Wednesday, 13 December, 2006

Bhartiya Chronology 3228 BC - 1947 AD

Descension of Bhagwan Krishn
3139 The Mahabharat WarStart of Brihadrath dynasty of MagadhStart of Yudhisthir dynasty of Hastinapur
3102 Ascension of Bhagwan Krishn, start of kaliyug
2139 End of Brihadrath dynasty
2139-2001 Pradyot dynasty
2001-1641 Shishunag dynasty
1894-1814 Gautam Buddh
1641-1541 Nandas
1541-1241 Maurya dynasty
1541-1507 Chandragupt Maurya
1507-1479 Bindusar
1479-1443 Ashokvardhan
1241-784 Shung and Kanau dynasty
784-328 Andhra dynasty
509-477 Jagadguru Shankaracharya
328-83 Gupt dynasty
328-321 Chandragupt Vijayaditya
326 Alexander’s invasion
321-270 Samudragupt Ashokaditya Priyadarshin, orAshok the Great
102BC-15AD Vikramaditya, established Vikram era in 57 BC

25-85 Shalivahan, established Shalivahan Shak era in 78 AD
85-1192 There were several kingdoms of Rajpoot kings all overIndia. They ruled for 1,107 years.
1192-1757 In 1192, Mohammad Gori invaded Delhi (Hastinapur) thesecond time, defeated and killed Prithviraj Chauhan, andbecame the king. Since then several dynasties ofMuslims ruled India for 565 years.
1757-1947 In 1757 English regime was established in Bengal.British ruled India for 190 years.
1947 India got Independence

3228 BC – Descension of Bhagwan Krishn
According to the “Surya Siddhant” the astrologers have calculated that kaliyug started on the afternoon of 17th February, 3102 BC. In the Bhagwatam, Brahma tells in round figures that Krishn remained on this earth planet for 125 years
Accordingly, if 125 years is added to February, 3102, it comes to February 3227 BC. But Krishn’s descension was in the Rohini nakc√Ěhatra (asterism) of the 8th waning moon midnight of bhadon (August) which is about seven months earlier. Thus, His descension date is 3228 BC and He stayed on the earth planet for 125 years and about 7 months.

3139 BC – The Mahabharat War
The Pandavas, after winning the Mahabharat war, ruled for 36 years and 8 months until the start of kaliyug in 3102 BC. Accordingly, the date of Mahabharat war comes to 3139 BC.

3102 BC – Beginning of Kaliyug
The famous Aihole inscription of glorious Chalulaya King Pulkeshi II of the 7th century says,
It means, “3,735 (30 + 3,000 + 700 + 5) years have already elapsed in kaliyug
after the Mahabharat war, and 556 (50 + 6 + 500) years of Shalivahan era is running (on this date of engraving this inscription).” The inscription says that 3,735 years of kaliyug had already elapsed. It means the 3,736th year of Kali era was running in the Shak era 556 AD which was 556 + 78 = 634 AD. Thus, deducting 634 from 3,736 comes to 3102 BC. (For more evidences read Mahabharat - 3139 BC).

3139–83 BC - The Magadh Dynasties
The authority of the Bhagwatam is taken to determine the dates of the dynasties of the kings of Magadh up to the Andhra dynasty. There were nine dynasties that ruled Magadh after the Mahabharat war (3139 BC). They were: 21 kings in Brihadrath dynasty (1,000 years), 5 in Pradyot (138 years), 10 in Shishunag (360 years), one King Mahapadm Nand along with his 8 sons (100 years), 10 Maurya (137 years), 10 Shung and 4 Kanva (457 years), and 30 kings of Andhra dynasty for 456 years (Bhagwatam 9/22/46-49, 12/1/1-28). The ninth one is Gupt dynasty. There were seven kings in the Gupt dynasty with total reigning period of 245 years (Kaliyug Rajvrittant, Bhavishyottar Puran).
The reigning period of 10 Maurya kings, shown only for 137 years, appears to be much less as compared to the reigning period of the other kings of Magadh. Apart from the Mauryas, 81 kings in the other seven dynasties ruled for 2,511 years which give an average of 31 years per king. It appears to be a copying mistake while writing the period of Maurya dynasty. Probably, instead of 317 it was mistakenly written as 137, because the Kaliyug Rajvrittant gives the figure of 316 years for 10 Maurya kings. However, 300 years in round figures for the Maurya dynasty has been taken.

1894–1814 BC – Gautam Buddh
According to the Buddhist records, Gautam Buddh was already 72 years old at the time of Ajatshatnu’s coronation; it means that he was in his 73rd year at that time. Shishunag dynasty’s period is 2001 BC to 1641 BC. The first five kings of Shishunag dynasty, Shishunag to Bindusar, ruled for (40+36+26+40+38) 180 years. Then Ajatshatnu became the king and ruled for 27 years. Accordingly, 2001 BC (-) 180 = 1821 BC is the coronation year of Ajatshatnu. Adding 73 years (the existing age of Gautam Buddh at that time) to 1821 BC comes to 1894 BC. Thus, the date of birth of Gautam Buddh is 1894 BC and his nirvan year is (1894-80) 1814 BC. He was born on Vaishakh full moon day which is March/April.

1541–1507 BC – Chandragupt Maurya
Magadh was the fourth dynasty after the Mahabharat war (3139 BC). Chandragupt Maurya was the first king of the Maurya dynasty. His mother’s name was Mur, so he was called Maurya in Sanskrit which means the son of Mur, and thus, his dynasty was called Maurya dynasty. A pious, learned and determined brahman, Chanakya, also known as Kautilya, who didn’t have a pleasant appearance but had an intelligent brain, managed to terminate the existing King Mahapadm Nand and his eight sons and made Chandragupt the King of Magadh who was also the legitimate heir of the throne. The total period of the four dynasties including the Nand dynasty after the Mahabharat war is 1,598 years (1,000 + 138 + 360 + 100). Thus, the coronation date of Chandragupt Maurya comes to 3139 - 1598 = 1541 BC. Chandragupt Maurya ruled for 34 years (1541-1507 BC), his son Bindusar ruled 28 years (1507-1479 BC) and his grandson Ashokvardhan ruled for 36 years (1479-1443 BC).

