Dream 2047: VP News

OCT2000

VPNEWS

PARTING THOUGHTS (II)

To continue from where we left off last time, Vigyan Prasar can really make a difference to the overall science popularisation scene in the country! It has all the essential ingredients, in the form of major programmes/efforts, I mentioned last time, to transform itself into a powerhouse of incredible and unimaginable strengths capable of delivering unheard of results!

Each one of the VP's major programmes referred to earlier, if handled appropriately, could develop into an independent, self-sustaining entity in its own right, under the overall VP umbrella, with greatly enhanced output. What would that mean in terms of the goals we are seeking to achieve?

It would certainly take us closer to achieving the goals with which VP was setup. Vigyan Prasar popular science publications on a variety of subjects and topics would become available in many more languages and throughout the country with the help of a widespread countrywide distribution system _ built upon a very strong back-bone provided by VIPNET, an active and vibrant network of several tens of thousands of science clubs around the country.

Vigyan Prasar books would also be distributed worldwide in many countries and also in many other non-Indian languages. Dream-2047 would develop into a very popular science magazine with a large circulation and several other language editions, besides Hindi and English. .

People around the world would increasingly turn to Vigyan Prasar's homepage on the internet for information on anything and everything on science (& technology) in India and there would be a large volume of sales of VP publications (and possibly other products and services) online. The homepage would only be a part of VIPRIS, Vigyan Prasar's Information System, which will put together a whole lot of useful, small and large, databases of interest to users. There will be several other components of VIPRIS (like the Clippings Service on S&T and environment).

Vigyan Prasar's audio visual productions will be regularly seen on DD1, satellite channels and regional DD Kendras as well as heard on all Akashwani Kendras in different Indian languages. News stories and regular television coverage by VP on S&T developments in Indian laboratories will not only regularly be seen on Indian TV channels, but would also be disseminated worldwide through foreign channels. .

. VIPNET, Vigyan Prasar's network of affiliated Science Clubs, once it crosses the 10,000 or so mark will gather a momentum of its own and the next ten thousand clubs _ and the next and subsequent ones _ would be achieved much faster than before. An essential condition for this would be to keep the clubs active and their members involved in activities and pursuits which they find exciting and worth their while to pursue for their present as well as for their future.

. . Apart from the four major efforts abovementioned, which are already taking good shape, there are several other initiatives of Vigyan Prasar which need to be pursued with greater vigour and more manpower. Among these, an important one relates to developing dedicated core-groups in all states/major language regions to look after, promote and add to VP activities in different languages _ ranging from joint VP-AIR and VP-DDK programmes; development, production and distribution of popular science publications in the local language; working with VIPNET Science Clubs in the state/region; organising activities with children, youth and other groups; and so on and on.

. Another very important area is dissemination of the VP (and NCSTC) software materials through a unique distribution system to be built up with the help of VIPNET clubs, VP core groups and existing VP distributors. I could go on and on but will not! Many more ideas have been discussed internally. The greatest hurdle has been humanware _ finding the right kind of people and hiring them for VP tasks within the framework of VP rules and regulations!

. I'll stop here and conclude by wishing my colleagues in the VP team many more achievements and successes in time to come. I do hope that VP and NCSTC would continue to work together _ like hand in glove _ and maximize the synergies that these two organisations are capable of generating. May the team spirit prevail over all else!

NKS

(To be continued)

Tycho Brahe (1546-1601)

Dr. Subodh Mahanti

Johannes Kepler (1571-1630), tells us that, Tycho Brahe, during the last few days of his life, would keep repeating the above words whenever he was in delirium. Kepler writes: "No doubt he (Brahe) wished that these words should be added to the title page of his works, thus dedicating them to the memory and uses of posterity." That Tycho's fear of being `lived in vain' was unfounded is evident from the fact that in the title page of Kepler's great work, Astronomia nova, the following words were written: "Founded on observations of the noble Brahe." It is true that Tycho could not build his own universe or provide any theoretical framework but as acknowledged by Kepler, the laws of planetary motion could not have been formulated without Brahe's enormous and meticulous data. Tycho produced more accurate data than anyone before him including Ptolemy (2nd century AD) and Nicolaus Copernicus (1473-1543).

p>Tycho was the greatest observational astronomer of the pre-telescopic age. He has been a very colourful personality in the history of astronomy. He was haughty and proud. He always asserted his own elevated position. He made all his astronomical observations waring his noble dress. He liked to lead a luxurious and comfortable life. He kept a dwarf to amuse him. He was the first to establish a completely outfitted observatory. He prepared a star catalogue of 777 stars with such accuracy that it provided a vital source of information for later astronomers. He became an international celebrity for making meticulous observations on the new star that appeared in 1572 now known as 'Tycho's Star'. He proved that comets are not objects in the atmosphere and that their orbits are far beyond the Moon. Earlier Aristotle had explained comets and meteors as atmospheric events that took place in the space between the Earth and the Moon. Tycho showed irregularities in the Moon's orbit. His many instruments became widely copied and led to improved stellar instruments. Tycho himself was an object of admiration and emulation. He occupies a special position in the history of astronomy. He was held in enormous respect by generations of astronomers. While copernicus left room for traditional solid spheres, Tycho clearly pronounced: "Now it is quite clear to me that there are no solid spheres in the heavens."

