Holy Pilgrimage – 29 (Ujjain -2)

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Bhāskara II

Bhāskara[1] (also known as Bhāskara II and Bhāskarāchārya ("Bhāskara the teacher"), (1114–1185), was an Indian mathematician and astronomer. He was born near Vijjadavida (Bijāpur in modern Karnataka). Bhāskara is said to have been the head of an astronomical observatory at Ujjain, the leading mathematical center of ancient India. He lived in the Sahyadri region (Patnadevi, Jalgaon, Maharashtra).[1]
Bhāskara and his works represent a significant contribution to mathematical and astronomical knowledge in the 12th century. He has been called the greatest mathematician of medieval India.[2] His main work Siddhānta Shiromani, (Sanskrit for "Crown of treatises,"[3]) is divided into four parts called Lilāvati, Bijaganita, Grahaganita and Golādhyāya.[4] These four sections deal with arithmetic, algebra, mathematics of the planets, and spheres respectively. He also wrote another treatise named Karan Kautoohal.
Bhāskara's work on calculus predates Newton and Leibniz by half a millennium.[5][6] He is particularly known in the discovery of the principles of differential calculus and its application to astronomical problems and computations. While Newton and Leibniz have been credited with differential and integral calculus, there is strong evidence to suggest that Bhāskara was a pioneer in some of the principles of differential calculus. He was perhaps the first to conceive the differential coefficient and differential calculus

Family

Bhaskaracharya was born into a family belonging to the Deshastha Brahmin community.[8] History records his great-great-great-grandfather holding a hereditary post as a court scholar, as did his son and other descendants. His father Mahesvara[1] was as an astrologer, who taught him mathematics, which he later passed on to his son Loksamudra. Loksamudra's son helped to set up a school in 1207 for the study of Bhāskara's writings.

Mathematics

Some of Bhaskara's contributions to mathematics include the following:
  • Solutions of indeterminate quadratic equations (of the type ax² + b = y²).[citation needed]
  • Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by the Renaissance European mathematicians of the 17th century[citation needed]
  • A cyclic Chakravala method for solving indeterminate equations of the form ax² + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method.[citation needed]
  • The first general method for finding the solutions of the problem x² − ny² = 1 (so-called "Pell's equation") was given by Bhaskara II.[9]
  • Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below.)[citation needed]

Arithmetic

Bhaskara's arithmetic text Lilavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.
Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include:
  • Indeterminate equations (Kuttaka), integer solutions (first and second order)[citation needed]. His contributions to this topic are particularly important[citation needed], since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century[citation needed], yet his work was of the 12th century. Bhaskara's method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians.[citation needed]
His work is outstanding for its systemisation, improved methods and the new topics that he has introduced.[citation needed] Furthermore the Lilavati contained excellent recreative problems and it is thought that Bhaskara's intention may have been that a student of 'Lilavati' should concern himself with the mechanical application of the method.[citation needed]

Algebra

His Bijaganita ("Algebra") was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root)[citation needed]. His work Bijaganita is effectively a treatise on algebra and contains the following topics:
  • Positive and negative numbers.
  • Zero.
  • The 'unknown' (includes determining unknown quantities).
  • Determining unknown quantities.
  • Surds (includes evaluating surds).
  • Kuttaka (for solving indeterminate equations and Diophantine equations).
  • Simple equations (indeterminate of second, third and fourth degree).
  • Simple equations with more than one unknown.
  • Indeterminate quadratic equations (of the type ax² + b = y²).
  • Solutions of indeterminate equations of the second, third and fourth degree.
  • Quadratic equations.
  • Quadratic equations with more than one unknown.
  • Operations with products of several unknowns.
Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax² + bx + c = y[citation needed]. Bhaskara's method for finding the solutions of the problem Nx² + 1 = y² (the so-called "Pell's equation") is of considerable importance.[9]

Trigonometry

The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara's knowledge of trigonometry, including the sine table and relationships between different trigonometric functions[citation needed]. He also discovered spherical trigonometry, along with other interesting trigonometrical results[citation needed]. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation[citation needed]. Among the many interesting results given by Bhaskara, discoveries first found in his works include the now well known results for \sin\left(a + b\right) and \sin\left(a - b\right) :[citation needed]

Calculus

His work, the Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier works[citation needed]. Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest.[citation needed]
Evidence[citation needed] suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement[citation needed]. Bhaskara also goes deeper into the 'differential calculus' and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of 'infinitesimals'.[10]
  • He gave the result that if x \approx ythen \sin(y) - \sin(x) \approx (y - x)\cos(y), thereby finding the derivative of sine, although he never developed the notion of derivatives.[11]
    • Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.
  • In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a ⅟33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time.[citation needed]
  • He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes.[citation needed] He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero.[citation needed] In this result, there are traces of the general mean value theorem[citation needed], one of the most important theorems in analysis[citation needed], which today is usually derived from Rolle's theorem[citation needed]. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara's Lilavati.[citation needed]
Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara's work and further advanced the development of calculus in India.[citation needed]

