Archive for the ‘Education’ Category

Tamil – Class: 3 / Teaching Tamil Through English

November 9, 2017

Class – 3

In class-1 we learnt Tamil Alphabets with their pronunciations. We learnt about basic vowels(6), extended vowels(6), consonants(18), and symbols for modifying these consonants. In class-2, we learnt a few words with their meanings. We also learnt about so called ‘northern alphbets’ to help us write and pronounce correctly, words from other languages. In this class-3, let us learn about some special features of Tamil phonetics.

First, let us see a few words where the hard consonants appear in there different vocal forms.

காரம் – KaaRaM – Spicy; ராகம் – RaaGaM – Melody, மேகம் – MEHum/MEGaM – Cloud. (Ka being used in three different vocal forms: Ka, Ga, Ha)

தங்கம் – ThaNGaM – Gold (here a soft consonant is explicitly used to soften ‘Ka’ to ‘Ga’)

சக்கை – ChaKKai – Remains of a fruit after Juice is extracted. Ka is doubled for harder accent.

சித்தி – ChiTThi – Mother’s younger sister, மோசம் – MoSaM – bad, பச்சை – PaChChai – Green, மஞ்சள் – MaNJaL –Yellow (here Cha is used in different vocal forms. Soft consonant again used in the last word, Tha is doubled for harder accent in the first word)

‘Ta, Tha, Pa, Rra’  (ட, த, ப, ற​)   also have different vocal forms as below

ட :  டீ – Tea – Tea, பாடு – PADu – Sing, பாட்டு – PATTu – Song, நண்டு – NaNDu – Crab

த : தங்கை – ThaNGai – Younger Sister, பாதி – PAdhi – Half, கத்தி – Katthi – Knife, பந்து – PaNDhu – Ball :

ப :  படம் – PaDaM – Picture, சுபம் – SuBaM – All well, கப்பல் – KaPPaL – Ship,  கம்பி – KaMBi – Metal Rod

ற : பறி – PaRri – Grab, வெற்றி – VeTRri – Victory, பன்றி – PaNDRri – Pig

It may be puzzling for some, to know which vocals to use. However in most of the cases meaning do not change even if we use a different vocal form. The words will be understood properly in its context.

There are some letters which even some Tamils do not pronounce correctly. They are La, (r)La and Zha; (i.e) ல, ள and ழ. Let us learn a few words involving these letters:

La (ல) is pronounced with the tip of the tongue just behind the upper teeth. (r)La (ள) is done with the tip of the tongue slightly behind in upper cavity. Zha (ழ) is done with the tip of the tongue still behind, deep in the upper cavity. The following words show their use. Sound bytes are included to help you pronounce them properly.

வலி, வளி, வழி – VaLi, Va(r)Li, VaZhi – Ache/Pain, Air(Atmosphere), Path

தலை, தளை, தழை – ThaLai, Tha(r)Lai, ThaZhai – Head, impediment/Bond, vegetation

பல்லி, பள்ளி, பழி – PaLLi, Pa(r)LLi, PaZhi – Lizard, School, Blame/revenge

வலம், வளம், பழம் – VaLaM, Va(r)LaM, PaZhaM – Right side, prosperity, Fruit

Ancient Tamil Literature

Tamil is one of the classical Languages of the world, along with Sanskrit. Tamil literary history is very ancient and rich. There were distinctly three periods of development of Tamil literature usually called as Sangam periods. Sangam means Academy and there were three Sangams. The last Sangam was from 400BC to 400AD and called as Kadai Sangam (கடைச்சங்கம், or the Last Academy). The literature of this period, known as Sangam Literature, are the only ones available from these ancient periods. The works of earlier two Sangams are many centuries older and now only known as just names. The literary history of Tamil records them as ‘lost in tsunami’. Sangam literatures, and even some ancient Sanskrit works, record a massive tsunami much before 400BC which destroyed a very big landscape known as Kumari Kandam (Continent of Kumari), also known as Lemuria. All the works of earlier two academies were lost forever as per this historical account. However modern history could not find much evidence of this tsunami and the Lost Land. The (3rd) Sangam literature is grouped into three parts – பத்துப்பாட்டு (Ten Anthologies), எட்டுத்தொகை (Eight Collections) and பதினெண்கீழ்கணக்கு (Eighteen Poetic Works).

