[[Mohan K.V 2014-10-20, 05:22:15 Source]]
सदास्वादः
68
**
**
शङ्कुकर्णो नगेश्वरः
(śaṅkukarṇo nageśvaraḥ)
**
**
Meaning
**
**
“The mountain-king has pointy ears.” What’s this about?
**
**
Context
**
**
Suppose you’re ambling by a riverbank on a lazy Sunday afternoon, and stumble into a stone with an inscription:
**
**
“रासिक्य-दं रञ्जकं हरिं जीवमधुना ऽनतोऽस्मि” इति मूलम् ।
“लोकात् भयं मा धर, तन् मूलमिदं धी-साधनम्” इति सूची ।
स्व-वर्गस्य एकाधिके या विनश्यति सा माया ।
मूलं प्रतिष्ठाप्य, तदुपरि च स-मायां सूचीं समारोप्य, तस्य एकाधिके निर्वाणम् आपन्नम् इति सर्वं समञ्जसम् ।
“rāsikya-dam rañjakam harim jīvam adhunā ānato asmi” iti mūlam |
“lokāt bhayaṃ mā dhara, tan mūlam idam dhī-sādhanam” iti sūcī |
sva-vargasya ekādhike yā vinaśyati sā māyā |
mūlaṃ pratiṣṭhāpya, tadupari ca sa-māyāṃ sūcīṃ samāropya, tasya ekādhike nirvāṇam āpannam iti sarvaṃ samañjasam |
**
**
After all those interminable essays on poetry, your first instinct is to read this too in the same light. The first line appears to be a kind of auspicious start: ““I bow down to Hari, the life-force, who grants both the power to enjoy as well as the joy itself."—this is the base.” Having the capacity for something and actually using it – we could get behind that!
**
**
The second appears to be life advice: ““Don’t fear the world; that is the root of any achievement”—this is the guide.” Yup, if we’ve learnt one thing from commencement speakers of all hues, it’s this.
**
**
The third seems to be a definition: “That which when exceeded by just one step vanishes—that is Maya.” Fair enough; reminds us of our experiences of the immense power of hindsight, even when looking back just one step or one minute.
**
**
All three then are combined to a conclusion: “After establishing the base and laying the guide on top together with Maya, one step forward will lead to dissolution (Nirvāṇa)—thus, everything is proper.” A somewhat contorted way of saying divine grace, courage and learning when combined will lead to good. We can nod at the truth of the lines, make a mental note about the contorted construction, and move on…
**
**
…Or can we?
**
**
**
**
Sometimes, a different perspective is all it takes to completely alter one’s valuation. This passage is not just philosophical pep talk, but also a mathematical equation! It encodes numbers in a method called the Ka-ṭa-pa-ya-ādi method, as explained in the table below:
**
**
[TABLE]
**
**
Each consonant represents a digit, namely the column in which it occurs. Each digit has multiple consonants corresponding to it. Vowels and half-consonants are simply ignored: ya, yi, yu, tya, pya, etc. all simply stand for ‘1’. Independent vowels are assigned to ‘0’. That’s it – the system is simple to describe, and because of the wide choice it offers for each digit, gives space for great creativity.
**
**
For example, the (full) consonants in “rā-si-kya-dam” are ra, sa, ya, da, which represent the digits 2, 7, 1, 8.
**
**
Looking at our passage again, and reading the quoted parts as a mathematical text, we see that it says:
**
**
“2 7 1 8 2 8 1 8 2 8 4 5 9 0 0 6 5”—this is the base.
“3 1 4 1 5 9 2 6 5 3 5 8 9 7 9 0”—this is is the sūcī.
That which when squared and exceeded by one vanishes—that is Māyā (‘varga’ in a mathematical sense means ‘square’; ‘ekādhika’ is another technical term for adding one) – clearly then, Māyā is i, √-1
“After establishing the base and laying the sūcī on top together with Māyā, adding one will lead to zero—thus, everything is proper.” In other words,
**
**
eiπ + 1 =0
**
**
Mathematically inclined readers will recognize Euler’s identity, which has been called ‘the most remarkable formula in mathematics’!