509–477 BC – Jagadguru Shankaracharya
The most efficient documentary evidence of Shankaracharya’s period is the carefully preserved date-wise list of all the succeeding Shankaracharyas who sat on that religious throne which was established by Adi (the original) Shankaracharya thousands of years ago; and that list goes back up to 477 BC. Adi Shankaracharya lived only 32 years so his birth date is 477 + 32 = 509 BC. He had established four maths. (Math is a religious throne, which is used as a center for propagating dharm, and whoever sits on that throne, holds the title of Shankaracharya.) In his last days, Adi Shankaracharya lived in Kanchi Kamkoti so it is also considered as a math. Dwarika Sharda Math and Kanchi Kamkoti Math both have the complete date-wise record of all the succeeding Shankaracharyas for the last 2,500 years, but the records of Kanchi Math are more detailed.Shankaracharya, after establishing the four maths and spreading the greatness of Sanatan Dharm, came back to South India and, for the last four to six years of his life, he lived in Kanchi Kamkoti. Thus, Kanchi Kamkoti is also called the fifth math. Thus, according to the records of Kanchi Kamkoti Math, Adi Shankaracharya was born on 2593 Kali era and left this earth planet on 2625 Kali era which comes to (3102 - 2593) 509 BC and (3102 - 2625) 477 BC.

328–83 BC – Gupt Dynasty
The Gupt dynasty was the ninth Magadh dynasty. There were seven kings in the Gupt dynasty: (1) Chandragupt Vijayaditya (ruling period 7 years), (2) Samudragupt Ashokaditya Priyadarshin or Ashok the Great (51 years), (3) Chandragupt II Vikramaditya (36 years), (4) Kumargupt Mahendraditya (42 years), (5) Skandgupt Parakramaditya (25 years), (6) Nrasinghgupt Baladitya (40 years) and (7) Kumargupt II Vikramaditya (44 years). The total reigning period was 245 years. After the downfall of Gupt dynasty the kingship of Magadh ended and it went under the subordination of Vikramaditya of Ujjain (Malva).

102 BC–15 AD – Vikramaditya and Start of Vikram era in 57 BC
Vikram era started in 57 BC by Vikramaditya the Great as a commemoration of his victory upon the Shaks. There is plentiful literature on Vikramaditya, and in the Bhavishya Puran itself there are descriptions of Vikramaditya in more than 40 chapters between Pratisarg Parv I and IV. Pratisarg Parv IV, chapter 1 of Bhavishya Puran says that after the elapse of a full 3,000 years in kaliyug (3102 - 3000 = 102 BC), a dynamic Divine personality was born who was named Vikramaditya. Bhavishya Puran further says that the great King Vikramaditya ruled for one hundred years. When he was only five years old he went into the jungles to worship God. After twelve years, when he came out, God Shiv sent for him a celestial golden throne which was decorated with thirty-two statues. According to the above descriptions Vikramaditya lived for (5 years + 12 years + 100 years) 117 years (102 BC - 15 AD).Kalidas, the greatest poet, writer and the literary figure of his time, living a pious life and sincerely devoted to his scholarly work, was one of the nine gems of King Vikram’s court. The “Jyotirvidabharnam” by Kalidas tells in its first chapter
that Vikram era started at the elapse of (agni 3, ambar 0, yug 4 and ved 4 = 3,0,4,4) 3,044 years of kaliyug. Thus, the 3,045th year of kaliyug was the beginning of Vikram era which is 57 BC. At the end of Jyotirvidabharnam, Mahakavi Kalidas mentions the exact date of his writing and says that
in the Kali era 3067 he had started to write this book. It means, he wrote that book when 3,067 years of kaliyug had passed. That comes to 35 BC (3102 - 3067 = 35), which is after the beginning of Vikram era.Thus, Vikramaditya was born in 102 BC (3102-3000), established his ‘era’ in 57 BC and left this earth planet in 15 AD.

25–85 AD – Shalivahan and Start of Shalivahan Shak era in 78 AD
Detail of the kings of Pramar dynasty up to King Bhojraj as given in the Bhavishya Puran (Pratisarg Parv IV chapter one) states Vikramaditya, son of Gandharv Sen, reigned up to 15 AD. Vikramaditya’s son Deobhakt ruled from 15-25 AD and his grandson, Shalivahan, ruled from 25-85 AD. Shalivahan established his era in 78 AD. It is called the Shalivahan Shak era.

Monday, 11 December, 2006

The Ancient Katapayadi Formula And The Modern Hashing Method


The essence of the modern hashing technique in Computer Science is the derivation of a number from a nonnumeric key to index into a table where the record containing the key is stored. It is believed that the idea of hashing was first seriously considered by H.P.Luhn of IBM in 1953.In this paper, an interestingly similar technique used in South Indian musicology in the 18th century is described and the question of whether it is an anticipation of the Hashing technique is briey addressed.

1 Introduction

The problem of retrieving a record from a table based upon a given key has been studied extensively. A survey of work in this area can be found in (Sev-erance 1974). In this paper I describe one particular approach to this problem Hashing, and also an interesting earlier development very similar to it. Itis generally believed that the idea of hashing was originated by H.P.Luhn, in an internal IBM memorandum in 1953 (Knuth 1973) and described first in the open literature by Arnold Dumey (Dumey 1956). But is it possible that the Katapayadi scheme of deriving numbers from names, in conjunction with the applications to which it had been put, especially in classical South Indian musicology, is an early anticipation of the hashing technique? We will look at this issue in more detail here.

2 Hashing

A hash-table is a data structure in which it takes on average constant time to find any given element. This constant time is the time taken to compute a function called the hash-function of the element being searched for. This is in contrast to a Binary Search Tree data structure, for example, in which the time taken to find an element is on average proportional to the log2N, an array or linked linear list data structures in which the time is proportional to N where N is the total number of elements. The following example illustrates the use of hashing where the marks of ten students need to be stored in a table. It is a trivial one, but it is sufficient to bring out the essential principle behind hashing.

