The Genius and the Myth

He ranks with Archimedes, Euclid, Isaac Newton and Leonard Euler as one of the greatest mathematicians of the world. He began a new epoch in Indian astronomy and mathematics, that continued for more than a millenium. His book Aryabhateeyam is a masterpiece of brevity and eloquence.

But what did Aryabhata actually do? Aryabhata did NOT invent zero; or gravity; or the heliocentric system. As I wrote in my first essay, even Indian mathematics and Sanskrit scholars are stunningly ignorant of Aryabhata’s actual accomplishments. Since we are equally ignorant of almost all of ancient India’s glories, this is not specifically galling; just generally abysmal. Only Bhaskara was perhaps as popular and admired, but unlike Newton’s apple or Watt’s tea kettle, or the anecdotes of Birbal or Tenali Raman, we don’t even have popular legends about him. But we are so creative, we blame the British for this situation, decades after they left.

Ever computed a square root? Aryabhata.
Cube root? Aryabhata.
Summed up a series of numbers? Aryabhata.
Series of squares? Aryabhata.
Divided by a fraction by multiplying by its inverse? Aryabhata.
Computed the areas of triangles, circles, trapeziums? Aryabhata.
Calculated sines? Aryabhata. 

And that’s just the simple mathematics we learn in school.

Wait! Did he invent ALL of these? Ah, that’s the question. Aryabhata himself claims not a single invention. He explicitly states that “by the grace of Brahma, the precious jewel of knowledge (jnana-uttama-ratnam) has been extracted from the sea of true and false knowledge (sat-asat-jnaana-samudraat), by the boat of my intellect (sva-mati-navaa).” As Euclid compiled five centuries of geometrical discoveries of the Greeks, Aryabhata compiled several centuries of mathematical and astronomical discoveries of Indians.

Sulba sutra and Jain mathematicians knew how to compute, square roots, but Aryabhata was the first to describe the algorithm. We don’t know if cube roots were calculated earlier, his algorithm is the oldest extant. His sine calculations are considered much superior to those listed by Varahamihira. His kuttakara algorithm to find solutions is considered ingenious even today.

It is not feasible to explain his mathematical and astronomical discoveries in a magazine article for the general reader. There are excellent translations, technical papers, books that do that. This essay’s purpose is to provoke you to read them, and marvel at Aryabhata’s sva-mati-navaa. And to place Aryabhata and his work in historical context.

Manuja Grantham

The eighteen siddhantas were attributed to rishis. But every jyotisha siddhanta after Aryabhata and Varahamihira, is attributed only to men, not rishis. These arose from commenting, understanding, questioning, correcting, improving existing siddhantas and inventing or discovering new concepts. There was no fear or taboo against criticizing a mere manuja like Aryabhata or Bhaskara, rather than a rishi. This era of Mathematics and Astronomy is called “Classical” by historians. I prefer VarahaMihira’s phrase Manuja Grantha.

मुनिविरचितमिदमिति यच्चिरन्तनं साधु न मनुजग्रथितम् ।
तुल्येऽर्थेऽक्षरभेदादमन्त्रके का विशेषोक्तिः ॥१–३॥ – बृहत्संहिता

muni-viracitam-idam-iti yat-cirantanam saadhu na manuja-grathitam
tulye-arthe-akshara-bhedaad-amantrake ko viSheshokti – BrihatSamhita 1-3

 

Translation This (idam) is muni-uttered (muni-viracitam) so sacred (cirantanam) and good (saadhu). Not (na) so manuja-grathitam (man-composed) it is said (iti). If it is not a mantra (amantraka), and meaning (artha) is equal (tulye) but words different(akshara-bhedaa), what’s wrong (vishesha) with it?

Philosophically, this verse by Varahamihira, is as insightful and expressive as Kalidasa’s verse puraanamityeva na saadhu sarvam(Not everything is excellent, simply because it is ancient). 

Aryabhateeyam

The phrase Kusumapure abhyaarcitam gnaanam (knowledge respected in Kusumapura), in Aryabhateeyam hints that he lived in Kusumapura (Pataliputra or Patna). No biography or portrait of any Indian astronomer exists. The pictures of Aryabhata pervading the internet, as well as his statue, are merely artists’ imaginations. Almost all we know about him comes from his books and those of his critics and commentators, like Brahmagupta and Bhaskara I, who mentions Pandurangasvami, Latadeva and Nishanku, as pupils of Arybhata.

