Wednesday, January 29, 2014

Vrikshayurveda or the science of plants



Vrikshayurveda or the science of plants is another achievement of the ancients, based on sound sense and practical knowledge. The Brihatsamhita of Varamahira devotes a chapter to the subject. One passage reads - 'Prantacchayavinirmukta na manojna jalasayh, yasmadato jalaprantesvaramanviniveseyet.' The gist is that parks and gardens are best situated along lakes and rivers. Another recommends mellow soil for plants, and suggests a means of acquiring this - 'Mrdui bhuh sarva-vrksanam hita tasyam tilan vapet, Puspitamstamsca mrdniyat karmaitat prathamam bhuvah.' Sesanum (tila) seeds should be planted and later, the flowering plant trampled into the soil. There is a verse dealing with the ideal season and stage of growth for cuttings - 'Ajatasakhan sisire Jatasakhan himagama, varsagame ca suskandha yathadiksthan praropayet' i.e. Mid-January to mid-March (the season of dews) is the season for making cuttings of plants that have not yet flowered or spread their branches. The plants with branches should be prepared for cuttings in the season of the mists, i.e, Winter (mid-November to mid-January), while those with well-grown branches should be transplanted during the rainy season. Also, the clones should be planted or mounted facing the same direction as they did on the parent tree. A medicament for the cloning material - "Ghrtosiralaksaudravidangaksiragomayaih amulaskandhaliptanam sankramananropanam.' The berries of sesanum, andropogon, and Embelia ribes (vidanga) together with cowdung are to be formed a paste of and applied. (This was a protection against fungal and other diseases). The chapter also suggests a way to ensure healthy germination and later fructification of the seed - the repeated application of oil followed by the drying of the seed in good sunlight. The oils recommended are of Alangium hexapetalum (Angola) or Cordia myxa (Slesmataka).
The hopes of tracing any independent text of Vrikshayurveda were given up by scholars, till Y L Nene (Chairman, Asian Agri-History Foundation) procured a manuscript of Vrikshayurveda of Surapala from the Bodleian Library, Oxford, UK. Sadhale undertook the translation of the text at Nene's request.
The manuscript is written in an old form of Nagari script. The script of the manuscript represents, most probably, the stage immediately preceding the modem form of Nagari. The script consists of sixty pages with margin on both sides. Each page contains six lines in general (occasionally five or seven). There are about thirty characters in each line written boldly with a thick pointed pen.

Brhatsamhita of Varahamihira of the sixth century also contains a chapter titled Vrikshayurveda. It also contains chapters on allied subjects such as divining groundwater, productivity and non-productivity of land as indicated by natural vegetation, etc. However, beyond establishing the antiquity of the sastra, it cannot give any definite clues to any full-fledged, independent texts onVrikshayurveda.

An anthological compilation of Sarngadharapaddhati (written by Sarngadhara), belonging to the thirteenth century, is yet another ancient text which in its chapter "Upavanavinoda" deals with an allied subject, viz., "arbori-horticulture". The chapter discusses such topics as planting, soil, nourishment of plants, plant diseases and remedies, groundwater resources, etc. Thus it shares withVrikshayurveda of Surapala almost all the topics. Many verses are identical and several others, although worded differently have an identical content. In spite of the striking resemblance between Upavanavinoda and Vrikshayurveda of Surapala, the former cannot be considered as a complete and independent text on Vrikshayurveda.

Surapal's Vrikshayurveda is a systematic composition starting with the glorification of trees and tree planting. It then proceeds to discuss various topics connected with the science of plant life such as procuring, preserving, and treating of seeds before planting; preparing pits for planting saplings; selection of soil; method of watering; nourishments and fertilizers; plant diseases and plant protection from internal and external diseases; layout of a garden; agricultural and horticultural wonders; groundwater resources; etc. The topics are neatly divided into different sections and are internally correlated. The author has expressed indebtedness to the earlier scholars but claims that in writing the present text he was guided by his own reason.

All these observations lead one to accept the text as an independent, full-fledged work on the subject of Vrikshayurveda. Sadhale informs that there are frequent references to this science in ancient Indian literature such as AtharvavedaBrhatsamhita of Varahamihira, Sarngadharapaddhati of Sarngadhara, etc. which bring out the botanical and agricultural aspects; works such as the Samhitas of Caraka and Susruta which bring out the medicinal aspect; and works such as GrhyasutrasManusmrtiArthasastra of Kautilya, SukranitiKrishisangraha of Parasara,Kamandakiya NitisaraBuddhist JatakasPuranas (Matsya, Varaha, Padma, Agni, etc.).

The colophon of the manuscript mentions Surapala as the writer of the text. He is described as a scholar in the court of Bhimapala. Surapala is stated to be "Vaidyavidyavarenya", a prominent physician.
Like several other Sanskrit texts the manuscript gives no clue to the date or place of the author. The subject deserves an in-depth study; however, any attempt at fixing a date of an author is bound to be at best a conjecture for want of definite proof.

Surapala's language, style, vocabulary, and expression also do not help much in providing any clue to his time or place. Interestingly, it is in Subandhu's Vasavadatta – a Sanskrit prose romance of the seventh century – that we come across the name Surapala. This might be a reference to some Surapala who through his writings or commentary could throw light on the plant. At least, there is a reasonable ground to accept such a proposition. An ancient work on plants mentioning Ganikarika may have existed on which Surapala might have written a vrtti and might have earned credit for identifying or throwing more light on the plant. Even though it is a reasonable conjecture, Sdahale thinks that the reference must have been to some other Surapala of the seventh century. Without going into the translators detailed arguments, Sadhale places Surpala in the 10th Century AD.

Sadhale sdays that the existence of the manuscript has solved some problems but it has also given rise to some new ones. The most important problems are:
How does one explain the overwhelming resemblance between Upavanavinoda and the present text of Vrikshayurveda?
The resemblance between Upavanavinoda and Vrikshayurveda may be explained by either proposing a theory that both have made use of texts of their predecessors or by revising our opinion regarding Surapala's date.

Surapala's merits as an author of a scientific work have been brought out incidentally in course of these discussions. Thus a systematic unfolding of the subject, a balanced treatment of various topics, neatly divided sections for the respective topics with clear demarcations of commencement and conclusion, a better and more logical expounding of various topics as compared with the other two texts, regard for predecessors combined with self-confidence and independent reasoning are some of the characteristics of his writing. However, in the description of the blossoming of some trees at the loving glance or a gentle kick of a charming young girl (as per conventions in literature), Surapala's poetic talent reveals itself fully and can match with the best of the classical poetry in Sanskrit (verses 147-151). Similarly, when he describes the plan and layout of a pleasure garden (verses 293-297), the poet in him automatically takes charge of his pen.
Below we quote some prescriptions from Vrikshayurveda; the stanza numbers refer to Sadhale's translation. Some of the prescriptions sound very unconventional and should be experimentally verified. Some agricultural institute should try these methods and if found successful, should be used in regular practice.

On Soil

35. Arid, marshy, and ordinary are the three types of land. It is further subdivided into six types by colour and savour.

36. Black, white, pale, dark, red, and yellow are the colours and sweet, sour, salty, pungent, bitter, and astringent are the tastes by which land is subdivided.

37. Land with poisonous element, abundance of stones, ant hills, holes, and gravel and having no accessibility to water is unfit for growing trees.

38. Bluish like saphire, soft like a parrot's feather, white like conch, jasmine, lotuses, or the moon, and yellow like heated gold or blooming champaka is the land recommended for planting.

39. Land, which is even, has accessibility to water, and is covered with green trees is good for growing all kinds of trees.

40. Arid and marshy land is not good. Ordinary land is good as all kinds of trees grow on it without fail.

41. Panasalakucatala, bamboo, jambeerajambutilakavatakadambaamratakharjurakadalitinisamrdviketakinarikela, etc. grow on a marshy land.

42. Sobhanjanasriphalasaptaparnasephalikaasokasamikarirakarkandhukesaranimba, and saka grow well on an arid land.

43. Bijapurakapunnagachampakaamraatimuktakapriyangudadima, etc. grow on an ordinary type of land.

On Propagation

45. Vanaspatidrumalata, and gulma are the four types of plants. They grow from seed, stalk, or bulb. Thus the planting is of three kinds.

46. Those which bear fruits without flowers are vanaspati (types); those which bear fruits with flowers are druma (types).

47. Those which spread with tendrils are lata (types) (creepers ). Those which are very short but have branches are gulma (types) (bushes).

4849. Jambuchampakapunnaganagakesaratamarindkapitthabadaribilvakumbhakaripriyangupanasaamramadhukakaramarda, etc. grow from seeds. Tambuli,sinduvaratagara, etc. grow from stalks.

50. Pataladadimiplaksakaraviravatamallikaudumbara kunda, etc. grow from seeds as well as from stalks.

51. Kumkumaardrarasonaalukanda, etc. grow from bulbs. Elapadmautpala, etc. grow from seeds as well as from bulbs.

52. Seed is extracted from dried fruits, which become ripe in the natural course and season. It is then sprinkled.

68. After the ash is naturally cooled and removed, kunapa water (liquid manure) should be sprinkled and the pits should be filled with good earth.

69. Sowing seeds for makandadadimakusmanda, and alambuka is good but planting is even better.

70. In fertile lands, which are used excessively, seeds of trapusa or of other vegetables are sown intermittently.

71. Here (in these fields), saffronmaruwaka, and damanaka are similarly grown in a small carry (?).

72. Large seeds should be sown singly but smaller ones should be sown in multiples. The seed of naranga should be sown in a slanting position with hand.

73. The seeds of phanijjhaka (maruwaka) should be mixed with earth and then water mixed with cow dung should be sprinkled gradually and gently.