2.1 Example

Examination marks for ten students Amy, Ben, Cho, Dan, Eva, Fan, Gus, Hal,Ian and Jim need to be stored in a table. We might additionally want to retrieve the mark of a student on demand, and optionally modify it. One way of doing this is to store the marks sequentially in a table of size 10, and perform a sequential search on it each time we want to retrieve a particular record. This would mean that on average, we can expect to scan half the table (5 elements)before ending the desired record. A more anecient storage technique will be to store the elements in sorted order by name. In this case, we would expect to search the table log210 (approximately 3.2) times on average for each retrieval, because at each examination, our search space is efectively halved as the element we want is either current, in the upper half or lower half depending on whether it is equal to, less than or greater than the current element.

In contrast to these techniques, the hashing scheme derives a unique number corresponding to each name which gives us the cell address of the element in the table. If we used a hash function H(x) = (ascii(x[0])-5) % 10 + 1, where x is the name or value being hashed, x[0] is the first letter of that name, ascii() is a function that returns the ASCII value of a given letter, and % stood for the modulus or remainder operator, then the following arrangement of elements inthe table would be seen.

Addr 0 1 2 3 4 5 6 7 8 9

Name Amy Ben Cho Dan Eva Fan Mark Gus Hal Ian Jim

To retrieve an element, we would not have to scan any part of the table, but go directly to the record’s location by computing its hash value. For example,If Eva wants to know what her mark was, since ascii(‘E’) = 69, we compute(69-5) % 10 which gives 4, the location of Eva’s record in the table.

Of course, there are other important considerations, such as the number of elements that can be stored at any given table location (called a bucket), and how to accommodate overflows and handle collisions (two or more elements withthe same hash value). It has been pointed out to me by a reviewer of this paper that such considerations are equally important as the derivation of the index.But it can be argued that these are secondary in nature given the motivation ofthe hashing technique. Its essence can be said to be the derivation of a number from a given key, which is then subsequently used to index into an array where the element is stored with the purpose of eliminating a scan of any part of the array.

3 The Katapayadi Scheme

In classical India, letters of the Sanskrit alphabet were initially used to represent numbers. The grammarian Panini (4th or 5th century BC) who is believed to have written the first generative grammar for a natural language(Asher 1994) assigned the values 1 through 9 and 0 to the Sanskrit vowels a, i, u, etc. For example,Sutra(rule) v.i.30 of his grammar,Ashtadhyayi, is marked with the letter i, which indicates that the rule applies to the next tworules (Datta and Singh 1962, p.63). It is also known that various synonymsfor the number words existed. In one system, words with meanings evoca-tive of the numbers they represented were used. For example, the words indu(moon), dhara (earth) etc. stood for the number one since there was onlyone of each, netra (eyes), paksha (wings), etc. stood for two and so on. Amore comprehensive list of such synonyms can be found in (Ifrah 1985, p.446)who also gives the following instance of its use by Bhaskara I who in 629A.D. wrote the number 4,320,000 as vijadambaraqkaqa uqnjajamaraqmavedaorsky/atmostphere/space/void/primordial couple/Rama/Veda= 0000234. The term has been transliterated from the Sanskrit using the International PhoneticAlphabet. The palatal sibilant, commonly transcribed ass is represented usingconforming to the guidelines in (Halle and Clements 1983).

The Katapayadi scheme was initially just another such system of expressingnumbers through the use of letters (Sanskrit consonants in this case), withmore than one synonym for each number. The consonants themselves wereunevocative of the values they represented unlike the earlier scheme, but theynow possessed the powerful ability to form easily memorisable words throughthe insertion of vowels between them. Meaningful and mnemonic words couldnow be formed using these letters in much the same way as mnemonic wordsare coined today to represent commercial telephone numbers. In this sense,the Katapayadi scheme could be seen as just a mnemonic technique to helpremember numbers, or at best, a coding scheme like ASCII to derive numericvalues from non-numeric tokens, but it is noteworthy that the scheme continuedto be used long after the invention of numeric symbols and during this time was put to several applications. It is the application of the scheme to the particularinstance described in the next section which is remarkably similar to that of modern hashing.

The following Sanskrit verse describes one version of the Katapayadi scheme.(Fleet 1911) quotes this from C.M.Whish (Trans. Lit. Soc. of Madras, Part 1,p.57, 1827) who quotes this from an unspecied source, but (Datta and Singh1962) state that it is found inSadratnamala, which is a treatise on astronomypublished in 1823 by Prince Sankaravarman of Katattanat in North Malabar.The prince was an acquaintance of Mr Whish who spoke of him in high terms asa very intelligent man and acute mathematician" (Raja 1963).


Value 1 2 3 4 5 6 7 8 9 0

Velar and Palatal Stops k kh g gh 8 c ch , ,h 7

Retroex and dental stops P Ph h 9 t th d dh n

Labial stops p ph b bh m

Fricatives & Glides j r l v L s h

Table 1: The Katapayadi translation table

was published with a commentary in the Malayalam monthly Kavanodayam,vol.16, 1898, Calicut.

na7avaca ca unjani samk hja kaPapajaqdajah j

mi re tuqpaqnta hal samk hja na ca cintjo halasvarah k

(7 and n denote zeroes; the letters (in succession) beginning with k, P, p andj (the palatal glide, y in non-phonetic representation) denote the digits. In aconjoint consonant, only the last one denotes a number; and a consonant notjoined to a vowel should be disregarded)

There are said to be four variations of this scheme, which is claimed as thereason for its not coming into general use. The transcription scheme is moreeasily understood from the table 1. It lists the Sanskrit consonants, with theirassociated numeric values as specied in the verse. Each of the lines exceptthe last consists of stops in the following sequence - unvoiced and unaspirated,unvoiced and aspirated, voiced and unaspirated, voiced and aspirated, and nasal.In the rst line the velars are followed by the palatals and in the second line,the retroexes are followed by the dentals. The last line consists of fricatives.

The following interesting verse also appearing in Sadratnamala, illustratesan application of the scheme:

bhadrambudhisiddha,anmaga9ita raddhaqmajadbhuqpagih

If we translate this using the procedure described earlier in the verse aboutthe scheme, we get

bh= 4 (from table)

dr = 2 (only the last part of the conjoint consonant, r, is considered)

mb = 3 (similarly, only the b of mb is considered), etc.

This gives the nal value 423979853562951413. Since it is known that tra-ditional Indian practice was to write number words in ascending powers of10 (Ifrah 1985, p.445) (Menninger 1969, pp.398{399), the number representedabove, properly, is 314159265358979324 which is recognisable to be just thedigits of pi to 17 places (except that the last digit is incorrectit must be 3).