He composed:

(1) Aryabhateeyam in 499AD when he was 23 years old. Multiple copies survive in full form.

(2) Aryabhata Siddhanta, which is lost, and known only by quotations from commentators. In this book, Arybhata advocated midnight as the starting hour of each day, instead of sunrise, perhaps based on Surya or Romaka Siddhanta. Aryabhateeyam uses sunrise as day-beginning.

I confine this essay to Aryabhateeyam. It consists of two parts. The first, Dasha Geetika (Ten Songs), lists astronomical constants:

·        Orbital periods and Diameters of Sun, Moon, Planets

·        Number of years in a yuga, yugas in a kalpa, kalpas in a manu

·        Deviation of planets from the ecliptic

·        Epicycles, in different quadrants

·        Table of Sine differences.

 

His first verse is a salutation to Brahma - he was a scientist, but not an atheist. Almost every jyotisha who followed him begins his work with a salutation to his favorite God. Jain mathematician Mahavira begins with an invocation to his namesake, the tirthankara Vardhamana Mahavira. It may also indicate that he was updating the Paitamaha (Brahma) siddhanta, some of whose data, had become obsolete.

The second part, called AryaAshataShatam (i.e The 108 Arya verses) consits of three chapters – Ganita (Mathematics), Kaala Kriyaa (Calculating Time), and Gola (Sphere – i.e. Celestial, Sphere meaning the visible universe).

The siddhantas of later jyotishas were each nearly a thousand verses long. What Aryabhata summaries in one or two verses is explained by them with whole chapters. So cryptic and compact was Aryabhateeyam, it was impossible to understand without bhashyaas (commentaries); such was its impact, that bhaashyaas were written on it centuries after others improved upon his methods. Telugu Marathi and Malayalam commentaries followed those in Sanskrit, Arabic etc; and English translations in the colonial period, which range in appreciation from astonishment to incredulity to calumny.

1.    Ganita - Mathematics

The mathematics set forth by Aryabhata is mostly practical, not theoretical: its primary purpose is astronomy. I mention only simpler concepts in this essay.

It also varies from extremely simple to extremely complex statements, hypotheses, and algorithms.

We must understand that mathematics was not taught to school children, then as it is today; it was perhaps the most advanced of technical subjects and confined to specialists.  Arithmetic symbols familiar to us like + - x ÷ = were only introduced in fifteenth century Europe. Mathematics was not expressed in equations, but in slokas.

Aryabhata gives two line slokas like this:

त्रिभुजस्य फल शरीरं समदलकोटी भुजार्ध संवर्गः

Tribhujasya phala shareeram samadalakoti bhujaardha samvargaH.

 

Bhuja means Arm. Tribhuja means three-armed or Triangle.

Translation “Multiplication (SamvargaH) of perpendicular(Samadalakoti) and half (ardha) the base(Bhuja) results (phala) in Triangle’s (Tribhuja-sya) area(Shareeram).”

A similar verse(sloka) defines the area of a circle as its half-perimeter (or half-circumference) multiplied by its half-diameter (radius) 

This is a simple algorithm, just a formula really, to calculate one value, based on known parameters. A more complex version is his algorithm for summation of a series, which includes several calculations, including for the mean of the series, and encoding an alternate algorithm! This way of stating multiple mathematical formulae is called muktaka by Bhaskara I.

Kaalakriyaa – Time

Aryabhata divided time and circles  with the same geometric units as earlier siddhantas. His major departure, was to define the four yugapadas namely krta, treta, dvaapara and kali, as of equal time; and as the time it took all the nine planets to align, or complete an integral number of revolutions around the earth. He included a biographical note, that 3600 years passed between the beginning of Kali yuga (end of Mahabharata war) and the twenty-third year of his birth. This implies that the constants in DashaGitike were based on his personal observations in that year.

This differed from the smriti definition of the first three yugapadas as four, three and two times as long as the kaliyuga, and offended the orthodox of everyone. Even his followers didn’t accept this division, but they followed his computations and algorithms, as they were significantly better than those of earlier siddhantas.

Gola – Celestial Sphere

Arybahata states that Solar and Lunar eclipses are shadows of the Moon on Earth and Earth on the Moon, respectively. He also stated that the  Sun is the only source of light, and not just planets, but even the stars only reflect sunlight.

