Indian Vedic Science & Medical Solution Through VEDAS

KAMBA RAMAYANA

2011-09-22T07:57:00.000-07:00

कंब रामायण तमिल साहित्य की सर्वोत्कृष्ट कृति एवं एक बृहत् ग्रंथ है (डा.आर.पी. सेतुपिल्लै, तमिल विभागाध्यक्ष, मद्रास विश्वविद्यालय का अंग्रेजी में "तमिल लिटरेचर" शीर्षक लेख) और इसके रचयिता कंबन "कविचक्रवर्ती" की उपाधि से प्रसिद्ध हैं।

[संपादित करें] परिचय

उपलब्ध ग्रंथ में 10,050 पद्य हैं और बालकांड से युद्धकांड तक छह कांडों का विस्तार इसमें मिलता है। इससे संबंधित एक उत्तरकांड भी प्राप्त है जिसके रचयिता कंबन के समसामयिक एक अन्य महाकवि "ओट्टककूत्तन" माने जाते हैं। पौराणिकों के कारण कंब रामायण में अनेक प्रक्षेप भी जुड़ गए हैं किंतु इन्हें बड़ी आसानी से पहचाना जा सकता है क्योंकि कंबन की सशक्त भाषा और विलक्षण प्रतिपादन शैली का अनुकरण शक्य नहीं है।
कंब रामायण का कथानक वाल्मीकि रामायण से लिया गया है, परंतु कंबन का मूल रामायण का अनुवाद अथवा छायानुवाद न करके, अपनी दृष्टि और मान्यता के अनुसार घटनाओं में सैकड़ों परिवर्तन किए हैं। विविध परिस्थितियों के प्रस्तुतीकरण, घटनाओं के चित्रण, पात्रों के संवाद, प्राकृतिक दृश्यों के उपस्थापन तथा पात्रों की मनोभावनाओं की अभिव्यक्ति में पदे-पदे मौलिकता मिलती है। तमिल भाषा की अभिव्यक्ति और संप्रेषणीयता को सशक्त बनाने के लिए भी कवि ने अनेक नए प्रयोग किए हैं। छंदोविधान, अलंकारप्रयोग तथा शब्दनियोजन के माध्यम से कंबन ने अनुपम सौंदर्य की सृष्टि की है। सीता-राम-विवाह, शूर्पणखा प्रसंग, बालिवध, हनुमान द्वारा सीता संदर्शन, इंद्रजीतवध, राम-रावण-युद्ध आदि प्रसंग अपने-अपने काव्यात्मक सौंदर्य के कारण विशेष आकर्षक हैं। लगता है, प्रत्येक प्रसंग अपने में पूर्ण है और नाटकीयता से ओतप्रोत है। घटनाओं के विकास के सुनिश्चित क्रम हैं। प्रत्येक घटना आरंभ, विकास और परिसमाप्ति में एक विशिष्ट शिल्पविधान लेकर सामने आती है।
वाल्मीकि ने राम के रूप में "पुरुष पुरातन" का नहीं, अपितु "महामानव का चित्र उपस्थित किया था, जबकि कंबन ने अपने युगादर्श के अनुरूप राम को परमात्मा के अवतार के साथ आदर्श महामानव के रूप में भी प्रतिष्ठित किया। वैष्णव भक्ति तत्कालीन मान्यताओं और जनता की भक्तिपूत भावनाओं से जुड़े रहकर इस महाकवि ने राम के चरित्र को महत्तापूरित एवं परमपूर्णत्व समन्वित ऐसे आयामों में प्रस्तुत किया जिनकी इयत्ता और ईदृक्ता सहज ग्राह्य होते हुए भी अकल्पनीय रूप से मनोहर किंवा मनोरम थी। यह निश्चित ही कंबन जैसा अनन्य सुलभ प्रतिभावान् महाकवि ही कर सकता था।
कंब रामायण का प्रचार प्रसार केवल तमिलनाडु में ही नहीं, उसके बाहर भी हुआ। तंजौर जिले में स्थित तिरुप्पणांदाल मठ की एक शाखा वाराणसी में है। लगभग 350 वर्ष पूर्व कुमारगुरुपर नाम के एक संत उक्त मठ में रहते थे। संध्यावेला में वे नित्यप्रति गंगातट पर आकर कंब रामायण की व्याख्या हिंदी में सुनाया करते थे। गोस्वामी तुलसीदास उन दिनों काशी में ही थे और संभवत: रामचरितमानस की रचना कर रहे थे। दक्षिण में जनविश्वास प्रचलित है कि तुलसीदास ने कंब रामायण से प्ररेणा ही प्राप्त नहीं की, अपितु मानस में कई स्थलों पर अपने ढंग से, उसकी सामग्री का उपयोग भी किया। यद्यपि उक्त विश्वास की प्रामाणिकता विवादास्पद है, तो भी इतना सच है कि तुलसी और कंबन की रचनाओं में कई स्थलों पर आश्चर्यजनक समानता मिलती है।
श्री वी.वी.एस. अय्यर (कंब रामायण - ए स्टडी) के अनुसार "कंब रामायण विश्वसाहित्य में उत्तम कृति है। इलियड, पैराडाइज़ लॉस्ट और महाभारत से ही नहीं, वरन् आदिकाव्य वाल्मीकि रामायण की तुलना में भी यह अधिक सुंदर है।"

वेद

2011-09-22T07:53:00.001-07:00

वेद शब्द संस्कृत भाषा के "विद्" धातु से बना है जिसका अर्थ है: जानना, ज्ञान इत्यादि। वेद हिन्दू धर्म के प्राचीन पवित्र ग्रंथों का नाम है । वेदों को श्रुति भी कहा जाता है, क्योकि पहले मुद्रण की व्यवस्था न होने से इनको एक दुसरे से सुन- सुनकर याद रखा गया इसप्रकार वेद प्राचीन भारत के वैदिक काल की वाचिक/श्रुति = श्रवण परम्परा की अनुपम कृति है जो पीढी दर पीढी पिछले चार-पाँच हजार वर्षों से चली आ रही है । वेद ही हिन्दू धर्म के सर्वोच्च और सर्वोपरि धर्मग्रन्थ हैं ।

अनुक्रम

[छुपाएँ]

[संपादित करें] वेदों का महत्व

भारतीय संस्कृति के मूल वेद हैं। ये हमारे सबसे पुराने धर्म-ग्रन्थ हैं और हिन्दू धर्म का मुख्य आधार हैं।
न केवल धार्मिक किन्तु ऐतिहासिक दृष्टि से भी वेदों का असाधारण महत्त्व है। वैदिक युग के आर्यों की संस्कृति और सभ्यता जानने का एकमात्र साधन यही है।
मानव-जाति और विशेषतः आर्य जाति ने अपने शैशव में धर्म और समाज का किस प्रकार विकास किया इसका ज्ञान वेदों से ही मिलता है।
विश्व के वाङ्मय में इनसे प्राचीनतम कोई पुस्तक नहीं है।
आर्य-भाषाओं का मूलस्वरूप निर्धारित करने में वैदिक भाषा बहुत अधिक सहायक सिद्ध हुई है।

[संपादित करें] वैदिक वाङ्मय का शास्त्रीय स्वरुप

वर्तमान काल में वेद चार माने जाते हैं। उनके नाम हैं-
(१) ऋग्वेद, (२) यजुर्वेद, (३) सामवेद तथा (४) अथर्ववेद
द्वापरयुग की समाप्ति के पूर्व वेदों के उक्त चार विभाग अलग-अलग नहीं थे। उस समय तो ऋक्, यजुः और साम - इन तीन शब्द-शैलियों की संग्रहात्मक एक विशिष्ट अध्ययनीय शब्द-राशि ही वेद कहलाती थी। विश्व में शब्द-प्रयोग की तीन शैलियाँ होती है; जो पद्य (कविता), गद्य और गानरुप से प्रसिद्ध हैं। पद्य में अक्षर-संख्या तथा पाद एवं विराम का निश्चित नियम होता है। अतः निश्चित अक्षर-संख्या तथा पाद एवं विराम वाले वेद-मन्त्रों की संज्ञा ‘ऋक्’ है। जिन मन्त्रों में छन्द के नियमानुसार अक्षर-संख्या तथा पाद एवं विराम ऋषिदृष्ट नहीं है, वे गद्यात्मक मन्त्र ‘यजुः’ कहलाते हैं। और जितने मन्त्र गानात्मक हैं, वे मन्त्र ‘साम’ कहलाते हैं। इन तीन प्रकार की शब्द-प्रकाशन-शैलियों के आधार पर ही शास्त्र एवं लोक में वेद के लिये ‘त्रयी’ शब्द का भी व्यवहार किया जाता है।
वेद के पठन-पाठन के क्रम में गुरुमुख से श्रवण एवं याद करने का वेद के संरक्षण एवं सफलता की दृष्टि से अत्यन्त महत्त्व है। इसी कारण वेद को ‘श्रुति’ भी कहते हैं। वेद परिश्रमपूर्वक अभ्यास द्वारा संरक्षणीय है, इस कारण इसका नाम ‘आम्नाय’ भी है।
द्वापरयुग की समाप्ति के समय श्रीकृष्णद्वैपायन वेदव्यास जी ने यज्ञानुष्ठान के उपयोग को दृष्टिगत उस एक वेद के चार विभाग कर दिये और इन चारों विभागों की शिक्षा चार शिष्यों को दी। ये ही चार विभाग ऋग्वेद, यजुर्वेद, सामवेद और अथर्ववेद के नाम से प्रसिद्ध है। पैल, वैशम्पायन, जैमिनि और सुमन्तु नामक - इन चार शिष्यों ने शाकल आदि अपने भिन्न-भिन्न शिष्यों को पढ़ाया। इन शिष्यों के द्वारा अपने-अपने अधीत वेदों के प्रचार व संरक्षण के कारण वे शाखाएँ उन्हीं के नाम से प्रसिद्ध हो रही है।

[संपादित करें] कर्मकाण्ड में वर्गीकरण

वेदों का प्रधान लक्ष्य आध्यात्मिक ज्ञान देना ही है। अतः वेद में कर्मकाण्ड और ज्ञानकाण्ड - इन दोनों विषयों का सर्वांगीण निरुपण किया गया है। वेदों का प्रारम्भिक भाग कर्मकाण्ड है और वह ज्ञानकाण्ड वाले भाग से अधिक है। जिन अधिकारी वैदिक विद्वानों को यज्ञ कराने का यजमान द्वारा अधिकार प्राप्त होता है, उनको ‘ऋत्विक’ कहते हैं। श्रौतयज्ञ में इन ऋत्विकों के चार गण हैं। (१) होतृगण, (२) अध्वर्युगण, (३) उद्गातृगण तथा (४) ब्रह्मगण। उपर्युक्त चारों गणों के लिये उपयोगी मन्त्रों के संग्रह के अनुसार वेद चार हुए हैं।
(१) ऋग्वेद- इसमें होतृवर्ग के लिये उपयोगी मन्त्रों का संकलन है। इसमें ‘ऋक्’ संज्ञक (पद्यबद्ध) मन्त्रों की अधिकता के कारण इसका नाम ऋग्वेद हुआ। इसमें होतृवर्ग के उपयोगी गद्यात्मक (यजुः) स्वरुप के भी कुछ मन्त्र हैं।
(२) यजुर्वेद- इसमें यज्ञानुष्ठान सम्बन्धी अध्वर्युवर्ग के उपयोगी मन्त्रों का संकलन है। इसमें ‘गद्यात्मक’ मन्त्रों की अधिकता के कारण इसका नाम ‘यजुर्वेद’ है। इसमें कुछ पद्यबद्ध, मन्त्र भी हैं, जो अध्वर्युवर्ग के उपयोगी हैं। यजुर्वेद के दो विभाग हैं- (क) शुक्लयजुर्वेद और (ख) कृष्णयजुर्वेद।
(३) सामवेद- इसमें यज्ञानुष्ठान के उद्गातृवर्ग के उपयोगी मन्त्रों का संकलन है। इसमें गायन पद्धति के निश्चित मन्त्र होने के कारण इसका नाम सामवेद है।
(४) अथर्ववेद- इसमें यज्ञानुष्ठान के ब्रह्मवर्ग के उपयोगी मन्त्रों का संकलन है। अथर्व का अर्थ है कमियों को हटाकर ठीक करना या कमी-रहित बनाना। अतः इसमें यज्ञ-सम्बन्धी एवं व्यक्ति सम्बन्धी सुधार या कमी-पूर्ति करने वाले मन्त्र भी है। इसमें पद्यात्मक मन्त्रों के साथ कुछ गद्यात्मक मन्त्र भी उपलब्ध है। इस वेद का नामकरण अन्य वेदों की भाँति शब्द-शैली के आधार पर नहीं है, अपितु इसके प्रतिपाद्य विषय के अनुसार है। इस वैदिक शब्दराशि का प्रचार एवं प्रयोग मुख्यतः अथर्व नाम के महर्षि द्वारा किया गया। इसलिये भी इसका नाम अथर्ववेद है।

[संपादित करें] वैदिक स्वर प्रक्रिया

वेद की संहिताओं में मंत्राक्षरॊं में खड़ी तथा आड़ी रेखायें लगाकर उनके उच्च, मध्यम, या मन्द संगीतमय स्वर उच्चारण करने के संकेत किये गये हैं। इनको उदात्त, अनुदात्त ऒर स्वारित के नाम से अभिगित किया गया हैं। ये स्वर बहुत प्राचीन समय से प्रचलित हैं और महामुनि पतंजलि ने अपने महाभाष्य में इनके मुख्य मुख्य नियमों का समावेश किया है ।

[संपादित करें] चार वेद

वेद के असल मन्त्र भाग को संहिता कहते हैं ।

ऋग्वेद (इसमें देवताओं का आह्वान करने के लिये मन्त्र हैं -- यही सर्वप्रथम वेद है)(यह वेद मुख्यतः ऋषि मुनियों के लिये होता है)
सामवेद (इसमें यज्ञ में गाने के लिये संगीतमय मन्त्र हैं)(यह वेद मुख्यतः गन्धर्व लोगो के लिये होता है)
यजुर्वेद (इसमें यज्ञ की असल प्रक्रिया के लिये गद्य मन्त्र हैं)(यह वेद मुख्यतः क्षत्रियो के लिये होता है)
अथर्ववेद (इसमें जादू, चमत्कार, आरोग्य, यज्ञ के लिये मन्त्र हैं)(यह वेद मुख्यतः व्यापारियो के लिये होता है)

[संपादित करें] चार भाग

हर वेद के चार भाग होते हैं । पहले भाग (संहिता) के अलावा हरेक में टीका अथवा भाष्य के तीन स्तर होते हैं । कुल मिलाकर ये हैं :

संहिता (मन्त्र भाग)
ब्राह्मण-ग्रन्थ (गद्य में कर्मकाण्ड की विवेचना)
आरण्यक (कर्मकाण्ड के पीछे के उद्देश्य की विवेचना)
उपनिषद (परमेश्वर, परमात्मा-ब्रह्म और आत्मा के स्वभाव और सम्बन्ध का बहुत ही दार्शनिक और ज्ञानपूर्वक वर्णन)

ये चार भाग सम्मिलित रूप से श्रुति कहे जाते हैं जो हिन्दू धर्म के सर्वोच्च ग्रन्थ हैं । बाकी ग्रन्थ स्मृति के अन्तर्गत आते हैं ।

यह लेख एक आधार है। इसे बढ़ाकर आप विकिपीडिया की सहायता कर सकते हैं।

[संपादित करें] वेदों का विभाजन

आधुनिक विचारधारा के अनुसार चारों वेदों की शब्द-राशि के विस्तार में तीन दृष्टियाँ पायी जाती है-

याज्ञिक,
प्रायोगिक और
साहित्यिक दृष्टि।

[संपादित करें] याज्ञिक दृष्टिः

इसके अनुसार वेदोक्त यज्ञों का अनुष्ठान ही वेद के शब्दों का मुख्य उपयोग माना गया है। सृष्टि के आरम्भ से ही यज्ञ करने में साधारणतया मन्त्रोच्चारण की शैली, मन्त्राक्षर एवं कर्म-विधि में विविधता रही है। इस विविधता के कारण ही वेदों की शाखाओं का विस्तार हुआ है। यथा-ऋग्वेद की २१ शाखा, यजुर्वेद की १०१ शाखा, सामवेद की १००० शाखा और अथर्ववेद की ९ शाखा- इस प्रकार कुल १,१३१ शाखाएँ हैं। इस संख्या का उल्लेख महर्षि पतञ्जलि ने अपने महाभाष्य में भी किया है। उपर्युक्त १,१३१ शाखाओं में से वर्तमान में केवल १२ शाखाएँ ही मूल ग्रन्थों में उपलब्ध हैः-

ऋग्वेद की २१ शाखाओं में से केवल २ शाखाओं के ही ग्रन्थ प्राप्त हैं-
1. शाकल-शाखा और
2. शांखायन शाखा।
यजुर्वेद में कृष्णयजुर्वेद की ८६ शाखाओं में से केवल ४ शाखाओं के ग्रन्थ ही प्राप्त है-
1. तैत्तिरीय-शाखा,
2. मैत्रायणीय शाखा,
3. कठ-शाखा और
4. कपिष्ठल-शाखा
शुक्लयजुर्वेद की १५ शाखाओं में से केवल २ शाखाओं के ग्रन्थ ही प्राप्त है-
1. माध्यन्दिनीय-शाखा और
2. काण्व-शाखा।
सामवेद की १,००० शाखाओं में से केवल २ शाखाओं के ही ग्रन्थ प्राप्त है-
1. कौथुम-शाखा और
2. जैमिनीय-शाखा।
अथर्ववेद की ९ शाखाओं में से केवल २ शाखाओं के ही ग्रन्थ प्राप्त हैं-
1. शौनक-शाखा और
2. पैप्पलाद-शाखा।

उपर्युक्त १२ शाखाओं में से केवल ६ शाखाओं की अध्ययन-शैली प्राप्त है-शाकल, तैत्तरीय, माध्यन्दिनी, काण्व, कौथुम तथा शौनक शाखा। यह कहना भी अनुपयुक्त नहीं होगा कि अन्य शाखाओं के कुछ और भी ग्रन्थ उपलब्ध हैं, किन्तु उनसे शाखा का पूरा परिचय नहीं मिल सकता एवं बहुत-सी शाखाओं के तो नाम भी उपलब्ध नहीं हैं।

[संपादित करें] प्रायोगिक दृष्टिः

इसके अनुसार प्रत्येक शाखा के दो भाग बताये गये हैं।

मन्त्र भाग- यज्ञ में साक्षात्-रुप से प्रयोग आती है।
ब्राह्मण भाग- जिसमें विधि (आज्ञाबोधक शब्द), कथा, आख्यायिका एवं स्तुति द्वारा यज्ञ कराने की प्रवृत्ति उत्पन्न कराना, यज्ञानुष्ठान करने की पद्धति बताना, उसकी उपपत्ति और विवेचन के साथ उसके रहस्य का निरुपण करना है।

[संपादित करें] साहित्यिक दृष्टि

इसके अनुसार प्रत्येक शाखा की वैदिक शब्द-राशि का वर्गीकरण-

संहिता,
ब्राह्मण,
आरण्यक और
उपनिषद् इन चार भागों में है।

[संपादित करें] वेद के अंग, उपांग एवं उपवेद

वेदों के सर्वांगीण अनुशीलन के लिये शिक्षा, कल्प, व्याकरण, निरुक्त, छन्द और ज्योतिष- इन ६ अंगों के ग्रन्थ हैं। प्रतिपदसूत्र, अनुपद, छन्दोभाषा (प्रातिशाख्य), धर्मशास्त्र, न्याय तथा वैशेषिक- ये ६ उपांग ग्रन्थ भी उपलब्ध है। आयुर्वेद, धनुर्वेद, गान्धर्ववेद तथा स्थापत्यवेद- ये क्रमशः चारों वेदों के उपवेद कात्यायन ने बतलाये हैं।

वेदः SANSKRIT

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चत्वारः वेदाः भवन्ति । ऋग्वेदः, यजुर्वेदः, सामवेदः, अथर्ववेदश्चेति । एकैकस्यापि संहिता, ब्राह्मणम्, आरण्यकम्, उपनिषत् इत्येवं विभागः अस्ति । वेदाः उत्कृष्टाः साहित्यकृतयः भवन्ति । तानि च सूक्तानि प्रतिभानवतां ऋषीणां यॊगदानानि भवन्ति । एकैकस्यापि सूक्तस्य ऋषिः, छन्दः, देवता इति त्रितयमस्ति ।
अनादिनिधनाः वेदाः ब्रह्मणः चतुर्भ्य मुखेभ्यः निःस्सृता इति प्राक्तनैः निरूपितम् ।

अन्तर्विषयाः

[गोपयतु]

[सम्पादयतु] ऋग्वेदः

मुख्य पृष्ठ :: ऋग्वेदः

Rigveda (padapatha) manuscript in Devanagari, early 19th century

वेदेषु आदिमः ऋग्वेदः हिन्दुधर्मस्य मूलग्रन्थः अस्ति । ऋग्वेदः ४५०० वर्षेभ्यः प्राक् संग्रथित: इति मन्यन्ते । अस्य १०१७ सूक्तानि सन्ति । तस्य श्लोकाः विविधदेवानां सम्बद्धा: - यथा इन्द्रः, अग्निः, वायुः इत्यादयः । ऋग्वेदस्य १०५८९ संहिताः, १०२८ सूक्तानि च १० मण्डलै: विभाजिता: सन्ति । महामुनॆ: व्यासस्य निर्देशे पैलः ऋग्वेदस्य संहितानां निर्माणम् अकरोत् ।

[सम्पादयतु] यजुर्वेदः

मुख्य पृष्ठ :: यजुर्वेदः

आर्याणां कुरुषु अधिनिवेशकाले संग्रथितो यजुर्वेद: इति अभिप्रायः । यजुर्वेदस्य अध्वरवेद इति नामान्तरमस्ति । यजुषः एकोत्तरशतं शाखाः सन्ति इति पतञ्जलिः प्रपञ्चहृदयकारः च प्रस्तौति । वाजसनेयापरनामा कृष्णयजु‍वेदः गद्यपद्यात्मक: । यदीया रचना विश्ववश्या देदीप्यते ।

[सम्पादयतु] सामवेदः

मुख्य पृष्ठ :: सामवेदः

साम- सान्त्वेन इति धातोः निष्पन्नं सामपदम् । सा इति ऋक्सूचकतया अम इति गानम् । हराबिव्यञ्गकतया च व्याख्यां केचिद् वदन्ति । सामवेदस्य एकसहस्रं शाखा असन् किल । प्रपञहृदयकारस्यकाले द्वादशशाखाभ्यः अन्याः नष्टाः इदानीन्तु कॆवलं तिस्रः शाखाः समुपलभ्यन्ते । सङ्गीतस्य उद्भवः सामगानात् इति विचक्षणा आचक्षते । सामगानेऽ पि सप्तस्वरा: एव भवन्ति । ते आधुनिकशास्त्रीयसङ्गीतसंविधानात् आरोहावरोहणक्रमे किञ्चिदिव व्यत्यस्ताः दृश्यन्ते । खरहरप्रियारागतुल्याः सामगानस्वराः कुष्ठं (प्रथमं), द्वितीयं, तृतीयं (मध्यमम्), चतुर्थं, मन्द्रं (पञ्चमम्), अतिस्वायं (षष्ठम्), अतिस्वरम् (अन्यं) एते सप्तस्वराः ।सामगानालापने गायकैः हस्ताङ्गुलिभिः मुद्राः अभिनीयन्ते एताभ्यः मुद्राभ्यः स्वरस्थानानि मात्राश्च प्रतीयन्ते ।

[सम्पादयतु] अथर्ववेदः

मुख्य पृष्ठ :: अथर्ववेदः

ब्रह्मपुत्रेण अथर्वेण समाहृतम् इति अथर्ववेदः । अथर्वाङ्गिराः, ब्रह्मवेदः इत्येते नामान्तरे । आथर्वसंहितायाः द्वे शाखे स्तः । शौनकीयशाखा, पैप्पलादशाखा चेति ।
भूर्जपत्रेषु शारदालिप्यां लिखितस्य अथर्ववेदस्य पुरातनं पुस्तकं काश्मीरेभ्यः सम्पादितम्। तद् अधुना ट्यूबि़ञ्जन् सर्वकलाशालायाः ग्रन्थशेखरे अस्ति ल ।

Vedas

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"Veda" redirects here. For other uses, see Veda (disambiguation).

"Vedic" redirects here. For other uses, see Vedic (disambiguation).

Hindu scriptures
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Vedas
Rigveda · Samaveda Yajurveda · Atharvaveda Divisions Samhita · Brahmana Aranyaka · Upanishad
Vedangas
Shiksha · Chandas Vyakarana · Nirukta Kalpa · Jyotisha
Upanishads
Rig vedic Aitareya Yajur vedic Brihadaranyaka · Isha Taittiriya · Katha Shvetashvatara Sama vedic Chandogya · Kena Atharva vedic Mundaka · Mandukya Prashna
Puranas
Brahma puranas Brahma · Brahmānda Brahmavaivarta Markandeya · Bhavishya Vaishnava puranas Vishnu · Bhagavata Naradeya · Garuda · Padma · Agni Shaiva puranas Shiva · Linga Skanda · Vayu
Itihasa
Ramayana Mahabharata (Bhagavad Gita)
Other scriptures
Manu Smriti Artha Shastra · Agama Tantra · Sūtra · Stotra Dharmashastra Divya Prabandha Tevaram Ramcharitmanas Yoga Vasistha
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Śruti · Smriti
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The Vedas (Sanskrit वेद véda, "knowledge") are a large body of texts originating in ancient India. Composed in Vedic Sanskrit, the texts constitute the oldest layer of Sanskrit literature and the oldest scriptures of Hinduism.^[1]^[2]
The class of "Vedic texts" is aggregated around the four canonical Saṃhitās or Vedas proper (turīya), of which three (traya) are related to the performance of yajna (sacrifice) in historical (Iron Age) Vedic religion:

The Rigveda, containing hymns to be recited by the hotṛ;
The Yajurveda, containing formulas to be recited by the adhvaryu or officiating priest;
The Samaveda, containing formulas to be sung by the udgātṛ.

The fourth is the Atharvaveda, a collection of spells and incantations, apotropaic charms and speculative hymns.^[3]
According to Hindu tradition, the Vedas are apauruṣeya "not of human agency",^[4] are supposed to have been directly revealed, and thus are called śruti ("what is heard").^[5]^[6] The four Saṃhitās are metrical (with the exception of prose commentary interspersed in the Black Yajurveda). The term saṃhitā literally means "composition, compilation". The individual verses contained in these compilations are known as mantras. Some selected Vedic mantras are still recited at prayers, religious functions and other auspicious occasions in contemporary Hinduism.
The various Indian philosophies and sects have taken differing positions on the Vedas. Schools of Indian philosophy which cite the Vedas as their scriptural authority are classified as "orthodox" (āstika). Other traditions, notably Buddhism and Jainism, which did not regard the Vedas as authorities are referred to by traditional Hindu texts as "heterodox" or "non-orthodox" (nāstika) schools.^[7]^[8] In addition to Buddhism and Jainism, Sikhism^[9]^[10] and Brahmoism,^[11] many non-Brahmin Hindus in South India ^[12] do not accept the authority of the Vedas. Certain South Indian Brahmin communities such as Iyengars consider the Tamil Divya Prabandham or writing of the Alvar saints as equivalent to the Vedas.^[13] In most Iyengar temples in South India the Divya Prabandham is recited daily along with Vedic Hymns.

