more on this subject. In order to create interest among our genext, I started to write a few blogs on this subject. The present one is on ancient Indian Mathematician Baudhayana (800BC) and his works on Circles. We seem to know him only through his Sulva-sutras. However by reading extensive material on him, I could make out a few narrations of his works. Of course this narration includes a few of my imaginations and intuitions. Ancient Indians in 800BC were well aware of the basic properties of the circle. Baudhayana’s sulvasutras I-58 and I-59 give ample proof of this. Baudhayana was also able to fix the value of Pi to be between 40/12 and 40/13 (i.e. between 3.33 and 3.08).  Baudhayana’s name is still uttered during many Hindu rituals. Even my own family is linked to Baudhayana through his line of disciples as mentioned often by us in our prayers as, Apasthamba, Aangirasa, Baragaspathya and Bharatwaja.

References:

1. A history of Ancient Indian Mathematics – C N Srinivasaiengar, The World Press Private Ltd. Calcutta. (1967)

2. Journey Through Genius – William Dunham, Penguin Books 1990.

3. S.G. Dani, Geometry in Sulvas_sutras, in ‘Studies in the history of Indian mathematics’, Cult. Hist. Math. 5, Hindustan Book Agency, New Delhi, 2010.

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

Ambition and Contentment

May 21, 2013

  

Many may wonder how one can connect the above two opposite qualities in one idea. At the same time, at least some of us would be very happy to find a way of achieving both Ambition and Contentment in their lives. Yes, both are great virtues as per normal standards of human values. Both are difficult to achieve, but, are they really opposite? I would rather think, they are complimentary, and not opposites.

As per standard dictionaries,

‘Ambition’ means :-  Strong Desire – to be successful – usually needing hard work

‘Contentment’ means :- To be happy with what one has – not wanting any more

But what is success? Success is generally evaluated in terms of money and marketability. If one strikes money early in his path of ambition, the ambition suffers a premature demise. I know of a doctor whose ambition was to do cancer research. After completing his MD, he set up a consulting room.  He was very successful, especially, in terms of money. His ambition to do cancer research just dissolved in thin air.

Real ambitions are those which involve sacrifices. Mahatma Gandhi’s ambition was to see free India. He sacrificed his carrier as a barrister. His ambition was social reforms. He sacrificed his political career. His ambition was communal harmony. He sacrificed his life. If you are ambitious in one aspect of life, you need to be content in many other aspects of life. Otherwise, you cannot succeed.

Contentment means to be happy with what one has. But this does not give anybody any justification to be lazy and unproductive. Contentment in one aspect of life should always lead one to follow a forgotten pursuit in some other area. Thus contentment can at the same time be a cause and a result of an ambition. Contentment leading to inactivity, or trivial activity, is not a virtue at all. It is this base level of contentment and inactivity that has led Uttar Pradesh and other BIMARU states in India to remain poor and backward, in spite of fertile land, suitable climate, abundant natural resources and a high level of political patronage.

Contentment without ambition will lead to laziness and inactivity, whereas, ambition without contentment will lead to frustration.

In trying to be ambitious, one should avoid the following two situations:

  1. Donkey and Carrot situations: where we keep our pursuit on and on, going after a target which is moving away in phase with our pursuit.
  2. Mirage situation: Where the target is an illusion and not real, or rather not realistic.

Contentment will help you avoid the above the above situations. But, in the same way, while trying to be contented you should avoid the following two situations:

  1. Lollypop situation: where you set yourself an easy target and rest happily after the same is achieved.
  2. Rocking Chair Situation: where you sit and relax with total inactivity

When thinking of ambition and contentment, one more factor we must consider is the AGE. It is always said, you are as young as you feel. But there is no denying the fact, age takes a toll on both your physical and mental capabilities. Your capacity to learn and remember, your stamina and tenacity reduce considerably, as you pass the age of, say, 50. You also know, at the back of your mind, you have much less time to live and work. With reduced capacity and limited time, achieving your ambitions is a real challenge. Failure to acknowledge this challenge, will lead to stress and related health problems. That is the time when one should scale down his ambitions, with the help of contentment.