Kadamba flower

Aryabhata used the metaphor of a kadamba-pushpa-grantha,  to explain how people and creatures in all parts of the world believe they are standing on top of the world. He introduced another metaphor, for Earth’s rotation: consider a boat-rider on the Ganga, who feels trees on the shore pass him by; whereas, in reality it is the boat that is moving. Similarly Aryabhata suggested, the earth actually rotates, and like trees on a river bank, the stars seem to revolve around it. But it was only a metaphor, not a proof.

He also explains such concepts as Ascencions of the Zodiac, Sine of Ecliptic etc. which are too technical for this essay.

The impact of Aryabhata was phenomenal. Even fervent critics could not ignore him or his works. But he launched an era of manuja grantham, and he was followed by a long line of brilliant scholars, whom we will discuss next.

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This essay was first published as part of a series in Swarajya
For the entire series click this link --> Indian Astronomy and Mathematics   

References

1.      The Aryabhateeyam by Walter Eugene Clark, University of Chicago, 1930.

2.      Aryabhatiyam, translated by KV Sarma and KC Sukla, Indian National Science Academy, New Delhi, 1976.

3.      Facets of Indian Astronomy, KV Sarma, Madras.

 Rishi Siddhaantas and Manuja Siddhaantas

The thousand years before Aryabhata were as rich in intellectual fervour and activity as the thousand years after him. This was the era of the composition of most of the Vedaangas, the creation of such seminal works like Bharata’s NaatyaShaastra, Chanakya’s ArthaShaastra, Vatsyayana’s KaamaSutra, and several magnificent treatises on various subjects. Among these were eighteen jyotisha siddhantas, all attributed to deva-s like Surya or rishi-s like Kashyapa, Atri, Mareechi as described in this sloka.

सूर्यः पितामहो व्यासो वसिष्ठोऽत्रि पराशरः ।
कश्यपो नारदो गर्गो मरीचिर्मनुरङ्गिराः।।
लोमशः पौलिशश्चैव च्यवनो यवनो भृगुः ।
शौनकोऽष्टादशश्चैते ज्योतिःशास्त्र प्रवर्तकः ।।

Surya pitaamaho vyaaso vashishto atri paraasharaH
Kashyapo naarado gargo mareechi-r manu-r angiraaH
lomashaH paulisha-shcaiva chyavano yavano bhrguH
shaunako ashtadasha-shchaite jyoti shaastra pravartakaH

 

This stands in stark contrast with the Siddhantas in the post-Aryabhata classical era, all of which are ascribed to scholarly astronomers, but not rishis. Varahamihira’s phrase manuja-grantham, succinctly describes this.

This was the period during which numerals, the place value system, angular units like degrees, minutes and seconds, trigonometry, and several such mathematical concepts must have been discovered. Instruments like shanku (gnomon), chakra (hoop), gola (armillary sphere), ghati yantra (copper pot) were used.

But all 18 siddhantas are now lost, except the Surya Siddhanta, which was modified and updated in the later centuries. Fortunately, Varahamihira, a contemporary of Aryabhata, wrote a treatise called Pancha Siddhantika, a comparative study of five of these eighteen siddhantas. He quoted and explained several verses from them. So, we understand some concepts of the era.

Types of Jyotisha texts

Jyotisha texts come in several categories. Siddanta-s are once in a century grand texts, composed by superlative scholars. A siddhanta may have several commentaries, called bhashya-s, in the succeeding centuries. For practical use, more compact books called karana-s were composed, which was used by pandits to prepare almanacs/calendars called panchaanga-s for public use. The latter tradition is still extant.

It is my belief that the various texts on astronomy and mathematics rival the commentaries and compositions on the Ramayana and Mahabharata. So rich and so widespread was the literature.

Pancha Siddhantika

The five siddhantas Varahamihira studied, those of Pitamaha (Brahma),  Vashishta, Surya, Romaka and Paulisha, explain motions of planets (in a geocentric model), prediction of eclipses, sine tables, celestial longitudes and latitudes. None of these are mentioned in Vedanga Jyotisha. They vary mostly in minor details, which Varahamihira explains. The small Vedic yuga of five years was dropped, and the humongous yuga of 432000 years used. We have no idea when or how this changed. A day count, ahargana, counting number of solar days (regardless of month or year) since the start of the Kaliyuga, which began when the Mahabharata war ended, came into vogue. Kaliyuga years are found inscribed in several royal inscriptions; for example, the Anamalai inscription of Maranjadayan Varaguna Pandyan in Madurai.