Tycho was born into a noble family in Knudstrup in southern Sweden (then under Danish rule). His father, Otto Brahe, was the Governor of Helsinborg Castle and his uncle, who brought him up, was a an admiral in the navy. The adoption of Tycho by his uncle was a highly dramatic affair. Tycho's father promised his childless brother that if he had a son he would allow Joergen to adopt him and bring him up as his own.

But then Tycho's father went back on his words when one of the twin sons that his wife bore him in 1546 was a still born. Joergen waited until another son was born to his brother and then kidnapped Tycho, the firstborn child. The Governor threatened to kill Joergen but afterwards cooled down, perhaps realising that Tycho would be well-looked after and inherit some of Joergen's fortune. Joergen died while saving the life of his king, Frederik II (1534-88), who while crossing the bridge joining Copenhagen and the royal castle had fallen into the river. Joergen valiantly jumped into the cold water to save the king. He could save his king but he himself died of pneumonia. This happened when Tycho was a student. This incident made Frederik II more sympathetic to Tycho, the adopted son of Joergen.

At the age of 18 Tycho fought a duel with another noble Danish youth over a trivial point: of the two Danish nobles who was the better mathematician? In the course of the duel Tycho got a part of his nose sliced off. The lost piece of the nose was replaced by a patch made of a gold and silver alloy. This fact has recently been confirmed when his grave was opened and the remains were inspected.

Following the family tradition, young Tycho was to take up the career of a statesman. For this purpose he was sent at thirteen to study rhetoric and philosophy at the University of Copenhagen. However, a partial solar eclipse, that he witnessed (on 21st August 1560) changed the whole future course of his life. The event itself was not at all spectacular but its impact on Brahe was because of its predictability. This solar eclipse, which had been predicted beforehand stuck young Tycho as "something divine that men could know the motions of the stars so accurately that they were able long time beforehand to predict their places and relative positions." From the very beginning Tycho had developed a passion for exact observation. Tycho chose to pursue astronomy, as Arthur Koestler wrote, `not as an escape or metaphysical life belt, but rather as a full-time hobby of an aristocrat in revolt against his milieu.'

After three years at the Copenhagen University Tycho was sent to the University of Leipzig accompanied by a tutor, Andreas Soersen Vedel. For Tycho's uncle astronomy was a profession unworthy of pursuit by a nobleman. So he instructed Vedel, who was just four years older than his pupil, to cure Tycho of this undignified preoccupation and persuade him to take up a profession befitting a nobleman. But Vedel was not successful in his mission. He found Tycho beyond remedy. Tycho was completely engrossed in his study of astronomy.

However, Tycho had to hide his celestial globe, which he had bought for learning the names of the constellations, from Vedel. He could use this only when Vedel was asleep. But finally Vedel accepted his defeat and they remained life-long friends. It may be noted here that Vedel later became famous as the first great Danish historian. From Leipzig Tycho moved to different universities of northern Europe namely Wittenberg, Rostock, Basle and Augsberg.

While moving from one university to another Tycho went on collecting bigger and better instruments. As we now know Tycho later designed his instruments himself. Among his early collections was a huge quadrant of brass and oak. The quadrant, which happened to be first among Tycho's fabulous instruments, was thirty-eight feet in diameter and turned by four handles. A quadrant, which is now obsolete, was a device for measuring angles. It consisted of an engraved arc of a quarter of a circle, with a plumb line suspended from the centre of the circle. Stars were sighted along one arm and their elevation was read off the scale against the plumb line.

After completing his study, he spent the next five years first on the family estate at Knudstrup and then with his uncle Steen Bille, the only one in Tycho's family who approved of Tycho's unseemly hobby.

On the evening of November 11, 1572, Tycho noticed a star brighter than Venus at her brightest, at a place where no star could be seen before. The star was visible in the Cassiopeia constellation. Initially Tycho did not believe his own eyes. So he called some of his servants and peasants who all confirmed the observation of the star at a place where no star was there before. The star remained in the same spot for eighteen months.

The star was so bright that people with sharp eyes could see it even in the middle of the day. This was an extraordinary event. No new star had been seen in Europe since the days of Hipparchus (second century BC). Pliny in his second book of Natural History, had noted that Hipparchus had seen a new star in the sky. It may be noted that new stars were seen in 1006 and 1054 by Japanese and Chinese astronomers. But these were not known to the european scientific community

Naturally Tycho was not the only one to witness this extraordinary event. Many other astronomers of the day also observed the star. But Tycho surpassed others in his meticulous observations. With his newly constructed sextant, with arms five and a half feet long, the result of Tycho's observations was unequivocal. Sextant is a navigation device for measuring the altitude of celestial objects above the horizon. It consists of a graduated 600 arc (one sixth of a circle) with a movable arm and sighting devices.

