Archive for the ‘Ancient India’ Category

Deluge – A Sequel?

October 19, 2016

A New Chapter

I am again returning to the Novel ‘Deluge – Agasthya Secrets’ by Dr Ramesh Babu. It is about a year since the Novel has been published. It is an ancient mystery novel set in modern times, in the style of Dan Brown. I thoroughly enjoyed the novel and had recommended the same to many of my friends. I had also written a review of the same, available elsewhere in my blog site. I felt it required a final chapter to tie up a few loose ends and also to give scope for a sequel, if ever, in the future. Initially I thought this added chapter, when read alone, will spoil the suspense element of the original narration and hence, I did not publish the same in my blog site. Now I have made a few changes which will hold the eventual suspense, and at the same time will induce the readers to read the whole novel. To begin with, I am introducing the important characters of the novel before going for the finale proposed by me.

Swathi: A medical intern starting on a career in medical research in micro biology for finding new antibiotics and pro-biotics. She teams up with an American professor in finding herbal antibiotics and takes the help of Ashwin, for locating herbal plants near Rameswaram.

Ashwin: An IIT(M) graduate in Marine Engineering and on internship at Indian Maritime University at Chennai, interested in Marine biology and Archeology.

Ravi: Director of Indian Maritime University at Chennai, a Marine Archeologist, guiding Ashwin in his internship.

MARCOS: Short form for Marine Commandos, a special operation unit of Indian Navy.

Shivani: Joined the team of Ashwin/Swathi, as a member of MARCOS, a marine commando unit, to prevent a possible terrorist attack from the sea, off the coast of Rameswaram.

Lee: Brings along a mini nuclear submarine Triton-1000 from China to help the team (!?!), in under water exploration.

 

THE FINALE

27th March 2016, 6.00 AM

After seeing off Ashwin/Swathi/Shivani trio, leaving for high seas by speed boat, Ravi was lucky to find a modest inn near Uvari itself and had a nap over night. But he got up early at 5 AM, with the village getting busy so early in the next morning. He tried to call Ashwin/Swathi, but their phones were out of range. He packed up his things and left, to take a tour of the shore temple as he had planned earlier. As he approached the Temple he again tried Ashwin/Swathi, as he was getting worried about them. He was looking at the sea intermittently for any sign of them. When he was trying his phone again, he sensed some movement in the sea. When he looked up with hope, he could only see a floating object slowly swaying towards the shore. It was semi-circular in shape and as it came nearer he realized it as a lifebuoy. But why is it floating vertically, as though some thing heavy is hanging from it? He approached the object and pulled the same to the shore. It was a lifebuoy with the inscription, TRITON 1000. Is it not the family of mini submarines, used for under water explorations, he suspected. As he pulled it completely, he saw a heavy object entangled with lifebuoy through some wild sea weeds. It was looking like a wheel guard of a heavy vehicle made of blue PVC material. It had a slit opening in which a black object was remaining stuck. On closer look he recognized it as the stone tablet Ashwin was handling the previous day. Now he was sure something serious has happened to Ashwin trio in the high seas. He again tried to contact them without success. He noticed the tablet now to have the inscriptions erased on both sides.

Sadly Ravi walks towards the shore temple ruins, after safe guarding his finds from the sea in his car. He went inside the temple ruins, to where sanctum sanctorum should have been. There was no Sivalinga – perhaps, it has been moved to the new Swayambu Linga temple nearby. The entrance to the sanctum was somewhat intact. He could see the carvings of Sages Patanjali and Vyagrapadar on either side of the entrance. But he was pleasantly surprised to see carvings of sage Agasthya below Patanjali and of sage Tirumoolar below Vyagrapadar. This carving of Agasthya was exactly the same as what they saw in Tiruchuli, again with the typical signature of Agasthiyar at the bottom, in Harappan script. Oh! What is this big space beneath the Agasthyar’s carving? It looks as though a tile of granite has been removed from this space. However on the other side there was a granite peace in place below Tirumoolar, though slightly damaged. Just as he was wondering about this, Sun’s rays started falling on these carvings. And in the sunlight he could see faintly a similar map in the space below Agasthya, as he has seen earlier on the tablet which he just now retrieved from the sea (with all inscriptions erased on both sides). But the Brahmi scripts on the map were not same. It did not take long for him to realize the map with the scripts was a mirror image of the original. Does it mean that the tablet originally belonged to this temple?

Ravi came hurrying to his car where he has stored the tablet and the lifebuoy. As he was about to return to the ruins with the tablet, he received a call from Swathi and was relieved to know they are safe. He asked them to come over to Uvari for the surprising revelation. As they arrived there in about twenty minutes along with Marcos team, they explained about what happened to them at the Spot-X. When Ravi showed them the tablet and lifebuoy of TRITON 1000, they all guessed that Shivani and Lee might have been involved in the mysterious explosion that occurred below the sea. Could they have ejected themselves and escaped? Or?

Ravi, Ashwin and Swathi approached the stone tablet in the car with all reverence. When they lifted it out into the sun, they could again see the map in the sun light at a particular angle. Ravi immediately reversed the tablet – Yes, they could see the inscription on the reverse also in the early morning sun light. Apparently the ancient chemical used for inscription is able to show up under ultra violet rays of the early morning sun!  They carried the Tablet as though it is a divine idol and tried to place it in the space below the Agasthya carving in the shore temple. Like a strong magnet the frame attracted the tablet with perfect fitting. It may require some force to take the same thing out again. Ravi observed a smile in the face of sage Agasthya. Or was it his imagination? But what is it about a similar tablet below Thirumoolar? Ravi made a mental note to come back soon to the site with ultraviolet lamp and other accessories to do further research.

With the sun still falling on the temple they could still read the inscription shining in the ultraviolet rays.