(Menninger 1969, p.275) also quotes an example1of the Indian name for the lu-nar cycle beinganantapura, which in addition to having semantic content itself,also gives the Katapayadi value 21600 (using the consonants n-n-t-p-r), whichis the number of minutes in the lunar half-month (15 25 60).

The originator of this scheme is not known, as with many other Indian inven-tions and discoveries, but it is believed that the scheme was probably familiarto the Indian mathematician and astronomer Aryabhata I in the 5th centuryA.D. and to Bhaskara I who lived in the 7th century A.D. (Sen 1971, p.175).The oldest datable text that employs the scheme isGrahacaranibandhana, writ-ten by Haridatta in 683 AD (Sarma 1972, pp.6{8). The scheme is said to havebeen used in a wide variety of contexts, including occultisms like numerology.A large number of South Indian chronograms have been composed using thisscheme (see for eg. Epigrahia Indica, 3: p.38, 4: pp.203{204, 11: pp.40{41,34: pp.205{206). It is also said that the Indian philosopher of the 7th Cen-tury, Sankara, was named such that the Katapayadi value of his name giveshis birthday215, indicating the fth day of the rst fortnight of the secondmonth in the Indian lunar calendar (Sambamurthy 1983). Not much else isknown about the status or application of this scheme since then. But in the18th century, we nd a novel revival of it in South Indian musicology which isarguably similar to modern hashing. This is described in the following section.

4 An Application of the Katapayadi Scheme

In classical South Indian music, the raga is roughly equivalent to the Western chord. These ragas are classied according to a unique scheme. What followsis a brief description of this classication as is pertinent to the subject of thispaper. A more comprehensive treatment of Indian musicology, its concepts andterms can be found in (Wade 1969).A raga can either be a Janaka(root) raga or a Janyaraga which is considered to be a descendant of one of the Janakaragas. The scale of a Janakaraga has seven notes in its ascent and the same seven notes in reverse in itsdescent. A Janyaraga is a modication of its parent Janakaraga through the insertion or deletion of one or more notes and/or possibly the re-ordering ofsome notes in either or both the ascent and descent of the scale. The seven notes are respectively called Sa (Shadjam), Ri (Rishabham), Ga (Gandharam),Ma (Madhyamam), Pa (Panchamam), Da (Dhaivatam) and Ni (Nishadam).These are the equivalents of the western solfa syllables Do, Re, Mi, Fa, So, La and Ti. The notes Sa and Pa (the fth) are considered xed, and must occur unchanged in all the Janakaragas. If we consider the octave to consist of the 12 notes C, C#, D, D#, E, F, F#, G, G#, A, A# and B, since C and G arexed, Ri and Ga can take any combination of two notes from C#, D, D# and E. Similarly Da and Ni can take any combination of two notes from G#, A,A# and B and Ma can take any of the two values F or F#. Thus there canbe a total of 24C24C2= 72 possible JanakaRagas. If we arrange these ragas systematically in a table, it is possible to derive the notes used by anyone of them from its index in the table. Accordingly, the table of 72 ragas is constructed as follows: The first 36 ragas in the table use F as the middle note Ma, and the second 36 use F#. In other respects they are identical. Each half of the table is further divided into 6 sections called Chakras, each of which has 6 ragas in it. Each of the 6 Chakras in each half use one of the 6 possible combi-nations of the notes Ri and Ga, while, within each Chakra, the notes Ri and Garemain constant, but Da and Ni take on each of their 6 possible combinations.Thus the arrangement in table 2 is arrived at.

This classication makes it easy for us to determine the notes of a raga givenits serial number in the table. For example if we were asked to play the scale ofraga number 65, we would know that it uses the note F# since 65 div 36+1=2.Since 65 % 36=29 and 29 div 6+1=5, we would know that it uses the fthpossible combination of Ri and Ga which is D & E. Also since 26 mod 6=5,we know it uses the fth possible combination of Da and Ni which is A and B.Thus the scale of Janaka Raga number 65 is: C, D, E, F#, G, A and B.

This means that given the name of a raga, one need only search for its raganumber. The notes can be mechanically derived from its number. However,an Indian raga has certain additional musical properties other than the notesit uses. Frequently, also, a janya raga which inherits some properties from itsjanaka raga is described in terms of the modications done to its parent whichresulted in that particular raga. These are usually discussed under a descriptionof the Janaka raga and its descendants, or in concise forms, given succinctlyalongside its name in a table. To get complete information about a JanakaRaga, then, a table search to nd its position given its name is presupposed.Things would be even simpler if we were able to derive the number of a ragadirectly from its name. This is precisely what was done by the South Indianmusicologists. Each raga was named in such a way that a Katapayadi translationof the rst two syllables of its name gives us its number in the table. Forexample, the raga Mechakalyani gives us the number 65 (derived from the rsttwo syllables Me and Cha) and Vanaspati gives 4. Thus it is now possible to godirectly to the raga’s position in its table from its name without having to do a search.

The exact person who coded the names of the ragas thus seems to be indispute, but it is fairly certain that such a codication was complete by the end ofthe 18th Century. (Aiyyangar 1972, p.189) states that although Venkatamakhilays a claim to this arrangement in 1660, it should really be credited to hisgrandson Muddu Venkatamakhi in the early 18th century who added it as asupplement to the former’s work Chaturdandi Prakasika.

5 Discussion

From an observation of the Katapayadi scheme, it seems that there are severalimportant dierences between it and modern hashing techniques. Notably, ahashing formula gives a valid bucket number for any given name, but the Kat-apayadi scheme only gives meaningful results for some names. For example, atrue hashing algorithm will never give a number greater than 72 in the aboveapplication, whatever the value hashed, but the Katapayadi scheme will.

A hashing algorithm can also take any input and return a number corre-sponding to its position in a table, whereas in the application of the Katapayadischeme above the names of the ragas have been carefully chosen for the purpose.Thus it seems more probable that the Katapayadi formula was intended as amnemonic technique to help people remember long numbers. Indeed, the versefromSadratnamalacoding the digits of pi seem to imply just that. In this sense,the scheme is an exact opposite of the modern hashing technique which aims toderive numbers from names, since it aims to derive names from numbers.