**
**
There can be no greater testament to the user-friendliness of the Kaṭapayādi system than the fact that we came up with this example ourselves! :-)
**
**
(Some notes: of course, our example only demonstrates a form of numerical encoding; many of the concepts necessary to understand this, like transcendental numbers, exponentiation and negative numbers (and what it means to take their square-roots) have to be known independently; also, the values of e and pi are accurate only to the 13th and 15th digits respectively, till the first zero appears; and lastly, in Indian classical mathematics, numbers were written in ‘reverse’, from lowest place value to the highest: e.g twenty-eight is encoded as ‘hari’ or ‘jarā’ i.e. ‘8 2’. Here, we have encoded in the modern form. Just giving ourselves an out if we find this quoted later as arising from some ancient tome. :-))
**
**
The Kaṭapayādi system has a long history, and is at least about 1500 years old per the available evidence. Its most familiar use today is in the melakartā system of naming rāgas in Carnatic music. Its use was mainly in astronomical texts, and has a strong association with the Kerala school of mathematics. For instance, one of the works of that tradition uses a pāda of an Indravajrā verse to specify 11 digits of π:
**
**
चण्डांशु-चन्द्राधम-कुम्भि-पालैर्
caṇḍāṃśu-candrādhama-kumbhi-pālair
[The consonants are ca-ḍa-śa-ca-ra-dha-ma-ka-bha-pa-la, encoding 6-3-5-6-2-9-5-1-4-1-3.]
**
**
More recently, in a book published in 1965, Swami Bharati Krishna Tirtha gives an Anuṣṭup verse whose 32 syllables encode the digits of π in the modern order (highest place value on the left). Although he writes about it with a vague air of antiquity, it seems clear that he composed the verse himself!
**
**
गोपी-भाग्य-मधु-व्रात शृङ्गिशोदधि-सन्धिग ।
खल-जीवित-खाताव गल-हाला-रसंधर ॥
gopī-bhāgya-madhu-vrāta śṛṅgiśodadhi-sandhiga |
khala-jīvita-khātāva gala-hālā-rasaṃ-dhara ||
[Encoding: 31415926 53589793 23846264 33832792 – accurate till the 31st digit inclusive. This verse can be read as a general-purpose prayer to Kṛṣṇa, and with some acrobatics, to Śiva. The verb is ‘ava’ ‘protect’ in the 3rd pāda, and all other words are simply vocatives. However, like with most citra-padyas, the meaning is rather a stretch and quite unsatisfying, so we won’t dwell on it here.]
**
**
This chapter’s line comes from a famous ‘table’ generated by the mathematician-astronomer Mādhava, who lived in near Thrissur in Kerala about 600 years ago. In a series of verses, Mādhava calculates the value of sine of various angles from 0 to 90. The results are expressed in colourful, mostly nonsensical anuṣṭup verses and encoded using the Kaṭapayādi method. For example, the third verse in the table encodes values for 33.75° to 45° in steps of 3.75°:
**
**
मूलं विशुद्धं नालस्य गानेषु विरला नराः ।
अशुद्धिगुप्ता चोरश्रीः शङ्कुकर्णो नगेश्वरः ॥
mūlaṃ viśuddhaṃ nālasya gāneṣu viralā narāḥ |
aśuddhiguptā coraśrīḥ śaṅkukarṇo nageśvaraḥ ||
**
**
[Translating only the 4th pāda] “The mountain-king has pointy ears”
encoding: 5 1 1 5 0 3 4 2
**
**
This represents a calculated angle, from which the sine of the original angle can be obtained with just a simple multiplication, yielding a number very close to the known value of 1/√2. We’ll also bet that you can’t think of 45 degrees any more without a image of a snake with pointy ears popping up!
**
**
The table is remarkably accurate, and it is speculated that Mādhava used the same methods that European mathematicians discovered several hundred years later. The very names ‘sine’ and ‘cosine’ arise out of Gupta-era work on trigonometry. Prof. David Joyce writes,
**
**
The Sanskrit word for chord-half was ‘jya-ardha’, which was sometimes shortened to ‘jiva’. This was brought into Arabic as ‘jiba’, and written in Arabic simply with two consonants ‘jb’, vowels not being written. Later, Latin translators selected the word ‘sinus’ to translate ‘jb’ thinking that the word was an arabic word ‘jaib’, which meant ‘breast’, and sinus had ‘breast’ and ‘bay’ as two of its meanings. In English, ‘sinus’ was imported as ‘sine’.