ஊழி அடைத்த உலக நாதன்

ஆழி வேலெறிந்த அய்யன் தலம்

Uuzhi adaitha Ulaga naathan

Aazhi vel erintha ayyan thalam

Ravi understood. Yes,

“Know this place from where the Swami

  Threw his spear to seal Tsunami”


 

(PS): I shared this proposed final chapter with the author Mr. Ramesh Babu. He liked this so much, he is proposing to include this finale in the next edition of the book. In his own words, “One reader Mr. L. Nagarajan has suggested an entire new chapter to my novel. It fits in so well just after the climax and also follows my style of narration! Hats off!” (https://plus.google.com/101148246018050836170). You may get this book ‘Deluge (Agasthya Secrets)’, at the following sites:

https://notionpress.com/read/deluge

https://store.kobobooks.com/en-US/ebook/deluge-20

http://bookstore.bookcountry.com/Products/SKU-001042017/Deluge.aspx

https://www.amazon.com/Deluge-Agasthya-Secrets-Ramesh-Babu-ebook/dp/B014XUEVR0

 

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Dan Brown’s Inferno, the hell

August 19, 2016

I just finished (Aug 2016) reading the novel Inferno by Dan Brown. When I finished reading The Amber Room by Steve Berry, I wrote a blog about similar mysteries that abound in ancient and medieval India. I invoked English language writers of Indian origin to write such mysteries a-la Dan-Brown-Style, but with Indian artifacts and mysteries. (Please refer to my blog.

https://lvnaga.wordpress.com/2014/01/28/ancient-mystery-thriller/). My thirst for the same was quenched somewhat by a novel ‘Deluge – Agasthya’s Secrets’, by Dr. Ramesh Babu of Chennai. (https://lvnaga.wordpress.com/2016/03/22/dr-ramesh-babu-indian-version-of-dan-brown/).

This new novel ‘Inferno’ by Dan Brown is based on a biological ‘terrorism’ of scientific age placed in the surroundings of medieval mysteries of Ottoman Empire covering present day Florence of Italy and Istanbul of Turkey.

This novel is heavily based on Italian poet Dante Alighieri’s master piece ‘The Devine Comedy’ consisting of three cantos – Inferno, Purgatory and Paradise, describing the path of the dead/soul towards hell, punishment and finally perhaps the heaven. There are many articles available in the net comparing this piece of literature with similar ideas represented in ancient Hindu scriptures by Saint Veda Vyasa in his Garuda Purana. Some of them even suggest that Dante was inspired by this description of Hell (and the travel of soul with its pseudo-body through the hell to the Paradise). I wonder if any other Indian  literature has given annotations of this work in Sanskrit or any other modern languages. I am vaguely familiar with a story in Mahabharat where King Yutishtra, with all his integrity and moral equipoise, is made to undergo a horrible view of the Hell, as a punishment for his abetment of a lie in killing Aswatthama in the war. Any extract of description such a view, is it available, I wonder.

(Those who have not read this novel ‘Inferno’ and are planning to read the same, may please avoid reading further, to retain the mystery and suspense of the Novel.)

In this novel there are some unexplained ambiguities as below

  • How can a single type of vector virus would do equal harm to the fertility of both men and women? Evidently, their reproductive systems are quite different.
  • Neither it is necessary to affect random one-third of both male and female population equally, to reduce the birth rate by a third.
  • Hence it is better to say that the vector-virus modifies the DNA of whole population but switches on at random only in one-third of male population. This will reduce the birth rate by one-third and gradually reduce the population by a third, as this DNA-Virus from even unaffected males gets inherited by the subsequent population. It will be again switched on at random in one third of males in every generation.
  • Though a lot of anxiety is expressed by all the characters in the novel about this biological ‘terror’, it appears to be a very humane way of controlling the population. It is same as vasectomy and tubectomy, which are of course, voluntary. This type of population control is normally adopted in animals and pests.
  • The characters in the novel, opposed to this type of ‘terrorism’ initially, come around and accept the same and think of making it reversible.

However the novel is quite interesting and highly readable. I understand it is also coming as a movie soon with Indian actor Irfan Khan in the role of Provost, the off-shore expediter and the secondary antagonist in the novel.

PS: I remember playing a board game in my village (India) on the nights of Gokul Ashtami and Shivratri known as Parama Pada Shobanam. It is a kind of a snake and ladder game where we start from hell and pass through several evil images and then on to happy images and finally to the heavenly images of Gods.(i.e. Inferno, Purgatory and Paradise).  On the way we encounter many snakes (named after villains of Hindu epics) and ladders of good conduct and behaviors. I wish somebody adds an image of this board to this blog. (LVN)

 

 

Dr. Ramesh Babu, Indian Version of DAN BROWN

March 22, 2016

Deluge

Agasthya Secrets

A novel by Dr. Ramesh Babu

A REVIEW by L V Nagarajan

An Indian version of Dan Brown has risen in the horizon. Many of us might have read Dan Brown’s Novels Da Vinci Code, Angels & Demons, Digital Fortress and the like. These novels are based on ancient mysteries being interpreted and used in the present days for good or bad deeds. These mysteries involve codes, puzzles and secrets that existed in the context of medieval European History. We always felt Indian history and culture, being much older, offers much more scope for such novels.

Please refer to my blog:

(https://lvnaga.wordpress.com/2014/01/28/ancient-mystery-thriller/) wherein I have given some ideas on such themes.

 

Dr. Ramesh Babu’s novel DELUGE is a very successful attempt to create a full length fiction of ancient mystery thriller set in modern India. He has made an excellent mix of several ancient intrigues like, Nadi jyothish, Siddha philosophy, mythology of Lord Siva as Tripurandaka, mystical land of Lemuria, ancient Tamil-Brahmi scripts etc. This novel is about events set to happen in the near future. The parallel story of devas and asuras of yore and merging it with the tsunami of Lemuria keep the reader quite engaged. The synthesis of Vedic Culture and Tamil Culture has been brought out very well in the narration. Extremist views in Politics and Religion has also been brought out convincingly. Narration is very absorbing and it is difficult to believe that it is author’s first full length novel.

I recommend this novel to all my friends for a good read.

Following are the details

Deluge

Agasthya Secrets

By Dr. Ramesh Babu, MS, MCh, FRCS Glas, FRCS Edin, FRCS Paed
Professor of Paediatric Urology,
SRMC, Chennai.

https://notionpress.com/read/deluge

Nagarajan

Rhythmatics

February 27, 2016

Rhythmatics

Fibonacci – Hemachandra Sequence

Some of my readers will m remember, one Krishnagiri Kittappa, the official percussionist of Oho Productions in the great Tamil romantic comedy of 1960s, Kathalikaa Neramillai (No time for romance). He was initially a self-taught mridangam (Drum) player. He wanted to learn to play Tabla also. He went to a Tabla player to learn the same. He was started on his first lesson, of course, in Teen Tal (or Triputa Tal in Carnatic music) of 8 beats.