But then its application in South Indian musicology, where there are only72 admissible root ragas is clearly directed at liberating the tablelookup op-eration from the constraints imposed on it by the size of the table. This is thebasic aim of a hashing technique. A good hashing algorithm seeks to performthe operations of insertion, deletion and lookup with constant time complex-ity. The insert and delete operations are irrelevant to the application outlinedabove since the raga names were deliberately coined and already inserted intothe table. But once the table had been constructed, lookup took a constanttime because of the application of the Katapayadi scheme. The motivation forthis must have been similar as for a situation that warrants the applicationof a hashing strategy nowconstant time table lookup. The result too is thesame. Here, it is obvious that it bears a strong similarity to the modern hash-ing technique. To be sure, the Katapayadi scheme was initially developed asa mnemonic technique given the oral culture of education in early India. In-deed, Sir Monier Williams remarks that even the grammar of Panini was mainlyintended to aid the memory of teachers than learners by the briefest possiblesuggestions (Williams 1969). Nevertheless, it is possible for such a mnemonictechnique to gradually evolve into a scheme that bears strong similarity to ourmodern hashing technique. It is relatively easy for one to look at this particularapplication of the Katapayadi scheme and to come up with a hashing strategyfor some modern requirement. But whether the scheme actually inuenced laterdevelopment of the hashing technique is in doubt. It is not certain whether anyIndian scholar with knowledge of this technique was a close associate of any ofthe proponents of early hashing. Nor is it likely that the proponents of hashingknew about the Katapayadi technique. Thus the most we can say at this stageis that the Katapayadi scheme can be thought of as an early precursor to themodern hash functions and its application in South Indian musicology bears inretrospect an interesting similarity to modern hash tables.

Friday, 8 December, 2006

Ancient City Found, Irradiated From Atomic Blast

The radiation still so intense, the area is highly dangerous
A heavy layer of radioactive ash in Rajasthan, India, covers a three-square mile area, ten miles west of Jodhpur. Scientists are investigating the site, where a housing development was being built.
For some time it has been established that there is a very high rate of birth defects and cancer in the area under construction. The levels of radiation there have registered so high on investigators' gauges that the Indian government has now cordoned off the region. Scientists have unearthed an ancient city where evidence shows an atomic blast dating back thousands of years, from 8,000 to 12,000 years, destroyed most of the buildings and probably a half-million people. One researcher estimates that the nuclear bomb used was about the size of the ones dropped on Japan in 1945.
The Mahabharata clearly describes a catastrophic blast that rocked the continent. "A single projectile charged with all the power in the Universe...An incandescent column of smoke and flame as bright as 10,000 suns, rose in all its was an unknown weapon, an iron thunderbolt, a gigantic messenger of death which reduced to ashes an entire race.
"The corpses were so burned as to be unrecognizable. Their hair and nails fell out, pottery broke without any apparent cause, and the birds turned white.
"After a few hours, all foodstuffs were infected. To escape from this fire, the soldiers threw themselves into the river."

A Historian Comments
Historian Kisari Mohan Ganguli says that Indian sacred writings are full of such descriptions, which sound like an atomic blast as experienced in Hiroshima and Nagasaki. He says references mention fighting sky chariots and final weapons. An ancient battle is described in the Drona Parva, a section of the Mahabharata. "The passage tells of combat where explosions of final weapons decimate entire armies, causing crowds of warriors with steeds and elephants and weapons to be carried away as if they were dry leaves of trees," says Ganguli.
"Instead of mushroom clouds, the writer describes a perpendicular explosion with its billowing smoke clouds as consecutive openings of giant parasols. There are comments about the contamination of food and people's hair falling out."

Archelogical Investigation Provides Information
Archeologist Francis Taylor says that etchings in some nearby temples he has managed to translate suggest that they prayed to be spared from the great light that was coming to lay ruin to the city. "It's so mid-boggling to imagine that some civilization had nuclear technology before we did. The radioactive ash adds credibility to the ancient Indian records that describe atomic warfare."
Construction has halted while the five member team conducts the investigation. The foreman of the project is Lee Hundley, who pioneered the investigation after the high level of radiation was discovered.
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Other Account

Ancient Nuclear Warfare
“Anybody not wearing 2 million sunblock is gonna have a real bad day. Get it?” --Sarah Connor, Terminator 2
"Thank you, India." --Alanis Morissette
There is evidence that the Rama empire (now India) was devastated by nuclear war. The Indus valley is now the Thar desert, and the site of the radioactive ash found west of Jodhpur is around there.
Consider these verses from the ancient (6500 BC at the latest) Mahabharata:

...a single projectile Charged with all the power of the Universe. An incandescent column of smoke and flame As bright as the thousand suns Rose in all its splendour... a perpendicular explosion with its billowing smoke clouds... ...the cloud of smoke rising after its first explosion formed into expanding round circles like the opening of giant parasols... was an unknown weapon, An iron thunderbolt, A gigantic messenger of death, Which reduced to ashes The entire race of the Vrishnis and the Andhakas. ...The corpses were so burned As to be unrecognisable. The hair and nails fell out; Pottery broke without apparent cause, And the birds turned white.
After a few hours All foodstuffs were infected... escape from this fire The soldiers threw themselves in streams To wash themselves and their equipment.

