ûû §&rûm ûûûû §&rûm ûû

Fascinating conventions in Vedic Fascinating conventions in Vedic scriptures: Concept of Variablesscriptures: Concept of Variables

(Talk at MNIT, Jaipur Dec. 15-16, (Talk at MNIT, Jaipur Dec. 15-16, 2007)2007)

Prof. Hansraj P JoshiProf. Hansraj P Joshi

Department of Mathematics and Department of Mathematics and StatisticsStatistics

York University, Toronto, Canada.York University, Toronto, Canada.

Introduction:-Introduction:- In Vedic period, knowledge was passed In Vedic period, knowledge was passed

on orally, and has survived as ’Shruti’. on orally, and has survived as ’Shruti’.

Our Rishi had to think of various ways of Our Rishi had to think of various ways of memorizing and teaching their pupilsmemorizing and teaching their pupils

I think one of the most innovative idea I think one of the most innovative idea they came up with was to weave they came up with was to weave (imbed) Scientific concepts in daily (imbed) Scientific concepts in daily prayers and daily ritualistic (religious) prayers and daily ritualistic (religious) activities. activities.

Result is that various concepts that Result is that various concepts that have survived are to be found in the have survived are to be found in the religion we practice. religion we practice.

A well known example is the Slok A well known example is the Slok

Ø pUïR md¶ pUïRimdm Ø pUïR md¶ pUïRimdm pUï·t pUïRmudCyte pUï·t pUïRmudCyte pUïRSy pUïRm·d·y pUïRSy pUïRm·d·y pUïRmev·vixWytepUïRmev·vixWyte

Purn madah, Purnam-idam Purnaat Purn-Purn madah, Purnam-idam Purnaat Purn-mudachchyate,mudachchyate,

Purnsy Purn MaadaayPurn-meva-vashishyate.Purnsy Purn MaadaayPurn-meva-vashishyate.

in which the concept of Zero and the in which the concept of Zero and the concept of Infinity, both are recited in our concept of Infinity, both are recited in our daily prayer.daily prayer.

We have lost most of what did not We have lost most of what did not become part of our daily religious practice. become part of our daily religious practice. I think we have lost a lot more than we I think we have lost a lot more than we know. know.

In this discussions we shall see some In this discussions we shall see some examples and interpretations of what we examples and interpretations of what we have in our existing scriptures and can have in our existing scriptures and can feel proud of our heritage. feel proud of our heritage.

I do not wish to go into murky field of I do not wish to go into murky field of whether we were the first or some other whether we were the first or some other civilization had an idea before us. civilization had an idea before us.

Personally I am convinced that that our Personally I am convinced that that our Vedic culture is more than 8000 years old Vedic culture is more than 8000 years old and therefore my ancestors were the first and therefore my ancestors were the first scientists on this planet, but I am not scientists on this planet, but I am not qualified to prove it. qualified to prove it.

I believe Paanini did not ‘create’ Sanskrit I believe Paanini did not ‘create’ Sanskrit grammar, but he analyzed then known grammar, but he analyzed then known Shrutis (scriptures) and wrote the ‘rules’ Shrutis (scriptures) and wrote the ‘rules’ as he understood them. Sanskrit literature as he understood them. Sanskrit literature existed many thousand years before existed many thousand years before Paanini’s time. (700 BC). Paanini’s time. (700 BC).

I enjoy discovering what is hidden in our I enjoy discovering what is hidden in our scriptures just for ‘Swaantah Sukhaay’ scriptures just for ‘Swaantah Sukhaay’

Background:Background:

Much has been written trying to prove to the world Much has been written trying to prove to the world that Geometry and Astronomy (and hence I say that Geometry and Astronomy (and hence I say some Mathematics) originated in Vedic period. some Mathematics) originated in Vedic period.

I am happy that some honest intellectuals are I am happy that some honest intellectuals are convinced but there are many more skeptics. convinced but there are many more skeptics.