Na Din Dinnah – Na Din Dinnah

Na Din Dinnah – Na Din Dinnah

Na Din Dinnah – Na Din Dinnah  . . . . . . .

This went on for quite some time. Our man got bored of playing the same rhythm. It is the same 1,1,2 – 1,1,2 all the time for the 8-beat cycle. Why not 1,2,1, – 1,1,2, he thought.

Din Dinnah Din – Na Dhin Dhinna

Din Dinnah Din – Na Dhin Dhinna

Then, why not 2,1,1-1,2,1

Dinnah Din Din – Din Dinnah Din

Good. Now he further thought about how many such combinations of 1 and 2 (Din and Dinnah), he can make in a cycle of 8 beats. Ancient Indians have already thought about this and so, I gave him the answer as 34 different combinations. He was surprised. So many? How did they calculate?

Ancient Indians always depended on recursive technique in solving such problems. They started from 1-beat, then to 2-beats, 3-beats etc.

No. of Beats (n) Syllables – 1 & 2 Combinations Total Combinations Kn
1 Din 1 1
2 Din, Din

Dinnah

 

2

 

2

3 Din Din Din

Dinnah Din

Din Dinnah

 

 

3

 

 

3

4 Dinnah (+ 2-beats)

Din  (+ 3-beats)

2

3

 

5

Now they generalized:
(n+1) Beats Dinnah +  (n-1) beats

Din + (n) beats

K(n-1)

K(n)

K(n+1) = K(n-1) + K(n)

Therefore,

K4 = K2 + K3 = 2 + 3 = 5

K5 = K3 + K4 = 3+ 5 = 8

K6 = K4 + K5 = 5 + 8 = 13

K7 = K5 + K6 = 8 + 13 = 21

K8 = K6 + K7 = 13 + 21 = 34
Hence we have 34 combinations of 1-2 in an 8-beat cycle.

Now look at the series K1, K2, ….. Kn:

1, 2, 3, 5, 8, 13, 21, 34, 55 ……

This is the famous Fibonacci series ‘invented’ by Fibonacci (alias Leonardo Pisano Bogollo) in 13th Century AD. Ancient Indians knew about this, at least, a thousand years before him. Fibonacci himself acknowledges this fact. Fibonacci also helped spread Hindu- Arabic Numerals (like our present numbers 0,1,2,3,4,5,6,7,8,9) through Europe in place of Roman Numerals (I, II, III, IV, V, etc). The olden day knowledge route was from India to Alexandria (Egypt) to Europe.

Susantha Goonatilake (Ref-2) writes that the development of the Fibonacci sequence ” is attributed in part to Pingala (200 BC), later being associated with Virahanka (c. 700 AD), Gopāla (c. 1135), and Hemachandra (c. 1150). Parmanand Singh cites Pingala’s cryptic formula misrau cha (“the two are made together”) and cites scholars who interpret it in the context as saying that the cases for ‘n’ beats (Kn+1) is obtained by adding [Short or 1] to Kn cases and [Long or 2] to the Kn−1 cases. He dates Pingala before 450 BC ”.

“However, the clearest exposition of the sequence arises in the work of Virahanka (c. 700 AD), whose own work is lost, but is available in a quotation by Gopala (c. 1135). The sequence is also discussed by Gopala (before 1135 AD) and by the Jain scholar Hemachandra (c. 1150) “. Fibonacci was born only in 1170 AD.

Prof. Manjul Bhargava (R Brandon Fradd Professor of Mathematics, Princeton University, USA) gave a Lec-Dem on Music & Mathematics at the Music Academy, Madras during their annual conference 2015, on 31st December 2015. Being a Tabla player himself, he dealt with the above aspect of rhythm variations in detail. His talk was the inspiration for me to write this blog.

References:

  1. https://www.mathsisfun.com/numbers/fibonacci-sequence.html
  2. https://en.wikipedia.org/wiki/Fibonacci_number
  3. Manjul Bhargava’s Lec-Dem at the Music Acedemy, Madras (2015)

L V Nagarajan

 

Melakartha Chart (Scales of Carnatic Music)

November 26, 2015

Melakartha Chart – (Scales of Carnatic Music)

L V Nagarajan

Prologue:

Some time back I was listening to a young upcoming Carnatic vocalist in Shanmukhananda Hall in Mumbai. After a few items, he started the alap of an unfamiliar raga. Some of the knowledgeable members of the audience were guessing what raga it could be. Somebody said it is like another rare raga but with prati-madyama. Another one said it is a janya of melakartha75(?!) or so. After a short alap he allowed the violinist to play the raga. Violinist started a little hesitantly. Vocalist leaned towards him and spoke a few words. Violinist nodded and continued more confidently. Then the vocalist proudly announced the rare raga as Gopriya, janya of Rishabapriya scale. Some in the audience were happy but I was not happy. If he had told us the Melakartha number I could have also understood the scale easily; but Rishabapriya? Luckily I happen to know something about Katapayadi Sankhya, a code into which these Melakartha names are encoded. Now let us see:

Ka (adi) nava – Ka, Kka, Ga Gga, Gna, Cha, Ccha Ja, Jja.

Ta (adi) nava – Ta, Tta, Da, dda, Nna, Tha, Ttha, Dha, Ddha.

Pa (adi) pancha – Pa, Ppa, Ba, Bba, Ma

Ya (adi) ashta – Ya, Ra, La, Va, Sya, Sha, Sa, Ha

Ri-Sha-bapriya is the Mela. Ri is second letter in Yaadiashta and Sha is 6th in the same series. Hence It is Mela number 62. (i.e) =10×6 +2, 11th cycle of the Melakartha scheme, 5th Cycle of Prathi-madhyama group and the second raga in the cycle. (i.e.) S R2 G3 M2 P D1 N2 S is the scale. QED.

How nice it could have been if the vocalist told us this number 62 as he did to the violinist.