Until the bombing of Hiroshima and Nagasaki, modern mankind could not imagine any weapon as horrible and devastating as those described in the ancient Indian texts. Yet they very accurately described the effects of an atomic explosion. Radioactive poisoning will make hair and nails fall out. Immersing oneself in water gives some respite, though it is not a cure.
When excavations of Harappa and Mohenjo-Daro reached the street level, they discovered skeletons scattered about the cities, many holding hands and sprawling in the streets as if some instant, horrible doom had taken place. People were just lying, unburied, in the streets of the city. And these skeletons are thousands of years old, even by traditional archaeological standards. What could cause such a thing? Why did the bodies not decay or get eaten by wild animals? Furthermore, there is no apparent cause of a physically violent death.
These skeletons are among the most radioactive ever found, on par with those at Hiroshima and Nagasaki. At one site, Soviet scholars found a skeleton which had a radioactive level 50 times greater than normal. Other cities have been found in northern India that show indications of explosions of great magnitude. One such city, found between the Ganges and the mountains of Rajmahal, seems to have been subjected to intense heat. Huge masses of walls and foundations of the ancient city are fused together, literally vitrified! And since there is no indication of a volcanic eruption at Mohenjo-Daro or at the other cities, the intense heat to melt clay vessels can only be explained by an atomic blast or some other unknown weapon. The cities were wiped out entirely.
While the skeletons have been carbon-dated to 2500 BC, we must keep in mind that carbon-dating involves measuring the amount of radiation left. When atomic explosions are involved, that makes then seem much younger.
Interestingly, Manhattan Project chief scientist Dr J. Robert Oppenheimer was known to be familiar with ancient Sanskrit literature. In an interview conducted after he watched the first atomic test, he quoted from the Bhagavad Gita: "'Now I am become Death, the Destroyer of Worlds.' I suppose we all felt that way." When asked in an interview at Rochester University seven years after the Alamogordo nuclear test whether that was the first atomic bomb ever to be detonated, his reply was, "Well, yes, in modern history."
Ancient cities whose brick and stonewalls have literally been vitrified, that is, fused together, can be found in India, Ireland, Scotland, France, Turkey and other places. There is no logical explanation for the vitrification of stone forts and cities, except from an atomic blast.
Another curious sign of an ancient nuclear war in India is a giant crater near Bombay. The nearly circular 2,154-metre-diameter Lonar crater, located 400 kilometres northeast of Bombay and aged at less than 50,000 years old, could be related to nuclear warfare of antiquity. No trace of any meteoric material, etc., has been found at the site or in the vicinity, and this is the world's only known "impact" crater in basalt. Indications of great shock (from a pressure exceeding 600,000 atmospheres) and intense, abrupt heat (indicated by basalt glass spherules) can be ascertained from the site.

Atlantis and Rama were not the only advanced civilizations in the world at that time. The Mediterranean was a large and fertile valley. This ancient civilization, pre-dating dynastic Egypt, was known as Osiris. The Nile river came out of Africa, as it does today, and was called the River Stix. However, instead of flowing into the Mediterranean Sea at the Nile Delta in northern Egypt, it continued into the valley, and then turned westward to flow in the deepest part of the Mediterranean Valley where it created a large lake and then flowed out between Malta and Sicily, and south of Sardinia into the Atlantic at Gibraltar (the Pillars of Hercules). When the sea level rose, it flooded the Mediterranean Basin, destroying the Osirians' great cities and forcing them to move to higher ground. This theory helps explain the strange megalithic remains found throughout the Mediterranean. It is an archaeological fact that there are more than 200 known sunken cities in the Mediterranean. That must be where the Pyramids came from.
Given that the ancients had nuclear capabilities, we know that they had the technology of 1945 at the earliest. Many things had to be discovered and invented before a nuclear explosion could take place. We do not know exactly what technology they had. However, since there are no people in space, we know they never colonized space, probably because they had no spacecraft. Even the wooden or plastic objects that weren't submerged would have disintegrated by now, so we have no idea what they had that was made out of those materials. But we do know that they used crystals. Aside from being used for clocks and radios, crystals can also be used to store information. A 1-cm cubic crystal could store 125 gigabytes, or as much as eight Pentium III processors. Maybe when we develop such technology (according to Moore's law, it will probably be by 2020), we will be able to access what they put on their crystal computers.
The Agastya Samhita, an ancient Indian text, gives directions on how to make a battery: "Place a well-cleaned copper plate in an earthenware vessel. Cover it first by copper sulfate and then moist sawdust. After that put a mercury-amalgamated-zinc sheet on top of an energy known by the twin name of Mitra-Varuna. Water will be split by this current into Pranavayu and Udanavayu. A chain of one hundred jars is said to give a very active and effective force." By the way, Mitra-Varuna is now called cathode-anode, and Pranavayu and Udanavayu are to us oxygen and hydrogen, respectively.
According to the Bible, anyone who touched the Ark of the Covenant, which contained the Ten commandments, would die. (Numbers 4:15) The only way touching an object can kill a person is if electricity is flowing through it.


Thursday, 7 December, 2006

The Cosmology of the Bhagavata Purana

The inquisitive human mind naturally yearns to understand the universe and man’s place within it. Today scientists rely on powerful telescopes and sophisticated computers to formulate cosmological theories. In former times, people got their information from traditional books of wisdom. Followers of India’s ancient culture, for example, learned about the cosmos from scriptures like the Srimad-Bhagavatam, or Bhagavata Purana. But the Bhagavatam’s descriptions of the universe often baffle modern students of Vedic literature. Here Bhaktivedanta Institute scientist Dr. Richard Thompson suggests a framework for understanding the Bhagavatam’s descriptions that squares with our experience and modern discoveries.

Jambudvipa: The Srimad-Bhagavatam describes that the universe lies within a series of spherical shells which is divided in two by an earth plane called Bhu-mandala. A series of dvipas, or ‘islands,’ and oceans make up Bhu-mandala. In the center of Bhu-mandala is the circular ‘island’ of Jambudvipa (inset), whose most prominent feature is the cone-shaped Mount Meru. The main illustration here shows a closer view of Jambudvipa and the base of Mount Meru.

The Srimad-Bhagavatam presents an earth-centered conception of the cosmos. At first glance the cosmology seems foreign, but a closer look reveals that not only does the cosmology of the Bhagavatam describe the world of our experience, but it also presents a much larger and more complete cosmological picture. I’ll explain.
The Srimad-Bhagavatam’s mode of presentation is very different from the familiar modern approach. Although the Bhagavatam’s "Earth" (disk-shaped Bhu-mandala) may look unrealistic, careful study shows that the Bhagavatam uses Bhu-mandala to represent at least four reasonable and consistent models: (1) a polar-projection map of the Earth globe, (2) a map of the solar system, (3) a topographical map of south-central Asia, and (4) a map of the celestial realm of the demigods.
Caitanya Mahaprabhu remarked, "In every verse of Srimad-Bhagavatam and in every syllable, there are various meanings." (Caitanya-caritamrita, Madhya 24.318) This appears to be true, in particular, of the cosmological section of the Bhagavatam, and it is interesting to see how we can bring out and clarify some of the meanings with reference to modern astronomy.

