"போதையனார் தேற்றம்" எனும் மாயை!

சமீப நாட்களாக இணையத்தில் உலாவரும் தமிழின் பெருமைகளைப் பேசும் மின்னஞ்சல்களில்/பதிவுகளில் ஒன்று "போதையனார் தேற்றம்" பற்றியது. "போதையனார் தேற்றத்தின்" சிறப்பம்சம் வர்க்கமூலம்(√) இல்லாமலேயே செம்பக்கத்தினை/கர்ணத்தினை கணிக்க முடிகின்றது என நீளுகின்றது அத்தகவல். ஆனால், இங்கே தமிழ் மொழியின் மீதான பற்றினைப் பயன்படுத்தித் தவறான தகவல் வழங்கப்படுகின்றது என்பதே உண்மை.
இந்த தகவலின் உண்மைத் தன்மையை அறிய கவனிக்கப்பட வேண்டியவை வருமாறு:
1. இங்கே கணிதவியலின் தர்க்க ரீதியிலான நிறுவுதல்கள் எதுவுமின்றி "தேற்றம்" என்று ஒன்று சொல்லப்படுகின்றது. அடிப்படையில் இங்கே குறிப்பிடப்படும் "தேற்றம்" எனும் சொல், அதன் அர்த்தத்தத்தினை இழந்து நிற்கின்றது.
2. உதாரணம் ஒன்றினை மட்டும் அடிப்படையாக வைத்து எந்த கணித சமன்பாட்டினையும்/கூற்றினையும் தேற்றம் என்று கூற முடியாது.
3. ஒரு தேற்றமானது சகல பொருத்தமான தரவுகளிட்கும் உண்மையாக இருத்தல் வேண்டும்.

இப்போது "போதையனார் தேற்றம்" என்ன சொல்ல விளைகின்றது என்று பார்ப்போம்.
"ஓடும் நீளம் தனை ஒரே எட்டுக்
கூறு ஆக்கி கூறிலே ஒன்றைத்
தள்ளி குன்றத்தில் பாதியாய்ச் சேர்த்தால்
வருவது கர்ணம் தானே"
இதற்குக் கொடுக்கப்படும் பொழிப்புரை வருமாறு:
ஒரு செங்கோண முக்கோணத்தில்:
கர்ணம் = செம்பக்கம் (செங்கோண முக்கோணத்தில் செங்கோணத்திட்கு எதிர அமைந்துள்ள மிக நீண்ட நீளமுடைய பக்கம்)
ஓடும் நீளம் = செம்பக்கத்திட்கு அடுத்ததாக நீளமாயுள்ள பக்கம்
குன்றம் = முக்கோணத்தின் மிகச்சிறிய நீளமுடைய பக்கம்

தரப்பட்டுள்ள உதாரணத்தின்படி பக்க நீளங்கள் (3, 4, 5) கொண்ட செங்கோண முக்கோணத்தில்:
ஓடும் நீளம், a = 4
குன்றம், b = 3
"போதையனார் தேற்றத்தின்படி",
c= (a - a/8) + (b/2)
= 4-(4/8) + (3/2)
= 5
கர்ணம், c= 5.
இந்த ஒரு உதாரணத்தை (அல்லது இவ்விலக்கங்களின் இருமடங்கான (6,8,10)) மட்டும் அடிப்படையாகக்கொண்டு இதனைத் தேற்றமென்று கூறப்படுகின்றது.

இப்போது "போதையனார் தேற்றத்தின்" உண்மைத்தன்மையினை மற்றைய பைதகரஸ் எண் கூட்டங்களிட்கும் பார்ப்போம்.

(5,12,13)
ஓடும் நீளம், a = 12
குன்றம், b = 5
"போதையனார் தேற்றத்தின்படி"
c= (a - a/8) + (b/2)
= 12-(12/8) + (5/2)
= 13
கர்ணம், c= 13.
இந்த உதாரணமும் சரி வருகின்றது.

(7,24,25)
ஓடும் நீளம், a = 24
குன்றம், b = 7
"போதையனார் தேற்றத்தின்படி",
c= (a - a/8) + (b/2)
= 24-(24/8) + (7/2)
= 24.5
கர்ணம், c= 24.5 ≠ 25.
இங்கே "போதையனார் தேற்றத்திற்குப்" போதையேறித் தடுமாற ஆரம்பிக்கின்றது.