The solar zodiac is used extensively. It was most probably borrowed from the Greeks or Babylonians. The solar zodiac is a popualar theme on ceiling sculptures of temples in Tamilnadu, like this one in Kudumiyan Malai, Pudukottai.

Romaka (also called Lomasha) and Yavana refer to a Roman and a Greek, Paulisha to a Paulus Alexandrinus, say historians of science. While some foreign ideas were obviously borrowed, there is a puzzling absence of inclusion of other ideas, including those of Euclid, Ptolemy, or Archimedes. Whereas the Greeks developed an epicyclic theory of planetary motion, Indians developed a theory based on air strings pulling the planets. Geometrically, these are simply different epicyclic model than those used by the Greeks. They involved extremely complicated geometry, trigonometry and algebra, but they were quite accurate in predicting eclipses, solar and lunar, the biggest challenges of Indian astronomy.

That Mercury and Venus had a different type of orbital movement, from the other planets, Mars Jupiter and Saturn, was realized. Siddhantas explain eight types of planetary movement.

A vocabulary of scientific and technical words developed, to describe both such astronomical concepts and mathematical ideas and theorems.

From the earlier knowledge of hypotenuses and circles, as found in Sulba Sutras, we can understand that the concepts of sine, cosine and other trigonometric ideas arose. The Indian sine was not the opposite/hypotenuse that we learn in school today, but the radial sine (abbreviated as R-sine), called the ardha-jyaa (half-bowstring). A chord connecting the ends of an arc looks like a bow (Sanskrit: chaapa or dhanush). When seen as part of a circle, the radius of the circle (CM )is the hypotenuse of the triangle (CMA) formed by the half-chord (MA), the radius touching the top (M) of this chord, and the segment (CA) of the radius dividing the chord into two equal halves. In Indian siddhantas, in the table of sines, expressed as a series, only the numerators are listed. Hence they are radial sines (multiplied by radius). The word for cosine is koti-jyaa.

The word jyaa and this concept of trigonometry traveled from India to Baghdad in the eighth century during the reign of Caliph al-Mamun, along with the zero, the decimal place value system, Indian numerals (now called Arabic numerals) and the works of Aryabhata and Brahmagupta. It transformed into the Arabic word jyaab or jeyb which means pocket. This then was taken to Europe by Leonardo Fibonacci, an Italian merchant, in the twelfth century, and translated into Latin as sinus, and later into English as sine. Then it came to India under English colonialism, making a full circle (pardon the pun) into our mathematical textbooks as sine. We learn trigonometry as the gift of the Europeans, not realizing its Indian origin.

Angular measurements called kalaa (degrees) liptaa (minutes) and viliptaa (seconds), were used, based on the sexagesimal system (Base-60) rather than decimal, which hints at a Babylonian origin. In addition, a sub division of the second into sixty parts and division of the cirlce into twelve parts (called raashi) also existed. Angles were often represented in karana texts with five aspects, not just the three we use today.

The division of time was also sexagesimal, with a day consisting of sixty naadis, each naadi of sixty vinaadis. Remember, the naadi existed in the Vedic period; was it indigienous or imported? It’s not one of several mysteries.

Step by step mathematical procedures (now called algorithms, after the Uzbek mathematician, Mohammad ibn Musa al-Khwarezmi) also emerged in the era of 18 Siddhantas. The place value system and zero were invaluable in developing algorithms for multiplication and division, square and cube roots, and several algebraic procedures solving indeterminate linear equations.

Ujjain Meridian

Two millennia before the world adopted the Greenwich meridian, Indian astronomers used the Ujjain meridian, as the prime meridian of longitude in India. This is the longitude that passed from north pole (Meru) to south pole (Vadavamukha). That the earth was a globe, not a flat plain was well understood by astronomers. They believed that Devas lived at Meru and Asuras at Vadavamukha, and Mankind in between.

गगनमुपैति शिखिशिखा क्षिप्तमपि क्षितिमुपैति गुरु किञ्चित् ।
यद्वदिह मानवानामसुराणां तद्वदेवाधः ॥ १३– ४ ॥  Pancha Siddhantika 13-4

Gaganam-upaiti shikhi-shikaa kshiptam-api kshitim-upaiti guru kincit
Yadvad-iha maanavaanaam-asuraaNaam tadvadeva-adaH 

The flame (shikhaa) of a lamp(shikhi) points skywards (gaganam) and a heavy (guru) object (kincit) thrown (kshiptam) skywards falls back to earth (kshiti); this happens in the lands of men (maanavaanaam) and asuras (asuraaNaam)

This was one concept of gravity, before Newton changed it.

