Vedānga: The limbs of Vedic knowledge
The meaning of Saṃskrta (Devanāgarī: संस्कृत) word Veda (Devanāgarī: वेद) means knowledge of primordial origins, which Rishi-s (Devanāgarī: ऋषि) gathered over the time through anubhava (Devanāgarī: अनुभव) (direct experience) and have been referred to as apauruṣeya (Devanāgarī: अपौरुषेय) (impersonal, authorless).63,64 To record and understand that knowledge, they designed several separate branches of knowledge dealing with understanding of several aspects of (Vedic) knowledge like language called Saṃskrta etc. and those branches are called Vedānga (Devanāgarī: वेदांग) i.e. the limbs of the (Vedic) knowledge, which are six in numbers. The design of those knowledge branches is the engineering marvel and is unparalleled in the entire world throughout the history and even today. The methodology is scientific and logical and possesses features not seen anywhere else.
Śikṣā (Devanāgarī: शिक्षा)
This branch of knowledge deals with phonetics and phonology aspect of the language. As per verse 1.2 of taittirīya upaniṣad (Devanāgarī: तैत्तिरीय उपनिषद्), śikṣāvallī (Devanāgarī: शिक्षावल्ली), the branch deals with – varṇa (Devanāgarī: वर्ण), svara (Devanāgarī: स्वर), mātrā (Devanāgarī: मात्रा), bala (Devanāgarī: बल), Sama (Devanāgarī: सम) and santana (Devanāgarī: सन्तन) aspect of sound.[1, 6] Rgveda (Devanāgarī: ऋग्वेद) in its two hymns 10.125 and 10.71 has revered sound as Goddess and links the development of thought to the development of speech. It defines various syllables based on different methodologies and contains rules for accent, quantity, stress, melody and saṃdhi (Devanāgarī: संधि) (euphonic combination of sounds) to make language pleasant and unambiguous to understand.[3, 5] These provides capabilities to individual sounds to have independent personalities where the reciter could develop their character and timbre. It is designed in such a way that it transforms a Saṃskrta text into a musical performance.
Different Vedic knowledge branches had their own way of recitation, sets of syllables etc. and these texts are called prātiśākhya (Devanāgarī: प्रातिशाख्य) (towards a branch), which contains Śikṣā, vyākaraṇa (Devanāgarī: व्याकरण) and sometimes chanda (Devanāgarī: छन्द) rules for that specific branch of knowledge. Apart from famous pāṇinīyaśikṣā (Devanāgarī: पाणिनीयशिक्षा) and nāradīyaśikṣā (Devanāgarī: नारदीयशिक्षा), there are many other texts, four of which are śaunakaprātiśākhya (Devanāgarī: शौनक प्रातिशाख्य) for Rig, taittirīyaprātiśākhya (Devanāgarī: तैत्तिरीय प्रातिशाख्य) for krṣṇayajur (Devanāgarī: कृष्ण यजुर्) and vājasaneyiprātiśākhya (Devanāgarī: वाजसनेयि प्रातिशाख्य) for śukla yajur (Devanāgarī: शुक्ल यजुर्), śaunakīyacaturadhyāyikā (Devanāgarī: शौनकीयचतुरध्यायिका) for atharva (Devanāgarī: अथर्व) and raka tantra (Devanāgarī: रक तन्त्र) for sāma (Devanāgarī: साम).
The syllables called akṣara (Devanāgarī: अक्षर) (non-destructible), also called varṇa (Devanāgarī: वर्ण), have been categorized into svara (Devanāgarī: स्वर) (vowel), also called prāṇaakṣara (Devanāgarī: प्राण अक्षर) and vyaṃjana (Devanāgarī: व्यंजन) (consonant), also called prāṇi akshara to indicate that they are something to which prāṇa is added.
Svara with total 15 in number could be classified in three ways, based on timing i.e. five hrasva (Devanāgarī: ह्रस्व) (short), eight dīrgha (Devanāgarī: दीर्घ) (long) and two pluta (Devanāgarī: प्लुत) (supporting), manner i.e. mukha (Devanāgarī: मुख) (oral) and nāsikā (Devanāgarī: नासिका) (nasal), and accent i.e. udātta (Devanāgarī: उदात्त) (high pitch), anudātta (Devanāgarī: अनुदात्त) (low pitch) and svarita (Devanāgarī: स्वरित) (descending pitch).
Vyaṃjana whose numbers varies with branch of Veda considered, are categorised into three types viz., sparsa (Devanāgarī: स्पर्स) (stops), antastha (Devanāgarī: अन्तस्थ) (semi-vowels) and usmāna (Devanāgarī: उस्मान) (spirants). The Samveda prātiśākhya organizes spars vyaṃjana into a 5X5 magic square to make the readings of letters from left to right and top to bottom symmetric and resonant and where the difference between sounds is preserved during recitation both horizontally or vertically. Further, they are arranged in the logical order where each row of vyaṃjana is mapped to anatomical nature of human sounds from throat to lips with the first row recited from back of oral cavity and second row from palate and so on.
It was first scientifically analysed how sound is generated and then based on that knowledge, the syllables were designed.