Etymology and Usage

The Sanskrit word véda "knowledge, wisdom" is derived from the root vid- "to know". This is reconstructed as being derived from the Proto-Indo-European root *u̯eid-, meaning "see" or "know".^[14]
As a noun, the word appears only in a single instance in the Rigveda, in RV 8.19.5, translated by Griffith as "ritual lore":

yáḥ samídhā yá âhutī / yó védena dadâśa márto agnáye / yó námasā svadhvaráḥ

"The mortal who hath ministered to Agni with oblation, fuel, ritual lore, and reverence, skilled in sacrifice."^[15]

The noun is from Proto-Indo-European *u̯eidos, cognate to Greek (ϝ)εἶδος "aspect", "form" . Not to be confused is the homonymous 1st and 3rd person singular perfect tense véda, cognate to Greek (ϝ)οἶδα (w)oida "I know". Root cognates are Greek ἰδέα, English wit, etc., Latin video "I see", etc.^[16]
In English, the term Veda is often used loosely to refer to the Samhitas (collection of mantras, or chants) of the four canonical Vedas (Rigveda, Yajurveda, Samaveda and Atharvaveda).
The Sanskrit term veda as a common noun means "knowledge", but can also be used to refer to fields of study unrelated to liturgy or ritual, e.g. in agada-veda "medical science", sasya-veda "science of agriculture" or sarpa-veda "science of snakes" (already found in the early Upanishads); durveda means "with evil knowledge, ignorant".^[17]

Chronology

Main article: Vedic period

The Vedas are among the oldest sacred texts. The Samhitas date to roughly 1500–1000 BCE, and the "circum-Vedic" texts, as well as the redaction of the Samhitas, date to c. 1000-500 BCE, resulting in a Vedic period, spanning the mid 2nd to mid 1st millennium BCE, or the Late Bronze Age and the Iron Age.^[18] The Vedic period reaches its peak only after the composition of the mantra texts, with the establishment of the various shakhas all over Northern India which annotated the mantra samhitas with Brahmana discussions of their meaning, and reaches its end in the age of Buddha and Panini and the rise of the Mahajanapadas (archaeologically, Northern Black Polished Ware). Michael Witzel gives a time span of c. 1500 BCE to c. 500-400 BCE. Witzel makes special reference to the Near Eastern Mitanni material of the 14th c. BCE the only epigraphic record of Indo-Aryan contemporary to the Rigvedic period. He gives 150 BCE (Patañjali) as a terminus ante quem for all Vedic Sanskrit literature, and 1200 BCE (the early Iron Age) as terminus post quem for the Atharvaveda.^[19]
Transmission of texts in the Vedic period was by oral tradition alone, preserved with precision with the help of elaborate mnemonic techniques. A literary tradition set in only in post-Vedic times, after the rise of Buddhism in the Maurya period, perhaps earliest in the Kanva recension of the Yajurveda about the 1st century BCE; however oral tradition predominated until c. 1000 CE.^[20]
Due to the ephemeral nature of the manuscript material (birch bark or palm leaves), surviving manuscripts rarely surpass an age of a few hundred years.^[21] The Benares Sanskrit University has a Rigveda manuscript of the mid-14th century; however, there are a number of older Veda manuscripts in Nepal belonging to the Vajasaneyi tradition that are dated from the 11th century onwards.

Categories of Vedic texts

The term "Vedic texts" is used in two distinct meanings:

Texts composed in Vedic Sanskrit during the Vedic period (Iron Age India)
Any text considered as "connected to the Vedas" or a "corollary of the Vedas"^[22]

Vedic Sanskrit corpus

The corpus of Vedic Sanskrit texts includes:

The Samhita (Sanskrit saṃhitā, "collection"), are collections of metric texts ("mantras"). There are four "Vedic" Samhitas: the Rig-Veda, Sama-Veda, Yajur-Veda, and Atharva-Veda, most of which are available in several recensions (śākhā). In some contexts, the term Veda is used to refer to these Samhitas. This is the oldest layer of Vedic texts, apart from the Rigvedic hymns, which were probably essentially complete by 1200 BC, dating to ca. the 12th to 10th centuries BC. The complete corpus of Vedic mantras as collected in Bloomfield's Vedic Concordance (1907) consists of some 89,000 padas (metric feet), of which 72,000 occur in the four Samhitas.^[23]
The Brahmanas are prose texts that discuss, in technical fashion, the solemn sacrificial rituals as well as comment on their meaning and many connected themes. Each of the Brahmanas is associated with one of the Samhitas or its recensions. The Brahmanas may either form separate texts or can be partly integrated into the text of the Samhitas. They may also include the Aranyakas and Upanishads.
The Aranyakas, "wilderness texts" or "forest treaties", were composed by people who meditated in the woods as recluses and are the third part of the Vedas. The texts contain discussions and interpretations of dangerous rituals (to be studied outside the settlement) and various sorts of additional materials. It is frequently read in secondary literature.
Some of the older Mukhya Upanishads (Bṛhadāraṇyaka, Chandogya, Kaṭha).^[24]^[25]
Certain Sūtra literature, i.e. the Shrautasutras and the Grhyasutras.

The Shrauta Sutras, regarded as belonging to the smriti, are late Vedic in language and content, thus forming part of the Vedic Sanskrit corpus.^[25]^[26] The composition of the Shrauta and Grhya Sutras (ca. 6th century BC) marks the end of the Vedic period , and at the same time the beginning of the flourishing of the "circum-Vedic" scholarship of Vedanga, introducing the early flowering of classical Sanskrit literature in the Mauryan and Gupta periods.
While production of Brahmanas and Aranyakas ceases with the end of the Vedic period, there is a large number of Upanishads composed after the end of the Vedic period. While most of the ten Mukhya Upanishads can be considered to date to the Vedic or Mahajanapada period, most of the 108 Upanishads of the full Muktika canon date to the Common Era.
The Brahmanas, Aranyakas, and Upanishads often interpret the polytheistic and ritualistic Samhitas in philosophical and metaphorical ways to explore abstract concepts such as the Absolute (Brahman), and the soul or the self (Atman), introducing Vedanta philosophy, one of the major trends of later Hinduism.
The Vedic Sanskrit corpus is the scope of A Vedic Word Concordance (Vaidika-Padānukrama-Koṣa) prepared from 1930 under Vishva Bandhu, and published in five volumes in 1935-1965. Its scope extends to about 400 texts, including the entire Vedic Sanskrit corpus besides some "sub-Vedic" texts.

Volume I: Samhitas

Volume II: Brahmanas and Aranyakas

Volume III: Upanishads

Volume IV: Vedangas

A revised edition, extending to about 1800 pages, was published in 1973-1976.

Shruti literature

Main article: Shruti

The texts considered "Vedic" in the sense of "corollaries of the Vedas" is less clearly defined, and may include numerous post-Vedic texts such as Upanishads or Sutra literature. These texts are by many Hindu sects considered to be shruti (Sanskrit: śruti; "the heard"), divinely revealed like the Vedas themselves. Texts not considered to be shruti are known as smriti (Sanskrit: smṛti; "the remembered"), of human origin. This indigenous system of categorization was adopted by Max Müller and, while it is subject to some debate, it is still widely used. As Axel Michaels explains:

These classifications are often not tenable for linguistic and formal reasons: There is not only one collection at any one time, but rather several handed down in separate Vedic schools; Upanişads ... are sometimes not to be distinguished from Āraṇyakas...; Brāhmaṇas contain older strata of language attributed to the Saṃhitās; there are various dialects and locally prominent traditions of the Vedic schools. Nevertheless, it is advisable to stick to the division adopted by Max Müller because it follows the Indian tradition, conveys the historical sequence fairly accurately, and underlies the current editions, translations, and monographs on Vedic literature."^[24]

The Upanishads are largely philosophical works in dialog form. They discuss questions of nature philosophy and the fate of the soul, and contain some mystic and spiritual interpretations of the Vedas. For long, they have been regarded as their putative end and essence, and are thus known as Vedānta ("the end of the Vedas"). Taken together, they are the basis of the Vedanta school.

Vedic schools or recensions

Main article: Shakha

Study of the extensive body of Vedic texts has been organized into a number of different schools or branches (Sanskrit śākhā, literally "branch" or "limb") each of which specialized in learning certain texts.^[27] Multiple recensions are known for each of the Vedas, and each Vedic text may have a number of schools associated with it. Elaborate methods for preserving the text were based on memorizing by heart instead of writing. Specific techniques for parsing and reciting the texts were used to assist in the memorization process. (See also: Vedic chant)
Prodigous energy was expended by ancient Indian culture in ensuring that these texts were transmitted from generation to generation with inordinate fidelity.^[28] For example, memorization of the sacred Vedas included up to eleven forms of recitation of the same text. The texts were subsequently "proof-read" by comparing the different recited versions. Forms of recitation included the jaṭā-pāṭha (literally "mesh recitation") in which every two adjacent words in the text were first recited in their original order, then repeated in the reverse order, and finally repeated again in the original order.^[29]
That these methods have been effective, is testified to by the preservation of the most ancient Indian religious text, the Ṛigveda, as a redacted into single text during the Brahmana period, without any variant readings.^[29]

The Four Vedas

Rigveda (padapatha) manuscript in Devanagari, early 19th century

Vedas and their Shakhas
Part of a series on Hindu scriptures

Rig Veda^[30][show] Shakala Bhashkala
Sama Veda^[30][show] Ranayana Shatyamukhya Vyasa Bhaguri Olundi Goulgulvi Bhanumanoupamayava Karati Mashaka Argya Varshgagavya Kuthuma Shakugitre Jaiminiya
Krishna Yajurveda^[30]^[31][show] Taittiriya Samhita Maitrayani Samhita Karaka Katha Samhita Kapisthala Kahta Samhita Kathaka
Shukla Yajurveda^[31][show] Kanava Madhyandin
Atharvaveda^[31][show] Shaunaka Paippalada Stauda Mauda Jajala Jalada Kuntap Brahmavada Devadarsa Caranavaidya

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The canonical division of the Vedas is fourfold (turīya) viz.,^[32]

Rigveda (RV)
Yajurveda (YV, with the main division TS vs. VS)
Sama-Veda (SV)
Atharva-Veda (AV)

Of these, the first three were the principal original division, also called "trayī vidyā", that is, "the triple sacred science" of reciting hymns (RV), performing sacrifices (YV), and chanting (SV).^[33]^[34] This triplicity is so introduced in the Brahmanas (ShB, ABr and others), but the Rigveda is the older work of the three from which the other two borrow, next to their own independent Yajus, sorcery and speculative mantras.
Thus, the Mantras are properly of three forms: 1. Ric, which are verses of praise in metre, and intended for loud recitation; 2. Yajus, which are in prose, and intended for recitation in lower voice at sacrifices; 3. Sāman, which are in metre, and intended for singing at the Soma ceremonies.
The Yajurveda, Samaveda and Atharvaveda are independent collections of mantras and hymns intended as manuals for the Adhvaryu, Udgatr and Brahman priests respectively.
The Atharvaveda is the fourth Veda. Its status has occasionally been ambiguous, probably due to its use in sorcery and healing. However, it contains very old materials in early Vedic language. Manusmrti, which often speaks of the three Vedas, calling them trayam-brahma-sanātanam, "the triple eternal Veda". The Atharvaveda like the Rigveda, is a collection of original incantations, and other materials borrowing relatively little from the Rigveda. It has no direct relation to the solemn Śrauta sacrifices, except for the fact that the mostly silent Brahmán priest observes the procedures and uses Atharvaveda mantras to 'heal' it when mistakes have been made. Its recitation also produces long life, cures diseases, or effects the ruin of enemies.
Each of the four Vedas consists of the metrical Mantra or Samhita and the prose Brahmana part, giving discussions and directions for the detail of the ceremonies at which the Mantras were to be used and explanations of the legends connected with the Mantras and rituals. Both these portions are termed shruti (which tradition says to have been heard but not composed or written down by men). Each of the four Vedas seems to have passed to numerous Shakhas or schools, giving rise to various recensions of the text. They each have an Index or Anukramani, the principal work of this kind being the general Index or Sarvānukramaṇī.

Rigveda

Main article: Rigveda

The Rigveda Samhita is the oldest extant Indic text.^[35] It is a collection of 1,028 Vedic Sanskrit hymns and 10,600 verses in all, organized into ten books (Sanskrit: mandalas).^[36] The hymns are dedicated to Rigvedic deities.^[37]
The books were composed by poets from different priestly groups over a period of several centuries, commonly dated to the period of roughly the second half of the 2nd millennium BCE (the early Vedic period) in the Punjab (Sapta Sindhu) region of the Indian subcontinent.^[38]
There are strong linguistic and cultural similarities between the Rigveda and the early Iranian Avesta, deriving from the Proto-Indo-Iranian times, often associated with the Andronovo culture; the earliest horse-drawn chariots were found at Andronovo sites in the Sintashta-Petrovka cultural area near the Ural Mountains and date to ca. 2000 BCE.^[39]

Yajurveda

Main article: Yajurveda

The Yajurveda Samhita consists of archaic prose mantras and also in part of verses borrowed and adapted from the Rigveda. Its purpose was practical, in that each mantra must accompany an action in sacrifice but, unlike the Samaveda, it was compiled to apply to all sacrificial rites, not merely the Somayajna. There are two major groups of recensions of this Veda, known as the "Black" (Krishna) and "White" (Shukla) Yajurveda (Krishna and Shukla Yajurveda respectively). While White Yajurveda separates the Samhita from its Brahmana (the Shatapatha Brahmana), the e Black Yajurveda intersperses the Samhita with Brahmana commentary. Of the Black Yajurveda four major recensions survive (Maitrayani, Katha, Kapisthala-Katha, Taittiriya).

Samaveda

Main article: Samaveda

The Samaveda Samhita (from sāman, the term for a melody applied to metrical hymn or song of praise^[40]) consists of 1549 stanzas, taken almost entirely (except for 78 stanzas) from the Rigveda.^[24] Like the Rigvedic stanzas in the Yajurveda, the Samans have been changed and adapted for use in singing. Some of the Rigvedic verses are repeated more than once. Including repetitions, there are a total of 1875 verses numbered in the Samaveda recension translated by Griffith.^[41] Two major recensions remain today, the Kauthuma/Ranayaniya and the Jaiminiya. Its purpose was liturgical, as the repertoire of the udgātṛ or "singer" priests who took part in the sacrifice.

Atharvaveda

Main article: Atharvaveda

The Artharvaveda Samhita is the text 'belonging to the Atharvan and Angirasa poets. It has 760 hymns, and about 160 of the hymns are in common with the Rigveda.^[42] Most of the verses are metrical, but some sections are in prose.^[42] It was compiled around 900 BCE, although some of its material may go back to the time of the Rigveda,^[43] and some parts of the Atharva-Veda are older than the Rig-Veda^[42] though not in linguistic form.
The Atharvaveda is preserved in two recensions, the Paippalāda and Śaunaka.^[42] According to Apte it had nine schools (shakhas).^[44] The Paippalada text, which exists in a Kashmir and an Orissa version, is longer than the Saunaka one; it is only partially printed in its two versions and remains largely untranslated.
Unlike the other three Vedas, the Atharvanaveda has less connection with sacrifice.^[45]^[46] Its first part consists chiefly of spells and incantations, concerned with protection against demons and disaster, spells for the healing of diseases, for long life and for various desires or aims in life.^[42]^[47]
The second part of the text contains speculative and philosophical hymns.^[48]
The Atharvaveda is a comparatively late extension of the "Three Vedas" connected to priestly sacrifice to a canon of "Four Vedas". This may be connected to an extension of the sacrificial rite from involving three types of priest to the inclusion of the Brahman overseeing the ritual.^[49]
The Atharvaveda is concerned with the material world or world of man and in this respect differs from the other three vedas. Atharvaveda also sanctions the use of force, in particular circumstances and similarly this point is a departure from the three other vedas.

Brahmanas

Further information: Brahmanas

The mystical notions surrounding the concept of the one "Veda" that would flower in Vedantic philosophy have their roots already in Brahmana literature, for example in the Shatapatha Brahmana. The Vedas are identified with Brahman, the universal principle (ŚBM 10.1.1.8, 10.2.4.6). Vāc "speech" is called the "mother of the Vedas" (ŚBM 6.5.3.4, 10.5.5.1). The knowledge of the Vedas is endless, compared to them, human knowledge is like mere handfuls of dirt (TB 3.10.11.3-5). The universe itself was originally encapsulated in the three Vedas (ŚBM 10.4.2.22 has Prajapati reflecting that "truly, all beings are in the triple Veda").

Vedanta

Veda Vyasa attributed to have compiled the Vedas

Further information: Vedanta, Upanishads, and Aranyakas

While contemporary traditions continued to maintain Vedic ritualism (Śrauta, Mimamsa), Vedanta renounced all ritualism and radically re-interpreted the notion of "Veda" in purely philosophical terms. The association of the three Vedas with the bhūr bhuvaḥ svaḥ mantra is found in the Aitareya Aranyaka: "Bhūḥ is the Rigveda, bhuvaḥ is the Yajurveda, svaḥ is the Samaveda" (1.3.2). The Upanishads reduce the "essence of the Vedas" further, to the syllable Aum (ॐ). Thus, the Katha Upanishad has:

"The goal, which all Vedas declare, which all austerities aim at, and which humans desire when they live a life of continence, I will tell you briefly it is Aum" (1.2.15)

In post-Vedic literature

Vedanga

Main article: Vedanga

Six technical subjects related to the Vedas are traditionally known as vedāṅga "limbs of the Veda". V. S. Apte defines this group of works as:

"N. of a certain class of works regarded as auxiliary to the Vedas and designed to aid in the correct pronunciation and interpretation of the text and the right employment of the Mantras in ceremonials."^[50]

These subjects are treated in Sūtra literature dating from the end of the Vedic period to Mauryan times, seeing the transition from late Vedic Sanskrit to Classical Sanskrit.
The six subjects of Vedanga are:

Phonetics (Śikṣā)
Ritual (Kalpa)
Grammar (Vyākaraṇa)
Etymology (Nirukta)
Meter (Chandas)
Astronomy (Jyotiṣa)

Parisista

Main article: Parisista

Pariśiṣṭa "supplement, appendix" is the term applied to various ancillary works of Vedic literature, dealing mainly with details of ritual and elaborations of the texts logically and chronologically prior to them: the Samhitas, Brahmanas, Aranyakas and Sutras. Naturally classified with the Veda to which each pertains, Parisista works exist for each of the four Vedas. However, only the literature associated with the Atharvaveda is extensive.

The Āśvalāyana Gṛhya Pariśiṣṭa is a very late text associated with the Rigveda canon.
The Gobhila Gṛhya Pariśiṣṭa is a short metrical text of two chapters, with 113 and 95 verses respectively.
The Kātiya Pariśiṣṭas, ascribed to Kātyāyana, consist of 18 works enumerated self-referentially in the fifth of the series (the Caraṇavyūha)
The Kṛṣṇa Yajurveda has 3 parisistas The Āpastamba Hautra Pariśiṣṭa, which is also found as the second praśna of the Satyasāḍha Śrauta Sūtra', the Vārāha Śrauta Sūtra Pariśiṣṭa and the Kātyāyana Śrauta Sūtra Pariśiṣṭa.
For the Atharvaveda, there are 79 works, collected as 72 distinctly named parisistas.^[51]

Puranas

Main article: Puranas

A traditional view given in the Vishnu Purana (likely dating to the Gupta period^[52]) attributes the current arrangement of four Vedas to the mythical sage Vedavyasa.^[53] Puranic tradition also postulates a single original Veda that, in varying accounts, was divided into three or four parts. According to the Vishnu Purana (3.2.18, 3.3.4 etc.) the original Veda was divided into four parts, and further fragmented into numerous shakhas, by Lord Vishnu in the form of Vyasa, in the Dvapara Yuga; the Vayu Purana (section 60) recounts a similar division by Vyasa, at the urging of Brahma. The Bhagavata Purana (12.6.37) traces the origin of the primeval Veda to the syllable aum, and says that it was divided into four at the start of Dvapara Yuga, because men had declined in age, virtue and understanding. In a differing account Bhagavata Purana (9.14.43) attributes the division of the primeval veda (aum) into three parts to the monarch Pururavas at the beginning of Treta Yuga. The Mahabharata (santiparva 13,088) also mentions the division of the Veda into three in Treta Yuga.^[54]

Upaveda

The term upaveda ("applied knowledge") is used in traditional literature to designate the subjects of certain technical works.^[55]^[56] Lists of what subjects are included in this class differ among sources. The Charanavyuha mentions four Upavedas:

Medicine (Āyurveda), associated with the Rigveda
Archery (Dhanurveda), associated with the Yajurveda
Music and sacred dance (Gāndharvaveda), associated with the Samaveda
Military science (Shastrashastra), associated with the Atharvaveda

But Sushruta and Bhavaprakasha mention Ayurveda as an upaveda of the Atharvaveda. Sthapatyaveda (architecture), Shilpa Shastras (arts and crafts) are mentioned as fourth upaveda according to later sources.

Buddhist and Jain views

Buddhism and Jainism do not reject the Vedas, but merely their absolute authority.^{[citation needed]}

Buddhism

In the Buddhist Vinaya Pitaka of the Mahavagga (I.245)^[57] section the Buddha declared that the Veda in its true form was declared to the Vedic rishis "Atthako, Vâmako, Vâmadevo, Vessâmitto, Yamataggi, Angiraso, Bhâradvâjo, Vâsettho, Kassapo, and Bhagu"^[58] but that it was altered by a few Brahmins who introduced animal sacrifices. The Vinaya Pitaka's section Anguttara Nikaya: Panchaka Nipata says that it was on this alteration of the true Veda that the Buddha refused to pay respect to the Vedas of his time.^[59]
Also in the "Brahmana Dhammika Sutta" (II,7)^[60] of the Suttanipata section of Vinaya Pitaka^[61] there is a story of when the Buddha was in Jetavana village and there were a group of elderly Brahmin ascetics who sat down next to the Buddha and asked him, "Do the present Brahmans follow the same rules, practise the same rites, as those in the more ancient times?" The Buddha replied, "No." The elderly Brahmins asked the Buddha that if it were not inconvenient for him, that he would tell them of the Brahmana Dharma of the previous generation. The Buddha replied: "There were formerly rishis, men who had subdued all passion by the keeping of the sila precepts and the leading of a pure life...Their riches and possessions consisted in the study of the Veda and their treasure was a life free from all evil...The Brahmans, for a time, continued to do right and received in alms rice, seats, clothes, and oil, though they did not ask for them. The animals that were given they did not kill; but they procured useful medicaments from the cows, regarding the as friends and relatives, whose products give strength, beauty and health." So in this passage also the Buddha describes when the Brahmins were studying the Veda but the animal sacrifice customs had not yet began.
The Buddha was declared to have been born as a Brahmin who was a knower of the Vedas and its philosophies in a number of his previous lives acording to Buddhist scriptures. Other Buddhas too were said to have been born as Brahmins that were trained in the Vedas.
The Mahasupina Jataka^[62] and Lohakumbhi Jataka^[63] declares that Brahmin Sariputra in a previous life was a Brahmin that prevented animal sacrifice by declaring that animal sacrifice was actually against the Vedas.

Jainism

A Jain sage intereprets the Vedic sacrifices as metaphorical:

"Body is the altar, mind is the fire blazing with the ghee of knowledge and burning the sacrificial sticks of impurities produced from the tree of karma;..."^[64]

Further, Jain Sage Jinabhadra in his Visesavasyakabhasya cites a numeber of passages from the Vedic Upanishads.^[65]
Jain are in conformity with the Vedas in reference to both the Vedas' and Jainism' acceptance of the 22 Tirthankaras:

Of Rishabha (1st Tirthankara Rishabha) is written:

"But Risabha went on, unperturbed by anything till he became sin-free like a conch that takes no black dot, without obstruction ... which is the epithet of the First World-teacher, may become the destroyer of enemies" (Rig Veda X.166)

Of Aristanemi (Tirthankara Neminatha) is written:

"So asmakam Aristanemi svaha Arhan vibharsi sayakani dhanvarhanistam yajatam visvarupam arhannidam dayase" (Astak 2, Varga 7, Rig Veda)

"Fifth" and other Vedas

Some post-Vedic texts, including the Mahabharata, the Natyasastra and certain Puranas, refer to themselves as the "fifth Veda".^[66] The earliest reference to such a "fifth Veda" is found in the Chandogya Upanishad. "Dravida Veda" is a term for canonical Tamil Bhakti texts.^{[citation needed]}
Other texts such as the Bhagavad Gita or the Vedanta Sutras are considered shruti or "Vedic" by some Hindu denominations but not universally within Hinduism. The Bhakti movement, and Gaudiya Vaishnavism in particular extended the term veda to include the Sanskrit Epics and Vaishnavite devotional texts such as the Pancaratra.^[67]

Western Indology

Further information: Sanskrit in the West

The study of Sanskrit in the West began in the 17th century. In the early 19th century, Arthur Schopenhauer drew attention to Vedic texts, specifically the Upanishads. The importance of Vedic Sanskrit for Indo-European studies was also recognized in the early 19th century. English translations of the Samhitas were published in the later 19th century, in the Sacred Books of the East series edited by Müller between 1879 and 1910.^[68] Ralph T. H. Griffith also presented English translations of the four Samhitas, published 1889 to 1899.

Sanskrit & Artificial Intelligence — NASA Knowledge Representation in Sanskrit and Artificial Intelligence

2011-09-14T07:32:00.000-07:00

Abstract

In the past twenty years, much time, effort, and money has been expended on designing an unambiguous representation of natural languages to make them accessible to computer processing. These efforts have centered around creating schemata designed to parallel logical relations with relations expressed by the syntax and semantics of natural languages, which are clearly cumbersome and ambiguous in their function as vehicles for the transmission of logical data. Understandably, 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 rigor.

But this dichotomy, which has served as a premise underlying much work in the areas of linguistics and artificial intelligence, is a false one. There is at least one language, Sanskrit, which for the duration of almost 1,000 years was a living spoken language with a considerable literature of its own. Besides works of literary value, there was a long philosophical and grammatical tradition that has continued to exist with undiminished vigor until the present century. Among the accomplishments of the grammarians can be reckoned a method for paraphrasing Sanskrit in a manner that is identical not only in essence but in form with current work in Artificial Intelligence. This article demonstrates that a natural language can serve as an artificial language also, and that much work in AI has been reinventing a wheel millenia old.

First, a typical Knowledge Representation Scheme (using Semantic Nets) will be laid out, followed by an outline of the method used by the ancient Indian Grammarians to analyze sentences unambiguously. Finally, the clear parallelism between the two will be demonstrated, and the theoretical implications of this equivalence will be given.

Semantic Nets
For the sake of comparison, a brief overview of semantic nets will be given, and examples will be included that will be compared to the Indian approach. After early attempts at machine translation (which were based to a large extent on simple dictionary look-up) failed in their effort to teach a computer to understand natural language, work in AI turned to Knowledge Representation.

Since translation is not simply a map from lexical item to lexical item, and since ambiguity is inherent in a large number of utterances, some means is required to encode what the actual meaning of a sentence is. Clearly, there must be a representation of meaning independent of words used. Another problem is the interference of syntax. In some sentences (for example active/passive) syntax is, for all intents and purposes, independent of meaning. Here one would like to eliminate considerations of syntax. In other sentences the syntax contributes to the meaning and here one wishes to extract it.

I will consider a "prototypical" semantic net system similar to that of Lindsay, Norman, and Rumelhart in the hopes that it is fairly representative of basic semantic net theory. Taking a simple example first, one would represent "John gave the ball to Mary" as in Figure 1. Here five nodes connected by four labeled arcs capture the entire meaning of the sentence. This information can be stored as a series of "triples":

give, agent, John

give, object, ball

give, recipient, Mary

give, time, past.

Note that grammatical information has been transformed into an arc and a node (past tense). A more complicated example will illustrate embedded sentences and changes of state:

John Mary

book past

Figure 1.

