Katapayadi Sankhya

गोपीभाग्यमधुव्रात-शृङ्गिशोदधिसन्धिग । 
खलजीवितखाताव गलहालारसंधर ॥


Gopibhagya madhuvrata srngisodadhisandhiga|
Khalajivitakhatava galahalarasandhara||



Oh Krishna, the fortune of the Gopis, the detoryer of the demon Madhu,
protector of cattle, the one who ventured the ocean-depths, destroyer of evildoers,
one with plough on the shoulder and the bearer of nectar, may (you) protect (us)!

The shloka above, seems to be one of the many written in praise of one of the most enigmatic and divine Lords of Hinduism, Krishna. The shloka, like many others, praises certain attributes of Krishna and requests him to protect the devotee.
So, what is so special about this shloka, that out of all the shlokas dedicated to Krishna, this particular one turned out to be the starting point of this post?

Simple enough, the shloka demonstrates one of the strongest and most intelligent examples of extreme knowledge and wisdom of our ancient men.

Grahacāraṇibandhana based in the year 683 CE and Laghubhāskariyavivarana based in the year 869 CE speak of a certain numerical notation which goes by the name of Katapayadi Sankhya.
Under this system, a number is ascribed to each and every alphabet of the script, a concept highly similar to the ASCII system in computers. The image below would better explain the relation between the alphabets and the numbers

So, based on the above method, if the letters in the shloka are replaced by the corresponding numbers, i.e. 'go' by 3, 'pi' by 1,  'bha' by 4, 'ya by 1' and so on, the following result is obtained:
31415926535897932384626433832792
The number, as obvious, is the decimal representation of pi upto 32 decimal places.
Who could have thought encrypting a mathematical concept in a devotional Shloka dedicated to Krishna?

So, what do they gain out of it by performing such extraordinary feat? The answer is two-fold.
Firstly, they used methodologies like this to check the correct usage and pronunciation of the verses, where every variation from the correct mantra or shloka would result in an incorrect rendition of the pi number(or any other system which they would have encoded)
Secondly, they used concepts like above to perform encryption and decryption so that the message would reach the intended receiver and no one would be able to catch the real meaning of the message.


Karanapaddhati: Written in the year 1733 CE by mathematician Puthumana Somayaji, the book consists of 10 chapters in total. Our  interest, however is in the 6th chapter.
In the 6th chapter, is given a shloka in Malayalam,


anūnanūnnānananunnanityai
ssmāhatāścakra kalāvibhaktoḥ
caṇḍāṃśucandrādhamakuṃbhipālair
vyāsastadarddhaṃ tribhamaurvika syāt

Not exactly Katapayadi system at play here, this is similar to encrypting formulae or constants in shlokas.
The above shloka, when translated simply means
"The circumference of a circle of diameter anūnanūnnānananunnanityai(10,000,000,000) is caṇḍāṃśucandrādhamakuṃbhipālair(31415926536), which, in turn, gives the value of pi till 10 decimal places.

Representation of dates:

palahāre pālu nallū, pularnnālo kalak kilāṃ

illā pālennu gopālan - āṃgḷamā sadinaṃ kramāl

Conceptualized in modern day India, based on Katapayadi Sankhya, the above shloka, which is in Malayalam, when translated, means that "Milk is best for breakfast, when it is morning, it should be stirred. But Gopālan says there is no milk - the number of days of English months in order"

True to its meaning, when the pairs of letters are substituted in accordance with Katapayadi system, yields

31 28 31 30 31 30 31 31 30 31 30 31

 The Vatsyayan Cipher:

Interestingly, the first written example of encryption in the entire history is found in the most unexpected source. The method, not that strong in that sense, is still used by army and security forces in a much more complex and stronger form.

As per Vatsyayan in his book Kamasutra, a girl needs to have certain attributes and learn certain arts and tricks, including how to cook, how to read and write, and how to send her lover secret messages which no one else would be able to decipher. Vatsyayan even goes on to give an example of such a cipher in the book.

The method is based on substitution cipher. Each alphabest would represent a certain another alphabet. The image below would explain the substitution cipher better:

So, the substitution cipher was invented in ancient India.