 

Dialogue

1. He! I understand all systems of music have seven notes. But, tell me, what is this scale, Mela or Melakartha?

The concept of present day scales in Carnatic Music has been imported from the western music. In the West an octave of eight notes are defined starting from a note of a specific frequency to its resonant note of double the frequency. This octave (set of eight notes) was initially divided into 7 frequency intervals between the notes, denoted as C-D-E-F-G-A-B-C. The musical scale of these major notes is known as Major scale. The frequency ratios selected for these intervals is same as Sankarbharanam (scale) of Carnatic music. However, a set of semi tones (or half intervals) were introduced, 2 between C-D-E and 3 between F-G-A-B.  In a piano or harmonium the white keys represent the major notes and the black keys(^) represent semi-tones – (C^D^EF^G^A^BC). The semi tone between C and D is called D-Flat or C-Sharp according the musical context. Same way the other semitones are also treated. For our convenience, we will call them as

C, D1, D2, E1, E2, F1, F2, G, A1, A2, B1, B2, C.

Frequency ratios of these semi-tones follow the rule of consonance and assonance. Many scales were developed using the 7 selected notes from these 12 semitones. Some of our main ragas (scales) like Karaharapriya and Mayamalavagowla matched these semitones.  Somewhere in the 17th century this concept of scale was adopted by theoreticians of carnatic music. They were also influenced by the keyed instruments of western music like piano and harmonium. Hence attempts were made to fit our ancient musical system into the keys of the above instruments. At this point scales were introduced in carnatic music, to group all the notes used in a raga. These attempts lead to the development of our own basic scales, Mela Karthas. Melas or new melodies were born out of these scales and hence the name Melakartha or melody-maker. (However, subsequently, sacrificing some amount of consonance and assonance between notes, Western music adopted 12 equal intervals for these 12 semitones. This system was called ‘equi-tempered’ as opposed to the earlier system called ‘just-tempered’ or ‘just intonations’ which takes care of consonance ratios. Western musical instruments are tuned accordingly.)

2. But how do these 12 semitones make 72 Melakarthas, I often hear about?

At this point it may be better to discuss about what constitutes a present day scale. A full scale constitutes seven notes or saptha swaras (sa, ri, ga, ma, pa, da, ni) in an octave. For convenience let us call these notes in short as S R G M P D N. These are selected from 12 basic nodes, which we may call as Sruthis’s, though they are called semitones in western music. The rules for selecting a basic-scale (a set of seven swaras from these 12 sruthis) can be summarized as below:

Swaras S       R       G      M P       D      N S
Semitones C D1 D2 E1 E2 F1 F2 G A1 A2 B1 B2 C
Sruthis 1 2 3 4 5 6 7 8 9 10 11 12 1
(12) Swara-sthanas S R1 R2 G1 G2 M1 M2 P D1 D2 N1 N2 S
(4) Vivadi-Swaras Gr Rg Nd Dn  
(16) Swara-sthanas S R1 R2 R3 M1 M2 P D1 D2 D3 S
G1 G2 G3 N1 N2 N3  

It may be noted above that notes nos. 3 and 4 are used as both Ri and Ga. Similarly notes 10 and 11 are used as both Da and Ni (though such usages are known as Vivadi Prayoga). In carnatic music system, these 16 swara-sthanas are called respectively: Shadjam, Suddha-Chatusruthi-Shatsruthi Rishabams, Suddha-Sadarana-Antara Gandharams, Suddha-Prati Madhyamams, Panchamam, Suddha-Chatusruthi-Shatsruthi Dhaivatams and Suddha-Kaisika-Kakali Nishadams. Here itself we can make out the attempt of reducing the number of sruthis from the original twenty-two to twelve. (Even in Western Music double-sharp and double-flat notes are used in a scale)

Any basic scale has two parts: Poorvanga, the notes upto and including M, (SRGM) and Uttaranga, the notes including P and above (PDNS). For each M (sudda-M1 and prati-M2), there are six types poorvangas based on six different combinations of notes R and G; namely R1-G1, R1-G2, R1-G3, R2-G2, R2-G3 and R3-G3. Similarly there are six types of uttarangas based on six different combinations of notes D and N. Hence we have 72 Melakartas – [(6+6) x 6].

3. Really ingenious! But what is this Katapayadi mumbo-jumbo being mentioned?

The above 72 melakarthas require some identification as most of them are not naturally available scales. They divided these in to twelve groups, each group having a specific R-G-M1 and R-G-M2 combinations with increasing order of R and G. Within each of this group, six scales with increasing order of D and N were included. Thus 36 scales or melakarthas using M1 were numbered 1 to 36. The others with M2 were numbered 37 to 72. Please refer to the chart at the end of this write-up. Each of this 12 groups known as chakras were named as Indu, Netra, ……, Rudra and Aditya. (Indu – one Moon, Netra – 2 eyes, ….., 11 Rudras and 12 Adityas). In the chart below each row represents two chakras. Since many of these scales were not in existence, these, as melakarthas, required some names also. The earlier day musicologists used a code to name these scales in such a way that one can find the number of a melakartha from the name itself. The code is known as Katapayadi Sankhya and is used in many applications of ancient Hindu science and mythology.

The numbers 1 to 9 are coded by using following letters: (Ka Ta Pa Ya)

Ka (adi) nava – 1 to 9 –  Ka, Kka, Ga Gga, Gna, Cha, Ccha Ja, Jja.

Ta (adi) nava – 1 to 9 – Ta, Tta, Da, dda, Nna, Tha, Ttha, Dha, Ddha.

Pa (adi) pancha – 1 to 5 – Pa, Ppa, Ba, Bba, Ma

Ya (adi) ashta – 1 to 8 – Ya, Ra, La, Va, Sya, Sha, Sa, Ha

Other letters are rendered as zero. In the scale Kamavardini, Ka decodes into number 1, and Ma decodes into number 5. Hence the number of the scale decodes into 5 tens plus one, 51. The same way other scales were also given such names. However, some scales are already established as popular and well known ragas from ancient times. Hence these names were modified, such as, DdheeraSankarabharanam (29), HanumaTodi (8), MechaKalyani (65), to fall into the Katapayadi scheme.