Astronomy

Using an astronomical model developed by Brahmagupta in the 7th century, Bhaskara accurately defined many astronomical quantities[citation needed], including, for example, the length of the sidereal year[citation needed], the time that is required for the Earth to orbit the Sun, as 365.2588 days[citation needed] which is same as in Suryasiddhanta. The modern accepted measurement is 365.2563 days, a difference of just 3.5 minutes.
His mathematical astronomy text Siddhanta Shiromani is written in two parts: the first part on mathematical astronomy and the second part on the sphere.
The twelve chapters of the first part cover topics such as:
The second part contains thirteen chapters on the sphere. It covers topics such as:

Legends

His book on arithmetic is the source of interesting legends that assert that it was written for his daughter, Lilavati[citation needed]. In one of these stories, which is found in a Persian translation of Lilavati[citation needed], Bhaskara II studied Lilavati's horoscope and predicted that her husband would die soon after the marriage if the marriage did not take place at a particular time. To alert his daughter at the correct time, he placed a cup with a small hole at the bottom of a vessel filled with water, arranged so that the cup would sink at the beginning of the propitious hour. He put the device in a room with a warning to Lilavati to not go near it. In her curiosity though, she went to look at the device and a pearl from her nose ring accidentally dropped into it, thus upsetting it. The marriage took place at the wrong time and she was soon widowed.
Bhaskara II conceived the modern mathematical convention that when a finite number is divided by zero, the result is infinity[citation needed]. In his book Lilavati, he reasons: "In this quantity also which has zero as its divisor there is no change even when many [quantities] have entered into it or come out [of it], just as at the time of destruction and creation when throngs of creatures enter into and come out of [him, there is no change in] the infinite and unchanging [Vishnu]".

Kālidāsa

Kālidāsa (Devanāgarī: कालिदास "servant of Kali") was a renowned Classical Sanskrit writer, widely regarded as the greatest poet and dramatist in the Sanskrit language. His floruit cannot be dated with precision, but most likely falls within 4th Century AD.[1]
His plays and poetry are primarily based on Hindu Puranas and philosophy

Life

One sunny day, Kalidasa was sitting on a branch of a tree, trying to saw it off. But the dimwitted man was sitting on the wrong end of the branch, so when he finally sawed through the branch, down he tumbled! This act of sheer stupidity was observed by some shrewd pundits minister passing by.
Now these pundits wanted to play a trick on the arrogant princess, to teach her a lesson. She was determined to marry someone who would defeat her in a debate about the scriptures. The princess had heaped considerable abuse on them over a period of time, and they were determined to extract their revenge. So, when they chanced upon Kalidas, they decided to present him to the queen as a suitable match for her.
In order to conceal his stupidity, the pundits asked Kalidas to pretend that he was a great sage, who was observing a vow of silence. Kalidas readily agreed, and they presented him to the queen, saying that Kalidas would only communicate by way of gestures. When the queen asked Kalidas a few questions to test his intelligence, Kalidas gesticulated wildly and the astute pundits 'interpreted' these gestures as extremely witty answers and retorts. The princess was suitably impressed, and the couple was married without much delay.
Kalidas's stupidity could be concealed for only so long, and the night of the wedding Kalidas blurted out something inane. The princess realized that she had married a prize fool. Furious, she threw him out of her palace, and her life.
The dejected Kalidas wandered around, till he came to the bank of the river. He contemplated taking his life when he suddenly saw some women washing clothes on the edge of the river bank. He observed that the stones which the women were pounding with clothes, were smooth and rounded, while the other stones were rough and ragged. This observation hit him like a thunderbolt, and it dawned upon him that if stones could be worn through and change their shape by being pounded upon by clothes, then why couldn't his thick brains change, by being pounded upon by knowledge!
Kalidas thus grew determined to become the wisest and most learned man in the country, and to achieve this end he started indulging in intellectual pastimes, reading, meditating and praying to his goddess Kali to grant him divine knowledge. His wish was fulfilled.
This is one of the most popular legends about Kalidas. There are several other stories but they lack authenticity. It is likely that "Kali" in his name refers not to goddess "Kali" (the dark one), but to God Shiva, also called Maha Kaal (the great destroyer. "Kaal" meaning time in Sanskrit refers to the end-time, as in your time has come). This is supported by the fact that all his works starts with an invocation to God Shiva and that Ujjain's most famous temple is the "Mahakaleshwar" temple, one of the Jyotirlinga temples dedicated to Siva.
It appears Kalidas was at the court of emperor Vikramaditya. The place and time of this king are also not definite. But it can be said with some certainty that Kalidas lived before the 6th century A.D., i.e., about 1400 years ago. But when exactly he lived before the 6th century is not firmly established. Though a deep affection for the city of Ujjain is discernible in his works, it cannot be said with certainty that he lived there. But we can assume that, wherever he may have been born, he had lived at Ujjain.
Kalidas, however, had good knowledge of the whole of Bharat. In his poem 'Meghaduta', his descriptions of mountains and rivers and cities and villages stretching from Ramagiri in Central India up to Alakanagari in the Himalayas are very beautiful. In another epic poem 'Raghuvamsha', Kalidas, while portraying the conquests of emperor Raghu, describes the places and peoples, their modes of living, food-habits and trades and professions, rivers and mountains in almost the whole country—Assam, Bengal and Utkal in the East; Pandya and Kerala in the South and Sind, Gandhara and other places in the North-west.
Reading these pen-pictures, one cannot help but conclude that the poet must have had a personal knowledge of these areas. In short, he must have traveled widely across the length and breadth of the land, seen those places, talked to the people and studied their modes of living.
Kalidas possessed that distinct intellect which makes one a great poet. He was a scholar and his works display his poetic genius as well as scholarship. Also they are marked by a belief of what is good in life and people's noble goals of life. He could describe the rich and wealthy life of a royal palace and the serene, simple and peaceful life at a hermitage with equal understanding. He could, likewise, describe the joys of the marital life of a man and his spouse as well as their pangs of separation. He creates scenes of a serious and thoughtful nature as also hilarious scenes of light comedy. In his works is found an excellent combination of art-consciousness, unmatched wordpower and an unparalleled capacity for vivid portrayals.


