What is scriptural source (preferably vedic) of the term
In which acaryas' writing do we find the description that there are six limbs of the vedas or six Vedangas and what they enlist?
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Vedangas literally mean the limbs of the Vedas. The Vedanga are six auxiliary disciplines traditionally associated with the study and understanding of the Vedas.
They are six in number. Just like the limbs of the body, they perform various supportive and augmenting functions in the study, preservation and protection of the Vedas and the vedic traditions.The six Vedangas are Siksha, Chhanda, Vyakarana, Nirukta, Jyotisha and Kalpa.
Jyotishya = astrology,
Chandas = metre,
Traditionally, Vyakarana and Nirukta are common to all four vedas, whilst each veda has its own shiksha, chandas, kalpa and jyotisha texts.
Panini in his text called Siksha Khanda 8 - Verse 41-42 is mentioning about the the six vedangas and their relation with Purusha .
छन्दः पादौ तु वेदस्य हस्तौ कल्पोऽथ पठ्यते ।
ज्योतिषामयनं चक्षुर्निरुक्तं श्रोत्रमुच्यते ।। 41 ।।
शिक्षा घ्राणं तु वेदस्य मुखं व्याकरणं स्मृतम् ।
तस्मात्साङ्गमधीत्येव ब्रह्मलोके महीयते ।। 42 ।।
Chandas are called as feet of Vedas, Kalpas as hands, Joytisha as eyes, Nirukta as ears, Shiksha as nose and Vyakrana as mouth.
The Mundaka Upanishad is describing these six limbs of vedas in Khanda 1 - Mundaka 1- Mantra 5 .Acharya Shankara also mentioned the above are Vedangas in his Commentary of the same mantra.
तत्रापरा ऋग्वेदो यजुर्वेदः सामवेदोऽथर्ववेदः शिक्षा कल्पो व्याकरणं निरुक्तं छन्दो ज्योतिषमिति । अथ परा यया तदक्षरमधिगम्यते ॥ ५ ॥
tatrāparā ṛgvedo yajurvedaḥ sāmavedo'tharvavedaḥ śikṣā kalpo vyākaraṇaṃ niruktaṃ chando jyotiṣamiti | atha parā yayā tadakṣaramadhigamyate || 5 ||
Of these, the Apara is the Rig Veda, the Yajur Veda, the Sama Veda, and the Atharva Veda, the siksha, the code of rituals, grammar, nirukta, chhandas and astrology. Then the para is that by which the immortal is known.
Shankara's Commentary of above Mantra- Of these, what Apara vidya is explained.
Rig Veda, Yajur Veda, Sama Veda, and the Atharva Veda, these four Vedas, the siksha, the code of rituals, grammar, nirukta, chhandas and astrology, these six angas (of Vedas), all this is knowledge called Apara;
Among these six Vedangas the definition and explanation of "Shiksha" part is provided in The Taittiriya Upanishad Sikshavalli - Anuvaka 2 - Mantra 1.
ॐ शीक्षां व्याख्यास्यामः । वर्णः स्वरः । मात्रा बलम् । साम सन्तानः । इत्युक्तः शीक्षाध्यायः ॥
oṃ śīkṣāṃ vyākhyāsyāmaḥ | varṇaḥ svaraḥ | mātrā balam | sāma santānaḥ | ityuktaḥ śīkṣādhyāyaḥ ||
Om! We shall treat of the phonetics: sound, rhythm, quantity, strength, modulation, union. Thus has been declared the lesson on phonetics.
Commentary - Phonetics (Śikṣā) is the science which treats of sounds and their pronunciation. Or, the word ‘śikṣā’ may here signify the sounds etc., which are treated of in that science. Sound: such as ‘ā’. Rhythm: such as udātta or high-pitched tone. Length: short, long, etc. Strength: intensity of effort. Modulation: pronunciation of sounds in the middle tone. Union: conjunction of several sounds.—These are the things to be learnt.
There are many sources actually.