The first way as per Pāniniya-Śikṣā, the soul having intellectually determined the object to be communicated to others, it urges the mind to give expression. The mind so stimulated acts upon the Agni which in its turn brings about a movement in the region of Prāna. The internal air thus moved gets upward till it reaches the vocal apparatus. Svara are produced by letting the air through vocal cord without obstruction while the vyaṃjana are produced by obstructing, suppressing or redirecting the air flow. The obstruction occurs between active articular i.e. one that is moving like tongue etc. and a passive articular i.e. one that is stationary like roof of the mouth etc. They recognised five passive articular viz., kaṃṭha (Devanāgarī: कंठ) (Velar), tālavya (Devanāgarī: तालव्य) (Palatal), mūrdhanya (Devanāgarī: मूर्धन्य) (Retroflex), danta (Devanāgarī: दन्त) (Dental) and oṣṭhya (Devanāgarī: ओष्ठ्य) (Labial). The other three passives articular are from combination of these five viz., Danta-oṣṭhya, kaṃṭha – tālavya and kaṃṭha – oṣṭhya. Similarly, there are four active articular viz., jihvāmūla (Devanāgarī: जिह्वामूल) (tongue root), jihavāmadhye (Devanāgarī: जिहवामध्ये) (tongue body), jihvāgra (Devanāgarī: जिह्वाग्र) (tip of tongue) and adhoṣṭh (Devanāgarī: अधोष्ठ्) (lower lip).
The second way to perceive how sounds are generated is in terms of prayatna uccāraṇa (Devanāgarī: प्रयत्न उच्चारण) (effort of articulation). Thus, there are two types of prayatna for vyaṃjana namely bāhya (Devanāgarī: बाह्य) (external) and abhyantara (Devanāgarī: अभ्यन्तर) (internal). The Bāhy is of three types viz., sprṣṭa (Devanāgarī: स्पृष्ट) (Plosive), īṣata sprṣṭa (Devanāgarī: ईषत स्पृष्ट) (Approximant) and īṣatasaṃvrta (Devanāgarī: ईषत संवृत) (Fricative). The abhyantara is of four types viz., alpa prāṇa (Devanāgarī: अल्प प्राण), mahā prāṇa (Devanāgarī: महा प्राण), śvāsa (Devanāgarī: श्वास) and nāda (Devanāgarī: नाद). The syllable could thus be defined in terms of logical combination of these two prayatna.
Thus, the syllable could be defined through any of these two methods like syllable ‘ka’ could be described as either kaṃṭha (jihvāmūla) through first method or as combination of Sprisht, Shwās and Alpa- prāṇa as per second method.
In addition, each syllable is mapped with mudrā (Devanāgarī: मुद्रा) (signs), by which it addresses those who are out of reach of speaker voice or those with hearing disability and it also provides additional cues in addition to sound to reinforce the learning. As both the mind and the body of the reciter are engaged, so it also adds aesthetic “sensuous, emotive” dimension to the communication, which foster thinking and intellectual skills in a participatory fashion. It is also part of Vedic dance traditions where the message is communicated through visual cues/signs.
The system is a phonetic where the words are spoken as the combination of sound of individual syllables so that one doesn’t need to separately learn the mappings between the words and their pronunciation unlike many other languages where the speaker won’t know how to speak a word without learning it beforehand.
The vowels and consonants has a particular and inalienable force which exists by the nature of things and not by development or human choice; these are the fundamental sounds which lie at the basis of the tāṃtrikabīja maṃtra (Devanāgarī: तांत्रिक बीज मंत्र) and constitute the efficacy of the maṃtra itself. Every vowel and every consonant has certain primary meanings which arose out of this essential Shakti or force and were the basis of other derivative meanings. 
There are many rules to recite the Saṃskrta texts called Pātha (Devanāgarī: पाठ) which helped it to orally transmit ancient knowledge from millenniums without slightest modification and adulteration. These rules of Pātha and their symmetric design helps remembering enormous knowledge and self-check the memory. Each text is recited in several ways to cross check with each other for preserving integrity and to help complete and perfect memorization. It works like Checksum technique used in computers. These techniques not only guaranteed the preservation of text but also the sound of the ancient knowledge. The pitch of the sound is also retained as sound plays a role in the recitation of maṃtra, as mentioned earlier. UNESCO proclaimed this tradition a Masterpiece of the Oral and Intangible Heritage of Humanity on November 7, 2003.
It provided eleven ways of pātha namely saṃhitā (Devanāgarī: संहिता), pāda (Devanāgarī: पाद), krama (Devanāgarī: क्रम), jāta (Devanāgarī: जात), mālā (Devanāgarī: माला), śikhā (Devanāgarī: शिखा), rekhā (Devanāgarī: रेखा), dhvaja (Devanāgarī: ध्वज), daṃḍa (Devanāgarī: दंड), ratha (Devanāgarī: रथ) and gaṇa (Devanāgarī: गण). The first three are called prakruti pāṭha (Devanāgarī: प्रक्रुति पाठ) (natural recitation styles) as they don’t involve reversing the word order while the remaining are called vikruti pāṭha (Devanāgarī: विक्रुति पाठ) (complex recitation styles). It is to be noted that the word order doesn’t alter the meaning of the sentence in this language! In saṃhitā technique, words are recited based on the phonetic rules of combination while in pāda technique, there is pause after each word and after special grammatical codes and ignores phonetic rules of combination. In Krama technique, words are combined successively and sequentially like “w1 w2 w3 w4” would be recited as “w1w2 w2w3 w3w4.” In the modified form, the phonetic rules of combination are ignored for each pair. The other eight are more complicated and requires reversing the word order. We mention three here, among which, the gaṇa technique is believed to be most difficult, in which words are repeated back and forth in a bell shape.
In jāta – pāṭha (mesh recitation) technique, every two adjacent words in the text are first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order as “word1word2, word2word1, word1word2; word2word3, word3word2, word2word3…” 
In dhvaja – pāṭha (flag recitation) technique, a sequence of N words are recited (and memorised) by pairing the first two and last two words and then proceeding as “word1word2, word(N−1)wordN; word2word3, word(N−3)word(N−2)..; word(N−1)wordN, word1word2..”.
