PARTING THOUGHTS (III)

Dear Readers... This is going to be my last piece for these editorial columns. In November, 2000, I superannuate on attaining the age of 60 years.

What does, or should, one write on such an occasion? Actually there are quite a few things one could write about! However, ecause of reasons of space and time, I think I will only offer thanks to you, dear readers, for bearing with me all these months

This newsletter actually began regular monthly publication in August 98. Prior to that there may have been some issues which came out in the preceding months. The need for a vehicle like "Dream-2047" was felt soon after our publications programme started taking shape, with the release of several of our publications and other items of software. But, before plunging into this endeavour, we wanted to ensure that, once started, it would continue coming out on time and regularly. This is our 28th issue and I am sure you will continue to receive it regularly and on time in future as well!

It has been our endeavour right from the beginning, to keep our readers fully informed about Vigyan Prasar's activities and programmes _ past, present and future _ through "Dream-2047". Not only that, very early on, we added a popular science section to the newsletter which has been very well received. The VP team working on the newsletter has been, and is, making conscious efforts to continually improve and enhance its contents and maintain a very high standard as far as printing, get-up and layout are concerned. The large number of appreciative letters we receive every month is indicative of the success of our efforts. I am sure that these efforts will continue to attract your attention through future issues. .

Communicating with you through this newsletter, and especially these editorial columns, has been a very satisfying and rewarding experience for me personally. In fact, I do receive a fair number of letters every month from readers in response to the editorial in particular and other items in general. It has always been letters from readers which made many a day for me and kept me enthused and motivated to keep writing month after month.

One knows that Indian readers are not in the habit of picking up the pen and expressing themselves on paper, in black and white. Those who did take the trouble of doing so, therefore, deserve my very special thanks. I really am grateful! We know on the basis of readers' feedback, requests for inclusion of new names in our mailing list and for change in address, orders for our publications and other software items and so on, that "Dream-2047" has in a very short span of time built up a large, strong and committed subscriber-base of its own in the country, covering all those who are interested in, or concerned with science communication and in promotion of scientific attitudes among people in different parts of the country. Part of this has undoubtedly been also due to this newsletter being regular, on time and bilingual. We know also that many of our readers have been preserving their copies.

. There have been occasions when readers' response to the editorial was particularly overwhelming _ and memorable! Some instances that readily come to mind are editorials that had to do with "Making counting in Hindi simpler", "Indians in India and abroad", "Personal code of conduct", "Impossible dreams", "Over-dependence on government", "What about hardware for information technology", and more. I only wish that more and more readers would pick up the pen and let us have it _ the benefit of their views, suggestions and what they don't like, or like _ at least once in year! Is that asking for too much? If yes, choose your own frequency. But please do write back.

. Many thanks once again, to all those who wrote in, and goodbye!

NKS

Frederick William Herschel (1738-1822)

Dr. Subodh Mahanti

"The rise of Herschel is one conspicuous anomaly in the otherwise somewhat quiet and prosy eighteenth century. It proved decisive of the course of events in the nineteenth. It was unexplained by anything that had gone before, yet all that came after hinged upon it. It gave a new direction to effort; it lent fresh impulse to thought. It opened a channel for the widespread public interest which was gathering towards astronomical subjects to flow in."

That is how Ms. Clerke described, as quoted in Pioneers of Science by the British Physicist Sir Oliver Joseph Lodge (1851-1940), the rise of Frederick William Herschel (born as Friedrich Wilhelm Herschel) on the horizon of the 18th century astronomy. The whole tendency of eighteenth century astronomy was to complete and round off the earlier works. After Isaac Newton (1642-1727) and Marquis Pierre Simon de Laplace (1749-1827) astronomy appeared to be devoid of any challenge. It was thought that whatever was to be explained had already been explained and whatever there was to know had also ready been discovered. All practical astronomers were busy calculating and cataloguing. Nobody seemed to bother about pursuing new and original lines. There was only one exception towards the end of the century. That was Herschel, a man who had no formal training in science and who was brought up as a musician. He suddenly made astronomy exciting again. With his sheer enthusiasm and love of nature William Herschel could infuse into astronomy a healthy spirit of fresh life and activity. Herschel showed that none of the stars was really fixed as was assumed to be earlier. They were moving in all manner and ways. Thus Herschel changed the perception of heavens for man for ever. He reviewed, described and gauged the entire portion of the sky from his location not once but four times. He had devised a careful method of "sweeping" the skies. Each night Herschel would work only a small portion of the sky, usually a strip of only about two degrees. Herschel would explore such a strip twice every night before he retired to sleep. The next night he would explore an adjacent portion. In this way he would explore the entire sky visible from his location. Each survey occupied him several years. He discovered nebulae, double stars, variable stars, comets and satellites. He discovered and catalogued 2,5000 nebulae and 806 double stars.