I will conclude with a well known prayer:

Oh Lord,

    Give me the courage to change the things I can

    Give me the humility to accept the things I cannot (change)

And please,

    Give me the wisdom to know one from the other 

L V Nagarajan

20 May 2013

         

The Electrical Power Grid

December 25, 2012

The  Electrical Power Grid

L V Nagarajan

Mr. Contractor and Mr. Engineer are great friends. They are running a reasonably successful construction business for the last twenty years. They decided to construct houses for their own use, but in a small village far away from maddening crowd. The village they chose, did not even have electricity supply as yet. They decided to start construction of their houses adjacent to each other. They had a small diesel generator for construction purposes. After completing the construction they promptly applied for electricity connection. Due to lack of infrastructure, the same was delayed for more than two years. However they wanted to live in their new houses, hence they commissioned two  diesel generators, one for each house and connected the same to their individual main switch boards. They both moved to their new houses

Mr. Contractor started the generator and switched on his lights, fans and heaters etc and as he switched them on, one by one, he observed the whine of the rotating generator getting louder and shriller. But there were other changes happening in the machine which he could not observe. As he put on one appliance after the other, the extra load slowed down the rotating speed of the machine. The speed governor sensed this speed drop and picked up the speed to normal by automatically increasing the fuel input. Just as the speed of the machine is an indicator of generation-load balance, speed of the machine also decides the frequency of AC-power output. The frequency should be maintained close to 50 Hz, as in all AC systems. As the load on the generator kept varying with the use of different appliances, the speed/frequency also followed the changes to maintain the generation-load balance.

There was a time once, when Mr. Contractor’s gen-set developed a problem and got shut down. He suffered a loss of power. He immediately called his neighbor Mr. Engineer, to look into the problem. Mr. Engineer’s gen-set was up and running. He suggested to connect his neighbor’s load also on to his machine, at least till the faulty machine is repaired. He installed a pair of cables to jumper the outputs of both the machines, but took care to put it through a circuit breaker. He requested Mr. Contractor to switch on only essential loads to keep the load on the single machine within its limits. Mr. Engineer checked the faulty machine and found it was a minor fault. He repaired the same and started the machine. He took the machine in service. But he forgot to remove the jumper cable. Then he decided to leave the same on, to increase the reliability of power supply to both of them. Thus was born a GRID, a Power Grid.

When both gen-sets operated as a GRID, they enjoyed the increased reliability and stability of power supply, but they also experienced a few problems:

i) Any load changes in one system affected both the gen-sets.

ii) The response of the two speed-governors being slightly different, there were oscillations in actual power sharing between the two gen-sets.

iii) The power flow through the jumper cable (normally called Tie-line in GRID terminology) was varying widely, some time very close to its full capacity.

Mr. Engineer being very smart, introduced some changes in the governor systems as below:

a) He made the speed governors less sensitive by introducing a speed-load droop response in their systems.

b) He introduced a secondary frequency-control equipment in both the gen-sets, which responds based on ‘Speed/Frequency plus Tie-line flow’, as seen from respective systems. This secondary control helped individual systems in keeping generation/load balance on their own systems. (This secondary control is called Tie-line Bias Control in GRID terminology).

Under normal conditions, the GRID was operating quite well. they derived following advantages:

– Any minor restrictions on the individual gen-sets could be easily managed

– Any gen-set could be released for routine maintenance easily, with only minor load restrictions

– Voltage and frequency were better regulated, even during load surges and drops

There were also a few new problems:

– lack of proper accounting of Tie-line power flows either way, which could enable proper sharing of energy/fuel expenses.

– Mr. Contactor’s gen-set was inherently more prone to frequent failures. This created problems for Mr. Engineer also. Can he be compensated in any way?

– Once, Mr. Contractor’s gen-set-1 started dropping load gradually due to some mechanical problem. But Mr. Contractor ignored to take action and did not switch of his non-essential loads. This lack of discipline resulted in tripping of both the gen-sets giving a total shut down to the GRID.

– Of course there were other times when Gen-set-1 one failed instantly with the same result. Mr. Contractor could not do anything to prevent this and Mr. Engineer had to accept it.

They together discussed these problems and decided on the following solutions:

1. Tie line power flow will be metered either way and the net flow of energy from either system to the other will be billed to the receiver at an agreed rate.