"John told Mary that the train moved out of the station at 3 o'clock."

As shown in Figure 2, there was a change in state in which the train moved to some unspecified location from the station. It went to the former at 3:00 and from the latter at 3:O0. Now one can routinely convert the net to triples as before.

The verb is given central significance in this scheme and is considered the focus and distinguishing aspect of the sentence. However, there are other sentence types which differ fundamentally from the above examples. Figure 3 illustrates a sentence that is one of "state" rather than of "event ." Other nets could represent statements of time, location or more complicated structures.

A verb, say, "give," has been taken as primitive, but what is the meaning of "give" itself? Is it only definable in terms of the structure it generates? Clearly two verbs can generate the same structure. One can take a set-theoretic approach and a particular give as an element of "giving events" itself a subset of ALL-EVENTS. An example of this approach is given in Figure 4 ("John, a programmer living at Maple St., gives a book to Mary, who is a lawyer"). If one were to "read" this semantic net, one would have a very long text of awkward English: "There is a John" who is an element of the "Persons" set and who is the person who lives at ADRI, where ADRI is a subset of ADDRESS-EVENTS, itself a subset of 'ALL EVENTS', and has location '37 Maple St.', an element of Addresses; and who is a "worker" of 'occupation 1'. . .etc."

The degree to which a semantic net (or any unambiguous, nonsyntactic representation) is cumbersome and odd-sounding in a natural language is the degree to which that language is "natural" and deviates from the precise or "artificial." As we shall see, there was a language spoken among an ancient scientific community that has a deviation of zero.

The hierarchical structure of the above net and the explicit descriptions of set-relations are essential to really capture the meaning of the sentence and to facilitate inference. It is believed by most in the AI and general linguistic community that natural languages do not make such seemingly trivial hierarchies explicit. Below is a description of a natural language, Shastric Sanskrit, where for the past millenia successful attempts have been made to encode such information.

Shastric Sanskrit

The sentence:

(1) "Caitra goes to the village." (graamam gacchati caitra)

receives in the analysis given by an eighteenth-century Sanskrit Grammarian from Maharashtra, India, the following paraphrase:

(2) "There is an activity which leads to a connection-activity which has as Agent no one other than Caitra, specified by singularity, [which] is taking place in the present and which has as Object something not different from 'village'."

The author, Nagesha, is one of a group of three or four prominent theoreticians who stand at the end of a long tradition of investigation. Its beginnings date to the middle of the first millennium B.C. when the morphology and phonological structure of the language, as well as the framework for its syntactic description were codified by Panini. His successors elucidated the brief, algebraic formulations that he had used as grammatical rules and where possible tried to improve upon them. A great deal of fervent grammatical research took place between the fourth century B.C and the fourth century A.D. and culminated in the seminal work, the Vaiakyapadiya by Bhartrhari. Little was done subsequently to advance the study of syntax, until the so-called "New Grammarian" school appeared in the early part of the sixteenth century with the publication of Bhattoji Dikshita's Vaiyakarana-bhusanasara and its commentary by his relative Kaundabhatta, who worked from Benares. Nagesha (1730-1810) was responsible for a major work, the Vaiyakaranasiddhantamanjusa, or Treasury of dejinitive statements of grammarians, which was condensed later into the earlier described work. These books have not yet been translated.

The reasoning of these authors is couched in a style of language that had been developed especially to formulate logical relations with scientific precision. It is a terse, very condensed form of Sanskrit, which paradoxically at times becomes so abstruse that a commentary is necessary to clarify it.

One of the main differences between the Indian approach to language analysis and that of most of the current linguistic theories is that the analysis of the sentence was not based on a noun-phrase model with its attending binary parsing technique but instead on a conception that viewed the sentence as springing from the semantic message that the speaker wished to convey. In its origins, sentence description was phrased in terms of a generative model: From a number of primitive syntactic categories (verbal action, agents, object, etc.) the structure of the sentence was derived so that every word of a sentence could be referred back to the syntactic input categories. Secondarily and at a later period in history, the model was reversed to establish a method for analytical descriptions. In the analysis of the Indian grammarians, every sentence expresses an action that is conveyed both by the verb and by a set of "auxiliaries." The verbal action (Icriyu- "action" or sadhyu-"that which is to be accomplished,") is represented by the verbal root of the verb form; the "auxiliary activities" by the nominals (nouns, adjectives, indeclinables) and their case endings (one of six).

The meaning of the verb is said to be both vyapara (action, activity, cause), and phulu (fruit, result, effect). Syntactically, its meaning is invariably linked with the meaning of the verb "to do". Therefore, in order to discover the meaning of any verb it is sufficient to answer the question: "What does he do?" The answer would yield a phrase in which the meaning of the direct object corresponds to the verbal meaning. For example, "he goes" would yield the paraphrase: "He performs an act of going"; "he drinks": "he performs an act of drinking," etc. This procedure allows us to rephrase the sentence in terms of the verb "to do" or one of its synonyms, and an object formed from the verbal root which expresses the verbal action as an action noun. It still leaves us with a verb form ("he does," "he performs"), which contains unanalyzed semantic information This information in Sanskrit is indicated by the fact that there is an agent who is engaged in an act of going, or drinking, and that the action is taking place in the present time.

Rather that allow the agent to relate to the syntax in this complex, unsystematic fashion, the agent is viewed as a one-time representative, or instantiation of a larger category of "Agency," which is operative in Sanskrit sentences. In turn, "Agency" is a member of a larger class of "auxiliary activities," which will be discussed presently. Thus Caitra is some Caitral or instance of Caitras, and agency is hierarchically related to the auxiliary activities. The fact that in this specific instance the agent is a third person-singular is solved as follows: The number category (singular, dual, or plural) is regarded as a quality of the Agent and the person category (first, second, or third) as a grammatical category to be retrieved from a search list, where its place is determined by the singularity of the agent.

The next step in the process of isolating the verbal meaning is to rephrase the description in such a way that the agent and number categories appear as qualities of the verbal action. This procedure leaves us with an accurate, but quite abstract formulation of the scntcnce: (3) "Caitra is going" (gacchati caitra) - "An act of going is taking place in the present of which the agent is no one other than Caitra qualified by singularity." (atraikatvaavacchinnacaitraabinnakartrko vartamaanakaa- liko gamanaanukuulo vyaapaarah:) (Double vowels indicate length.)

If the sentence contains, besides an agent, a direct object, an indirect object and/or other nominals that are dependent on the principal action of the verb, then in the Indian system these nominals are in turn viewed as representations of actions that contribute to the complete meaning of the sentence. However, it is not sufficient to state, for instance, that a word with a dative case represents the "recipient" of the verbal action, for the relation between the recipient and the verbal action itself requires more exact specification if we are to center the sentence description around the notion of the verbal action. To that end, the action described by the sentence is not regarded as an indivisible unit, but one that allows further subdivisions. Hence a sentence such as: (4) "John gave the ball to Mary" involves the verb Yo give," which is viewed as a verbal action composed of a number of auxiliary activities. Among these would be John's holding the ball in his hand, the movement of the hand holding the ball from John as a starting point toward Mary's hand as the goal, the seizing of the ball by Mary's hand, etc. It is a fundamental notion that actions themselves cannot be perceived, but the result of the action is observable, viz. the movement of the hand. In this instance we can infer that at least two actions have taken place:

(a) An act of movement starting from the direction of John and taking place in the direction of Mary's hand. Its Agent is "the ball" and its result is a union with Mary's hand.

(b) An act of receiving, which consists of an act of grasping whose agent is Mary's hand.

It is obvious that the act of receiving can be interpreted as an action involving a union with Mary's hand, an enveloping of the ball by Mary's hand, etc., so that in theory it might be difficult to decide where to stop this process of splitting meanings, or what the semantic primitives are. That the Indians were aware of the problem is evident from the following passage: "The name 'action' cannot be applied to the solitary point reached by extreme subdivision."

The set of actions described in (a) and (b) can be viewed as actions that contribute to the meaning of the total sentence, vix. the fact that the ball is transferred from John to Mary. In this sense they are "auxiliary actions" (Sanskrit kuruku-literally "that which brings about") that may be isolated as complete actions in their own right for possible further subdivision, but in this particular context are subordinate to the total action of "giving." These "auxiliary activities" when they become thus subordinated to the main sentence meaning, are represented by case endings affixed to nominals corresponding to the agents of the original auxiliary activity. The Sanskrit language has seven case endings (excluding the vocative), and six of these are definable representations of specific "auxiliary activities." The seventh, the genitive, represents a set of auxiliary activities that are not defined by the other six. The auxiliary actions are listed as a group of six: Agent, Object, Instrument, Recipient, Point of Departure, Locality. They are the semantic correspondents of the syntactic case endings: nominative, accusative, instrumental, dative, ablative and locative, but these are not in exact equivalence since the same syntactic structure can represent different semantic messages, as will be discussed below. There is a good deal of overlap between the karakas and the case endings, and a few of them, such as Point of Departure, also are used for syntactic information, in this case "because of". In many instances the relation is best characterized as that of the allo-eme variety.

To illustrate the operation of this model of description, a sentence involving an act of cooking rice is often quoted: (5) "Out of friendship, Maitra cooks rice for Devadatta in a pot, over a fire."

Here the total process of cooking is rendered by the verb form "cooks" as well as a number of auxiliary actions:

1. An Agent represented by the person Maitra

2. An Object by the "rice"

3. An Instrument by the "fire"

4. A Recipient by the person Devadatta

5. A Point of Departure (which includes the causal relationship) by the "friendship" (which is between Maitra and Devadatta)

6. The Locality by the "pot"

So the total meaning of the sentence is not complete without the intercession of six auxiliary actions. The action itself can be inferred from a change of the condition of the grains of rice, which started out being hard and ended up being soft.

Again, it would be possible to atomize the meaning expressed by the phrase: "to cook rice": It is an operation that is not a unitary "process", but a combination of processes, such as "to place a pot on the fire, to add fuel to the fire, to fan", etc. These processes, moreover, are not taking place in the abstract, but they are tied to, or "resting on" agencies that are associated with the processes. The word used for "tied to" is a form of the verbal root a-sri, which means to lie on, have recourse to, be situated on." Hence it is possible and usually necessary to paraphrase a sentence such as "he gives" as: "an act of giving residing in him." Hence the paraphrase of sentence (5) will be: (6) "There is an activity conducive to a softening which is a change residing in something not different from rice, and which takes place in the present, and resides in an agent not different from Maitra, who is specified by singularity and has a Recipient not different from Devadatta, an Instrument not different from.. .," etc.

It should be pointed out that these Sanskrit Grammatical Scientists actually wrote and talked this way. The domain for this type of language was the equivalent of today's technical journals. In their ancient journals and in verbal communication with each other they used this specific, unambiguous form of Sanskrit in a remarkably concise way.

Besides the verbal root, all verbs have certain suffixes that express the tense and/or mode, the person (s) engaged in the "action" and the number of persons or items so engaged. For example, the use of passive voice would necessitate using an Agent with an instrumental suffix, whereas the nonpassive voice implies that the agent of the sentence, if represented by a noun or pronoun, will be marked by a nominative singular suffix.

Word order in Sanskrit has usually no more than stylistic significance, and the Sanskrit theoreticians paid no more than scant attention to it. The language is then very suited to an approach that eliminates syntax and produces basically a list of semantic messages associated with the karakas.

An example of the operation of this model on an intransitive sentence is the following:

(7) Because of the wind, a leaf falls from a tree to the ground."

Here the wind is instrumental in bringing about an operation that results in a leaf being disunited from a tree and being united with the ground. By virtue of functioning as instrument of the operation, the term "wind" qualifies as a representative of the auxiliary activity "Instrument"; by virtue of functioning as the place from which the operation commences, the "tree" qualifies to be called "The Point of Departure"; by virtue of the fact that it is the place where the leaf ends up, the "ground" receives the designation "Locality". In the example, the word "leaf" serves only to further specify the agent that is already specified by the nonpassive verb in the form of a personal suffix. In the language it is rendered as a nominative case suffix. In passive sentences other statements have to be made. One may argue that the above phrase does not differ in meaning from "The wind blows a leaf from the tree," in which the "wind" appears in the Agent slot, the "leaf" in the Object slot. The truth is that this phrase is transitive, whereas the earlier one is intransitive. "Transitivity" can be viewed as an additional feature added to the verb. In Sanskrit this process is often accomplished by a suffix, the causative suffix, which when added to the verbal root would change the meaning as follows: "The wind causes the leaf to fall from the tree," and since English has the word "blows" as the equivalent of "causes to fall" in the case of an Instrument "wind," the relation is not quite transparent. Therefore, the analysis of the sentence presented earlier, in spite of its manifest awkwardness, enabled the Indian theoreticians to introduce a clarity into their speculations on language that was theretofore un- available. Structures that appeared radically different at first sight become transparent transforms of a basic set of elementary semantic categories.

It is by no means the case that these analyses have been exhausted, or that their potential has been exploited to the full. On the contrary, it would seem that detailed analyses of sentences and discourse units had just received a great impetus from Nagesha, when history intervened: The British conquered India and brought with them new and apparently effective means for studying and analyzing languages. The subsequent introduction of Western methods of language analysis, including such areas of research as historical and structural linguistics, and lately generative linguistics, has for a long time acted as an impediment to further research along the traditional ways. Lately, however, serious and responsible research into Indian semantics has been resumed, especially at the University of Poona, India. The surprising equivalence of the Indian analysis to the techniques used in applications of Artificial Intelligence will be discussed in the next section.

Equivalence

A comparison of the theories discussed in the first section with the Indian theories of sentence analysis in the second section shows at once a few striking similarities. Both theories take extreme care to define minute details with which a language describes the relations between events in the natural world. In both instances, the analysis itself is a map of the relations between events in the universe described. In the case of the computer-oriented analysis, this mapping is a necessary prerequisite for making the speaker's natural language digestible for the artificial processor; in the case of Sanskrit, the motivation is more elusive and probably has to do with an age-old Indo-Aryan preoccupation to discover the nature of the reality behind the the impressions we human beings receive through the operation of our sense organs. Be it as it may, it is a matter of surprise to discover that the outcome of both trends of thinking-so removed in time, space, and culture-have arrived at a representation of linguistic events that is not only theoretically equivalent but close in form as well. The one superficial difference is that the Indian tradition was on the whole, unfamiliar with the facility of diagrammatic representation, and attempted instead to formulate all abstract notions in grammatical sentences. In the following paragraphs a number of the parallellisms of the two analyses will be pointed out to illustrate the equivalence of the two systems.

Consider the sentence: "John is going." The Sanskrit paraphrase would be

"An Act of going is taking place in which the Agent is 'John' specified by singularity and masculinity."

If we now turn to the analysis in semantic nets, the event portrayed by a set of triples is the following:

1. "going events, instance, go (this specific going event)"

2. "go, agent, John"

3. "go, time, present."

The first equivalence to be observed is that the basic framework for inference is the same. John must be a semantic primitive, or it must have a dictionary entry, or it must be further represented (i.e. "John, number, 1" etc.) if further processing requires more detail (e.g. "HOW many people are going?"). Similarly, in the Indian analysis, the detail required in one case is not necessarily required in another case, although it can be produced on demand (if needed). The point to be made is that in both systems, an extensive degree of specification is crucial in understanding the real meaning of the sentence to the extent that it will allow inferences to be made about the facts not explicitly stated in the sentence

The basic crux of the equivalence can be illustrated by a careful look at sentence (5) noted in Part II.

"Out of friendship, Maitra cooks rice for Devadatta in a pot over a fire "

The semantic net is supplied in Figure 5. The triples corresponding to the net are:

cause, event, friendship

friendship, objectl, Devadatta

friendship, object2, Maitra

cause, result cook

cook, agent, Maitra

cook, recipient, Devadatta

cook, instrument, fire

cook, object, rice

cook, on-lot, pot.

The sentence in the Indian analysis is rendered as follows:

The Agent is represented by Maitra, the Object by "rice," the Instrument by "fire," the Recipient by "Devadatta," the Point of Departure (or cause) by "friendship" (between Maitra and Devadatta), the Locality by "pot."

Since all of these syntactic structures represent actions auxiliary to the action "cook," let us write %ook" uext to each karakn and its sentence representat(ion:

cook, agent, Maitra

cook, object, rice

cook, instrument, fire

cook, recipient, Devadatta

cook, because-of, friendship

friendship, Maitra, Devadatta

cook, locality, pot.

The comparison of the analyses shows that the Sanskrit sentence when rendered into triples matches the analysis arrived at through the application of computer processing. That is surprising, because the form of the Sanskrit sentence is radically different from that of the English. For comparison, the Sanskrit sentence is given here: Maitrah: sauhardyat Devadattaya odanam ghate agnina pacati.

Here the stem forms of the nouns are: Muitra-sauhardya- "friendship," Devadatta -, odana- "gruel," ghatu- "pot," agni- "fire' and the verb stem is paca- "cook". The deviations of the stem forms occuring at the end of each word represent the change dictated by the word's semantic and syntactic position. It should also be noted that the Indian analysis calls for the specification of even a greater amount of grammatical and semantic detail: Maitra, Devadatta, the pot, and fire would all be said to be qualified by "singularity" and "masculinity" and the act of cooking can optionally be expanded into a number of successive perceivable activities. Also note that the phrase "over a fire" on the face of it sounds like a locative of the same form as "in a pot." However, the context indicates that the prepositional phrase describes the instrument through which the heating of the rice takes place and, therefore, is best regarded as an instrument semantically. cause

Of course, many versions of semantic nets have been proposed, some of which match the Indian system better than others do in terms of specific concepts and structure. The important point is that the same ideas are present in both traditions and that in the case of many proposed semantic net systems it is the Indian analysis which is more specific.

A third important similarity between the two treatments of the sentence is its focal point which in both cases is the verb. The Sanskrit here is more specific by rendering the activity as a "going-event", rather than "ongoing." This procedure introduces a new necessary level of abstraction, for in order to keep the analysis properly structured, the focal point ought to be phrased: "there is an event taking place which is one of cooking," rather than "there is cooking taking place", in order for the computer to distinguish between the levels of unspecified "doing" (vyapara) and the result of the doing (phala).

A further similarity between the two systems is the striving for unambiguity. Both Indian and AI schools en-code in a very clear, often apparently redundant way, in order to make the analysis accessible to inference. Thus, by using the distinction of phala and vyapara, individual processes are separated into components which in term are decomposable. For example, "to cook rice" was broken down as "placing a pot on the fire, adding fuel, fanning, etc." Cooking rice also implies a change of state, realized by the phala, which is the heated softened rice. Such specifications are necessary to make logical pathways, which otherwise would remain unclear. For example, take the following sentence:

"Maitra cooked rice for Devadatta who burned his mouth while eating it."

The semantic nets used earlier do not give any information about the logical connection between the two clauses. In order to fully understand the sentence, one has to be able to make the inference that the cooking process involves the process of "heating" and the process of "making palatable." The Sanskrit grammarians bridged the logical gap by the employment of the phalu/ vyapara distinction. Semantic nets could accomplish the same in a variety of ways:

1. by mapping "cooking" as a change of state, which would involve an excessive amount of detail with too much compulsory inference;

2. by representing the whole statement as a cause (event-result), or

3. by including dictionary information about cooking. A further comparison between the Indian system and the theory of semantic nets points to another similarity: The passive and the active transforms of the same sentence are given the same analysis in both systems. In the Indian system the notion of the "intention of the speaker" (tatparya, vivaksa) is adduced as a cause for distinguishing the two transforms semantically. The passive construction is said to emphasize the object, the nonpassive emphasizes the agent. But the explicit triples are not different. This observation indicates that both systems extract the meaning from the syntax.

Finally, a point worth noting is the Indian analysis of the intransitive phrase (7) describing the leaf falling from the tree. The semantic net analysis resembles the Sanskrit analysis remarkably, but the latter has an interesting flavor. Instead of a change from one location to another, as the semantic net analysis prescribes, the Indian system views the process as a uniting and disuniting of an agent. This process is equivalent to the concept of addition to and deletion from sets. A leaf falling to the ground can be viewed as a leaf disuniting from the set of leaves still attached to the tree followed by a uniting with (addition to) the set of leaves already on the ground. This theory is very useful and necessary to formulate changes or statements of state, such as "The hill is in the valley."

In the Indian system, inference is very complete indeed. There is the notion that in an event of "moving", there is, at each instant, a disunion with a preceding point (the source, the initial state), and a union with the following point, toward the destination, the final state. This calculus-like concept fascillitates inference. If it is stated that a process occurred, then a language processor could answer queries about the state of the world at any point during the execution of the process.

As has been shown, the main point in which the two lines of thought have converged is that the decomposition of each prose sentence into karalca-representations of action and focal verbal-action, yields the same set of triples as those which result from the decomposition of a semantic net into nodes, arcs, and labels. It is interesting to speculate as to why the Indians found it worthwhile to pursue studies into unambiguous coding of natural language into semantic elements. It is tempting to think of them as computer scientists without the hardware, but a possible explanation is that a search for clear, unambigous understanding is inherent in the human being.

Let us not forget that among the great accomplishments of the Indian thinkers were the invention of zero, and of the binary number system a thousand years before the West re-invented them.

Their analysis of language casts doubt on the humanistic distinction between natural and artificial intelligence, and may throw light on how research in AI may finally solve the natural language understanding and machine translation problems.

Physics to Metaphysics - A Vedic Paradigm

2011-09-14T07:26:00.000-07:00

For India's great realizers, the primary evidence in support of their theses is revealed scripture (sastra), such as the Vedanta-sutras. This evidence is considered to originate beyond the limits of human reasoning. Yet, especially for Westerners, as an introduction to the virtues of scriptural evidence, it may be prudent to first discuss the concept of a transcendental personal Godhead in the context of modern science and quantum mechanics in particular. Following the transition from Newtonian classical physics to quantum mechanics, several scientists have explored the possibility of a connection between physics and transcendence.

This may be due to the more abstract nature of quantum mechanics as opposed to classical physics. For example, classical physics attempts to describe the physical reality in concrete, easily understandable terms, while quantum mechanics deals in probabilities and wave functions. Quantum mechanics, however, is much more rigorous in its attempt to describe reality and explains phenomena that classical physics fails to account for. The "quantum leap" has given several physicists the hope that the transcendentalist's experience of consciousness can also be explained by the quantum mechanical theory. Although the quantum theory does not account for consciousness, it has become popular to attempt to bridge the gap between the transcendentalist's experience and the quantum mechanic world view. Some people have loosely called this the "new physics."

The rational spiritually-minded community cheered the appearance of Fritjof Capra's The Tao of Physics and Gary Zukav's Dancing Wu Li Masters. Later, David Bohm's The Implicate Order was similarly praised. Although there is good reason to applaud their work and the work of others like them, their theories, scientifically speaking, do not quite bridge the gap between physics and transcendence. However, these scientists have to some extent become "believers" and that is a major breakthrough. Furthermore, the theories have turned many educated persons in the spiritual direction.
Of all the recent attempts to show the "oneness" in what physicists and transcendentalists speak of, Bohm's implicate order theory is the most worthy of consideration. In comparison, Capra's "realization" that the dance of Shiva and the movement of atomic particles is one and the same — although profoundly beautiful — falls more in the realm of poetry than science.
Of course any attempt to find harmony between the scientific world view and the mystic's vision will be incomplete unless we adjust the scientific world view through an interface with the many realities it fails to account for (subtle bodies, consciousness, etc.). Taking that liberty, as Bohm has, Richard L. Thompson, Dr. of Mathematics and author of the book Mechanistic and Non-mechanistic Science, has postulated a new theory of "creation through sound" using what he calls The Vedic Paradigm.
Thompson advocates the philosophy of achintya bhedabheda, a transcendental conception which, interestingly enough, fits well with the example of the hologram (often used to illustrate Bohm's implicate order theory). This transcendental conception is different than the one Bohm advocates. Thompson attempts to show in his upcoming book, End of Physics, how some of the holes in Bohm's theory can be filled using an alternative view of transcendence, namely acintya bedhabedha.
Simply stated acintya bedhabedha means that reality is ultimately, inconceivably one and different at the same time. Bohm is an adherent of advaita vedanta or non-dualism. Non-dualists percieve reality as one homogenious substance. In their view all forms of variety and individuality are products of illusion. Acintya bedhabedha, holds that the world of material variety is illusory but not altogether false. It insists that there is a transcendental variety and spiritual individuality that lies beyond illusion. Acintya bedhabedha is a theistic conception and advaita vedanta is monistic or atheistic.
Thompson is a practicing scientist who has been pursuing transcendental disciplines for the last thirteen years. This kind of combination is rare. It is hard to find someone who is thoroughly familiar with science as well as with spirituality. In order to appreciate his theory of creation by sound it will be helpful to first briefly explain Bohm's theory of the implicate order and then proceed to further elaborate on the philosophy of acintya bedhabedha. Such explanations will serve as a peface to the discussion of creation, all of which shed new light on the nature of reality, helping to harmonize physics and metaphysics.
THE IMPLICATE ORDER
Bohm's explanation of reality involves an "implicate" and "explicate" order, with vague references to love, compassion, and other similar attributes that may lie beyond both the implicate and explicate. The implicate order is an ultimate physical substrate which underlies our present perception of reality. The reality that we perceive is what Bohm calls the explicate order. All order and variety, according to Bohm, are stored at all times in the implicate order in an enfolded or unmanifested state. Information continually unfolds or becomes manifest from the implicate order as the explicate order of our experience.

Of course any attempt to find harmony between the scientific world view and the mystic's vision will be incomplete unless we adjust the scientific world view through an interface with the many realities it fails to account for.