4. Looks neat! Why then people prefer melakartha numbers instead of names?

DdheeraSankarabharanam and Sulini are two different scales, the first one being a well known raga. Can you spot how these scales compare? Yes, only when one compares the numbers 29 and 35. we can now say immediately: Sulini uses R3 instead of R2 in Sankarabharanam. And Kosalam (71) is immediately identified as M2 of Sulini, (or R3, M2 of Sankarabharanam). With names these relationships are not immediately evident. Hence, the preference for numbers. Of course there are many who have completely memorized the Melakartha table

5. Is there a way to name these scales in a way to reflect these inter relationships?

To my knowledge, nobody has tried so far. However, I propose, in the following paragraphs, a way of encoding such relationships in the name itself.

Earlier the scale was split into two parts Poorvanga (SRGM) and Uttaranga (PDNS). Let us now split the scale into three parts: SRG, M, PDN. If we consider the variations in these parts, we get SRG(6) x M(2) x PDN(6) = 72 Scales. Let us now name these parts along with their variations:

S R1 G1 – athi M1- Daya P D1 N1- Vathi
S R1 G2 – Sakala P D1 N2 – Nidhi
S R1 G3 – Sarva P D1 N3 – Kari
S R2 G2 – Parama M2 – Kripa P D2 N2 – Varshini
S R2 G3 – Poorna P D2 N3 – Varithi
S R3 G3 – Poojya

P D3 N3 – Sagari

With these code names, Sankarabharanam (29) will be called as Poorna Daya Varithi. And Kalyani (65) will be known as Poorna Kripa Varithi. It is clear from the name that Kalyani scale is same as Sankarabharanam, but with M2. Poojya Daya Varithti is Sankarabharanam with R3, i.e. Shulini (35). Here I take the liberty of using the Melakarta Chart as developed by Dr. Mukund (http://www.carnaticcorner.com/articles/mukund_chart.htm). This chart is prepared on the same principle as mentioned above. I have superimposed the above code names on this chart, as shown below. We may use this chart to get used to the proposed new names. The melakarthas using Athi, Poojya, Vathi and Sagari codes are the 40 Nos. of Vivadi melas – the outer most squares of the chart.

Mukund-Chart

6. Umm…! I really need time to study and understand this.

Please take your time and do write your comments.

Epilogue:

  1. The material discussed in this blog has been extracted from my more detailed write-up on Melakartha scheme done in 2006. However I did not have a blog-spot then, to publicize it. I sent it to a few musicians and musicologists but there was not any response. I was enthused to write this note again, when I recently came across Mukund’s Chart. (http://www.carnaticcorner.com/articles/mukund_chart.htm).
  2. The rasikas of Chennai Music Festival (2015) may use this chart while attending the concerts.
  3. The above suggested codes/names can definitely be improved by musicologists like Dr. Mukund. It is only an idea to be adopted, or improved (or rejected, if found not suitable).
  4. While there are numerous advantages of this Melakartha scheme, there are a few disadvantages also. Though Melakartha formulation helps us to classify, formulate and document our music, it has limitations in formulating such nutpa-sruthis as Madhyama of Varali or Rishaba of Saveri. Possibly this was the reason why Harmonium (or Piano) was not preferred as a part of carnatic music ensemble. More so due to the equi-tempered tuning of Harmonium, where the 12 notes of the octave are tuned with equal frequency intervals. Here-in, even Sa-Pa consonance is sacrificed. We should not allow the Melakartha scheme to restrict our music to just 12 flat notes and 72 basic scales. We may discuss more about this later.  The author of this note is only a normal listener and follower of Indian music and whatever knowledge (or lack of it) displayed here are purely incidental. Let our Music and its traditions live forever.

Phonetics of Devanagari and Tamil scripts

October 14, 2015

Phonetics of Devanagari and Tamil scripts

These two scripts are phonetically related in a very odd way. Many of the north Indian languages closely follow the Devanagari script as used by Sanskrit presently. Other south Indian languages (I am consciously avoiding the term Dravidian languages) follow their own independently evolved scripts. I found some interesting contrast between the scripts of Sanskrit and Tamil which I want to share with my readers. These are all my own original ideas and purposely I have not read or referred any previous work or study on this aspect (which I will eventually do after I publish this write-up).

Like the scripts of all languages, these two scripts also have vowels and consonants. The Devanagari script has thirteen vowels as below. (I am giving only the phonetic value of these vowels);

Vowels : a, Aa, i, Ee, u, Uu, ru, eay, ai, oh, ou, amm, aha.

Actual Devanagari script is not of our interest as of now. Let us now look at the vowels in Tamil language:

Vowels : a, Aa, i, Ee, u, Uu, eay, Eaay, ai, oh, Oh, ou.

As we compare these phonetics we see, they are essentially similar, but with significant differences. When ‘a’, ‘i’ and ‘u’ have their extended phonetic as ‘Aa’, ‘Ee’ and ‘Uu’, the same extension is not given to ‘eay’ and ‘oh’ in Devanagari script. But Tamil script has extended phonetics for them also. Tamil seems to need them too. Look at the Tamil words –

mattu, mAattu, mittu, meettu, muttu, mUuttu, mettu, meayttu, mottu, mOhttu

– meaning respectively,

less, hang, earth/land, pluck a string of a musical instrument, butt or bang with your head, enflame (or bone joint), musical tune, raised level, flower bud and bulging. (Phew, look at the strength of Tamil vocabulary).

If we assume that Tamil phonetics were derived from Devanagari script, then Tamils have surely added two more vowels to meet their phonetic requirements. Let us wait till we see the consonants.

The primary consonants in Devanagari are:

Ka, kha, ga, gha, nga

Cha, chha, ja, jha, ngya

Ta, tha, da, dha, Nna

Tha, Thha, dha, dhha, na

Pa, pha, ba, bha, ma

The corresponding consonants in Tamil are:

(Ka, nga) – (Cha, ngya) – (Ta, Nna) – (Tha, na) – (Pa, ma)

Tamils added (perhaps after a later thought) one more set of similar consonant in

Rra – na, (literally at the end of the list of alphabets).

While an emphasized Rra (or tra) is required to handle a few important Tamil words, I always found a second ‘na’ superfluous, although the two ‘na’s are surely used in different contexts in the written language.