74-75. Smeared with the pulp of a plantain ripened naturally and dried in the sun, a rope of the stalk of sastika (a rice variety that matures in 60 days) should be laid in the pits intermittently. Sprinkled with little water continuously in the hot days, it yields without fail sprouts blue like tamala.

76. The stalk should be eighteen angula, not too tender nor too hard. Half of it should be smeared with plenty of cow dung and then (it) should be planted with three-fourth part in the pit and should be sprinkled with water mixed with soft sandy mud.

77. The lower part of the stalks of satapatrika should be half-ripened and then in the month of Kartika (post-rainy season) should be planted in a carry and drenched with water for about two months.

78. When they are covered with leaves they should be uprooted and transplanted wherever desired in the month of Asadha (beginning of rains).

79-80. The branches of dadima and karavira should be bent and planted applying enough cow dung at the root. They should be watered regularly for two months. After the leaves start growing they should be cut in the middle.

81. Bulbs should be planted in pits measuring one forearm-length, breadth, and depth-and filled with mud mixed with thick sand.

82. Kadali should be planted after smearing the root profusely with cow dung. It should be planted in the pit along with the root and should be watered well.

83. Small trees should be transplanted by daytime at the proper directions when they are one forearm tall. The roots should be smeared with honey, lotus-fibre, ghee, and bidanga and then planted in proper pits along with the earth.

84. Big trees should be similarly transplanted with their roots covered during evening after reciting the following mantra the previous day.

87. Ksirikatutadadimibakula, etc. should be planted in the month of Sravana (midst of rainy season). Rajakosaamralakuca, etc. should be planted in the month of Bhadrapada (when rains are receding).

On Treatment

187. The diseases of the kafa type can be overcome with bitter, strong, and astringent decoctions made out of panchamula (roots of five plant species – sriphalasarvatobhadrapatala,ganikarika, and syonaka) with fragrant water.

188. For warding off all kafa type of diseases, the paste of white mustard should be deposited at the root and the trees should be watered with a mixture of sesame and ashes.

189. In case of trees affected by the kafa disease, earth around the roots of the trees should be removed and fresh, dry earth should be replaced for curing them.

190. A wise person should treat all types of trees affected by the pitta type of diseases with cool and sweet substances.

191. When watered by the decoction of milk, honey, yastimadhu, and madhuka, trees suffering from pitta type of diseases get cured.

192. Watered with the decoctions of fruits, triphala, ghee, and honey the trees are freed of all diseases of the pitta type.

193. To remove insects both from the roots and branches of the trees, wise men should water the trees with cold water for seven days.

194. The worms can be overcome by the paste of milk, kunapa water, and cow dung mixed with water and also by smearing the roots with the mixture of white mustard, vacakusta, andativisa.

195. The worms accumulated on trees can be treated quickly by smoking the tree with the mixture of white mustard, ramathavidangavacausana, and water mixed with beef, horn of a buffalo, flesh of a pigeon, and the powder of bhillata (bhallataka ?).

196. Anointing with vidanga mixed with ghee, watering for seven days with salt water, and (applying) ointment made out of beef, white mustard, and sesame destroy the worms, insects, etc.

197. Creepers eaten away by insects should be sprinkled with water mixed with oil cake. The insects on the leaves can be destroyed by sprinkling the powder of ashes and brick-dust.

198. A wound caused by insects heals if sprinkled with milk after being anointed with a mixture of vidanga, sesame, cow's urine, ghee, and mustard.

199. Trees suffering from (damage due to) frost or scorching heat should be externally covered. Sprinkling with kunapa water and milk is also advisable.

200-201. The broken trees should be smeared with the paste of the bark of plaksa and udumbara mixed with ghee, honey, wine, and milk and the broken parts should be firmly tied together with the rope of a rice stalk. Fresh soil should then be filled in the basin around the trees, sprinkled immediately with the milk of buffalo and flooded with water. Thus they recover.

203. If the branches fall off, the particular spot should be anointed with the mixture of honey and ghee and sprinkled over by milk and water so that the tree will have its branches reaching the sky.

204. If the branches are burnt they should be cut off and the particular spots should be sprinkled with water and grape, crystalline sugar, and barley (and then watered with the same ?).

239. The white flowers of a tree turn into a golden colour if the tree is watered with the mixture of turmeric powder, kimsuka, cotton seed, manjista, and lodhra.

240. The white flowers of a tree turn into a golden colour if it is smeared at the roots with the mixture of manjistadarada, milk, kanksi (kind of fragrant earth), and flesh of a pigeon.

241. Trees watered continuously with the liquid of triphala, barley, mango seed, and indigo; and also filled at the root with the powder of the same mixture produce fruits resembling collyrium (see anjana).

242. Trees treated with water and paste containing the mixture of barley, kimsukamanjista, turmeric, and sesame and also smeared with the same paste bear red fruits.

243. Trees watered and smeared at roots with the mixture of the bark of the salmali tree, turmeric, indigo, triphalakusta, and liquor bear fruits having the shades of a parrot.

244. Trees watered after being sprinkled at the root with the mixture of indigo, turmeric, lodhravara (triphala), sesame, asanakasisa and yasti – all powdered together – produce fruits of golden colour.

245. Bakula trees blossom forth producing lots of champaka flowers if continuously fed with fresh water after filling the bottom with plenty of mud mixed with kalaaya and the skin of a python or snake.

246. Plantain trees create wonder by producing pomegranate fruits if fed by water mixed with the urine of a hog and ankolha.

247. A castor tree produced from a seed cultured by the marrow of a boar, treated further by the process in the previous verse, produces karavella fruits.

248. Fragrance of the blossom can be changed by filling (the base near) the roots of the trees with the earth scented with the desired fragrance and then fed with water mixed with jaladamura,natavalaka, and patraka.

249. All types of flowering plants produce excellent fragrance if earth strongly scented by their own flowers is filled around the base (of the trees) and then fed with water mixed with musta,muranata leaves, and wine.

250. The same treatment used in the evening at their blossoming time along with fat, milk, blood, and kusta intensifies the natural fragrance of the blossoms of punnaganagabakula, etc.

251. A big and strong mud pot should be filled with the mixture of mud and plenty of beef; and the karavira plant should be grown there with effort by watering profusely with cow dung and good quality beef.

252. The above stated plant of karavira should then be shifted to a pit, previously prepared by filling with cow bones, well-burnt ashes and then wetted by water mixed with beef. Thereafter, the plant should be fed with plenty of water mixed with beef. So treated, it is transformed into a creeper to blossom profusely and perennially.

253. A tamarind plant is grown into an excellent creeper if fed with water, mixed with the powder of triphala.
ecology
Thanks for ancient ind tech,

DISTANCE OF EARTH IN HANUMAN CHALISA BY TULSIDAS Ji.


It is a poem written by Tulsidas in the Awadhi language, and is his best known Hindu text apart from the Ramcharitmanas. The word "chālisā" is derived from "chālis" in Hindi, which means 40, as the Hanuman Chalisa has 40 verses. Hanuman Chalisa (Hindi: हनुमान चालीसा "Forty chaupais on Hanuman") , composed by great devotee and sage Goswami Tulsidas, is a devotional prayer recited daily by Millions Hindus in morning prayers. Devotees also recite at the times when they have to ask protection and help from Lord Hanuman who is servant of Ram. Hanuman is the Param-Bhagavat, the topmost devotee in this cosmos. Hanuman Ji is himself Lord Shiva and thus also comes in Ishwara-Koti.

While describing greatness of lord Hanuman in Hanuman Chalisa, Goswami Tulsidas, the greatest devotee of lord Ram in Kaliyuga, mentioned the distance between SUN and EARTH very correctly in simple words. This shows not only spiritual greatness but also scientific knowledge and enlightement of Goswami Tulsidas.
Thank you for ancient indian technolgy.

The Samrat yantra


The Samrat yantra or the 'Supreme Instrument' is Jai Singh's most important creation. The instrument is basically an equinoctial sundial, which had been in use in one form or the other for hundreds of years in different parts of the world. In India, Brahmagupta (AD 598) describes Kartari yantra, an equinoctial sundial, which operates on the same principle as the Samrat. Jai Singh however, turned the simple equinoctial sundial into a tool of great precision for measuring time and the coordinates of a celestial object.

The Jantar Mantar was the largest of the five Astronomical Observatories built by Jai Singh. The Samrat Yantra is the world's largest sundial, standing 27 meters (73 feet) high.The Samrat Yantra, pictured far right and in illustrations, below, is the largest sundial in the world. It’s gnomon rises over 73 feet above its base, and the marble faced quadrants, 9 feet in width, create an arc that reaches 45 feet in height.The primary object of a Samrat is to indicate the apparent solar time or local time of a place. On a clear day, as the sun journeys from east to west, the shadow of the Samrat gnomon sweeps the quadrant scales below from one end to the other. At a given moment, the time is indicated by the shadow’s edge on a quadrant.

The Jaipur observatory is the largest and best preserved of these. It has been inscribed on the World Heritage List as "an expression of the astronomical skills and cosmological concepts of the court of a scholarly prince at the end of the Mughal period"


The above One shows the principle of a Samrat yantra.
The instrument consists of a meridian wall ABC, in the shape of a right triangle, with its hypotenuse or the gnomon CA pointing toward the north celestial pole and its base BC horizontal along a north-south line. The angle ACB between the hypotenuse and the base equals the latitude lambda of the place. Projecting upward from a point S near the base of the triangle, are two quadrants SQ1 and SQ2 of radius DS. These quadrants are in a plane parallel to the equatorial plane. The center of the two 'quadrant arcs' lies at point D on the hyptenuse. The length and radius of the quadrants are such that, if put together, they would form a semicircle in the plane of the equator.


The tangential scale over Samrat gnomon. The scale indicates angle of declination.

