The Katapayadi Scheme - Mathomathis Mathomathis: Connect God By Mathematical Science

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 two rules (Datta and Singh 1962, p.63).

It is also known that various synonyms for the number words existed. In one system, words with meanings evocative of the numbers they represented were used.For example, the words indu(moon), dhara (earth) etc. stood for the number one since there was only one of each, netra (eyes), paksha (wings), etc. stood for two and so on.

A more 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


vijadambaraqkaqa uqnjajamaraqmavedaorsky/atmostphere/space/void/primordial couple/Rama/Veda = 0000234.

The term has been transliterated from the Sanskrit using the International Phonetic alphabet. The palatal sibilant, commonly transcribed ass is represented using conforming to the guidelines in (Halle and Clements 1983).The Katapayadi scheme was initially just another such system of expressing numbers through the use of letters (Sanskrit consonants in this case), with more than one synonym for each number. The consonants themselves were evocative of the values they represented unlike the earlier scheme, but they now possessed the powerful ability to form easily memorialize words through the insertion of vowels between them. Meaningful and mnemonic words could now be formed using these letters in much the same way as mnemonic words are coined today to represent commercial telephone numbers.

In this sense,the Katapayadi scheme could be seen as just a mnemonic technique to help remember numbers, or at best, a coding scheme like ASCII to derive numeric values from non-numeric tokens, but it is noteworthy that the scheme continued to 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 particular instance 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 unspecified source, but (Datta and Singh1962) state that it is found in Sadratnamala, which is a treatise on astronomy published 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 as a very intelligent man and acute mathematician” (Raja 1963).
Sadratnamala

 
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 and j (the palatal glide, y in non-phonetic representation) denote the digits. In a conjoint consonant, only the last one denotes a number; and a consonant not joined to a vowel should be disregarded)There are said to be four variations of this scheme, which is claimed as the reason for its not coming into general use. The transcription scheme is more easily understood from the table 1. It lists the Sanskrit consonants, with their associated numeric values as specified in the verse. Each of the lines except the last consists of stops in the following sequence – unvoiced and unaspirated,unvoiced and aspirated, voiced and un - aspirated, voiced and aspirated, and nasal.
In the rst line the velars are followed by the palatals and in the second line,the retro exes are followed by the dentals. The last line consists of fricatives.
The following interesting verse also appearing in Sadratnamala, illustrates an application of the scheme:

 
"bhadrambudhisiddha,anmaga9ita raddhaqmajadbhuqpagih"
If we translate this using the procedure described earlier in the verse about the 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 traditional Indian practice was to write number words in ascending powers of10 (Ifrah 1985, p.445) (Menninger 1969, pp.398{399), the number represented above, properly, is 314159265358979324 which is recognizable to be just the digits of pi to 17 places (except that the last digit is incorrect it must be 3).

 
(Menninger 1969, p.275) also quotes an example1of the Indian name for the lunar cycle being anantapura, which in addition to having semantic content itself,also gives the Katapayadi value 21600 (using the consonants n-n-t-p-r), which is 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 inventions and discoveries, but it is believed that the scheme was probably familiar to the Indian mathematician and astronomer Aryabhata I in the 5th century A.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 is Grahacaranibandhana, writ-ten by Haridatta in 683 AD (Sarma 1972, pp.6{8).

 
The scheme is said to have been used in a wide variety of contexts, including occultism like numerology.A large number of South Indian chronograms have been composed using this scheme (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 Century, Sankara, was named such that the Katapayadi value of his name gives his birthday215, indicating the 5th day of the 1st fortnight of the second month in the Indian lunar calendar (Sambamurthy 1983).

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