Tirukkural (திருக்குறள்), by a saint poet Tiruvalluvar is one of the works in the last group of eighteen and is widely translated in almost all major languages of the world. I am giving below the first couplet of this great work consisting 1330 couplets, divided into 133 chapters of ten each

அகர முதல எழுத்தெல்லாம் – ஆதி

பகவன் முதற்றே உலகு

Akara Mudala ezhutthellam – Aadhi

Bhagavan Mudatre’ Ulagu

Meaning:

‘A’ is the start of all alphabets (of all languages) – (Just as)

GOD is the Origin of the world (of this whole Universe)

You may like to listen the audio of this verse given below:

 

Here is a Tamil proverb which states the importance of ‘Letters and Numbers’.

எண்ணும் எழுத்தும் கண் எனத் தகும்.

Ennum Ezhutthum Kann ena Thahum.

Meaning:

Numbers and Letters are rendered as the eyes (for obtaining knowledge)

 

With this thought let us conclude our Tamil class – 3


 

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Toilets for Multitude

October 24, 2017

Toilet – Ek Prem Katha

L V Nagarajan

On 2nd Oct 2017, Gandhi Jayanthi (the birth anniversary of Mahatma Gandhi), I saw the Hindi Movie ‘Toilet – Ek Prem Katha’. It is about ‘Open Defecation’ prevalent in India and about the public and private efforts to eradicate the same. While the movie is mainly telling the story form ladies’ point of view – about their privacy, hygiene and safety, the social aspects are also discussed.

This movie reminded me about an incident and the subsequent interaction I had when I was a 10-year old boy in my village, Sholavandan, near Madurai (India). It was summer vacation time for the schools when several of my cousins visit us and spend the vacations with us. One of my city cousins (no name please) visiting us, elder to me by 5 years, called me one morning to accompany him for a walk. I went along happily with him. We went along the railway line a distance of about 500 meters near a small canal and a bridge. I understood his purpose when he asked me to take care of his wrist watch and purse. After he finished ‘it’ and when we were walking back home. I asked him ‘Why here? We have a toilet at home’. The answer he gave me opened my eyes of conscience. He said, ‘Rajoo, the toilet in our house is an open type dry lavatory which is not very hygienic. Also, I don’t like manual scavenging’.

Yes. Here is a problem. Why many of us want others to clean our toilets, even the modern sanitary ones? Why public toilets are so unclean? Even now don’t we avoid using a public toilet unless it is an emergency? Whenever we stay in a hotel, first thing we inspect are the toilets, whether they are clean and hygienic. Don’t we?

The multitude of people in India cannot afford space for their own toilets. Hence the need arises for common toilets and public toilets. People who do not want to clean their own toilets, how will they ever keep common and public toilets clean? In addition, will these common facilities like flush-out and water closet be kept well maintained, in working condition? This is the area where we have to impart training to our people on basic hygiene and co-operation in handling such common facilities.

Even sanitary toilets require two septic tanks which should be alternately emptied and cleaned at least once in five years. How often have you seen it done? (almost never). Eventually, it gets choked up and blocked and soon becomes unusable and becomes a major health hazard. These are all special problems of a densely populated country like ours.

As shown in the above movie, at least for the male population in many thousands of villages in India, ‘Field Defecation’ seems to be a very practical solution. We may perhaps think of finding ways and means of making this practice, private and hygienic. In my school days there used to be a class known as ‘Citizenship Training’. In one of those books, I remember to have seen a design of a mobile toilet perfectly suited for our population. It was somewhat similar to what is given in the following link. http://akvopedia.org/wiki/Dry_Toilet.

It consists of a pit over which a pedestal or a squatting slab is provided. A pile of sand or saw dust or dry earth nearby can be used to pour into the pit after every use. A second pit may be used over which the whole facility as above will be moved to enable hygienic emptying and cleaning of the first pit. A batch of such toilets can be made mobile and moved over different pits, specially prepared in the fields away from the village. Similar common toilets (or home toilets), within the village precincts, may be used by seniors, ladies and children. They can also be used by others, during unfair weather conditions and during nightly periods. This precludes the need for mechanised scavenging for periodically cleaning the pits. There are many designs available for producing bio-fuel just as gobar-gas plants. Such initiatives, of using appropriate technologies, must be encouraged to be undertaken by municipalities and gram panchayats, instead of forcing down a uniform policy and design by state and central governments.

Jai Ho to Swacch Bharat

 Victory to Clean India

_________________________________________________________

 

Teaching Tamil through English

September 8, 2017

Many parents of Tamil origin may not have learnt Tamil as a language anytime in their life. Some of them may regret it now and may want to learn Tamil, at least as a language of conversation and understanding. They may also want to teach their children Tamil, as they are not learning the same in their schools. Once children learn basic Tamil, they, depending on their interest, may pick-up deeper knowledge of Tamil on their own at a later stage in life. Unfortunately, not much work is done on such teaching of Tamil, and the regular pedagogy kills even their initial interest and more often than not, they discontinue learning and teaching Tamil. I have two lovely granddaughters (Mili and Tara) who have also started learning Tamil recently in California. I am sure the Tamil classes there is adequately interesting and enthusing for them to continue. With my small experience of teaching Tamil to my daughter and son in early 80’s, I thought of putting together my way of teaching Tamil in a series of blogs, which could be useful to Tamil loving parents in the US and elsewhere. Let me start straight away.