Validating the Shlokas:
Almost every major deity of Hinduism has a certain "Shata-Namavaali", or a list of 108 names dedicated to that deity. Shiva and Vishnu  have a list of 108 names dedicated to them. 
Venkateswara Ahtottara Shatanamavali is a list of 108 names dedicated to Lord Venkatesha.
Needless to say, if any of the name is forgotten by a devotee, the hymn would never be complete. 
It is always easier to remember a certain list of they are in a certain order.
For the same reason, and for validating the occurrence of all the 108 names in the order in which they are written in the text, a certain Anustup Chanda was composed, which consists of 4 shlokas, which are made up of the first letters of all the 108 names, in that order, thus validating the 108 names.


अनुष्टुप्छन्दसि चत्वारः पादाः भवन्ति प्रत्येकपादे च अष्ट अक्षराणि।श्लोके षष्ठं गुरु ज्ञेयं सर्वत्र लघु पञ्चमम्।
द्विचतुष्पादयोर्ह्रस्वं सप्तमं दीर्घमन्ययोः॥
अस्य छन्दसः षष्ठम् अक्षरं गुरु पञ्चमम् च लघु। सप्तमम् अक्षरं प्रथमे तृतीये च पादे गुरु, द्विचतुष्पदयोः सप्तमम् अक्षरं लघु भवति। सप्तमम् अक्षरं यथाक्रमम् परिवर्तते, प्रथमपादे गुरु द्वितीयपादे लघु तृतियपादे गुरु चतुष्पादे लघु।

Encryption or Miracle?
The most interesting part of this entire post, Sri-Raghava-Yadaveeyam deserves a distinct status altogether. Not exactly a work of the ancient Vedic men, this short text of 30 verses, which was compiled in the 17th century, actually deserves a separate blog post altogether. 
The name of the text itself, is quite curious. While the first part of the name of the text denotes Lord Rama, the latter, Yadava, denotes Krishna.

The 30 shlokas in the text, describe glimpses of Rama's life and works. Sample a verse from the text:

वंदेSहं देवं तं श्रीतं रन्तारं कालं भासा य:।

रामो रामाधीराप्यागो लीलामारायोध्ये वासे॥

The above text translates to
I pay my obeisance to Lord Sri Rama, who with his heart pining for Sita, travelled across the Sahyadri, returned to Ayodhya after killing Ravana, and sported with his consort, Sita, in Ayodhya for a long time.

The translation justifies the term "Sri-Raghava" in the name of the text. But, why the reference to Lord Krishna, as in the name of the text?

Now, the above text is read exactly in the reversed order:

सेवायेध्यो रामालाली गोप्याराधी मारामोरा:।

यस्साभालंकारं तारं तं श्रीतं वन्दे अहं देवं॥1

Interestingly, the above translation means:

I bow to Lord Sri Krishna, whose chest is the sporting resort of Sri Lakshmi who is fit to be contemplated through penance and sacrifice, who fondles Rukmani and his other consorts, who is worshipped by the Gopikas, and who is decked with jewels radiating splendor.

Astonishing, isnt it? The 30 shlokas when read in the order which they are composed in, describe the story of Rama, while when they are read in reverse order, describe the story of Krishna. Needless to say, the strong computational attributes of Sanskrit are at play here. And though I have been able to take up only one such Shloka here, the entire composition is an astounding work in itself. 

The text projects the strong computational, and hence encryption prowess of Sanskrit and also how smartly Sanskrit could be played with to create magnificent compositions

With that, I wish to bring an end to my post. The purpose of this post was to bring forward the same familiar concept, that not only were our ancestors religious, but all the concepts around religion and their lifestyle, overall was highly scientific and analytic in nature. Encryption was already in use and in place very smartly, the reason why most of the ancient texts is still considered unsolvable by our neo-modern intellectuals. What people claim to have invented or solved only lately, has been in place in this country since ages.

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Bidding Adieu for now.

