The Vedaangas are six subjects created to assist the study of the Vedas. Four of the Vedaangas are about linguistics; the other two have significant mathematical sections.
|
Subject |
English name |
|
Shiksha |
Phonetics |
|
Vyaakarna |
Grammar |
|
Chandas |
Poetic
metre |
|
Nirukta |
Etymology |
|
Kalpa |
Ritual |
|
Jyotisha |
Astronomy |
Mathematical patterns in Akshara, Vyaakarana and
Chandas
As we saw in
chapter three, the Sanskrit alphabet is a masterpiece of analysis and
organization - and this may predate its
written form. The sounds are classified as svara
(vowels) and vyanjana (consonants),
then grouped by origin in the mouth, voicing and aspiration. This may seem
unremarkable, until we learn a language which practically don’t use vowels,
like Hebrew or Arabic. Pertinent to mathematics, though is the differentiation
of vowels as short (hrsva) and long (dheerga) based on duration of pronunciation,
called maatra (measure). This is
standardized in every Indian literary language, including the Dravidian family,
but missing from European languages. Even after hearing every name, city,
object creatively uttered by foreigners, rarely do we appreciate this. Ah, for
an Oxbridge accent!
There is a
separate classification of syllables, rather than letters, as laghu (light) or guru (heavy), which is the basis of generation and classification
of chandas (prosody). Each poetic meter
has a different number of syllables. The
binary nature of the syllables, and the various possible combinations for
letters led to the development of combinatorics; Meru-prastara, a precursor to Pascal’s triangle; and algorithms to
find metrical patterns or their various aspects based on the number of
syllables.
The most
common chandas or metre called anushTubh has four paadams ( quarters) of eight
syllables each.
A longer meter
called mandakraanta has four quarters of seventeen syllables each.
For example,
consider a three syllable meter. There are eight possible combinations of guru
(G) and laghu (L) syllables.
|
1 |
L |
L |
L |
|
2 |
L |
L |
G |
|
3 |
L |
G |
L |
|
4 |
L |
G |
G |
|
5 |
G |
L |
L |
|
6 |
G |
L |
G |
|
7 |
G |
G |
L |
|
8 |
G |
G |
G |
Table Prastara for three syllables
Pingala
provides six different pratyayas (mathematical procedures or algorithms),
summarized in this table.
Is this
mathematics or linguistics, one might ask? Even the linguistics texts have a
mathematical structure.
Sanskrit
literature is in one of three forms: chandas (verse), champu (text) or sutra
(brief text). All the Vedangas, composed in the Vedic era, except Jyotisha are
in sutra form, in very cryptic
notation, akin to modern mathematical notation. Ironically all the ganita
siddhantas of post Vedic period are in chandas, poetic form. What an irony:
Grammar and poetry books in mathematical notation, but mathematics books in
verse!
In fact, we
find the oldest mentions of zero, Shunyam
and Lopah, in Panini’s Grammar Ashtadhyaayi and Pingala’s prosody ChandaSutra, not in the Jyotisha text!
Panini’s Ashtadhyayi
Long before
Panini, Sanskrit had eleven books on grammar, by Indra, Galava, Gautama etc.,whom
Panini himself mentioned. The Siva Sutra,
a brilliant arrangement of letters into 14 groups, in algebraic notation, also
pre-dates Panini and was fully exploited by him. Panini’s grammar is so
algebraic in its notation, it has been an inspiration for later mathematicians,
say experts. This was centuries before algebra developed! His grammar is generative
rather than an analytical or descriptive grammar. No other language in the
world had such a grammar book, until
mathematicians developed similar notations for computer languages in the
20th century.
Let us look at
one sample – sandhi rules for combination of letters and sounds.
First if you
look at English words, what rules govern spelling when words combine?
Work + ing = working fly+able = flyable
Bowl
+ ing = bowling ply+able
= pliable
use +
ing = using note+able
= notable
bat
+ ing = batting hit+able = hittable
run
+ ing = running free+able = freeable
There is no
English grammar book with such formal rules as is seen in the books of Sanskrit
grammar. English spelling and pronunciation have undergone drastic changes over
the last few centuries, unlike Sanskrit whose rules of grammar havent changed
much in millennia. But most people accept these rules at least in an informal
way.
Sanskrit,
though has very specific rules for how letters merge, when letters drop, add or
transform
|
a + a =
aa |
अ + अ = आ |
அ + அ = ஆ |
|
aa + a =
aa |
आ + अ = आ |
ஆ + அ = ஆ |
|
a + aa
= aa |
अ + आ = आ |
அ + ஆ = ஆ |
|
aa + aa =
aa |
आ + आ = आ |
ஆ + ஆ = ஆ |
Examples
veda + anga = vedaanga
kamalaa + ambaa =
kamalaamba
simha + aasanam = simhaasanam
padmaa + aasani = padmaasani
|
a +
i = e |
अ + इ = ए |
அ + இ = ஏ |
|
a +
e = ai |
अ + ए = ऐ |
அ + எ = ஐ |
|
a +
u =
o |
अ + उ = ओ |
அ + உ = ஓ |
|
a +
o = au |
अ + ओ = औ |
அ + ஓ = ஔ |
Examples
raaja + indra =
raajEndra
chola + eeshvara =
cholEshvara
loka + eka =
lokAIka
purusha + uttama = purushOttama
kula + uttunga = kulOttunga
SivaSutras
also called Maheshvara Sutras, is a set of shlokas, that organized the Sanskrit
alphabet into 14 subsets. Each subset consists of some letters, not in the
original alphabetical order, and a terminal letter. A two letter notation, the
first of which is the a letter that indicates the first letter of the subset,
te second of which is a terminal letter, is used to indicate any desired
subset.