In gaṇa – pāṭha (dense recitation) technique, it takes the form as “word1word2, word2word1, word1word2word3, word3word2word1, word1word2word3; word2word3, word3word2, word2word3word4, word4word3word2, word2word3word4”. 
The Saṃskrta texts were passed on orally throughout millenniums but later some of the texts were written down mainly in nāgarī (Devanāgarī: नागरी) script from which scripts like devanāgarī (Devanāgarī: देवनागरी) and nandināgarī (Devanāgarī: नन्दिनागरी) came. Paṇini also mentioned the existence of these scripts in his work.
Mendeleev, the creator of Periodic Table of Elements that groups elements based on similar properties, was strongly influenced by Devanāgarī. But during his time, many elements weren’t discovered. So, he suggested that there were gaps where new elements, yet to be discovered, belonged and named them ek-boron, or dvi-silicon or tri-carbon, Saṃskrta words for one, two, three etc. Later, those elements were indeed discovered and were given new names.
“Four are its horns, three its feet, two its heads, and seven its hands, roars loudly the threefold-bound bull, the great god enters mortals”
Rig-Veda, iv. 58, 3
Vyākaraṇa (Devanāgarī: व्याकरण)
This branch deals with the grammar aspect of the language to properly express ideas. It is the world’s first formal system. The most popular text in this area is aṣṭādhāyī (Devanāgarī: अष्टाधायी) of Panini. Its grammar is perfect without exceptions, so you would not have sentences with ambiguous meanings. The text is also noted for its systematically arranging the sutras in algorithmic fashion so that the grammar rules also apply in the same order. 
Verbs are categorized into ten classes, which can be joined to form various verb forms with ending conveying person, number, and voice. Nouns have three genders, three numbers, and eight cases. Nominal compounds could include over 10 word stems. Word order doesn’t matter, though subject–object–verb word order is preferred, which is also the original system of Vedic prose. A sentence is a collection of words; a word is a collection of phonemes. As per Panini, Saṃskrta passages must be understood through context, purpose stated, subject matter being discussed, what is stated, how, where and when.
It helped along with techniques from other branches to ensure that the message of Shabda Brahm that Rishi-s had realized through their efforts, remains available in its pristine form. Their texts also suggest that this branch had competing schools of thoughts and one school, for example, held that all nouns have verbal roots. It is traceable to the RigVeda, in hymns attributed to sage Sakalya.
Panini observes that a proper sentence has a single purpose. It must have proper syntax so that it has ākāṃśā (Devanāgarī: आकांशा) (mutual expectancy) of the words and sannidhi (Devanāgarī: सन्निधि) (phonetic contiguity) of construction. He accepts that a sentence can be grammatically correct even if it is semantically inappropriate. Panini asserts that grammar is about the means of semantically connecting a word with other words to express and understand meaning.
As per Bhartrihari, the author of another famous grammar text vākyapadīya (Devanāgarī: वाक्यपदीय), all thoughts and knowledge are “words”, every word has an outward expression and inward meaning. A word has meaning only in the context of a sentence. Word is considered a form of energy, with the potential to transform a mind.,  Grammar is a science, which is expressed as relations between words, but ultimately internally understood as reflecting relations between the different levels of reality. Language at lower level is used to express material world, and thereon to express feelings, and ultimately the human desire for meaning in life and the spiritual inner world.
In his sphoṭa (Devanāgarī: स्फोट) (spurt) theory, he described the production of speech in three stages viz., paśyanti (Devanāgarī: पश्यन्ति), madhyamā (Devanāgarī: मध्यमा) and vaikharī (Devanāgarī: वैखरी) i.e. idea, medium and utterance. He views sphoṭa as the language capability of man, revealing his consciousness.
The Panini grammar has context-free feature, which is also exhibited by computer languages, and this shouldn’t be surprising as while the designing of the computer language compilers, the compiler designers took the assistance of the Saṃskrta grammar for the concept of formal rules in language and which is also reflected from the many of the theories proposed by modern “fathers” of linguistics and other linguistics scholars like Ferdinand de Saussure, Noam Chomsky, Franz Bopp, Leonard Bloomfield, Roman Jakobson, Frits Staal etc. in their theories likes Backus–Naur Form etc. which seems exact rip-off from work of Indian scholars and same observation is also noted by some scholars like Frits Staal etc.. Some honest scholars like Frits Staal, Ferdinand de Saussure etc. openly acknowledge their work to ancient Indian scholars while some others project themselves as original thinkers. Also for example, Panini used the method of “auxiliary symbols”, in which new affixes are designated to mark syntactic categories and the control of grammatical derivations. This technique, “rediscovered” (or copied from Saṃskrta texts) by the logician Emil Post, became a standard method in the design of computer programming languages. These features of the language are the reason that the Saṃskrta language and computer languages are used across different domains ranging from computations, music, architecture etc. The systematic grammar rules help composing highly intuitive structure like contemporary machine languages. Panini’s work was the forerunner to modern formal language theory (mathematical linguistics) and formal grammar, and a precursor to computing. The use of meta-rules, transformations, and recursion together make it as rigorous as a modern Turing machine. The approach for deriving complex structures and sentences represent modern finite state machines. The work of Panini is used in several theories like in Optimality Theory, the hypothesis about the relation between specific and general constraints is known as “Panini’s Theorem on Constraint Ranking”. Behaghel’s law of increasing terms, which is proposed to be called Panini’s Law by Cooper and Ross, who found it to be taken from Panini’s work.