One of them is the Narada Purana for example.
Quoting from the book "Essence of Narada Purana" (hosted @Kamakoti.org).
In the process of ‘Siksha Nirupana’, Maharshi Sanandana explained to Brahmarshi Narada that six Vedangas constituted the ‘Sadhanas’or the means to accomplish Mukti, viz. Siksha, Kalpa, Vyakarana, Nirukti, Chhandas and Jyotisha. Maha Shuka Muni was a glorious example of ‘Anuchan’ or an outstanding expert in all the Six Vedangas, besides being an epitome of Dharma and an unparalelled ‘Adhyayi’ or a Master of the Four Vedas of Rig, Yajur, Sama and Atharva.
So, as you can see the 6 Angas or limbs of Vedas are - Shiksha (Phonetics), Kalpa (process of how rituals are to be performed), Vyakarana (grammar), Nirukti (etymology), Chandahs (meters) and Jyotisha (astrology).
Further details are found on the same linked page in the same book that I've quoted. But that's too long to quote here.
From the Devi Bhagavata Purana (DBP) too, we are getting the mentions of the six limbs of the Vedas.
From the Devî Gâyatrî has originated first the Rig veda. Brahmâ worships the virgin Gâyatrî; on the idea of S’rî Parames’varî Gâyatrî has four feet. The Rig Veda is one; the Yajurveda is the second, the Sâmaveda is the third and the Atharva veda is the fourth foot. The Gâyatrî has eight bellies; the east side is the one; the south is the second; the west is the third; the north is the fourth; the zenith is the fifth; the nadir is the sixth; the intermediate space is the seventh and all the corners are the eighth belly. Gâyatrî has seven S’iras (heads); Vyâkaranam (Grammar) is one; S’iksâ is the second (that Amga of the Veda, the science which teaches the proper pronunciation of words and laws of euphony); Kalpa is the third (the Vedânga which lays down the ritual and prescribes rules for ceremonial and sacrificial acts); Nirukta is the fourth (the Vedânga that contains glossarial explanation of obscure words, especially those occurring in the Vedas); Jyotish or astronomy is the fifth; Itahâsa (history) and Purânas is the sixth head; and Upanisadas is the seventh head
The source and definition is already covered by Swift Pushkar, I'm here dealing with the second question that looks for bit description from work of Acharyas.
Madhusudana Saraswati, an Advaita Acharya discussed them in his work PrasthanaBheda:
In the introduction to 18 Vidyas, Vedanga are mentioned.
शिक्षा कल्पो व्याकरणं निरुक्तं छन्दो ज्योतिषमिति वेदाङ्गानि षट् ।
The vedAngas are six—shikShA (science of pronunciation), yyAkaraNam (grammar), niruktam (etymology), Chandas (metre), jyoutiSham (astronomy and astrology), and kalpa.
Then he described all of them:
The purpose as well as importance/significance and work on each Vedanga are also discussed.