As can be seen above, Devanagari script appears to be an expanded version of Tamil phonetics. If it is still believed that Tamil phonetics were derived from Devanagari, what could have been the reason for dropping the different variations of Ka, Cha etc? When writing purely Tamil words, Tamils don’t feel the need for these emphasized consonants, as they are intended to be pronounced with such variations as per the context. Tamil also uses double letters for emphasized consonants such as KKa for Kha. However, when writing words of other languages in Tamil script, one feels handicapped by this deficiency in the script. Some authors use subscripts like 1, 2, 3 and 4 in the script for marking these variations. As you can see in the next section, in later days, a few Devanagari phonetics were also adopted in to Tamil script, even though purists do not use them.

Now let us write down the secondary consonants of Devanagari script, as below:

Secondary consonants: ya, ra, la, va, sha, sya, sa, ha

Tamil has similar consonants as below:

Secondary consonants: ya, ra, la, va, zha, Lla

Earlier Tamils did not adopt the latter half of the above set of Devanagari phonetics in to their script. As need arose to read and write words of foreign roots, they adopted the others also in to a set of special (Northern!) scripts – ‘sha, ksha, sa, ha and Sri’. Here again the purists do not use them. The consonants zha, Lla, Rra, na are special phonetics of Tamil, (especially the zha), and significantly they at the end of the list of Tamil alphabets.

From the above account it appears that the phonetic scripts of Devanagari and Tamil could have had their roots in a more ancient common script. Critical comments on the above are most welcome.

Meru Prastarah (or Pascal’s Triangle ?!?)

October 21, 2014

Meru Prastarah (or Pascal’s Triangle ?!?)

Let me start with an ancient (1000 CE) Sanskrit text as below:

Anena ekadvyaadilaghukriyasiddhyartham, yaavadabhimatam prathama prastaravat meruprastaram darsayati, uparistadekam chaturasrakoshtam likhitva, tasya adhastat ubhayatordhaniskrantam koshtadwayam likhet, tasyapiadhastatrayam tasyapiadhastatccaturtyamevam yaavadabhimatam sthanamiti meruprastarah tasya prathame koshte ekasamkhyam vyasthapyalakshanamidam pravarttayet, tatra dvikoshtaayaampanktau ubhayo koshtayorekaikamankam dadyaat, tatastritiyaayaam panktau, paryantakoshtayorekaikamankam dadyaat, madhyamakoshtethuparikoshtadvayaankamekikrtya purnam nivesayediti purnasabdarthah, chaturtyampanktavapi, paryantakoshtayorekaikamankam sthapayet, madhyamakoshtayothuparakoshtadvayaankamekikrtya purnam trisamkhya rupam sthapayeth,  uttaraataraapyevameva nyaasah, tatra dwikoshtaayaam pankatauekaakshrasya prastaarah,……. tritiyayaam pankatau, dviakshrasya prastaarah, chaturtyaam pankatau, triakshrasya prastaarah, ….

The above is not in praise of any god of Jain, Budha or Hindu religion. It is not a religious text at all. It is a text describing a method for constructing a mathematical table. Ancient Indian Mathematician Pingala (200 BC) in his Chandahsutra had given the rules for formation of different chandahs (≈ musical meters) for Sanskrit prosody. Another ancient Indian mathematician Halayudha (1000 CE) has given the explanation and commentary on this work by Pingala. Given above is a selected portion of his commentary. For some reasons unknown to me, ancient Sanskrit texts always use composite words very frequently. These words need to be broken into individual words properly to obtain the intended meaning of these words. Here is an attempt to translate the above text into English with proper separation of words.

Anena ekadvyaadi laghu kriya siddhyartham, yaavadabhi matam

(To get every combination of one, two, etc. syllables as required)

Prathama prastaravat meru prastaram darsayati.

(from first row onwards , the meru tabulation will be shown below)

Uparihi tad ekam Chaturasrakoshtam likhitva,

(At the top itself one square cell is drawn)

Tasya adhah tat ubhayato ardhani skrantam

(Below this row let us have a pair, half over lapping)

Koshtadwayam likhet.

(Two cells are drawn)

Tasyapi adhah tat trayam

(Again the row below will have three)

Tasyapi adhah tat chaturtyam,

(Again its next line will have four)

Evam yaavadabhi matam sthanam

(same way, up to the  required stage, cells are constructed)

iti meru prastarah.

(This is called Meru Prastara or Meru-Tabulation)

Tasya prathame koshte eka samkhyam

(Its first stage-cell will hold the number 1)

Vyvasthapya lakshanamidam pravarttayet

(From here on, the following is the way it grows)

Tatra dvikoshtaayaam panktau

(in its twin-cell row)

ubhayo Koshtayoh eka ekam ankam dadyaat

(the pair of cells holds numbers 1,1)

Tatah tritiyaayaam panktau, paryanta Koshtayoh Eka ekam ankam dadyaat

(then in the 3rd row, the extreme cells will hold numbers 1,1)

Madyama koshteth, upari koshtadvayah ankam eki krtya purnam nivesayeth

(middle cell takes the added value of the two cells above)

Iti purnasabdarthah

 (Thus completes the table for 2nd power)

Chaturtyam panktau api, paryanta Koshtayoh eka ekam ankam sthapayet

(then in the 4th row also, the extreme cells will hold numbers 1,1)

Madyama koshtayoth, upara koshtadvayah ankam eki krtya purnam

(middle cells take the added values of the two cells above each)

Trisamkhya rupam sthapayeth

(this completes the 3rd power)

Uttara utaaro api evameva nyaasah,

(next and next stages also follow the same rule)

tatra dwikoshtaayaam pankatau, eka akshrasya prastaarah

(Here the twin-cell row gives one syllable table)

tritiyaam pankatau, dvi akshrasya prastaarah

(the 3rd row gives two syllables table)

chaturtyaam pankatau, tri akshrasya prastaarah

(Thus 4th row gives three syllables table)

And so on.

Meru

If we follow the above step by step construction given so clearly by Halayudha (1000 CE), we get the above pyramid or Meru in Sanskrit, (stands for a mountain with a peak). What do we have here? It is the same as Pascal’s Triangle, “discovered” by Blaise Pascal (1623-1662 CE).

This table gives in every nth line the coefficients (a+b)**(n-1). i.e. the second line gives coefficients of (a+b) as 1,1; the second line gives 1,2,1, as coefficients of (a+b)2.; the third line gives 1,3,3,1 as coefficients of (a+b)3 and so on.