Location

Scholars have speculated that Kālidāsa may have lived either near the Himalayas or in the vicinity of Ujjain or in Kalinga. The three speculations are based respectively on Kālidāsa's detailed description of the Himalayas in his Kumārasambhava, the display of his love for Ujjain in Meghadūta and his highly eulogistic quotes for Kalingan emperor Hemāngada in Raghuvaṃśa (sixth sarga).

Works

Kālidāsa wrote three plays. Among them, Abhigñānaśākuntalam ("Of Shakuntala recognised by a token") is generally regarded as a masterpiece. It was among the first Sanskrit works to be translated into English, and has since been translated into many languages.
  • Mālavikāgnimitram ("Mālavikā and Agnimitra") tells the story of King Agnimitra, who falls in love with the picture of an exiled servant girl named Mālavikā. When the queen discovers her husband's passion for this girl, she becomes infuriated and has Mālavikā imprisoned, but as fate would have it, Mālavikā is in fact a true-born princess, thus legitimizing the affair.
  • Abhigñānaśākuntalam ("Of Shakuntala recognised by a token") tells the story of King Dushyanta who, while on a hunting trip, meets Shakuntalā, the adopted daughter of a sage, and marries her. A mishap befalls them when he is summoned back to court: Shakuntala, pregnant with their child, inadvertently offends a visiting sage and incurs a curse, by which Dushyanta will forget her completely until he sees the ring he has left with her. On her trip to Dushyanta's court in an advanced state of pregnancy, she loses the ring, and has to come away unrecognized. The ring is found by a fisherman who recognizes the royal seal and returns it to Dushyanta, who regains his memory of Shakuntala and sets out to find her. After more travails, they are finally reunited.
  • Vikramōrvaśīyam ("Pertaining to Vikrama and Urvashi") tells the story of mortal King Pururavas and celestial nymph Urvashi who fall in love. As an immortal, she has to return to the heavens, where an unfortunate accident causes her to be sent back to the earth as a mortal with the curse that she will die (and thus return to heaven) the moment her lover lays his eyes on the child which she will bear him. After a series of mishaps, including Urvashi's temporary transformation into a vine, the curse is lifted, and the lovers are allowed to remain together on the earth.
Kālidāsa is the author of two epic poems, Raghuvaṃśa ("Dynasty of Raghu") and Kumarasambhava (Birth of 'Kumara' or Subrahmanya)
  • Raghuvaṃśa is an epic poem about the kings of the Raghu dynasty.'Raghuvamsha' depicts Indian ancient, historical culture and tradition. Indian ancestors had discussed in detail about such matters as to who could be a good ruler, who is a man of 'tapas' (penance), how one should lead a good, purposeful life and the like. The poet has portrayed diverse characters like Vashishta, Dileepa, Raghu, Aja and others. Agnivarna is an example of a king who could be termed as 'depraved'.
  • Kumarasambhava One of Kalidas's greatest works is 'Kumarasambhava'. Critics maintain that Kalidas wrote only the first eight chapters of the epic poem. The work describes the marriage of Lord Shiva and his consort Parvati. It begins with a fine description of that giant among mountains, the Himalaya.
He has also written two Khanda Kayva's
  • Ṛtusaṃhāra describes the six seasons by narrating the experiences of two lovers in each of the seasons.
  • Meghadūta or Meghasāndesa is the story of a Yaksha trying to send a message to his lover through a cloud. Kalidasa set this poem to the 'mandākrānta' meter known for its lyrical sweetness. It is one of Kalidasa's most popular poems and numerous commentaries on the work have been written.
Kalidas's poem gives us a vivid picture of what a good, meaningful life a man could and should lead as propounded by our learned ancestors.