(8,15,17)
ஓடும் நீளம், a = 15
குன்றம், b = 8
"போதையனார் தேற்றத்தின்படி",
c= (a - a/8) + (b/2)
= 15-(15/8) + (8/2)
= 17.125
கர்ணம், c= 17.125 ≠ 17.
இதற்கப்பால் எல்லாமே தப்பான முடிவுகள்தான். அதனைவிட கர்ணமானது ஓடும்நீளத்தினை விட சிறிதான இலக்கமாக வருவதும் குறிப்பிடத்தக்கது.

(9,40,41)
"போதையனார் தேற்றத்தின்படி",
கர்ணம், c= 39.5 ≠ 41 (அத்துடன் 39.5 < 40).

(11,60,61)
"போதையனார் தேற்றத்தின்படி",
கர்ணம், c= 58 ≠ 61 (அத்துடன் 58 < 60).

(12,35,37)
"போதையனார் தேற்றத்தின்படி",
கர்ணம், c= 36.625 ≠ 37.

(13,84,85)
"போதையனார் தேற்றத்தின்படி",
கர்ணம், c= 80 ≠ 85 (அத்துடன் 80 < 84).

இப்படியே (15,112,113), (16,63,65), (17,144,145), (19,180,181), (20,21,29), (20,99,101), (21,220,221), (23,264,265), ...... போன்ற இலக்கங்களுக்கும் தொடர்கின்றது துல்லியமற்ற முடிவுகள். ஆக மொத்தத்தில் போதையனார் கூறியது இரண்டு சந்தர்ப்பங்களுக்கு மட்டுமே பொருந்தும் ஒரு கூற்று. "தேற்றம்" கிடையாது. ஆக மொத்தத்தில் ஒரு செம்பக்க முக்கோணத்திற்கு பைதகரஸ் தேற்றத்திற்கு பதிலாக "போதையனார் தேற்றத்தினை"ப் பயன்படுத்தினால் பெறப்படும் விடைகளானது மேற்குறிப்பிட்ட இரண்டு சந்தர்ப்பக்கங்கள் ((3,4,5), (5,12,13), அல்லது அதன் மடங்குகள்) தவிர்ந்த அனைத்து சந்தர்ப்பங்களிலும் துல்லியமற்ற முடிவுகளாகவே இருக்கும்.

Regards, BMS 

Some hidden information contained in famous logos :)

Amazon.com
This logo doesnt seem to hide much at first sight, but it gives you a little insight in the philosophy behind the brand. First of all, the yellow swoosh looks like a smile: Amazon.com want to have the best customer satisfaction. The swoosh also connects the letters a and z, meaning that this store has everything from a to z.

Fedex
This is probably one of the best known logos with a hidden meaning. If you look closely, youll see an arrow thats formed by the letters E and x. This arrow symbolizes speed and precision, two major selling points of this company.

Continental
Continental is a manufacturer of tyres. You could actually see this in their logo, because the first two letters create a 3-dimensional tyre.

Toblerone
Toblerone is a chocolate-company from Bern , Switzerland . Bern is sometimes called The City Of Bears. They have incorporated this idea in the Toblerone logo, because if you look closely, youll see the silhouette of a bear.

Baskin Robins
The old logo of Baskin Robbins had the number 31 with an arc above it. The new logo took this idea to the next level. The pink parts of the BR still form the number 31, a reference to the 31 flavours.

Sony Vaio
Sony Vaio is a well known brand of laptops. But did you know that the name Vaio logo also had a hidden meaning? Well, the first two letters represent the basic analogue signal. The last two letters look like a 1 and 0, representing the digital signal.

Carrefour
Carrefour is one of the biggest European retailers, and its also French for crossroads. The logo symbolizes this word via two opposite arrows. They also added the first letter of the name, because if you look closely youll see the letter C in the negative space between the two arrows.

Unilever
Unilever is one of the biggest producers of food, beverages, cleaning agents and personal care products. They produce a huge amount of different products and they wanted to reflect this in their logo. Each part of the logo has a meaning. For example: the heart represents love, care and health - feeling good, a bird is a symbol of freedom. Relief from daily chores “ getting more out of life.

Formula 1
At first, this logo might not make much sense. But if you look closely, youll see the number 1 in the negative space between the F and the red stripes. I also love how this logo communicates a feeling of speed.

Sun Microsystems
The Sun logo is one of the most famous ambigrams in the world. You can read the brand name in every direction; both horizontally and vertically. This logo was designed by professor Vaughan Pratt of the Stanford University .

NBC
The NBC (National Broadcasting Company) is one of the biggest American television networks. I think most of you have already seen the peacock in this logo. The peacock has 6 different tail feathers, referring to the six divisions at the time that this logo was created. The peacocks head is flipped to the right to suggest it was looking forward, not back.

