Archive for February, 2012

DID YOU KNOW | Evolution of Algebra & Geometry

DID YOU KNOW that Greek and Romans were idol worshippers and Geometry developed to a great extent in their country. Arabs were not idol worshippers and Algebra developed in their country. India was a combination of both philosophies where some people follow idol worship and some didn’t. Could that be one of the reasons why both Algebra and Geometry developed in India?

The ancient texts of Mathematics, Sulba Sutras, contain geometrical constructions pertaining to religious sacrificial rituals. When there was a need, the branch of geometry developed. Temples & rituals can be seen in Greek & Roman history which might have forced them to develop geometry in their country.

Be it a coincidence or a fact, it is surely a good point to think upon. Do religious faiths factor the development of mathematics in a culture?

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MATHEMATICS IN KALIDASA’s KUMARASAMBHAVAM – Part 3

Ananta ratna prabhavasya yasya Hime na saubhagya vilopi jaatam |
Eko hi dosho guna sannipaate Nimajjatindoho kiraneshvivaankah ||

We now proceed to the next mathematical observation by Dr. Kannan in Kumarasambhavam. There are more observations that he mentioned but has not been included in these notes.

Second Mathematical Finding:
Pra bha va sya
2 4 4 1

Right to left – 1442 — (3)
Adding (1) & (3) we get
1442.0260002600…
= 1442 + 2600/9999

= 14418558 + 2600/9999

= 14421158/9999

Numerator is 1 4 4 2 1 1 5 8 and in the table is 1 4 4 2 1 1 8 5

pra bha va sya ya sya hi me
2 4 4 1 1 1 8 5

Vilopi means reverse order. 53 becomes 35 when expressed in hexadecimal number system (here the rule of hexadecimal number system is hinted. Only the genius ones like Dr. Kannan can read between these lines). And 53 is the only number like that the digits of which gets interchanged when converted to hexadecimal number system. This is the same thing happening in the case of 58, it has become 85; “and that is my only mistake (Eko hi doshah)” admits Kalidasa in the next line.

ey ko hi do sho
0 1 8 8 6

Eko hi doshah – There is only one mistake (at the place of hi – 8) in the earlier explanation (where 85 becomes 58 in the numerator)

gu na sa nni pa te
3 5 7 0 1 6
6 1 0 7 5 3

Numbers right to left – 6 1 0 7 5 3

Divide 53 by 16, we get Quotient as 3 and Remainder as 5. So, 53 in hexadecimal number is 35.
First Astronomical Finding:

Eko hi doshah – There is only one mistake (at the place of hi – 8) in the earlier explanation (where 85 becomes 58 in the numerator)

do sho gu na
8 6 3 5
Writing the number right to left we get 5368. Here too, 5 should be in the place of 8. Now 368 becomes 365 which is the number of days in a year.

Second Astronomical Finding:
ni ma jja ti
0 5 8 6
Writing the number right to left we get 6850. Again, eko hi doshah, 5 should be in the place of 8. Hence, the number becomes 6580* which is approximately equal to the SAROS CYCLE. Saros Cycle in astronomy is an 18-year period after which the stars’ get back to the position which it attained 18 years & 10 days before. That is why, our date of birth and star of birth (according to lunar Hindu calendar) fall on the same day at every 19 years.
The calculation is 6580 days = 365 days x 18 years + 10 days. In some cases, we find that the date of birth and star of birth come in one day of difference, there we have to add 11 instead of 10 which is because it falls on a leap year**.

Conclusion:
The above interesting observations were the result of contemplation and interpretation by Dr. Kannan. After the day’s session, three of us went to meet him to ask a few doubts. One of the doubts was, “Did Kalidasa write the poem keeping the Mathematical & Astronomical facts in mind? Because, that was too hard to believe.” Dr. Kannan’s answer was interesting and convincing to us. He said, “Ordinary people think and they speak, only then it makes sense. In the case of great visionaries, due to their greatness, whatever they speak will contain thousand meanings. The way different people see it, according to it, the people will interpret different meanings from it.” Kalidasa was a great poet. Owing to the greatness of the poet, Sanskrit language, Katapayadi number system, etc., these meanings are interpreted by Dr. Kannan. Interpretations of different people can be right only when it is true in all angles. The sole intention of this article was for the readers to have a delight and look at the wonderful way in which Dr. Kannan looked at Kumarasambhava.

