जैसा कि मैने पहले भी बताया है कि युग व्यवस्था बहुत ही भ्रामक है और अलग अलग विद्वानों ने इसकी व्याख्या अपने तरीके से की है| (पढ़ें- युग-युग की बातें (भारतीय युग सिद्धान्त))इस लेख का उद्देश्य भारतीय युग व्यवस्था पर एक सकारात्मक चर्चा कर उसकी प्रामाणिकता की खोज करनी है| मैं यहाँ पर अपनी कोई व्याख्या न देकर केवल उन विद्वानों की व्याख्या दे रहा हूँ| यह चर्चा पहले भी की जा चुकी है|(पढ़ें- भारतीय युग सिद्धान्त – Indian Yuga Theory: Some Ideas ) आज इसी कड़ी में प्रस्तुत है श्री युक्तेश्वर गिरि के विचार|

श्री युक्तेश्वर गिरि – 1 युग चक्र = 12000 वर्ष

परमहंस योगानन्द के गुरु श्री युक्तेश्वर गिरि के अनुसार हमारा पारम्परिक दृष्टिकोण ज्योतिषियो और खगोलशास्त्रियो के द्वारा किये गये गलत गणनाओ पर आधारित है। अपनी पुस्तक The Holy Science मे वह प्रत्येक युग चक्र को संधिकाल के साथ 12000 वर्षो का बताते हैं। प्रत्येक संधिकाल मुख्य काल का 10% होता है तथा आरम्भ एवं अन्त दोनो मे होता है। उदाहरण के लिये, कलियुग 1000 वर्षो का तथा इसकी 100 वर्षो की दो संधियां, इस प्रकार कुल 1200 वर्षो का होता है। इसी प्रकार द्वापरयुग 2000 वर्षो का तथा इसकी 200 वर्षो की दो संधियां, इस प्रकार कुल 2400 वर्षो का होता है। त्रेतायुग 3600 वर्षो (संधियों सहित) का तथा सत्ययुग 4800 वर्षो (संधियों सहित) का होता है। इस प्रकार कुल 12000 वर्षो का युग चक्र हुआ। श्री युक्तेश्वर गिरि के अनुसार यह युग चक्र भी जोड़ियो मे आते है। पहले अवरोही युगो मे सत्ययुग से त्रेतायुग, द्वापरयुग तथा कलियुग, फिर आरोही युगो मे कलियुग से द्वापरयुग, त्रेतायुग, तथा सत्ययुग का आगमन होगा। इन आरोही युगो और अवरोही युगो की संधि 499AD मे हुयी थी। कलियुग का आरोहण सितम्बर 499AD मे आरम्भ हुआ। इस गणना के अनुसार हम सितम्बर 1699AD से द्वापरयुग के आरोहण मे है।

श्री युक्तेश्वर के अनुसार कोई भी विद्वान अवरोही कलियुग के आरम्भ की बुरी घोषणा नही करना चाहता था, अतः वह द्वापर को ही बढ़ाते गये। जैसे ही कलियुग का आरोहण शुरु हुआ, तब उन्हे लगा की काल गणना मे उनसे गलती हो गयी है। समाधान के लिये उन्होने एक नयी व्यवस्था की। कलियुग को 1200 वास्तविक वर्षो के बजाय 1200 दिव्य वर्षो का बताया। प्रत्येक दिव्य वर्ष मे 12 दिव्य महीने और प्रत्येक दिव्य महीना 30 दिव्य दिवसो का होता है। एक दिव्य दिवस पृथ्वी के एक वास्तविक (सौर) वर्ष के बराबर होता है। तो इस प्रकार कलियुग 1200x12x30=432000 वर्षो का हुआ।
श्री युक्तेश्वर बताते है कि जिस प्रकार दिन-रात्रि का चक्र एक celestial गति (पृ्थ्वी का अपने अक्ष पर घूमना) के कारण होता है और मौसमो का चक्र भी एक celestial गति ( पृ्थ्वी का सूर्य के चारो ओर घूमना ) के कारण् होता है, उसी प्रकार युग चक्र भी एक celestial गति के कारण होता है। यह celestial गति हमारे सौर-मण्डल के किसी अन्य बिन्दु के चारो ओर घूमने के कारण होती है। इस मूल-बिन्दु को उन्होने “विष्णुनाभि” कहा है।

परन्तु 12000 वर्षो के चक्र के लिये यह बिन्दु अन्तरिक्ष मे कहां है?

डेविड फ्राले, जो कि एक खगोलशास्त्री और वैदिक परम्परा के बहुत से पुस्तको के लेखक है, भी समान व्याख्या प्रस्तुत करते है। इनकी व्याख्या मनु-संहिता पर आधारित है, जिसमे  बहुत छोटे, 2400 वर्षो का युग चक्र बताया गया है। मनु का युग चक्र लगभग उसी लम्बाई का है जैसा कि खगोलशास्त्री Precession of the Equinoxes के लिये बताते है। Precession of the Equinoxes का समय 2580 वर्ष माना जाता है। उनका कहना है कि छोटे युग चक्र के द्वारा हम रामायण और महाभारत तथा अन्य ऎतिहासिक तथ्यो के समय को दूसरे तरीको से बेहतर तरीके से निकाल सकते है। अन्य तरीको से इन घटनाओ का समय लाखो वर्ष पहले निकलता है जो कि धरती पर मानव इतिहास के परिप्रेक्ष्य मे अस्वीकार करने योग्य है।

Proposal of Sri Yukteshwar—

Ascending Dwapara Yuga proper	1900 – 3900 AD
Satya subyuga	3100 – 3900 AD
Treta subyuga	2500 – 3100 AD
Dwapara subyuga	2100 – 2500 AD
Kali subyuga	1900 – 2100 AD

Dwapara Yuga sandhi	1700 – 1900 AD
Kali Yuga sandhi	1600 – 1700 AD

Ascending Kali Yuga proper	600 – 1600 AD
Satya subyuga	1200 – 1600 AD
Treta subyuga	900 – 1200 AD
Dwapara subyuga	700 -   900 AD
Kali subyuga	600 -  700 AD

Ascending Kali Yuga sandhi	500 -  600 AD
Descending Kali Yuga sandhi	400 -  500 AD

Descending Kali Yuga proper	600 BCE -  400 AD
Kali subyuga	300 –   400 AD
Dwapara subyuga	100 -   300 AD
Treta subyuga	200 BCE – 100 AD
Satya subyuga	600  –   200 BCE

Kali Yuga sandhi	700 -   600 BCE
Dwapara Yuga sandhi	900 –   700 BCE

Descending Dwapara Yuga proper	2900 –   900 BCE
Kali subyuga	1100 –   900 BCE
Dwapara subyuga	1500 – 1100 BCE
Treta subyuga	2100 – 1500 AD
Satya subyuga	2900 – 2100 BCE

Dwapara Yuga sandhi	3100 – 2900 BCE
Treta Yuga sandhi	3400 – 3100 BCE

Descending Treta Yuga proper	6400 – 3400 BCE
Kali subyuga	3700 – 3400 BCE
Dwapara subyuga	4300 – 3700 BCE
Treta subyuga	5200 – 4300 AD
Satya subyuga	6400 – 5200 BCE

Treta Yuga sandhi	6700 – 6400 BCE
Satya Yuga sandhi	7100 – 6700 BCE

Descending Satya Yuga proper	11,100 – 7100 BCE
Kali subyuga	7500 – 7100 BCE
Dwapara subyuga	8300 – 7500 BCE
Treta subyuga	9500 – 8300 BCE
Satya subyuga	11,100 – 9500 BCE

[ref: Sri Yukteswar, Swami (1949). The Holy Science. Yogoda Satsanga Society of India.]