उदयो यो लङ्कायां सोऽस्तमयः सवितुरेव सिद्धपुरे ।
मध्याह्नो यमकोट्यां रोमक विषयेऽर्धरात्रं स॥ Pancha Siddhantika 15-23

Udayo yo lankaayaam sa-astamaya savitur-eva siddhapure
Madhyaahno yamakotyaam romaka-vishaye arddha-raatram saH

Translation When it is Sunrise (udaya) in Lanka, it is Sunset (astamaya) in Siddhapura, Noon (madhyaahna) at Yamakoti,  Midnight (arddha-raatra) in Romaka-vishaya 

Lanka is not the Sri Lanka we know, but the point where the Ujjain meridian intersects the equator. Ujjain was a major center of learning in ancient India, and is also perhaps closes to the Tropic of Cancer (Karkata). We don’t know what places Yamakoti and Siddhapura signify, perhaps they are also place marker names like the equatorial Lanka.

While all the other Siddhantas determine time with Ujjain as the prime meridian, Romaka Siddhanta says the days starts with sunset at Yavanapura, which is not Athens or Rome, but Alexandria in Egypt.

The logical thought process which inspired the use of Ujjain and Lanka for calculations is simple, but brilliant. Longitude and latitude determine local time. So, the times of sunrise, sunset, moonrise, eclipses, will vary from place to place. Once the calculations are made for a prime meridian like Ujjain, local panchaangam-s can be prepared with only minor changes applied for local longitude and latitude – these are called deshantara, Each Siddhanta has a section about it. 

Celestial longitudes and latitudes were easier to calculate, than those on earth. The Surya Siddhanta lists Rohitaka (Rohtak, Haryana) and Kurukshetra and other cities on the Ujjain meridian. Others list such places as Kanyakumari, Malavanagar, Sthaneshvar, Vatsyagulma, Mahishmati, Vananagara as cities on the Ujjain meridian.

Some  trivia : Ujjain passed on its torch to Madras, briefly. Today, Indian standard time is set on longitude 82.5E,  based on Greenwich meridian. But for about a century, the Madras meridian was used as the prime meridian, especially for railway clocks.

For the entire series click this link --> Indian Astronomy and Mathematics   

References

1.      Surya Siddhanta, by Phanindralal Gangooly

2.      Pancha Siddhantika, edited by KV Sarma

3.      Pancha Siddhantika, edited by G Thibaut, Sudhakara Dwivedi

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.

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.

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 20^th 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

 

In some cases, two words simply merge together, with no change to either. In some cases, as in hittable and running, the last letter of the first word doubles. In some cases, as in using and notable, the last letter of the first word is dropped. In some cases as in pliable the last letter of the first is transformed from y to i.

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

 

Examples

veda + anga   =  vedaanga
kamalaa + ambaa    =  kamalaamba
simha + aasanam = simhaasanam
padmaa + aasani = padmaasani

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. 

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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

Vedic Mathematics is a more popular catchphrase than Vedic Astronomy. The adjective Vedic has become associated with anything Sanskrit! The four Vedas, though, are primarily slokas about ritual, prayer, philosophy. But numbers are mentioned, even revered; 28 stars, new and full moon, eclipses seasons, days, periods, years, etc are mentioned. From these we can glean some facts and deduct concepts.

All arts and sciences, of whatever variety that existed, had their own separate texts. Invariably, older texts or anthologies were discarded or forgotten over time, because newer compositions replaced archaic knowledge. Panini’s grammar, for example, made eleven previous Sanskrit grammars obsolete. Even scripts of written languages underwent this change, as Brahmi, Kharoshti inscriptions in India and Sumerian Cuneiform and Egyptian heiroglyphs show us. As an aside: we rarely hear of Sumerian commercial mathematics or Pharaonic mathematics.