Bohm uses the example of the hologram to help explain his theory. A hologram is a photographic plate on which information is recorded as a series of density variations. Because holography is a method of lensless photography, the photographic plate appears as a meaningless pattern of swirls. When a coherent beam of light -- typically the laser -- interacts with the plate, the resultant emerging light is highly ordered and is perceived as an image in three dimensions. The image has depth and solidity, and by looking at it from different angles, one will see different sides of the image. Any part of the hologram will reproduce the whole image (although with less resolution). Bohm would say that the three-dimensional form of the image is enfolded or stored in the pattern of density variations on the hologram.
A further understanding of the nature of Bohm's implicate order is somewhat more difficult to grasp. In the transition from the classical description of physical objects to a quantum mechanical description, one is forced to use mutually incompatible descriptions. That is, to understand the behavior of electrons, it is necessary to describe them as point-like particles and extended waves. This concept of complementarity, devised in the 1920's by the physicist Niels Bohr, leads naturally to the thought that electrons, or their ultimate substrate, may not actually be fully describable in mathematical terms. Thus the ultimate physical reality may be an undefinable "something" which is only partially describable but not fully, because some of the partial descriptions will inevitably contradict each other. This is Bohm's idea regarding the nature of his implicate order.
Although Bohm accepts the reality of a whole containing distinguishable parts, he maintains that ultimately, reality at its most fundamental level is devoid of variety or individuality. Bohm believes that individuality is a temporal or illusory state of perception. According to his theory, although the parts appear to be distinct from the whole, in fact, because they "enfold" or include the whole, they are identical with the whole.
The intuitive basis behind this idea of wholeness is that when information is enfolded into a physical system, it tends to become distributed more or less uniformly throughout the system.
The hologram provides an easily understandable example. If portions of a hologram are blocked off, the resultant image remains basically the same. This, perhaps metaphorically, helps to illustrate the concept that the whole is present in each of its parts. Consider then a continuum in which all patterns ever manifested in any part of the continuum are represented equally in all parts. Loosely speaking, then one could say that the whole of the continuum in both space and time is present in any small part of the continuum. If we invoke the precedent of quantum mechanical indefinability, we could leap to the idea of a unified entity encompassing all space and time in which each part contains the whole and thus is identical to it. Because wholes are made up of parts, such an entity could not be fully described mathematically, although mathematical descriptions could be applied to the parts.
THOMPSON'S OBSERVATIONS
Although Bohm's theory of the implicate order is partially based on the standard methodology of physics, it is also apparent that it involves ideas that are not found in traditional science. Most of these ideas are clearly the influence of a preconceived notion of non-dualism.
Bohm's theory is sorely in need of a logical source of compassion which provides inspiration enabling finite beings to know the infinte. Ironically while Bohm emphatically states that it is not possible for unaided human thought to rise above the realm of manifest matter (explicate order) he proceeds to carry on a lengthy discussion about the unmanifest (implicate order). Although he speaks of compassion it is only in a vague reference to an abstract attribute. The logical necessity for an entity possesed of compassion is avoided by Bohm (although he almost admits the need). He retreats from this idea because the standard notions of a personal God are dualistic and thus undermine the sense that reality at the most fundamental plane is unified.
Bohm's idea that the parts of the implicate order actually include the whole is not fully supported by his physical examples alone. Indeed this is impossible to demonstrate mathematically. The part of the hologram is not fully representative of the whole. The part suffers from lack of resolution. It is qualitatively one but quantitatively different.
Bohm's account for the corruption in human society is also a short coming in an otherwise profound theory. The theory alleges that evil arises from the explicate order -- which is a contradiction of the basis of the theory which states that everything in the explicate order unfolds from the implicate order. This means that evil and human society at large or something at least resembling it must be originally present in the implicate order. But what would lead us to believe that an undifferentiated entity would store anything even remotely resembling human society? Or how could there be evil in or beyond the implicate order which is the source of love and compassion?
Bohm states that the totality of all things is timeless and unitary and therefore incapable of being changed. Later on he proposes that through collective human endeavor the state of arrairs can be changed. This is similar to the contradiction of advaita vedanta in which ultimate oneness is thought to be attained even though it is beyond time and forever uninfluenced by our actions.
These are some of the scientific and philosophical problems with the theory of the implicate order pointed out by Thompson. They are resolved by Thompson by replacing advaita vedanta with achintya bedhabedha.
ACHINTYA BHEDABHEDA
The history of philosophy bears evidence that neither the concepts of oneness (non-dualism) or difference (dualism) are adequate to fully describe the nature of being. Exclusive emphasis on oneness leads to the denial of the world and our very sense of self as an individual -- viewing them as illusion. Exclusive emphasis on difference divides reality, creating an unbridgeable gap between man and God. Both concepts at the same time seem necessary inasmuch as identity is a necessary demand of our reason while difference is an undeniable fact of our experience. Therefore a synthesis of the two can be seen as the goal of philosophy. In the theory of achintya bhedabheda, the concepts of both oneness and difference are transcended and reconciled in this higher synthesis, and thus they become associated aspects of an abiding unity in the Godhead. The word achintya is central to the theory. It can be defined as the power to reconcile the impossible. Achintya is that which is inconceivable on account of the contradictory notions it involves, yet it can be appreciated through logical implication.
Achintya, inconceivable, is different from anirvacaniya, or indescribable, which is said to be the nature of transcendence in the non-dualistic school. Anirvacaniya involves the joining of the opposing concepts of reality and illusion, producing a canceling effect -- a negative effect. Achintya, on the other hand, signifies a marriage of opposite concepts leading to a more complete unity -- a positive effect.

Just as the eye cannot see the mind but can be in connection with it if the mind chooses to think about it, so similarly the finite can know about the infinite only by the grace of the infinite.

It may be helpful to draw upon a reference from Vedic literature. Actually, the example of the hologram is similar to an explanation of the basis of reality recorded in the Brahma Samhita. There we find a verse in which, ironically, Godhead has been described as personal and individual and Who, at the same time one with and different from His energies.

He is an undifferentiated entity as there is no distinction between potency and possessor thereof. In His work of creation of millions of worlds, His potency remains inseparable. All the universes exist in Him and He is present in His fullness in every one of the atoms that are scattered throughout the universe at one and the same time. Such is the primeval Lord whom I adore. (Brahma Samhita 5.35)

In the material conception of form, the whole can be reduced to a mere juxtaposition of the parts. This makes the form secondary. In this verse the material conception of form is transcended. The supreme entity is fully present in all of the parts which make up the total reality and thus the supreme is one unified principle underlying all variegated manifestations. Yet He is personal and in this feature different from his parts or energies at the same time. The Brahma Samhita goes on to say that each of the parts of the Godhead's form are equal to each other and to the whole form as well. At the same time each of the parts remains a part. This is fundamental to the philosophical outlook of achintya bhedabheda. It allows for the eternal individuality of all things without the loss of oneness or harmony. It also allows for the possibility that man, even while possessed of limited mind and senses, can come to know about the nature of transcendence. The infinite, being so, can and does reveal Himself to the finite. Just as the eye can not see the mind but can be in connection with it if the mind chooses to think about it, so similarly the finite can know about the infinite by the grace of the infinite. The concept of non-dualism however allows for neither of these things.
In the Bhagavad Gita we find the following verse: (9.4)

By Me in My unmanifested form this entire universe is
pervaded. All beings are in Me, but I am not in them.

Although this is inconceivable -- achintya -- an example drawn from material nature may help us to understand this concept (logical implication). We cannot think of fire without the power of burning; similarly, we cannot think of the power of burning without fire. Both are identical. While fire is nothing but that which burns; the power of burning is but fire in action. Yet at the same time, fire and its burning power are not absolutely the same. If they were absolutely the same, there would be no need to warn our children that "fire burns." Rather it would be sufficient to say "fire." Furthermore, if they were the same, it would not be possible to neutralize the burning power in fire through medicine or mantra without causing fire to disappear altogether. In reality the fire is the energetic source of the energy which is the power to burn. From this example drawn from the world of our experience, we can deduce that the principle of simultaneous oneness and difference is all pervading, appearing even in material objects.
Just as there is neither absolute oneness nor absolute difference in the material example of fire and burning power, there is neither absolute oneness nor absolute difference between Godhead and His energies. Godhead consists of both the energetic and the energy, which are one and different. Godhead is also necessarily complete without His various emanations. This is absolute completeness. No matter how much energy He distributes, He remains the complete balance.
In this theory the personal form of God exists beyond material time in a trans-temporal state, There eternality and the passage of time are harmonized by the same principle of simultaneous oneness and variegatedness that applies to transcendental form. Thus within Godhead there may very well be something that resembles human society which could unfold as the explicate order.

The individual self is a minute particle of will or consciousness -- a sentient being -- endowed with a serving tendency. This self is transcendental to matter and qualitatively one with Godhead, while quantitatively different.

A personal, "human-like" Godhead replete with abode and paraphernalia is a perennial notion. In this conception the explicate order becomes in effect a perverted reflection of the ultimate reality existing in the transcendental realm. The reflection of that realm, appearing as the explicate order, amounts to the kingdom of God without God. It would be without God inasmuch as God, being the center of the ultimate reality, when expressed in reflected form no longer appears as the center. This produces illusion and the necessity for corruption. The basis of corruption is the misplaced sense of proprietorship resulting in the utterly false notions of "I" and "mine.
According to achintya bhedabheda,the individual self is a minute particle of will or consciousness -- a sentient being -- endowed with a serving tendency. This self is transcendental to matter and qualitatively one with Godhead, while quantitatively different. The inherent defect of smallness in size in the minute self in contrast to the quantitative superiority of Godhead makes the individual minute particle of consciousness prone to the influence of illusion. This is analogous to the example of the hologram in which only a portion of the holographic plate is illuminated with a coherent light source. The resultant image, although apparently complete, is slightly fuzzy and does not give the total three-dimensional view from all directions which one would observe when the entire holographic plate is illuminated.
Living in illusion, the atomic soul sees himself as separate from the Godhead. As a result of imperfect sense perception he is caused to make false distinctions such as good and bad, happiness and distress. The minute self can also live in an enlightened state in complete harmony with the Godhead by the latter's grace -- which is attracted by sincere petition or devotion. The very nature of devotion is that it is of another world, and for it to be devotion in the full sense, it must be engaged in for its own sake and nothing else. This act of devotion is the purified function of the inherent serving tendency of the self. It makes possible a communion with Godhead. In this communion the self becomes one in purpose with the one reality and eternally serves that reality with no sense of any separateness from Godhead. If we accept this theory then there is scope for action from within the explicate order, such as prayer or meditation, to have influence upon the whole. At least it would appear so, inasmuch as, in reality, the inspiration for such action has its origin in Godhead. Of course this idea is also found in varying degrees in many perennial theistic philosophies. It is perhaps most thoroughly dealt with, however, in the doctrine of achintya bhedabheda.
Although it is true that the human mind cannot possibly demonstrate the truth of this conception, this does not provide sufficient justification for rejecting the notion in favor of something more abstract, such as non-dualism. The fact is that any conception of the Godhead that is generated from the finite mind is subject to the same criticism. If we are limited to our mundane mind and senses for acquiring transcendental knowledge, then we may as well forego any speculation about transcendence and turn our attention exclusively to the manifest mundane world. The achintya bhedabheda theory of transcendence, however, at least allows for the possibility of the finite entity to approach the plane of transcendence through the acquisition of transcendental "grace." This conception provides for us something we can do in relation to Godhead (such as prayer or meditation) whereby our understanding can be enhanced. Alternatively, the non-dualistic approach really affords no method of approach.
Finally it must be emphasized that both the doctrines of non-dualism and achintya bhedabheda are quite extensive and impossible to deal with thoroughly in this short article. At least it should be clear that insistence on the non-dual conception of the ultimate reality creates problems for the theory of the implicate order. At the same time the theistic doctrine of inconceivable simultaneous oneness and difference at the very least deals with these problems adequately.

CREATION THROUGH SOUND

Thompson points out that the purely physical observations on which Bohm's theory is based provide insight as to how physics can be linked with transcendence. Thompson suggests that, scientifically speaking, the implicate order is limited to the observation that "organized macroscopic forms can arise by natural physical transformations from patterns of minute fluctuations that look indistinguishable from random noise." Such patterns could appear in many different forms such as electromagnetic fields (light waves) or the matter waves of quantum mechanics. These patterns which may later produce distinct macroscopic events can either be all-pervading or localized, and two such patterns could even occupy the same volume of space.
Thompson uses the philosophy of the Bhagavad Gita and other Vedic literatures as a source of metaphysical ideas. He offers a tentative proposal of a synthesis of physical and spiritual knowledge by introducing the necessary element of divine revelation.
He states that, "According to the Srimad Bhagavatam, the material creation is brought about and maintained through the injection of divinely ordered sound vibrations into a primordial material substrate called pradhana. According to this idea, the pradhana is an eternally existing energy of the supreme which is potentially capable of manifesting material space and time, the material elements, and their various possible combinations." In the absence of external influences no manifestations would take place. However, the pradhana will indeed produce various manifestations under the influence of intelligently directed sound vibrations generated by the Godhead. Thompson explains the meaning of "sound," coming from the Sanskrit word shabda, as "any type of propagating vibration, however subtle."
Keeping in mind that creation is a very complex affair, let's look at the final stages of creation in which organized forms are generated and controlled in a setting made up of the physical elements as we know them. According to the Vedic paradigm, at this stage, transcendental sound is introduced into the material continuum on the most subtle level. As a result, grosser elements are agitated, and finally organized structures such as the bodies of living organisms are produced.
Consider the phenomenon of optical phase conjugation -- a process that can reverse the motion of a beam of light and cause an image scrambled by frosted glass to return to its original, undistorted form. In a typical experiment, light is reflected from an object and passes through a pane of frosted glass. It then reflects from a device called a phase conjugate mirror and passes back through the glass. When the light enters the eye, one perceives a clear, undistorted image of the original object. This can be contrasted with the garbled blur one would observe if the light were reflected back through the glass by an ordinary mirror. See Figure left.
The explanation of this phenomenon is that the light on its first pass through the frosted pane is distorted in a complicated way by irregularities in the glass. The phase conjugate mirror reverses the distorted beam, and as it passes back through the glass it precisely retraces its steps and thus returns to its original undistorted form.
The beam reflected from the phase conjugate mirror has the curious property that it encodes information for the original image in a distorted, unrecognizable form, and as time passes, the distortion is reduced and the information contained by the beam becomes clearly manifest. Normally, we expect to see just the opposite -- a pattern containing meaningful information will gradually degrade until the information is irretrievably lost.
Thompson further elucidates the connection between the material and transcendental levels of existence with an example similar to that of optical phase conjugation. Suppose we have an arrangement in which pictures are being transmitted through a sheet of frosted glass. On one side of the glass we would see a series of images but we would not be able to determine the source of the images on the other side of the opaque glass. But in thinking about it, one would expect that the light coming through the frosted glass would become distorted. The fact that it does not seems to indicate that there is some sort of intelligence which is organizing or ordering the transmitted images. This is a simplified example of optical phase conjugation. Similarly, the order and complexity we find in matter must have intelligence behind it, although we at present cannot directly see that intelligence. The Vedic conception states that a veil of illusion called maya prevents living beings in the material domain from directly perceiving their origin, Godhead -- the supreme intelligent being. The Vedas further maintain that although God predominates the material nature, He is manipulating it in such an expert way that His influence cannot be detected; as Bohm states, "Complex patterns of events seem to unfold simply by material action and reaction."
As Thompson progresses in the formulation of his Vedic paradigm, a number of questions arise. How are the postulated organized vibrations introduced into the known physical continuum? How can some outside influence be accommodated? This would seem to involve violations of certain basic laws of physics such as conservation of energy, the second law of thermodynamics, and statistical laws of quantum mechanics.
In response to these objections, Thompson postulates a model involving levels of physical reality more subtle than quantum fields. "One can readily imagine a hierarchy of subtler and subtler levels culminating in an ultimate substrate which is transcendental and not amenable to mathematical description. Organized wave patterns could propagate through this hierarchy from the transcendental level to the level of gross matter. In such models the quantum fields will be reducible to these subtler levels, and phenomena on these levels will have effects on the level of the quantum fields." In the transition from Newtonian physics to quantum mechanics and further to quantum field theory, the conceptual framework diverges from the domain of familiar mechanical imagery. Thompson suggests that "The degree of subtlety of a level of reality corresponds to the degree of novelty and unfamiliarity of the concepts needed to adequately comprehend it. On the subtlest (or transcendental) level, the materially inconceivable principle of achintya bheda bheda tattva becomes applicable."

According to the Vedic paradigm, the conscious self is transcendental and has the same qualitative nature as the Godhead. Thus the link between conscious will and the initiation of physical action by the brain should also entail the transmission of patterns of information from transcendental to gross physical levels of reality.

The introduction of wave patterns into the gross material realm from an outside independent source should produce detectable violations of the conservation laws of physics. It would not be surprising to find violations of known laws if such subtler levels of material energy do exist. Indeed, the existence of the neutrino was postulated by Enrico Fermi in the 1930's because of an apparent violation of the principle of conservation of momentum in the radioactive decay of certain atomic nuclei. The discovery of the neutrino showed the existence of a subtle level which was previously unknown. It is therefore entirely reasonable to speak of the existence of more subtle levels which are as yet undiscovered. Also, in his forthcoming book, Thompson shows that models which receive influences from more subtle levels without undergoing any detectable change in momentum or energy may be constructed.
Thompson suggests, "Let us suppose for the moment that organized wave patterns are continually being injected into the known physical continuum perhaps from subtler levels of physical reality. Such patterns will appear to be random, especially if they encode information for many different macroscopic forms and sequences of events. For this reason they will be very difficult to distinguish from purely random patterns by experimental observation."
Consider a two-dimensional wave field -- exemplified by the surface of a body of water. This is illustrated in Figure 2, left. A two-dimensional wave field is capable of propagating waves which can be expressed by what is called the classical wave equation. In the first frame of Figure 2 we see the wave field moving in an apparently random way. As time passes it becomes apparent that this pattern of waves contains hidden information. This is illustrated in successive frames, where first in frame 2 we see that a letter "A" has appeared in the field. This form quickly takes shape and dissipates (frame 3), and it is replaced in frame 4 by the similar rapid appearance and disappearance of the symbol "" (Aum). Actually the information for both symbols is present in all 4 frames of the figure. This example is discussed in detail by Thompson in his forthcoming book:

Thus much of the random noise that surrounds us may consist of information for patterns that will 'unfold' in the future to produce macroscopic results, while the rest consists of the 'enfolded' or 'refolded' remnants of past macroscopic patterns.

Because the original source of these patterns is the inaccessible transcendental level, it is not possible to produce them at will. A thorough investigation of this phenomenon would necessarily depend on the analysis of observed spontaneous events.
Thompson believes that this type of study might be fruitful in the field of cognitive science. "According to the Vedic paradigm, the conscious self is transcendental and has the same qualitative nature as the Godhead. Thus the link between conscious will and the initiation of physical action by the brain should also entail the transmission of patterns of information from transcendental to gross physical levels of reality."
The concept of the unfolding of information is also useful in the field of natural history. The predominating scientific viewpoint is that the origin of living species can be explained by Darwin's theory of evolution by natural selection and random variation. Included in the group of those who have always dissented from this view is Alfred Russell Wallace, the co-inventor of Darwin's theory. Wallace felt that certain biological phenomenon, such as the brain, could not be accounted for properly without the action of some higher intelligence. Similarly, Bohm feels that "Natural selection is not the whole story, but rather that evolution is a sign of the creative intelligence of matter." Thompson has pointed out that "Bohm regards this intelligence as emanating either from his implicate order or from beyond."
The Vedic paradigm proposes that the supreme intelligent being can create or modify the forms of living beings by the transmission of organized wave patterns into the physical realm. Of course both this theory of creation by sound and the Darwinian theory of evolution are very difficult to verify. Thompson states, "The theory of creation by sound vibration involves transcendental levels of reality not accessible to the mundane senses, and thus in one sense it is more unverifiable than the purely physical Darwinian theory. However, if a purely physical theory turns out to be empirically unverifiable, then there is nothing further one can do to be sure about it. In contrast, a theory that posits a supreme intelligent being opens up the possibility that further knowledge may be gained through internal and external revelation brought about by the will of that being."
This entire approach is in line with the oft-mentioned need for a new paradigm, a new world view which is said to be in the making. Although the mechanistic world view founded by Descartes, Galileo, Newton, and Bacon has dominated thought since the seventeenth century -- now, as we approach the twenty-first century, the severe limitations of this view have become apparent. The mechanistic approach must be replaced with a holistic approach. Rather than torturing nature for her secrets, Thompson's idea calls for a reverence for nature and a humble appeal to Godhead for divine service.
Finally, in Thompson's own words, "This approach to knowledge and to life also constitutes one of the great perennial philosophies of mankind, but it has tended to be eclipsed in this age of scientific empiricism. To obtain the fruits of this path to knowledge, one must be willing to follow it, and one will be inclined to do this only if one thinks the world view on which it is based might possibly be true. Establishing this possibility constitutes the ultimate justification for constructing theories such as the one considered here: linking physics and metaphysics."

Vedic Mathemacs and the Spiritual Dimension

2011-09-14T07:22:00.000-07:00

Vedic Mathemacs
and the
Spiritual Dimension

by
Swami B. B. Visnu

The diagonal chord of the rectangle makes both the squares that the horizontal and vertical sides make separately.
— Sulba Sutra
(8th century B.C.)

The square of the hypotenuse of a right angle triangle is equal to the sum of the squares of the other two sides.
— Pythagorean Theorem
(6th century B.C.)

The Samrat Yantra, at Jaipur, designed by Jai Singh, measuring 147' at its base and 90' high could calculate time within two seconds accuracy per day.
		I remember the time my father pulled me aside and said, "Son, you can explain everything with math." He was a rationalist, and for him God existed only in the sentiments of the uneducated. At the time I believed him, and I think his advice had a lot to do with my decision to pursue a degree in physics. Somewhere along the way, however, in 1969, something happened (something many people are still trying to figure out) which drew me away from the spirit of that fatherly advice and subsequently my once promising career. Vedic Mathematics Continues

Vedic Mathematics Index

Vedic Mathematics Intro
Vedic Mathematics and the Spiritual Dimension
Archimedes and Pythagoras
Shulba Sutras
Evolution of Roman Numerals from India
Equations and Symbols
Poetry in Math
Vedic Mathematical Sutras
Bibliography
Home

Vedic Mathematics and the Spiritual Dimension

(cont.)

Unfortunately, I think I went too far to the other side. I threw reason to the wind, so to speak, and unceremoniously became a self-ordained "spiritual person." Science, the foundation of which is mathematics, as I saw it, had nothing to offer. It was only years later, when the cloud of my sentimentalism was dissipated by the sun of my soul's integrity, that I was able to separate myself from yet another delusion-the first being the advice of my father, and the second being the idea that I could wish myself into a more profound understanding of the nature of reality.
Math cannot take the mystery out of life without doing away with life itself, for it is life's mystery, its unpredictability-the fact that it is dynamic, not static-that makes it alive and worth living. We may theoretically explain away God, but in so doing we only choose to delude ourselves; I = everything is just bad arithmetic.
However, before we can connect with our heart of hearts, our real spiritual essence, we cannot cast reason aside. With the help of the discriminating faculty we can know at least what transcendence is not. Withdrawing our heart from that is a good beginning for a spiritual life.
Mathematics has only recently risen to attempt to usurp the throne of Godhead. Ironically, it originally came into use in human society within the context of spiritual pursuit. Spiritually advanced cultures were not ignorant of the principles of mathematics, but they saw no necessity to explore those principles beyond that which was helpful in the advancement of God realization. Intoxicated by the gross power inherent in mathematical principles, later civilizations, succumbing to the all-inviting arms of illusion, employed these principles and further explored them in an attempt to conquer nature. The folly of this, as demonstrated in modern society today, points to the fact that "wisdom" is more than the exercise of intelligence. Modern man's worship of intelligence blinds him from the obvious: the superiority of love over reason.

Archimedes and Pythagoras

A common belief among ancient cultures was that the laws of numbers have not only a practical meaning, but also a mystical or religious one. This belief was prevalent amongst the Pythagoreans. Prior to 500 B.C.E., Pythagoras, the great Greek pioneer in the teaching of mathematics, formed an exclusive club of young men to whom he imparted his superior mathematical knowledge. Each member was required to take an oath never to reveal this knowledge to an outsider. Pythagoras acquired many faithful disciples to whom he preached about the immortality of the soul and insisted on a life of renunciation. At the heart of the Pythagorean world view was a unity of religious principles and mathematical propositions.
In the third century B.C.E. another great Greek mathematician, Archimedes, contributed considerably to the field of mathematics. A quote attributed to Archimedes reads, "There are things which seem incredible to most men who have not studied mathematics." Yet according to Plutarch, Archimedes considered "mechanical work and every art concerned with the necessities of life an ignoble and inferior form of labor, and therefore exerted his best efforts only in seeking knowledge of those things in which the good and the beautiful were not mixed with the necessary." As did Plato, Archimedes scorned practical mathematics, although he became very expert at it.
The Abacus: A mechanical counting device
The Greeks, however, encountered a major problem. The Greek alphabet, which had proved so useful in so many ways, proved to be a great hindrance in the art of calculating. Although Greek astronomers and astrologers used a sexagesimal place notation and a zero, the advantages of this usage were not fully appreciated and did not spread beyond their calculations. The Egyptians had no difficulty in representing large numbers, but the absence of any place value for their symbols so complicated their system that, for example, 23 symbols were needed to represent the number 986. Even the Romans, who succeeded the Greeks as masters of the Mediterranean world, and who are known as a nation of conquerors, could not conquer the art of calculating. This was a chore left to an abacus worked by a slave. No real progress in the art of calculating nor in science was made until help came from the East.

Shulba Sutra

In the valley of the Indus River of India, the world's oldest civilization had developed its own system of mathematics. The Vedic Shulba Sutras (fifth to eighth century B.C.E.), meaning "codes of the rope," show that the earliest geometrical and mathematical investigations among the Indians arose from certain requirements of their religious rituals. When the poetic vision of the Vedic seers was externalized in symbols, rituals requiring altars and precise measurement became manifest, providing a means to the attainment of the unmanifest world of consciousness. "Shulba Sutras" is the name given to those portions or supplements of the Kalpasutras, which deal with the measurement and construction of the different altars or arenas for religious rites. The word shulba refers to the ropes used to make these measurements.

Math cannot take the mystery out of life without doing away with life itself, for it is life's mystery, its unpredictability — the fact that it is dynamic, not static — that makes it alive and worth living.

Although Vedic mathematicians are known primarily for their computational genius in arithmetic and algebra, the basis and inspiration for the whole of Indian mathematics is geometry. Evidence of geometrical drawing instruments from as early as 2500 B.C.E. has been found in the Indus Valley. [1] The beginnings of algebra can be traced to the constructional geometry of the Vedic priests, which are preserved in the Shulba Sutras. Exact measurements, orientations, and different geometrical shapes for the altars and arenas used for the religious functions (yajnas), which occupy an important part of the Vedic religious culture, are described in the Shulba Sutras. Many of these calculations employ the geometrical formula known as the Pythagorean theorem.

This theorem (c. 540 B.C.E.), equating the square of the hypotenuse of a right angle triangle with the sum of the squares of the other two sides, was utilized in the earliest Shulba Sutra (the Baudhayana) prior to the eighth century B.C.E. Thus, widespread use of this famous mathematical theorem in India several centuries before its being popularized by Pythagoras has been documented. The exact wording of the theorem as presented in the Sulba Sutras is: "The diagonal chord of the rectangle makes both the squares that the horizontal and vertical sides make separately." [2] The proof of this fundamentally important theorem is well known from Euclid's time until the present for its excessively tedious and cumbersome nature; yet the Vedas present five different extremely simple proofs for this theorem. One historian, Needham, has stated, "Future research on the history of science and technology in Asia will in fact reveal that the achievements of these peoples contribute far more in all pre-Renaissance periods to the development of world science than has yet been realized." [3]
The Shulba Sutras have preserved only that part of Vedic mathematics which was used for constructing the altars and for computing the calendar to regulate the performance of religious rituals. After the Shulba Sutra period, the main developments in Vedic mathematics arose from needs in the field of astronomy. The Jyotisha, science of the luminaries, utilizes all branches of mathematics.
The need to determine the right time for their religious rituals gave the first impetus for astronomical observations. With this desire in mind, the priests would spend night after night watching the advance of the moon through the circle of the nakshatras (lunar mansions), and day after day the alternate progress of the sun towards the north and the south. However, the priests were interested in mathematical rules only as far as they were of practical use. These truths were therefore expressed in the simplest and most practical manner. Elaborate proofs were not presented, nor were they desired.

Evolution of Arabic (Roman) Numerals from India

A close investigation of the Vedic system of mathematics shows that it was much more advanced than the mathematical systems of the civilizations of the Nile or the Euphrates. The Vedic mathematicians had developed the decimal system of tens, hundreds, thousands, etc. where the remainder from one column of numbers is carried over to the next. The advantage of this system of nine number signs and a zero is that it allows for calculations to be easily made. Further, it has been said that the introduction of zero, or sunya as the Indians called it, in an operational sense as a definite part of a number system, marks one of the most important developments in the entire history of mathematics. The earliest preserved examples of the number system which is still in use today are found on several stone columns erected in India by King Ashoka in about 250 B.C.E. [4 ] Similar inscriptions are found in caves near Poona (100 B.C.E.) and Nasik (200 C.E.). [5] These earliest Indian numerals appear in a script called brahmi.
After 700 C.E. another notation, called by the name "Indian numerals," which is said to have evolved from the brahmi numerals, assumed common usage, spreading to Arabia and from there around the world. When Arabic numerals (the name they had then become known by) came into common use throughout the Arabian empire, which extended from India to Spain, Europeans called them "Arabic notations," because they received them from the Arabians. However, the Arabians themselves called them "Indian figures" (Al-Arqan-Al-Hindu) and mathematics itself was called "the Indian art" (hindisat).