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In the Hindu texts, a year was taken to comprise 12 months (360 days). Multiply these two numbers to get 360×12=4320. Now, suffix this number with the requisite number of zeroes to produce long structured time-spans. One Mahayuga equals 4,320,000 years. A Kalpa, is defined as equal to 1000 Mahayugas. A Mahayuga, in turn, is composed of four ages or yugas. The other yugas are Satya or Krita-yuga, Treta-yuga, Dvapara-yuga and Kaliyuga. The smallest yuga is Kaliyuga, which is 1/10th of Mahayuga.

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But are they real counts? Are yugas really of such a long time period? Let see some ideas.

Traditional concept is miscalculated: Sri Yukteswar Giri

According to Sri Yukteswar Giri, guru of Paramahansa Yogananda, the traditional view is based on miscalculations made by astronomers and astrologers. In his book, The Holy Science, Sri Yukteswar proposes a four Yuga cycle of 12,000 years with transitional periods called ‘Sandhis’ coming in between. Each Sandhi is 10% the length of the main period and occurs at both its beginning and end. Kali Yuga of 1000 years has two sandhis of 100 years for a total of 1200 years. Dwapara Yuga of 2000 years has two sandhis of 200 years for a total of 2400 years. Treta Yuga of 3000 years has two sandhis of 300 years for a total of 3600 years. Satya Yuga of 4000 years has two sandhis of 400 years for a total of 4800 years. This makes for a total of 12,000 years for the cycle of the four yugas. Yukteswar has the yuga cycles coming in pairs. One set of ‘descending Yugas’ from Satya to Treta, Dwapara and Kali is followed by another set of ‘ascending Yugas’ from Kali to Dwapara, Treta and Satya. The midpoint of the descending and ascending Yuga cycles occurred around 499 AD. The ascending phase of Kali Yuga began in September of 499 C.E. So it follows that we have been in the ascending phase of Dwapara Yuga since September of 1699.

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He says that at the end of the last descending Dvapara Yuga (about 700 BC), “Maharaja Yudhisthira, noticing the appearance of the dark Kali Yuga, made over his throne to his grandson [and]…together with all of his wise men…retired to the Himalaya Mountains… Thus there was none in the court…who could understand the principle of correctly accounting the ages of the several Yugas.”

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According to Sri Yukteswar, nobody wanted to announce the bad news of the beginning of the descending Kali Yuga, so they kept adding years to the Dvapara date (at that time 2400 Dvapara) only retitling the epoch to Kali. As the Kali began to ascend again, scholars of the time recognized that there was a mistake in the date (then being called 3600+ Kali, even their texts said Kali had only 1200 years). “By way of reconciliation, they fancied that 1200 years, the real age of Kali, were not the ordinary years of our earth, but were so many daiva (or deva) years (“years of the gods”), consisting of 12 daiva months of 30 daiva days each, with each daiva day being equal to one ordinary solar year of our earth. Hence according to these men 1200 years of Kali Yuga must be equal to 432,000 years of our earth.”

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Sri Yukteswar explained that just as the cycle of day and night is caused by a celestial motion (the earth spinning on its axis in relation to the sun), and just as the cycle of the seasons are caused by a celestial motion (the earth with tilted axis orbiting the sun) so too is the yuga cycle (seen as the precession of the equinox), caused by a celestial motion. He explained this celestial motion as the movement of the whole solar system around another star. As our sun moves through this orbit, it takes the solar system (and earth) closer to and then further from a point in space known as the “grand centre” also called ‘Vishnunabhi‘, which is the seat of the creative power, ‘Brahma’, [which]…regulates…the mental virtue of the internal world.” He implied that it is the proximity of the earth and sun to this grand centre that determines which season of man or yuga it is.

A deeper look on Shri Yukteshwar’s theory by David Frawley

David Frawley, an astrologer and author of many books on Vedic traditions, provides a similar revision of the traditional timescale. His reinterpretation is based on the writings of Manu, who, in his Manhu Samhita, posits a much shorter Yuga cycle of 2,400 years. Manu’s Yuga cycle happens to correspond roughly to the same length of time that astronomers attribute to the Precession of the Equinoxes. As with Sri Yukteswar, Frawley’s interpretation of scripture suggests that we are currently near the beginning of a Dwapara Yuga cycle that will last a total of 2,400 years. He further points out that that the traditional 432,000 year cycle is questionable, based on Vedic and Puranic historical records. He explains that the shorter yuga theory offers better proof of the age of Rama and Krishna and other important historical Indian figures than other dating methods, which conceive of some of these figures to be millions of years old; far too old to place them within the accepted chronology of human history on Earth.

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In addition to Sri Yukteshwar’s scheme of Yugas and Sandhis, he propose dividing up each Yuga itself into four sections or subyugas of Satya, Treta, Dwapara and Kali in the same relative proportions. He divides up the 1000 year yuga of Kali into four yugas, 400 years of Satya Kali, 300 years of Treta Kali, 200 years of Dwapara Kali and 100 years of Kali Kali. He would divide up Dwapara Yuga of 2000 years into 800 years of Satya Dwapara, 600 years of Treta Dwapara, 400 years of Dwapara Dwapara and 200 years of Kali Dwapara. He is sure that in this manner, we can bring more specificity into the kind of historical changes that we can expect during the long period of each Yuga.

An astrological view by N P Ramadurai

N P Ramadurai concentrates on astronomical views. The fundamental points that N P Ramadurai uses as his foundation for his findings are four in number:

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1. The planets, including Uranus, Neptune and Pluto, complete one full rotation (in respect of their mutual positions) and all come back to their ‘starting’ point every 12160 years. Precession of the equinoxes is taken as 53.2894736842 seconds of arc per year

The modern astronomers take precession as about 50”. The difference of 3” has a significant effect on the massive calculations that Ramadurai makes, but this is purely a technical matter of practical astronomy. Similarly the number 12160, which Ramadurai uses significantly, is a matter for astronomers to decide.

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2. The unit for duration of each Yuga mentioned in scriptural texts is not to be taken as the divine year (equal to 360 earthly years) but should be taken as ordinary earthly years. Ramadurai declares that every time the duration of a Yuga is mentioned in terms of divine years, the statement should be interpreted as a ‘decorative’ statement and not as an actual statement of facts.

At first glance this seems logical. But there are some statements in puranas, which clearly states like “One hundred human years” When puranas want the text to show ‘earthly years’ they mention it invariably. When they want to refer to ‘human’ years, they say so and they do not leave it to our imagination whether the reference is to ‘human’ or ‘divine’.

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3. Regarding the Saptarishi era Ramadurai assumes that the stay of the saptarishis in each nakshatras is only 81 years and not 100 as traditionally accepted. Again he takes the number 100 as a rounding-up approximation and not a precise number.