Namaste

References:

1.) http://books.google.co.in/books?id=sEX11ZyjLpYC&pg=PA22&lpg=PA22&dq=encryption+methods+ancient+india&source=bl&ots=8-8qoGBlbD&sig=F9dEDJ8xrwEyZpBXauooNqhCOv0&hl=en&sa=X&ei=IbKdUdGJFMzhrAeOjYDAAg&ved=0CIYBEOgBMAk#v=onepage&q=encryption%20methods%20ancient%20india&f=false.
2.) http://ektariovee.blogspot.in/2012/12/sri-raghava-yadaveeyam.html
3.) http://www.cryptool-online.org/index.php?option=com_content&view=article&id=148&Itemid=386&lang=en
4.) http://en.wikipedia.org/wiki/Katapayadi_system
5.) http://en.wikipedia.org/wiki/Karanapaddhati

1,2,3

"The world owes most to India in the realm of mathematics, which was developed in the Gupta period to a stage more advanced than that reached by any other nation of antiquity. The success of Indian mathematics was mainly due to the fact that Indians had a clear conception of the abstract number as distinct from the numerical quantity of objects or spatial extension."
A.L. Basham, noted Australian historian

In the year 1911, noted Vedic scholar Bharati Krishna Tirthaji, a scholar of history, mathematics, Sanskrit and philosophy. and had made a comprehensive study of the four Vedas, Rigveda, Atharvaveda, Yajurveda, and Samaveda. Studying these texts for years, he was able to reconstruct a series of mathematical formulae, called Sutras. He continued the study for 7 years and compiled them together in a major single volume, which got published five years after his death, Vedic Mathematics, in the year 1965.

Repeated attempts of a few British mathematicians finally bore results duting the years 1981 to 1987, when Vedic mathematics started being taken seriously. A number of London and Indian schools took it up as subjects, and intellectuals started delving more time and energy into the same.

The Content:
The entire Vedic mathematics is divided into 16 Sutras or formulae, which when translated to english and analyzed, makes up 19 major aphorisms and 14 sub-aphorisms. It would be a bit too ambitious to include each one of them here, I would like to pick up a few simpler ones  which would strengthen my point..


Mental Multiplication:
So, what do you do when you are to multiply, say, 78X92?
You jot down the numbers hurrily on a notepad, do a quick 8X2, write down 6, carry over 1...blah blah blah...something like this
    78
    92
  156
702X
7176
Because thats what was taught in schools. Now, what Vedic Maths asks you to do is, think beyond the obvious(am I sounding like some management Guru?)
The nearest 10s base to 78 (and 92) is 100
Step 1) We subtract 100 from both numbers
78 - 100 = -22
92 - 100 = -8
Step2) Add 78 and 92 and subtract 100 from it 78+92-100=70
Step3) Multiply -22 and -8  -22 X -8 = 176
Step4) Carry the 1 from 176 and add it to the result obtained in Step 2 70+1 = 71
Step5)Concatenate the two 71 and 76=7176
And this can be done mentally, which is almost impossible in the first approach. A similar approach could be taken up for two digits or three digit numbers

Mental Squaring:
What would you do if someone asks you to calculate the area of a 42m X 42 m land? Again take out your notepad and scribble?
Or maybe take the above approach and solve it out much quicker?
Or maybe take another better and much interesting approach?
Step 1)Round of 42 to the nearest 10s, ie 40.
Step 2)Now add the remaining amount, ie 2 to 42 ie 44
Step 3) Multiply 40 and 44(much easier), ie 1760
Step4)Now take the number from step 2, ie 2 and square it, ie 4. Add the number to result of Step 3, ie 1764

Cubing a two digit number:
What would you do if someone asks you to find the volume of a 25X25X25 box?
Consider ab=25(where a=2, b=5)
Step 1)Cube a, is 8
Step 2)Divide b by a, ie 5/2=2.5
Step 3) Now write four subsequent products of a with the result of Step 2 and prepare a matrix
8    20     50     125
Step 4) Double the number at 2nd and 3rd place and add them at their subsequent places. Carry over the leftmost digit if the sum is of two digits
8    20     50     125
      40     100
8    60     150   125
8    60     162    5
8    76     2        5
15     6    2         5
The answer is 15625

Multiply and two 2-digit numbers:
If the base 100 method(the first in this blog), still looks tougher to compute mentally, or if you want to explore some more of it, here is another
Consider you have to multiply 12X34. Prepare a matrix of the two numbers.
1     2
3     4
Step1) Multiply ac, ie 1X3=3
Step 2)Multiply bd, ie 2X4=8
Step 3)Add ad and bc, ie 4 and 6=10
Step 4)Prepare another matrix mentally with the results of Step1, 3 and 2 respetively
3     10     8
Step 5) Carry over any extra digit to the left
4   0   8
The Answer is 408