Evolution of "Arabic numerals" from Brahmi
(250 B.C.E.) to the 16th century.

Mastery of this new mathematics allowed the Muslim mathematicians of Baghdad to fully utilize the geometrical treatises of Euclid and Archimedes. Trigonometry flourished there along with astronomy and geography. Later in history, Carl Friedrich Gauss, the "prince of mathematics," was said to have lamented that Archimedes in the third century B.C.E. had failed to foresee the Indian system of numeration; how much more advanced science would have been.
Prior to these revolutionary discoveries, other world civilizations-the Egyptians, the Babylonians, the Romans, and the Chinese-all used independent symbols for each row of counting beads on the abacus, each requiring its own set of multiplication or addition tables. So cumbersome were these systems that mathematics was virtually at a standstill. The new number system from the Indus Valley led a revolution in mathematics by setting it free. By 500 C.E. mathematicians of India had solved problems that baffled the world's greatest scholars of all time. Aryabhatta, an astronomer mathematician who flourished at the beginning of the 6th century, introduced sines and versed sines-a great improvement over the clumsy half-cords of Ptolemy. A.L. Basham, foremost authority on ancient India, writes in The Wonder That Was India,

Medieval Indian mathematicians, such as Brahmagupta (seventh century), Mahavira (ninth century), and Bhaskara (twelfth century), made several discoveries which in Europe were not known until the Renaissance or later. They understood the import of positive and negative quantities, evolved sound systems of extracting square and cube roots, and could solve quadratic and certain types of indeterminate equations." [6] Mahavira's most noteworthy contribution is his treatment of fractions for the first time and his rule for dividing one fraction by another, which did not appear in Europe until the 16th century.

Equations and Symbols

B.B. Dutta writes: "The use of symbols-letters of the alphabet to denote unknowns, and equations are the foundations of the science of algebra. The Hindus were the first to make systematic use of the letters of the alphabet to denote unknowns. They were also the first to classify and make a detailed study of equations. Thus they may be said to have given birth to the modern science of algebra." [7] The great Indian mathematician Bhaskaracharya (1150 C.E.) produced extensive treatises on both plane and spherical trigonometry and algebra, and his works contain remarkable solutions of problems which were not discovered in Europe until the seventeenth and eighteenth centuries. He preceded Newton by over 500 years in the discovery of the principles of differential calculus. A.L. Basham writes further, "The mathematical implications of zero (sunya) and infinity, never more than vaguely realized by classical authorities, were fully understood in medieval India. Earlier mathematicians had taught that X/0 = X, but Bhaskara proved the contrary. He also established mathematically what had been recognized in Indian theology at least a millennium earlier: that infinity, however divided, remains infinite, represented by the equation oo /X = oo." In the 14th century, Madhava, isolated in South India, developed a power series for the arc tangent function, apparently without the use of calculus, allowing the calculation of pi to any number of decimal places (since arctan 1 = pi/4). Whether he accomplished this by inventing a system as good as calculus or without the aid of calculus; either way it is astonishing.
Spiritually advanced cultures were not ignorant of the principles of mathematics, but they saw no necessity to explore those principles beyond that which was helpful in the advancement of God realization.
By the fifteenth century C.E. use of the new mathematical concepts from India had spread all over Europe to Britain, France, Germany, and Italy, among others. A.L. Basham states also that

The debt of the Western world to India in this respect [the field of mathematics] cannot be overestimated. Most of the great discoveries and inventions of which Europe is so proud would have been impossible without a developed system of mathematics, and this in turn would have been impossible if Europe had been shackled by the unwieldy system of Roman numerals. The unknown man who devised the new system was, from the world's point of view, after the Buddha, the most important son of India. His achievement, though easily taken for granted, was the work of an analytical mind of the first order, and he deserves much more honor than he has so far received.

Unfortunately, Eurocentrism has effectively concealed from the common man the fact that we owe much in the way of mathematics to ancient India. Reflection on this may cause modern man to consider more seriously the spiritual preoccupation of ancient India. The rishis (seers) were not men lacking in practical knowledge of the world, dwelling only in the realm of imagination. They were well developed in secular knowledge, yet only insofar as they felt it was necessary within a world view in which consciousness was held as primary.
In ancient India, mathematics served as a bridge between understanding material reality and the spiritual conception. Vedic mathematics differs profoundly from Greek mathematics in that knowledge for its own sake (for its aesthetic satisfaction) did not appeal to the Indian mind. The mathematics of the Vedas lacks the cold, clear, geometric precision of the West; rather, it is cloaked in the poetic language which so distinguishes the East. Vedic mathematicians strongly felt that every discipline must have a purpose, and believed that the ultimate goal of life was to achieve self-realization and love of God and thereby be released from the cycle of birth and death. Those practices which furthered this end either directly or indirectly were practiced most rigorously. Outside of the religio-astronomical sphere, only the problems of day to day life (such as purchasing and bartering) interested the Indian mathematicians.

Poetry in Math

One of the foremost exponents of Vedic math, the late Bharati Krishna Tirtha Maharaja, author of Vedic Mathematics, has offered a glimpse into the sophistication of Vedic math. Drawing inspiration from the Atharva-veda, Tirtha Maharaja points to many sutras (codes) or aphorisms which appear to apply to every branch of mathematics: arithmetic, algebra, geometry (plane and solid), trigonometry (plane and spherical), conics (geometrical and analytical), astronomy, calculus (differential and integral), etc.

It must be pointed out that these sutras given by Tirtha Maharaja are created by the author himself, as stated in the introduction to his book, "Vedic Mathematics" (published posthumously) and are therefore not actually Vedic.
These mathematical sutras are Vedic only in the sense that they are inspired by the Vedas in the mind of one dedicated to the Vedas. Thus the title "Vedic Mathematics" is not correct.

Utilizing the techniques derived from these sutras, calculations can be done with incredible ease and simplicity in one's head in a fraction of the time required by modern means. Calculations normally requiring as many as a hundred steps can be done by this method in one single simple step. For instance the conversion of the fraction 1/29 to its equivalent recurring decimal notation normally involves 28 steps. Utilizing this method it can be calculated in one simple step. (see the next section for examples of how to utilize Vedic sutras)
In order to illustrate how secular and spiritual life were intertwined in Vedic India, Tirtha Maharaja has demonstrated that mathematical formulas and laws were often taught within the context of spiritual expression (mantra). Thus while learning spiritual lessons, one could also learn mathematical rules.
Tirtha Maharaja has pointed out that Vedic mathematicians prefer to use the devanagari letters of Sanskrit to represent the various numbers in their numerical notations rather than the numbers themselves, especially where large numbers are concerned. This made it much easier for the students of this math in their recording of the arguments and the appropriate conclusions.
Tirtha Maharaja states, "In order to help the pupil to memorize the material studied and assimilated, they made it a general rule of practice to write even the most technical and abstruse textbooks in sutras or in verse (which is so much easier-even for the children-to memorize). And this is why we find not only theological, philosophical, medical, astronomical, and other such treatises, but even huge dictionaries in Sanskrit verse! So from this standpoint, they used verse, sutras and codes for lightening the burden and facilitating the work (by versifying scientific and even mathematical material in a readily assimilable form)!" [8] The code used is as follows:

The Sanskrit consonants

ka, ta, pa, and ya all denote 1;
kha, tha, pha, and ra all represent 2;
ga, da, ba, and la all stand for 3;
Gha, dha, bha, and va all represent 4;
gna, na, ma, and sa all represent 5;
ca, ta, and sa all stand for 6;
cha, tha, and sa all denote 7;
ja, da, and ha all represent 8;
jha and dha stand for 9; and
ka means zero.

Vowels make no difference and it is left to the author to select a particular consonant or vowel at each step. This great latitude allows one to bring about additional meanings of his own choice. For example kapa, tapa, papa, and yapa all mean 11. By a particular choice of consonants and vowels one can compose a poetic hymn with double or triple meanings. Here is an actual sutra of spiritual content, as well as secular mathematical significance.

gopi bhagya madhuvrata
srngiso dadhi sandhiga
khala jivita khatava
gala hala rasandara

While this verse is a type of petition to Krishna, when learning it one can also learn the value of pi/10 (i.e. the ratio of the circumference of a circle to its diameter divided by 10) to 32 decimal places. It has a self-contained master-key for extending the evaluation to any number of decimal places.
The translation is as follows:

O Lord anointed with the yogurt of the milkmaids' worship (Krishna), O savior of the fallen, O master of Shiva, please protect me.

At the same time, by application of the consonant code given above, this verse directly yields the decimal equivalent of pi divided by 10: pi/10 = 0.31415926535897932384626433832792. Thus, while offering mantric praise to Godhead in devotion, by this method one can also add to memory significant secular truths.
This is the real gist of the Vedic world view regarding the culture of knowledge: while culturing transcendental knowledge, one can also come to understand the intricacies of the phenomenal world. By the process of knowing the absolute truth, all relative truths also become known. In modern society today it is often contended that never the twain shall meet: science and religion are at odds. This erroneous conclusion is based on little understanding of either discipline. Science is the smaller circle within the larger circle of religion.
We should never lose sight of our spiritual goals. We should never succumb to the shortsightedness of attempting to exploit the inherent power in the principles of mathematics or any of the natural sciences for ungodly purposes. Our reasoning faculty is but a gracious gift of Godhead intended for divine purposes, and not those of our own design.

Mathematical Sutras

Consider the following three sutras:
1. "All from 9 and the last from 10," and its corollary: "Whatever the extent of its deficiency, lessen it still further to that very extent; and also set up the square (of that deficiency)."
2. "By one more than the previous one," and its corollary: "Proportionately."
3. "Vertically and crosswise," and its corollary: "The first by the first and the last by the last."
The first rather cryptic formula is best understood by way of a simple example: let us multiply 6 by 8.
1. First, assign as the base for our calculations the power of 10 nearest to the numbers which are to be multiplied. For this example our base is 10.
2. Write the two numbers to be multiplied on a paper one above the other, and to the right of each write the remainder when each number is subtracted from the base 10. The remainders are then connected to the original numbers with minus signs, signifying that they are less than the base 10.

6-4
8-2

3. The answer to the multiplication is given in two parts. The first digit on the left is in multiples of 10 (i.e. the 4 of the answer 48). Although the answer can be arrived at by four different ways, only one is presented here. Subtract the sum of the two deficiencies (4 + 2 = 6) from the base (10) and obtain 10 - 6 = 4 for the left digit (which in multiples of the base 10 is 40).

6-4
8-2
4

4. Now multiply the two remainder numbers 4 and 2 to obtain the product 8. This is the right hand portion of the answer which when added to the left hand portion 4 (multiples of 10) produces 48.

6-4
8-2
----
4/8

Another method employs cross subtraction. In the current example the 2 is subtracted from 6 (or 4 from 8) to obtain the first digit of the answer and the digits 2 and 4 are multiplied together to give the second digit of the answer. This process has been noted by historians as responsible for the general acceptance of the X mark as the sign of multiplication. The algebraical explanation for the first process is

(x-a)(x-b)=x(x-a-b) + ab

where x is the base 10, a is the remainder 4 and b is the remainder 2 so that

6 = (x-a) = (10-4)
8 = (x-b) = (10-2)

The equivalent process of multiplying 6 by 8 is then

x(x-a-b) + ab or
10(10-4-2) + 2x4 = 40 + 8 = 48

These simple examples can be extended without limitation. Consider the following cases where 100 has been chosen as the base:

97 - 3 93 - 7 25 - 75
78 - 22 92 - 8 98 - 2
______ ______ ______
75/66 85/56 23/150 = 24/50

In the last example we carry the 100 of the 150 to the left and 23 (signifying 23 hundred) becomes 24 (hundred). Herein the sutra's words "all from 9 and the last from 10" are shown. The rule is that all the digits of the given original numbers are subtracted from 9, except for the last (the righthand-most one) which should be deducted from 10.
Consider the case when the multiplicand and the multiplier are just above a power of 10. In this case we must cross-add instead of cross subtract. The algebraic formula for the process is: (x+a)(x+b) = x(x+a+b) + ab. Further, if one number is above and the other below a power of 10, we have a combination of subtraction and addition: viz:

108 + 8 and 13 + 3
97 - 3 8 - 2
_______ ______

105/-24 = 104/(100-24) = 104/76 11/-6 = 10/(10-6) = 10/4

The Sub-Sutra: "Proportionately" Provides for those cases where we wish to use as our base multiples of the normal base of powers of ten. That is, whenever neither the multiplicand nor the multiplier is sufficiently near a convenient power of 10, which could serve as our base we simply use a multiple of a power of ten as our working base, perform our calculations with this working base and then multiply or divide the result proportionately.
To multiply 48 by 32, for example, we use as our base 50 = 100/2, so we have

Base 50 48 - 2
32 - 18
______
2/ 30/36 or (30/2) / 36 = 15/36

Note that only the left decimals corresponding to the powers of ten digits (here 100) are to be effected by the proportional division of 2. These examples show how much easier it is to subtract a few numbers, (especially for more complex calculations) rather than memorize long mathematical tables and perform cumbersome calculations the long way.
Squaring Numbers
The algebraic equivalent of the sutra for squaring a number is: (a+-b)² = a² +- 2ab + b² . To square 103 we could write it as (100 + 3 )² = 10,000 + 600 + 9 = 10,609. This calculation can easily be done mentally. Similarly, to divide 38,982 by 73 we can write the numerator as 38x³ + 9x² +8x + 2, where x is equal to 10, and the denominator is 7x + 3. It doesn't take much to figure out that the numerator can also be written as 35x³ +36x² + 37x + 12. Therefore,

38,982/73 = (35x³ + 36x² +37x + 12)/(7x + 3) = 5x² + 3x +4 = 534

This is just the algebraic equivalent of the actual method used. The algebraic principle involved in the third sutra, "vertically and crosswise," can be expressed, in one of it's applications, as the multiplication of the two numbers represented by (ax + b) and (cx + d), with the answer acx² + x(ad + bc) + bd. Differential calculus also is utilized in these sutras for breaking down a quadratic equation on sight into two simple equations of the first degree. Many additional sutras are given which provide simple mental one or two line methods for division, squaring of numbers, determining square and cube roots, compound additions and subtractions, integrations, differentiations, and integration by partial fractions, factorisation of quadratic equations, solution of simultaneous equations, and many more.

The Universe Through Vedas

2011-05-21T08:15:00.001-07:00

In the early part of this century, two opposing theories about the origin of the universe were postulated. (1) The Steady State theory, which says the universe is never born, never dies, and is always like what it is. (2) The Big bang theory, which says the universe began with a point of energy exploding in a "big-bang". All the matter came into being from energy continuously expanding and changing form. Ultimately the expansion will stop and it will start contracting, ending into nothingness with a "big-crunch". What is before big-bang or after big crunch, the theory doesn’t know.

In reality, both the theories are correct. The universe begins from a point with a bang and ends in a point with a crunch. This duration we call one Kalpa (cosmos) or Brahma Diwas (eternal day). It is preceded and succeeded by an equal period during which matter lies in a dormant, inert state and that is called a Brahma Ratri i.e. a divine night (for the nature that sleeps as it were). All the souls also remain in a dormant state, a sort of hibernation, during this period. The evolution of cosmos from dormant state may be called a ‘creation’ or ‘srishti’, and its involution back into inert state is called dissolution (pralaya). As days and nights succeed each other, so do cosmos and divine nights in this eternal sinusoidal cycle of evolutions and involutions 3 (Figure 1).

All matter, i.e. nature, has three basic attributes/forces – satva, rajasa and tamasa. During brahma ratri, these forces remain in a balanced state. After the big bang, the three forces get realigned to form elementary particles called Mahat or Aapah, which combine further to form other basic particles, atoms and so on. 4

Figure 1

A – "Big Bang"

B – "Big Crunch"

A to B – One "Kalpa"

Age of the Universe

The age of each Kalpa (eternal day) is 4.32 billion years (4,320,000,000 years). According to Hindu scriptures this is further subdivided as below:

1 Kalpa	=	1000 Chaturyugis
	=	14 Manvantars + Buffer Periods of 6 Chaturyugis
1 Manvantar	=	71 Chaturyugis
1 Chaturyugi	=	4,320,000 years

Of the 14 manvantars, the universe expands for the first seven, and contracts for the next seven.

Each chaturyugi is subdivided into four Yugas:

Krit yuga = 1,728,000 years
Treta yuga = 1,296,000 years
Dwapar yuga = 864,000 years
Kali yuga = 432,000 years

At present, kaliyuga of the 28th chaturyugi of the 7th manvantar is in progress. According to this calculation, 1,972,949,100 years have elapsed since the evolution of present cosmos began, and it has 2,347,050,900 years still to go before the "big-crunch". 5

Vedic Mathematics

2011-05-11T07:37:00.000-07:00

What is Vedic Mathematics?

Vedic Mathematics is the name given to the ancient system of Mathematics which was rediscovered from the Vedas between 1911 and 1918 by Sri Bharati Krsna Tirthaji (1884-1960). According to his research all of mathematics is based on sixteen Sutras or word-formulae. For example, 'Vertically and Crosswise` is one of these Sutras. These formulae describe the way the mind naturally works and are therefore a great help in directing the student to the appropriate method of solution.

Perhaps the most striking feature of the Vedic system is its coherence. Instead of a hotch-potch of unrelated techniques the whole system is beautifully interrelated and unified: the general multiplication method, for example, is easily reversed to allow one-line divisions and the simple squaring method can be reversed to give one-line square roots. And these are all easily understood. This unifying quality is very satisfying, it makes mathematics easy and enjoyable and encourages innovation.

In the Vedic system 'difficult' problems or huge sums can often be solved immediately by the Vedic method. These striking and beautiful methods are just a part of a complete system of mathematics which is far more systematic than the modern 'system'. Vedic Mathematics manifests the coherent and unified structure of mathematics and the methods are complementary, direct and easy.

The simplicity of Vedic Mathematics means that calculations can be carried out mentally (though the methods can also be written down). There are many advantages in using a flexible, mental system. Pupils can invent their own methods, they are not limited to the one 'correct' method. This leads to more creative, interested and intelligent pupils.

Interest in the Vedic system is growing in education where mathematics teachers are looking for something better and finding the Vedic system is the answer. Research is being carried out in many areas including the effects of learning Vedic Maths on children; developing new, powerful but easy applications of the Vedic Sutras in geometry, calculus, computing etc.

But the real beauty and effectiveness of Vedic Mathematics cannot be fully appreciated without actually practising the system. One can then see that it is perhaps the most refined and efficient mathematical system possible.

The Vedic Mathematics Sutras

This list of sutras is taken from the book Vedic Mathematics, which includes a full list of the sixteen Sutras in Sanskrit, but in some cases a translation of the Sanskrit is not given in the text and comes from elsewhere.

This formula 'On the Flag' is not in the list given in Vedic Mathematics, but is referred to in the text.

The Main Sutras

By one more than the one before.

All from 9 and the last from 10.

Vertically and Cross-wise

Transpose and Apply

If the Samuccaya is the Same it is Zero

If One is in Ratio the Other is Zero

By Addition and by Subtraction

By the Completion or Non-Completion

Differential Calculus

By the Deficiency

Specific and General

The Remainders by the Last Digit

The Ultimate and Twice the Penultimate

By One Less than the One Before

The Product of the Sum

All the Multipliers

The Sub Sutras

Proportionately

The Remainder Remains Constant

The First by the First and the Last by the Last

For 7 the Multiplicand is 143

By Osculation

Lessen by the Deficiency

Whatever the Deficiency lessen by that amount and
set up the Square of the Deficiency

Last Totalling 10

Only the Last Terms

The Sum of the Products

By Alternative Elimination and Retention

By Mere Observation

The Product of the Sum is the Sum of the Products

On the Flag

Try a Sutra

Mark Gaskell introduces an alternative
system of calculation based on Vedic philosophy
At the Maharishi School in Lancashire we have developed a course on Vedic mathematics for key stage 3 that covers the national curriculum. The results have been impressive: maths lessons are much livelier and more fun, the children enjoy their work more and expectations of what is possible are very much higher. Academic performance has also greatly improved: the first class to complete the course managed to pass their GCSE a year early and all obtained an A grade.
Vedic maths comes from the Vedic tradition of India. The Vedas are the most ancient record of human experience and knowledge, passed down orally for generations and written down about 5,000 years ago. Medicine, architecture, astronomy and many other branches of knowledge, including maths, are dealt with in the texts. Perhaps it is not surprising that the country credited with introducing our current number system and the invention of perhaps the most important mathematical symbol, 0, may have more to offer in the field of maths.
The remarkable system of Vedic maths was rediscovered from ancient Sanskrit texts early last century. The system is based on 16 sutras or aphorisms, such as: "by one more than the one before" and "all from nine and the last from 10". These describe natural processes in the mind and ways of solving a whole range of mathematical problems. For example, if we wished to subtract 564 from 1,000 we simply apply the sutra "all from nine and the last from 10". Each figure in 564 is subtracted from nine and the last figure is subtracted from 10, yielding 436.

This can easily be extended to solve problems such as 3,000 minus 467. We simply reduce the first figure in 3,000 by one and then apply the sutra, to get the answer 2,533. We have had a lot of fun with this type of sum, particularly when dealing with money examples, such as £10 take away £2. 36. Many of the children have described how they have challenged their parents to races at home using many of the Vedic techniques - and won. This particular method can also be expanded into a general method, dealing with any subtraction sum.
The sutra "vertically and crosswise" has many uses. One very useful application is helping children who are having trouble with their tables above 5x5. For example 7x8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10.

The whole approach of Vedic maths is suitable for slow learners, as it is so simple and easy to use.
The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply
32 by 44. We multiply vertically 2x4=8.
Then we multiply crosswise and add the two results: 3x4+4x2=20, so put down 0 and carry 2.
Finally we multiply vertically 3x4=12 and add the carried 2 =14. Result: 1,408.

We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. By the end of Year 8, I would expect all students to be able to do a "3 by 2" long multiplication in their heads. This gives enormous confidence to the pupils who lose their fear of numbers and go on to tackle harder maths in a more open manner.
All the techniques produce one-line answers and most can be dealt with mentally, so calculators are not used until Year 10. The methods are either "special", in that they only apply under certain conditions, or general. This encourages flexibility and innovation on the part of the students.
Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92. 96 is 4 below the base and 92 is 8 below.
We can cross-subtract either way: 96-8=88 or 92-4=88. This is the first part of the answer and multiplying the "differences" vertically 4x8=32 gives the second part of the answer.

This works equally well for numbers above the base: 105x111=11,655. Here we add the differences. For 205x211=43,255, we double the first part of the answer, because 200 is 2x100.
We regularly practise the methods by having a mental test at the beginning of each lesson. With the introduction of a non-calculator paper at GCSE, Vedic maths offers methods that are simpler, more efficient and more readily acquired than conventional methods.
There is a unity and coherence in the system which is not found in conventional maths. It brings out the beauty and patterns in numbers and the world around us. The techniques are so simple they can be used when conventional methods would be cumbersome.
When the children learn about Pythagoras's theorem in Year 9 we do not use a calculator; squaring numbers and finding square roots (to several significant figures) is all performed with relative ease and reinforces the methods that they would have recently learned.
For many more examples, try elsewhere on this page, the Vedic Maths Tutorial
Mark Gaskell is head of maths at the Maharishi School in Lancashire
'The Cosmic Computer'
by K Williams and M Gaskell,
(also in an bridged edition),
Inspiration Books, 2 Oak Tree Court,
Skelmersdale, Lancs WN8 6SP. Tel: 01695 727 986.
Saturday school for primary teachers at
Manchester Metropolitan University on
October 7. See website. www.vedicmaths.org
19th May 2000 Times Educational Supplement (Curriculum Special)
http://www.tes.co.uk/
copyright to the ACADEMY OF VEDIC MATHEMATICS
_{_______________________}
Books on Vedic Maths
VEDIC MATHEMATICS
Or Sixteen Simple Mathematical Formulae from the Vedas The original introduction to Vedic Mathematics.
Author: Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaja,
1965 (various reprints).
Paperback, 367 pages, A5 in size.
ISBN 81 208 0163 6 (cloth)
ISBN 82 208 0163 4 (paper)/p
MATHS OR MAGIC?
This is a popular book giving a brief outline of some of the Vedic Mathematics methods.
Author: Joseph Howse. 1976
ISBN 0722401434
Currently out of print./p

VEDIC MATHEMATICS
Master Multiplication tables, division and lots more!
We recommed you check out this ebook, it's packed with tips,
tricks and tutorials that will boost your math ability, guaranteed!
www.vedic-maths-ebook.com