Class – 1

It is always said that Tamil Alphabet has 247+ characters. Any learner who hears this, immediately loses interest to some extent.  In its actual sense, Tamil language has only 26+ characters. Of course it has a dozen more symbols (and a few special characters for writing words from Sanskrit and other foreign languages). Let us first have a look at all these characters.

Five Basic Vowels: The following are the basic vowels in Tamil. Their pronunciation is also given right below.

A as in ‘Avatar’ I as in ‘In’ U as in ‘pUt’ E as in ‘End’

O as in ‘One’

 

Two Composite Vowels: There are two composite vowels as below:

Ai as in ‘Aisle’

Ou as in ‘Out’

The first one above is a combination of: அ  and  இ   =    ஐ

Second one above is a combination of:   அ  and   உ = ஔ

 

Five Extended Vowels: The five basic vowels as above have their extended versions with slightly elongated pronunciation as compared to

A as in ‘Avatar’ – I as in ‘In’ –   U as in ‘pUt’ –  E as in ‘End’ –   O as in ‘One’

(As below)

Au as in ‘Aunt’ Ea as in ‘Easy’ Oo as in ‘Ooze’ A as in ‘Area’

Ow as in ‘Own’

                   

Special Character:

ஃ     —    Akh

This is a special character grouped along with vowels to add a specific accent to a few consonants. We will list it here but learn about it later.

 

Eighteen Consonants: There are eighteen consonants in Tamil. They are as below:

Ka

Ga

Cha

Sa

Ta

Da

Tha

Dha

Pa

Ba

Rra

Tra

The above six are called Hard consonants. However same letters are used for softer pronunciations also as shown.

 

Nga Gnya Rn as in ‘BoRn’ Na Ma

Na

The above six letters are known as Soft consonants. The two ‘Na’s are used in different contexts. They are also used to soften the corresponding hard consonants shown earlier. We will learn about them later.

 

Ya Ra La Va Zha

Rl as in Pearl

The above six letters are known as Medium consonants. The letter ‘Zha’ is very special for Tamil language and its pronunciation presents some difficulty even for some Tamils. La (ல) is pronounced with the tip of the tongue just behind the upper teeth. Rla (ள) is done with the tip of the tongue slightly behind in upper cavity. Zha (ழ) is done with the tip of the tongue still behind.

 

Symbols: The following table shows the symbols used to add the vowels to the above consonants. A few typical consonants are shown with symbols added.

Basic Vowels:

Symbols

_

ி 3 types  ெ

 ொ

Consonants with symbol added:

கி கு கெ

கொ

Ka

Ki Ku Ke

Ko

சி சு செ சொ
தி து தெ

தொ

 

Composite Vowels:

Symbols:

Consonants:

கை

கௌ க்
kai kau

k

as in ‘Park’

சை

சௌ ச்
தை தௌ

த்

 

Extended Vowels

Symbols

 ா

 ீ 3 types  ே

Consonants

கா

கீ கூ கே கோ
kaa kee koo kay

koe

சா

சீ சூ சே சோ
தா தீ தூ தே

தோ

The letters with a dot above them are known as ‘Otru’ – that is, it sounds without any vowel, like, ‘ch’ and ‘th’.

 

There are a few more special characters and symbols which we can learn later.

The complete list of alphabets as per the above scheme is given in

http://tamilcube.com/learn-tamil/tamil-alphabets-chart.aspx

This is enough for class-1. In Class -2 we will learn a few words using some of these alphabets

Bye for now.

 

Pink Poem by Tanveer Ghazi

August 14, 2017

Movie – Pink (16 Sept 2016)

Poem –  Tu Khud Ki Khoj Mein Nikal

Lyrics – Tanveer Ghazi

Rendering by – Amitabh Bachchan

 

In an earlier blog (Feminism and Humanism), I had expressed my views on aggressive feminism displayed in a write-up on CNN-IBN web site. I give below a summary of my views expressed therein.