Various resources have been spent without any considerable success on designing an unambiguous representation of natural languages to make them accessible to computer processing through creating schemata designed to parallel logical relations and after all the effort, there is a widespread belief that natural languages are unsuitable for the transmission of many ideas that artificial languages can render with great precision and mathematical rigour.  But in case of Saṃskrta, with a long philosophical and grammatical tradition, the method for paraphrasing Saṃskrta in a manner that is identical not only in essence but in form with current work in AI and it can serve as an artificial language also, and that much work in AI has been reinventing a wheel millennia old.
The same observation has been made many times by many scholars like In July 1987, Forbes magazine published an article on how Saṃskrta is the most convenient language for computer software programming. A paper by Rick Briggs talks about using Saṃskrta in natural language processing and in 1980, NASA declared it the most unambiguous of all human speech and best suited for Computer processing and artificial intelligence., 
Panini’s work also includes use of Boolean logic, null operator among other things and it’d be quite difficult to disapprove that the same ideas weren’t taken from these texts as we’ll see how plagiarism of ideas from Saṃskrta texts is much more common than generally known and openly acknowledged.
These ancient works on Saṃskrta grammar is praised and appreciated by many scholars like George Cardona, who described it as greatest monuments of human intelligence while Sir Monier Williams mentioned it as wondrous capacity of the human brain, which till today no other country has been able to produce except India. Another academician Bilas Sarda mentioned it as one of the most remarkable literary works that the world has ever seen, and which no other country can produce any grammatical system at all comparable to it, either for originality of plan or analytical subtlety.
Nirukta (Devanāgarī: निरुक्त)
This branch of knowledge deals with etymology and morphology aspect of the language i.e. the systematic creation of words and how to understand them through the use of stems, root words, prefixes, suffixes etc. There are set of rules to compose new words or to understand the meaning of the word, which hasn’t been seen before without referring the dictionary and to create new words as and when need arises. However, this branch is not to be confused with a dictionary, a genre of texts called kośa (Devanāgarī: कोश). The texts of this branch are also called nirvācanaśāstra (Devanāgarī: निर्वाचन शास्त्र) and the two famous ones being nirkuta by Yāska and more ancient one nighaṃṭu (Devanāgarī: निघंटु).
The central idea of Yāska, the author of text Nirukta, is that we need new words to conceptualize and describe actions, i.e. nouns often have verbal roots. The meaning and etymology of words is context dependent. Words are created around object-agent to express external or internal reality, which are one of six modifications of kriyā (Devanāgarī: क्रिया) and bhāva (Devanāgarī: भाव), namely being born, existing, changing, increasing, decreasing and perishing.
As per Yāska, Veda, can be interpreted in three ways – from the perspective of adhiyajña (Devanāgarī: अधियज्ञ), from the perspective of the adhideva (Devanāgarī: अधिदेव), and from the perspective of the adhyātma (Devanāgarī: अध्यात्म). It is quite common in Saṃskrta texts that poets have often embedded and expressed double meanings, ellipses and novel ideas in their composition.11 So, one single text often has many alternate embedded meanings, and this is quite common in all ancient Saṃskrta works. Nirukta helps one to identify alternate embedded meanings in the texts.
Aṣṭadhyāyī (Devanāgarī: अष्टध्यायी) describes numerous usage of words, and how the meaning of a word is driven by overall context of the sentences and composition it is found in. The popular usage and meaning of a word at the time the text was composed supersedes the historical or etymologically derived meanings of that word, but it is not so when it is quoted (cited or referred to) from another prior art text. In the latter case, the Saṃskrta word is suffixed with iti (Devanāgarī: इति) (thus), wherein it means what the prior text meant it to be.
By combination with the vowels, the consonants, and, without any combination, the vowels themselves formed number of primary roots, out of which secondary roots were developed by the addition of other consonants. All words were formed from these roots, simple words by the addition again of pure or mixed vowel and consonant terminations with or without modification of the root and more complex words by the principle of composition.
The words are formed through precise method. The algorithms take input from dhātupāṭha (Devanāgarī: धातुपाठ) and combine them with upasarga (Devanāgarī: उपसर्ग) and pratya (Devanāgarī: प्रत्य) to output well-formed words. So, to understand some words, which was never seen before, it needs to be broken down into Dhātu, upasarga and pratya and then the meaning of the word could be figured out and opposite for creating new words. E.g. the meaning of Saṃskrta itself is superior art (Saṃs+Krta). So, by learning the algorithm to combine these elements and learning the list of these words, the person could develop the vocabulary of thousands of words in no time without the need of pain to learn them manually. Additionally, the words describe themselves i.e. there is no mapping of arbitrary choosing a word and then mapping it to some meaning and to figure out the meaning of that word would require a dictionary, but here, just by breaking the word, the meaning could be understood without referring to any other text. That also helps in preservation of knowledge and maintaining its integrity, which prevent it from being corrupted and which otherwise could be easily corrupted.
As noted by Max Muller, the number of roots necessary to account for the whole wealth of the English Dictionary, which is said to amount to 250,000 words then, is smaller than that of Panini’s roots, even after they have been reduced to their proper limits. There is no sentence in English of which every word cannot be traced back to the 800 roots, and every thought to the 121 fundamental concepts, which remained after a careful sifting of the materials supplied to us by Panini. All that we admire/pride, our thoughts, whether poetical, philosophical or religious, our whole literature and our whole intellectual life, is built up with about 121 bricks. The sum total of all dictionaries with hundreds of thousands of the words could be reduced to lesser number of roots that are provided in texts, to about 1,000 roots. The Science of Thought goes beyond this, and assures us that every thought that ever crossed the mind of man can be traced back to about 121 simple concepts.