A PEEP INTO VEDIC MATHEMATICS
Mainly on recurring decimals.
Author: B R Baliga, 1979.
Pamphlet./p
INTRODUCTORY LECTURES ON VEDIC MATHEMATICS
Following various lecture courses in London an interest arose for printed material containing the course material. This book of 12 chapters was the result covering a range topics from elementary arithmetic to cubic equations.
Authors: A. P. Nicholas, J. Pickles, K. Williams, 1982.
Paperback, 166 pages, A4 size./p
DISCOVER VEDIC MATHEMATICS
This has sixteen chapters each of which focuses on one of the Vedic Sutras or sub-Sutras and shows many applications of each. Also contains Vedic Maths solutions to GCSE and 'A' level examination questions.
Author: K. Williams, 1984, Comb bound, 180 pages, A4.
ISBN 1 869932 01 3./p
VERTICALLY AND CROSSWISE
This is an advanced book of sixteen chapters on one Sutra ranging from elementary multiplication etc. to the solution of non-linear partial differential equations. It deals with (i) calculation of common functions and their series expansions, and (ii) the solution of equations, starting with simultaneous equations and moving on to algebraic, transcendental and differential equations.
Authors: A. P. Nicholas, K. Williams, J. Pickles
first published 1984), new edition 1999. Comb bound, 200 pages, A4.
ISBN 1 902517 03 2./p
TRIPLES
This book shows applications of Pythagorean Triples (like 3,4,5). A simple, elegant system for combining these triples gives unexpected and powerful general methods for solving a wide range of mathematical problems, with far less effort than conventional methods use. The easy text fully explains this method which has applications in trigonometry (you do not need any of those complicated formulae), coordinate geometry (2 and 3 dimensions) transformations (2 and 3 dimensions), simple harmonic motion, astronomy etc., etc.
Author: K. Williams (first published 1984), new edition 1999. Comb bound.,168 pages, A4.
ISBN 1 902517 00 8/p
VEDIC MATHEMATICAL CONCEPTS OF SRI VISHNU SAHASTRANAMA STOTRAM
Author: S.K. Kapoor, 1988. Hardback, 78 pages, A4 size./p
ISSUES IN VEDIC MATHEMATICS
Proceedings of the National workshop on Vedic Mathematics
25-28 March 1988 at the University of Rajasthan, Jaipur.
Paperback, 139 pages, A5 in size.
ISBN 81 208 0944 0/p
THE NATURAL CALCULATOR
This is an elementary book on mental mathematics.
It has a detailed introduction and each of the nine chapters covers one of the Vedic formulae. The main theme is mental multiplication but addition, subtraction and division are also covered.
Author: K. Williams, 1991. Comb bound ,102 pages, A4 size.
ISBN 1 869932 04 8./p.
VEDIC MATHEMATICS FOR SCHOOLS BOOK 1
Is a first text designed for the young mathematics student of about eight years of age, who have mastered the four basic rules including times tables. The main Vedic methods used in his book are for multiplication, division and subtraction. Introductions to vulgar and decimal fractions, elementary algebra and vinculums are also given.
Author: J.T,Glover, 1995. Paperback, 100 pages + 31 pages of answers, A5 in size.
ISBN 81-208-1318-9./p
JAGATGURU SHANKARACHARYA
SHRI BHARATI KRISHNA TEERTHA
An excellent book giving details of the life of the man
who reconstructed the Vedic system.
Dr T. G. Pande, 1997
B. R. Publishing Corporation, Delhi-110052
INTRODUCTION TO VEDIC MATHEMATICS
Authors T. G. Unkalkar, S. Seshachala Rao, 1997
Pub: Dandeli Education Socety, Karnataka-581325
THE COSMIC COMPUTER COURSE
This covers Key Stage 3 (age 11-14 years) of the
National Curriculum for England and Wales. It consists of three books each of which has a Teacher's Guide and an Answer Book. Much of the material in Book 1 is suitable for children as young as eight and this is developed from here to topics such as Pythagoras' Theorem and Quadratic Equations in Book 3. The Teacher's Guide contains a Summary of the Book, a Unified Field Chart (showing the whole subject of mathematics and how each of the parts are related), hundreds of Mental Tests (these revise previous work, introduce new ideas and are carefully correlated with the rest of the course), Extension Sheets (about 16 per book) for fast pupils or for extra classwork, Revision Tests, Games, Worksheets etc.
Authors: K. Williams and M. Gaskell, 1998.
All Textbooks and Guides are A4 in size, Answer Books are A5.
GEOMETRY FOR AN ORAL TRADITION
This book demonstrates the kind of system that could have existed before literacy was widespread and takes us from first principles to theorems on elementary properties of circles. It presents direct, immediate and easily understood proofs. These are based on only one assumption (that magnitudes are unchanged by motion) and three additional provisions (a means of drawing figures, the language used and the ability to recognise valid reasoning). It includes discussion on the relevant philosophy of mathematics and is written both for mathematicians and for a wider audience.
Author: A. P. Nicholas, 1999. Paperback.,132 pages, A4 size.
ISBN 1 902517 05 9
THE CIRCLE REVELATION
This is a simplified, popularised version of "Geometry for an Oral Tradition" described above. These two books make the methods accessible to all interested in exploring geometry. The approach is ideally suited to the twenty-first century, when audio-visual forms of communication are likely to be dominant.
Author: A. P. Nicholas, 1999. Paperback, 100 pages, A4 size.
ISBN 1902517067
VEDIC MATHEMATICS FOR SCHOOLS BOOK 2
The second book in this series.
Author J.T. Glover , 1999.
ISBN 81 208 1670-6
Astronomica; Applications of Vedic Mathematics
To include prediction of eclipses and planetary positions,
spherical trigonometry etc.
Author Kenneth Williams, 2000.
ISBN 1 902517 08 3
Vedic Mathematics, Part 1
We found this book to be well-written, thorough and easy to read.
It covers a lot of the basic work in the original book by B. K. Tirthaji
and has plenty of examples and exercises.
Author S. Haridas
Published by Bharatiya Vidya Bhavan, Kulapati K.M. Munshi Marg, Mumbai - 400 007, India.
INTRODUCTION TO VEDIC MATHEMATICS – Part II
Authors T. G. Unkalkar, 2001
Pub: Dandeli Education Socety, Karnataka-581325
VEDIC MATHEMATICS FOR SCHOOLS BOOK 3
The third book in this series.
Author J.T. Glover , 2002.
Published by Motilal Banarsidass.
THE COSMIC CALCULATOR
Three textbooks plus Teacher's Guide plus Answer Book.
Authors Kenneth Williams and Mark Gaskell, 2002.
Published by Motilal Banarsidass.
TEACHER’S MANUALS – ELEMENTARY & INTERMEDIATE
Designed for teachers (of children aged 7 to 11 years,
9 to 14 years respectively)who wish to teach the Vedic system.
Author: Kenneth Williams, 2002.
Published by Inspiration Books.
TEACHER’S MANUAL – ADVANCED
Designed for teachers (of children aged 13 to 18 years)
who wish to teach the Vedic system.
Author: Kenneth Williams, 2003.
Published by Inspiration Books.
FUN WITH FIGURES (subtitled: Is it Maths or Magic?)
This is a small popular book with many illustrations, inspiring quotes and amusing anecdotes. Each double page shows a neat and quick way of solving some simple problem. Suitable for any age from eight upwards.
Author: K. Williams, 1998. Paperback, 52 pages, size A6.
ISBN 1 902517 01 6.
Please note the Tutorial below is based on material from this book 'Fun with Figures'
Book review of 'Fun with Figures'
From 'inTouch', Jan/Feb 2000, the Irish National Teachers Organisation (INTO) magazine.
"Entertaining, engaging and eminently 'doable', Williams' pocket volume reveals many fascinating and useful applications of the ancient Eastern system of Vedic Maths. Tackling many number operations encountered between First and Sixth class, Fun with Figures offers several speedy and simple means of solving or double-checking class activities. Focusing throughout on skills associated with mental mathematics, the author wisely places them within practical life-related contexts." "Compact, cheerful and liberally interspersed with amusing anecdotes and aphorisms from the world of maths, Williams' book will help neutralise the 'menace' sometimes associated with maths.
It's practicality, clear methodology, examples, supplementary exercises and answers may particularly benefit and empower the weaker student." "Certainly a valuable investment for parents and teachers of children aged 7 to 12." Reviewed by Gerard Lennon, Principal, Ardpatrick NS, Co Limerick. The Tutorial below is based on material from this book 'Fun with Figures'

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_________________
Vedic Maths Tutorial
Vedic Maths is based on sixteen Sutras or principles. These principles are general in nature and can be applied in many ways. In practice many applications of the sutras may be learned and combined to solve actual problems. These tutorials will give examples of simple applications of the sutras, to give a feel for how the Vedic Maths system works.
These tutorials do not attempt to teach the systematic use of the sutras.For more advanced applications and a more complete coverage of the basic uses of the sutras, we recommend you study one of the texts available atwww.vedicmaths.org

N.B. The following tutorials are based on examples and exercises given in the book 'Fun with figures' by Kenneth Williams, which is a fun introduction to some of the applications of the sutras for children.

Tutorial 1
Tutorial 2
Tutorial 3
Tutorial 4
Tutorial 5
Tutorial 6
Tutorial 7
Tutorial 8 (By Kevin O'Connor)

Tutorial 1
Use the formula ALL FROM 9 AND THE LAST FROM 10 to
perform instant subtractions.
For example 1000 - 357 = 643
We simply take each figure in 357 from 9 and the last figure from 10.

So the answer is 1000 - 357 = 643
And thats all there is to it!
This always works for subtractions from numbers consisting of a 1 followed by noughts: 100; 1000; 10,000 etc.
Similarly 10,000 - 1049 = 8951

For 1000 - 83, in which we have more zeros than
figures in the numbers being subtracted, we simply
suppose 83 is 083.
So 1000 - 83 becomes 1000 - 083 = 917

Exercise 1 Tutorial 1
Try some yourself:

1) 1000 - 777       =
2) 1000 - 283      =
3) 1000 - 505       =
4) 10,000 - 2345 =
5) 10,000 - 9876 =
6) 10,000 - 1011 =
7) 100 - 57            =
8) 1000 - 57          =
9) 10,000 - 321   =
10) 10,000 - 38     =
Answers to exercise 1 Tutorial 1    < click

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Tutorial 2
Using VERTICALLY AND CROSSWISE you do
not need the multiplication tables beyond 5 X 5.
Suppose you need 8 x 7
8 is 2 below 10 and 7 is 3 below 10.
Think of it like this:

The answer is 56.
The diagram below shows how you get it.

You subtract crosswise 8-3 or 7 - 2 to get 5,
the first figure of the answer.
And you multiply vertically: 2 x 3 to get 6,
the last figure of the answer.
That's all you do:
See how far the numbers are below 10, subtract
one number's deficiency from the other number,
and multiply the deficiencies together.
7 x 6 = 42
Here there is a carry: the 1 in the
12 goes over to make 3 into 4.
Exercise 1 Tutorial 2
Multply These:
1) 8 x 8 =
2) 9 x 7 =
3) 8 x 9 =
4) 7 x 7 =
5) 9 x 9 =
6) 6 x 6 =
Answers to exercise 1 tutorial 2

Here's how to use VERTICALLY AND CROSSWISE
for multiplying numbers close to 100.
Suppose you want to multiply 88 by 98.
Not easy,you might think. But with
VERTICALLY AND CROSSWISE
you can give the answer immediately,
using the same method as above
Both 88 and 98 are close to 100.
88 is 12 below 100 and 98 is 2 below 100.
You can imagine the sum set out like this:

As before the 86 comes from subtracting crosswise:
88 - 2 = 86 (or 98 - 12 = 86: you can subtract either way,
you will always get the same answer).
And the 24 in the answer is just 12 x 2: you
multiply vertically.
So 88 x 98 = 8624
Exercise 2 Tutorial 2
This is so easy it is just mental arithmetic.
Try some:
1) 87 x 98 =
2) 88 x 97 =
3) 77 x 98 =
4) 93 x 96 =
5) 94 x 92 =
6) 64 x 99 =
7) 98 x 97 =
Answers to Exercise 2 Tutorial 2 < click
Multiplying numbers just over 100.
103 x 104 = 10712
The answer is in two parts: 107 and 12,
107 is just 103 + 4 (or 104 + 3),
and 12 is just 3 x 4.
Similarly 107 x 106 = 11342
107 + 6 = 113 and 7 x 6 = 42
Exercise 3 Tutorial 2
Again, just for mental arithmetic
Try a few:
1) 102 x 107 =
2) 106 x 103 =
3) 104 x 104 =
4) 109 x 108 =
5) 101 x123 =
6) 103 x102 =
Answers to exercise 3 Tutorial 2 < click

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Tutorial 3
The easy way to add and subtract fractions.
Use VERTICALLY AND CROSSWISE
to write the answer straight down!

Multiply crosswise and add to get the top of the answer:
2 x 5 = 10 and 1 x 3 = 3. Then 10 + 3 = 13.
The bottom of the fraction is just 3 x 5 = 15.
You multiply the bottom number together.
So:

Subtracting is just as easy: multiply
crosswise as before, but the subtract:

Exercise 1 Tutorial 3
Try a few:

Answers to Exercise 1 Tutorial 3 < click

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Tutorial 4
A quick way to square numbers that end in 5 using
the formula BY ONE MORE THAN THE ONE BEFORE.
75² = 5625
75² means 75 x 75.
The answer is in two parts: 56 and 25.
The last part is always 25.
The first part is the first number, 7, multiplied
by the number "one more", which is 8:
so 7 x 8 = 56

Similarly 85² = 7225 because 8 x 9 = 72.
Exercise 1 Tutorial 4
Try these:
1) 45² =
2) 65² =
3) 95² =
4) 35² =
5) 15² =
Answers to Exercise 1 Tutorial 4 < click
Method for multiplying numbers where the first
figures are the same and the last figures add up to 10.
32 x 38 = 1216
Both numbers here start with 3 and the last figures (2 and 8) add up to 10.
So we just multiply 3 by 4 (the next number up)
to get 12 for the first part of the answer.
And we multiply the last figures: 2 x 8 = 16 to
get the last part of the answer.
Diagrammatically:

And 81 x 89 = 7209
We put 09 since we need two figures as in all the other examples.
Exercise 2 Tutorial 4
Practise some:
1) 43 x 47 =
2) 24 x 26 =
3) 62 x 68 =
4) 17 x 13 =
5) 59 x 51 =
6) 77 x 73 =
Answers to Exercise 2 Tutorial 4

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Tutorial 5
An elegant way of multiplying numbers using a simple pattern
21 x 23 = 483
This is normally called long multiplication but actually
the answer can be written straight down using the
VERTICALLY AND CROSSWISEformula.
We first put, or imagine, 23 below 21:

There are 3 steps:
a) Multiply vertically on the left: 2 x 2 = 4.
This gives the first figure of the answer.
b) Multiply crosswise and add: 2 x 3 + 1 x 2 = 8
This gives the middle figure.
c) Multiply vertically on the right: 1 x 3 = 3
This gives the last figure of the answer.
And thats all there is to it.
Similarly 61 x 31 = 1891

6 x 3 = 18; 6 x 1 + 1 x 3 = 9; 1 x 1 = 1
Exercise 1 Tutorial 5
Try these, just write down the answer:
1) 14 x 21
2) 22 x 31
3) 21 x 31
4) 21 x 22
5) 32 x 21
Answers to Exercise 1 Tutorial 5
Exercise 2a Tutorial 5
Multiply any 2-figure numbers together by mere mental arithmetic!
If you want 21 stamps at 26 pence each you can
easily find the total price in your head.
There were no carries in the method given above.,/p>
However, there only involve one small extra step.
21 x 26 = 546

The method is the same as above
except that we get a 2-figure number, 14, in the
middle step, so the 1 is carried over to the left
(4 becomes 5).
So 21 stamps cost £5.46.
Practise a few:
1) 21 x 47
2) 23 x 43
3) 32 x 53
4) 42 x 32
5) 71 x 72
Answers to Exercise 2a Tutorial 5
Exercise 2b Tutorial 5
33 x 44 = 1452
There may be more than one carry in a sum:

Vertically on the left we get 12.
Crosswise gives us 24, so we carry 2 to the left
and mentally get 144.
Then vertically on the right we get 12 and the 1
here is carried over to the 144 to make 1452.

6) 32 x 56
7) 32 x 54
8) 31 x 72
9) 44 x 53
10) 54 x 64
Answers to Exercise 2b Tutorial 5
Any two numbers, no matter how big, can be
multiplied in one line by this method.

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Tutorial 6
Multiplying a number by 11.
To multiply any 2-figure number by 11 we just put
the total of the two figures between the 2 figures.
26 x 11 = 286
Notice that the outer figures in 286 are the 26
being multiplied.
And the middle figure is just 2 and 6 added up.
So 72 x 11 = 792
Exercise 1 Tutorial 6
Multiply by 11:
1) 43 =
2) 81 =
3) 15 =
4) 44 =
5) 11 =
Answers to Exercise 1 Tutorial 6
77 x 11 = 847
This involves a carry figure because 7 + 7 = 14
we get 77 x 11 = 7₁47 = 847.
Exercise 2 Tutorial 6
Multiply by 11:
1) 11 x 88 =
2) 11 x 84 =
3) 11 x 48 =
4) 11 x 73 =
5) 11 x 56 =
Answers to Exercise 2 Tutorial 6
234 x 11 = 2574
We put the 2 and the 4 at the ends.
We add the first pair 2 + 3 = 5.
and we add the last pair: 3 + 4 = 7.
Exercise 3 Tutorial 6
Multiply by 11:
1) 151 =
2) 527 =
3) 333 =
4) 714 =
5) 909 =
Answers to Exercise 3 Tutorial 6

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Tutorial 7
Method for dividing by 9.
23 / 9 = 2 remainder 5
The first figure of 23 is 2, and this is the answer.
The remainder is just 2 and 3 added up!
43 / 9 = 4 remainder 7
The first figure 4 is the answer
and 4 + 3 = 7 is the remainder - could it be easier?
Exercise 1a Tutorial 7
Divide by 9:
1) 61 / 9 =        remainder
2) 33 / 9 =        remainder
3) 44 / 9 =       remainder
4) 53 / 9 =       remainder
5) 80 / 9 =       remainder
Answers to Exercise 1a Tutorial 7
134 / 9 = 14 remainder 8
The answer consists of 1,4 and 8.
1 is just the first figure of 134.
4 is the total of the first two figures 1+ 3 = 4,
and 8 is the total of all three figures 1+ 3 + 4 = 8.
Exercise 1b Tutorial 7
Divide by 9:
6) 232 =         remainder
7) 151 =         remainder
8) 303 =        remainder
9) 212 =         remainder
10) 2121 =      remainder
Answers to Exercise 1b Tutorial 7
842 / 9 = 8₁2 remainder 14 = 92 remainder 14
Actually a remainder of 9 or more is not usually
permitted because we are trying to find how
many 9's there are in 842.
Since the remainder, 14 has one more 9 with 5
left over the final answer will be 93 remainder 5
Exercise 2 Tutorial 7
Divide these by 9:
1) 771 / 9 =         remainder
2) 942 / 9 =         remainder
3) 565 / 9 =          remainder
4) 555 / 9 =       remainder
5) 2382 / 9 =        remainder
6) 7070 / 9 =        remainder
Answers to Exercise 2 Tutorial 7

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Answers
Answers to exercise 1 Tutorial 1
1) 223
2) 717
3) 495
4) 7655
5) 0124
6) 8989
7) 43
8) 943
9) 9679
10) 9962
Return to Exercise 1 Tutorial 1
Answers to exercise 1 tutorial 2
1) 64
2) 63
3) 72
4) 49
5) 81
6)2₁6= 36
Return to Exercise 1 Tutorial 2
Answers to Exercise 2 Tutorial 2
1) 8526
2) 8536
3) 7546
4) 8928
5) 8648
6) 6336
7) 9506 (we put 06 because, like all the others,
we need two figures in each part)
Return to Exercise 2 Tutorial 2

Answers to exercise 3 Tutorial 2
1) 10914
2) 10918
3) 10816
4) 11772
5) 12423
6) 10506 (we put 06, not 6)
Return to Exercise 3 Tutorial 2
Answers to Exercise 1 Tutorial 3
1) 29/30
2) 7/12
3) 20/21
4) 19/30
5) 1/20
6) 13/15
Return to Exercise 1 Tutorial 3
Answers to Exercise 1 Tutorial 4
1) 2025
2) 4225
3) 9025
4) 1225
5) 225
Return to Exercise 1 Tutorial 4
Answers to Exercise 2 Tutorial 4
1) 2021
2) 624
3) 4216
4) 221
5) 3009
6) 5621
Return to Exercise 2 Tutorial 4
Answers to Exercise 1 Tutorial 5
1) 294
2) 682
3) 651
4) 462
5) 672
Return to Exercise 1 Tutorial 5

Answers to Exercise 2a Tutorial 5
1) 987
2) 989
3) 1696
4) 1344
5) 5112
Return to Exercise 2a Tutorial 5
Answers to Exercise 2b Tutorial 5
6) 1792
7) 1728
8) 2232
9) 2332
10) 3456
Return to Exercise 2b Tutorial 5
Answers to Exercise 1 Tutorial 6
1) 473
2) 891
3) 165
4) 484
5) 121
Return to Exercise 1 Tutorial 6
Answers to Exercise 2 Tutorial 6
1) 968
2) 924
3) 528
4) 803
5) 616
Return to Exercise 2 Tutorial 6
Answers to Exercise 3 Tutorial 6
1) 1661
2) 5797
3) 3663
4) 7854
5) 9999
Return to Exercise 3 Tutorial 6
Answers to Exercise 1a Tutorial 7
1) 6 r 7
2) 3 r 6
3) 4 r 8
4) 5 r 8
5) 8 r 8
Return to Exercise 1a Tutorial 7
Answers to Exercise 1b Tutorial 7
1) 25 r 7
2) 16 r 7
3) 33 r 6
4) 23 r 5
5) 235 r 6 (we have 2, 2 + 1, 2 + 1 + 2, 2 + 1 + 2 + 1)
Return to Exercise 1b Tutorial 7
Answers to Exercise 2 Tutorial 7
1) 714 r15 = 84 r15 = 85 r6
2) 913 r 15 = 103 r15 = 104 r6
3) 516 r16 = 61 r16 = 62 r7
4) 510 r15 = 60 r15 = 61 r6
5) 714 r21 = 84 r21 = 86 r3
6) 2513 r15 = 263 r15 = 264 r6
7) 7714 r14 = 784 r14 = 785 r5
Return to Exercise 2 Tutorial 7
copyright to the
ACADEMY OF VEDIC MATHEMATICS

____________________________________

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Tutorial 8
Vedic Maths - Tips & Tricks
Courtesy www.vedic-maths-ebook.com
By Kevin O'Connor
* Copyright Notice

Is it divisible by four?

This little math trick will show you whether
a number is divisible by four or not.
So, this is how it works.
Let's look at 1234
Does 4 divide evenly into 1234?
For 4 to divide into any number we have
to make sure that the last number is even
If it is an odd number, there is no way it will go in evenly.
So, for example, 4 will not go evenly into 1233 or 1235
Now we know that for 4 to divide evenly into any
number the number has to end with an even number.
Back to the question...
4 into 1234, the solution:
Take the last number and add it to 2 times the second last number
If 4 goes evenly into this number then you know
that 4 will go evenly into the whole number.
So
4 + (2 X 3) = 10
4 goes into 10 two times with a remainder of 2 so it does not go in evenly.
Therefore 4 into 1234 does not go in completely.
Let’s try 4 into 3436546
So, from our example, take the last number, 6 and add it to
two times the penultimate number, 4
6 + (2 X 4) = 14
4 goes into 14 three times with two remainder.
So it doesn't go in evenly.
Let's try one more.
4 into 212334436
6 + (2 X 3) = 12
4 goes into 12 three times with 0 remainder.
Therefore 4 goes into 234436 evenly.
So what use is this trick to you?
Well if you have learnt the tutorial at Memorymentor.com
about telling the day in any year, then you can use it in
working out whether the year you are calculating is a leap year or not.

Multiplying by 12 - shortcut

So how does the 12's shortcut work?
Let's take a look.
12 X 7
The first thing is to always multiply the 1 of the twelve by the
number we are multiplying by, in this case 7. So 1 X 7 = 7.
Multiply this 7 by 10 giving 70. (Why? We are working with BASES here.
Bases are the fundamentals to easy calculations for all multiplication tables.
To find out more check out our
Vedic Maths ebook at www.vedic-maths-ebook.com
Now multiply the 7 by the 2 of twelve giving 14. Add this to 70 giving 84.
Therefore 7 X 12 = 84
Let's try another:
17 X 12
Remember, multiply the 17 by the 1 in 12 and multiply by 10
(Just add a zero to the end)
1 X 17 = 17, multiplied by 10 giving 170.
Multiply 17 by 2 giving 34.
Add 34 to 170 giving 204.
So 17 X 12 = 204
lets go one more
24 X 12
Multiply 24 X 1 = 24. Multiply by 10 giving 240.
Multiply 24 by 2 = 48. Add to 240 giving us 288
24 X 12 = 288 (these are Seriously Simple Sums to do aren’t they?!)

Converting Kilos to pounds

In this section you will learn how to convert Kilos to Pounds, and Vice Versa.
Let’s start off with looking at converting Kilos to pounds.
86 kilos into pounds:
Step one, multiply the kilos by TWO.
To do this, just double the kilos.
86 x 2 = 172
Step two, divide the answer by ten.
To do this, just put a decimal point one place in from the right.
172 / 10 = 17.2
Step three, add step two’s answer to step one’s answer.
172 + 17.2 = 189.2
86 Kilos = 189.2 pounds
Let's try:
50 Kilos to pounds:
Step one, multiply the kilos by TWO.
To do this, just double the kilos.
50 x 2 = 100
Step two, divide the answer by ten.
To do this, just put a decimal point one place in from the right.
100/10 = 10
Step three, add step two's answer to step one's answer.
100 + 10 = 110
50 Kilos = 110 pounds

Adding Time

Here is a nice simple way to add hours and minutes together:
Let's add 1 hr and 35 minutes and 3 hr 55 minutes together.
What you do is this:
make the 1 hr 35 minutes into one number, which will give us 135 and do
the same for the other number, 3 hours 55 minutes, giving us 355
Now you want to add these two numbers together:
135
355
___
490
So we now have a sub total of 490.
What you need to do to this and all sub totals is
add the time constant of 40.
No matter what the hours and minutes are,
just add the 40 time constant to the sub total.
490 + 40 = 530
So we can now see our answer is 5 hrs and 30 minutes!

Temperature Conversions

This is a shortcut to convert Fahrenheit to Celsius and vice versa.
The answer you will get will not be an exact one, but it will
give you an idea of the temperature you are looking at.
Fahrenheit to Celsius:
Take 30 away from the Fahrenheit, and then divide the answer by two.
This is your answer in Celsius.
Example:
74 Fahrenheit - 30 = 44. Then divide by two, 22 Celsius.
So 74 Fahrenheit = 22 Celsius.
Celsius to Fahrenheit just do the reverse:
Double it, and then add 30.
30 Celsius double it, is 60, then add 30 is 90
30 Celsius = 90 Fahrenheit
Remember, the answer is not exact but it gives you a rough idea.

Decimals Equivalents of Fractions

With a little practice, it's not hard to recall
the decimal equivalents of fractions up to 10/11!
First, there are 3 you should know already:
1/2 = .5
1/3 = .333...
1/4 = .25
Starting with the thirds, of which you already know one:
1/3 = .333...
2/3 = .666...
You also know 2 of the 4ths, as well, so there's only one new one to learn:
1/4 = .25
2/4 = 1/2 = .5
3/4 = .75
Fifths are very easy. Take the numerator (the number on top),
double it, and stick a decimal in front of it.
1/5 = .2
2/5 = .4
3/5 = .6
4/5 = .8
There are only two new decimal equivalents to learn with the 6ths:
1/6 = .1666...
2/6 = 1/3 = .333...
3/6 = 1/2 = .5
4/6 = 2/3 = .666...
5/6 = .8333...
What about 7ths? We'll come back to them
at the end. They're very unique.
8ths aren't that hard to learn, as they're just
smaller steps than 4ths. If you have trouble
with any of the 8ths, find the nearest 4th,
and add .125 if needed:
1/8 = .125
2/8 = 1/4 = .25
3/8 = .375
4/8 = 1/2 = .5
5/8 = .625
6/8 = 3/4 = .75
7/8 = .875
9ths are almost too easy:
1/9 = .111...
2/9 = .222...
3/9 = .333...
4/9 = .444...
5/9 = .555...
6/9 = .666...
7/9 = .777...
8/9 = .888...
10ths are very easy, as well.
Just put a decimal in front of the numerator:
1/10 = .1
2/10 = .2
3/10 = .3
4/10 = .4
5/10 = .5
6/10 = .6
7/10 = .7
8/10 = .8
9/10 = .9
Remember how easy 9ths were? 11th are easy in a similar way,
assuming you know your multiples of 9:
1/11 = .090909...
2/11 = .181818...
3/11 = .272727...
4/11 = .363636...
5/11 = .454545...
6/11 = .545454...
7/11 = .636363...
8/11 = .727272...
9/11 = .818181...
10/11 = .909090...
As long as you can remember the pattern for each fraction, it is
quite simple to work out the decimal place as far as you want
or need to go!
Oh, I almost forgot! We haven't done 7ths yet, have we?
One-seventh is an interesting number:
1/7 = .142857142857142857...
For now, just think of one-seventh as: .142857
See if you notice any pattern in the 7ths:
1/7 = .142857...
2/7 = .285714...
3/7 = .428571...
4/7 = .571428...
5/7 = .714285...
6/7 = .857142...
Notice that the 6 digits in the 7ths ALWAYS stay in the same
order and the starting digit is the only thing that changes!
If you know your multiples of 14 up to 6, it isn't difficult to,
work out where to begin the decimal number. Look at this:
For 1/7, think "1 * 14", giving us .14 as the starting point.
For 2/7, think "2 * 14", giving us .28 as the starting point.
For 3/7, think "3 * 14", giving us .42 as the starting point.
For 4/14, 5/14 and 6/14, you'll have to adjust upward by 1:
For 4/7, think "(4 * 14) + 1", giving us .57 as the starting point.
For 5/7, think "(5 * 14) + 1", giving us .71 as the starting point.
For 6/7, think "(6 * 14) + 1", giving us .85 as the starting point.
Practice these, and you'll have the decimal equivalents of
everything from 1/2 to 10/11 at your finger tips!
If you want to demonstrate this skill to other people, and you know
your multiplication tables up to the hundreds for each number 1-9, then give them a
calculator and ask for a 2-digit number (3-digit number, if you're up to it!) to be
divided by a 1-digit number.
If they give you 96 divided by 7, for example, you can think,
"Hmm... the closest multiple of 7 is 91, which is 13 * 7, with 5 left over.
So the answer is 13 and 5/7, or: 13.7142857!"