According to traditional opinion, the Saptarishis stay in each nakshatra for 100 years and when they stayed in Magha for 100 years that cycle is called the Magha cycle. Thus, each of the Saptarishi cycles was named after the nakshatra in which they stayed. After the Magha cycle there was Purva Phalguni cycle and so on. Ramadurai’s ’81 years’ for the stay of saptarishis in one nakshatra  comes from 1000 lunar months converted into solar years.

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4. The significance of the Magha cycle

The Magha cycle of 160 sidereal years which Ramadurai uses, is an original idea. Ramadurai proceeds as if there is another cycle called Magha cycle and he estimates it as 160 sidereal years. While it might be true that the concept of these 160 sidereal years  may be useful to ‘relate full moon with the respective stars during Tamil months’, to associate it with the Magha cycle of the saptharishi movement, has yet to be accepted by scholars.  This is a point which must be left to scholars for further investigation.

A mathematical approach by R.R. Karnik

R.R. Karnik concentrates on the mathematics presented in Hindu Text. According to him “In the hightech uses of computer technology there is an area called  simulation, in which time is used in two different manners. One is called “real time” and the other, “non-real time” — either slow motion or fast motion. The surprising thing is that the Surya Siddhanta makes use of this concept of time. All the arithmetics in the Surya Siddhanta is in terms of integers for which some times “real time” is not suitable.”

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Some people believe that these huge number of years correspond to an L.C.M. of the periods of revolution of the planets around the sun. There have been attempts in Vedanga Jyotisha, where a period of five years has been used and by Varahamihira in Romaka Siddhanta using 2,850 years.

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Cicero gives 12,954 years in the thirty-fifth fragment of Hortensius by the name of Magnun Annum. Neither of these numbers can divide 4,32,000. In fact no number however large can give an integer multiple of the number of years required.

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The answer is simple and elementary. The least count used in Surya  Siddhanta for general computations is one Kala or minute of the arc of which there are 21,600 in the whole  circle. If one has to specify the movement of planetary element to the accuracy of one Kala in a year a period  of 21,600 years will have to be taken. This is the period of a Mahayuga and its tenth part which is 2,160 years is the period of Kaliyuga. Dwaparyuga is twice this or 4,320 years, Tretayuga is thrice this or 6,480  years and  Kritayuga (Satyayuga) is four times this or 8,640 years, all four totaling up to 21,600 years.

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The accuracy of one Kala in a year is not adequate. To therefore obtain an accuracy of one two-hundredth of a Kala in a year, a period that is two hundred times 21,600 or 4,320,000 years is necessary. The first number of 21,600 years is the real time and the other number of 4,320,000 years taken for Mahayuga is the frame time or the non-real time.

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According to the above, the real time calculation will be like this:

1 kalpa = 1000 X Mahayuga = 21,600,000 years = 1day of Brahma
1 Mahayuga = 10 X kaliyuga = 21,600 years = One cycle of 4 yugas
1 Satyayuga = 4 X Kaliyuga = 8640 years
1 Tretayuga = 3 X Kaliyuga = 6480 years
1 Dwaparyuga = 2 X Kaliyuga =4320 years
1 Kaliyuga = 2160 years

According to Valmiki, Ramayana starts in the beginning of Tretayuga, so approximately 12960 years before.

A broader study of Ramayana and Mahabharata by Dr.P.V.Vartak

Dr P. V. Vartak used the Hindu texts Ramayan, and Mahabharat, and dated the events astronomically. As we know Ramayan happened in the beginning phase of Tretayuga, and Mahabharat happened in the last phase of Dwaparyuga, so, by knowing the time of these events, the mystery can be solved enough.

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It should be noted that the ancient Indians had a prefect method of time measurement. They recorded the ‘tithis’, days according to the nakshatra on which the moon prevailed, the months, the seasons and even the different Solstices. By therefore noting a particular arrangement of the astronomical bodies, which occur once in many thousand years, the dates of the events can be calculated. Dr. P.V. Vartak has thus attempted to calculate the dates of important incidents that occured during the Ramayanic Era. The correct astronomical records goes to show that Valmiki’s has chronicled an account of a true story and also, that the an advanced time measurement system was known to the Hindus (Indians) atleast 9000 years ago. Please refer to Dr. Vartak’s celebrated book “Vastav Ramayan” for further reading.

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The Mahabharat Era has already been dated by Dr. Vartak to 5561 B.C. [Refer to Dr. Vartak's book "Swayambhu"].

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Genealogical links available from the Mahabharat and Puranas, Yuga calculations and some archaelogical findings also provide clues to the dating of the Ramayanic era. Also, literary references to the characters from the Ramayanic Era provide limits after which the Ramayan could not have occured. For example, Guru Valmiki (the author of Ramayana) is refered to in the Taittiriya Brahmana (dated to 4600 B.C) and therefore Ramayana must have before the Brahmana was composed. However, archaeological and literary methods can only provide approximate datelines and for determining the precise time of the Ramayanic events, astronomical calculations may alone be useful.

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Mahabharat states that Sage Vishwamitra started counting nakshatras from Shravana (Aadiparva A.71 and Ashwamedha A.44) and a new reference to time measurement thus initiated. According to the old tradition, the first place was assigned to the nakshatra prevelant on the Vernal Equinox. Vishwamitra modified this and started measuring from the nakshatra at the Autumnal Equinox. Sharvan was at this juncture at about 7500 B.C, which is therefore the probable period when Vishwamitra existed and also that of the Ramayanic Era.

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Formerly, the year initiated with the Varsha-Rutu (season) and therefore was termed “Varsha”. Ramayan shows that the flag was being hoisted to celebrate the new year on Ashwin Paurnima (Kishkindha 16/37, Ayodhya 74/36). Ayodhya 77 mentions that the flags were defaced and damaged due to heat and showers. These descriptions point to the fact that their new year started on the Summer Solstice when heat and rain simultaneously exist. The Summer Solstice fell on Ashwin Full Moon, so the Sun was diagonally opposite at Swati nakshatra. This astral configuration can be calculated to have occured around 7400 B.C.

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Kishkindha 26-13 describes the commencement of the rainy season. In shloka 14, refers to Shravan as “Varshika Poorva Masa”. Kishkindha 28/2 clearly shows that the rainy season began in Bhadrapada Masa. Further description “Heated by the Sun and showered by new waters, the earth is expelling vapours” (Kish.26/7) points to Bhadrapada as premonsoon. Kish.28/17 tells that there was alternate sun-shine and shadowing by the clouds. Kish.28/14 describes the on-coming rainy season. Thus Bhadrapada was the month of pre-monsoon, that is before 21st June or Summer Solstice. Naturally, months of Ashwin and Kartika formed the rainy season. It is therefore concluded that Ashwin Full Moon coincided with Summer Solstice, that year being 7400 B.C.

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Rama started forest-exile in Chaitra and ended it in Chaitra. He was coronated in the same month and one month later, proceeded to Ashokavan with Seeta (Uttar 41/18) when the Shishira Rutu terminated. So it seems that Vaishakha Masa coincided with Shishira. So the Winter Solstice was at Vaishakha with the Sun at Ashwini. At present, the Winter Solstice takes place at Moola. Thus a shift of 10 nakshatras has occured since the Ramayanic Era. Precession has a rate of 960 years per nakshatra. Therefore, Ramayan must have occured 9600 years ago, which is 7600 B.C approximately.