The quadrants are graduated into equal-length divisions of time measuring units, such as ghatikas and palas, according to the Hindu system, or hours, minutes and seconds, according to the Western system. The upper two ends Q1 and Q2 of the quadrants indicate either the 15-ghatika marks for the Hindu system, or the 6 A.M. and the 6 P.M. marks according the Western system. The bottom-most point of both quadrants, on the other hand, indicates the zero ghatika or 12 noon. The hypotenuse of the gnomon edge AC is graduated to read the angle of declination. The declination scale is a tangential scale in which the division lenghts gradually increase according to the tangent of the declination angle as illustrated in figure 2. The zero marking of this scale is at point D. Further, the gnomon scale AC is divided into two sections, such that the section DA reads angle of declination to the north of the celestial equator, and the section DC reads the declination to the south, as illustrated in the figure 2. 


Samrat Yantra perspectives. The primary objective of a Samrat is to indicate the apparent solar time or local time of a place. On a clear day, as the sun journeys from east to west, the shadow of the Samrat gnomon sweeps the quadrant scales below from one end to the other. At a given moment, the time is indicated by the shadow's edge on a quadrant scale.
The right ascension (RA) of an object is determined by simultaneously measuring the hour angle of the object and the hour angle of a reference star. From the measurements, the difference between the right ascension of the two is calculated. By adding or subtracting this difference to the right ascension of the reference star, the RA of the object is determined.


To measure the declination of the sun with a Samrat, the observer moves a rod over the gnomon surface AC up or down until the rod's shadow falls on a quadrant scale below. The location of the rod on the gnomon scale then gives the declination of the sun. Declination measurements of a star or a planet require the collaboration of two observers. One observer stays near the quadrants below and, sighting the star through a sighting device, guides the assistant, who moves a rod up or down along the gnomon scale. The assistant does this until the vantage point V on a quadrant edge below, the gnomon edge above where the rod is placed, and the star - all three - are in one line. The location of the rod on the gnomon scale then indicates the declination of the star. 

भारत का स्वर्णिम अतीत खो गया...

 खो गया वह स्वर्णिम अतीत.

Photo: ^ कहाँ खो गया वह स्वर्णिम अतीत.

भारत में ७ लाख ३२ हज़ार गुरुकुल एवं विज्ञान की २० से अधिक शाखाए थी

"भारत का स्वर्णिम अतीत" से आगे : अब बात आती है की भारत में विज्ञान पर इतना शोध किस प्रकार होता था, तो इसके मूल में है भारतीयों की जिज्ञासा एवं तार्किक क्षमता, जो अतिप्राचीन उत्कृष्ट शिक्षा तंत्र एवं अध्यात्मिक मूल्यों की देन है। "गुरुकुल" के बारे में बहुत से लोगों को यह भ्रम है की वहाँ केवल संस्कृत की शिक्षा दी जाती थी जो की गलत है। भारत में विज्ञान की २० से अधिक शाखाएं रही है जो की बहुत पुष्पित पल्लवित रही है जिसमें प्रमुख १. खगोल शास्त्र २. नक्षत्र शास्त्र ३. बर्फ़ बनाने का विज्ञान ४. धातु शास्त्र ५. रसायन शास्त्र ६. स्थापत्य शास्त्र ७. वनस्पति विज्ञान ८. नौका शास्त्र ९. यंत्र विज्ञान आदि इसके अतिरिक्त शौर्य (युद्ध) शिक्षा आदि कलाएँ भी प्रचुरता में रही है। संस्कृत भाषा मुख्यतः माध्यम के रूप में, उपनिषद एवं वेद छात्रों में उच्चचरित्र एवं संस्कार निर्माण हेतु पढ़ाए जाते थे।

थोमस मुनरो सन १८१३ के आसपास मद्रास प्रांत के राज्यपाल थे, उन्होंने अपने कार्य विवरण में लिखा है मद्रास प्रांत (अर्थात आज का पूर्ण आंद्रप्रदेश, पूर्ण तमिलनाडु, पूर्ण केरल एवं कर्णाटक का कुछ भाग ) में ४०० लोगो पर न्यूनतम एक गुरुकुल है। उत्तर भारत (अर्थात आज का पूर्ण पाकिस्तान, पूर्ण पंजाब, पूर्ण हरियाणा, पूर्ण जम्मू कश्मीर, पूर्ण हिमाचल प्रदेश, पूर्ण उत्तर प्रदेश, पूर्ण उत्तराखंड) के सर्वेक्षण के आधार पर जी.डब्लू.लिटनेर ने सन १८२२ में लिखा है, उत्तर भारत में २०० लोगो पर न्यूनतम एक गुरुकुल है। माना जाता है की मैक्स मूलर ने भारत की शिक्षा व्यवस्था पर सबसे अधिक शोध किया है, वे लिखते है "भारत के बंगाल प्रांत (अर्थात आज का पूर्ण बिहार, आधा उड़ीसा, पूर्ण पश्चिम बंगाल, आसाम एवं उसके ऊपर के सात प्रदेश) में ८० सहस्त्र (हज़ार) से अधिक गुरुकुल है जो की कई सहस्त्र वर्षों से निर्बाधित रूप से चल रहे है"।

उत्तर भारत एवं दक्षिण भारत के आकडों के कुल पर औसत निकलने से यह ज्ञात होता है की भारत में १८ वी शताब्दी तक ३०० व्यक्तियों पर न्यूनतम एक गुरुकुल था। एक और चौकानें वाला तथ्य यह है की १८ शताब्दी में भारत की जनसंख्या लगभग २० करोड़ थी, ३०० व्यक्तियों पर न्यूनतम एक गुरुकुल के अनुसार भारत में ७ लाख ३२ सहस्त्र गुरुकुल होने चाहिए। अब रोचक बात यह भी है की अंग्रेज प्रत्येक दस वर्ष में भारत में भारत का सर्वेक्षण करवाते थे उसे के अनुसार १८२२ के लगभग भारत में कुल गांवों की संख्या भी लगभग ७ लाख ३२ सहस्त्र थी, अर्थात प्रत्येक गाँव में एक गुरुकुल। १६ से १७ वर्ष भारत में प्रवास करने वाले शिक्षाशास्त्री लुडलो ने भी १८ वी शताब्दी में यहीं लिखा की "भारत में एक भी गाँव ऐसा नहीं जिसमें गुरुकुल नहीं एवं एक भी बालक ऐसा नहीं जो गुरुकुल जाता नहीं"।

राजा की सहायता के अपितु, समाज से पोषित इन्ही गुरुकुलों के कारण १८ शताब्दी तक भारत में साक्षरता ९७% थी, बालक के ५ वर्ष, ५ माह, ५ दिवस के होते ही उसका गुरुकुल में प्रवेश हो जाता था। प्रतिदिन सूर्योदय से सूर्यास्त तक विद्यार्जन का क्रम १४ वर्ष तक चलता था। जब बालक सभी वर्गों के बालको के साथ निशुल्कः २० से अधिक विषयों का अध्यन कर गुरुकुल से निकलता था। तब आत्मनिर्भर, देश एवं समाज सेवा हेतु सक्षम हो जाता था।

इसके उपरांत विशेषज्ञता (पांडित्य) प्राप्त करने हेतु भारत में विभिन्न विषयों वाले जैसे शल्य चिकित्सा, आयुर्वेद, धातु कर्म आदि के विश्वविद्यालय थे, नालंदा एवं तक्षशिला तो २००० वर्ष पूर्व के है परंतु मात्र १५०-१७० वर्ष पूर्व भी भारत में ५००-५२५ के लगभग विश्वविद्यालय थे। थोमस बेबिगटन मैकोले (टी.बी.मैकोले) जिन्हें पहले हमने विराम दिया था जब सन १८३४ आये तो कई वर्षों भारत में यात्राएँ एवं सर्वेक्षण करने के उपरांत समझ गए की अंग्रेजो पहले के आक्रांताओ अर्थात यवनों, मुगलों आदि भारत के राजाओं, संपदाओं एवं धर्म का नाश करने की जो भूल की है, उससे पुण्यभूमि भारत कदापि पददलित नहीं किया जा सकेगा, अपितु संस्कृति, शिक्षा एवं सभ्यता का नाश करे तो इन्हें पराधीन करने का हेतु सिद्ध हो सकता है। इसी कारण "इंडियन एज्यूकेशन एक्ट" बना कर समस्त गुरुकुल बंद करवाए गए। हमारे शासन एवं शिक्षा तंत्र को इसी लक्ष्य से निर्मित किया गया ताकि नकारात्मक विचार, हीनता की भावना, जो विदेशी है वह अच्छा, बिना तर्क किये रटने के बीज आदि बचपन से ही बाल मन में घर कर ले और अंग्रेजो को प्रतिव्यक्ति संस्कृति, शिक्षा एवं सभ्यता का नाश का परिश्रम न करना पड़े।

उस पर से अंग्रेजी कदाचित शिक्षा का माध्यम अंग्रेजी नहीं होती तो इस कुचक्र के पहले अंकुर माता पिता ही पल्लवित होने से रोक लेते परंतु ऐसा हो न सका। हमारे निर्यात कारखाने एवं उत्पाद की कमर तोड़ने हेतु भारत में स्वदेशी वस्तुओं पर अधिकतम कर देना पड़ता था एवं अंग्रेजी वस्तुओं को कर मुक्त कर दिया गया था। कृषकों पर तो ९०% कर लगा कर फसल भी लूट लेते थे एवं "लैंड एक्विजिशन एक्ट" के माध्यम से सहस्त्रो एकड़ भूमि भी उनसे छीन ली जाती थी, अंग्रेजो ने कृषकों के कार्यों में सहायक गौ माता एवं भैसों आदि को काटने हेतु पहली बार कलकत्ता में कसाईघर चालू कर दिया, लाज की बात है वह अभी भी चल रहा है। सत्ता हस्तांतरण के दिवस (१५-८-१९४७ ) के उपरांत तो इस कुचक्र की गोरे अंग्रेजो पर निर्भरता भी समाप्त हो गई, अब तो इसे निर्बाधित रूप से चलने देने के लिए बिना रीढ़ के काले अंग्रेज भी पर्याप्त थे, जिनमें साहस ही नहीं है भारत को उसके पूर्व स्थान पर पहुँचाने का |