Tycho demonstrated that the new star showed no large parallax or 'proper motion' and which meant that the new star truly belonged to the sphere of the fixed starts. (Parallax is defined as angular difference between an object's direction as seen from two points of observation). Tycho did not comment on the nature of the star or how it was created. Today we know that it was a supernova. In 1573 Tycho published his first book De stella Nova (On the New Star).

In those days the observation of a new star had a special significance. It contradicted the basic doctrine _ Aristotelian, Ptolemaic and Christian. Aristotle (384-322BC), the Greek philosopher, based his geocentric model of the universe on the system of concentric spheres originally proposed by Eudoxus of Cnindus (c.480-c.350 BC), the Greek mathematician and astronomer. Eudoxus had developed a model of planetary motion in which the Sun, Moon and planets were carried around the Earth on a series of 27 Earth-centred spheres, with axes at different angles and rotating at different speeds.

Callippus (c.370-c.300BC), also a Greek mathematician and astronomer, modified the Eudoxus scheme by adding extra spheres for the Sun, Moon and some other planets. Callippus brought the total number of spheres to 34. This system was further modified by Aristotle who increased the number of spheres to 49 to account for the movement of all celestial bodies. The outermost spheres, in which the fixed stars were located, controlled the motion of the other spheres. The outermost sphere itself was thought to be controlled by a supernatural agency. And this sphere was thought to be immutable (that is no change takes place there) from the day of creation to eternity. Changes can take place only in the immediate vicinity of the Earth, the sublunary sphere. Ptolemy modified Aristotle's world-view by replacing the spheres by epicycles but otherwise kept its basic tenets intact. The church accepted the Ptolemaic system which was essentially the Aristotellian system. The stars were called 'fixed stars' as they did not show amy detectable 'proper motion'. It only participated in the daily rotation of the firmament as a whole. So the sight of new star and if it was really a star with no 'proper motion', would force one to think the world afresh

The whole europe was excited to know the cosmological and astrological significance of the new star. Most of the people considered it a sinister omen. However, the serious astronomers of the day with few exceptions tried to explain the unprecedented phenomenon by calling the new star a tailless comet with slow motion.

Thus proving the existence of a new star Tycho demonstrated a basic flaw in the Aristotellian world view of the unchangeability of the heavens. His observation of the comet of 1577 and five subsequent comets also convinced him that their orbits were far beyond the moon. There is no doubt that Tycho's discovery and his meticulous observations of planetary motions, laid a firm basis for the breakthrough of the Copernican world view in the 17th century. .

In 1575 Tycho, whose reputation was already established, made a tour of europe visiting his astronomer friends. After completing his tour when Tycho returned to Denmark, the king Frederik II offered him various castles to choose from. But Tycho had no intention settling in Denmark and so he declined the offer made by the king. He was planning to settle in Basle, a city in Switzerland where the French, German and Swiss borders meet. The city became a major literary centre during the Reformation. Desiderius Erasmus (1466-1536) taught at the University of Basel. Among other great thinkers and scholars settled in Basle was Phillippus Aureolus Paracelsus (1493-1541). Frederik II was determined to induce Tycho to stay in Denmark. So he offered Tycho an Island, the Island of Hveen. The island was three miles in length and extending over 200 acres of flat tableland. He was promised enough money to build an observatory on the island and an annual grant plus various positions that would provide regular income with little or no work. After a week's hesitation Tycho accepted the offer

A royal instrument, signed on May 23, 1576 decreed that :

"We, Frederik the Second, & c. make known to all men, that we of our special favour and grace have conferred and granted in fee, to our beloved Tyge Brahe, Otto's son, of Knudstrup, our man and servant, our land of Hveen, with all our and the crown's tenants and servants, who thereon live, with all rent and duty which comes from that, and is given to us and to the crown, to have use and hold quit and free, without any rent all the days of his life and as long as he lives and likes to continue and follow his studia mathematicus ...."

In 1577 Tycho moved to the Island of Hveen and built his observatory, Uraniborg (Castle of the Heavens). The observatory was built by a German architect under Tycho's supervision. Tycho combined his meticulous precision with fantastic extravagance. It had an onion-shaped dome, flanked by cylindrical towers. Each tower had a removable top housing Tycho's instruments. In the basement Tycho had his own printing press, an alchemist's furnace and a private prison for punishing his servants and the peasants living on his lands when they broke one of his strict rules. Tycho had his own paper mill, pharmacy, game preserves and artificial fish ponds. He gathered or built some of the most fabulous astronomical instruments of his day. Tycho created a remarkable range of instruments from armillary spheres after the manner of Ptolemy to large quadrants of bold and original design. Most of his instruments were built in his own workshop. His largest celestial globe was five feel in diameter. It was made of brass. Tycho engraved the fixed stars one by one on this celestial globe which was placed in his library, after their correct positions were determined by him and his assistant. Tycho's largest quadrant was fourteen feet in diameter. It was fastened to the wall and the space inside the arc was filled with a life-sized fresco of Tycho himself in the midst of his instruments.