2. The above is fine for inadvertent exchanges of power, happening due to system dynamics. For scheduled assistance from one system to the other, the charges could be agreed at a higher rate.

3. For unscheduled short-term emergency assistance of power, the charges could be agreed at a still higher rate.

4. Proper discipline should be followed during abnormal situations, to keep the loads to be within the limits of  ‘own generation plus agreed assistance’.

5. They also agreed to commission automatic load shedding (on under frequency and reverse power) and system separation (i.e. tripping the tie-line) in cases where such load-generation balance could not be achieved by self-discipline.

Both Mr. Engineer and Mr. Contractor (somewhat reluctantly) agreed and implemented these ideas. Surely, there were frequent bickering between the two, but the GRID basically worked well. Before the GRID could be affected by this bickering, fortunately they received the electrical connection from the power utility. Yes, now they are connected to a common grid managed by the utility. Mr. Contractor’s gen-set was disposed off. Mr. Engineer’s gen-set was retained as a stand-by supply till their connections were fully stabilized.

Though the above GRID was a small one, the principles of grid operations are the same even when two bigger systems are connected as a super GRID. To ensure that these principles are always upheld, usually a separate GRID operation department, (or a Load Dispatch Centre) is established, common and neutral to all the systems interconnected. It monitors the grid operation on a 24×7 basis and instructs the component systems to adhere to the principles of grid operation.  Failure by one or many of these connected systems to adhere to these principles will lead to major system black-outs, as occurred in the Northern Grid of India on 31 July 2012, which affected 500 million people. But out of these large population, even 0.01% would not have understood what exactly led to this shut down. With this note, I sincerely hope, at least 0.02% may now understand how any lack of discipline as described above could lead to failure of GRID operation and eventually lead to a black-out. (which means at least 50,000 people should read this blog !). I have kept this write-up intentionally simple to help even non engineers to understand.

(Refer: http://en.wikipedia.org/wiki/2012_northern_India_power_grid_failure).

25 Dec 2012

Sine of an angle – by Hindu Maths

December 3, 2012

Sine of an angle – by Hindu Maths

L V Nagarajan

Aryabhata (AD 476-550) was the first in the line of great mathematician-astronomers known to us from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Āryabhaṭīya and the Arya-siddhanta. The following stanza in Āryabhaṭiya gives a series of 24 numbers and calls them as Ardha-Jya differences.

मखि भखि फखि धखि णखि ञखि ङखि हस्झ स्ककि किष्ग श्घकि किघ्व |
घ्लकि किग्र हक्य धकि किच स्ग झश ङ्व क्ल प्त फ छ कला-अर्ध-ज्यास् ||

Makhi Bhakhi Phakhi Dhakhi Nnakhi Nyakhi

Ngakhi Hasjha Skaki Kishga Sghaki Kighva

Ghlaki Kigra Hakya Dhaki Kicha Sga

Jhasa Ngava Kla Ptha Pha Cha kala-ardha-jyas

When decoded into numbers it reads thus:

225, 224, 222, 219, 215, 210; 

                               205, 199, 191, 183, 174, 164

154, 143, 131, 119, 106, 93;  

                               79, 65, 51, 37, 22, 7 : Ardha Jya Differences.

You may imagine a bow. The string tying the two ends of the bow is called Jya (or rope) in Sanskrit. (This sanskrit word Jya, for rope, is the root for Geometry, Geology, Geography, as we know them now). Jya-ardha is half of this length.

Please see the figures below. In Figure-1, ABC is an arc of a circle. AC is its Jya. AM is ardha-Jya or half-Jya. It is seen clearly that half-Jya, AM  is nothing but Sine of angle AOB, multiplied by radius OA.

Considering the Figure-2 above, all the vertical steps from bottom to top add up to respective Jya-ardhas or half-jyas of the increasing angles. In the above diagram, the angles are in steps of 15 degrees. Hence, Step 1 = R Sin 15; step1+ Step2 = R Sin 30; Step 1 +2 +3 = R Sin 45 and so on up to 90 degrees. These steps are called half-Jya differences. The above sanskrit verse gives the step-sizes or half-Jya differences for 24 steps of 3.75 degrees each to add up to 90 degrees. Thus it gives a table RSines for 0 to 90 degrees in steps of 3.75 degrees, with R= R sin90 = Sum of all steps 1 to 24 = 3438. These values are found to be highly accurate with the present day values of Sines, as shown in the table and chart given at the end.