We don’t know what other texts or anthologies, existed during the Vedic era. Also, the Vedas were transmitted from one generation to the next entirely orally. If any other books were written or even transmitted orally, they are mostly lost to us. The Vedas, alone, because they were considered sacred, were passed on thousands of years later, unmodified. No Indian astronomer, from Aryabhata in 5^th century to Pathani Chandrasekar in 19^th century refers to the Vedas, as a source of their astronomy or mathematics – they only refer to jyotisha siddhantas of earlier astronomers. Nevertheless some historical information may be gleaned from the Vedas.

Remember also, the Vedas are a compilation of various slokas of rishis – we don’t truly know the period of their composition. The puranas, and Vedic commentaries, attribute yugas, millions of years to history. Geology, archaeology, biology, anthropology consider Indian civilization to be a few thousand years old; how many thousands is subjective. Since it is least controversial, I call the period before the 5^th century BC as Vedic, without an upper bound.

Divisions of Day

The concept of solar (savana or divasa), lunar(tithi), stellar(nakshatra) days co-existed, as we saw earlier. 28 Nakshatras are listed, but Abhijit was dropped, with no loss to astronomy. The 12 zodiac series from Mesha (Aries) to Mina (Pisces) was a later import, probably from Babylonia.

The day was divided into two major parts: poorvaahna (forenoon) and aparaahna(afternoon). Poorvahna itself divided into two parts, praatha and samgava. Aparaahna into three parts, madhyaahna, aparaahna and saayaahna. Even today, sandhyaavandanam and maadhyaahnam are important Vedic rituals, performed by millions daily.

This five part division was at some point superceded by a division of the day into thirty muhurtha-s, fifteen each for daytime and nighttime. Today we number the hours, from 1 to 12 or 1 to 24 (this is an import from Egyptian astronomy!) But Vedic muhurthas were named, not numbered. Much later, during the 18 Siddhantas era, and naadi-s supplemented muhurta-s, as we saw in Aryabhata’s sloka. A naadi equals two muhurta-s.

We use numbered tithis : prathama, dvitiya, trithiya, chaturthi, panchami (first, second, third, fourth, fifth) upto trayaodashi, chaturdashi. But the Vedas had names, not numbers, for each tithi. In fact the tithi-s had different names for the day versus nighttime, and different names during the waxing (shukla paksha) versus waning (krishna paksha) phase. So there were sixty tithi names for a lunar month. Similarly there were sixty muhurthas. All these are named in Taittreya Brahmana (3.10). A sample of these names are shown in the accompanying tables.

Divisions of the Year
Just as the day was divided into parts, the year also divided into days(Dina), months(Maasa), seasons(Rtu), Ayana(half-years). The months were named Madhu, Madhava, Shukra etc, not the Chaitra, Vaishaka, series we use today. Each rtu is a season of two months; India experiences six seasons, not four like Europe. The ayana is the apparent traversal of the Sun between northern and southern latitudes; uttaraayana (also called devaayana) is the traversal from south to north; dakshinaayana (also called pitraayana) is traversal from north to south.

Disparity threatened; the months were lunar; 12 months came to 354 days; but the solar year was eleven days longer. To resolve this disparity which threw off the calendar, the concept of adhika maasa was introduced. Adhika means extra. Basically, it was a leap month, similar to a leap day.

A five year cycle after which five solar years and 62 lunar months aligned again, was called a yuga (which means conjunction).

These demonstrate that the concept of time evolved even during the Vedic period.

When the yuga transformed to a tremendous period of 4,32,000 years and a chaturyuga ten times as long is unclear; this is the period the puranas mention. The five year yuga is not even referred to by astronomers of the Siddhanta or Classical period, except Brahmagupta, who rejects it, and the entire Vedanga Jyotisha, without explanation. A fine historical and philosophical anomaly - the very Brahmagupta who is dismissed as orthodox for rejecting Aryabhata’s alleged heresies, is not himself considered a heretic for rejecting an entire Vedanga!

The invocatory sloka of Vedanga Jyotisha mentions all these divisions of the year, saluting Prajapati, the Creator, as one whose limbs are days, seasons, months etc.

पञ्चसंवत्सरमयं  युगाद्यक्षम् प्रजापतिम् ।
दिनर्तोयनमासाङ्गं प्रणम्य शिरसा शुचिः ॥

Pancha-samvatsam-ayam yugaadi-aksham prajaapatim
Dina-rtu-ayana-maasa-angam praNamya shirasaa shuchi

 

This is a common theme throughout the Vedas and Hindu philosophy: not only the Sun, Moon and planets, besides the Earth are considered manifestations of divinity, all celestial phenomena described as parts of their divine bodies. Even numbers are considered divine, and offered salutations. Everything was sacred, which rails against our perception of science today, when considering anything sacred is unscientific.