Tycho published a detailed account of his observatory and instruments in a book titled. Astronomice instauratae machanice. This became a model for others.

Tycho highlighted the importance of precise and continuous observational data. For him meticulous and precise observation was a form of worship. Tycho never believed in the Copernican system. At the same time he discarded the Ptolemaic system, because it was incompatible with his observations of planetary motions. Tycho sought a compromise between the two systems. Thus he proposed a new model for planetary motions. According to the Tychonic system the Earth is the centre of the solar system, the Sun and the Moon revolve around the Earth, and the other planets revolve around the Sun. One reason for his quest for precise and continuous observation was to substantiate his own system. Tycho was attracted to astronomy by the predictability of a partial solar eclipse. But then on August 17, 1563 he noticed that Saturn and Jupiter were so close together as to be almost indistinguishable from each other. Tycho found that the Alphonsine tables were almost one month in error regarding this event and the Copernican table by several days. This discovery was so shocking to Tycho that he took unto himself to correct the existing inconsistencies.

Frederik II died in 1588 and he was succeeded by his son Christian IV (1577-1648). Tycho who had already acquired the image of a despotic ruler of his island, quarrelled with the young king. Christian IV, though well disposed towards Tycho, did not ignore Tycho's despotism in the Island of Hveen. This was added to Tycho's arrogance. Tycho did not bother to answer the king's letters. He disobeyed provincial as well as high court's decision by holding a tenant and all his family in chains. Still Christian IV did not initiate any direct action against Tycho but he did reduce Tycho's fantastic income assured by Frederik II to a reasonable level. This measure prompted Tycho, who was becoming restless and bored on his Island, to resume his wanderings. In 1597 Tycho left the Island of Hveen. The last recorded observation made at Hveen was on 15th March, 1596. His luggage consisted of the printing press, library, furniture and all his instruments except the four largest ones which followed later. From the very beginning Tycho had ensured that all his instruments were made in such a way that they could be dismantled and taken from one place to another. Tycho was of the opinion that, "An astronomer must be cosmopolitan, because ignorant statesmen cannot be expected to value his services." Before leaving the Danish territory Tycho wrote to Christian IV. He bitterly complained about the way his ungrateful country treated him. He intended "to look for help and assistance from other princes and potentates". But Tycho also wrote that he would be willing to return to his country "if it could be done on fair condition and without injury to myself." Christian IV in his letter to Tycho refuted every complaint made by Tycho and made it clear that condition of Tycho's return to Denmark was "to be respected by you (Tycho) in a different manner if you are to find a gracious lord and king".

As there was no chance of reaching a compromise with the king of Denmark, Tycho continued his wanderings for two years before reaching Prague in June 1599. In Prague Tycho found a new patron. Ruldolph II (1552-1612), king of Hungary and Bohemia and Holy Roman Emperor, appointed Tycho as imperial mathematicus of the Holy Roman Empire at his court in Prague. Tycho was again given a castle of his choice (the castle at Benatake outside Prague) and an annual salary of three thousand florins. The magnitude of Tycho's salary could be guessed if we compare it with Kepler's salary of two hundred florins a year in Gratz. Tycho took possession of the Castle in August 1599. It was in Prague that Kepler met Tycho and their association heralded a new era in astronomy.

Throughout his life Tycho also practised alchemy. He had prepared a wonderful patent medicine to cure all disorders. This was widely circulated in Europe in its time as Holloway's pills. Among many ingredients it contained a little of antimony, a well-known sudorific.

Tycho died on 24 October 1601. He was buried with great pomp in Prague. Tycho's coffin was carried by twelve imperial gentlemen-at-arms, preceded by his coat of arms, his golden spur and favorite horse. After Tycho's death Kepler was appointed as Tycho's successor and inherited Tycho's large collection of astronomical observations from which Kepler deduced his famous laws of planetary motion

Tycho wrote in his Mechanica : "The person who cultivates divine Astronomy ought not to be influenced by ignorant judgements, but rather look upon them from his elevated position considering the cultivation of his studies the most precious of all things, and remaining indifferent to the coarseness of others. And when statesmen or others bother him too much, then he should leave with his possessions."

Great Scientists of Ancient India

B R A H M A G U P T A

Gunakar Muley

Of all the sciences cultivated in ancient India, the contribution of mathematics to world science was the most significant. This the Western historians of mathematics also admit. The scope and power of modern calculation derives directly from the invention of decimal place-value notation - the principle of position in writing numbers - the biggest achievement of ancient Indian science. In his famous work History of Mathematics, Florian Cajori writes : "The grandest achievement of the Hindoos and the one which, of all mathematical investigations, has contributed most to the general progress of intelligence, is the invention of the principle of position in writing numbers."