The genius of Aryabhata defined length-MB (refer to Figure-1) as Utkrama-Jya, reverse-Sine or Versine. Aryabhata proposed accumulation of the above Jya-differences in the reverse order to get the successive Utkrama-Jyas, as it is obvious from the above step-diagram where steps are symmetrical about 45 degrees. Hence Koti-Jya or (Cos x) was defined by Aryabhata as (1 – Utkrama-Jya).

Aryabhata has actually devised an algorithm to develop this Sine table. The second section of Āryabhaṭiya, titled Ganitapāda, contains the following stanza indicating a method for the computation of the sine table.

rasi lipthashtamo bhaga: prathamam jya-arda muchyathe

thath dwibhakta labdhon mishritham thath dwitheeyakam

aadyenaivam kramaath pindaan bhaktwa labdhon samyutha:

khandaka: syu: chaturvimsa jya-ardha pinda: kramadami.

There are several ambiguities in correctly interpreting the meaning of this verse. For example, the following is a translation of the verse (by Katz) wherein the words in square brackets are insertions of the translator and not translations of texts in the verse.

“When the second half-chord partitioned is less than the first half-chord, which is [approximately equal to] the [corresponding] arc, by a certain amount, the remaining [sine-differences] are less [than the previous ones] each by that amount of that divided by the first half-chord.”

With my highly limited knowledge of Sanskrit, I can guess a few parts of the above Sutra.

Rasi – 12th part of a full circle (30 degrees).

Ashtomo Bhaga – 8th part (of Rasi), 3.75 degrees

prathamam – First; Jya-ardha – Sine; Muchyate – obtained

Dwibhakta – Double or  Add to itself;

Labdhon – Profit, Dividend, Quotient;

Mishritam – together;  Dwiteeyakam -The second;

aadyenaivam – In a similar way

kramath – Successive; Pindan – difference

chaturvimsa – twenty four; Jya-ardha Pinda – Sine differences;

Kramadami; recursively

Now let me try to give a context-based English translation of the above Sutra:

Arc of an eighth of a rasi gives you the first Jya;

That doubled and divided – together gives the second;

Same way successive Jya-differences, together with quotient,

recurrently give all the twenty four Jya-differences.

There are 12 Rasis in Earth’s trajectory around the Sun. Hence each Rasi is 30 degrees. A quadrant arc of a circle subtends an angle of 90 degrees at the centre, or 3 rasis of 30 degrees each. 8th part of rasi will be 3.75 degrees. There will be twenty four such 3.75 degree arc-sectors in a quadrant, totaling to 90 degrees. An arc (or a bow) of 2 x 3.75 degrees will have a Jya (or a rope). Half of this, is Jya-ardha and corresponds to Sine of 3.75 degrees; which will be same as this arc of 3.75 degrees, as per the Sutra. Yes, in modern mathematics, it only means Sin(x) = x, when the angle is as small as 3.75 degrees. Evidently, a basic circle of certain radius must have been considered for this purpose and hence Jya really means Rsin(x). For some reason (to be explained later) the first Jya is taken as 225 :– 4 x 24 x 225 gives, the total circumference of the circle considered as 21,600, with corresponding radius of 3438.  The value of Jya of the first angle having been initiated as 225, the Jyas of all the twenty four angles can be found following the above Sutra. Interestingly the Sutra above gives different rules for the first two Jyas and prescribes the recursive rule only from Jya3. (Jya is used synonymously with Jya-ardha, which actually represents Sine).

Jya1 = Jya-Diff 1 = 225; Jya2 = (225+225) – (225/225) = 449; Jya-Diff 2 = 224

Jya-Diff 3 (as per the Sutra) =  Jya-Diff 2 – Jya2/225 = 224 – (449/225) = 222.0044; Jya3 = 671.0044

Jya-Diff 4 =  Jya-Diff 3 – Jya3/225 = 222.0044 – (671.0044/225) = 219.0222; Jya4 = 819.0266

The whole table of 24 Jya-diff’s, as developed by using the above sutra is given in a table below. Even this table gives good values for RSines, though not as accurate as the earlier table as could be seen from the plot.