“Don’t memorize, seek the meaning, what has been taken [learned] but not understood, is uttered by mere [memory] recitation,
It never flares up, like dry firewood without fire. The meaning of Speech, is its fruit and flower.”
— Yaska, Nirukta 1.18-1.20
Chanda (Devanāgarī: छन्द)
Chanda, which means “esteemed to please or feel pleasant or gratified or celebrated”, is a branch of knowledge that deals with the poetic metres. These are not reserved for only specific passages, but entire texts are written in different Chanda. It includes both linear and non-linear systems meters, many of which are based on fixed number of syllables and others are based on repeating numbers of morae (matr per foot/pād). The system has seven major meters, called the “seven birds” or “seven mouths of Brihspati“, each with its own rhythm, movements and aesthetics wherein a non-linear structure (aperiodicity) was mapped into a four verse polymorphic linear sequence.
The seven-major ancient Saṃskrta meters are the three 8-syllable gāyatri (Devanāgarī: गायत्री), the four 8-syllable anuṣṭupa (Devanāgarī: अनुष्टुप), the four 11-syllable tristubha (Devanāgarī: त्रिस्तुभ), the four 12-syllable jagatī (Devanāgarī: जगती), and the mixed pād (Devanāgarī: पाद) meters named uṣṇiha (Devanāgarī: उष्णिह), brhatī (Devanāgarī: बृहती) and paṃkti (Devanāgarī: पंक्ति). Beyond these seven meters, Saṃskrta scholars developed numerous other syllable-based meters (akṣaracanda (Devanāgarī: अक्षरचन्द)) with examples including atijagati (Devanāgarī: अतिजगति) (13×4, in 16 varieties), śakkarī (Devanāgarī: शक्करी) (14×4, in 20 varieties), atiśakkarī (Devanāgarī: अतिशक्करी) (15×4, in 18 varieties), aṣṭi (Devanāgarī: अष्टि) (16×4, in 12 varieties), atyaṣṭi (Devanāgarī: अत्यष्टि) (17×4, in 17 varieties), dhrti (Devanāgarī: धृति) (18×4, in 17 varieties), atidhrti (Devanāgarī: अतिधृति) (19×4, in 13 varieties), krti (Devanāgarī: कृति) (20×4, in 4 varieties) and so on.27 Saṃskrta scholars also developed hybrid class of Saṃskrta meters, which combined features of the syllable-based meters and morae-based meters called Matr-chanda.27 Examples of this group of meters include vaitālīya (Devanāgarī: वैतालीय), mātrasamaka (Devanāgarī: मात्रसमक) and gītayarya (Devanāgarī: गीतयर्य). About 150 treatises on Saṃskrta prosody are known, in which some 850 meters were defined. This whole branch clearly indicates that poetry is more a mathematical system than an art as per Saṃskrta scholars and same is the case with other branches like Music etc.
The Saṃskrta texts were composed in a manner where a change in meters was an embedded code to inform the reciter and audience that it marks the end of a section or chapter. The verse perfection in the Saṃskrta texts have led some scholars to identify suspect portions of texts where a line or sections are off the expected meter. Significant changes in meter can also serve as an indication of likely later interpolations or insertions to the text or that the text is a compilation of works of different authors and time periods.
The idea is to keep knowledge in poetic language if we’re anyway keeping it. The attempt to identify the most pleasing sounds and perfect compositions led ancient Indian scholars to study permutations and combinatorial methods of enumerating musical meters.
Pāda (stanza) is defined in Chanda as a group of four verses, also called as four quarters, which is of two types of stanzas viz., vrtta (Devanāgarī: वृत्त) and jātī (Devanāgarī: जाती). Vrtta pāda are those that are crafted with precise number syllables, while Jāti stanza are those that are designed based on syllabic instants (morae, matr). The vrtta pād are further recognized in three forms, with Sāma- vrtta (Devanāgarī: साम) where the four verses are similar in its embedded mathematical pattern, ardhasāmavrtta (Devanāgarī: अर्धसामवृत्त) where alternate verses keep similar syllabic structure, and viṣamavrtta (Devanāgarī: विषमवृत्त) where all four quarters are different.
The meters are alternatively classified into three kinds. akṣaravrtta (Devanāgarī: अक्षरवृत्त) (Syllabic verse) meters depend on the number of syllables in a verse, with relative freedom in the distribution of light and heavy syllables. Varṇavrtta (Devanāgarī: वर्णवृत्त) (Syllabo-quantitative verse) meters depend on syllable count, but the light-heavy patterns are fixed. Mātravrtta (Devanāgarī: मात्रवृत्त) (Quantitative verse) meters depend on duration, where each verse-line has a fixed number of morae, usually grouped in sets of four.