* The number 6580 for SAROS Cycle is according to the Hindu calendar and might change depending in other non-Hindu calendars.
** This explanation was rapidly noted down while taking notes. The writer he does not have more clarity on the topic. Hence, it’s suggested that this article be taken only as an interesting one to read and not to study in detail.

MATHEMATICS IN KALIDASA’s KUMARASAMBHAVAM – Part 2

Ananta ratna prabhavasya yasya Hime na saubhagya vilopi jaatam |
Eko hi dosho guna sannipaate Nimajjatindoho kiraneshvivaankah ||

After proving that Lord Murugan was praised in the third sloka, Dr. Kannan went further to explain his interpretation of Mathematical findings in the same sloka. He kept chanting the verse many times in between so that we could memorise the verse as much as possible. He said that without memorizing, it’s not possible to analyse. Every time he picked up a few words from the stanzas, he would chant the stanzas in a very melodious tone. His involvement in the chanting described his feeling about the subject. It is his feeling that went inside the audience, more than anything else.

About Katapayadi
Katapayadi is one of the numerical codes that were used in India for representing numbers by writing them as words. In this system numbers can be written in a coded form with alphabets and only those who know the code can decode it. It was used for remembering oft-used numbers (e.g. dates, era, etc) by converting it into words using a code. Katapayadi was also used in Vedic Astrology, Astronomy, Carnatic Music (Indian Classical Music), Vedic rituals for performing sacrifices and preparation of its altars, Poetry, Code language among the soldiers and in many other areas.

In this system, every number (from 0 – 9) can be represented using Sanskrit alphabets. Needless to say, to use this number system knowledge of Sanskrit is required. The system is so named because the consonants ‘ka क्, ta ट्, pa प्, ya य्’ are used to represent the number 1. The complete table is shown below:

1 2 3 4 5 6 7 8 9 0
ka क kha ख ga ग gha घ nga ङ ca च cha छ ja ज jha झ nya ञ
ṭa ट ṭha ठ ḍa ड ḍha ढ ṇa ण ta त tha थ da द dha ध na न
pa प pha फ ba ब bha भ ma म – – – – –
ya य ra र la ल va व śha श sha ष sa स ha ह – –

Rules for using Katapayadi:-
• Every consonant is given a number.
• If any vowel is used separately it is considered as zero.
• The consonants which are not combined with a vowel do not have any weightage and are simply ignored.
• The rule applied in writing the code was ‘Ankanaam Vamatogati’ अन्कनाम् अमतोगति. Meaning: Numbers move in the left direction. Hence, when a word was decoded into a number, it is read from right to left.
E.g. Mukha represents 25, ParA denote 21, and so on. Mu denotes 5, Kha & Ra denote 2, Pa denotes 1. Hence reversing the two digits we get 25 and 21 respectively.

First Mathematical Finding:
Use of Katapayadi can be seen in Kumarasambhavam as well, is what Dr. Kannan is trying to establish.
Let’s decode the verse using Katapayadi.
A Na Nta ra Tna pra bha va sya ya sya hi me na sau bha gya vi lo pi jaa tam
0 0 6 2 0 2 4 4 1 1 1 8 5 0 7 4 1 4 3 1 8 6
ey Ko Hi do sho gu na sa nni pa te ni ma jja ti ndoho ki ra ne shvi vaa nkah
0 1 8 8 6 3 5 7 0 1 6 0 5 8 6 8 1 2 5 4 4 1

A Na Nta ra tna
0 0 6 2 0

Read it from right to left – 0 2 6 0 0

Anantara means ‘After’
Ananta means Infinity

Repeating it infinite number of times we get
x = 0.2600,02600,02600,…. —- (1)

Multiplying (1) by 10000 we get
10000x = 2600.02600,02600… —- (2)

Subtracting (1) from (2) we get,
9999x = 2600.0260002600…- 0.0260002600..
9999x = 2600
Hence, x = 2600/9999

This proves that if we divide a number by that the number of nines as the number of digits in the given number, then the number goes in a pattern like this infinitely. E.g. 153/999 = 0.153153153…. 28384/99999 = 0.283842838428384…

More of Dr. Kannan’s findings are mentioned in the next article.