अथाङ्गानामुच्यते । तत्र शिक्षाया उदात्तानुदात्तस्वरितह्रस्वदीर्घप्लुतादिविशिष्ट- स्वरव्यञ्जनात्मकवर्णोच्चारणविशेषज्ञानं प्रयोजनम् । तदभावे मन्त्राणामनर्थकत्वात् । तथाचोक्तम् (शि- ५२)
मन्त्रो हीनः स्वरतो वर्णतो वा मिथ्याप्रयुक्तो न तमर्थमाह ।
स वाग्वज्रो यजमानं हिनस्ति यथेन्द्रशत्रुः स्वरतोऽपराधात् ॥ इति ।
तत्र सर्व वेदसाधारणी शिक्षा । अथ शिक्षां प्रवक्ष्यामीत्यादिपङ्चखण्डात्मिका पाणिनिना प्रकाशिता । प्रतिवेदशाखं च भिन्नरूपा प्रातिशाख्यसंज्ञिताऽन्यैरेव मुनिभिः प्रदर्शिता ।
Now the differences among the vedAngas are being stated. The purpose of the study of shikShA is the attainment of the knowledge of the svaras such as udAtta, anudAtta, svarita, and the pronunciation of vowels and consonants as short, long and pluta. If the mantras are not correctly pronounced they can be disastrous. So it has been said - “If a mantra is wrong in svara or in any letter, or wrongly used, it does not have the intended meaning. It is a verbal thunder-bolt that destroys the user, like Indra’s enemy because of mistake in svara”. shikShA which is common to all the Vedas has been expounded by pANini in five parts beginning with, “Now I shall expound shikShA”. Other munis have expounded separately for each branch of the Vedas under the name ‘prAtishAkhya’
एवं वैदिकपदसाधुत्वज्ञानेनोहादिकं व्याकरणस्य प्रयोजनम् । तच्च वृद्धिरादैजित्याद्यध्यायाष्टकात्मकं महेश्वरप्रसादेन भगवता पाणिनिनैव प्रकाशितम् । तत्र कात्यायनेन मुनिना पाणिनीयसूत्रेषु वार्तिकं विरचितम् । तद्वार्तिकस्योपरि च भगवता मुनिना पतञ्जलिना महाभाष्यमारचितम् । तदेतत्त्रिमुनि व्याकरणं वेदाङ्गं माहेश्वरमित्याख्यायते कौमारादिव्याकरणानि तु न वेदाङ्गानि किंतु लौकिकप्रयोगमात्रज्ञानार्थानीत्यवगन्तव्यम् ।
vyAkaraNa (grammar) is useful for performing rites by knowing the correct vedic word. It was expounded by bhagavAn pAnIni with the grace of maheshvara and consists of eight chapters beginning with the sUtra, “vRiddhirAdaic”. KatyAyana muni composed a vArtika on the pANini sUtras. On that vArtika the mahAbhAShya was composed by bhagavAn patanjali muni. This grammar composed by the three munis which is a vedAnga is called mAheshvaram. It should be noted that the grammars such as ‘kaumAra’ are not vedAngas but are meant only for knowledge of worldly usages.
एवं शिक्षाव्याकरणाभ्यां वर्णोच्चारणपदसाधुत्वे ज्ञाते वैदिकमन्त्रपदानामर्थज्ञानाकाङ्क्षायां तदर्थ भगवता यास्केन समाम्नायः समाम्नातः स व्याख्यातव्य इत्यादित्रयोदशाध्यायकं निरुक्तमाराचितम् । तत्र च नामाख्यातनिपातोपसर्गभेदेन चतुर्विधं पदजातं निरूप्य वैदिकमन्त्रपदानामर्थः प्रकाशितः । मन्त्राणां चानुष्ठेयार्थप्रकाशनद्वारेणैव करणत्वात्पदार्थज्ञानाधीनत्वाच्च वाक्यार्थज्ञानस्य मन्त्रस्थपदार्थज्ञानाय निरुक्तमवश्यमपेक्षितमन्यथाऽनुष्ठानासंभवात् । सृण्येव जर्भरी तुर्फरी तू इत्यादि दुरूहाणां (निरु.१३-५) प्रकारान्तरेणार्थज्ञानस्यासंभवनीयत्वाच्च । एवं निघण्टवोऽपि वैदिकद्रव्यदेवतात्मकपदार्थपर्यायशब्दात्मका निरुक्तान्तर्भूता एव । तत्रापि निघण्टुसंज्ञकः पञ्चाध्यायात्मको ग्रन्थो भगवता यास्केनैव कृतः ।
When the correct pronunciation of letters and the correctness of words is learnt through shikShA and vyAkaraNa, the desire to know the meaning of the words in the vedic mantras arises. For that purpose bhagavAn yAska composed niruktam with thirteen chapters, beginning with ‘samaamnAyaH samAmnAtaH’, ‘sa vyAkhyAtavyaH’, etc. There the meaning of the words of the vedic mantras has been expounded with the help of the four terms ‘noun’, verb’, ‘indeclinable’ and ‘prefix’. Since the mantras become instruments in a vedic rite by giving the knowledge of what is to be performed and since the meaning of a sentence can be known only by knowing the meaning of the words (of the sentence), nirukta is necessary for knowing the meaning of the words in the mantra; otherwise performance of the rite will not be possible. It is not poss ible to know the meanings of difficult passages such as, “sRiNyeva jarbharI turpharI tU” by any other means. The nikhaNDu which gives the synonyms of the words for vedic materials and deities is included in nirukta. The work ‘nikhaNDU’ consisting of five chapters was composed by bhagavAn Yaska himself.