"दुर्भाग्य है की भारत में हम अपने श्रेष्ठतम सृजनात्मक पुरुषों को भूल चुके है। इसका कारण विदेशियत का प्रभाव और अपने बारे में हीनता बोध की मानसिक ग्रंथि से देश के बुद्धिमान लोग ग्रस्त है" – डॉ.कलाम, "भारत २०२० : सहस्त्राब्दी"


आप सोच रहे होंगे उस समय अमेरिका यूरोप की क्या स्थिति थी, तो सामान्य बच्चों के लिए सार्वजानिक विद्यालयों की शुरुआत सबसे पहले इंग्लैण्ड में सन १८६८ में हुई थी, उसके बाद बाकी यूरोप अमेरिका में अर्थात जब भारत में प्रत्येक गाँव में एक गुरुकुल था, ९७ % साक्षरता थी तब इंग्लैंड के बच्चों को पढ़ने का अवसर मिला। तो क्या पहले वहाँ विद्यालय नहीं होते थे? होते थे परंतु महलों के भीतर, वहाँ ऐसी मान्यता थी की शिक्षा केवल राजकीय व्यक्तियों को ही देनी चाहिए बाकी सब को तो सेवा करनी है।
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"जन-जागरण लाना है तो पोस्ट को Share करना है।"

Ojasvi Hindustan पेज के साथ जुड़े। जानीये हमसे जुड़ने का तरीका ।
लिंक :- http://goo.gl/G0ZZIu
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भारत में ७ लाख ३२ हज़ार गुरुकुल एवं विज्ञान की २० से अधिक शाखाए थी

"भारत का स्वर्णिम अतीत" से आगे : अब बात आती है की भारत में विज्ञान पर इतना शोध किस प्रकार होता था, तो इसके मूल में है भारतीयों की जिज्ञासा एवं तार्किक क्षमता, जो अतिप्राचीन उत्कृष्ट शिक्षा तंत्र एवं अध्यात्मिक मूल्यों की देन है। "गुरुकुल" के बारे में बहुत से लोगों को यह भ्रम है की वहाँ केवल संस्कृत की शिक्षा दी जाती थी जो की गलत है। भारत में विज्ञान की २० से अधिक शाखाएं रही है जो की बहुत पुष्पित पल्लवित रही है जिसमें प्रमुख १. खगोल शास्त्र २. नक्षत्र शास्त्र ३. बर्फ़ बनाने का विज्ञान ४. धातु शास्त्र ५. रसायन शास्त्र ६. स्थापत्य शास्त्र ७. वनस्पति विज्ञान ८. नौका शास्त्र ९. यंत्र विज्ञान आदि इसके अतिरिक्त शौर्य (युद्ध) शिक्षा आदि कलाएँ भी प्रचुरता में रही है। संस्कृत भाषा मुख्यतः माध्यम के रूप में, उपनिषद एवं वेद छात्रों में उच्चचरित्र एवं संस्कार निर्माण हेतु पढ़ाए जाते थे।

थोमस मुनरो सन १८१३ के आसपास मद्रास प्रांत के राज्यपाल थे, उन्होंने अपने कार्य विवरण में लिखा है मद्रास प्रांत (अर्थात आज का पूर्ण आंद्रप्रदेश, पूर्ण तमिलनाडु, पूर्ण केरल एवं कर्णाटक का कुछ भाग ) में ४०० लोगो पर न्यूनतम एक गुरुकुल है। उत्तर भारत (अर्थात आज का पूर्ण पाकिस्तान, पूर्ण पंजाब, पूर्ण हरियाणा, पूर्ण जम्मू कश्मीर, पूर्ण हिमाचल प्रदेश, पूर्ण उत्तर प्रदेश, पूर्ण उत्तराखंड) के सर्वेक्षण के आधार पर जी.डब्लू.लिटनेर ने सन १८२२ में लिखा है, उत्तर भारत में २०० लोगो पर न्यूनतम एक गुरुकुल है। माना जाता है की मैक्स मूलर ने भारत की शिक्षा व्यवस्था पर सबसे अधिक शोध किया है, वे लिखते है "भारत के बंगाल प्रांत (अर्थात आज का पूर्ण बिहार, आधा उड़ीसा, पूर्ण पश्चिम बंगाल, आसाम एवं उसके ऊपर के सात प्रदेश) में ८० सहस्त्र (हज़ार) से अधिक गुरुकुल है जो की कई सहस्त्र वर्षों से निर्बाधित रूप से चल रहे है"।

उत्तर भारत एवं दक्षिण भारत के आकडों के कुल पर औसत निकलने से यह ज्ञात होता है की भारत में १८ वी शताब्दी तक ३०० व्यक्तियों पर न्यूनतम एक गुरुकुल था। एक और चौकानें वाला तथ्य यह है की १८ शताब्दी में भारत की जनसंख्या लगभग २० करोड़ थी, ३०० व्यक्तियों पर न्यूनतम एक गुरुकुल के अनुसार भारत में ७ लाख ३२ सहस्त्र गुरुकुल होने चाहिए। अब रोचक बात यह भी है की अंग्रेज प्रत्येक दस वर्ष में भारत में भारत का सर्वेक्षण करवाते थे उसे के अनुसार १८२२ के लगभग भारत में कुल गांवों की संख्या भी लगभग ७ लाख ३२ सहस्त्र थी, अर्थात प्रत्येक गाँव में एक गुरुकुल। १६ से १७ वर्ष भारत में प्रवास करने वाले शिक्षाशास्त्री लुडलो ने भी १८ वी शताब्दी में यहीं लिखा की "भारत में एक भी गाँव ऐसा नहीं जिसमें गुरुकुल नहीं एवं एक भी बालक ऐसा नहीं जो गुरुकुल जाता नहीं"।

राजा की सहायता के अपितु, समाज से पोषित इन्ही गुरुकुलों के कारण १८ शताब्दी तक भारत में साक्षरता ९७% थी, बालक के ५ वर्ष, ५ माह, ५ दिवस के होते ही उसका गुरुकुल में प्रवेश हो जाता था। प्रतिदिन सूर्योदय से सूर्यास्त तक विद्यार्जन का क्रम १४ वर्ष तक चलता था। जब बालक सभी वर्गों के बालको के साथ निशुल्कः २० से अधिक विषयों का अध्यन कर गुरुकुल से निकलता था। तब आत्मनिर्भर, देश एवं समाज सेवा हेतु सक्षम हो जाता था।

इसके उपरांत विशेषज्ञता (पांडित्य) प्राप्त करने हेतु भारत में विभिन्न विषयों वाले जैसे शल्य चिकित्सा, आयुर्वेद, धातु कर्म आदि के विश्वविद्यालय थे, नालंदा एवं तक्षशिला तो २००० वर्ष पूर्व के है परंतु मात्र १५०-१७० वर्ष पूर्व भी भारत में ५००-५२५ के लगभग विश्वविद्यालय थे। थोमस बेबिगटन मैकोले (टी.बी.मैकोले) जिन्हें पहले हमने विराम दिया था जब सन १८३४ आये तो कई वर्षों भारत में यात्राएँ एवं सर्वेक्षण करने के उपरांत समझ गए की अंग्रेजो पहले के आक्रांताओ अर्थात यवनों, मुगलों आदि भारत के राजाओं, संपदाओं एवं धर्म का नाश करने की जो भूल की है, उससे पुण्यभूमि भारत कदापि पददलित नहीं किया जा सकेगा, अपितु संस्कृति, शिक्षा एवं सभ्यता का नाश करे तो इन्हें पराधीन करने का हेतु सिद्ध हो सकता है। इसी कारण "इंडियन एज्यूकेशन एक्ट" बना कर समस्त गुरुकुल बंद करवाए गए। हमारे शासन एवं शिक्षा तंत्र को इसी लक्ष्य से निर्मित किया गया ताकि नकारात्मक विचार, हीनता की भावना, जो विदेशी है वह अच्छा, बिना तर्क किये रटने के बीज आदि बचपन से ही बाल मन में घर कर ले और अंग्रेजो को प्रतिव्यक्ति संस्कृति, शिक्षा एवं सभ्यता का नाश का परिश्रम न करना पड़े।

उस पर से अंग्रेजी कदाचित शिक्षा का माध्यम अंग्रेजी नहीं होती तो इस कुचक्र के पहले अंकुर माता पिता ही पल्लवित होने से रोक लेते परंतु ऐसा हो न सका। हमारे निर्यात कारखाने एवं उत्पाद की कमर तोड़ने हेतु भारत में स्वदेशी वस्तुओं पर अधिकतम कर देना पड़ता था एवं अंग्रेजी वस्तुओं को कर मुक्त कर दिया गया था। कृषकों पर तो ९०% कर लगा कर फसल भी लूट लेते थे एवं "लैंड एक्विजिशन एक्ट" के माध्यम से सहस्त्रो एकड़ भूमि भी उनसे छीन ली जाती थी, अंग्रेजो ने कृषकों के कार्यों में सहायक गौ माता एवं भैसों आदि को काटने हेतु पहली बार कलकत्ता में कसाईघर चालू कर दिया, लाज की बात है वह अभी भी चल रहा है। सत्ता हस्तांतरण के दिवस (१५-८-१९४७ ) के उपरांत तो इस कुचक्र की गोरे अंग्रेजो पर निर्भरता भी समाप्त हो गई, अब तो इसे निर्बाधित रूप से चलने देने के लिए बिना रीढ़ के काले अंग्रेज भी पर्याप्त थे, जिनमें साहस ही नहीं है भारत को उसके पूर्व स्थान पर पहुँचाने का |