  1. A woman can decide to take time to internalize and process an incident. Outward expression may hide internal trauma. But In case of serious crimes such as rape and sexual assault one should not hide her internal trauma. She should express her internal trauma as quickly as possible after any such crime. Otherwise you are risking yourself of mal-intent.
  2. A woman can choose to file a complaint at any time she deems fit – even a month after the incident if she so chooses. However for any crime, the complaint should be made immediately after the crime. Surely efforts will be made by the criminal to stall the same. Any undue delay will only aid the criminal in such efforts.
  3. Even if it started as a consensual affair, a woman can say ‘no’ at any time. When you start any activity jointly, it is always difficult to walk out in the middle. This is very much true in consensual sex. Think thousand times before your consent, either by intent or by default. Otherwise, say ‘no’ at the earliest.
  4. A woman can have multiple sexual partners. What she chooses to do in her own time isn’t anybody’s concern. It is immoral for both men and women to have multiple sex partners, but may be not illegal. Anyone has a right to be immoral. Having any kind of expertise, or lack of it, does not enhance or diminish this right.
  5. A woman’s clothes aren’t testimony to her character. True. Indecent people do parade in decent clothes. Some time, very decent people do come in rather revealing clothes. But decent people, both men and women, are expected to attire themselves decently in public and they also expect others to do the same. Revealing clothes expose people, especially women, to some risks.
  6. Even in a feminist world, men have to be courteous to women. Women value generosity in a man. Similarly, men value modesty in a woman.
  7. Don’t exploit woman’s emotions as leverage for bargaining for her freedom and choices she makes. Many women are also seen to exploit such emotions against men. Any such exploitation either way is despicable.
  8. You wouldn’t like to be told how to live your life. Don’t tell woman how to live hers. Woman sometimes need advice of friends and close relatives, even on some private matters. She should not hesitate to ask. Any unsolicited advice does irritate you, I agree.
  9. Her freedom – to wear what she wants, to go where she wants, to choose her friends – isn’t yours to bestow. Any youngster will sometimes need the advice and acceptance of his/her seniors on such matters. Outsiders definitely do not have any say on this.

Having said all this purely in the interest of safety of my wife, sisters, mothers, daughters, colleagues and friends, I sincerely wish for more space for all women to grow, to move about, to progress, to enjoy and to achieve as per their wish and aspirations. It is going to be about a year since the release of the Hindi film PINK, where Amitabh Bachan plays the part of an advocate for the victimized girls and makes many significant statements supporting freedom and safety for women. He also advices girls some responsible behavior while demanding and enjoying such freedom. At the end of the film he celebrates women freedom with a poem rendered very convincingly, in his sonorous voice.

On this Independence Day of India (15th Aug 2017), we celebrate the independence India obtained from Britain. We also celebrate this as a day of freedom from many other ills of our society which we got rid off during this 70 years of Independence.

Let me celebrate this Independence Day 2017 as a day for Women’s Freedom, by translating the PINK Poem into English and dedicating the same to Women’s Freedom.

 

Translation by L V Nagarajan

 

You decide your path and depart

Why fear? And hesitate for what?

Go! Even time is on your side, Start

Yes time is on your side, Start.

Decide your path and depart

 

Folks who restrict; bend them as a bow

Break the restricts to pieces

And use them as arrows,

Make them as arrows

Decide your path and depart

 

Your conduct so pure, why hardships to endure 

With sins in their mind,

Who allows them to judge you?

Why allow them to judge you?

Decide your path and depart

 

Those tricks of cruelty, burn them to ashes

The wick in your lamp can become

The big torch of your anger

Light the torch of your anger

Decide your path and depart

 

Raise your scarf as banner; for skies to shudder

If ever your scarf falls,

A quake should occur. 

Yes, a quake will occur

Decide your path and depart

 

Original Hindi Version

 

Tu khud ki khoj mein nikal   Tu kisliye hatash hai

Tu chal, Tere Wajood ki  Samay ko bhi talash hai.

samay ko bhi talash hai

(Tu khud ki khoj mein nikal)

Jo tujhse lipti bediayan…Samajhna inko vastra tu

Ye bediyan pighal ke..Bana le inko shastra tu..

Bana le inko shastra tu

(Tu khud ki khoj mein nikal)

Charitra jab pavitra hai..Toh kyoun hai ye dasha teri

Ye papiyon ko hak nahi..Ki lein pariksha teri.

Ki lein pariksha teri

(Tu khud ki khoj mein nikal)

Jala ke bhasm kar use jo krurta ka jal hai

Tu Aarati ki lau nahi..Tu krodh ki mashal hai..

Tu krodh ki mashal hai

(Tu khud ki khoj mein nikal)

Chunar ko udaa dhwaj bana gagan bhi kap-kapayegaa

Agar teri chunar geeri..Toh ek bhukamp ayegaa.

.ek bhukamp ayegaa

(Tu khud ki khoj mein nikal)

Happy Independence day to all.

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