Figure 1

Figure 2

When one structure is used to represent several things in a composite map, there are bound to be contradictions. But these do not cause a problem if we understand the underlying intent. We can draw a parallel with medieval paintings portraying several parts of a story in one composition. For example, Masaccio’s painting "The Tribute Money" (Figure 1) shows Saint Peter in three parts of a Biblical story. We see him taking a coin from a fish, speaking to Jesus, and paying a tax collector. From a literal standpoint it is contradictory to have Saint Peter doing three things at once, yet each phase of the Biblical story makes sense in its own context.
A similar painting from India (Figure 2) shows three parts of a story about Krishna. Such paintings contain apparent contradictions, such as images of one character in different places, but a person who understands the story line will not be disturbed by this. The same is

true of the Bhagavatam, which uses one

model to represent
different features of the cosmos.

The Bhagavatam Picture at First Glance

The Fifth Canto of the Srimad-Bhagavatam tells of innumerable universes. Each one is contained in a spherical shell surrounded by layers of elemental matter that mark the boundary between mundane space and the unlimited spiritual world.
The region within the shell (Figure 3) is called the Brahmanda, or "Brahma egg." It contains an earth disk or plane—called Bhu-mandala—that divides it into an upper, heavenly half and a subterranean half, filled with water. Bhu-mandala is divided into a series of geographic features, traditionally called dvipas, or "islands," varshas, or "regions," and oceans.
In the center of Bhu-mandala (Figure 4) is the circular "island" of Jambudvipa, with nine varsha subdivisions. These include Bharata-varsha, which can be understood in one sense as India and in another as the total area inhabited by human beings. In the center of Jambudvipa stands the cone-shaped Sumeru Mountain, which represents the world axis and is surmounted by the city of Brahma, the universal creator.
To any modern, educated person, this sounds like science fiction. But is it? Let’s consider the four ways of seeing the Bhagavatam’s descriptions of the Bhu-mandala.

Figure 3

Figure 4

We begin by discussing the interpretation of Bhu-mandala as a planisphere, or a polar-projection map of the Earth globe. This is the first model given by the Bhagavatam. A stereographic projection is an ancient method of mapping points on the surface of a sphere to points on a plane. We can use this method to map a modern Earth globe onto a plane, and the resulting flat projection is called a planisphere (Figure 5). We can likewise view Bhu-mandala as a stereographic projection of a globe (Figure 6).

Figure 5

Figure 6

In India such globes exist. In the example shown here (Figure 7), the land area between the equator and the mountain arc is Bharata-varsha, corresponding to greater India. India is well represented, but apart from a few references to neighboring places, this globe does not give a realistic map of the Earth. Its purpose was astronomical, rather than geographical.

Figure 7

Although the Bhagavatam doesn’t explicitly describe the Earth as a globe, it does so indirectly. For example, it points out that night prevails diametrically opposite to a point where it is day. Likewise, the sun sets at a point opposite where it rises. Therefore, the Bhagavatam does not present the naive view that the Earth is flat.

We can compare Bhu-mandala with an astronomical instrument called an astrolabe, popular in the Middle Ages. On the astrolabe, an off-centered circle represents the orbit of the sun—the ecliptic. The Earth is represented in stereographic projection on a flat plate, called the mater. The ecliptic circle and important stars are represented on another plate, called the rete. Different planetary orbits could likewise be represented by different plates, and these would be seen projected onto the Earth plate when one looks down on the instrument.
The Bhagavatam similarly presents the orbits of the sun, the moon, planets, and important stars on a series of planes parallel to Bhu-mandala.
Seeing Bhu-mandala as a polar projection is one example of how it doesn’t represent a flat Earth.

Bhu-mandala as a Map of the Solar System

Here’s another way to look at Bhu-mandala that also shows that it’s not a flat-Earth model.
Descriptions of Bhu-mandala have features that identify it as a model of the solar system. In the previous section I interpreted Bhu-mandala as a planisphere map. But now, we’ll take it as a literal plane. When we do this, it looks at first like we’re back to the naive flat Earth, with the bowl of the sky above and the underworld below.
The scholars Giorgio de Santillana and Hertha von Dechend carried out an intensive study of myths and traditions and concluded that the so-called flat Earth of ancient times originally represented the plane of the ecliptic (the orbit of the sun) and not the Earth on which we stand. Later on, according to de Santillana and von Dechend, the original cosmic understanding of the earth was apparently lost, and the Earth beneath our feet was taken literally as a flat plate. In India, the earth of the Puranas has often been taken as literally flat. But the details given in the Bhagavatam show that its cosmology is much more sophisticated.
Not only does the Bhagavatam use the ecliptic model, but it turns out that the disk of Bhu-mandala corresponds in some detail to the solar system (Figure 8). The solar system is nearly flat. The sun, the moon, and the five traditionally known planets—Mercury through Saturn—all orbit nearly in the ecliptic plane. Thus Bhu-mandala does refer to something flat, but it’s not the Earth.

Figure 8

One striking feature of the Bhagavatam’s descriptions has to do with size. If we compare Bhu-mandala with the Earth, the solar system out to Saturn, and the Milky Way galaxy, Bhu-mandala matches the solar system closely, while radically differing in size from Earth and the galaxy.
Furthermore, the structures of Bhu-mandala correspond with the planetary orbits of the solar system (Figure 9).

Figure 9

Figure 10
If we compare the rings of Bhu-mandala with the orbits of Mercury, Venus (Figure 10), Mars, Jupiter, and Saturn, we find several close alignments that give weight to the hypothesis that Bhu-mandala was deliberately designed as a map of the solar system.
Until recent times, astronomers generally underestimated the distance from the earth to the sun. In particular, Claudius Ptolemy, the greatest astronomer of classical antiquity, seriously underestimated the Earth-sun distance and the size of the solar system. It is remarkable, therefore, that the dimensions of Bhu-mandala in the Bhagavatam are consistent with modern data on the size of the sun’s orbit and the solar system as a whole.
[See BTG, Nov./Dec. 1997.]