"दुर्भाग्य है की भारत में हम अपने श्रेष्ठतम सृजनात्मक पुरुषों को भूल चुके है। इसका कारण विदेशियत का प्रभाव और अपने बारे में हीनता बोध की मानसिक ग्रंथि से देश के बुद्धिमान लोग ग्रस्त है" – डॉ.कलाम, "भारत २०२० : सहस्त्राब्दी"


आप सोच रहे होंगे उस समय अमेरिका यूरोप की क्या स्थिति थी, तो सामान्य बच्चों के लिए सार्वजानिक विद्यालयों की शुरुआत सबसे पहले इंग्लैण्ड में सन १८६८ में हुई थी, उसके बाद बाकी यूरोप अमेरिका में अर्थात जब भारत में प्रत्येक गाँव में एक गुरुकुल था, ९७ % साक्षरता थी तब इंग्लैंड के बच्चों को पढ़ने का अवसर मिला। तो क्या पहले वहाँ विद्यालय नहीं होते थे? होते थे परंतु महलों के भीतर, वहाँ ऐसी मान्यता थी की शिक्षा केवल राजकीय व्यक्तियों को ही देनी चाहिए बाकी सब को तो सेवा करनी है।


Tuesday, January 28, 2014

The Bhimbetka cave painting-CIVILIZATION MORE THAN >100,000 YRS AGO.

Photo: The Bhimbetka cave painting..


The Bhimbetka shelters exhibit the earliest traces of human life in India. A number of analyses suggest that some of these shelters are more than 100,000 years ago. Some of the Stone Age rock paintings found among the Bhimbetka rock shelters are approximately 30,000 years old 

 Bhimbetka owes its name to the character from the epic Mahabharata. It is believed that when the five brothers, called Pandavas, were banished from their kingdom, they came here and stayed in these caves, the massive rocks seating the gigantic frame of Bhima, the second Pandava. However, these claim still remains to be corroborated with concrete evidence.

The Rock Shelters of Bhimbetka is a World Heritage Site. Bhimbetka was first mentioned in Indian archeological records in 1888 as a Buddhist site, based on information gathered from local tribes. The caves were eventually discovered in 1957-58 by accident. An archaeologist from Ujjain, Dr. Vishnu Wakankar, strayed too far from the beaten path and found himself amidst this prehistoric treasure trove.
Since then more than 700 such shelters have been identified, of which 243 are in the Bhimbetka group and 178 in the Lakha Juar group. Archeological studies revealed a continuous sequence of Stone Age cultures (from the late Acheulian to the late Mesolithic ). It also has the world’s oldest stonewalls and floors. The earliest paintings on the cave walls are believed to be of the Mesolithic  period. A broad chronology of the finds has been done, but a detailed chronology is yet to be created.

 Executed mainly in red and white, with the occasional use of green and yellow with themes taken from the everyday events, the scenes usually depict hunting, childbirth, communal dancing, drinking, religious rites, burials, horse and elephant riders, animal fights, honey collection, decoration of bodies, disguises, masks and different type of animals etc. It depicts the detail of social life during the long period of time, when man used to frequent these rock shelters. Animals such as bison, tiger, rhinoceros, wild boar, elephants, monkeys, antelopes, lizards, peacocks etc. have been abundantly depicted. One rock, popularly referred to as “Zoo Rock”, depicts elephants, sambar, bison and deer.

 It is a marvel that the paintings have not faded even after thousands of years. The colors used by the cave dwellers were prepared by combining manganese, hematite, soft red stone and wooden charcoal. Perhaps, animal fat and extracts of leaves, vegetables, and roots were also used in the mixture. Brushes were made of pieces of fibrous plants. The natural pigments have endured through time because the drawings are generally made deep inside a niche or on inner walls. The oldest paintings are considered to be 30,000 years old, but some of the geometric figures date to as recently as the medieval period.

The rock art of Bhimbetka has been classified into various groups on the basis of the style and subject. The superimposition of paintings shows that different people used the same canvas at different times. The drawings and paintings can be classified under seven different periods.

http://www.sandhyamanne.com/blogs/bhimbetka-cave-paintings

http://www.thehindu.com/features/magazine/rocks-of-ages/article5075148.ece
https://www.facebook.com/ANCIENTINDIANTECHNOLOGY
 
 
The Bhimbetka cave painting..
 
BHIMBEDKA CAVE

The Bhimbetka shelters exhibit the earliest traces of human life in India. A number of analyses suggest that some of these shelters are more than 100,000 years ago. Some of the Stone Age rock paintings found ...among the Bhimbetka rock shelters are approximately 30,000 years old

Bhimbetka owes its name to the character from the epic Mahabharata. It is believed that when the five brothers, called Pandavas, were banished from their kingdom, they came here and stayed in these caves, the massive rocks seating the gigantic frame of Bhima, the second Pandava. However, these claim still remains to be corroborated with concrete evidence.

The Rock Shelters of Bhimbetka is a World Heritage Site. Bhimbetka was first mentioned in Indian archeological records in 1888 as a Buddhist site, based on information gathered from local tribes. The caves were eventually discovered in 1957-58 by accident. An archaeologist from Ujjain, Dr. Vishnu Wakankar, strayed too far from the beaten path and found himself amidst this prehistoric treasure trove.
Since then more than 700 such shelters have been identified, of which 243 are in the Bhimbetka group and 178 in the Lakha Juar group. Archeological studies revealed a continuous sequence of Stone Age cultures (from the late Acheulian to the late Mesolithic ). It also has the world’s oldest stonewalls and floors. The earliest paintings on the cave walls are believed to be of the Mesolithic period. A broad chronology of the finds has been done, but a detailed chronology is yet to be created.

Executed mainly in red and white, with the occasional use of green and yellow with themes taken from the everyday events, the scenes usually depict hunting, childbirth, communal dancing, drinking, religious rites, burials, horse and elephant riders, animal fights, honey collection, decoration of bodies, disguises, masks and different type of animals etc. It depicts the detail of social life during the long period of time, when man used to frequent these rock shelters. Animals such as bison, tiger, rhinoceros, wild boar, elephants, monkeys, antelopes, lizards, peacocks etc. have been abundantly depicted. One rock, popularly referred to as “Zoo Rock”, depicts elephants, sambar, bison and deer.

It is a marvel that the paintings have not faded even after thousands of years. The colors used by the cave dwellers were prepared by combining manganese, hematite, soft red stone and wooden charcoal. Perhaps, animal fat and extracts of leaves, vegetables, and roots were also used in the mixture. Brushes were made of pieces of fibrous plants. The natural pigments have endured through time because the drawings are generally made deep inside a niche or on inner walls. The oldest paintings are considered to be 30,000 years old, but some of the geometric figures date to as recently as the medieval period.

The rock art of Bhimbetka has been classified into various groups on the basis of the style and subject. The superimposition of paintings shows that different people used the same canvas at different times. The drawings and paintings can be classified under seven different periods.
bhimbetka
 

 

Aryabhata-GREAT INDIAN MATHMATICIAN,ASTROLOGER

Photo: Ancient Indian Technology


Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed to believe that there were two different mathematicians called Aryabhata living at the same time. He therefore created a confusion of two different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.

We know the year of Aryabhata's birth since he tells us that he was twenty-three years of age when he wrote Aryabhatiya which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.

We do know that Aryabhata wrote Aryabhatiya in Kusumapura at the time when Pataliputra was the capital of the Gupta empire and a major centre of learning, but there have been numerous other places proposed by historians as his birthplace. Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although the author accepted that he lived most of his life in Kusumapura in the Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the Aryabhatiya.


Continuation :

http://www-history.mcs.st-and.ac.uk/Printonly/Aryabhata_I.html


https://www.facebook.com/ANCIENTINDIANTECHNOLOGY

Aryabhata is also known as Aryabhata I to distinguish him from the later mathematician of the same name who lived about 400 years later. Al-Biruni has not helped in understanding Aryabhata's life, for he seemed t...o believe that there were two different mathematicians called Aryabhata living at the same time. He therefore created a confusion of two different Aryabhatas which was not clarified until 1926 when B Datta showed that al-Biruni's two Aryabhatas were one and the same person.

We know the year of Aryabhata's birth since he tells us that he was twenty-three years of age when he wrote Aryabhatiya which he finished in 499. We have given Kusumapura, thought to be close to Pataliputra (which was refounded as Patna in Bihar in 1541), as the place of Aryabhata's birth but this is far from certain, as is even the location of Kusumapura itself. As Parameswaran writes in [26]:-

... no final verdict can be given regarding the locations of Asmakajanapada and Kusumapura.

We do know that Aryabhata wrote Aryabhatiya in Kusumapura at the time when Pataliputra was the capital of the Gupta empire and a major centre of learning, but there have been numerous other places proposed by historians as his birthplace. Some conjecture that he was born in south India, perhaps Kerala, Tamil Nadu or Andhra Pradesh, while others conjecture that he was born in the north-east of India, perhaps in Bengal. In [8] it is claimed that Aryabhata was born in the Asmaka region of the Vakataka dynasty in South India although the author accepted that he lived most of his life in Kusumapura in the Gupta empire of the north. However, giving Asmaka as Aryabhata's birthplace rests on a comment made by Nilakantha Somayaji in the late 15th century. It is now thought by most historians that Nilakantha confused Aryabhata with Bhaskara I who was a later commentator on the Aryabhatiya.
 Aryabhata

Madhava--GREAT INDIAN MATHMATICIAN

Madhava of Sangamagramma was born near Cochin on the coast in the Kerala state in southwestern India. It is only due to research into Keralese mathematics over the last twenty-five years that the remarkable cont...ributions of Madhava have come to light. In Rajagopal and Rangachari put his achievement into context when they write:-

[Madhava] took the decisive step onwards from the finite procedures of ancient mathematics to treat their limit-passage to infinity, which is the kernel of modern classical analysis.