Converting Kilometres to Miles

This is a useful method for when travelling between imperial
and metric countries and need to know what kilometres to miles are.
The formula to convert kilometres to miles is number of (kilometres / 8 ) X 5
So lets try 80 kilometres into miles
80/8 = 10
multiplied by 5 is 50 miles!
Another example
40 kilometres
40 / 8 = 5
5 X 5= 25 miles
Vedic Mathematics
Master Multiplication tables, division and lots more!
We recommed you check out this ebook, it's packed with tips,
tricks and tutorials that will boost your math ability, guaranteed!
www.vedic-maths-ebook.com

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This publication is protected by international copyright laws. You have the author’s permission to transmit this ebook and use it as a gift or as part of your advertising campaign. However you CANNOT charge for it, nor can you edit its contents. The author’s contact details must stay
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By Kevin O'Connor_________________

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Vedic Mathematics
By Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj (1884-1960)
Book ref: ISBN 0 8426 0967 9 Published by Motilal Banarasidas
From the Introduction by Smti Manjula Trivedi 16-03-1965. An extract:
Revered Guruji used to say that he had reconstructed the sixteen mathematical formulae from the Atharvaveda after assiduous research and ‘Tapas’ (austerity) for about eight years in the forests surrounding Sringeri. Obviously these formulae are not to be found in the present recensions of Atharvaveda. They were actually reconstructed, on the basis of intuitive revelation, from materials scattered here and there in the Atharvaveda.
From the Preface by the author Jagadguru Swami Sri Bharati Krsna Tirthaji Maharaj
Extracts:
We may however, at this point draw the earnest attention of every one concerned to the following salient items thereof:
The Sutras (aphorisms) apply to and cover each and every part of each and every chapter of each and every branch of mathematics (including Arithmetic, Algebra, Geometry – plane and solid, Trigonometry – plane and spherical, Conics – geometrical and analytical, Astronomy, Calculus – differential and integral etc.) In fact, there is no part of mathematics, pure or applied, that is beyond their jurisdiction.
The Sutras are easy to understand, easy to apply and easy to remember, and the whole work can be truthfully summarised in one word ‘Mental’!
Even as regards complex problems involving a good number of mathematical operations (consecutively or even simultaneously to be performed), the time taken by the Vedic method will be a third, a fourth, a tenth, or even a much smaller fraction of the time required according to modern (i.e. current) Western methods.
And in some very important and striking cases, sums requiring 30, 50, 100 or even more numerous and cumbrous ‘steps’ of working (according to the current Western methods) can be answered in a single and simple step of work by the Vedic method! And little children (of only 10 or 12 years of age) merely look at the sums written on the blackboard and immediately shout out and dictate the answers. And this is because, as a matter of fact, each digit automatically yields its predecessor and its successor! And the children have merely to go on tossing off (or reeling off) the digits one after another (forwards or backwards) by mere mental arithmetic (without needing pen or pencil, paper, slate etc.).
On seeing this kind of work actually being performed by the little children, the doctors, professors and other ‘big-guns’ of mathematics are wonder-struck and exclaim: ‘Is this mathematics or magic’? And we invariably answer and say: ‘It is both. It is magic until you understand it; and it is mathematics thereafter’. And then we proceed to substantiate and prove the correctness of this reply of ours!
As regards the time required by the students for mastering the whole course of Vedic Mathematics as applied to all its branches, we need merely state from our actual experience that 8 months (or 12 months) at an average rate of 2 or 3 hours per day should suffice for completing the whole course of mathematical studies on these Vedic lines instead of 15 or 20 years required according to the existing systems of the Indian and also of foreign universities.
And we were agreeably astonished and intensely gratified to find that exceedingly tough mathematical problems (which the mathematically most advanced present day Western scientific world had spent huge amount of time, energy, and money on and which even now it solves with the utmost difficulty and that also after vast labour involving large numbers of difficult, tedious and cumbersome ‘steps’ of working) can be easily and readily solved with the help of these ultra-easy Vedic Sutras (or mathematical aphorisms) contained in the Parisista (the appendix portion) of the Atharvaveda in a few simple steps and by methods that can be conscientiously described as mere ‘mental arithmetic’.
It is thus in the fitness of things that the Vedas include 1. Ayurveda (anatomy, physiology, hygiene, sanitary science, medical science, surgery etc.), not for the purpose of achieving perfect health and strength in the after-death future but in order to attain them here and now in our present physical bodies. 2.Dhanurveda (archery and other military sciences), not for fighting with one another after our transportation to heaven but in order to quell and subdue all invaders from abroad and all insurgents from within. 3. Gandharva Veda (the science of art and music) and 4. Sthapatya Veda (engineering, architecture etc. and all branches of mathematics in general). All these subjects, be it noted, are inherent parts of the Vedas i.e., are reckoned as ‘spiritual’ studies and catered for as such therein.
Similar is the case with Vedangas (i.e., grammar, prosody, astronomy, lexicography etc.) which according to the Indian cultural conceptions, are also inherent parts and subjects of Vedic (i.e. religious) study.
From the Foreward by Swami Pratyagatmananda Saraswati Varanasi, 22-03-1965
An extract:
Vedic Mathematics by the late Shankaracharya (Bharati Krsna Tirtha) of Govardhan Pitha is a monumental work. In his deep-layer explorations of cryptic Vedic mysteries relating especially to their calculus of shorthand formulae and their neat and ready application to practical problems, the late Shankaracharya shows the rare combination of the probing insight of revealing intuition of a Yogi with the analytic acumen and synthetic talent of a mathematician.
With the late Shankaracharya we belong to a race, now fast becoming extinct, of diehard believers who think that the Vedas represent an inexhaustible mine of profoundest wisdom in matters of both spiritual and temporal; and that this store of wisdom was not, as regards its assets of fundamental validity and value at least, gathered by the laborious inductive and deductive methods of ordinary systemic enquiry, but was direct gift of revelation to seers and sages who in their higher reaches of Yogic realisation were competent to receive it from a source, perfect and immaculate.
Whether or not the Vedas are believed as repositories of perfect wisdom, it is unquestionable that the Vedic race lived not as merely pastoral folk possessing a half or a quarter developed culture and civilisation. The Vedic seers were, again, not mere ‘navel-gazers’ or ‘nose-tip gazers’. They proved themselves adepts in all levels and branches of knowledge, theoretical and practical. For example, they had their varied objective science both pure and applied.
Let us take a concrete illustration. Suppose in a time of drought we require rains by artificial means. The modern scientist has his own theory and art (technique) for producing the result. The old seer scientist had his both also, but different from these now availing. He had his science and technique, called Yajna, in which Mantra, Yantra, and other factors must co-operate with mathematical determinateness and precision. For this purpose, he had developed the six auxiliaries of the Vedas in each of which mathematical skill and adroitness, occult or otherwise, play the decisive role. The Sutras lay down the shortest and surest lines. The correct intonation of the Mantra, the correct configuration of the Yantra (in the making of the Vedi etc., e.g. the quadrate of a circle), the correct time or astral conjunction factor, the correct rhythams etc. All had to be perfected so as to produce the desired results effectively and adequately. Each of these required the calculus of mathematics. The modern technician has his logarithmic tables and mechanic’s manuals. The old Yajnik had his Sutras.
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Pages from the history of the Indian sub-continent:
Science and Mathematics in India
History of Mathematics in India
In all early civilizations, the first expression of mathematical understanding appears in the form of counting systems. Numbers in vey early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (as in India) or were designated by alphabetic letters (such as in Rome). Although today, we take our decimal system for granted, not all ancient civilizations based their numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60) system was in use.
The Decimal System in Harappa
In India a decimal system was already in place during the Harappan period, as indicated by an analysis of Harappan weights and measures. Weights corresponding to ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500 have been identified, as have scales with decimal divisions. A particularly notable characteristic of Harappan weights and measures is their remarkable accuracy. A bronze rod marked in units of 0.367 inches points to the degree of precision demanded in those times. Such scales were particularly important in ensuring proper implementation of town planning rules that required roads of fixed widths to run at right angles to each other, for drains to be constructed of precise measurements, and for homes to be constructed according to specified guidelines. The existence of a gradated system of accurately marked weights points to the development of trade and commerce in Harappan society.
Mathematical Activity in the Vedic Period
In the Vedic period, records of mathematical activity are mostly to be found in Vedic texts associated with ritual activities. However, as in many other early agricultural civilizations, the study of arithmetic and geometry was also impelled by secular considerations. Thus, to some extent early mathematical developments in India mirrored the developments in Egypt, Babylon and China . The system of land grants and agricultural tax assessments required accurate measurement of cultivated areas. As land was redistributed or consolidated, problems of mensuration came up that required solutions. In order to ensure that all cultivators had equivalent amounts of irrigated and non-irrigated lands and tracts of equivalent fertility - individual farmers in a village often had their holdings broken up in several parcels to ensure fairness. Since plots could not all be of the same shape - local administrators were required to convert rectangular plots or triangular plots to squares of equivalent sizes and so on. Tax assessments were based on fixed proportions of annual or seasonal crop incomes, but could be adjusted upwards or downwards based on a variety of factors. This meant that an understanding of geometry and arithmetic was virtually essential for revenue administrators. Mathematics was thus brought into the service of both the secular and the ritual domains.
Arithmetic operations (Ganit) such as addition, subtraction, multiplication, fractions, squares, cubes and roots are enumerated in the Narad Vishnu Purana attributed to Ved Vyas (pre-1000 BC). Examples of geometric knowledge (rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (600 BC) which describe techniques for the construction of ritual altars in use during the Vedic era. It is likely that these texts tapped geometric knowledge that may have been acquired much earlier, possibly in the Harappan period. Baudhayana's Sutra displays an understanding of basic geometric shapes and techniques of converting one geometric shape (such as a rectangle) to another of equivalent (or multiple, or fractional) area (such as a square). While some of the formulations are approximations, others are accurate and reveal a certain degree of practical ingenuity as well as some theoretical understanding of basic geometric principles. Modern methods of multiplication and addition probably emerged from the techniques described in the Sulva-Sutras.
Pythagoras - the Greek mathematician and philosopher who lived in the 6th C B.C was familiar with the Upanishads and learnt his basic geometry from the Sulva Sutras. An early statement of what is commonly known as the Pythagoras theorem is to be found in Baudhayana's Sutra: The chord which is stretched across the diagonal of a square produces an area of double the size. A similar observation pertaining to oblongs is also noted. His Sutra also contains geometric solutions of a linear equation in a single unknown. Examples of quadratic equations also appear. Apasthamba's sutra (an expansion of Baudhayana's with several original contributions) provides a value for the square root of 2 that is accurate to the fifth decimal place. Apasthamba also looked at the problems of squaring a circle, dividing a segment into seven equal parts, and a solution to the general linear equation. Jain texts from the 6th C BC such as the Surya Pragyapti describe ellipses.
Modern-day commentators are divided on how some of the results were generated. Some believe that these results came about through hit and trial - as rules of thumb, or as generalizations of observed examples. Others believe that once the scientific method came to be formalized in the Nyaya-Sutras - proofs for such results must have been provided, but these have either been lost or destroyed, or else were transmitted orally through the Gurukul system, and only the final results were tabulated in the texts. In any case, the study of Ganit i.e mathematics was given considerable importance in the Vedic period. The Vedang Jyotish (1000 BC) includes the statement: "Just as the feathers of a peacock and the jewel-stone of a snake are placed at the highest point of the body (at the forehead), similarly, the position of Ganit is the highest amongst all branches of the Vedas and the Shastras."
(Many centuries later, Jain mathematician from Mysore, Mahaviracharya further emphasized the importance of mathematics: "Whatever object exists in this moving and non-moving world, cannot be understood without the base of Ganit (i.e. mathematics)".)
Panini and Formal Scientific Notation
A particularly important development in the history of Indian science that was to have a profound impact on all mathematical treatises that followed was the pioneering work by Panini (6th C BC) in the field of Sanskrit grammar and linguistics. Besides expounding a comprehensive and scientific theory of phonetics, phonology and morphology, Panini provided formal production rules and definitions describing Sanskrit grammar in his treatise called Asthadhyayi. Basic elements such as vowels and consonants, parts of speech such as nouns and verbs were placed in classes. The construction of compound words and sentences was elaborated through ordered rules operating on underlying structures in a manner similar to formal language theory.
Today, Panini's constructions can also be seen as comparable to modern definitions of a mathematical function. G G Joseph, in The crest of the peacock argues that the algebraic nature of Indian mathematics arises as a consequence of the structure of the Sanskrit language. Ingerman in his paper titled Panini-Backus form finds Panini's notation to be equivalent in its power to that of Backus - inventor of the Backus Normal Form used to describe the syntax of modern computer languages. Thus Panini's work provided an example of a scientific notational model that could have propelled later mathematicians to use abstract notations in characterizing algebraic equations and presenting algebraic theorems and results in a scientific format.
Philosophy and Mathematics
Philosophical doctrines also had a profound influence on the development of mathematical concepts and formulations. Like the Upanishadic world view, space and time were considered limitless in Jain cosmology. This led to a deep interest in very large numbers and definitions of infinite numbers. Infinite numbers were created through recursive formulae, as in the Anuyoga Dwara Sutra. Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Permutations and combinations are listed in the Bhagvati Sutras (3rd C BC) and Sathananga Sutra (2nd C BC).
Jain set theory probably arose in parallel with the Syadvada system of Jain epistemology in which reality was described in terms of pairs of truth conditions and state changes. The Anuyoga Dwara Sutra demonstrates an understanding of the law of indeces and uses it to develop the notion of logarithms. Terms like Ardh Aached , Trik Aached, and Chatur Aached are used to denote log base 2, log base 3 and log base 4 respectively. In Satkhandagama various sets are operated upon by logarithmic functions to base two, by squaring and extracting square roots, and by raising to finite or infinite powers. The operations are repeated to produce new sets. In other works the relation of the number of combinations to the coefficients occurring in the binomial expansion is noted.
Since Jain epistemology allowed for a degree of indeterminacy in describing reality, it probably helped in grappling with indeterminate equations and finding numerical approximations to irrational numbers.
Buddhist literature also demonstrates an awareness of indeterminate and infinite numbers. Buddhist mathematics was classified either as Garna (Simple Mathematics) or Sankhyan (Higher Mathematics). Numbers were deemed to be of three types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite).
Philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD. Although scholars are divided about how early the symbol for zero came to be used in numeric notation in India, (Ifrah arguing that the use of zero is already implied in Aryabhatta) tangible evidence for the use of the zero begins to proliferate towards the end of the Gupta period. Between the 7th C and the 11th C, Indian numerals developed into their modern form, and along with the symbols denoting various mathematical functions (such as plus, minus, square root etc) eventually became the foundation stones of modern mathematical notation.
The Indian Numeral System
Although the Chinese were also using a decimal based counting system, the Chinese lacked a formal notational system that had the abstraction and elegance of the Indian notational system, and it was the Indian notational system that reached the Western world through the Arabs and has now been accepted as universal. Several factors contributed to this development whose significance is perhaps best stated by French mathematician, Laplace: "The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. It's simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions."
Brilliant as it was, this invention was no accident. In the Western world, the cumbersome roman numeral system posed as a major obstacle, and in China the pictorial script posed as a hindrance. But in India, almost everything was in place to favor such a development. There was already a long and established history in the use of decimal numbers, and philosophical and cosmological constructs encouraged a creative and expansive approach to number theory. Panini's studies in linguistic theory and formal language and the powerful role of symbolism and representational abstraction in art and architecture may have also provided an impetus, as might have the rationalist doctrines and the exacting epistemology of the Nyaya Sutras, and the innovative abstractions of the Syadavada and Buddhist schools of learning.
Influence of Trade and Commerce, Importance of Astronomy
The growth of trade and commerce, particularly lending and borrowing demanded an understanding of both simple and compound interest which probably stimulated the interest in arithmetic and geometric series. Brahmagupta's description of negative numbers as debts and positive numbers as fortunes points to a link between trade and mathematical study. Knowledge of astronomy - particularly knowledge of the tides and the stars was of great import to trading communities who crossed oceans or deserts at night. This is borne out by numerous references in the Jataka tales and several other folk-tales. The young person who wished to embark on a commercial venture was inevitably required to first gain some grounding in astronomy. This led to a proliferation of teachers of astronomy, who in turn received training at universities such as at Kusumpura (Bihar) or Ujjain (Central India) or at smaller local colleges or Gurukuls. This also led to the exchange of texts on astronomy and mathematics amongst scholars and the transmission of knowledge from one part of India to another. Virtually every Indian state produced great mathematicians who wrote commentaries on the works of other mathematicians (who may have lived and worked in a different part of India many centuries earlier). Sanskrit served as the common medium of scientific communication.
The science of astronomy was also spurred by the need to have accurate calendars and a better understanding of climate and rainfall patterns for timely sowing and choice of crops. At the same time, religion and astrology also played a role in creating an interest in astronomy and a negative fallout of this irrational influence was the rejection of scientific theories that were far ahead of their time. One of the greatest scientists of the Gupta period - Aryabhatta (born in 476 AD, Kusumpura, Bihar) provided a systematic treatment of the position of the planets in space. He correctly posited the axial rotation of the earth, and inferred correctly that the orbits of the planets were ellipses. He also correctly deduced that the moon and the planets shined by reflected sunlight and provided a valid explanation for the solar and lunar eclipses rejecting the superstitions and mythical belief systems surrounding the phenomenon. Although Bhaskar I (born Saurashtra, 6th C, and follower of the Asmaka school of science, Nizamabad, Andhra ) recognized his genius and the tremendous value of his scientific contributions, some later astronomers continued to believe in a static earth and rejected his rational explanations of the eclipses. But in spite of such setbacks, Aryabhatta had a profound influence on the astronomers and mathematicians who followed him, particularly on those from the Asmaka school.
Mathematics played a vital role in Aryabhatta's revolutionary understanding of the solar system. His calculations on pi, the circumferance of the earth (62832 miles) and the length of the solar year (within about 13 minutes of the modern calculation) were remarkably close approximations. In making such calculations, Aryabhatta had to solve several mathematical problems that had not been addressed before, including problems in algebra (beej-ganit) and trigonometry (trikonmiti).
Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as the longitudes of the planets; conjunctions of the planets with each other and with bright stars; risings and settings of the planets; and the lunar crescent. Again, these studies required still more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta, and like Aryabhatta correctly assessed pi to be an irrational number. Amongst his most important contributions was his formula for calculating the sine function which was 99% accurate. He also did pioneering work on indeterminate equations and considered for the first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.
Another important astronomer/mathematician was Varahamira (6th C, Ujjain) who compiled previously written texts on astronomy and made important additions to Aryabhatta's trigonometric formulas. His works on permutations and combinations complemented what had been previously achieved by Jain mathematicians and provided a method of calculation of nCr that closely resembles the much more recent Pascal's Triangle. In the 7th century, Brahmagupta did important work in enumerating the basic principles of algebra. In addition to listing the algebraic properties of zero, he also listed the algebraic properties of negative numbers. His work on solutions to quadratic indeterminate equations anticipated the work of Euler and Lagrange.
Emergence of Calculus
In the course of developing a precise mapping of the lunar eclipse, Aryabhatta was obliged to introduce the concept of infinitesimals - i.e. tatkalika gati to designate the infinitesimal, or near instantaneous motion of the moon, and express it in the form of a basic differential equation. Aryabhatta's equations were elaborated on by Manjula (10th C) and Bhaskaracharya (12th C) who derived the differential of the sine function. Later mathematicians used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
Applied Mathematics, Solutions to Practical Problems
Developments also took place in applied mathematics such as in creation of trigonometric tables and measurement units. Yativrsabha's work Tiloyapannatti (6th C) gives various units for measuring distances and time and also describes the system of infinite time measures.
In the 9th C, Mahaviracharya ( Mysore) wrote Ganit Saar Sangraha where he described the currently used method of calculating the Least Common Multiple (LCM) of given numbers. He also derived formulae to calculate the area of an ellipse and a quadrilateral inscribed within a circle (something that had also been looked at by Brahmagupta) The solution of indeterminate equations also drew considerable interest in the 9th century, and several mathematicians contributed approximations and solutions to different types of indeterminate equations.
In the late 9th C, Sridhara (probably Bengal) provided mathematical formulae for a variety of practical problems involving ratios, barter, simple interest, mixtures, purchase and sale, rates of travel, wages, and filling of cisterns. Some of these examples involved fairly complicated solutions and his Patiganita is considered an advanced mathematical work. Sections of the book were also devoted to arithmetic and geometric progressions, including progressions with fractional numbers or terms, and formulas for the sum of certain finite series are provided. Mathematical investigation continued into the 10th C. Vijayanandi (of Benares, whose Karanatilaka was translated by Al-Beruni into Arabic) and Sripati of Maharashtra are amongst the prominent mathematicians of the century.
The leading light of 12th C Indian mathematics was Bhaskaracharya who came from a long-line of mathematicians and was head of the astronomical observatory at Ujjain. He left several important mathematical texts including the Lilavati and Bijaganita and the Siddhanta Shiromani, an astronomical text. He was the first to recognize that certain types of quadratic equations could have two solutions. His Chakrawaat method of solving indeterminate solutions preceded European solutions by several centuries, and in his Siddhanta Shiromani he postulated that the earth had a gravitational force, and broached the fields of infinitesimal calculation and integration. In the second part of this treatise, there are several chapters relating to the study of the sphere and it's properties and applications to geography, planetary mean motion, eccentric epicyclical model of the planets, first visibilities of the planets, the seasons, the lunar crescent etc. He also discussed astronomical instruments and spherical trigonometry. Of particular interest are his trigonometric equations: sin(a + b) = sin a cos b + cos a sin b; sin(a - b) = sin a cos b - cos a sin b;
The Spread of Indian Mathematics
The study of mathematics appears to slow down after the onslaught of the Islamic invasions and the conversion of colleges and universities to madrasahs. But this was also the time when Indian mathematical texts were increasingly being translated into Arabic and Persian. Although Arab scholars relied on a variety of sources including Babylonian, Syrian, Greek and some Chinese texts, Indian mathematical texts played a particularly important role. Scholars such as Ibn Tariq and Al-Fazari (8th C, Baghdad), Al-Kindi (9th C, Basra), Al-Khwarizmi (9th C. Khiva), Al-Qayarawani (9th C, Maghreb, author of Kitab fi al-hisab al-hindi), Al-Uqlidisi (10th C, Damascus, author of The book of Chapters in Indian Arithmetic), Ibn-Sina (Avicenna), Ibn al-Samh (Granada, 11th C, Spain), Al-Nasawi (Khurasan, 11th C, Persia), Al-Beruni (11th C, born Khiva, died Afghanistan), Al-Razi (Teheran), and Ibn-Al-Saffar (11th C, Cordoba) were amongst the many who based their own scientific texts on translations of Indian treatises. Records of the Indian origin of many proofs, concepts and formulations were obscured in the later centuries, but the enormous contributions of Indian mathematics was generously acknowledged by several important Arabic and Persian scholars, especially in Spain. Abbasid scholar Al-Gaheth wrote: " India is the source of knowledge, thought and insight”. Al-Maoudi (956 AD) who travelled in Western India also wrote about the greatness of Indian science. Said Al-Andalusi, an 11th C Spanish scholar and court historian was amongst the most enthusiastic in his praise of Indian civilization, and specially remarked on Indian achievements in the sciences and in mathematics. Of course, eventually, Indian algebra and trigonometry reached Europe through a cycle of translations, travelling from the Arab world to Spain and Sicily, and eventually penetrating all of Europe. At the same time, Arabic and Persian translations of Greek and Egyptian scientific texts became more readily available in India.
The Kerala School
Although it appears that original work in mathematics ceased in much of Northern India after the Islamic conquests, Benaras survived as a center for mathematical study, and an important school of mathematics blossomed in Kerala. Madhava (14th C, Kochi) made important mathematical discoveries that would not be identified by European mathematicians till at least two centuries later. His series expansion of the cos and sine functions anticipated Newton by almost three centuries. Historians of mathematics, Rajagopal, Rangachari and Joseph considered his contributions instrumental in taking mathematics to the next stage, that of modern classical analysis. Nilkantha (15th C, Tirur, Kerala) extended and elaborated upon the results of Madhava while Jyesthadeva (16th C, Kerala) provided detailed proofs of the theorems and derivations of the rules contained in the works of Madhava and Nilkantha. It is also notable that Jyesthadeva's Yuktibhasa which contained commentaries on Nilkantha's Tantrasamgraha included elaborations on planetary theory later adopted by Tycho Brahe, and mathematics that anticipated work by later Europeans. Chitrabhanu (16th C, Kerala) gave integer solutions to twenty-one types of systems of two algebraic equations, using both algebraic and geometric methods in developing his results. Important discoveries by the Kerala mathematicians included the Newton-Gauss interpolation formula, the formula for the sum of an infinite series, and a series notation for pi. Charles Whish (1835, published in the Transactions of the Royal Asiatic Society of Great Britain and Ireland) was one of the first Westerners to recognize that the Kerala school had anticipated by almost 300 years many European developments in the field.
Yet, few modern compendiums on the history of mathematics have paid adequate attention to the often pioneering and revolutionary contributions of Indian mathematicians. But as this essay amply demonstrates, a significant body of mathematical works were produced in the Indian subcontinent. The science of mathematics played a pivotal role not only in the industrial revolution but in the scientific developments that have occurred since. No other branch of science is complete without mathematics. Not only did India provide the financial capital for the industrial revolution (see the essay on colonization) India also provided vital elements of the scientific foundation without which humanity could not have entered this modern age of science and high technology.
Notes:
Mathematics and Music: Pingala (3rd C AD), author of Chandasutra explored the relationship between combinatorics and musical theory anticipating Mersenne (1588-1648) author of a classic on musical theory.
Mathematics and Architecture: Interest in arithmetic and geometric series may have also been stimulated by (and influenced) Indian architectural designs - (as in temple shikaras, gopurams and corbelled temple ceilings). Of course, the relationship between geometry and architectural decoration was developed to it's greatest heights by Central Asian, Persian, Turkish, Arab and Indian architects in a variety of monuments commissioned by the Islamic rulers.
Transmission of the Indian Numeral System: Evidence for the transmission of the Indian Numeral System to the West is provided by Joseph (Crest of the Peacock):
Quotes Severus Sebokht (662) in a Syriac text describing the "subtle discoveries" of Indian astronomers as being "more ingenious than those of the Greeks and the Babylonians" and "their valuable methods of computation which surpass description" and then goes on to mention the use of nine numerals.
· Quotes from Liber abaci (Book of the Abacus) by Fibonacci (1170-1250): The nine Indian numerals are ...with these nine and with the sign 0 which in Arabic is sifr, any desired number can be written. (Fibonaci learnt about Indian numerals from his Arab teachers in North Africa)
Influence of the Kerala School: Joseph (Crest of the Peacock) suggests that Indian mathematical manuscripts may have been brought to Europe by Jesuit priests such as Matteo Ricci who spent two years in Kochi (Cochin) after being ordained in Goa in 1580. Kochi is only 70km from Thrissur (Trichur) which was then the largest repository of astronomical documents. Whish and Hyne - two European mathematicians obtained their copies of works by the Kerala mathematicians from Thrissur, and it is not inconceivable that Jesuit monks may have also taken copies to Pisa (where Galileo, Cavalieri and Wallis spent time), or Padau (where James Gregory studied) or Paris (where Mersenne who was in touch with Fermat and Pascal, acted as an agent for the transmission of mathematical ideas).
References:
1.Studies in the History of Science in India (Anthology edited by Debiprasad Chattopadhyaya)
2.A P Juskevic, S S Demidov, F A Medvedev and E I Slavutin: Studies in the history of mathematics, "Nauka" (Moscow, 1974), 220-222; 302.
3. B Datta: The science of the Sulba (Calcutta, 1932).
4.G G Joseph: The crest of the peacock (Princeton University Press, 2000).
5. R P Kulkarni: The value of pi known to Sulbasutrakaras, Indian Journal Hist. Sci. 13 (1) (1978), 32-41.
6. G Kumari: Some significant results of algebra of pre-Aryabhata era, Math. Ed. (Siwan) 14 (1) (1980), B5-B13.
7. G Ifrah: A universal history of numbers: From prehistory to the invention of the computer (London, 1998).
8. P Z Ingerman: 'Panini-Backus form', Communications of the ACM 10 (3)(1967), 137.
9.P Jha: Contributions of the Jainas to astronomy and mathematics, Math. Ed. (Siwan) 18 (3) (1984), 98-107.
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13. K Shankar Shukla: Bhaskara I, Bhaskara I and his works III. Laghu-Bhaskariya (Sanskrit) (Lucknow, 1963).
14. K S Shukla: Hindu mathematics in the seventh century as found in Bhaskara I's commentary on the Aryabhatiya, Ganita 22 (1) (1971), 115-130.
15. R C Gupta: Varahamihira's calculation of nCr and the discovery of Pascal's triangle, Ganita Bharati 14 (1-4) (1992), 45-49.
16. B Datta: On Mahavira's solution of rational triangles and quadrilaterals, Bull. Calcutta Math. Soc. 20 (1932), 267-294.
17. B S Jain: On the Ganita-Sara-Samgraha of Mahavira (c. 850 A.D.), Indian J. Hist. Sci. 12 (1) (1977), 17-32.
18. K Shankar Shukla: The Patiganita of Sridharacarya (Lucknow, 1959).
19. H. Suter: Mathematiker
20. Suter: Die Mathematiker und Astronomen der Araber
21. Die philosophischen Abhandlungen des al-Kindi, Munster, 1897
22. K V Sarma: A History of the Kerala School of Hindu Astronomy (Hoshiarpur, 1972).
23. R C Gupta: The Madhava-Gregory series, Math. Education 7 (1973), B67-B70
24. S Parameswaran: Madhavan, the father of analysis, Ganita-Bharati 18 (1-4) (1996), 67-70.
25. K V Sarma, and S Hariharan: Yuktibhasa of Jyesthadeva : a book of rationales in Indian mathematics and astronomy - an analytical appraisal, Indian J. Hist. Sci. 26 (2) (1991), 185-207
26. C T Rajagopal and M S Rangachari: On an untapped source of medieval Keralese mathematics, Arch. History Exact Sci. 18 (1978), 89-102.
27. C T Rajagopal and M S Rangachari: On medieval Keralese mathematics, Arch. History Exact Sci. 35 (1986), 91-99.
28. A.K. Bag: Mathematics in Ancient and Medieval India (1979, Varanasi)
29. Bose, Sen, Subarayappa: Concise History of Science in India, (Indian National Science Academy)
30. T.A. Saraswati: Geometry in Ancient and Medieval India (1979, Delhi)
31.N. Singh: Foundations of Logic in Ancient India, Linguistics and Mathematics ( Science and technology in Indian Culture, ed. A Rahman, 1984, New Delhi, National Instt. of Science, Technology and Development Studies, NISTAD)
32. P. Singh: "The so-called Fibonacci numbers in ancient and medieval India, (Historia Mathematica, 12, 229-44, 1985)
33. Chin Keh-Mu: India and China: Scientific Exchange (History of Science in India Vol 2.)
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Another view on Indian Mathematics:

Indic Mathematics: India and the Scientific Revolution (1)

Dr. David Gray writes:
"The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the "Western" scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages."
Dr Gray goes on to list some of the most important developments in the history of mathematics that took place in India, summarizing the contributions of luminaries such as Aryabhatta, Brahmagupta, Mahavira, Bhaskara and Maadhava. He concludes by asserting that "the role played by India in the development (of the scientific revolution in Europe) is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of its greatest contributions to world civilization."
-Indic Mathematics
Related Essays:
The following essays can be found athttp://india_resource.tripod.com/indianhistory.html
Development of Philosophical Thought and Scientific Method in Ancient india
Philosophical Development from Upanishadic Theism to Scientific Realism
History of the Physical Sciences in India
_________________
Indic Mathematics
India and the Scientific Revolution(2)
By David Gray, PhD
1. Math and Ethnocentrism
The study of mathematics in the West has long been characterized by a certain ethnocentric bias, a bias which most often manifests not in explicit racism, but in a tendency toward undermining or eliding the real contributions made by non-Western civilizations. The debt owed by the West to other civilizations, and to India in particular, go back to the earliest epoch of the "Western" scientific tradition, the age of the classical Greeks, and continued up until the dawn of the modern era, the renaissance, when Europe was awakening from its dark ages. This awakening was in part made possible by the rediscovery of mathematics and other sciences and technologies through the medium of the Arabs, who transmitted to Europe both their own lost heritage as well as the advanced mathematical traditions formulated in India.
George Ghevarughese Joseph, in an important article entitled "Foundations of Eurocentrism in Mathematics," argued that "the standard treatment of the history of non-European mathematics is a product of historiographical bias (conscious or otherwise) in the selection and interpretation of facts, which, as a consequence, results in ignoring, devaluing or distorting contributions arising outside European mathematical traditions." (1987:14)
Due to the legacy of colonialism, the exploitation of which was ideologically justified through a doctrine of racial superiority, the contributions of non-European civilizations were often ignored, or, as Joseph argued, even distorted, in that they were often misattributed as European, i.e. Greek, contributions, and when their contributions were so great as to resist such treatment, they were typically devalued, considered inferior or irrelevant to Western mathematical traditions.
This tendency has not only led to the devaluation of non-Western mathematical traditions, but has distorted the history of Western mathematics as well. In so far as the contributions from non-Western civilizations are ignored, there is the problem of accounting for the development of mathematics purely within the Western cultural framework. This has led, as Sabetai Unguru has argued, toward a tendency to read more advanced mathematical concepts into the relatively simplistic geometrical formulations of Greek mathematicians such as Euclid, despite the fact that the Greeks lacked not only mathematic notation, but even the place-value system of enumeration, without which advanced mathematical calculation is impossible. Such ethnocentric revisionist history resulted in the attribution of more advanced algebraic concepts, which were actually introduced to Europe over a millennium later by the Arabs, to the Greeks. And while the contributions of the Greeks to mathematics was quite significant, the tendency of some math historians to jump from the Greeks to renaissance Europe results not only in an ethnocentric history, but an inadequate history as well, one which fails to take into account the full history of the development of modern mathematics, which is by no means a purely European development.
2. Vedic Altars and the "Pythagorean theorem"
A perfect example of this sort of misattribution involves the so-called Pythagorean theorem, the well-known theorem which was attributed to Pythagoras who lived around 500 BCE, but which was first proven in Greek sources in Euclid's Geometry, written centuries later. Despite the scarcity of evidence backing this attribution, it is not often questioned, perhaps due to the mantra-like frequency with which it is repeated. However, Seidenberg, in his 1978 article, shows that the thesis that Greece was the origin of geometric algebra was incorrect, "for geometric algebra existed in India before the classical period in Greece." (1978:323) It is now generally understood that the so-called "Pythagorean theorem" was understood in ancient India, and was in fact proved in Baudhayana's Shulva Sutra, a text dated to circa 600 BCE. (1978:323).
Knowledge of mathematics, and geometry in particular, was necessary for the precise construction of the complex Vedic altars, and mathematics was thus one of the topics covered in the brahmanas. This knowledge was further elaborated in the kalpa sutras, which gave more detailed instructions concerning Vedic ritual. Several of these treat the topic of altar construction. The oldest and most complete of these is the previously mentioned Shulva Sutra of Baudhaayana. As this text was composed about a century before Pythagoras, the theory that the Greeks were the source of Geometric algebra is untenable, while the hypothesis that India was have been a source for Greek geometry, transmitted via the Persians who traded both with the Greeks and the Indians, looks increasingly plausible. On the other hand, it is quite possible that both the Greeks and the Indians developed geometry. Seidenberg has argued, in fact, that both seem to have developed geometry out of the practical problems involving their construction of elaborate sacrificial altars. (See Seidenberg 1962 and 1983
3. Zero and the Place Value System
Far more important to the development of modern mathematics than either Greek or Indian geometry was the development of the place value system of enumeration, the base ten system of calculation which uses nine numerals and zero to represent numbers ranging from the most minuscule decimal to the most inconceivably large power of ten. This system of enumeration was not developed by the Greeks, whose largest unit of enumeration was the myriad (10,000) or in China, where 10,000 was also the largest unit of enumeration until recent times. Nor was it developed by the Arabs, despite the fact that this numeral system is commonly called the Arabic numerals in Europe, where this system was first introduced by the Arabs in the thirteenth century.
Rather, this system was invented in India, where it evidently was of quite ancient origin. The Yajurveda Samhitaa, one of the Vedic texts predating Euclid and the Greek mathematicians by at least a millennium, lists names for each of the units of ten up to 10 to the twelfth power (paraardha). (Subbarayappa 1970:49) Later Buddhist and Jain authors extended this list as high as the fifty-third power, far exceeding their Greek contemporaries, who lacking a system of enumeration were unable to develop abstract mathematical concepts.
The place value system of enumeration is in fact built into the Sanskrit language, where each power of ten is given a distinct name. Not only are the units ten, hundred and thousand (daza, zata, sahasra) named as in English, but also ten thousand, hundred thousand, ten million, hundred million (ayuta, lakSa, koti, vyarbuda), and so forth up to the fifty-third power, providing distinct names where English makes use of auxillary bases such as thousand, million, etc. Ifrah has commented that:
By giving each power of ten an individual name, the Sanskrit system gave no special importance to any number. Thus the Sanskrit system is obviously superior to that of the Arabs (for whom the thousand was the limit), or the Greeks and Chinese (whose limit was ten thousand) and even to our own system (where the names thousand, million etc. continue to act as auxillary bases). Instead of naming the numbers in groups of three, four or eight orders of units, the Indians, from a very early date, expressed them taking the powers of ten and the names of the first nine units individually. In other words, to express a given number, one only had to place the name indicating the order of units between the name of the order of units immediately immediately below it and the one immediately above it. That is exactly what is required in order to gain a precise idea of the place-value system, the rule being presented in a natural way and thus appearing self-explanatory. To put it plainly, the Sanskrit numeral system contained the very key to the discovery of the place-value system. (2000:429)
As Ifrah has shown at length, there is little doubt that our place-value numeral system developed in India (2000:399-409), and this system is embedded in the Sanskrit language, several aspects of which make it a very logical language, well suited to scientific and mathematical reasoning. Nor did this system exhaust Indian ingenuity; as van Nooten has shown, Pingala, who lived circa the first century BCE, developed a system of binary enumeration convertible to decimal numerals, described in his Chandahzaastra. His system is quite similar to that of Leibniz, who lived roughly fourteen hundred years later. (See Van Nooten)
India is also the locus of another closely related an equally important mathematical discovery, the numeral zero. The oldest known text to use zero is a Jain text entitled the Lokavibhaaga, which has been definitely dated to Monday 25 August 458 CE. (Ifrah 2000:417-1 9) This concept, combined by the place-value system of enumeration, became the basis for a classical era renaissance in Indian mathematics.
The Indian numeral system and its place value, decimal system of enumeration came to the attention of the Arabs in the seventh or eighth century, and served as the basis for the well known advancement in Arab mathematics, represented by figures such as al-Khwarizmi. It reached Europe in the twelfth century when Adelard of Bath translated al-Khwarizmi's works into Latin. (Subbarayappa 1970:49) But the Europeans were at first resistant to this system, being attached to the far less logical roman numeral system, but their eventual adoption of this system led to the scientific revolution that began to sweep Europe beginning in the thirteenth century.
4. Luminaries of Classical Indian Mathematics
Aryabhata
The world did not have to wait for the Europeans to awake from their long intellectual slumber to see the development of advanced mathematical techniques. India achieved its own scientific renaissance of sorts during its classical era, beginning roughly one thousand years before the European Renaissance. Probably the most celebrated Indian mathematicians belonging to this period was Aaryabhat.a, who was born in 476 CE.
In 499, when he was only 23 years old, Aaryabhat.a wrote his Aaryabhat.iiya, a text covering both astronomy and mathematics. With regard to the former, the text is notable for its for its awareness of the relativity of motion. (See Kak p. 16) This awareness led to the astonishing suggestion that it is the Earth that rotates the Sun. He argued for the diurnal rotation of the earth, as an alternate theory to the rotation of the fixed stars and sun around the earth (Pingree 1981:18). He made this suggestion approximately one thousand years before Copernicus, evidently independently, reached the same conclusion.
With regard to mathematics, one of Aaryabhat.a's greatest contributions was the calculation of sine tables, which no doubt was of great use for his astronomical calculations. In developing a way to calculate the sine of curves, rather than the cruder method of calculating chords devised by the Greeks, he thus went beyond geometry and contributed to the development of trigonometry, a development which did not occur in Europe until roughly one thousand years later, when the Europeans translated Indian influenced Arab mathematical texts.
Aaryabhat.a's mathematics was far ranging, as the topics he covered include geometry, algebra, trigonometry. He also developed methods of solving quadratic and indeterminate equations using fractions. He calculated pi to four decimal places, i.e., 3.1416. (Pingree 1981:57) In addition, Aaryabhat.a "invented a unique method of recording numbers which required perfect understanding of zero and the place-value system." (Ifrah 2000:419)
Given the astounding range of advanced mathematical concepts and techniques covered in this fifth century text, it should be of no surprise that it became extremely well known in India, judging by the large numbers of commentaries written upon it. It was studied by the Arabs in the eighth century following their conquest of Sind, and translated into Arabic, whence it influenced the development of both Arabic and European mathematical traditions.
Brahmagupt
Born in 598 CE in Rajastan in Western India, Brahmagupta founded an influential school of mathematics which rivaled Aaryabhat.a's. His best known work is the Brahmasphuta Siddhanta, written in 628 CE, in which he developed a solution for a certain type of second order indeterminate equation. This text was translated into Arabic in the eighth century, and became very influential in Arab mathematics. (See Kak p. 16)
Mahavira
Mahaaviira was a Jain mathematician who lived in the ninth century, who wrote on a wide range of mathematical topics. These include the mathematics of zero, squares, cubes, square-roots, cube-roots, and the series extending beyond these. He also wrote on plane and solid geometry, as well as problems relating to the casting of shadows. (Pingree 1981:60)
Bhaaskara
Bhaaskara was one of the many outstanding mathematicians hailing from South India. Born in 1114 CE in Karnataka, he composed a four-part text entitled the Siddhanta Ziromani. Included in this compilation is the Biijagan.ita, which became the standard algebra textbook in Sanskrit. It contains descriptions of advanced mathematical techniques involving both positive and negative integers as well as zero, irrational numbers. It treats at length the "pulverizer" (kut.t.akaara) method of solving indeterminate equations with continued fractions, as well as the so-called "Pell's equation (vargaprakr.ti) dealing with indeterminate equations of the second degree. He also wrote on the solution to numerous kinds of linear and quadratic equations, including those involving multiple unknowns, and equations involving the product of different unknowns. (Pingree 1981, p. 64)
In short, he wrote a highly sophisticated mathematical text that proceeded by several centuries the development of such techniques in Europe, although it would be better to term this a rediscovery, since much of the Renaissance advances of mathematics in Europe was based upon the discovery of Arab mathematical texts, which were in turn highly influenced by these Indian traditions.
Maadhava
The Kerala region of South India was home to a very important school of mathematics. The best known member of this school Maadhava (c. 1444-1545), who lived in Sangamagraama in Kerala. Primarily an astronomer, he made history in mathematics with his writings on trigonometry. He calculated the sine, cosine and arctangent of the circle, developing the world's first consistent system of trigonometry. (See Hayashi 1997:784-786) He also correctly calculated the value of p to eleven decimal places. (Pingree 1981:490)
This is by no means a complete list of influential Indian mathematicians or Indian contributions to mathematics, but rather a survey of the highlights of what is, judged by any fair, unbiased standard, an illustrious tradition, important both for its own internal elegance as well as its influence on the history of European mathematical traditions. The classical Indian mathematical renaissance was an important precursor to the European renaissance, and to ignore this fact is to fail to grasp the history of latter, a history which was truly multicultural, deriving its inspiration from a variety of cultural roots.
There are in fact, as Frits Staal has suggested in his important (1995) article, "The Sanskrit of Science", profound similarities between the social contexts of classical India and renaissance Europe. In both cases, important revolutions in scientific thought occurred in complex, hierarchical societies in which certain elite groups were granted freedom from manual labor, and thus the opportunity to dedicate themselves to intellectual pursuits. In the case of classical India, these groups included certain brahmins as well as the Buddhist and Jain monks, while in renaissance Europe they included both the monks as well as their secular derivatives, the university scholars.
Why, one might ask, did Europe take over thousand years to attain the level of abstract mathematics achieved by Indians such as Aaryabhat.a? The answer appears to be that Europeans were trapped in the relatively simplistic and concrete geometrical mathematics developed by the Greeks. It was not until they had, via the Arabs, received, assimilated and accepted the place-value system of enumeration developed in India that they were able to free their minds from the concrete and develop more abstract systems of thought. This development thus triggered the scientific and information technology revolutions which swept Europe and, later, the world. The role played by India in the development is no mere footnote, easily and inconsequentially swept under the rug of Eurocentric bias. To do so is to distort history, and to deny India one of it's greatest contributions to world civilization.
Works Cited
Hayashi, Takao. 1997. "Number Theory in India". In Helaine Selin, ed. Encyclopaedia of the History of Science, Technology, and Medicine in Non-Western Cultures. Boston: Kluwer Academic Publishers, pp. 784-786.
Ifrah, Georges. 2000. The Universal History of Numbers: From Prehistory to the Invention of the Computer. David Bellos, E. F. Harding, Sophie Wood and Ian Monk, trans. New York: John Wiley & Sons, Inc.
Joseph, George Ghevarughese. 1987. "Foundations of Eurocentrism in Mathematics". In Race & Class 28.3, pp. 13-28.
Kak, Subhash. "An Overview of Ancient Indian Science". In T. R. N. Rao and Subhash Kak, eds. Computing Science in Ancient India, pp. 6-21.
van Nooten, B. "Binary Numbers in Indian Antiquity". In T. R. N. Rao and Subhash Kak, eds. Computing Science in Ancient India, pp. 21-39.
Pingree, David. Jyotih.zaastra: Astral and Mathematical Literature, Wiesbaden: Otto Harrassowitz, 1981, p. 4.
Seidenberg, A. 1962. "The Ritual Origin of Geometry". In Archive for History of Exact Sciences 1, pp. 488-527.
______. 1978. "The Origin of Mathematics". In Archive for History of Exact Sciences 18.4, pp. 301-42.
______. 1983. "The Geometry of Vedic Rituals". In Frits Staal, ed. Agni: The Vedic Ritual of the Fire Altar. Delhi: Motilal Banarsidass, 1986, vol. 2, pp. 95-126.
Unguru, Sabetai. 1975. "On the Need to Rewrite the History of Greek Mathematics". In Archive for History of Exact Sciences 15.1, pp. 67-114.
Staal, Frits. 1995. "The Sanskrit of Science". In Journal of Indian Philosophy 23, pp. 73-127.
Subbarayappa, B. V. 1970. "India's Contributions to the History of Science". In Lokesh Chandra, et al., eds. India's Contribution to World Thought and Culture. Madras: Vivekananda Rock Memorial Committee, pp. 47-66.

Maths in Vedas

2011-05-11T07:18:00.000-07:00

The oldest known document in the world is the Rgveda, which is otherwise known simply as the Vedas (plural because it is split into four parts). It was orally passed down until around 5 BC, when writing either started or became more commonplace in India. It came from the Vedic people, who lived between the Ganga and Sindhu rivers. Unlike other civilizations, the Vedas are the only remaining evidence left from this society -- so far no archeological discoveries have been made that shed more light on them. This means that there are no brick altars or similar remains that testify to their level of science and technology.
The Vedas are not mathematical texts; they are merely hymns to the Vedic gods. However, the word Veda means "knowledge", and when analyzed closely it actually contains many mathematical references, especially in the section on Jyotisa, or "the constellations". Unfortunately, almost all of these references are implied, so much of the interpretation is largely guesswork. Another reason that the Vedas are hard to interpret is that because it was an oral document, there are no symbols for numbers or operations -- only words. It is highly likely, however, that they did use symbols, because without them math becomes very tedious. For example, consider doing a multiplication problem using "four thousand six hundred and thirty-seven times two hundred and eighty-eight." You would most likely convert the words to symbols, do the math on a piece of paper, and then probably only take the time to convert the answer back into words.
And what makes it even harder is that the Vedas were written in verse¹, probably for ease in memorization (because it was passed down orally). This means that not only were ancient mathematicians poets as well, but more importantly the math they were working with had to be written to fit in verse. Most people consider it hard to do long division using symbols and well worked out methods. Imagine trying to complete this process with words and in poetry! And then, consider the difficulty of decoding your work 4000 years later!
First of all, a scholar has to determine the name of every number, i.e. "three" or "forty-seven". But besides this, there are words meaning "some", "many", "both", "few", etc., and ordinal numbers ("first", "second", "third", etc.) These also have to be deciphered to even begin to get into the math. And yet another oddity of the Sanskrit language involves what happens with compound numbers, numbers with more than one "digit" (like "thirty-four"). In normal Sanskrit, compound words (like "servant of the king") came from left to right in order of prevalence (so our example would be "king-servant"; "servant-king" would mean a servant who was treated well). However, compound numbers are written the opposite way, with the higher digits on the right. (Our number 529 would be written "nine-two-five".) It is always like this, and there is even a rule included: ankanam vamato gatih, which literally means "the understanding of the numbers in the reverse way."²
And then, the whole thing is in the form of stories and myths, which have to be closely analyzed for mathematical content. A good example of a story from which we can extract mathematics is one about a man named Manu.³ Manu had ten wives, who had one, two, three, four, etc. sons each (the first wife had one son, the second wife two, etc.). The one son allied with the nine sons, and the two sons allied with the eight, and so on until the five sons were left by themselves. They asked Manu for help, and so he gave them each a samidh or "oblation-stick". The five sons then used these sticks to defeat all of the other sons.
On the surface, this is just a silly fable, but it shows several things about Vedic mathematics. Because the ten sons did not ally with anyone, and the nine did with the one, eight with two, and so on, the mathematicians must have been thinking that nine plus one, and eight plus two equal ten. This obviously shows that they practiced addition, and it also implies that they used a base 10, or decimal, system. For the second part of the story, the authors probably added the tens up to find that there were 50 allied sons. When the five remaining sons asked their father for help, it is likely that he gave them just enough mathematical power to defeat the others. This would mean that each stick equaled the strength of 10 men, for a total of 50. With the five sons added to that, they were able to defeat the 50. (Or maybe the father gave them 50 men worth of sticks, thinking it would be an equal battle, but not realizing that the five sons would throw off the balance.) But this 50 business implies both multiplication and division as well, because there were five groups of ten sons allied, or five times ten. Then, when the father went to decide the power of the sticks, he would have had to divide that 50 by five, to split the power equally among the five sons.⁴
Going even further, the story can be shown to symbolize the idea of positional notation -- the idea of place values in numerals. (For an example of positional notation, 218 is the same as 200 + 10 + 8, or 2 x 10² plus 1 x 10¹ plus 8 x 10⁰. In summary, the order of numerals tells how big the numbers are.) The "oblation-sticks" are obviously thought of as very powerful, just as 10 might be thought of as more "powerful" than a lowly 1. So when the 5 "lowly" sons were "added" to the 5 "powerful" sticks, this could have symbolized the 50 and 5 making 55, which is a bigger number (and therefore more powerful) than 50. This view of things also gives further evidence to the fact that they used base 10.
So this simple story shows examples of addition, multiplication, division, base 10, and even positional notation. The Vedas are full of these stories, and many more examples are given throughout of all these concepts, along with subtraction, fractions, and squares.⁵ There are even instances of arithmetic and geometric sequences, which are series of numbers that increase by adding or multiplying a certain number (arithmetic sequence: 2 (+3 =) 5, 8, 11, 14, ... ; geometric sequence: 2 ( x 3 =) 6, 18, 54, ...).
The Vedas give us many examples of arithmetic in ancient India, but to find out about the geometry of the time we turn to a collection of Hindu religious documents called the sulva-sutras.

Indian Vedic Science & Medical Solution Through VEDAS

KAMBA RAMAYANA

[संपादित करें] परिचय

वेद

अनुक्रम

[संपादित करें] वेदों का महत्व

[संपादित करें] वैदिक वाङ्मय का शास्त्रीय स्वरुप

[संपादित करें] कर्मकाण्ड में वर्गीकरण

[संपादित करें] वैदिक स्वर प्रक्रिया

[संपादित करें] चार वेद

[संपादित करें] चार भाग

[संपादित करें] वेदों का विभाजन

[संपादित करें] याज्ञिक दृष्टिः

[संपादित करें] प्रायोगिक दृष्टिः

[संपादित करें] साहित्यिक दृष्टि

[संपादित करें] वेद के अंग, उपांग एवं उपवेद

वेदः SANSKRIT

अन्तर्विषयाः

[सम्पादयतु] ऋग्वेदः

[सम्पादयतु] यजुर्वेदः

[सम्पादयतु] सामवेदः

[सम्पादयतु] अथर्ववेदः

Vedas

Contents

Etymology and Usage

Chronology

Categories of Vedic texts

Vedic Sanskrit corpus

Shruti literature

Vedic schools or recensions

The Four Vedas

Rigveda

Yajurveda

Samaveda

Atharvaveda

Brahmanas

Vedanta

In post-Vedic literature

Vedanga

Parisista

Puranas

Upaveda

Buddhist and Jain views

Buddhism

Jainism

"Fifth" and other Vedas

Western Indology

Sanskrit & Artificial Intelligence — NASA Knowledge Representation in Sanskrit and Artificial Intelligence

Abstract

Equivalence

Physics to Metaphysics - A Vedic Paradigm

Of course any attempt to find harmony between the scientific world view and the mystic's vision will be incomplete unless we adjust the scientific world view through an interface with the many realities it fails to account for.

Just as the eye cannot see the mind but can be in connection with it if the mind chooses to think about it, so similarly the finite can know about the infinite only by the grace of the infinite.

The individual self is a minute particle of will or consciousness -- a sentient being -- endowed with a serving tendency. This self is transcendental to matter and qualitatively one with Godhead, while quantitatively different.

CREATION THROUGH SOUND

Vedic Mathemacs and the Spiritual Dimension

Vedic Mathemacsand theSpiritual Dimension

bySwami B. B. Visnu

Vedic Mathematics Index

Vedic Mathematics and the Spiritual Dimension

Archimedes and Pythagoras

Shulba Sutra

Evolution of Arabic (Roman) Numerals from India

Equations and Symbols

Poetry in Math

It must be pointed out that these sutras given by Tirtha Maharaja are created by the author himself, as stated in the introduction to his book, "Vedic Mathematics" (published posthumously) and are therefore not actually Vedic.

These mathematical sutras are Vedic only in the sense that they are inspired by the Vedas in the mind of one dedicated to the Vedas. Thus the title "Vedic Mathematics" is not correct.

The Sanskrit consonants

Mathematical Sutras

The Universe Through Vedas

Age of the Universe

Vedic Mathematics

Maths in Vedas

Vedic Mathemacs
and the
Spiritual Dimension

by
Swami B. B. Visnu