|
अइउण् |
a i uN |
|
Examples |
|
|
ऋऌक् |
R L k |
|
|
|
|
एओङ् |
E O ng |
अक् a k |
अ इ उ ऋ ऌ a i u R L |
|
|
ऐऔच् |
ai au c |
यण् ya N |
य व र ल ya va
ra la |
|
|
हयवरट् |
ha ya va ra T |
तय् ta
y |
त क प ta ka pa |
|
|
लण् |
laN |
|
|
|
|
ञमङणनम् |
nya ma nga Na na m |
अच् ac |
All vowels |
|
|
झभञ् |
jha bha ny |
हल्
hal |
All consonants |
|
|
घढधष् |
gha Dha dha SH |
|
|
|
|
जबगडदश् |
ja ba ga Da da sh |
|
|
|
|
खफछठथचटतव् |
kha
pha cha Tha tha caTata v |
|
|
|
|
कपय् |
ka pa y |
|
|
|
|
शषसर् |
Sha1 SHa2 sa r |
|
|
|
|
हल् |
ha l |
|
|
|
SivaSutras
Panini and Backus-Naur Form
In the 1950s,
IBM Corp developed the ForTran computer programming language, the first
high-level language for computer programming. The team’s leader John Backus
developed a notation called Backus-Normal Form, later altered to Backus-Naur
Form, to notate grammar for ForTran.
Peter Zilahy
Ingerman, Manager of Langauge Systems Standards and Research, Radio Corporation
of America, in 1967, wrote a letter to the journal Communications of the ACM,
suggesting that this notation be renamed Panini-Backus Form. He opined that,
“Panini had invented a notation, which is equivalent in power to that of Backus
and has many similar properties.”
Linguists
since then have hotly debated the virtues and pitfalls of Panini’s system. But
it is most proabable that this is the origin of the rumour that Sanskrit is the
most suitable language for computer programming. That is not true; but surely,
Panini’s grammatical notation has the greatest similarity to the grammatical
notation of computer languages.
Sulba sutras
Egyptians and
Greeks developed geometry for earth (geo) measurement (metry). We may say
Indians developed vedi-metry: the sulba sutras are manuals for configuring
vedis(altars) of different shapes for performing yajnas(Vedic sacrifices). They
were measured using ropes (rajju or sulba). Baudhayana, Kaatyaayana,
Apastamba, Maanava, Maitraayana, Vaaraaha, Vaadhula are seven Sulbasutras named
after their composers, that survive. Their contents are very similar.
विहारयोगान्व्याख्यास्यामः । Aapastambha
Sulbasutra 1.1
Vihaara-yoga-vyaakhyaasyaamaH.
Translation We make known (vyaakhyaasyaama) constructing figures (vihaara-yoga) 1.1
रज्जुसमासं वक्ष्यामः । Kaatyaayana Sulbasutra 1.1
Rajju-samaasam vakshyaaamaH.
Translation We explain (vakshyaama) by combination (samaasam) of ropes (rajju)
Unlike
axiomatic Greek geometry, which is very theoretical, Sulbasutras are practical
or applied. They specify rules and methods for construction of vedi-s, finding true east, calculating
areas, diagonals, choosing clay, making bricks, calculating number of bricks,
etc.
Praachi – Finding
true east
समे शङ्कु निखाय शङ्कुसम्मितया रज्जवा मण्डलं परिलिख्य यत्र लेखयोः शङ्क्वग्रच्छाया निपतति तत्र शङ्कू निहन्ति सा प्राची || कात्यायन शुल्बसूत्र 1–2
Translation
Plant a shanku (gnomon), on level
ground.
Draw a circle with rope measured by
shanku.
Where shadows fall on circle, fix a
stick (morning, evening)
That line is the praachi (line
connecting the two sticks), the east west line
As the concept of ayana was well understood, this explains that true east is not merely the direction the sun rises, but its midpoint over the full year. The method of finding true east, using simple devices like a stick, rope, pegs is very typical of Indian astronomy.
Diagonal of a
square
चतुरश्रयाक्ष्णयारज्जुर्द्विस्तावतीं
भूमीं करोति ।
समस्य द्विकरणी ॥
Aapastambha 1.5
chaturashraya
akshnayaa rajju dvis-taavatim bhoomi karoti । samasya
dvikarani ॥
Translation The rope (rajju) on the diagonal (akshnayaa) of a square (chatur-ashra) produces (karoti) double (dvi) the area (bhumi) of the square. It equals (sama) √2 (dvikaraNi) of the side of the square.