All the mathematical writings of Madhava have been lost, although some of his texts on astronomy have survived. However his brilliant work in mathematics has been largely discovered by the reports of other Keralese mathematicians such as Nilakantha who lived about 100 years later.

Madhava discovered the series equivalent to the Maclaurin expansions of sin x, cos x, and arctan x around 1400, which is over two hundred years before they were rediscovered in Europe. Details appear in a number of works written by his followers such as Mahajyanayana prakara which means Method of computing the great sines. In fact this work had been claimed by some historians such as Sarma (see for example [2]) to be by Madhava himself but this seems highly unlikely and it is now accepted by most historians to be a 16th century work by a follower of Madhava. This is discussed in detail in [4].

Jyesthadeva wrote Yukti-Bhasa in Malayalam, the regional language of Kerala, around 1550. In [9] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:-

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. Perhaps we should write down in modern symbols exactly what the series is that Madhava has found. The first thing to note is that the Indian meaning for sine of θ would be written in our notation as r sin θ and the Indian cosine of would be r cos θ in our notation, where r is the radius. Thus the series is

r θ = r(r sin θ)/1(r cos θ) - r(r sin θ)3/3r(r cos θ)3 + r(r sin θ)5/5r(r cos θ)5- r(r sin θ)7/7r(r cos θ)7 + ...

putting tan = sin/cos and cancelling r gives

θ = tan θ - (tan3θ)/3 + (tan5θ)/5 - ...

which is equivalent to Gregory's series

tan-1θ = θ - θ3/3 + θ5/5 - ...

Now Madhava put q = π/4 into his series to obtain

π/4 = 1 - 1/3 + 1/5 - ...

and he also put θ = π/6 into his series to obtain

π = √12(1 - 1/(3×3) + 1/(5×32) - 1/(7×33) + ...

We know that Madhava obtained an approximation for π correct to 11 decimal places when he gave

π = 3.14159265359

which can be obtained from the last of Madhava's series above by taking 21 terms. In [5] Gupta gives a translation of the Sanskrit text giving Madhava's approximation of π correct to 11 places.

Perhaps even more impressive is the fact that Madhava gave a remainder term for his series which improved the approximation. He improved the approximation of the series for π/4 by adding a correction term Rn to obtain

π/4 = 1 - 1/3 + 1/5 - ... 1/(2n-1) ± Rn

Madhava gave three forms of Rn which improved the approximation, namely

Rn = 1/(4n) or
Rn = n/(4n2 + 1) or
Rn = (n2 + 1)/(4n3 + 5n).

There has been a lot of work done in trying to reconstruct how Madhava might have found his correction terms. The most convincing is that they come as the first three convergents of a continued fraction which can itself be derived from the standard Indian approximation to π namely 62832/20000.

Madhava also gave a table of almost accurate values of half-sine chords for twenty-four arcs drawn at equal intervals in a quarter of a given circle. It is thought that the way that he found these highly accurate tables was to use the equivalent of the series expansions

sin θ = θ - θ3/3! + θ5/5! - ...

cos θ = 1 - θ2/2! + θ4/4! - ...

Jyesthadeva in Yukti-Bhasa gave an explanation of how Madhava found his series expansions around 1400 which are equivalent to these modern versions rediscovered by Newton around 1676. Historians have claimed that the method used by Madhava amounts to term by term integration.

Rajagopal's claim that Madhava took the decisive step towards modern classical analysis seems very fair given his remarkable achievements. In the same vein Joseph writes in :-

We may consider Madhava to have been the founder of mathematical analysis. Some of his discoveries in this field show him to have possessed extraordinary intuition, making him almost the equal of the more recent intuitive genius Srinivasa Ramanujan, who spent his childhood and youth at Kumbakonam, not far from Madhava's birthplace.

PANINI- GREAT INDIANS/BHARTIYA


Panini was born in Shalatula, a town near to Attock on the Indus river in present day Pakistan. The dates given for Panini are pure guesses. Experts give dates in the 4th, 5th, 6th and 7th century BC and there is... also no agreement among historians about the extent of the work which he undertook. What is in little doubt is that, given the period in which he worked, he is one of the most innovative people in the whole development of knowledge. We will say a little more below about how historians have gone about trying to pinpoint the date when Panini lived.

Panini was a Sanskrit grammarian who gave a comprehensive and scientific theory of phonetics, phonology, and morphology. Sanskrit was the classical literary language of the Indian Hindus and Panini is considered the founder of the language and literature. It is interesting to note that the word "Sanskrit" means "complete" or "perfect" and it was thought of as the divine language, or language of the gods.

A treatise called Astadhyayi (or Astaka ) is Panini's major work. It consists of eight chapters, each subdivided into quarter chapters. In this work Panini distinguishes between the language of sacred texts and the usual language of communication. Panini gives formal production rules and definitions to describe Sanskrit grammar. Starting with about 1700 basic elements like nouns, verbs, vowels, consonants he put them into classes. The construction of sentences, compound nouns etc. is explained as ordered rules operating on underlying structures in a manner similar to modern theory. In many ways Panini's constructions are similar to the way that a mathematical function is defined today. Joseph writes in [2]:-

[Sanskrit's] potential for scientific use was greatly enhanced as a result of the thorough systemisation of its grammar by Panini. ... On the basis of just under 4000 sutras [rules expressed as aphorisms], he built virtually the whole structure of the Sanskrit language, whose general 'shape' hardly changed for the next two thousand years. ... An indirect consequence of Panini's efforts to increase the linguistic facility of Sanskrit soon became apparent in the character of scientific and mathematical literature. This may be brought out by comparing the grammar of Sanskrit with the geometry of Euclid - a particularly apposite comparison since, whereas mathematics grew out of philosophy in ancient Greece, it was ... partly an outcome of linguistic developments in India.

Joseph goes on to make a convincing argument for the algebraic nature of Indian mathematics arising as a consequence of the structure of the Sanskrit language. In particular he suggests that algebraic reasoning, the Indian way of representing numbers by words, and ultimately the development of modern number systems in India, are linked through the structure of language.

Panini should be thought of as the forerunner of the modern formal language theory used to specify computer languages. The Backus Normal Form was discovered independently by John Backus in 1959, but Panini's notation is equivalent in its power to that of Backus and has many similar properties. It is remarkable to think that concepts which are fundamental to today's theoretical computer science should have their origin with an Indian genius around 2500 years ago.

At the beginning of this article we mentioned that certain concepts had been attributed to Panini by certain historians which others dispute. One such theory was put forward by B Indraji in 1876. He claimed that the Brahmi numerals developed out of using letters or syllables as numerals. Then he put the finishing touches to the theory by suggesting that Panini in the eighth century BC (earlier than most historians place Panini) was the first to come up with the idea of using letters of the alphabet to represent numbers.

There are a number of pieces of evidence to support Indraji's theory that the Brahmi numerals developed from letters or syllables. However it is not totally convincing since, to quote one example, the symbols for 1, 2 and 3 clearly do not come from letters but from one, two and three lines respectively. Even if one accepts the link between the numerals and the letters, making Panini the originator of this idea would seem to have no more behind it than knowing that Panini was one of the most innovative geniuses that world has known so it is not unreasonable to believe that he might have made this step too.

There are other works which are closely associated with the Astadhyayi which some historians attribute to Panini, others attribute to authors before Panini, others attribute to authors after Panini. This is an area where there are many theories but few, if any, hard facts.

We also promised to return to a discussion of Panini's dates. There has been no lack of work on this topic so the fact that there are theories which span several hundreds of years is not the result of lack of effort, rather an indication of the difficulty of the topic. The usual way to date such texts would be to examine which authors are referred to and which authors refer to the work. One can use this technique and see who Panini mentions.

There are ten scholars mentioned by Panini and we must assume from the context that these ten have all contributed to the study of Sanskrit grammar. This in itself, of course, indicates that Panini was not a solitary genius but, like Newton, had "stood on the shoulders of giants". Panini must have lived later than these ten but this is absolutely no help in providing dates since we have absolutely no knowledge of when any of these ten lived.

What other internal evidence is there to use? Well of course Panini uses many phrases to illustrate his grammar any these have been examined meticulously to see if anything is contained there to indicate a date. To give an example of what we mean: if we were to pick up a text which contained as an example "I take the train to work every day" we would know that it had to have been written after railways became common. Let us illustrate with two actual examples from the Astadhyayi which have been the subject of much study. The first is an attempt to see whether there is evidence of Greek influence. Would it be possible to find evidence which would mean that the text had to have been written after the conquests of Alexander the Great? There is a little evidence of Greek influence, but there was Greek influence on this north east part of the Indian subcontinent before the time of Alexander. Nothing conclusive has been identified.

Another angle is to examine a reference Panini makes to nuns. Some argue that these must be Buddhist nuns and therefore the work must have been written after Buddha. A nice argument but there is a counter argument which says that there were Jaina nuns before the time of Buddha and Panini's reference could equally well be to them. Again the evidence is inconclusive.

There are references by others to Panini. However it would appear that the Panini to whom most refer is a poet and although some argue that these are the same person, most historians agree that the linguist and the poet are two different people. Again this is inconclusive evidence.