Pingala described any meter as a sequence of gaṇa-s (Devanāgarī: गण), or triplets of syllables (trisyllabic feet), plus the excess, if any, as single units. There being eight possible patterns of laghu (Devanāgarī: लघु) (light) and guru (Devanāgarī: गुरु) (heavy) syllables in a sequence of three, a notion similar to “Morse code”. He also associated a letter to each gana, allowing the meter to be described compactly as an acronym. If we assign 0 to laghu and 1 to guru syllable then Pingal’s order of the gaṇa-s, viz. m-y-r-s-t-j-bh-n, corresponds to a standard enumeration in binary system from 000 to 111. This text presents the first known description of a binary numeral system in connection with the systematic enumeration of meters with fixed patterns of short and long syllables with the discussion of the combinatorics of meter corresponds to the binomial theorem. It also includes a discussion of binary system rules to calculate permutations of Saṃskrta meters.33 He also used the Saṃskrta word śūnya (Devanāgarī: शून्य) explicitly to refer to zero. He and other ancient scholars also developed the art of mātrameru (Devanāgarī: मात्रमेरु), which is the field of counting sequences such as 0, 1, 1, 2, 3, 5, 8 and so on which is famously known as “Fibonacci series”. Halāyudha’s (Devanāgarī: हलायुध) commentary on this text, developed meruprastāra (Devanāgarī: मेरुप्रस्तार), which mirrors the “Pascal’s triangle” in the west, and now also called as the Halāyudha’s triangle in books on mathematics. [32, 33] Pingala’s work also include binomial coefficients, combinatorial identity etc.
Ratnākaraśāṃti’s (Devanāgarī: रत्नाकरशांति) caṃdoratnākara (Devanāgarī: चंदोरत्नाकर) describes algorithms to enumerate binomial combinations of meters through pratyaya (Devanāgarī: प्रत्यय).
As per Sheldon Pollock, Chanda have been one of the five categories of Saṃskrta literary knowledge with other four being Guna-s (expression forms), rītī (Devanāgarī: रीती) & mārga (Devanāgarī: मार्ग) (the ways or styles of writing), alaṃkāra (Devanāgarī: अलंकार) (tropology), and rāsa (Devanāgarī: रास) & bhāva (Devanāgarī: भाव) (aesthetic moods and feelings).
With the Gāytri, he measures a song; with the song – a chant, with the Tristubh – a recited stanza, With the stanza of two feet and four feet – a hymn; with the syllable, they measure the seven voices.
— Rigveda 1.164.24, Translated by Tatyana J. Elizarenkova
Kalpa (Devanāgarī: कल्प)
This branch deals with procedures, ceremonies associated with rituals, duties, responsibilities and ethics to oneself, to family and as a member of society among other things with different sections like śrautasūtra (Devanāgarī: श्रौतसूत्र) (use of Veda-s in rituals and their proper performance), grhyasūtra (Devanāgarī: गृह्यसूत्र) (householder rituals), śulbasūtra (Devanāgarī: शुल्बसूत्र) (mathematics for altar building describing geometric formulae and constants), dharmasūtra (Devanāgarī: धर्मसूत्र) (views on right and wrong) (not to be confused with Dharma-shastra-s which are of recent origin) etc.
The starting verses 1-2 of baudhāyana śulbasūtra (Devanāgarī: बौधायन शुल्बसूत्र) state that the squares of any rectangle’s width and length add up to the square of its diagonal, which is also popularly known today as the “Pythagorean theorem”. It also mentioned “Pythagorean triples”, calculating the square root of two up to five decimal places among other things.
Similarly, kātyāyana śulbasūtra (Devanāgarī: कात्यायन शुल्बसूत्र) mentions that the rope which is stretched along the length of the diagonal of a rectangle produces an area which the vertical and horizontal sides make together.
Sulbasūtra also contains methods for calculating square roots, recognition of incommensurables, transformations of figures like transforming the circle into square etc.
Jyotiṣa (Devanāgarī: ज्योतिष)
This branch deals with tracking and predicting the movements of astronomical bodies for time keeping. Like Rishi Lagadha (Devanāgarī: लगध) mentioned that since time is of utmost importance in the performance of yajña (Devanāgarī: यज्ञ) given in Veda-s, so it becomes quite important to understand it. The same is also the case with today where we need to track the passage of time in all kind of experiments, computers etc.
The need to do large number of complicated and complex calculations for time keeping, altar formation etc. is the reason of strong basis of mathematics in India. The ancient Indian scholars’ contributions have been enormous that spread throughout the word in the past but unfortunately no proper credit is given to them today. Some of the contributions are as below to give a general idea.
It is well known that the decimal number system that is used throughout the world and which is also the foundation of our modern mathematics came from India and then transmitted to the Islamic world and to Europe.41 The use of binary system is already mentioned under the works of Panini and Pingala as mentioned earlier. Mathematicians such as Aryabhata, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Bhaskara II, Madhava, Nilakantha Somyaji etc. also wrote their works, as per the Indian tradition where scholars from time to time try to simplify the understanding of the underlying ancient concepts through summaries, aggregations, commentaries etc. like Panini, Pingala Yaska, etc. and later their contributions also spread throughout the world, first to Asia and then the Middle East, and eventually to Europe.