Dvikarani literally means double-maker. To construct a square twice the size of a given square, the latter’s diagonal should be used as the side for the bigger square. The word karani means square root, diagonal of a square, producer etc.
Square root of
two
प्रमाणं तृतीयेन वर्धयेत्तच्च चतुथेनात्मचतुस्त्रिंशोनेन सविशेषः ॥
Aapastambha 1.6
pramaaNam
truteeyena vardhayet-tacca caturthena-aatma-catus-trimsha-unena sa-visheshaH ||
Translation The measure (pramaaNam) is to be increased by its third (truteeyena) and this (aatma) again
by its own fourth (caturthena) less (unena) the thirty-fourth
part(catus-trimsha) approximately (sa-visheshaH) 1.6
√2 = 1
+ 1 + 1 ( 1 - 1 )
3 3*4 34
This is an
excellent approximation; but also an indication that they didn’t understand the
irrational numbers. visheshaH means
special; here the special nature is interpreted as the approximate nature of
the result. A similar sutra exists for square root of three, which demonstrates
the general concept.
Squaring a Circle
मण्डलं चतुरश्रं चिकीर्षन् विष्कम्भं पञ्चदशभागान्कृत्वा द्वावुद्धरेत् । त्रयोदशवशिष्यन्ते । सानित्या चतुरश्रं ॥ 3.3 ॥
Translation Mandalam
catur-ashram cikirshan vishkambham panca-dasha-bhaagaan-krtvaa dvaar-uddharet |
trayo-dasha-avavashishyante | sanityaa catur-ashram ||
To transform a circle (mandalam) into
a square(catur-ashram), the diameter(vishkambham) is divided into fifteen parts
(pancha-dasha-bhaagam) and two (dvaa) of them are removed(uddharet), leaving
(avashishyante) thirteen parts (trayaodasha). This gives the approximate (side
of the desired) square.
But this shows not only an understanding of equation famously called Pythagoras theorem, but also an understanding that it applies not only to whole numbers. As Baudhayaana is oldest Sulbasutra with this sutra, Fields medalist Manjul Bhargava is among many mathematicians who thinks Pythagoras theorem should be renamed after Baudhaayana.
In the
Classical era, Aryabhata onwards, the phrase bhuja-koti-karna nyaaya denotes this relationship of hypotense to a
right triangle, or diagonal to a square, with bhuja, koti and karna meaning,
base, perpendicular and hypotenuse. The
SulbaSutra words for these are paarshva-maani, paryanka-maani and akshnayaa.
Such a vast period of time elapsed between the Sulba sutras and Aryabhata, that
even the scientific vocabulary transformed.
Vedanga jyotisham
I explained
some of the ideas in the Rg and Yajur Vedanga Jyotishas in the previous essay.
Ill touch upon some other concepts here. One was the categorization of some of
the twenty seven stars into fierce, and cruel; this is not found in later siddhantas.
A sloka referncing each star by a single letter name is another unique,
unrepeated highlight.
Another is the
use water clocks. Pots of different sizes, called कुडव, आढक,
द्रोण kuDava, aaDhaka, drona were used. They represented the time
consumed by the complete draining of water in a pot. 16 kuDava-s is a aaDhaka, 4
aaDhakas is a drona. A naadi is the time it took for 45 kuDavas
of water to drain away. No explanation is given; these must have been taken
from common usage.
Other time
measures like kala, muhurtha, parva, lagna are introduced. Most of the slokas
in Vedanga Jyotisha are procedures for calculating various time based
phenonmenon. These include calculating the equinoxes, increase in day and night
lenghts, differences between lunar and solar days etc. They are too complicated
for this essay, so I will omit them. Remember the purpose of the subject was
explained as kaala-vidhaana-shaastram, the
science of calculating time.
The simple
mathematical calculation of proportions called thrai-raashika, (Rule of Three)
is seen here. If x tasks take y of units time, how many units of time does z
tasks take, is an example. Given any three of these, the fourth can be
calculated. This arithmetic, perhaps the simplest algorithm for calculating an
unknown based on known quantities, which we learn in our primary schools, is as
old as the Vedangas.
The astronomical phenomenon not discussed are eclipses, their calculation or prediction, celestial longitudes and latitudes, orbits of planets etc. These subjects were studied in the subsequent era of the 18 Siddhantas.
------------------------
References
Pre-Siddhantic Astronomy, Lecture by Prof RN Iyengar, Seminar on Contributions of KV Sarma, KV Sarma Library Chennai, 2018
NPTEL Lectures on Indian Mathematics by Profs MD Srinivas, MS Sriram, K Ramasubramanian
Boks
Apastambha, Baudhayana Sulba sutras
Vedanga Jyotisha, TS Kuppana Sastry, edited by KV Sarma, 1984