History credits Thales as creator of geometry and his History credits Thales as creator of geometry and his student Pythagoras (ca. 580-500 B.C.) as ‘organizer’ student Pythagoras (ca. 580-500 B.C.) as ‘organizer’ of geometry. Pythagoras did live in India for some of geometry. Pythagoras did live in India for some time, went back to Greece, and then formally wrote time, went back to Greece, and then formally wrote ‘proofs’. Thales was in Egypt for some time and ‘proofs’. Thales was in Egypt for some time and traveled extensively; possibly to India as well. traveled extensively; possibly to India as well.

Democritus (ca. 460-370 B.C), a founder of Atomic Democritus (ca. 460-370 B.C), a founder of Atomic theory of matter and follower of Pythagoras also theory of matter and follower of Pythagoras also came to India. Therefore, I believe Greeks were came to India. Therefore, I believe Greeks were aware of and learned geometry from religious aware of and learned geometry from religious practices in Indiapractices in India

The modern idea of Rank ( or hierarchy or The modern idea of Rank ( or hierarchy or well orderedness) was and still is part of well orderedness) was and still is part of our religious practice. our religious practice.

For example the statements For example the statements (a) (a) Gurudevobhav, Gurudevobhav,

Maatrudevobhavah, Pitrudevobhav Maatrudevobhavah, Pitrudevobhav and Atithee Devobhav. and Atithee Devobhav. (b)(b) Prithavi, Prithavi, Jal, Tej, Vaayu and AakaashJal, Tej, Vaayu and Aakaash. This . This specific order conveys proper scientific specific order conveys proper scientific and logically connected sequential and logically connected sequential meaningmeaning

Reading history of Mathematics we Reading history of Mathematics we understand that Greek philosopher Thales understand that Greek philosopher Thales (ca. 625-547 B.C), had idea of whole (ca. 625-547 B.C), had idea of whole numbers, the smallest being numeral 2. numbers, the smallest being numeral 2. They had no concept of one or zero or They had no concept of one or zero or fractionsfractions. .

Whereas in Yajurvedeey Ashtaadhyaa-ee Whereas in Yajurvedeey Ashtaadhyaa-ee Rudree, chapter 8, we find not only concept Rudree, chapter 8, we find not only concept

of integers, but in Slok 24 we have Odd of integers, but in Slok 24 we have Odd Integers 1 through 33Integers 1 through 33

Slok 25 has even integers (as multiples Slok 25 has even integers (as multiples of 4)of 4)

and in Slok 26, Rishi is asking for various and in Slok 26, Rishi is asking for various halveshalves

So in Yajurved, we have the rudiments So in Yajurved, we have the rudiments of concepts of addition, subtraction, of concepts of addition, subtraction, multiplication and square (cube) rootsmultiplication and square (cube) roots..

1.1. ..

In Rgaved itself (X. 62.7) which refers In Rgaved itself (X. 62.7) which refers to the writing of number eight as to the writing of number eight as “Sahasraani Me“Sahasraani Me Dadaato Dadaato AshtakaranyohAshtakaranyoh”, ”, sh[Üûin me ddûto sh[Üûin me ddûto a¸$krNyo¶a¸$krNyo¶ Rishi is asking his pupil to Rishi is asking his pupil to ‘sort’ cows on whose ear there is a ‘sort’ cows on whose ear there is a mark for numeral 8. So we had our mark for numeral 8. So we had our symbols and numerals that timesymbols and numerals that time

ConventionsConventionsThe conventions of using the letters of the The conventions of using the letters of the alphabet and or word, to denote numbers alphabet and or word, to denote numbers can be traced back to Paanini (700 BC) can be traced back to Paanini (700 BC) More than one such systems were employed More than one such systems were employed over a long period of time. over a long period of time. Most of these system were never meant for Most of these system were never meant for use by common people or for purpose of use by common people or for purpose of making calculations; there knowledge was making calculations; there knowledge was strictly confined to the learned and their use strictly confined to the learned and their use to the expression of ideas and numbers in to the expression of ideas and numbers in verses, which helped oral transmission of verses, which helped oral transmission of knowledge and scriptures. knowledge and scriptures. Historian have not found prevalence of any Historian have not found prevalence of any one system from Paanini’s time till the time one system from Paanini’s time till the time of Aaryabhat I (499 AD). of Aaryabhat I (499 AD).