Numbers

The Taittiriya Aranyakam 4.69 of the Krishna Yajur Veda has this sloka

सकृते अग्ने नमः । द्विस्ते नमः। त्रिस्ते नमः। दशकृत्वस्ते नमः । शतकृत्वस्ते नमः । आसहस्रकृत्वस्ते नमः ।अपरिमितकृत्वस्ते नमः । 

Sakrte agne namaH. Dvis-te namaH. Tris-te namaH. Dasha-krtvas-te namaH. Shata-krtvas-te namaH. Aa-sahasra-krtvas-te namaH. Aparimita-krtvas-te namaH. 

Translation NamaH (salutations) to Agni. NamaH twice. Namah thrice. Ten times namaH. Hundred times namaH. Thousand times namaH. Unlimited times namaH.

The Taittiriya Samhita 7.2.20 has this sloka

शताय स्वाहा सहस्राय स्वाहा अयुताय स्वाहा नियुताय स्वाहा प्रयुताय स्वाहा अर्बुदाय स्वाहा न्यर्बुदाय स्वाहा समुद्राय स्वाहा…… परार्धाय स्वाहा

Shataaya svaahaa sahasraaya svaahaa ayutaaya svaahaa niyutaaya svaahaa prayutaaya svaahaa arbudaaya svaahaa nyarbudaaya svaahaa samudraaya svaahaa … paraardhaaya svaahaa

Translation Svaahaa to a hundred, thousand, ten thousand, lakh, etc. up to one trillion.

This prayer or offering follows a series of prayers, to the first twenty natural numbers, odd numbers, even numbers, multiples of four, five, ten, twenty, forty and fifty. The religious significance is not my scope. What they counted a trillion of, is an enigma. But these slokas demonstrate the use and understanding of decimal (base 10) numbers and large numbers. 

Note the absence of the familiar laksha (lakh) and koti (crore). By the classical era, new names for some larger numbers emerged. Aryabhata, for example, uses koti for arbuda, and vrndam for samudra.

The fractions paada, shapha, kushTha and kalaa respectively 1/4, 1/8, 1/12 and 1/16 are mentioned in the Vedas, too.

These are all names of numbers only, not mathematical operations or algorithms, which are explained in Vedangas.

Other stars, planets, constellations

Five planets are mentioned, but only Brhaspati by name, and Venus as Vena rather than Shukra. The most famous constellation was Sapta Rishi, (a section of Ursa Minor), stars in the polar regions, which revolved around Dhruva, the Pole star. There is a beatiful poem, comparing stars rising after Vrkam(Sirius), as birds of heaven chasing the wolf.

The brightest star of the southern hemisphere suddenly became visible during this era. This star was called Agastya (Canopus) and linked with the story of rishi who headed south, requesting the growing Vindhyas to stop until he returned. Perhaps the legend captures the period of a tilt in the earth’s axis, when the night sky at that latitude became on Aryavrata. Another southern star is called Vishvamitra, and three surrounding stars, Trishanku.

Comets, Meteors, Eclipses

A prayer in Atharva veda mentions comets (ulkaani) and meteors (dhoomaketu). Strangely, except for Varahamihira, no classical astronomer mentioned these objects.

अत्रि: सूर्यस्य दिवि चक्षुर आधत् स्वर्भानोर अप माया  अगुक्षत ||
यं वै सूर्यं स्वर्भानुस् तमसा विध्यद आसुर: |
अत्रयस् तं अन्वविन्धन् न हि अन्ये अशक्नुवन् || Rg Veda 5.40.8-9 

This sloka refers to the rishi Atri “who set the press stone, revered the Gods, dispelled the illusions of Svarbhanu and restored the Sun’s light”, a reference to a solar eclipse. “Atri and his sons alone could restore the Sun when Svarbhanu had covered him with darkness”, it continues. By the era of the 18 Siddhantas, Indian astronomers understood eclipses, and could predict them. We will come to that after a look at the Vedangas.

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References

Facets of Indian Astronomy, by KV Sarma
Vedanga Jyotisha, TS Kuppana Sastry, edited by KV Sarma, 1984
NPTEL Lectures on Indian Mathematics by Profs MD Srinivas, MS Sriram, K Ramasubramanian