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Dr. P. V. Vartak goes beyond the approximate dating, and give exact dating with detailed analysis.

Conclusion

It cannot be concluded by me. Each analysis given above has some evidences in the support of it and some opposing it. One of them may be correct, or may be all the wrong. But in my opinion, the best way to decipher the confusion, is by astronomy. We have to read the texts very carefully, and then date the events astronomically. In this manner Dr. P.V. Vartak may be write. Even the dates are defer as dated by the others, for example Acharya Chatursan, in his book, “Vayam Rakshamah” date the Ramayan approximately 7000 years before today, Aryabhat date the Mahabharat war on 3102 BC. So, as the mystery is not solved yet now, the discussion can not be concluded by me.

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Ref:

Sri Yukteswar, Swami (1949). The Holy Science. Yogoda Satsanga Society of India.

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http://www.geocities.com/kvforp/VK1/Review_NPRamadurais_work.html

http://www.hindunet.org/hindu_history/ancient/ramayan/rama_vartak.html

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“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.”

– Laplace, French mathematician.

लाप्लास का यह कथन कितना तर्कसंगत लगता है जब हम वर्तमान वैज्ञानिक प्रगति को देखते हैं| गणित और विज्ञान के वर्तमान स्वरूप की कल्पना शून्य के बिना असंभव है| सत्य ही कहा है-

“जब ज़ीरो दिया मेरे भारत ने, दुनिया को तब गिनती आयी|”

भारत मे शून्य का प्रथम वास्तविक प्रमाण 876 AD के एक मन्दिर के शिलालेख मे मिलता है| यह शिलालेख ग्वालियर के चतुर्भुज मन्दिर मे मिला है| यहाँ पर 187 गुणे 270 हस्त का एक बागीचा लगाया गया था, जिससे 50 पुष्पमालाये प्रतिदिन मन्दिर को भेजी जाती थी| शिलालेख पर 270 और 50 (२७० और ५०) लगभग उसी प्रकार से लिखे गये है जिस प्रकार आज लिखे जाते है|

परन्तु ऐसा नही है की 876 AD मे शून्य का प्रथम प्रयोग किया गया हो| शून्य के संदर्भ मे सबसे प्राचीन एक जैन पुस्तक “लोकविभाग” (458 BC) मानी जाती है| इसमे संख्याओं के लिये संस्कृत के शब्दो का प्रयोग किया गया था| कुछ अन्य प्राचीन भारतीय पाण्डुलिपियों मे शून्य के लिये बिन्दु के प्रयोग के भी प्रमाण मिले है, परन्तु उन्ही पाण्डुलिपियो मे अज्ञात के लिये भी बिन्दु का प्रयोग किया गया है| आर्यभट्ट वह प्रथम गणितज्ञ थे जिन्होने शून्य का गणित मे व्यापक प्रयोग किया| लगभग 498 AD मे आर्यभट्ट ने विश्व को दाशमिक संख्या पद्धति दी| वे लिखते है- “स्थानम स्थानम दश गुणम्|” इसमे उन्होने शून्य के लिये एक शब्द “ख” का प्रयोग किया था|

परन्तु कथानक यहीं समाप्त नही होता है| अभी तो शून्य का गणित विकसित करना बाकी था| गणित मे शून्य एक संख्या है, तथा दाशमिक संख्या पद्धति का आधार है, परन्तु इसके गणितीय गुणो की विवेचना अभी बाकी थी| शून्य के साथ साथ ऋणात्मक संख्याओ के गणित के विकास मे तीन गणितज्ञो ने प्रमुख योगदान दिया| वे थे- ब्रह्मगुप्त, महावीर, और भास्कर|

ब्रह्मगुप्त ने 628 AD मे लिखित अपने पुस्तक “ब्रह्मगुप्त सिद्धांत” मे अंकगणित के कुछ नियम दिये गये है| वे कहते है कि किसी संख्या को उसी संख्या से घटाने पर शून्य मिलता है| शून्य से संबन्धित योग के लिये निम्नलिखित नियम देते है-

शून्य और एक ऋणात्मक संख्या का योग एक ऋणात्मक संख्या होती है| शून्य और एक धनात्मक संख्या का योग एक धनात्मक संख्या होती है| शून्य और शून्य का योग शून्य होता है|

घटाने की क्रिया थोड़ी जटिल है-

किसी ऋणात्मक संख्या को शून्य से घटाने पर धनात्मक संख्या मिलती है| किसी धनात्मक संख्या को शून्य से घटाने पर ऋणात्मक संख्या मिलती है| किसी ऋणात्मक संख्या से शून्य को घटाने पर वही ऋणात्मक संख्या मिलती है| किसी धनात्मक संख्या से शून्य को घटाने पर वही धनात्मक संख्या मिलती है| शून्य से शून्य को घटाने पर शून्य मिलता है| एक धनात्मक और ऋणात्मक संख्या का योग उनके अन्तर के बराबर होता है, और यदि उन संख्याओं का निरपेक्ष मान बराबर हो तो योग शून्य होता है|

ब्रह्मगुप्त आगे बताते है -

किसी संख्या को शून्य से गुणा करने पर शून्य मिलता है|

परन्तु भाग की क्रिया थोड़ी जटिल है-

किसी धनात्मक अथवा ऋणात्मक संख्या को शून्य से भाग देने पर एक भिन्न संख्या (fractional number) मिलती है जिसका हर (denominator) शून्य होता है| शून्य को किसी धनात्मक अथवा ऋणात्मक संख्या से भाग देने पर निष्कर्ष शून्य अथवा एक भिन्न संख्या होती है जिसका अंश (numerator) शून्य और हर (denominator) वह संख्या होती है| शून्य को शून्य से भाग देने पर शून्य मिलता है|

निस्सन्देह ब्रह्मगुप्त भाग कि क्रियाओं मे थोड़े भ्रमित थे, परन्तु उनका प्रयास सराहनीय था| वह पहले ऐसे गणितज्ञ थे जिसने शून्य और ऋणात्मक संख्याओं का अंकगणित प्रस्तुत किया|

ब्रह्मगुप्त से लगभग 200 वर्षो के बाद, 830 AD मे भारतीय गणितज्ञ महावीर ने “गणित सार-संग्रह “ लिखा| यह पुस्तक ब्रह्मगुप्त सिद्धांत का ही एक विकसित रूप लगती है| महावीर यहाँ स्पष्टतः लिखते है-

…किसी संख्या को शून्य से गुणा करने पर शून्य प्राप्त होता है, किसी संख्या से शून्य घटाने पर वह अपरिवर्तित रहती है|

हालांकि ब्रह्मगुप्त की कमियों को दूर करने मे वह असफल रहे है-

किसी संख्या को शून्य शून्य से भाग देने पर वह अपरिवर्तित रहती है|

ब्रह्मगुप्त से लगभग 500 वर्षो के बाद भास्कर भी शून्य के बारे मे बहुत कुछ सिद्धान्त देते है, परन्तु इतना समय बीत जाने पर भी वह शून्य से भाग देने की क्रिया को नही समझा पाये| वह लिखते हैं-