Indian mathematicians' contribution to arithmetic and algebra was also substantial. The German historian of mathematics, Hermann Hankel, has said : "Indeed, if one understands by algebra the application of arithmetic operations to composite magnitudes of all kinds, whether they be rational or irrational numbers or space magnitudes, then the learned Brahmins of Hindostan are the true inventors of algebra." And, F. Cajori confirms : "It is remarkable to what extent Indian mathematics enters into the science of our time. Both the form and the spirit of the arithmetic and algebra of modern times are essentially Indian and not Grecian."

Why is it then that our students and teachers are not well-informed about our own achievements, particularly in the field of mathematics? In modern textbooks of mathematics we do not find even the names of Aryabhata, Brahmagupta or Bhaskaracarya - who were associated with at least some principles or theorems of mathematics. . The reason is not difficult to find out. One example is enough - that of the Indeterminate Equations (If the number of unknown quantities is greater than the number of independent equations, they are called indeterminate). The topic, taught in higher secondary classes, is discussed in Hall and Night's Higher Algebra, a standard textbook since 1887 till recent times. After treating the subject, the authors state : "Problems of this kind are sometimes called Diophantine Problems because they were first investigated by the Greek mathematician Diophantus about the middle of the fourth century."

Students of higher mathematics know the name and contribution of Diophantus, because the whole subject of indeterminate equations is known as Diophantine Equations. But the correct name for them should be Brahmagupta's Equations, even according to Western historians of mathematics. In his A Concise History of Mathematics, the Dutch-American mathematician Dirk J. Struik points out : "The first general solution of indeterminate equations of the first degree ax + by = c (a, b, c integers) is found in Brahmagupta. It is therefore, strictly speaking, incorrect to call linear indeterminate equations Diophantine equations. Where Diophantus still accepted fractional solutions, the Hindus were only satisfied with integer solutions."

In fact, the first Indian mathematician to solve the indeterminate equation of the first degree was Aryabhata (499 AD). Brahmagupta (628 AD) followed him in this important field and solved the famous indeterminate quadratic equation : ax 2 + 1 = y 2. About this equation D. E. Smith in his History of Mathematics (vol. 2) writes : "This form is commonly attributed to John Pell (1668) but is really due to Fermat (c.1640) and Lord Brouncker (1657). The problem itself is apparently much older than this, however, for it seems involved in various ancient approximations to the square roots of numbers. ... The general problem may have been discussed in the lost books of Diophantus, perhaps in the form x 2 - Ay 2 = 1, and its equivalent is clearly stated in the works of Brahmagupta (c. 628). ... It was Euler (1730) who, through an error, gave to the general type the name of the Pell Equation." (italics mine) Thus, the so-called Pell's Equation ax 2 + 1 = y 2 , erroneously attributed to John Pell, really owns its origin to Brahmagupta. The equation and its solution is clearly stated in the works of Brahmagupta. For solving this equation Brahmagupta established two important lemmas. This is an outstanding contribution of the mathematical genius of Brahmagupta.

Bhillamalakacarya

In ancient India the most neglected subject was chronology; it is very rarely that we get some information regarding the time and life of a great personality. However, mathematicians and astronomers are an exception to this otherwise appalling situation. In their works we at least find one concrete date regarding their time, and sometimes also some information about their life. The first Indian mathematician-astronomer to give his date was sryabha˜a (born in 476 AD). Of Brahmagupta we know little more than about sryabha˜a. Brahmagupta's "autobiography", comprising two couplets, is given at the end of his major work, the Brahma-sphuta siddhanta (BSS

.e., In the reign of Sri Vyaghramukha, of the Sri Capa dynasty, five hundred and fifty years after the raka king having passed (or in 628 AD), Brahmagupta, the son of Jisu, at the age of thirty, composed the Brahma-sphuta-siddhanta, for the gratification of mathematicians and astronomers. Brahmagupta was thus born in 598 AD, and was a contemporary of king Harsavardhana and the famous Sanskrit poet Banabhatta. According to al-Biruni_ (1030 AD), Brahmagupta "belonged to the town of Bhillamala, between Multana and Anhilwada". At that time Bhillamala (modern Bhinamala orSrimala, Jodhapur Dist., Rajasthan) was the capital of northern Gujarat. Hiuen Tsiang, the famous Chinese traveller, who was in India during 629 to 645 AD, also indicates that Bhinamala (Pi - l o - mo - lo) was the capital of Gurjara country. Now a small village situated on the river Luni, Bhinamala is some 75 kms to the north-west of Mt. Abu. It was once a flourishing city, well-known for its contribution to literature and the arts. The great Sanskrit poet Magha (second half of the seventh century), who composed the epic poem Sisupalavadha, was a resident of Bhillamala. Brahmagupta is also known as "Bhillamalakacarya". Though we have no information regarding the year of Brahmagupta's death, from his other work, the Khanda-khadyaka, we learn that he composed it in 665 AD at the ripe age of 67. This is all that we know about the life of this great mathematician and astronomer.