Lagadha in his work Vedanga Jyotiṣa details astronomical calculations, calendrical studies, rules for empirical observation, several important aspects of the time and seasons, including lunar months, solar months, and their adjustment by a lunar leap month of adhimāsa (Devanāgarī: अधिमास). rtu-s (Devanāgarī: ऋतु) as yugama-s (Devanāgarī: युगम), twenty-seven constellations, eclipses, seven planets, and twelve signs of the zodiac among other things.[42, 43]
Aryabhata’s works which also became the basis of development of Islamic astronomy and adoption of Hindu numerals, which later spread to Europe, presents many ideas like the Moon, planets, and asterisms shine by reflected sunlight, the causes of eclipses of the Sun and the Moon and length the size and extent of the Earth’s shadow, implicit knowledge of zero in its place value system, values for π and it being irrational through the word āsanna (Devanāgarī: आसन्न) and the length of the sidereal year at 365 days 6 hours 12 minutes 30 seconds, the diameter of the Earth, described a geocentric model of the solar system and also the heliocentric model in the śīghrocca (Devanāgarī: शीघ्रोच्च) model, in which the Sun and Moon are each carried by epicycles, motions of the planets are each governed by two epicycles, a smaller maṃda (Devanāgarī: मंद) (slow) and a larger śīghra (Devanāgarī: शीघ्र) (fast) with order of the planets in terms of distance from earth is Moon, Mercury, Venus, the Sun, Mars, Jupiter, Saturn, and the asterisms, mentioned that the earth rotates about its axis and not the sky, thereby causing what appears to be an apparent westward motion of the stars among other things like arithmetic, algebra including solutions of simultaneous quadratic equations, linear equations, intermediate linear equations, formula of sum of cubes, plane trigonometry including definition of sine, cosine, inverse sine, and methods to calculate their approximate values, Aryabhata‘s sine table and table of their values, trigonometric formulae like sin(n + 1)x − sin nx = sin nx − sin(n − 1)x − (1/225) sin nx, spherical trigonometry, continued fractions, quadratic equations, sums-of-power series, table of sines, summation of series of squares and cubes, etc. 
Varahmihira made many important contributions including the following in trigonometry
sin^2(x) + cos^2(x) = 1
sin(x) = cos(pi/2-x)
(1-\cos(2x))/2 = sin^2(x)
Bhaskara I mentions sine approximation, which is also known as “Bhaskara sine approximation formula”, assertion that if p is a prime number, then 1 + (p–1)! is divisible by p which was also mentioned by Fibonacci, and is now known as “Wilson’s theorem”, theorems about the solutions of today so called “Pell equations” among other things.
Baṭeśvara, in his work baṭeśvara siddhāṃta (Devanāgarī: बटेश्वर सिद्धांत) devised methods for determining the parallax in longitude directly, the motion of the equinoxes and the solstices, and the quadrant of the sun at any given time etc.
Bhaskara II, the author of the work siddhāṃta śiromaṇi (Devanāgarī: सिद्धांत शिरोमणि), mentioned the principles of differential calculus and its application to astronomical problems and computations, the differential coefficient and differential calculus after discovering an approximation of the derivative and differential coefficient (not surprisingly attributed to Leibniz and Newton half a millennium later), zero, infinity, positive and negative numbers, and indeterminate equations including (the now called) Pell’s equation and solving it using a kuṭṭaka (Devanāgarī: कुट्टक) method (not to be surprised that the exact is mentioned by European mathematicians of the 17th century), solved the case of 61x^2+1=y^2, instantaneous speeds of planets with also stating that at its highest point a planet’s instantaneous speed is zero, proof of so called “Pythagorean theorem”, solutions of quadratic, cubic and quartic indeterminate equations, integer solutions of linear and quadratic indeterminate equations through kuṭṭaka algorithm, cyclic cakravāla (Devanāgarī: चक्रवाल) method for solving indeterminate equations (not surprisingly attributed to William Brouncker of 1657), solutions of “Diophantine equations” of the second order, solving quadratic equations with more than one unknown with negative and irrational solutions, concepts of mathematical analysis, concepts of infinitesimal calculus, along with integral calculus, mean value theorem, stated so called Rolle’s theorem, derivatives of trigonometric functions and formulae, spherical trigonometry along with a number of other trigonometric results, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, combinations, Inverse rule of three, determining unknown quantities, surds, equations of second, third and fourth degree and solutions, equations with more than one unknown, Indeterminate quadratic equations, computation of sines of angles of 18 and 36 degrees, and the now well-known formulae for sin(a+b) and sin(a-b), reference to a perpetual motion machine with description a wheel that he claimed would run forever, yasti yaṃtra (Devanāgarī: यस्ति यंत्र) for determining angles with the help of a calibrated scale among other things like mean and true longitudes and latitudes of the planets, problem of diurnal rotation, sunrise equation, conjunctions of stars, planets, Eccentric epicyclic model of the planets, ellipse calculations etc.47,48,49,50 He has also mentioned the rules of gravitation under the name of mādhyakarṣaṇatatva (Devanāgarī: माध्यकर्षणतत्व) in his book golādhyāya (Devanāgarī: गोलाध्याय). He also reasoned that a quantity which has zero as its divisor there is no change even when many quantities have entered it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [Him, there is no change in] the infinite and unchanging [Vishnu (one who is present in every anu or particle)].
Nīlakaṇṭha (Devanāgarī: नीलकण्ठ) in his commentary on āryabhaṭīya (Devanāgarī: आर्यभटीय) had discussed infinite series expansions of trigonometric functions among other topics. In his work, taṃtrasaṃgraha (Devanāgarī: तंत्रसंग्रह), and in a commentary on this work, called taṃtrasaṃgrahavyākhyā (Devanāgarī: तंत्रसंग्रहव्याख्या), there is a description of series expansion for trigonometric functions, power series, geometric series like (1-x)-1, Sigma (in), i * Sin(x/i), i * (1-Cos(x/i)), i * Arctan(y/x), differentiation of some trigonometric functions, approximation of the error for the finite sum of their series, fluxional forms and series, theory that the area under a curve is its integral, use of mathematical induction, notion of limit some etc. with some of these series also known as “Taylor series”, “Gregory series” etc. 