More practical usage of Vedic Maths: Convert kilo to pounds:
Often people switch to Google in the blank of an eye when asked to fill up a form with weight as pounds(when they are more familiar with kilo). Or they would switch to their cellphones.
Well, its much easier to do it mentally, and a better way to keep the mind healthy.
If we have to convert 85 kilo to pounds,
Step 1) Double the kilos, 85X2=170
Step 2) Divide the answer by 10, ie 170/10=17
Step 3) Add the two numbers, 170+17=187, and thats the answer.(to the nearest integer)

Adding to time:
Whats the fastest way to calculate if the current time is 4:15, and someone asks you to calculate the time 2 hours 55 minutes later?
You first add the minutes and then the hours, which is quite simple as such, but then what would you say about this
Step1) Add 415 and 255, ie 670
Step 2) Add 40 to the answer, ie 670+40=710,and 7:10 is the answer

Concept of infinity and zero:
Two concepts highly useful in almost every practical application of maths in daily lives and even for astronomical calculations.
Infinity finds its first usage in the Yajurveda. Vedic men had several terms to describe what is infinity, ananta, purnam, asamkhya being few of them. Consider the following shloka out of the Yajur Veda,


पूर्णमदः पूर्णमिदम पूर्णत पूर्णमुदच्यते
पूर्णस्य पूर्णमादाय पुर्नामेवावासिश्यते 


From infinity is born infinity. When infinity is taken out of infinity, only infinity is left.
Till then, the Greeks used to distinguish between numbers with positional systems, like 27,207,270 would be represented as 27, 2 7, 27 . this system did not provide much flexibility to them to write large numbers with regular instances of 0.
Similarly, Romans didnt have the concept of 0, and had to write numbers like 101,000 as 101MMM.
AtharvaVeda described how the vale of a single digit number increases by 10 by writing 0 in front of it.

Trigonometry:
A fact iterated in the movie Namaste London, Trigonometry even got its name from Vedic mathematics, the word being Tri(three)-kon(angle)-mati(perimeter). Though, now the concepts have already been discovered and have been in use since years, The circular nature of Sin, Cos and Tan finds its mention in the Vedas.
The fact that Sinx/Cosx = Tanx finds its mention in the Vedas.

Currently, as you are reading all of this, researches are being carried on, and undertaken, including the effects of Vedic mathematics on children. Major research is being done on how to develop more powerful and simpler application of Vedic mathematics in geometry, calculus, etc.
What makes Vedic mathematics different from other schools and branches of theoretical and practical mathematics, is the flexibility of Vedic mathematics in providing them freedom with creating their own methods and breaking the "correct answer" jinx.
Vedic Mathematics provides us with quick and fast one-line formulae which speeds up mental calculation. It is a mental tool which not only helps a person solve complex problems quickly, but with repeated practice, also helps a person concentrate better.

Its absolute pity, that in spite of it being such a great marvel as "Vedic Mathematics", some elitist, from within the Indian sub-continent itself, claim it to be a farce. They say that they dont find any such reference in Vedas, and that it was a result of imagination of Tirthaji himself.
The Vedas, primarily, have been known to reveal the true knowledge and meaning of the verses, the hidden secrets to those who are worthy. While this may appear to be a line straight out of a Dan Brown book, it is a mighty known fact that Vedas are written in a cryptic language and a mere translation would mean nothing or may produce some relevant results. The Vedas are believed not to reveal the true meaning to the reader, but rather help the reader find certain answers, which others wont be able to see or even validate.
The irony is, that while the western world is embracing and opening up itself to the charms and secrets of Vedic Mathematics, there are Indians themselves, who claim the Vedic Mathematics to be a lie.
Nothing is greater a tragedy than to not believe in your achievements while the world celebrates them

Realists, as always, would claim that Vedas do not actually teach mathematics on the face of it, but the fact is Vedic mathematics is gradually making a stronghold in the education system of the entire world and is soon becoming a force to reckon.

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