एवमृङ्मन्त्राणां पादबद्धच्छन्दोविशेषविशिष्टत्वात्तदज्ञाने च निन्दाश्रवणाच्छन्दोविशेषनिमित्तानुष्ठानविशेषविधानाच्च । छन्दोज्ञानाकाङ्क्षायां तत्प्रकाशनाय धीः श्रीः स्त्रीमित्याद्यष्टाध्यायात्मिका छन्दोविवृतिर्भगवता पिङ्गलेन विरचिता । तत्राप्यलौकिकमित्यन्तेनाध्यायत्रयेण गायत्र्युष्णिगनुष्टुब्बृहती पङ्क्तिस्त्रिष्टुब्जगतीति सप्त च्छन्दांसि सावान्तरभेदानि निरूपितानि । अथ लौकिकमित्यारभ्याध्यायपञ्चकेन पुराणेतिहासादावुपयोगीनि लौकिकानि च्छन्दांसि प्रसङ्गान्निरूपितानि व्याकरणे लौकिकपदनिरूपणवत् ।
Thus, since the Rg mantras are metrical in form with each mantra in a particular metre, and ignorance of the metres becomes an object of ridicule, and performance of rites is prescribed on the basis of particular metres, the knowledge of metres is desired. For explaining them cchandovivRiti consisting of eight chapters and beginng with “dhIH shrIH strIm” was composed by bhagavAn pingala. In it, by the three chapters commencing with ‘alaukikam’ seven metres along with their internal variations, namely, gAyatrI, uShNik, anuShTub, bRihatI, pangktiH, triShTub, jagatI, have been described. In five chapters beginning with “athalaukikam” the laukika metres which are useful for purANa and itihAsa, etc., have been explained contextually, like laukika words being explained in grammar.
एवं वैदिककर्माङ्गदर्शादिकालज्ञानाय ज्यौतिषं भगवताऽऽदित्येन गर्गादिभिश्च प्रणीतं बहुविधमेव ।
In order to know the time such as new-moon for the performance of vedic rites, jyotiSha was composed by bhagavAn Aditya and garga and others and it is of many kinds.
शाखान्तरीयगुणोपसंहारेण वैदिकानुष्ठानक्रमविशेषज्ञानाय कल्पसूत्राणि तानि च प्रयोगत्रयभेदात्त्रिविधानि । तत्र हौत्रप्रयोगप्रतिपादकान्याश्वलायनशाङ्खायनादिप्रणीतानि । आध्वर्यवप्रयोगप्रतिपादकानि बौधायनापस्तम्बकात्यायनादिप्रणीतानि । औद्गात्रप्रयोगप्रतिपादकानि लाट्यायनद्राह्व्यायणादिप्रणीतानि ।
For conveying knowledge about the manner in which vedic rites we re to be performed, kalpasUtras were composed, combining the traits of the various shAkhas. They are of three kinds, based on differences in method. The method to be adopted by the hota (Rgvedin) is given in the sUtras of sages AshvalAyana, shAnkhAyana, etc. Those for the adhvaryu (yajurvedin) are in the sUtras of bodhAyana, Apastamba, kAtyAyana, etc. For the udgAta (sAmavedin) the sUtras of lATyAyana, drAhyAyaNa, etc., are applicable.