Convention of letters representing Convention of letters representing numbers became more usable. numbers became more usable. Katapayaadi system is a variation of one Katapayaadi system is a variation of one found in Aaryabhattiyfound in Aaryabhattiy

There seems to be some disagreement There seems to be some disagreement about the origin of this system. about the origin of this system.

According to Datta and Singh, in History According to Datta and Singh, in History of Hindu Mathematics (1938), “The of Hindu Mathematics (1938), “The origin of this system can be traced back origin of this system can be traced back to the fifth century A D, … system was to the fifth century A D, … system was known to Aaryabhat I (499)”. known to Aaryabhat I (499)”.

Pandit Gaurishankar Ojha in his book Pandit Gaurishankar Ojha in his book The Paleography of India (1918) says, The Paleography of India (1918) says, a reason for adopting such convention a reason for adopting such convention was that using word numeral was not was that using word numeral was not convenient in Astrology. convenient in Astrology.

In this system, consonants of In this system, consonants of Sanskrit alphabet have been used in Sanskrit alphabet have been used in the place of the numbers 1 through 9 the place of the numbers 1 through 9 and zero to express numbers as and zero to express numbers as listed in the Appendix A as described listed in the Appendix A as described in the Sadrain the Sadrattnamaala:namaala:

nÑ·vcªc xuNy·in sNrVy· nÑ·vcªc xuNy·in sNrVy· k$py·dy¶k$py·dy¶

im§e tUp·Nt hl sN<y· n c im§e tUp·Nt hl sN<y· n c icNTyo hlSvr¶icNTyo hlSvr¶ In practice the convention is as follows:In practice the convention is as follows:

The vowels n (The vowels n (NaNa) and Ñ () and Ñ (InyIny) denote zero; ) denote zero; Letters in succession beginning with Letters in succession beginning with k, $, p, k, $, p,

yy (i.e. (i.e. Ka, Ta, Pa YaKa, Ta, Pa Ya ) each denote the digits ) each denote the digits 1 onwards, 1 onwards,

In a conjoint consonant, only the last one In a conjoint consonant, only the last one denotes a number.denotes a number.

Consonant not joined to a vowel should be Consonant not joined to a vowel should be discarded.discarded.

Vowels themselves each stand for zero.Vowels themselves each stand for zero.

Further, the consonants with vowels are used in Further, the consonants with vowels are used in place of the numerical figures just as in the place place of the numerical figures just as in the place value notation i.e. a right to left arrangement is value notation i.e. a right to left arrangement is employed in the formation of chronograms. employed in the formation of chronograms.

The letter denoting the unit figure is written first, The letter denoting the unit figure is written first, then follows the letter denoting the tens figure then follows the letter denoting the tens figure and so on. In traditional Sanskrit we would follow and so on. In traditional Sanskrit we would follow the opposite, namely the opposite, namely a’k·n·’ v·mto git¶a’k·n·’ v·mto git¶ ((Ankaanaam Vaamato GatihiAnkaanaam Vaamato Gatihi))

The following examples further explain the The following examples further explain the convention:convention:

Now we come to the Now we come to the purpose of the talk and my purpose of the talk and my

journey.journey. We have been taught to believe and We have been taught to believe and

some of us really do live by it, that some of us really do live by it, that Bhagavaan Raam and Krishn both are Bhagavaan Raam and Krishn both are incarnation of Parabhrahm paramaatma incarnation of Parabhrahm paramaatma and are the and are the samesame. .

All Puraan teach us that and ask us to All Puraan teach us that and ask us to believe in. believe in.

In the History of Hindu Mathematics by In the History of Hindu Mathematics by Dutta and Singh, on page 71 of volume 1, Dutta and Singh, on page 71 of volume 1, they state “The following examples taken they state “The following examples taken from inscriptions, grant plates and from inscriptions, grant plates and manuscripts will illustrate the system (i.e. manuscripts will illustrate the system (i.e. Katapayaadi scheme)” There the first Katapayaadi scheme)” There the first example quoted is the word: example quoted is the word: rû`vûy rû`vûy (2441)=2441. (2441)=2441.

The date was either 21st or 22nd Dec. The date was either 21st or 22nd Dec. 2003 and place was the BHU guest house. 2003 and place was the BHU guest house.