“किसी संख्या को शून्य से भाग देने पर एक भिन्न संख्या (fractional number) मिलती है जिसका हर (denominator) शून्य होता है| इस भिन्न संख्या को अनंत कहा जाता है| शून्य को हर के स्थान पर रखने वाली यह संख्या इसमे से कुछ भी निकाले या इसमे कुछ भी जोड़े जाने पर भी अपरिवर्तनीय है, ठीक उसी प्रकार जैसे इस संसार के निर्माण और नष्ट होने से उस अनंती ईश्वर पर कोई प्रभाव नही पड़ता है|”

इस प्रकार भास्कर ने इस समस्या को n/0 = ∞ लिख कर हल करने का प्रयास किया| परन्तु हम जानते है कि यह भी सही नही है| वस्तुतः n/0 अपरिभाषित है, किसी भी संख्या को शून्य से भाग नही दिया जा सकता| हालांकि भास्कर ने शून्य के अन्य गुणो को ठीक लिखा है, जैसे- 0X0 = 0, और √0 = 0

भारतीयो का यह उद्दीप्त कार्य मुस्लिम और अरबी गणितज्ञो ने संसार मे पहुँचाया | 9वीं शताब्दी मे एक पारसी गणितज्ञ मुहम्मद इब्न् मूसा अल्-ख़्वारिज़्मी ने एक पुस्तक Algoritmi de numero Indorum (“al-Khwārizmī on the Hindu Art of Reckoning”) लिखी जिसमे उसने दस प्रतीको 1,2,3,4,5,6,7,8,9 और 0 पर आधारित भारतीय स्थान-मान संख्या-पद्धति का वर्णन किया था| आज जहाँ पर इराक है, वहाँ पर इसी पुस्तक ने शून्य का प्रयोग शुरु किया| इस पुस्तक का यह नाम इटली के एक इतिहासकार Baldassarre Boncompagni द्वारा दिया गया, मूल अरबी नाम संभवतः “किताब्-अल्-ज़ाम्-व्-ल्-तफरीक़ बी-हिसाब् अल्-हिन्द्” (“The Book of Addition and Subtraction According to the Hindu Calculation”) था| 12वीं शताब्दी मे अब्राहम-बेन्-मीर्-इब्न इर्ज़ा ने भारतीय अंको और दाशमिक भिन्न संख्याओ पर तीन पुस्तके लिखी जिसने युरोप के विद्वानो का ध्यान भी आकर्षित किया| The Book of the Number मे पूर्णांको के लिये बांये से दांये स्थान-मान संख्या-पद्धति का वर्णन किया गया है| इब्न इर्ज़ा ने इस पुस्तक मे शून्य को गलगल कहा है, जिसका अर्थ होता है वृत्त| कुछ समय पश्चात 12वीं शताब्दी मे ही  एक अरबी गणितज्ञ इब्न्-याह्या अल्-मग़रीबी अल्-समवाल ने बीजगणित की एक पुस्तक लिखी जिसका नाम था- अल्-बहिर् फि-ल्-जब्र (The brilliant in algebra)| इसमे शून्य के बारे मे विस्तार से चर्चा की गयी है|

भारतीय गणित पश्चिम मे इस्लामिक देशो के साथ-साथ पूर्व मे चीन मे भी फैली| 1247AD मे चीनी विद्वान चिन् चियु-शाउ ने अपनी पुस्तक Shushu Jiuzhang (Mathematical Treatise in Nine Sections) मे शून्य के लिये O का प्रयोग किया है|

भारतीय गणित को यूरोप मे पहुँचाने का महत्वपूर्ण कार्य प्रसिद्ध इटैलियन गणितज्ञ लियोनार्दो पिसानो फिबोनक्कि (Leonardo Pisano Fibonacci) ने किया| 1202 AD मे लिखी अपनी पुस्तक “लिबर अबेकि” (Liber abaci) मे फिबोनक्कि ने दसो भारतीय अंको (1, 2, …9 और 0 ) का प्रयोग किया है| गौरतलब है कि फिबोनक्कि ने 1,2,3,4,5,6,7,8,9 को अंक कहा है, परन्तु 0 को केवल एक प्रतीक|

पूरे संसार के लोग ध्यान देते गये और इस प्रकार धीरे धीरे पूरे संसार मे भारतीय संख्या पद्धति अपना ली गयी| आज भी संख्याओं पर भारतीय प्रभाव देखा जा सकता है| भारतीय अंको और वैश्विक अंको मे समानता स्पष्ट दिखती है| १ = 1, २ = 2, ३ = 3, ६ = 6, ० = 0| नीचे दी गयी सूची मे संस्कृत्, लैटिन्, इटैलियन और फ्रेन्च संख्याये तुलनात्मक रूप से लिखे है|

स्पष्ट है कि लैटिन शब्दो को संस्कृत से ही लिया गया है| यहाँ तक कि संख्याओं का क्रम भी अपरिवर्तित रहता है| अंग्रेजी का शब्द ज़ीरो (zero) भी संस्कृत से ही लिया गया है| संस्कृत मे इसे शून्य कहते है, अरबी मे अशिफ्र् (बाद मे शिफ्र् sifra) कहा जाने लगा| लैटिन मे सिफ्र (cifra), तत्पश्चात इटैलियन मे ज़ेफिरो (zefiro), ज़ेफ्रो (zefro), और बाद मे ज़ेवेरो (zevero) कहा जाने लगा, और अन्त मे अंग्रेजी मे ज़ीरो (zero)|

कुछ वर्षो पहले नयी सहस्त्राब्दि की शुरुआत 1 जनवरी 2000 को मनायी गयी| इसे देख कर लगता है कि संसार को अभी भी शून्य पूरी तरह से समझ नही आया है| 31 दिसम्बर 1999 की रात 12 बजे इसवी कैलेण्डर के 1999 वर्ष पूरे हुये| 2000 वर्ष तो 31 दिसम्बर 2000 को पूरे हुये थे| तो नयी सहस्त्राब्दि की शुरुआत 1 जनवरी 2000 को नही बल्कि 1 जनवरी 2001 को हुयी थी|

Ref:

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Zero.html

http://en.wikipedia.org/wiki/0_(number)

http://www.timelineindex.com/content/view/1452

http://mathforum.org/k12/mayan.math/

http://mathforum.org/k12/mayan.math/mayan2.html

http://mathforum.org/k12/mayan.math/mayan3.html

http://en.wikipedia.org/wiki/Numeral_system

http://en.wikipedia.org/wiki/Al-Khwarizmi#Arithmetic

http://en.wikipedia.org/wiki/Abraham_ibn_Ezra

http://www.gap-system.org/~history/Mathematicians/Ezra.html

http://en.wikipedia.org/wiki/Al-Samawal

Popularity: 3%

मनोज कुमार की प्रसिद्ध फिल्म (चलचित्र) का यह गीत तो सभी ने सुना होगा| भारत के गौरवगाथा को कहता यह गीत सबसे पहले ज़ीरो की महत्ता बताता है| अब यह तो सभी मानते है कि ज़ीरो (शून्य) भारत ने ही संसार को दिया| परन्तु कब? शून्य का अविष्कार कब हुआ था? कैसे इसका प्रचार सारे संसार मे हुआ? क्या केवल भारतीय ही थे जिन्हे रिक्त को व्यक्त करने का विचार सूझा? आइये कुछ चर्चा करें इस बात पर|