Magnum Opus

Brahmagupta's magnum opus, Brahma-sphuta-siddhanta (BSS), comprising little more than 1000 Sanskrit slokas, is divided in 24 chapters; the last chapter gives the table of contents and the brief "autobiography" of the author. The first ten chapters deal with the main astronomical topics in the following order : (1) mean planetary motions; (2) true planetary motions; (3) problems of time, space and distance; (4) lunar eclipses; (5) solar eclipses; (6) risings and settings of planets; (7) the moon's cusps; (8) the moon's shadows; (9-10) conjunctions of planets. .

Some of these topics are further discussed in chapters 16 to 17 and 19 to 21. Chapter 22 (Yantradyaya) is devoted to astronomical instruments. Chapter 12 (Ganitadyaya) and 18 (Kuttakadyaya) deal with arithmetic and algebra and reveal the originality of Brahmagupta as an excellent mathematician

In chapter 11, called Tantra-pariksadyaya (examination of other astronomical systems), Brahmagupta criticized the views and methods presented by other astronomers, particularly Aryabhata (499 AD), whom he wrongly attacked for upholding the diurnal motion of the earth and for not accepting the traditional theory of eclipses as the work of demons Rahu and Ketu. He was also very critical of foreign astronomers and their methods.

From all this it appears that Brahmagupta, though a genius, was bound to and was a supporter of the orthodox views. Al-Biruni criticizes Brahmagupta for being unduly harsh and hostile to Aryabhata, and states : "The truth is entirely with the followers of Aryabhata who give us the impression of really being men of great scientific attainments." Apart from the 24 chapters of the BSS, there is, generally attached to it, a monograph of 72 Slokas called Dhyanagrahopadyaya

. It gives simple methods for calculating tithis, naksatras etc. Being basically an observational science, astronomy needs instruments to ascertain the positions and motions of heavenly bodies and to measure the duration of time.

the Yantradyaya (chapter 22) of his BSS, Brahmagupta states : "One who knows mathematics knows spherics (gola), and one who knows spherics understands the motion of planets. If one is ignorant of mathematics and spherics, how can he know the motion of planets?" Brahmagupta has described several astronomical instruments : Gola (armillary sphere), Sanku (gnomon), Dhanus (arc), Cakra (circle), Yasti (staff), Ghatika (water-clock or clepsydra) etc.

I The Golayantra, used mainly for demonstration, was a wooden model of the celestial sphere showing the various great circles used in astronomy. The great circles represented the horizon, the meridian, the prime vertical and so on. In a separate section, called Golabandha, Brahmagupta has described the construction of the armillary sphere.

The Sanku (gnomon), in its simplest form a vertical rod, was used by all ancient nations for determining the east-west direction as well as knowing time. Perhaps the most popular instrument with the Indian astronomers, it was also used for the determination of the solstices, the equinoxes and the geographical latitudes. Brahmagupta has described a conical gnomon. The staff (yasti), described in detail by Brahmagupta, represented the radius of the celestial sphere and was used for determination of the position of heavenly bodies, and also for terrestrial surveying

The cakra-yantra (circle), according to our astronomer, is graduated with degrees and zodiacal signs on its circumference. A plumb is hung from its centre. It was used for determining the sun's zenith distance. The dhanur-yantra was half a cakra, and graduated with 180 degrees. The kapala-yantra (bowl instrument) was a hemispherical sundial. Of all the water instruments (clepsydra) the floating type (ghatika-yantra) was the most popular astronomical instrument in India till recently. In the BSS, it is described as follows : "The ghatika is a copper cup, half a pot in shape, and has a small hole at its bottom. It is made in such a way that it sinks into water 60 times a day and night." Brahmagupta also described a self-rotating instrument in his BSS. But his suggestion to rotate it by using mercury is practically impossible.

THE GEM

Asryabhata (499 AD) was the first to include a section on mathematics in his Siddhanta (Aryabhatiya). After him it became a regular feature. Chapter 12 and 18 of the BSS, as mentioned earlier, deal with mathematics. The former is called Ganiitadhaya and is concerned with 20 operations or logistics (parikarma) of arithmetic and 8 determinations (vyavahara). In the very first sloka Brahmagupta states : "He who distinctly and severally knows the twenty logistics, additions etc., and eight determinations including (measurement by) shadow is a mathematician." Addition, multiplication, square-root, cube-root, rule of three, barter etc. are the logistics, and mixture, series, stock (citi), mound (rasi) etc. are the determinations. Based on the concept of zero, the decimal place-value system of numeration was invented in India about the beginning of the Christian era. The earliest treatment of zero in algebra is found in the BSS. Brahmagupta gives the definition of zero as a - a = 0, and states :