Raam-jee kee prerna se prashn uthaRaam-jee kee prerna se prashn utha: :

WhichWhich Katapayaadi number is Katapayaadi number is rûmrûm ??

Just for Swaantah Sukhaay, I would quote one Just for Swaantah Sukhaay, I would quote one example from Maanas, Doha number 198 in example from Maanas, Doha number 198 in Baal-Kaand reads:Baal-Kaand reads:

Byaapak Byaapak BrahmBrahm Niranjan, Nirgun Bigat Binod, Niranjan, Nirgun Bigat Binod,So Aj Prem Bhagati Bas, Kausalya Ke God.So Aj Prem Bhagati Bas, Kausalya Ke God.

Then come the 12 Cahaupaa-ee describing Then come the 12 Cahaupaa-ee describing Baalak Raam-swaroopBaalak Raam-swaroop. The twelfth being :. The twelfth being :

Roop Sakahi Nahin Kahi Roop Sakahi Nahin Kahi SrutiSruti Sesha, Sesha, So Jaana-i Sapanehu Jehin Dekha So Jaana-i Sapanehu Jehin Dekha

Here word Shruti is for Ved. Thus Raam of Here word Shruti is for Ved. Thus Raam of Raamaayan is the Brahm of Ved.Raamaayan is the Brahm of Ved.

Several questions come to Several questions come to mind.mind.

Muni Vashishth-jee gave the name Muni Vashishth-jee gave the name Raam. He believed that Raam was Raam. He believed that Raam was Brahm (avataar). Was Vashishth-jee Brahm (avataar). Was Vashishth-jee aware of Katapayaadi like convention?aware of Katapayaadi like convention?

Name Krishn was given by Aachaary Name Krishn was given by Aachaary Garg Muni, in Dwaapar Yug, and he Garg Muni, in Dwaapar Yug, and he was also aware that Krishn was was also aware that Krishn was Parabrahm Parmatma. Was he too Parabrahm Parmatma. Was he too aware of any such convention?aware of any such convention?

Did the person or persons who adopted Did the person or persons who adopted these conventions, deliberately did so to hide these conventions, deliberately did so to hide

the truth the truth [[B·ÈB·È== rûm rûm == kò¸ï kò¸ï from from common people? common people?

Certainly this is not one of those ‘lucky’ Certainly this is not one of those ‘lucky’ coincidences of our Hindu heritage, as Prof. coincidences of our Hindu heritage, as Prof. Subhash Kak says “The Indian texts are Subhash Kak says “The Indian texts are either full of the most astonishingly lucky either full of the most astonishingly lucky guesses or we do not understand their guesses or we do not understand their knowledge framework” knowledge framework”

Katapayaadi number for the word Katapayaadi number for the word x’krx’kr in in

the name the name x’krûcûyRx’krûcûyR is 215. We know is 215. We know that that x’krûcûyR jay’itx’krûcûyR jay’it is on fifth day of is on fifth day of the first half of the second month. He was the first half of the second month. He was born in the Veekram Samvat 730 born in the Veekram Samvat 730 (Vaisaakh, ShuklaPax, Panchamee). I (Vaisaakh, ShuklaPax, Panchamee). I believe he was named according to the believe he was named according to the horoscope. Is there any connection in horoscope. Is there any connection in deciding deciding rûixrûix, in general and this , in general and this convention? convention?

Conclusion:Conclusion:

I would like to say that origin of I would like to say that origin of Algebra is in Katapayaadi Algebra is in Katapayaadi convention. Certainly here we have convention. Certainly here we have used letters of an alphabet as used letters of an alphabet as variables in a restricted sense and variables in a restricted sense and that is where Algebra began.that is where Algebra began.

A gift from Canada to all retiring A gift from Canada to all retiring Mathematician of BhaaratMathematician of Bhaarat

Just keep looking for numbers and Just keep looking for numbers and concepts hidden in our religious practices concepts hidden in our religious practices and scriptures and scriptures with faithwith faith, and you shall , and you shall experience divine pleasure as long as you experience divine pleasure as long as you live.live.