शून्य का प्रयोग दो तरीको से किया जाता है| प्रथमतः यह एक संख्या है जो कि “कुछ नहीं” को प्रदर्शित करती है| यदि आपके पास 5 रुपये है और वह आपने मुझे दे दिये तो आपके पास कितने रुपये बचे? कुछ नही| गणित की भाषा मे इसे शून्य कहा जाता है| शून्य का दूसरा प्रयोग दाशमिक स्थानमान संख्या पद्धति मे खाली स्थान को दिखाने के लिये किया जाता है| उदाहरण के लिये एक संख्या 304 को देखते है| इकाई स्थान पर 4 है, और सैकड़ा के स्थान पर 3 है, परन्तु दहाई के स्थान पर कुछ नही है| 304 को हम 3X10X10+0X10+4 भी लिख सकते है| अतः हम कह सकते है कि शून्य को दाशमिक आधार दिखाने के लिये प्रयोग किया जाता है|

अब आते है इतिहास पर| अंग्रेजी मे शून्य को ज़ीरो (zero) कहते है, जो कि अरबी शब्द शिफ्र से आया है| शून्य का इतिहास कुछ ऐसा नहीं है कि किसी को इसका विचार आया और फिर सभी उसका प्रयोग करने लगे| इसका इतिहास बहुत ही अस्पष्ट और विस्तृत है| प्रथम विचार भारतीय धर्म-ग्रन्थो मे मिलता है, जहाँ बताया गया है कि सृष्टि की रचना शून्य से हुई है| परन्तु यह इसका आध्यात्मिक रूप है, गणितीय नही| ऐसा भी नही है कि पहले स्थानमान संख्या पद्धति अस्तित्व मे आया फिर शून्य| क्योकि स्थानमान संख्या पद्धति का अस्तित्व तो बिना शून्य के भी विभिन्न रूपो मे था|

सबसे पहले स्थानमान संख्या-पद्धति का आविष्कार लगभग 3100 BC मे बेबीलोन मे माना जाता है| बेबीलोन मे आधार-60 (sexagesimal) वाली एक संख्या पद्धति विकसित थी| आरम्भ मे 60 से उपर की संख्याओ के लिये इसमे कुछ समस्यायें थी| आधार-60 प्रदर्शित करने के लिये कोई चिन्ह नही था, अतः 23, 23X60, 23X60X60, इत्यादि संख्यायें एक ही प्रतीत होती थी| लगभग 400 BC के आसपास बेबीलोन के विद्वानो ने इसके लिये एक चिन्ह का उपयोग करना आरम्भ किया| लगभग 700 BC मे मेसोपोटामिया मे भी इस रिक्त स्थान को दिखाने के लिये चिन्ह का प्रयोग किया जाता था| बेबीलोन की यह आधार-60 वाली संख्या पद्धति का उपयोग आज भी किया जाता है, कोण के माप मे| एक अंश को o से, 1/60 अंश को ” से तथा (1/60)” को ‘ से प्रदर्शित करते है|

लगभग इसी समय ग्रीक विद्वानो ने भी खाली स्थान को एक संकेत से दिखाना आरम्भ किया, परन्तु उन्होने बेबीलोन कि स्थानमान संख्या पद्धति को नही लिया, क्योंकि उनका मुख्य ध्यान संख्या पद्धति पर नही बल्कि ज्यामिति पर था| ग्रीक विद्वानो ने शून्य के लिये जिस चिन्ह का प्रयोग किया वह था “O” | यह संकेत कहाँ से लिया गया, इसके बारे मे अलग-अलग मत है| कुछ का कहना है कि यह ग्रीक शब्द “ouden” (जिसका अर्थ होता है “कुछ नही”) का पहला अक्षर “omicron” है, और कुछ का कहना है कि यह लगभग शून्य मूल्य वाले सिक्के “obol” के लिये है| ऐसे भी संकेत मिलते है कि ग्रीक दार्शनिक शून्य की स्थिति के बारे मे भ्रमित थे| वह स्वयं से पूछते थे “कैसे कुछ नहीं कुछ हो सकता है?”|

दक्षिण-मध्य मैक्सिको और मध्य अमेरिका मे विकसित मेसोमेरिकन कैलेन्डर मे भी शून्य का प्रयोग किया गया था| इसमे आधार-20 (vigesimal) माया संख्या पद्धति का प्रयोग किया गया था| हालांकि इस पद्धति का उदगम स्थल कहाँ है, यह अभी विवाद का विषय है|

भारतीय गणित मे 400 BCE के आसपास गणनाओं के लिये एक गणना पटल का प्रयोग किया जाता था| इसमे भी शून्य के उपयोग के प्रमाण मिलते है| चीन मे 400BC से गणना के लिये दण्डो (counting rods) का प्रयोग किया जाता था| चीनी गणित मे शून्य के साथ-साथ ऋणात्मक संख्याओ के भी प्रमाण मिलते है, परन्तु शून्य के लिये वे किसी चिन्ह का प्रयोग नही करते थे| 13वीं

शताब्दी मे चीनी विद्वानो ने शून्य के लिये “O” का प्रयोग शुरु किया| कुछ विद्वानो का मत है यह चिन्ह भारत से 718 AD मे एक विद्वान गौतम सिद्ध (Chinese: Qútán Xīdá, fl. 8th century) द्वारा चीन मे पहुँचा| यह ध्यान देने की बात है कि चीनी संख्या पद्धति भी दाशमिक संख्या पद्धति थी, परन्तु उतनी परिष्कृत नही जितनी भारतीय संख्या पद्धति|

रोमन

अंको मे शून्य के लिये किसी चिन्ह का नही बल्कि एक शब्द null (जिसका अर्थ “कुछ नहीं” होता है) का प्रयोग करते थे|

भारत मे इसका इतिहास बहुत प्राचीन है| जैसा कि ऊपर बताया गया है, भारतीय धर्म-ग्रन्थो मे लिखा मिलता है कि सृष्टि की रचना शून्य से हुई है| जैसा कि श्री मुखर्जी (Discovery of Zero and its impact on India के लेखक) कहते है-

… the mathematical conception of zero … was also present in the spiritual form from 17 000 years back in India.