Let us end with an evaluation of Panini's contribution by Cardona in [1]:-

Panini's grammar has been evaluated from various points of view. After all these different evaluations, I think that the grammar merits asserting ... that it is one of the greatest monuments of human intelligence.
 

JYESTHADEVA- GREAT INDIAN MATHMATICIAN

Jyesthadeva lived on the southwest coast of India in the district of Kerala. He belonged to the Kerala school of mathematics built on the work of Madhava, Nilakantha Somayaji, Paramesvara and others.

Jyesthadeva... wrote a famous text Yuktibhasa which he wrote in Malayalam, the regional language of Kerala. The work is a survey of Kerala mathematics and, very unusually for an Indian mathematical text, it contains proofs of the theorems and gives derivations of the rules it contains. It is one of the main astronomical and mathematical texts produced by the Kerala school. The work was based mainly on the Tantrasamgraha of Nilakantha.

The Yuktibhasa is a major treatise, half on astronomy and half on mathematics, written in 1501. The Tantrasamgraha on which it is based consists of 432 Sanskrit verses divided into 8 chapters, and it covers various aspects of Indian astronomy. It is based on the epicyclic and eccentric models of planetary motion. The first two chapters deal with the motions and longitudes of the planets. The third chapter Treatise on shadow deals with various problems related with the sun's position on the celestial sphere, including the relationships of its expressions in the three systems of coordinates, namely ecliptic, equatorial and horizontal coordinates.

The fourth and fifth chapters are Treatise on the lunar eclipse and On the solar eclipse and these two chapters treat various aspects of the eclipses of the sun and the moon. The sixth chapter is On vyatipata and deals with the complete deviation of the longitudes of the sun and the moon. The seventh chapter On visibility computation discusses the rising and setting of the moon and planets. The final chapter On elevation of the lunar cusps examines the size of the part of the moon which is illuminated by the sun and gives a graphical representation of it.

The Yuktibhasa is very important in terms of the mathematics Jyesthadeva presents. In particular he presents results discovered by Madhava and the treatise is an important source of the remarkable mathematical theorems which Madhava discovered. Written in about 1550, Jyesthadeva's commentary contained proofs of the earlier results by Madhava and Nilakantha which these earlier authors did not give. In [4] Gupta gives a translation of the text and this is also given in [2] and a number of other sources. Jyesthadeva describes Madhava's series as follows:-

The first term is the product of the given sine and radius of the desired arc divided by the cosine of the arc. The succeeding terms are obtained by a process of iteration when the first term is repeatedly multiplied by the square of the sine and divided by the square of the cosine. All the terms are then divided by the odd numbers 1, 3, 5, .... The arc is obtained by adding and subtracting respectively the terms of odd rank and those of even rank. It is laid down that the sine of the arc or that of its complement whichever is the smaller should be taken here as the given sine. Otherwise the terms obtained by this above iteration will not tend to the vanishing magnitude.

This is a remarkable passage describing Madhava's series, but remember that even this passage by Jyesthadeva was written more than 100 years before James Gregory rediscovered this series expansion. To see how this description of the series fits with Gregory's series for arctan(x) see the biography of Madhava. Other mathematical results presented by Jyesthadeva include topics studied by earlier Indian mathematicians such as integer solutions of systems of first degree equation solved by the kuttaka method, and rules of finding the sines and the cosines of the sum and difference of two angles.

Not only does the mathematics anticipate work by European mathematicians a century later, but the planetary theory presented by Jyesthadeva is similar to that adopted by Tycho Brahe.
 

SAGE CHARAK AND CHARAK SAMHITA

Photo: Acharya Charak

 Acharya Charak was one of the principal contributors to the ancient art and science of Ayurveda, a system of medicine and lifestyle developed in Ancient India. Acharya Charak has been crowned as the Father of Medicine. His renowned work, the “Charak Samhita“, is considered as an encyclopedia of Ayurveda. His principles, diagoneses, and cures retain their potency and truth even after a couple of millennia. When the science of anatomy was confused with different theories in Europe, Acharya Charak revealed through his innate genius and enquiries the facts on human anatomy, embryology, pharmacology, blood circulation and diseases like diabetes, tuberculosis, heart disease, etc.

The following statements are attributed to Acharya Charak:

    “A physician who fails to enter the body of a patient with the lamp of knowledge and understanding can never treat diseases. He should first study all the factors, including environment, which influence a patient’s disease, and then prescribe treatment. It is more important to prevent the occurrence of disease than to seek a cure.”

These remarks appear obvious today, though they were often not heeded, and were made by Charak, in his famous Ayurvedic treatise Charak Samhita. The treatise contains many such remarks which are held in reverence even today. Some of them are in the fields of physiology, etiology and embryology.

In the “Charak Samhita” he has described the medicinal qualities and functions of 100,000 herbal plants. He has emphasized the influence of diet and activity on mind and body. He has proved the correlation of spirituality and physical health contributed greatly to diagnostic and curative sciences. He has also prescribed and ethical charter for medical practitioners two centuries prior to the Hippocratic oath. Through his genius and intuition, Acharya Charak made landmark contributions to Ayurvedal. He forever remains etched in the annals of history as one of the greatest and noblest of rishi-scientists.

Under the guidance of the ancient physician Atreya, Agnivesa had written an encyclopedic treatise in the eighth century B.C. However, it was only when Charaka revised this treatise that it gained popularity and came to be known as Charakasamhita. For two millennia it remained a standard work on the subject and was translated into many foreign languages, including Arabic and Latin.

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Acharya Charak was one of the principal contributors to the ancient art and science of Ayurveda, a system of medicine and lifestyle developed in Ancient India. Acharya Charak has been crowned as the Father of Medicine. His ...renowned work, the “Charak Samhita“, is considered as an encyclopedia of Ayurveda. His principles, diagoneses, and cures retain their potency and truth even after a couple of millennia. When the science of anatomy was confused with different theories in Europe, Acharya Charak revealed through his innate genius and enquiries the facts on human anatomy, embryology, pharmacology, blood circulation and diseases like diabetes, tuberculosis, heart disease, etc.

The following statements are attributed to Acharya Charak:

“A physician who fails to enter the body of a patient with the lamp of knowledge and understanding can never treat diseases. He should first study all the factors, including environment, which influence a patient’s disease, and then prescribe treatment. It is more important to prevent the occurrence of disease than to seek a cure.”

These remarks appear obvious today, though they were often not heeded, and were made by Charak, in his famous Ayurvedic treatise Charak Samhita. The treatise contains many such remarks which are held in reverence even today. Some of them are in the fields of physiology, etiology and embryology.

In the “Charak Samhita” he has described the medicinal qualities and functions of 100,000 herbal plants. He has emphasized the influence of diet and activity on mind and body. He has proved the correlation of spirituality and physical health contributed greatly to diagnostic and curative sciences. He has also prescribed and ethical charter for medical practitioners two centuries prior to the Hippocratic oath. Through his genius and intuition, Acharya Charak made landmark contributions to Ayurvedal. He forever remains etched in the annals of history as one of the greatest and noblest of rishi-scientists.

Under the guidance of the ancient physician Atreya, Agnivesa had written an encyclopedic treatise in the eighth century B.C. However, it was only when Charaka revised this treatise that it gained popularity and came to be known as Charakasamhita. For two millennia it remained a standard work on the subject and was translated into many foreign languages, including Arabic and Latin.
 SAGE CHARAK

Bhaskaracharya- The mathmatician , astrologer.

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Bhaskaracharya – The Crown Jewel of Mathematics and Astronomy


A Glance at the Astronomical Achievements of Bhaskaracharya

1. The Earth is not flat, has no support and has a power of attraction.

2. The north and south poles of the Earth experience six months of day and six months of night.

3. One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.

4. Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars. There is slight difference between the orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

5. Earth’s atmosphere extends to 96 kilometers and has seven parts.

6. There is a vacuum beyond the Earth’s atmosphere.

There have been several unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara's work, but this once again seems like an attempt by European scholars to claim European influence on (all) the great works of mathematics. These claims should be ignored. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.

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Bhaskaracharya – The Crown Jewel of Mathematics and Astronomy


A Glance at the Astronomical Achievements of Bhaskaracharya...

1. The Earth is not flat, has no support and has a power of attraction.

2. The north and south poles of the Earth experience six months of day and six months of night.

3. One day of Moon is equivalent to 15 earth-days and one night is also equivalent to 15 earth-days.

4. Bhaskaracharya had accurately calculated apparent orbital periods of the Sun and orbital periods of Mercury, Venus, and Mars. There is slight difference between the orbital periods he calculated for Jupiter and Saturn and the corresponding modern values.

5. Earth’s atmosphere extends to 96 kilometers and has seven parts.

6. There is a vacuum beyond the Earth’s atmosphere.

There have been several unscrupulous attempts to argue that there are traces of Diophantine influence in Bhaskara's work, but this once again seems like an attempt by European scholars to claim European influence on (all) the great works of mathematics. These claims should be ignored. Particularly in the field of algebra, Diophantus only looked at specific cases and did not achieve the general methods of the Indians.
 The period between 500 and 1200 AD was the golden age of Indian Astronomy. In this long span of time Indian Astronomy flourished mainly due to emi...nent astronomers like Aryabhat, Lallacharya, Varahamihir, Brahmagupta, Bhaskaracharya and others. Bhaskaracharya’s Siddhanta Shiromani is considered as the pinnacle of all the astronomical works of those 700 hundred years. It can be aptly called the “essence” of ancient Indian Astronomy and mathematics. In the ninth century Brahmagupta’s Brahmasphutasiddhanta was translated in Arabic. The title of the translation was ‘Sind Hind’. This translation proved to be a watershed event in the history of numbers. The Arabs quickly grasped the importance of the Indian decimal system of numbers. They played a key role in transmitting this system of numbers to Europeans. For a long time Europeans were using Roman Numerals, which were very tedious to handle. After accepting the decimal system of numbers, European mathematicians made a remarkable progress in mathematics, but that was about 500 years after Bhaskaracharya.
From 750 AD Onwards India was engulfed in waves of foreign attacks. In 1205 AD Bakhtiyar Khilji destroyed the magnificent Nalanda University, which was a renowned center of knowledge for about 800 years. India was in utter chaotic state till the country was colonized by British. All universities and learning centers in India were destroyed, knowledge was lost and hardly any progress was made in mathematics and astronomy. A few scholars like Keshav Daivadnya, Ganesh Daivadnya Madhav, Sawai Jai Singh and others tried to keep the flame of knowledge burning in that dark period.