MATHEMATICS IN KALIDASA’s KUMARASAMBHAVAM – Part 1

Presented by: Dr. N Kannan – M.A. Sanskrit 1st ranker, M.Phil in Mathematics, Phd in Mathematics (subject of thesis – Mathematics in Sanskrit works), studied Yajurveda & Rigveda in the traditional style; H.O.D. of Oriental Studies & Research Shastra University, Thanjavur

Presented at: Rashtriya Sanskrit Vidyapeetha, Tirupati during the National Workshop on Ancient Indian Mathematics with special focus on Vedic Mathematics and Astronomy (24th – 28th February 2012)

A note by the writer of these notes:
This is just an attempt to encapsulate the great knowledge that Dr. N Kannan shared in the presentation. However hard we try, it’s not going to come anywhere near the original presentation. Yet, for the sake of those who could not attend it, we are attempting to pen.

Prof. P.V.Arunachalam, former Vice Chancellor of Dravidian University, Kuppam, introduced Dr. N Kannan to us. Dr. Kannan was a medium statured middle-aged man with wheatish complexion. He was ready with a collar mike and a PPT to present his subject. We all sat anxiously waiting to hear what was to come out. To surprise us, he chanted a prayer from Vedic literature in the typical vedic tune and that too in his full volume. The audience was zapped because they never expected such a start. He started his addressing (the dignitaries on and off the dais) in Sanskrit. This came as our second shock. Because how much ever we adored Sanskrit, we didn’t wish a talk in Sanskrit. To bring down our levels of anxiety, he changed his language immediately to English after the first two lines.

Kumarasambhavam was one of the two epic poems the greatest poet in Sanskrit, Kalidasa, has composed. The ring finger is called ‘Anamika’ in Sanskrit, which means, ‘without a name’, in Sanskrit. There is a story behind how the name Anamika came for the ring finger. They say, great scholars started counting the great poets. They counted Kalidasa first on the little finger. To their surprise, they could not name any other poet who can be compared to the level of Kalidasa. Hence the next finger, the ring finger, went unnamed (or named as Anamika). So, we can imagine to what heights Kalidasa was held in the minds of people of ancient times.

It is generally said that Kumarasambhavam (literal meaning: birth of Kumara, i.e. Karthikeyan or whom some call as Murugan, Subrahmanyam, etc) does not include any praise of Lord Murugan. But Dr. N Kannan, who has been studying and contemplating over the poem, says that the third verse does praise about Subrahmaniyam. The verse goes as under:

Ananta ratna prabhavasya yasya Hime na saubhagya vilopi jaatam |
Eko hi dosho guna sannipaate Nimajjatindoho kiraneshvivaankah ||

Meaning given by M.R.Kale: Snow could not be a destroyer of beauty in the case of him who is the source of countless jewels; for one blemish is lost in a host of virtues, like the spot on the moon in her rays.

Dr. Kannan has done a lot of research and found out that Kalidasa’s verses not only contained poetry but also great deal of Mathematics & Astronomy.

Ananta – Infinity
Eko – One
Guna – Multiplication
Ankah – Apparently it means numbers. But it also refers to 9. Here, we should understand that Mathematics or Science encoded in Sanskrit can be understood only by those who have mastered both Sanskrit and the other subject. The reason why this encoding was done was because the knowledge in subjects like Quantum Physics, which involved making of Atom bombs, etc, was not be revealed easily to everybody to ensure safety of mankind.

Nipaate (in the second line) – ni + paate means falling down. Falling where? To find this out, we need to split all the letters of the first and second lines.
a na Nta ra tna pra bha va sya ya sya
ey ko Hi do sho gu na sa nni pa te

Here, bha falls down and na goes up in the place of bha to form pranava instead of prabhava. The source of Pranava (also another name for Murugan) is Bhava (another name for Shiva). This proves that Subrahmaniyam was praised by Kalidasa in Kumarasambhavam.

(Note: The explanation given in talk was brief due to time constraint. More explanation is given in the paper submitted by Dr. N Kannan)