हड्प्पा की सभ्यता मे भी दाशमिक पद्धति के संकेत मिले है| हड़प्पा मे प्राप्त वस्तुओं के भारो के अनुपात 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, और 50 मिले है, यह सभी दाशमिक भाज्य है|

शून्य का वर्तमान रूप भारत द्वारा दिया गया है| हालांकि चीनी भी दाशमिक आधार वाले संख्या पद्धति का प्रयोग कर रहे थे, उनकी पद्धति मे बहुत कमियां थी| यह भारतीय संख्या पद्धति है जो कि अरब के रास्ते पश्चिम मे पहुँची और आज वैश्विक रूप से स्वीकार की गयी है| प्रसिद्ध फ्रांसीसी गणितज्ञ लाप्लास के शब्दो मे- “प्रत्येक संभव संख्या को केवल दस प्रतीको के समूह से दर्शाने (प्रत्येक प्रतीक का एक स्थानीय मान तथा एक निरपेक्ष मान होता है) की सरल विधि भारत मे विकसित हुई| देखने मे यह विचार इतना सरल लगता है कि आज इसके महत्व की सराहना नही की जा रही है| इसकी सरलता इस बात मे है कि इसके कारण गणनाये सरल हो गयी और अंकगणित  आविष्कारो मे अग्रणी हो गया|”

जहाँ पश्चिम मे बोझिल रोमन अंक प्रणाली मे संख्याओ की अधिकता एक प्रमुख बाधा के रूप में सामने थी, वहीं चीन के सचित्र संख्याओं की जटिलता एक अन्य बाधा थी| परन्तु भारत की संख्या पद्धति मे सब कुछ व्यवस्थित और सरल था| शून्य के व्यापक उपयोग के साथ दाशमिक संख्या पद्धति ने वैज्ञानिक प्रगति मे बहुत बड़ा योगदान दिया|

भारत मे किन-किन गणितज्ञो ने शून्य के विकास मे योगदान दिया तथा यह संसार मे किस प्रकार फैली, इसकी चर्चा हम इस लेख के अगले अंक मे करेंगे| अपने विचारो से हमे अवश्य अवगत करायें|

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Geometry

The Shulva Sutras explain a large number of simple geometrical constructions — constructions of squares, rectangles, parallelograms and trapezium. These and others involve the following theorems

• The diagonal of a rectangle divides it into two equal parts.
• The diagonals of a rectangle bisect each other and the opposite areas are equal.
• The perpendicular through the vertex of an isosceles triangle on the base divides the triangle into equal halves.
• A rectangle and a parallelogram on the same base and between the same parallels are equal in area.
• The diagonals of a rhombus bisect each other at right angles.
• The famous theorem known after the name of Pythagoras.
• Properties of similar rectilinear figures.

These cover roughly the first two books and the sixth book of Euclid. How these theorems were actually obtained is a matter for which no definite answer is available. The Baudhayana Shulva sutra gives the construction of geometric shapes such as squares and rectangles. It also gives sometimes approximate, geometric area-preserving transformations from one geometric shape to another. These include transforming a square into a rectangle, an isosceles trapezium, an isosceles triangle, a rhombus, and a circle, and transforming a circle into a square. In these texts approximations, such as the transformation of a circle into a square, appear side by side with more accurate statements.

Pythagoras theorem before Pythagoras

“Pythagorean theorem” is the rule for the lengths of sides of right triangles (now written as a²+b²=c²) . The sutras contain discussion and non-axiomatic demonstrations of cases of the Pythagorean theorem and Pythagorean triples. It is also implied and cases presented in the earlier work of Apastamba and Baudhayana, although there is no consensus on whether or not Apastamba’s rule is derived from Mesopotamia.

In Shulva Sutra, it was expressed as “the cord stretched in the diagonal of a rectangle produces both areas which the cords forming the longer and the shorter side of a rectangle produce separately.” (Baudhayana 74). In other words, the sum of the squares of the two different sides of the rectangle equals the square of its diagonal. (“Squares” probably meant the area of actual squares drawn on the three sides of the triangle formed.). In Baudhayana, the other rules related to this right angle theorem are given as follows:

• 1.9. The diagonal of a square produces double the area [of the square].
• 1.12. The areas [of the squares] produced separately by the lengths of the breadth of a rectangle together equal the area [of the square] produced by the diagonal.
• 1.13. This is observed in rectangles having sides 3 and 4, 12 and 5, 15 and 8, 7 and 24, 12 and 35, 15 and 36.

“Pythagorean triplets”, sets of side lengths that can be expressed as integers (for example 3,4,5 and 5,12,13), were used as a sort of proof of that proposition, and were also used to make right angles. Pythagorean triples are found in Apastamba’s rules for altar construction. They were used for the construction of right angles. The complete list is:

• (3,4,5)
• (5,12,13)
• (8,15,17)
• (7,24,25)
• (12,35,37).

However, since these triples are easily derived from an old Babylonian rule, Mesopotamian influence is not unlikely.

Carl B Boyer says “we find rules for the construction of right angles by means of triples of cords the lengths of which form Pythagorean triages, such as 3, 4, and 5, or 5, 12, and 13, or 8, 15, and 17, or 12, 35, and 37. However all of these triads are easily derived from the old Babylonian rule; hence, Mesopotamian influence in the Shulva sutras is not unlikely. Aspastamba knew that the square on the diagonal of a rectangle is equal to the sum of the squares on the two adjacent sides, but this form of the Pythagorean theorem also may have been derived from Mesopotamia. [...] So conjectural are the origin and period of the Shulva sutras that we cannot tell whether or not the rules are related to early Egyptian surveying or to the later Greek problem of alter doubling. They are variously dated within an interval of almost a thousand years stretching from the eighth century B.C. to the second century of our era.” [ref: Boyer (1991). "China and India". p. 207.]

The Satapatha Brahmana and the Taittiriya Samhita were probably also aware of the Pythagoras theorem. Seidenberg (1983) argued that either “Old Babylonia got the theorem of Pythagoras from India or that Old Babylonia and India got it from a third source”.[6] Seidenberg suggested that this source might be Sumerian and may predate 1700 BC.

We see various applications of this theorem in Shulva Sutras for different geometrical constructions. For example- construction of a square equal (in area) to the sum, difference, of two given squares, or to a rectangle, or to the sum of n squares. These construction unconditionally involve application of algebraic identities, such as (a±b)²=a²+b²±2ab, a²-b²=(a+b)(a-b), ab=((a+b)/2)²-((a-b)/2)², and na²=((n+1)/2)2a²-((n-1)/2)2a² etc.

value of pi

Another of their most famous problems is that of getting a square and circle to be equal in area. Two authors used very different constructions (neither of which actually uses pi) which both lead to a value of pi very close to 3.08831 (pi is actually about 3.14159). What’s remarkable is the similarity of the two values, but one must wonder how this particular value was reached. (It might be noted that only one of the authors showed knowledge that his method was an approximation.)

As an example, the statement of circling the square is given in Baudhayana as:

2.9. If it is desired to transform a square into a circle, [a cord of length] half the diagonal [of the square] is stretched from the centre to the east [a part of it lying outside the eastern side of the square]; with one-third [of the part lying outside] added to the remainder [of the half diagonal], the [required] circle is drawn.

Fig: Circling a square

and the statement of squaring the circle is given as:

2.10. To transform a circle into a square, the diameter is divided into eight parts; one [such] part after being divided into twenty-nine parts is reduced by twenty-eight of them and further by the sixth [of the part left] less the eighth [of the sixth part].

2.11. Alternatively, divide [the diameter] into fifteen parts and reduce it by two of them; this gives the approximate side of the square [desired].