Regards,
B M S

DIGITAL WATCH :)

இதுக்கு பேர்தாங்க DIGITAL WATCH :)

DIFFERENT THINGS MOTIVATE DIFFERENT PEOPLE !

How to Regulate the Mind? - 9 Easy Steps!

What is Mind?:

What is Mind?  Is it part of the Brain?  The brain has got physical characteristics which can be visualized, touched, handled and is located inside the skull. There are certain brain control techniques. But what about the mind?  It has no physical dimensions.  But there appears to be some kind of relationship between the brain and the mind.  When the brain is non functional, (say when the brain activity is blocked by deep anesthesia or coma), the mind also does not seem to function.

Mind is not a gross matter - visible and tangible. The mind can not be seen. The magnitude of the mind cannot be measured. Mind does not require a space to exist.

Definition of Mind:

How to define the Mind?  The mind is nothing but a “bundle of desires, thoughts, wants and expectations”.  As long as a person is alive, he will have the thoughts, desires, wants and expectations.  And these are continually changing, minute to minute and day by day.

If your desires and expectations do not fructify, you are under stress, leading to anxiety, mental tension, frustration and depression.  If you want to enjoy stress free life, you need to exercise control over mind.  You cannot under-estimate the mind powers.

Mind Control Techniques:

But how to control your mind?  How to control your negative thoughts and desires?  Why there is a necessity to control the negative thoughts and desires, thereby attempting to control the mind?  Because, the thoughts and desires transform into words and words result in actions.  When a person has a lot of negative thoughts, it may lead him to stress, anxiety, depression, negative actions and violent behavior.    There are certain mind control techniques / mind control tricks widely practiced, such as Suppression technique and/or Outburst technique, for controlling the mind.

In the Suppression technique, you are asked to suppress your negative thoughts and desires and not to manifest them.  But the fact is, your thoughts and desires are like coiled springs.  The more you suppress the coiled spring, the more pressure is built upon it.  The moment you let go of your hand even slightly, the coiled spring jumps across much more vigorously.  Similarly, the moment you relapse in suppressing your negative thoughts and desires, they come out violently.  The situation becomes from bad to worse.

In the Outburst technique, you are asked to let out your negative thoughts with full force in the open.  If you are angry with somebody, you are asked to outburst your anger, say in front of a mirror or in an open park.  Once your outburst is over, you may temporarily feel relaxed.  But ultimately, this technique leads you to the guilt feeling, which in turn generates a lot of negative thoughts.  It is a vicious cycle.  These methods employ negative techniques and hence the relief becomes temporary, ultimately leading to more and more negative thoughts, stress, anxiety, cynicism and mental imbalances.

Other methods involve the techniques called “still the mind” and/or “be a witness to your thoughts”.  But for majority of the people (including me), it is very difficult to practice in day to day life, as only a few holy men / yogis can master these techniques.

Substitution Technique:

What I recommend and as well as I practice, is the “Substitution” technique.  We should understand the nature of mind.  As I mentioned in the beginning of this article, mind is the bundle of desires, thoughts and wants.  As long as you are alive, your mind will always be with you.  Your mind is always filled with thoughts and desires.  You cannot empty your mind in your day to day life.

Suppose you enter your bedroom in the night.  The room is dark and you can not see anything.  Once you switch on the light bulb, the entire room is filled with light and you do not see the darkness.  What happened to the darkness?  As soon as the light came in, the darkness vanished automatically.  You did not attempt to drive away nor suppress the darkness.  You only put on the light switch.  You enabled the light to come in, which drove away the darkness automatically.  Similar is the case with your mind.  Bring in positive thoughts to your mind, which will automatically drive away your negative thoughts.  But the mind can not be empty.  Instead of mind full of negative thoughts, substitute them with positive thoughts.  When the mind is full of positive thoughts, there is no place for negative thoughts to enter.

In fact, the phrase “how to control the mind” itself is wrong.  It is almost impossible to “control” the mind.  The right phrase is “how to regulate the mind”.  By practicing the “substitution” technique properly, we can regulate our mind in the day to day life.

How to practice the “substitution” technique in our daily life?  It is not difficult, provided you make a commitment to practice them on a daily basis.  There are 9 Steps to practice the “substitution” technique, which I will explain below.