However Halayudha gives credit for this table to Pingala (200 BC). He claims to have derived this table from Pingala’s cryptic clue which he translates to a set of rules, as below (with a and b as the two syllables to be combined, in any n-syllable chandah):

  1. First write down all (‘n’ number of)  b’s as the first combination
  2. In the next line, replace the first ‘b’ with an ‘a
  3. At the same line, replace all letters to the left of this new ‘a’ with ‘b
  4. For the next and the subsequent lines repeat the steps 2 & 3.
  5. Continue as above till we arrive at a line with all a’s,

This can be clearly seen as a binomial expansion (a+b)n staring with bn and ending in an. Halayudha later puts these results on a table known as Meru Prasthara. He later gives a step-by-step method as above, for constructing this table without specifically going through the above rules. This Meru Prastarah traveled to China and the Chinese mathematician Yang Hui reported it in the thirteenth century, although his work was unknown in Europe until relatively recent times. The Meru Prastarah traveled to Europe a little later through Arabia, Egypt and Greece and gets “discovered” by Pascal in 17th century CE, 600 years after Halayudha. We are blaming all the time ‘the lack of scientific temper’ among Indians.

Ref:

  1. Binomial Theorem in Ancient India – By Amulya Kumar Bag, History of Science, Ancient Period Unit II, No.1, Park Street, Calcutta-16 (1966)
  2. Journey Through Genius – The Great Theorems Of Mathematics – by William Dunham – Wiley Science Editions, John Wiley & Sons Inc.(1990)
  3. Probability in Ancient India, by C K Raju., ckraju.net, 2011.

Connected Topics:

Meru Prastarah

Baudhayana’s Circles

Square Root of Two

Sine of an Angle

LVN/ Oct, 2014

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

Ancient Mystery Thriller

January 28, 2014

Ancient Mystery Thriller

(by L V Nagarajan)

I have just finished a novel by Steve Berry titled The Amber Room. It is a novel about a mysterious art piece known as Amber Room, apparently pilfered by Germans from Russia during World War II.  It is a whole room paneled all over by Amber pieces carrying exquisite carvings in dazzling colours. The novel describes the hard chase by legal and illegal art collectors of the world to find this missing piece of art. This book is recommended by none other than Dan Brown, the author of Da Vinci Code, Angels and Demons etc. After reading these novels involving ancient mysteries, intrigues, puzzles and secrets, mostly of European culture and history, I felt why such novels are not written with Indian theme. There are enormous amount of such material in ancient and very ancient Indian history and culture. I wish authors of Indian origin like, Amitav Ghosh, Jhumpa Lahiri or Arvind Adiga would take up such themes in their future works. In Tamil, Indra Soundar Rajan could make an excellent narration. I have attempted here to give a few ingredients for such a novel.

Ancient Treasures

As many Indians know, Anantha Padmanabha Temple in Trivandram, Kerala is one of the richest shrines in the world. It has got treasures of art and other riches stored in their underground vaults. They were recently opened for judicial inventory. Even so, one of the underground vaults could not be opened as there are no visible locks for the strong art-full door. The legend is: the vault is locked by a special mantra recitation known as Naga Bandham (Cobra Lock). It can be opened by only a counter recitation of another mantra known as Garuda Moksham (Falcon key), if at all anybody knows it these days. The vault is believed to be protected by the divine Anantha himself who is the multi-headed cobra serving as a recliner for Lord Padmanabha, a form of Vishnu. If anybody forces open this vault many calamities are forecast. Hence even the Supreme Court of India left this vault alone. This vault apparently contains highly valuable treasures and secrets.

When India was under Mughal invasion and occupation, many Hindu dynasties hid their treasures in such temple vaults. Later on these vaults were locked and sealed in various ways, like camouflaging, walling-up, coded locks etc. There are legends of many such treasure vaults in many temples especially in South India. During the Mughal rule and the later British rule, some of these vaults were pilfered and treasures moved to different parts of the world. We only know about a few like Mayurasan (Peacock Throne), Kohinoor Diamond and Sivapuram Nataraja. Many other vaults are still remaining unopened and un-explored.

(Please refer: http://www.boydom.com/2013/03/21/top-10-alleged-indian-treasures/)

Secret passages and escapes

There are many legends about secret caves, tunnels and escape routes used by ancient rulers of India to hide their treasures from their enemies, citizens, even descendents and other claimants. They even used them as hiding places with all basic necessities provided for. Unfortunately the present day Indians are not good explorers. Neither, they take sufficient pride of, or care for, their ancient treasures and monuments. Due to this, these tunnels and caves remain totally unexplored. And some of them are even demolished for other purposes.  We have heard of some tunnels as below:

From Nayakar Mahal Madurai to Tiruvedagam Temple 10 miles away

From Srirangam Temple to Rockfort temple in Trichy

From Mysore palace to Chamundi Hills

From Amber Palace, Jaipur to uphill Jai Garh Fort. (Barely a small portion of this tunnel is uncovered)

Secret Codes          

There are many works of literature and poetry which have different secret messages encoded in them.  The words of poetry are split and combined in different ways to encode messages. Other methods include numerical codes, reverse reading etc. There are poems which can be read entirely in the reverse to give a different message. There are formal coding procedures like Katapayadi Sankhya to encode messages. As children we have used what was called Ka-language where every word in a sentence will be preceded by ‘Ka’ syllable. There were people who were very good at that. Another way was to interchange the first two syllables of every word, (e.g) Mantra will be pronounced as Namtra or Tranma. There were also extensive sign and symbolic languages used, where symbols of Swastika, Om, Trishul, Conch are employed. Hasta mudras (hand gestures) convey many ideas and emotions without any spoken word and they are employed even today in Indian dance forms.