Kim Plofker say, “The “circulature” and quadrature techniques in 2.9 and 2.10, the first of which is illustrated in figure 4.4, imply what we would call a value of π of 3.088, [...] The quadrature in 2.11, on the other hand, suggests that π = 3.004 (where s = 2r·13/15), which is already considered only “approximate.” In 2.12, the ratio of a square’s diagonal to its side (our √2is considered to be 1 + 1/3 + 1/(3·4) – 1/(3·4·34) = 1.4142 .” [ref: Plofker, Kim (2007). p. 391.&392]

arithmetic

The essentially arithmetical background of the Shulva mathematics must be contrasted with the essentially geometrical background characteristic of Greek mathematics. Simple fractions and operations on them are available in the Shulvas.

Along with the “simple” mathematics involved in altar-building, there were very complicated problems to decide what math to use where. One of these problems can be understood just by the fact that the most basic altar was in the form of a falcon! However, all of the bricks retained a square, rectangular, triangular, or circular shape. So the precise dimensions and area of every brick had to be calculated, after the specific design was decided upon. Usually, the total area was 7-1/2 units, because that was convenient for the falcon shape. But to make matters exponentially “worse”, when two or more altars were made for the same purpose, subsequent ones usually had to be one unit bigger than the last; and the areas of each brick had to stay proportional. As you can infer, this was one heck of a math problem! And it had to be done twice for each altar, because a different design was needed for alternating layers, so that spaces between bricks weren’t directly over each other.[Ref: Thibaut, George. Mathematics in the Making in Ancient India. Calcutta: K.P. Bagchi, 1984.]

Fractions

We meet with fractions like 3/8 (Thri Ashtama), 2/7 (Dwi Saptama), 3/4 (Chaturbhagona). These are not unit fractions only, as were used in ancient Egypt, Babylonia and China. Apasthamba gives the area of a square of side 1-1/2 purushas as 2-1/4, and that of a square of side 2-1/2 as 6-1/4.. If the area is 7-1/9 sq. purushas, the side of the square is 2-2/3 (Bodhayana).

Surds

Surds of the form √2, √3 etc. are called Karanis, thus √2 is dwi-karani, √3 = trikarani, √1/3=triteeya karani, √1/7=saptama karani, √18 = ashtadasa karani.

area of – a trapezium

The shape of the Ashwamedhiki Vedika is an isosceles trapezium whose head, foot and altitude are respectively 24√2, 30√2, 36√2 prakramas. Its area is stated to be 1944 prakramas (sq. is to be understood).

Area=36√2 x 1/2*(24√2 + 30√2) = 1944

This indicates a knowledge of the method of finding the area of – a trapezium, and simple operations on surds.

approximation to √2 and irrational numbers

Altar construction also led to an estimation of the square root of 2 as found in three of the sutras. In the Baudhayana sutra it appears as:

2.12. The measure is to be increased by its third and this [third] again by its own fourth less the thirty-fourth part [of that fourth]; this is [the value of] the diagonal of a square [whose side is the measure].[9]

This approximation to √2 occurs also in Shulvas Apasthamba and Katyayana.

√2 = 1 + 1/3 + 1/(3*4) – 1/(3*4*34)=577/408

This gives √2 = 1.4142156 , whereas the true value is 1.414213. The approximation is thus correct to five decimal places, and is expressed by means of simple unit fractions. The problem evidently arises in the construction of a square double a given square in area.

It is also interesting to note that three approximations of √2 are given.

[latex]\sqrt 2 = 7 / 5[/latex]
[latex]\sqrt 2 = 17 / 12[/latex]
[latex]\sqrt 2 = 577 / 408[/latex]

Now the continued fraction for √2 = 1 + 1/2 + 1/2+ …

The third, fourth and eight convergent of this are exactly the approximations given above. This gives no clue to the method used in Shulva Sutras, but the coincidence is noteworthy.

The Shulvas contain no clue at all as to the manner in which this remarkable approximation was arrived at. Many theories or plausible explanations have been proposed.

One conjecture about how such an approximation was obtained is that it was taken by the formula:

√(a²+r)≈a+{r/(2a)}-[{r/(2a)²}/{2(a+r/(2a))}] with a = 4 / 3 and r = 2 / 9. which is an approximation that follows a rule given by the twelfth century Muslim mathematician Al-Hassar. The result is correct to 5 decimal places. This formula is also similar in structure to the formula found on a Mesopotamian tablet from the Old Babylonian period (1900-1600 BCE): √2 = 1 + 24/60 + 51/(60)² – 10 /(60)³=1.41421297 which expresses √2 in the sexagesimal system, and which too is accurate up to 5 decimal places (after rounding). Indeed an early method for calculating square roots can be found in some Sutras, the method involves the recursive formula: √x≈√(x-1)+1/(2√(x-1)) for large values of x, which bases itself on the non-recursive identity √(a²+r) ≈ a+r/(2a) for values of r extremely small relative to a.

The Manava Shulva gives the following:

40² + 40² = 56²
4² + 4² = (5-2/3)²
36² + 90² = 97²
5² + 6² = (7-5/6)²

The above facts make it clear that the Indians were the first to use irrational numbers. The Greeks also used irrational numbers. If AB is a given segment, Pythagoras and others described the methods of constructing segments of length √2 AB, √3 AB,√5 AB, etc. But no rational approximations to √2, √3 etc., are found in Greek mathematics, nor are there any problems involving arithmetical operations on irrational numbers. This is easily explained, because the requisite knowledge of arithmetic was not available to the Greeks. It will also be borne in mind that according to unprejudiced estimates, the Shulva Sutras are about two or three centuries prior to Pythagoras.

Conclusion

We all know that Euclid’s geometry is based upon certain axioms and postulates, and the proofs involve a strict logical application of these. The logical methods of Greek geometry are certainly not discernible in Hindu geometry. No book on Hindu mathematics explains the system of axioms and postulates assumed, and this itself should go some way in refuting the concocted claim that Hindu mathematics is borrowed from the Greeks. At the same time, it may not be correct to conclude that the above theorems were asserted as a matter of experience and measurement. The people who could make out and solve complicated problems of arithmetic, algebra and spherical trigonometry should be credited with some amount of logic in their work. The Sulvas are not formal mathematical treatises. They are only adjuncts to certain religious works.

Further Readings

· Seidenberg, A. 1983. “The Geometry of the Vedic Rituals.” In The Vedic Ritual of the Fire Altar. Ed. Frits Staal. Berkeley: Asian Humanities Press.
· Sen, S.N., and A.K. Bag. 1983. The Sulbasutras. New Delhi: Indian National Science Academy.
· A Seidenberg, The Origin of Mathematics in Archives for History of Exact Science
· Plofker, Kim (2007). “Mathematics in India”. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. ISBN 9780691114859.
· Boyer, Carl B. (1991). A History of Mathematics (Second Edition ed.). John Wiley & Sons, Inc. ISBN 0471543977.
· Cooke, Roger (1997). The History of Mathematics: A Brief Course. Wiley-Interscience. ISBN 0471180823.
· Cooke, Roger (2005), written at New York, The History of Mathematics: A Brief Course, Wiley-Interscience, 632 pages, ISBN 0471444596
· Joseph, G. G. 2000. The Crest of the Peacock: The Non-European Roots of Mathematics. Princeton University Press. 416 pages. ISBN 0691006598. page 229.
· [ref: Thibaut, George. Mathematics in the Making in Ancient India. Calcutta: K.P. Bagchi, 1984]

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