Birth and Education of Bhaskaracharya :

Ganesh Daivadnya has bestowed a very apt title on Bhaskaracharya. He has called him ‘Ganakchakrachudamani’, which means, ‘a gem among all the calculators of astronomical phenomena.’ Bhaskaracharya himself has written about his birth, his place of residence, his teacher and his education, in Siddhantashiromani as follows,
‘ A place called ‘Vijjadveed’, which is surrounded by Sahyadri ranges, where there are scholars of three Vedas, where all branches of knowledge are studied, and where all kinds of noble people reside, a brahmin called Maheshwar was staying, who was born in Shandilya Gotra (in Hindu religion, Gotra is similar to lineage from a particular person, in this case sage Shandilya), well versed in Shroud (originated from ‘Shut’ or ‘Vedas’) and ‘Smart’ (originated from ‘Smut’) Dharma, respected by all and who was authority in all the branches of knowledge. I acquired knowledge at his feet’.
From this verse it is clear that Bhaskaracharya was a resident of Vijjadveed and his father Maheshwar taught him mathematics and astronomy. Unfortunately today we have no idea where Vijjadveed was located. It is necessary to ardently search this place which was surrounded by the hills of Sahyadri and which was the center of learning at the time of Bhaskaracharya. He writes about his year of birth as follows,
‘I was born in Shake 1036 (1114 AD) and I wrote Siddhanta Shiromani when I was 36 years old.’
Bhaskaracharya has also written about his education. Looking at the knowledge, which he acquired in a span of 36 years, it seems impossible for any modern student to achieve that feat in his entire life. See what Bhaskaracharya writes about his education,
‘I have studied eight books of grammar, six texts of medicine, six books on logic, five books of mathematics, four Vedas, five books on Bharat Shastras, and two Mimansas’.
Bhaskaracharya calls himself a poet and most probably he was Vedanti, since he has mentioned ‘Parambrahman’ in that verse.
 

SIDDHANTASHIROMANI
...
Bhaskaracharya wrote Siddhanta Shiromani in 1150 AD when he was 36 years old. This is a mammoth work containing about 1450 verses. It is divided into four parts, Lilawati, Beejaganit, Ganitadhyaya and Goladhyaya. In fact each part can be considered as separate book. The numbers of verses in each part are as follows,
Lilawati has 278, Beejaganit has 213, Ganitadhyaya has 451 and Goladhyaya has 501 verses.
One of the most important characteristic of Siddhanta Shiromani is, it consists of simple methods of calculations from Arithmetic to Astronomy. Essential knowledge of ancient Indian Astronomy can be acquired by reading only this book. Siddhanta Shiromani has surpassed all the ancient books on astronomy in India. After Bhaskaracharya nobody could write excellent books on mathematics and astronomy in lucid language in India. In India, Siddhanta works used to give no proofs of any theorem. Bhaskaracharya has also followed the same tradition.
Lilawati is an excellent example of how a difficult subject like mathematics can be written in poetic language. Lilawati has been translated in many languages throughout the world. When British Empire became paramount in India, they established three universities in 1857, at Bombay, Calcutta and Madras. Till then, for about 700 years, mathematics was taught in India from Bhaskaracharya’s Lilawati and Beejaganit. No other textbook has enjoyed such long lifespan.

BHASKAR’S MATHEMATICS

Lilawati and Beejaganit together consist of about 500 verses. A few important highlights of Bhaskar’s mathematics are as follows,
Terms for numbers
In English, cardinal numbers are only in multiples of 1000. They have terms such as thousand, million, billion, trillion, quadrillion etc. Most of these have been named recently. However, Bhaskaracharya has given the terms for numbers in multiples of ten and he says that these terms were coined by ancients for the sake
of positional values.

Bhaskar’s terms for numbers are as follows:
eka(1), dasha(10), shata(100), sahastra(1000), ayuta(10,000), laksha(100,000), prayuta (1,000,000=million), koti(107), arbuda(108), abja(109=billion), kharva (1010), nikharva (1011), mahapadma (1012=trillion), shanku(1013), jaladhi(1014), antya(1015=quadrillion), Madhya (1016) and parardha(1017).

Kuttak

Kuttak is nothing but the modern indeterminate equation of first order. The method of solution of such equations was called as ‘pulverizer’ in the western world. Kuttak means to crush to fine particles or to pulverize. There are many kinds of Kuttaks. Let us consider one example.
In the equation, ax + b = cy, a and b are known positive integers. We want to also find out the values of x and y in integers. A particular example is,
100x +90 = 63y
Bhaskaracharya gives the solution of this example as, x = 18, 81, 144, 207… And y=30, 130, 230, 330…
Indian Astronomers used such kinds of equations to solve astronomical problems. It is not easy to find solutions of these equations but Bhaskara has given a generalized solution to get multiple answers.

Chakrawaal

Chakrawaal is the “indeterminate equation of second order” in western mathematics. This type of equation is also called Pell’s equation. Though the equation is recognized by his name Pell had never solved the equation. Much before Pell, the equation was solved by an ancient and eminent Indian mathematician, Brahmagupta (628 AD). The solution is given in his Brahmasphutasiddhanta. Bhaskara modified the method and gave a general solution of this equation. For example, consider the equation 61x2 + 1 = y2. Bhaskara gives the values of x = 22615398 and y = 1766319049
There is an interesting history behind this very equation. The Famous French mathematician Pierre de Fermat (1601-1664) asked his friend Bessy to solve this very equation. Bessy used to solve the problems in his head like present day Shakuntaladevi. Bessy failed to solve the problem. After about 100 years another famous French mathematician solved this problem. But his method is lengthy and could find a particular solution only, while Bhaskara gave the solution for five cases. In his book ‘History of mathematics’, see what Carl Boyer says about this equation,
‘In connection with the Pell’s equation ax2 + 1 = y2, Bhaskara gave particular solutions for five cases, a = 8, 11, 32, 61, and 67, for 61x2 + 1 = y2, for example he gave the solutions, x = 226153980 and y = 1766319049, this is an impressive feat in calculations and its verifications alone will tax the efforts of the reader’
Henceforth the so-called Pell’s equation should be recognized as ‘Brahmagupta-Bhaskaracharya equation’.

Simple mathematical methods

Bhaskara has given simple methods to find the squares, square roots, cube, and cube roots of big numbers. He has proved the Pythagoras theorem in only two lines. The famous Pascal Triangle was Bhaskara’s ‘Khandameru’. Bhaskara has given problems on that number triangle. Pascal was born 500 years after Bhaskara. Several problems on permutations and combinations are given in Lilawati. Bhaskar. He has called the method ‘ankapaash’. Bhaskara has given an approximate value of PI as 22/7 and more accurate value as 3.1416. He knew the concept of infinity and called it as ‘khahar rashi’, which means ‘anant’. It seems that Bhaskara had not notions about calculus, One of his equations in modern notation can be written as, d(sin (w)) = cos (w) dw.

PATANJALI YOGA

ACHARYA PATANJALI FATHER OF YOGA
Photo: Ancient Indian Technology

ACHARYA PATANJALI FATHER OF YOGA



The Science of Yoga is one of several unique contributions of India to the world. It seeks to discover and realize the ultimate Reality through yogic practices. Acharya Patanjali , the founder, hailed from the district of Gonda (Ganara) in Uttar Pradesh . He prescribed the control of prana (life breath) as the means to control the body, mind and soul. This subsequently rewards one with good health and inner happiness. Acharya Patanjali ‘s 84 yogic postures effectively enhance the efficiency of the respiratory, circulatory, nervous, digestive and endocrine systems and many other organs of the body. Yoga has eight limbs where Acharya Patanjali shows the attainment of the ultimate bliss of God in samadhi through the disciplines of: yam, niyam, asan, pranayam, pratyahar, dhyan and dharna. The Science of Yoga has gained popularity because of its scientific approach and benefits. Yoga also holds the honored place as one of six philosophies in the Indian philosophical system. Acharya Patanjali will forever be remembered and revered as a pioneer in the science of self-discipline, happiness and self-realization.


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 The Science of Yoga is one of several unique contributions of India to the world. It seeks to discover and realize the ultimate Reality through yogic practices. Acharya Patanjali , the founder, hailed from the district of Gonda (Ganara) in Uttar Pradesh . He prescribed the control of prana (life breath) as the means to control the body, mind and soul. This subsequently rewards one with good health and inner happiness. Acharya Patanjali ‘s 84 yogic postures effectively enhance the efficiency of the respiratory, circulatory, nervous, digestive and endocrine systems and many other organs of the body. Yoga has eight limbs where Acharya Patanjali shows the attainment of the ultimate bliss of God in samadhi through the disciplines of: yam, niyam, asan, pranayam, pratyahar, dhyan and dharna. The Science of Yoga has gained popularity because of its scientific approach and benefits. Yoga also holds the honored place as one of six philosophies in the Indian philosophical system. Acharya Patanjali will forever be remembered and revered as a pioneer in the science of self-discipline, happiness and self-realization.