Now let us check this recursive calculation

Difference, D(n) = Sin (n+1)x – Sin nx

= Sin nx Cos x + Cos nx Sin x – Sin nx

Similarly, D(n -1) = Sin nx – Sin (n -1)x

= Sin nx – Sin nx Cos x + Cos nx Sin x

D(n) – D(n-1) = 2 Sin nx Cos x – 2 Sin nx = -2 Sin nx (1 – cos x)

We know,  Jya(n) = J(n) = R Sin nx

Hence Jya Difference, DJ(n) = R * D(n)

Similarly, DJ(n -1) = R * D(n-1)

Hence, DJ(n) – DJ(n-1) = R * [D(n) – D(n-1)] = – 2 J(n) (1- cos x)

i.e., DJ(n) – DJ(n-1) = -J(n)/K,

K being a constant and equals [1/2(1-cosx)]

With x = 3.75 degrees, It works out that K = 233.5374.

If we start the recursive process with J(1) = K = 233.5374, we will get a very accurate table of Jyas, but with R = K x 48/Pi = 3568. However with values of Pi and (Cos x) as available in ancient Hindu period, values of 225 and 3438 for K and R were good enough, as can be seen from the plot below. Or was there a reason to choose these values? The whole circle is 360 degrees. Each degree can be divided into 60 minutes. Now the whole circle is 360×60 = 21600 minutes. Hence perhaps, the radius of the circle was taken as 21600/2Pi = 3438. An arc of 3.75 degrees will be 225 units long.

The ancient Hindus however knew the exact values of Jya for angles of 30, 45, 60 and 90 degrees and Aryabhata could have very well used them to apply corrections to the above table as required. Hence the value of R may not be very critical. Aryabhatta’s table of Sines, given earlier, is the corrected and improved version of the table as developed by his own formula, and hence is much closer to actual values, especially at 30, 45, 60, 90 degrees.

The whole table looks as below:

As per the recursive fromula

Aryabhata’s  Final Table

Actual

Angle   x

Quotient   Jx/225

 Ardha Jya Diff RSine(n)  Jya-n  Ardha Jya Diff RSine(n)  Jya-n

R=3438   RSine(n)

0

0.00     0   0 0

1

3.75 1.0000 225 225 225 225 224.86
2 7.50 1.9956 224 449 224 449

448.75

3

11.25 2.9822 222.004 671.004 222 671 670.72

4

15.00 3.9557 219.022 890.027 219 890 889.82
5 18.75 4.9115 215.067 1105.093 215 1105

1105.11

6

22.50 5.8455 210.155 1315.248 210 1315 1315.67
7 26.25 6.7536 204.309 1519.558 205 1520

1520.59

8

30.00 7.6316 197.556 1717.114 199 1719 1719.00
9 33.75 8.4757 189.924 1907.038 191 1910

1910.05

10 37.50 9.2822 181.449 2088.486 183 2093

2092.92

11

41.25 10.0473 172.166 2260.653 174 2267 2266.83

12

45.00 10.7679 162.119 2422.772 164 2431 2431.03
13 48.75 11.4405 151.351 2574.123 154 2585

2584.83

14

52.50 12.0624 139.911 2714.033 143 2728 2727.55
15 56.25 12.6306 127.848 2841.882 131 2859

2858.59

16 60.00 13.1427 115.218 2957.099 119 2978

2977.40

17

63.75 13.5963 102.075 3059.174 106 3084 3083.45

18

67.50 13.9896 88.479 3147.653 93 3177

3176.30

19 71.25 14.3206 74.489 3222.142 79 3256

3255.55

20

75.00 14.5880 60.168 3282.310 65 3321

3320.85

21 78.75 14.7906 45.580 3327.891 51 3372

3371.94

22

82.50 14.9275 30.790 3358.681 37 3409

3408.59

23 86.25

14.9980

15.862 3374.543 22 3431

3430.64

24 90.00

15.0018

0.864 3375.407 7 3438

3438.00

The above is a complete table of Sines as per ancient Hindu mathematicians. The last column gives the value of Rsine, (i.e. Jya) as calculated using the current accurate values. The closeness of the values can be observed in the following chart.