Style

Kālidāsa's poetry is celebrated for its beautiful imagery and use of similes. The following are some specimen verses from his works.
One celebrated example occurs in the Kumārasambhava. Umā (Parvati) has been meditating even throughout the summer, and as the monsoon arrives, the first raindrop falls on her:


The beauty of this verse is held to result from "the association through suggestion of numerous harmonious ideas". Firstly (as described in Mallinatha's commentary), the description suggests signs of her physical beauty: long eyelashes, pouting lower lip, hard breasts large enough to touch each other, deep navel, and so on.[7] Secondly (as described in Appayya Dikshita's commentary[8]), it suggests her pose as a perfect yoginī: her motionlessness through pain and pleasure, her posture, and so on. Finally, and more subtly, in comparing the mother goddess to the mother earth, and the rain coursing down her as it courses over the surface of the earth, it suggests earthly fertility. Thus the verse harmoniously brings to mind beauty, self-restraint, and fertility.[9][10]


Ujjain Climate

Ujjain experiences a warm sub-tropical climate, typical of the North Central India. Summer starts in late March with temperatures rising to 45°C at its peak in May. In addition, hot winds (called loo) may blow in the afternoons, adding to the discomfort. The monsoon arrives in the middle of June and continues till early October. About 870 mm (35 inches) of precipitation is received during those months, though some years may see the figure climb up to 40 inches.
The rest of October generally is very warm and with high humidity. Winter starts in the middle of November and is pleasant and cool with daytime temperatures typically 20°C, though temperatures can drop significantly in the night. January takes chilli winds (called Sheet-lehar) with it, and sometimes daytime temperature rise only up to 15-17°C, and minimum temperature fall to 3C, 0°C is the lowest temperature recorded in the city in January 1982 and 1965. The morning dew in the outskirts of the city, and fields and farms turns ice because of some nights of freezing points, which results in damaged crops almost every year

Transport

Ujjain is well-connected by rail and road. It is on the Western Railway and is connected by direct train to most major Indian cities. The road network is developed with other parts of Madhya Pradesh. Ujjain is connected to Indore through SH-27 and SH-18 Dewas-Badnawar passes through it. Unfortunately Ujjain is the largest city that has no National Highway connectivity.[10]

Airports

1.   A private airstrip, situated on Dewas road is being used as a pilot training institute. It has no commercial scheduled air services.
2.   The nearest airport is the Devi Ahilyabai Holkar Airport at Indore (50 km away)

Railway stations

There are four railway stations:
1.   Ujjain Junction main (Back side of this station is known as Madhav Nagar Rly.Stn.)
2.   Vikram Nagar
4.   Pingleshwar

Bus stations

1.   Dewas Gate (Shaheed Raja Bhau Mahakal bus stand)
2.   Nanakheda (Pt. Deendayal Upadhyay bus stand)

Major roads

Indore Road, Dewas Road, Kota Road,Badnagar Road, Maksi Road, Nagda Road, Tarana Road Via Undasa Dam, Chintaman Road which is connected to four-lane state highway from Badnagar to Sanwer

Local transport

Ujjain City Transport Services Limited (UCTSL) runs the city bus service that operates 40 buses plying on all important routes in the city. Besides the bus service, auto rickshaws, taxis, and other transport vehicles - locally referred to as 'tempo' and 'Tata Magic' - are also easily available for travelling within the city

















Om Tat Sat
                                                        
(Continued...)                                                                                                                              



(My humble  salutations to the great devotees ,  wikisources  and Pilgrimage tourist guide for the collection
Posted by gopalakrishna Friday, November 23, 2012
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