A famous secret is kept as a secret even today – the Secret of Chidambaram or Chidambara Rahasyam. This secret is interpreted in three different levels, but still remaining a mystery, yet to be solved. Beyond the statue of Nataraja at Chidambaram Temple is hidden, the real sanctum sanctorum. On occasions this sanctum is exposed to perform special prayers, by unveiling the curtain covering the same. What is inside – NOTHING! According to Hindu tradition there is a fifth element in addition to the four known to other cultures, Earth, Water, Wind and Fire. The fifth element is Akash, or Ether, i.e., the space containing the entire universe. The god of Chidambaram is said to represent this Space element, just like the other four known temples dedicated to the other four elements. Hence only Space is worshiped in the sanctum sanctorum of this temple. This is a religious explanation of the Secret. Philosophically, it is explained as a representation of the formless GOD. But there is a scientific explanation also. In Hindu tradition Sakthi or energy is represented by a Goddess. In her formless representation, she is known as a DOT or a Bindu, wherein the whole energy of the universe is concentrated. In Chidambaram, the male, ‘SIVA, the SPACE’ interacts with his consort ‘Sakthi, the DOT’, to create all the matter in this universe, by creating a big explosion or the Big-Bang. That is the secret message hidden behind the veil! People are still looking beyond these explanations to find the real secret hidden. Will it ever be exposed?

Religious rites and procedures

Sacred Hindu Texts like vedas and upanishads are known to encode great scientific principles, some of which are still not fully decoded.  There is an entire sect of veda known as Atharvana veda, which is reported to contain many procedures used for psychotherapy and black magic (and white magic!). It also has chapters on various armory (empowered by mantras) used in warfare. The Tantric applications of Hindu religious rites are sometimes very mysterious and frightful. There are also some folk-arts and practices which are meant for certain special occasions – to induce or remove emotions such as anger, fear, sleep, madness etc. “Inception” of thoughts and ideas in human mind have been tried successfully in ancient days using dedicated mantras and procedures. Garuda Purana another Hindu text describes the aspects of Gemology. In another part it describes the punishments to be meted out for various crimes committed by humans.

Mantras, Yantras and Tantras

Mantras, Yantras and Tantras are a few of the Hindu religious practices. There are apparently hundreds of mantras (recitations) to invoke the divine and natural forces, either in favour of yourself or against your adversary. It is believed that there are also counter-mantras to cancel the effects of earlier mantras. Yantras (symbols) are also used to invoke the divine (and demonic) forces to enter and stay in those symbols. They are used for meditation and prayer and also for ill effects. These yantras are believed to protect the people and places against evil forces. Yantras vary from simple to very complex geometrical patterns either drawn on paper, or on the floor, or imprinted on copper plates. Some of them are even said to contain secret texts and messages. In Indian religious tradition Kolams (or rangolis, elaborate decorative patterns drawn on the floor or on a pedestal) play an important role. Tantras (Mysticals) are special practices involving the way a mantra is recited, the hand and bodily gestures, special anointments and accompaniments of sounds of Conch, bells, drums and other musical instruments. It also uses aspects of sex and sexual symbols for special effects. In the 21st century, we may ignore these things as no more than superstitions. However, we cannot deny that they were widely prevalent till the end of 19th century in India. Even now, they are practiced at least as rituals in many temples, religious centres and social and religious festivals. There were many yogic and mystic experiences of transplantation of souls between bodies and out-of-body experiences.

A Bija-Akshara is a seed-letter and is a very powerful Mantra. Every Deity or Devata has his or her own Bija-Akshara. The greatest of all Bija-Aksharas is OM or Pranava. It is the symbol of the (supreme soul) Paramatman Himself. Generally a Bija-Mantra consists of a single letter but sometimes it constitutes several syllables. Some Bija-Mantras are made up of compound letters, such as the Mantra ‘Hreem’. Their meaning is subtle and mystical. Ham, Yam, Ram, Vam and Lam are the Bija aksharas of the five elements, namely Ether, Air, Fire, Water and Earth, respectively.

(Ref: http://www.dhyanfoundation.com/bija-akshara.php)

Great Tsunamis and lost City

In very ancient literary history, there are mentions of a great tsunami in which a large track of land south of the present India was lost to the sea. This land was known as Lemuria. However there is no geological or archeological evidence to prove the existence of this hypothetical  land mass referred to frequently in ancient Tamil literature. More recently around 400-900 AD, another tsunami was blamed for disappearance of Poompuhar, a coastal capital city of great Chola kingdom. Archeological excavation do support this fact of history.

Sectarian and religious conflicts       

Throughout the ancient history of India there were regional, sectarian and religious conflicts between different dynasties, communities and religious sects. These led to destruction and abduction of many symbols and treasures of the victims, which are even to the present day, not traceable. Some major conflicts are as below:

Conflicts among  kingdoms – such as Chera, Chola, Pandyas, pallavas, Chalukyas and rajaputs

Conflicts among religious sects – such as Shaivas, Vaishnavas, Kaapalikas, Jains and Buddhists

Conflicts beyond the seas – Cholas towards Srilanka and Indonesian islands, Yavanas (Romans on the Indian ocean)

A possible theme for a Novel (as seen by L V Nagarajan)

Chidambaram Nataraja Temple is under the total control a special sect of Brahmins known as Dikshits. According to religious history God Siva himself has ordained this religious order and established the sect known as Thillai-3000, a set of 3000 priests. Their religious practices are special and different from others. Due to mis-use and mis-rule, this sect has now reduced to just 300 people, barely fighting to save this tradition. Many of them may not even know the full texts of their religious order, let alone their obvious and hidden meanings. Under such circumstances, our protagonist is the second son of a very senior Dikshit, who is no more. Both brothers were very bright students of their father and had learned many religious texts from him and others. They still have in their possession their father’s religious work books. Elder brother becomes a traditional Dikshit in the Nataraja Temple. But the younger one pursues his science education and obtains his Masters degree in Astro Physics. He joins the National Physical Laboratory and starts doing research on the origin of universe. The kind of scientific literature he gets access to, simply amazes him. Many of the postulates and ideas remind him of some of his father’s texts and explanations. He decides to get back to Chidambaram and to continue his religious research with the help of his more learned bother. Some of his later correspondences with the world scientific community evince a great amount of interest. He is awarded a grant from Ford Foundation to continue his research. With the help of his brother he is able to penetrate the secrecy of several other Dikshits and their hold on some ancient religious literature. He gets access to several secret vaults in the Chidambaram Temple and in nearby areas in the town of Chidambaram. When he is on the threshold of a big discovery, perhaps, a discovery of the SECRET itself, a major conspiracy from a terrorists group comes to light … … … … And so goes the story!

I hope some international authors of Indian origin will enhance and evolve this theme into full length novel.

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.