a - 0 = a,

a x 0 = 0,

0 / 0 = 0

- a - 0 = - a,

- 0 x 0 = 0

Brahmagupta says that x/0 and 0/x should be written as x/0 and 0/x respectively. To x/0 he calls taccheda (i.e., the quantity with zero as denominator), which probably he means 'infinity'. However, his statement that 0/0 = 0 (zero) is incorrect. In Brahmagupta's time, mathematics in India had developed so much that it needed subdivisions. It was Brahmagupta who, for the first time, divided mathematics into arithmetic and algebra. However, he did not coin the modern name BijagaSita for algebra. The word BijagaSita is found for the first time in the work (a commentary on the BSS) of Prthudakasvami (860 AD). Brahmagupta calls algebra kuttka-ganita or kuttakara or simply kuttaka, meaning "pulveriser", a process of continued division adopted for solving the indeterminate equations. The early Indian mathematicians attached great importance to algebra. In the opening verse of the Kuttakadyaya (chapter 18), Brahmagupta observes : "Since questions can scarcely be solved without algebra (kuttakara), therefore, I shall speak of algebra with examples. By knowing the kuttaka (pulveriser), zero, negative and positive quantities, unknowns, ... one becomes the learned professor (acarya) amongst the learned." As noted, Brahmagupta was the world's first mathematician to solve satisfactorily the indeterminate equations. For solving the indeterminate quadratic equation of the type Nx 2 + 1 = y 2, technically known as varga-prakrti (square-nature), Brahmagupta established two important lemmas. The word prakrti here means coefficient, and refers to the coefficient N in the indeterminate equation Nx 2 + 1 = y 2, where N is a positive integer. This subject was further elaborated by later Indian mathematicians and was thoroughly treated by Bhaskaracarya (1150 AD), who gave Brahmagupta the title Ganakaracakra-cudamani (the gem of the circle of mathematicians). The earliest Indian geometry occurs in the Suulva-sutras (c. 800 BC) in connection with the construction of the altars for the Vedic sacrifices. The famous Theorem of Pythagoras (c. 540 BC), without proof, is given in the Sulvasutras of Baudhayana, Katyayana etc. in the following almost identical words: "The diagonal of a rectangle produces both areas which its length and breadth produce separately." Brahmagupta's most important contribution to geometry is the theorem: The area (A) of an inscribed quadrilateral whose sides are a, b, c, d, gives the following formula :

A = v(s - a) (s - b) (s - c) (s - d)

where 2s = a + b + c + d

This formula is true only for cyclic (inscribed in circle) quadrilaterals. But Brahmagupta and some later writers failed to mention this limitation, though it might have been contemplated by them. There are several ramifications of this theorem, which were later worked our by the mathematicians Mahavira (c. 850 AD) and Bhaskaracarya (1150 AD).

In the very early period p (the ratio of circumference to diameter) was calculated as 3 as whole number and no fraction. Baudhayana in his srulva-sutras used this rough value and also a formula that yielded p = 3.088. The early canonical works of the Jainas employed the value :

? = v (10). Aryabhata (499 AD) gave a much better value : ? = 3.1416. This value, correct up to four decimal places, was the most accurate till his time. But Brahmagupta, whatever the reasons, did not adopt it. He used 3 as 'practical value' and v (10) as 'neat value'. In ancient India Trigonometry was called Jyotpatti-ganita (jya = sine, utpatti = construction). The earliest mention of this word is found in the BSS of Brahmagupta. Sometimes that name was simplified to Jya-ganita. The modern name TrikoSamiti is a literal as well as phonetic rendering of the Greek word Trigonometry.

The Indian mathematicians usually employed three trigonometrical functions : jya, koti-jya and utkrama-jya. It should be noted that they are functions of an arc of a circle, but not of an angle. Thus, in the adjoining figure : If AP is an arc of a circle with centre at O, then jya = PM, koti-jya = OM and utkrama-jya = OA - OM. Hence their relation with modern trigonometrical functions will be : jya AP = R sin q, koti-jya AP = R cos ?, utkrama-jya AP = R - R cos ? = R versin q, where R is the radius of the circle and q the angle subtended at the centre by the arc AP

The Surya-siddhanta is the earliest Indian treatise in which these trigonometrical functions are found recorded. This work gives a table of Rsines and versed Rsines for every arc of 30 45' (or 225') of a circle of radius 3438'. sryabha˜a (499 AD) followed almost the same method in computing his trigonometrical tables. Indian mathematicians generally calculated tables of trigonometrical functions for every arc of 30 45', or twenty-four Rsines in a quadrant. Brahmagupta follows the method but takes the radius arbitrarily to be 3270'. The Indian methods of trigonometry were first adopted by the Arab mathematicians and later from them by the Europeans. This can easily be demonstrated by taking the term sine as an example. The Sanskrit term j_va (half-chord) was adopted by the early Arab mathematicians and was rendered as j_ba. Later it was corrupted in their tongue into jaib. The early Latin translators confused this word with the pure Arabic word jeba (pocket), which used to be made into a shirt in front of the 'bosom'. This 'bosom' was literally rendered into the Latin word sinus, which ultimately became sine. Similarly, the Sanskrit word ko˜i-jya, also written as kojya in abbreviated form, became ko-sinus or co-sinus, which finally became cosine.