**
**
This word history for sine is interesting because it follows the path of trigonometry from India, through the Arabic language from Baghdad through Spain, into western Europe in the Latin language, and then to modern languages such as English and the rest of the world.
**
**
[It struck us that a more plausible bridge from ‘jaib’ to ‘sinus’ would be the common meaning of ‘cavity’ or ‘pocket’; ‘jeb’ means pocket in many Indian languages as well, and ‘sinus’ is used in the same way in much of anatomy.]
**
**
There was recently a fascinating article, titled ‘Notation, notation, notation: a brief history of mathematical symbols’; only in the 16th century or so did writing mathematics in symbols become the norm! The Kaṭapayādi method may seem circuitous and difficult to us, but for the largest part of history, methods like it were the natural way to write down numbers in many contexts. How deeply rooted our modern systems feel to us, and yet, how recent their innovation!
**
**
Of course, poetry and wonder are but inevitable when a field as universal as mathematics is involved. We’ll leave the reader with a timeless article by Prof. Scott Aaronson , ‘Who Can Name the Bigger Number?’, which is also on notation in mathematics. It starts off with an arresting ‘hook’:
**
**
You have fifteen seconds. Using standard math notation, English words, or both, name a single whole number—not an infinity—on a blank index card. Be precise enough for any reasonable modern mathematician to determine exactly what number you’ve named, by consulting only your card and, if necessary, the published literature.
:-)
**
**
Parting Thought
**
**
There were many other encoding systems that developed in India, but the Kaṭapayādi system was superior to them because of the combination of freedom as well as precision that it offered. For example, a system attributed to Āryabhaṭa assigned a number to each consonant, and an order-of-magnitude multiplier to each vowel. This gave it great conciseness, but made it terribly difficult to use: e.g, to represent the population of India, 1.2 billion, there is only one syllable: ठॄ ṭhṝ. Not ठृ ṭhṛ mind you, that’d be 0.12 billion. Not even टॄ tṝ, that would be 1.1 billion. Tough luck if you heard तॄ tṝ — that would take you away to 1.6 billion. Clearly, it was too sensitive to be useful.
**
**
Another system was called the Bhūta-saṃkhyā system which allowed for more creative space. There, numbers were represented using the abstractions behind commonly understood words: any synonym of ‘earth’ – ‘bhū’, ‘mahī’, ‘dharaṇī’, etc., represented ‘1’ because there was only one earth. Any synonym of ‘eyes’ represented 2, and so on. There’s a lovely old joke that runs on similar lines:
**
**
A certain paṇḍit in a village grew very arrogant about his vast learning, and started annoying everyone else. His friends decided to bring him down a notch, and formed a plan for a prank. They found an utter bumpkin, and told him to keep absolutely quiet and use only his hands to communicate. Then they went to the paṇḍit, and told him that a very great mauna-yogi had come to the village, and that he would very selectively grant an audience to the great paṇḍit only – but owing to his vow, he wouldn’t speak. The paṇḍit agreed, and they met.
**
**
To start the conversation, the paṇḍit held up one finger. In response, the bumpkin held up two. The paṇḍit found this very profound, and after much thought held up all five fingers. The bumpkin promptly responded with a fist. At this the paṇḍit surrendered and declared his opponent to be greater than him, and accepted him as his guru.
**
**
His friends asked him what happened. The paṇḍit replied, “I held up one finger, saying Brahman is one. To this, the great man held up two fingers, obviously meaning that the jīvātman (individual soul) is different from the paramātman (Supreme soul) — surely he has studied all the debates surrounding this. I tried another track, and held up five fingers, meaning that the body is composed of five elements. He countered with a fist, clearly telling me that all five are in fact unified by consciousness. He knows both the metaphysical and physical; I was cornered, and I surrendered.”
The friends then asked the bumpkin to relate his version. “This man sat down and held up one finger. Obviously, he was telling me he’ll poke me in the eye. I held up two to tell him I’ll poke both his eyes. He then opened his palm, threatening to slap me. I made a fist to tell him I’ll punch him right back. The coward couldn’t take it anymore, and surrendered!”
Please join the Google Group to subscribe to these (~ weekly) postings: <https://groups.google.com/group/sadaswada/subscribe?hl=en