Step 1 – Keeping Good Company:

There is a famous quote- “Tell me who your friends are and I’ll tell you who you are”.  Seeking and remaining in good company will keep your mind with positive thoughts.  Consider a small example.  You are all familiar with mineral water.  Imagine that you have a 1-litre bottle which is full of filthy water.  You are required to replace the filthy water with clean drinking water, but there is a condition.  You can not pour out the filthy water first, wash the bottle and then fill up the bottle with clean water.  In such a situation, how can you replace the filthy water with good water?  It is very easy.  Go to a stream where clean water is flowing.  Just keep the water bottle, containing the filthy water, immersed in the stream.  You will find that after a few minutes, the entire bottle is filled with clean water.  How has this happened?  The clean water from the stream has entered the bottle and driven away the filthy water.  You have not done anything extra-ordinary.  You have simply brought the filthy water bottle in contact with the stream water.  That’s all.  Now, compare the water bottle with your Mind.  The filthy water can be compared to the negative thoughts and negative desires in your mind.  The water stream can be compared with good company (in Sanskrit language, “good company” is called “satsang”). When you remain in the good company, the positive thoughts enter your mind and drive away the negative thoughts.  It is similar to the light driving away the darkness in the room.

Step 2 – Repeat God’s Name Often:

Choose the name of God you love the most.  Whenever you find time, irrespective of where you are, keep on repeating your God’s name mentally.  (In Sanskrit, it is called “nama smaran”). God loves music.  If you can repeat the name of your God in a musical tune (as per your liking), it is wonderful.  This process will drive away your negative thoughts and emotions.

Step 3 – See Good Sights:

We should understand one important aspect.  Mind can be compared to the CPU of the computer.  Whatever we feed in, CPU processes it and gives the output.  Garbage in – Garbage out.  Quality data in – Quality reports out. The data to the CPU can be fed in through the FIVE input devices, such as the Keyboard, Mouse, Floppy Drive, CD Drive and Pen Drive.  Similarly, our FIVE sensory organs, such as the Eyes, Ears, Nose, Mouth and the Hands are the input devices to the Mind.  Based on the input information fed by these sensory organs, our Mind reacts.  If you see only good sights and positive videos / movies, positive thoughts are developed in the Mind.  If you see horror films and negative / filthy programs on the TV (most of TV programs have a lot of these), negative and violent thoughts are developed in the Mind. Read only good and holy books and the books on personality development.  Avoid reading horror stories and negative news (newspapers have plenty of them).

Step 4 – Hear Good Music:

Tune your ears to hear only good and soothing music and sounds.  As far as possible, avoid hearing bad and filthy words, heated arguments and violent shouts.  It is better to move away from such places.

Step 5 – Speak Softly and Positively:

Always try to speak softly and positively.  Don’t hate others. Avoid speaking harsh words.  Don’t enter into heated arguments with anybody.  It is pertinent to remember Mahatma Gandhi’s reference to the Three Monkeys – First Monkey closing his both EYES with both hands – Don’t see bad sights; the Second Monkey closing his both EARS with both hands – Don’t hear bad sounds; and the Third Monkey closing his MOUTH with both hands – Don’t speak bad words / untruth.

Step 6 – Eat Moderately:

Another important aspect which influences your thoughts and the Mind, is the type and quantity of food you eat. A subtler part of the food digested, influences the Mind and the thought process.  Always choose nutritious food and eat in moderate quantity.  Too much of food not only leads to obesity, but also leads to dullness of the Mind. Avoid alcoholic drinks, cigarettes and drugs, which not only spoil your health, but also adversely affect your Mind.  Also avoid stale / pungent food.

Step 7 – Smell Good Fragrance:

Try to smell good fragrances, particularly the natural flowers.  Avoid / move away from the place where there is a bad / filthy smell.

Step 8 – Do Good Things:

Always try to do good things in life.  Whenever and wherever you get a chance to help others, however tiny it is, please extend your help.  Avoid doing wrong things against your Conscience, as it will lead to guilt feelings in your Mind.

Step 9 – Go to Good / Holy Places:

Visit good and holy places.  Whenever possible, make a tour to the country-side and / or Mountain ranges, where you can enjoy the natural surroundings.  They will refresh your Mind and Heart.  Avoid going to gamblers’ dens and other unholy places.

“Hands that Help are Holier, than the Lips that Pray”.

Regards,
B M S  

இனிய பொங்கல் நல்வாழ்த்துக்கள்!

உழுதுண்டு வாழ்வாரே வாழ்வார்
மற்றெல்லாம் தொழுதுண்டு பின்செல்பவர்

~1033:

Who ploughing eat their food, they truly live:
The rest to others bend subservient, eating what they give.

They alone live who live by agriculture; all others lead a cringing, dependent life.

அனைவருக்கும் இனிய பொங்கல் நல்வாழ்த்துக்கள்!!!!

Regards, BMS