Dedication:

When I was researching for this blog, I came across the works of late Sri T S Kuppanna Sastri, an expert in Sanskrit and Ancient astronomy. I was naturally feeling proud, since I have met him about 30 years back. He is an uncle of my wife and he is the father of my friend Dr. T K Balasubramanian, a retired scientist of BARC. Sri Kuppanna Sastri was a professor of Sanskrit and Astronomy in several colleges. His renowned major works are two books namely Pañcasiddhāntikā of Varāhamihira and Vedāṅga jyotiṣa of Lagadha.   I dedicate this blog to the memory of late Sri T S Kuppanna Sastri.

References:

1. http://en.wikipedia.org/

2. History of Ancient Indian Mathematics, C N Srinivasiengar, The World Press Private Ltd. Calcutta (1967)

LVN/26 Nov 2012

Evolution of Indo/Arabic Numerals

October 24, 2012

Evolution of Indo/Arabic Numerals

L V Nagarajan

I am not attempting here to give an account of history about development of Number Systems. This is only to wonder how the present Indo/Arabic numerals came in to being. Arabs, the early residents of Arabian Peninsula, are always known to be the link between Europe and ancient India especially in carrying the ancient Indian thought and culture to the elite European community. This is not to say that Arabians themselves were devoid of higher thought and culture. It is the brighter Arabian minds which appreciated the importance of Indian and Oriental contributions and took them to the world along with their own achievements in similar fields. The Arabic numerals are one such great contribution to the world in general, and scientific community in particular. Later day research on ancient India resulted in naming these numerals as Indo/Arabic (or Hindu/Arabic) numerals.

Roman numerals are the earliest system of numerals known to the world. The Romans were active in trade and commerce, and from the time of learning to write they needed a way to indicate numbers. The system they developed lasted many centuries, and still sees some specialized uses today.

seven symbols/letters were used in Roman numerals to indicate following numbers

Roman Numeral Number
I One
V Five
X Ten
L Fifty
C Hundred
D Five Hundred
M Thousand

The list below illustrates how other numbers were constructed using the above 7 symbols/letters:

1 – 10              I,  II,    III,    IV,  V, VI,  VII,    VIII,    IX,   X

10 – 100        X, XX, XXX, XL, L, LX, LXX,  LXXX, XC, C

100 – 1000   C, CC,  CCC, CD, D, DC, DCC, DCCC, CM, M

A string of letters means that their values should be added together. For example, XXX = 10 + 10 + 10 = 30, and LXI = 50 + 10 + 1 = 61. If a smaller value is placed before a larger one, we subtract instead of adding. For instance, IV = 5 – 1 = 4. This is a major difference compared to the modern system. In addition there is no separate symbol for Zero.

The biggest Roman numeral is M, for 1000, so one easy way to write large numbers is to line up the M’s: MMMMMMM would be 7000, for instance. This system gets cumbersome quickly.

The system of numeration employed throughout the greater part of the world today was probably developed in India, but because it was the Arabs who transmitted this system to the West, these numerals have come to be called Arabic. After extending Islam throughout the Middle East, the Arabs began to assimilate the cultures of the peoples they had subdued. One of the great centers of learning was Baghdad, where Arab, Greek, Persian, Jewish, and other scholars pooled their cultural heritages and where in 771CE  an Indian scholar appeared, bringing with him a treatise on astronomy using the Indian numerical system.

Until that time the Egyptian, Greek, and other cultures used their own numerals in a manner similar to that of the Romans. Thus the number 321 was expressed like this:

Egyptian – I nn 999  (Right to left, the Arabic way)

Greek – HHH ÆÆ I

Roman – CCC XX I

The Egyptians actually wrote them from right to left. (Presently with the Indo/Arabic numerals they write numbers only from left to right, may be because of its non-Arabic origin)

The Indian contribution was to substitute a single sign for each cluster of similar signs. In this manner the Indians would render Roman CCC XX I as: 3 2 1. But however CCC I should mean 301, and not 31. Hence the  scholars perceived that a sign representing “nothing” or “naught” was required and Indians are credited with filling this need by inventing the symbol ‘zero’.