Jambudvipa as a Topographical Map of South-Central Asia

Jambudvipa, the central hub of Bhu-mandala, can be understood as a local topographical map of part of south-central Asia. This is the third of the four interpretations of Bhu-mandala. In the planisphere interpretation, Jambudvipa represents the northern hemisphere of the Earth globe. But the detailed geographic features of Jambudvipa do not match the geography of the northern hemisphere. They do, however, match part of the Earth.

Figure 11
Six horizontal and two vertical mountain chains divide Jambudvipa into nine regions, or varshas (Figure 11). The southernmost region is called Bharata-varsha. Careful study shows that this map corresponds to India plus adjoining areas of south-central Asia. The first step in making this identification is to observe that the Bhagavatam assigns many rivers in India to Bharata-varsha. Thus Bharata-varsha represents India. The same can be said of many mountains in Bharata-varsha. In particular, the Bhagavatam places the Himalayas to the north of Bharata-varsha in Jambudvipa (Figure 11).
A detailed study of Puranic accounts allows the other mountain ranges of Jambudvipa to be identified with mountain ranges in the region north of India. Although this region includes some of the most desolate and mountainous country in the world, it was nonetheless important in ancient times. For example, the famous Silk Road passes through this region. The Pamir mountains can be identified with Mount Meru and Ilavrita-varsha, the square region in the center of Jambudvipa. (Note that Mount Meru does not represent the polar axis in this interpretation.)
Other Puranas give more geographical details that support this interpretation.

Bhu-mandala as a Map of the Celestial Realm of the Devas

We can also understand Bhu-mandala as a map of the celestial realm of the demigods, or devas. One curious feature of Jambudvipa is that the Bhagavatam describes all of the varshas other than Bharata-varsha as heavenly realms, where the inhabitants live for ten thousand years without suffering. This has led some scholars to suppose that Indians used to imagine foreign lands as celestial paradises. But the Bhagavatam does refer to barbaric peoples outside India, such as Huns, Greeks, Turks, and Mongolians, who were hardly thought to live in paradise. One way around this is to suppose that Bharata-varsha includes the entire Earth globe, while the other eight varshas refer to celestial realms outside the Earth. This is a common understanding in India.
But the simplest explanation for the heavenly features of Jambudvipa is that Bhu-mandala was also intended to represent the realm of the devas. Like the other interpretations we have considered, this one is based on a group of mutually consistent points in the cosmology of the Bhagavatam.
First of all, consider the very large sizes of mountains and land areas in Jambudvipa. For example, India is said to be 72,000 miles (9,000 yojanas) from north to south, or nearly three times the circumference of the Earth. Likewise, the Himalayas are said to be 80,000 miles high.

Figure 12

People in India in ancient times used to go in pilgrimage on foot from one end of India to the other, so they knew how large India is. Why does the Bhagavatam give such unrealistic distances? The answer is that Jambudvipa doubles as a model of the heavenly realm, in which everything is on a superhuman scale. The Bhagavatam portrays the demigods and other divine beings that inhabit this realm to be correspondingly large. Figure 12 shows Lord Siva in comparison with Europe, according to one text of the Bhagavatam.

Figure 13
Why would the Bhagavatam describe Jambudvipa as both part of the earth and part of the celestial realm? Because there’s a connection between the two. To understand, let’s consider the idea of parallel worlds. By siddhis, or mystic perfections, one can take shortcuts across space. This is illustrated by a story from the Bhagavatam in which the mystic yogini Citralekha abducts Aniruddha from his bed in Dvaraka and transports him mystically to a distant city (Figure 13).
Besides moving from one place to another in ordinary space, the mystic siddhis enable one to travel in the all-pervading ether or to enter another continuum. The classical example of a parallel continuum is Krishna’s transcendental realm of Vrindavana, said to be unlimitedly expansive and to exist in parallel to the finite, earthly Vrindavana in India.

Figure 14

The Sanskrit literature abounds with stories of parallel worlds. For example, the Mahabharata tells the story of how the Naga princess Ulupi abducted Arjuna while he was bathing in the Ganges River (Figure 14). Ulupi pulled Arjuna down not to the riverbed, as we would expect, but into the kingdom of the Nagas (celestial snakelike beings), which exists in another dimension.
Mystical travel explains how the worlds of the devas are connected with our world. In particular, it explains how Jambudvipa, as a celestial realm of devas, is connected with Jambudvipa as the Earth or part of the Earth. Thus the double model of Jambudvipa makes sense in terms of the Puranic understanding of the siddhis.

Concluding Observations:
The Vertical Dimension in Bhagavata Cosmology

For centuries the cosmology of the Bhagavatam has seemed incomprehensible to most observers, encouraging many people either to summarily reject it or to accept it literally with unquestioning faith. If we take it literally, the cosmology of the Bhagavatam not only differs from modern astronomy, but, more important, it also suffers from internal contradictions and violations of common sense. These very contradictions, however, point the way to a different understanding of Bhagavata cosmology in which it emerges as a deep and scientifically sophisticated system of thought. The contradictions show that they are caused by overlapping self-consistent interpretations that use the same textual elements to expound different ideas.
Each of the four interpretations I’ve presented deserves to be taken seriously because each is supported by many points in the text that are consistent with one another while agreeing with modern astronomy. I’ve applied the context-sensitive or multiple-aspect approach, in which the same subject has different meanings in different contexts. This approach allows for the greatest amount of information to be stored in a picture or text, reducing the work required by the artist or writer. At the same time, it means that the work cannot be taken literally as a one-to-one model of reality, and it requires the viewer or reader to understand the different relevant contexts. This can be difficult when knowledge of context is lost over long periods of time.
In the Bhavagatam, the context-sensitive approach was rendered particularly appropriate by the conviction that reality, in the ultimate issue, is avak-manasam, or beyond the reach of the mundane mind or words. This implies that a literal, one-to-one model of reality is unattainable, and so one may as well pack as much meaning as possible into a necessarily incomplete description of the universe. The cosmology of the Bhagavata Purana is a sophisticated system of thought, with multiple layers of meaning, both physical and metaphysical. It combines practical understanding of astronomy with spiritual conceptions to produce a meaningful picture of the universe and reality.
Richard L. Thompson earned his Ph.D. in mathematics from Cornell University. He is the author of several books, of which Mysteries of the Sacred Universe is the most recent.