Jay Seeya RaamJay Seeya Raam

Another Convention:Another Convention:

A system (see appendix B) of A system (see appendix B) of notation in which sixteen vowels notation in which sixteen vowels a – a – aa¶,¶, are assigned numbers 1 to 16 are assigned numbers 1 to 16 and consonants and consonants k – Dk – D are assigned are assigned numbers 1 to 36, the conjoint letter numbers 1 to 36, the conjoint letter

++ being 35 (and some variations of being 35 (and some variations of this), is found in certain manuscripts this), is found in certain manuscripts from Southern India. from Southern India.

A similar convention was used by Aaryabha A similar convention was used by Aaryabha I. There are some interesting and I. There are some interesting and informative revelations when we use this informative revelations when we use this convention:convention: The word (Brahm) The word (Brahm) [BûÁ[BûÁ is made up of four letters, is made up of four letters,

b, r, h, m, (b, r, h, m, (Ba, Ra, Ha, MaBa, Ra, Ha, Ma). ). They are respectively 23rd , 27th , 33rd 25th letters They are respectively 23rd , 27th , 33rd 25th letters

of the Sanskrit alphabet system. of the Sanskrit alphabet system. The sum is 23+27+33+25=108, The sum is 23+27+33+25=108, and hence to and hence to

attain the attain the a[=r-BûÁa[=r-BûÁ (Axar-Brahm) (Axar-Brahm) perform perform Mantr Jaap with 108 beads Mantr Jaap with 108 beads maala.maala.

The words SeetaaRaam (The words SeetaaRaam (s&tûrûms&tûrûm) and ) and RaadhaaKrishn (RaadhaaKrishn (rûñûkò¸ïrûñûkò¸ï) when interpreted as ) when interpreted as made up of letters, made up of letters, s, IR, t, aû, r, aû, ms, IR, t, aû, r, aû, m and and r, aû, ñ, aû, k, Å, W, ïr, aû, ñ, aû, k, Å, W, ï add up to a very add up to a very fascinating and revealing number. fascinating and revealing number.

Following the convention of Appendix Following the convention of Appendix BB

((s IR t aûs IR t aû) + () + (r aû mr aû m) = ) = (32 + 4 + 16 + 2) + (27 + 2 + 25)(32 + 4 + 16 + 2) + (27 + 2 + 25)

= 54 + 54=108= 54 + 54=108 ((r aû ñ aûr aû ñ aû) + () + (k Å W ïk Å W ï)=)= (27 + 2 +19 +2) + (1 + 11 + 39 + 15)(27 + 2 +19 +2) + (1 + 11 + 39 + 15)

= 50 + 58=108= 50 + 58=108

Some thoughts (Some thoughts (ChintanChintan) on this ) on this

numbernumber:: Hundred and eight beads of a Maala are Hundred and eight beads of a Maala are

indicative of both, Aadishakti and Brahm..indicative of both, Aadishakti and Brahm.. Raam and Krishn are not Poorn-Brahm unless Raam and Krishn are not Poorn-Brahm unless

accompanied by their Aadyaashakti.accompanied by their Aadyaashakti. Seeta and Raam both add up to 54, Seeta and Raam both add up to 54,

therefore, now we have mathematical base in therefore, now we have mathematical base in believing that Seeta-jee is His Ardhangini. believing that Seeta-jee is His Ardhangini.

Raadha is only 50 where as Krishn is 58, why Raadha is only 50 where as Krishn is 58, why so? One explanation is that Raadha-jee is not so? One explanation is that Raadha-jee is not His Ardhaangini, they never married, but He His Ardhaangini, they never married, but He is not Poorn-Brahm unless accompanied by is not Poorn-Brahm unless accompanied by Raadha, one would not like to see Him alone. Raadha, one would not like to see Him alone. ((Raadha ke bina Krishn adhura hai)Raadha ke bina Krishn adhura hai)

A gift from Canada to all retiring A gift from Canada to all retiring Mathematician of Bhaarat:Mathematician of Bhaarat:

Just keep looking for numbers and Just keep looking for numbers and concepts hidden in our religious concepts hidden in our religious practices and scriptures with faith, practices and scriptures with faith, and you shall experience divine and you shall experience divine pleasure as long as you live.pleasure as long as you live.

Jay Seeya RaamJay Seeya Raam