If the origin of this new method was Indian, it is not at all certain that the original shapes of the Arabic numerals also were Indian. In fact, it seems quite possible that the Arab scholars used their own numerals but manipulated them in the Indian way. The Indian way had the advantage of using much smaller clusters of symbols and greatly simplifying written computations. Their adoption in Europe began in the tenth century after an Arabic mathematical treatise was translated by a scholar in Spain and spread throughout the West.

Most of the ancient communities traditionally assigned numerical values to their letters and used them as numerals. This alphabetical system is still used by many, much as Roman numerals are used in the West for outlines and in enumerating kings, emperors, and popes. This part of evolution of numerals from letters is not apparently researched and discussed enough. Recently I read an essay about the way the letters of Tamil language were widely used to represent numbers, till as late as 19th century. This might have been the system that existed, since perhaps (not sure?) early Chola period,  9th Century.  This Tamil system of numerals is remarkable not only for using the letters for numerals but also for serving as a precursor for the evolution of modern numerals.

This system of numerals were in use till 19th century in most of the Tamil documents and  records. Tamils used 12 symbols or letters to denote whole numbers 1-9, 10, 100 and 1000. These symbols or letters are given below:.

Numbers Present Digits Tamil Symbols Unicode
One        1        ௧ &#3047
Two        2        ௨ &#3048
Three        3        ௩ &#3049
Four        4        ௪ &#3050
Five        5        ௫ &#3051
Six        6        ௬ &#3052
Seven        7         ௭ &#3053
Eight        8         ௮ &#3054
Nine        9         ௯ &#3055
Ten       10         ௰ &#3056
Hundred      100         ௱ &#3057
Thousand      1000         ௲ &#3058

As these symbols themselves are not important in the present context, I am dealing with Tamil’s number system with known symbols as below

Numerals 1 to 9, X for 10,  C for 100 and M for 1000.

With the above symbols let us see how Tamils wrote other numbers

27 was written as 2X7  –  (More precisely as ௨௰௭)

327 was written as 3C2X7

5327 was written as 5M3C2X7

5307 was written as 5M3C7 – (Symbol for Zero is not used)

234 021 was written as 2C3X4M2X1

1 Million will be  1MM or MM – (Just 2000 in Roman system)

3, 234, 521 will be  3M2C3X4M5C2X1 or (3M 2C3X4)M 5C2X1

1 Billion was written as 1MMM or MMM

and so on.

First let us look at the numbers without a zero in between. If you remove the symbols X,C and M from the sequence it exactly coincides with the present system. This may exactly be the reason why a symbol for zero was invented by Indians because they needed it the most. This symbol zero (0) helped them to totally remove X, C and M from their numerals (but still using their implied presence) . This is perhaps the way the place system of numerals was evolved and gifted to rest of the world. Tamils were definitely a part of the larger Indian Science and Culture and hence this could have been the number system that existed all over ancient India. The ancient Tamil inscriptions just give the evidence of the same. The reason this system was used till 19th century could be the same why English people still call their Queen as Elizabeth II. They find the older systems as authentic in recording history.

It may also be interesting to know that Tamils had letters even for representing about 15 fractions. There are also combination of symbols to represent fractions as low as 1/1,838,400. There are separate names for each of these fractions. After the advent of decimal systems in coinage, weights and measures since 1960s, most of these fractions have gone out of use.

In a lighter vein, there is a poem attributed to Avvayar(?) which mocks at perhaps another inferior poet thus:

௭ட்டேகால் லக்ஷணமே, எமனேறும் வாகனமே

Ettekaal Lakshaname, Emanerum Vahaname

Ettekaal = 8 and 1/4. In Tamil, the symbols are அ and வ

Avalakshaname – Means ‘the ugly one’

Emanerum Vahaname – Lord Yama’s mount, the buffalo

Such pun on number/symbols are aplenty in Tamil Language literature

Ref:

1. http://www.novaroma.org/via_romana/numbers.html

2. http://mathforum.org/dr.math/faq/faq.roman.html

3. http://www.islamicity.com/education/ihame/default.asp?Destination=/education/ihame/22.asp

4. http://ramanchennai.wordpress.com/category/archaeology-history/

5. http://tamilelibrary.org/teli/numeral.html

LVN / 24 Oct 2012