SWEETMEAT

Brahmagupta's another work, the Khanda-khadyaka ('Sweetmeat'), a strange name, was written in 665 AD. Though the work does not deal with mathematical topics, it reveals a mature mathematician at work, and that when he was 67 years old. It is purely astronomical in nature. Divided in two parts and fourteen chapters, the work, a small tract, contains 265 verses. Whatever the reason, in this work Brahmagupta does not attack the methods of Aryabhata. On the contrary, at the very beginning he declares that he is going to produce results similar to Aryabhata, and for this he followed the ardharatrika (mid-night) system. For finding the trigonometrical functions of an arc, other than those whose values have been tabulated, the Indian mathematicians generally followed the principle of proportional increase. In the Khanda-khadyaka, Brahmagupta discusses a method of obtaining from a given table of sines, the sines of intermediate angles. But this process yields results correct only to a first degree approximation. More accurate results will be obtained by taking into consideration the second differences. Brahmagupta is the earliest Indian mathematician to do so. This more correct method of interpolation does not occur in his bigger work, the BSS, but in his monograph Dyanagrahopadhyaya and the later work Khanda-khadyaka. It can be said that a new branch of mathematics - Interpolation Theory - was initiated by Brahmagupta

IN ARABIC

In his own lifetime Brahmagupta's contribution might not have attracted the attention of scholars, but in less than two hundred years later his fame spread far and wide, beyond the borders of India. He was the first Indian mathematician-astronomer to be translated into Arabic. When Sindha came under the rule of Khalif al-Mansur (753-774 AD), envoys went from there to Bagdad, and among them were panditas who took with them two works of Brahmagupta, the Brahma-sphuta-siddhanta and the Khanda-khadyaka. Under al-Mansur's orders both these were translated into Arabic by Muhammad ibn Ibrahim al-Fazar_ and Ya'qub ibn Tariq with the help of Indian panditas. In Arabic they were named as Sindhind and Arkand respectively. Dr. Sachau, the translator of al-Biruni's Kitab al-Hind (Account of India), writes : "It was on this occasion that the Arabs first became acquainted with scientific system of astronomy. They learned from Brahmagupta earlier than from Ptolemy." Next important figure to propagate Indian astronomy and mathematics among the Arabs was al-Khwarizm_ (783 - c. 850 AD), one of the greatest mathematicians of his time. Al-Khwarizmi learnt Sanskrit, prepared an abridged edition of Sindhind (Brahma-sphuta-siddhanta) and wrote a treatise explaining the Indian system (decimal place-value system) of numeration. Again it was Brahmagupta's work that drew the attention of modern Western scholars to the rich mathematical heritage of ancient India. In 1817 Henry Thomas Colebrooke (1765-1837), who spent 32 years in India and is considered the first great Sanskrit scholar of Europe, published the translation of Bhaskaracarya's (1150 AD) BijagaSita and Lilavati and the two mathematical chapters (Ganitadhyaya and Kuttakadhyaya) of the Brahma-sphuta-siddhanta composed by Brahmagupta. Since then the original contribution of Brahmagupta, specially in the field of indeterminate analysis, has been acknowledged by all historians of mathematics and he is regarded as one of the greatest algebraists of the world. George Sarton, the noted historian of science, called Brahmagupta as 'one of the greatest scientists of his race and the greatest of his time'. In evaluating Brahmagupta's contribution, we should always keep in mind that he belonged to the seventh century AD

Sources

1. Brahma-sphuta-siddhanta of Brahmagupta, Edited with commentary by Pt. Sudhakara Dvivedin, Benares, 1902.

2. Ganaka-tarangni (Sanskrit), Pt. Sudhakara Dvivedi, Benares, 1933.

3. Alberuni's India, Edited by Edward C. Sachau, Delhi, 1964.

4.Diksita, Sankarar Balakrsna, Bharatiya jyotisa (Hindi), Lucknow, 1963.

5. Muley, Gunakar, Samsara ke Mahana Ganitajna (Hindi), New Delhi, 1994.

6. Bose, Sen and Subbarayappa, A Concise History of Science in India, New Delhi, 1989.

7. Srinivasiengar, C.N., The History of Ancient Indian Mathematics, Calcutta,1967

8. Smith, D. E., History of Mathematics (2 Vols.), New York, 1953.

9. Datta and Singh, History of Hindu Mathematics, Bombay, 1962.

10.Moritz, R. E., On Mathematics and Mathematicians, New York, 1943

11.. Sen, S. N. and Shukla, K. S. (Ed.), History of Astronomy in India, New Delhi, 1985.

12.Struik, Dirk J., A Concise History of Mathematics, London, 1959.

13.Volumes of the Indian Journal of History of Science, INSA, New Delhi.

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