Brahmasphuṭasiddhāṃta (Devanāgarī: ब्रह्मस्फुटसिद्धांत) by Brahmagupta (Devanāgarī: ब्रह्मगुप्त ), includes a good understanding of the role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta‘s identity, Brahmagupta‘s Theorem on rational triangles, Brahmagupta formula for the area of a cyclic quadrilateral (a generalisation of “Heron’s formula”), Brahmagupta‘s Theorem on rational triangles, solution of the general linear equation, two equivalent solutions to the general quadratic equation, solution of simultaneous indeterminate equations, operations on fractions including cube and cube-root of an integer, computation of squares and square roots, sum of different kind of series, concept and usage of positive negative and zero, so called “Pythagorean” triples, Brahmagupta’s theorem, Brahmagupta‘s interpolation formula, Brahmagupta–“Fibonacci” identity, Brahmagupta matrix among other things.[52,53,54]
Virsen, gave rules for binary logarithm when applied to powers of two, three and four, derivation of the volume of a frustum by a sort of infinite procedure among other things.[55,56]
Mahavira Acharya, gave sum of a series whose terms are squares of an arithmetical progression, rules for area and perimeter of an ellipse, general solutions of the higher order polynomial equations and solved some quintic equations and higher-order polynomials, indeterminate quadratic, cubic and indeterminate higher order equations, formula for nCr, asserted that the square root of a negative number did not exist, rules for decomposing fractions among other things.[58,59]
Shridhara gave rule for finding the volume of a sphere, methods of summation of different arithmetic and geometric series, Munjul gave elaboration of Aryabhata’s differential equation etc.
Shripati Mishra worked mainly on Permutations and combinations among other things.
Parmeshvara’s work līlāvatībhāṣya (Devanāgarī: लीलावती भाष्य), contains a version of the mean value theorem.
Citrabhanu gave integer solutions to 21 types of systems of two simultaneous algebraic equations in two unknowns. These types are all the possible pairs of equations of the following seven forms:
x + y = a, x – y = b, xy = c, x^2 + y^2 = d, x^2 – y^2 = e, x^3 + y^3 = f, x^3 – y^3 = g
The text sūrya siddhāṃta (Devanāgarī: सूर्य सिद्धांत), discusses how to use the movement of planets, sun and moon to keep time and calendar. It also has trigonometry formulae to represents orbits, predict planetary positions and calculate relative mean positions of celestial nodes and apsides. It also contains formulae to calculate the distance between earth and moon and the distance between earth and Sun along with the time occurrence of solar/lunar eclipses. It includes many formulae to calculate the duration of day, the position of planets at various times among many other things. It structured time into Yuga, divided into multiple lunisolar intervals such as 60 solar months, 61 sāvana (Devanāgarī: सावन) months, 62 synodic months and 67 sidereal months. A Yug had 1,860 tithi (Devanāgarī: तिथि) (dates), and it defined a sāvana-day (civil day) from one sunrise to another. It mentioned formulae to predict the length of day time, sun rise and moon cycles. For example,
The length of daytime = (12 + n * (2/61)) muhūrta (Devanāgarī: मुहूर्त)
where n is the number of days after or before the winter solstice, and one muhurta equals 1⁄30 of a day (~48 minutes)
The Kalpa and Jyotiṣa have been the reason for strong base of Mathematics in India as there was strong need of mathematics to calculate and measure various things. The same knowledge spread throughout the world. Many Scholars have noted how many aspects of our present day mathematics have their origins in India.
Florian Cajori, for example, suggested that Diophantus got his algebraic knowledge from India. P. S. Laplace mentioned that Indians gave the ingenious method of expressing all numbers by means of ten symbols with each symbol receiving a value of position as well as an absolute value. Sir Mountstuart Elphinstone noted the Indians knowledge of various fields of Mathematics like Algebra much before they were known in the West or Arab world. Francois Marie Arouet Voltaire, Pierre Sonnerat, Sir William Temple, H. G. Rawlinson, noted that Pythagoras went to the Ganges to learn Indian sciences. Leopold von Schroeder noted that all the philosophical and mathematical doctrines attributed to Pythagoras are derived from India. Charles Whish, claimed that Indians heralded the discovery of the calculus. Pingree, noted that Indian mathematicians discovered an alternate and powerful solutions to many of our mathematical problems. Joseph and Plofker, noted that the concept of differentiation was understood in India and Indians being founders of mathematical analysis and precursor of Newton and Leibniz in the discovery of the principle of the differential calculus. Also, G. G. Joseph among many other scholars have noted the manipulation and suppression of facts regarding the source of modern European knowledge. It is of common knowledge that many interested students from various countries used to come to India to the city of Kashi and other ancient Indian universities to learn Indian sciences, as Varāhmihira notes in his work brhatsaṃhitā (Devanāgarī: बृहत्संहिता) verse 2.15. Thus, there was a lot of flow of mathematical and astronomical knowledge from India to the West.
[The current year] minus one, multiplied by twelve, multiplied by two, added to the elapsed [half months of current year], increased by two for every sixty [in the sun], is the quantity of half-months (syzygies).
— Rigveda Jyotiṣa-Vedānga 4 Translator: Kim Plofker
In this article, we have briefly summarized important aspects of different Vedangas. They are called the limbs of Vedas because without their assistance, it is impossible to properly understand the Vedic knowledge. The Rishis developed these Vedangas as tools using which the meaning and essence of Vedas were preserved and transmitted over many millennia. Thus, understanding Vedas in a true sense, not only include reading or chanting the verses, but constitutes having a thorough understanding about grammar, etymology, astronomy, mathematics, ritual procedures, prosody, and laws of righteous conduct, among other things. Vedangas provide a wholesome understanding of the Vedas, which goes far beyond mere theoretical interpretation and plunges into the practical, ethical, and spiritual aspects of the Vedic truths and a facilitates a direct realization of those truths.
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