Herschel's most startling discovery was the planet Uranus in 1781, which he originally thought to be a comet. The importance of this discovery could be gauged when we realise that since antiquity the existence of only five planets viz., Mercury, Venus, Mars, Jupiter and Saturn was known. There was no increase in the number of planets till Herschel discovered Uranus. Galileo and others did discover satellites but no primary planets. So the discovery of a new primary planet was an utterly unexpected novelty. Herschel discovered two satellites (Mimus and Enceladus) of Saturn and two (Oberon and Titania) of Uranus .

Herschel's contributions to astronomy were really astounding. He defined the discipline of Stellar Astronomy and led astronomers to focus their eyes far beyond the Solar System. Of course, he was fortunate to be alive at a time when prolonged viewing with a large reflector could not but be amply rewarded. Herschel utilised this opportunity to the fullest extent. His work is characterised by unbelievable comprehensiveness with which he extended the observations of other astronomers. For example following the publication of Charles Joseph Messier's (1730-1817) (catalogue known as Messier Catalogue) of 103 nebulae, star clusters and galaxies in 1781, Herschel began a systematic search for the non-stellar objects. His systematic effort revealed 2,500 such objects which he listed in three catalogues (published in 1786, 1789 and 1802). He not only began the study of double stars but catalogued over 800 of them. It was Herschel's work on double stars that provided the first demonstration of the assertion that Newton's laws of gravitation are also applicable outside the solar system.

Herschel's contribution to astronomy was not confined to the observational aspect alone. He made theoretical contributions on the structure of the Universe. He was the first to establish the motion of the Sun alongwith its planets and their satellites and other objects in the Solar System. He found that the Sun is moving in space relative to its stellar neighbours towards a point in the Hercules Constellation not far from the bright star Vega. He began to study the structure of the galaxy. His famous paper On the Construction of the Heavens, published in 1784, produced a model of the Milky Way (the Galaxy which contains our Sun) as a non-uniform aggregation of stars. Herschel concluded from his star counts (he had counted over 90,000 stars) that the Milky Way had the shape of a disk, like a grindstone. Herschel placed the Sun near the centre of the Milky Way. Later studies confirmed Herschel's deduction that our Galaxy is disk-shaped but the Sun was found far from the centre and the Galaxy was found to be much larger than Herschel originally supposed. The Sun is 30,000 light years away from the centre of the Milky Way. In 1800, using a thermometer and prism, he discovered infrared radiation. In 1802 Herschel coined the term 'Asteroids' for the new celestial objects discovered by Giuseppe Piazzi (1746-1826) and Heinrich Wilhelm Matthans Olbers (1758-1840). In 1821 Herschel was elected president of the Royal Astronomical Society. Herschel was responsible for launching the field of solar influence on Earth's atmosphere.

Herschel was born on November 15, 1738 at Hanover, Germany, His father, Isaac Herschel, a military musician and a cultured man, was an oboe player in a military regiment. His mother Anna Ilse Moritzen had no education. Isaac and his wife had ten children. However, four of them died very young. William Herschel and the other five surving children were brought up as musicians. Isaac also instilled an interest in astronomy among his children by his fireside talks and visual observations. At the age of seventeen Herschel became oboist to the Hanoverian guards the same military musical orchestra on which his father played. In 1756 the regiment was sent to England for some months. Herschel learnt the English language quickly. He served the regiment only for two years. In 1757 Herschel and his brother Anton Jakob left for England on a ship from Hamburg. They planned to earn a living as musicians. After Jacob's return to Hanover, Herschel joined the Orchestra of the Earl of Darlington at Richmond. In 1762 Herschel moved to Leeds. After staying four years at Leeds Herschel moved to Halifax where he stayed for some months. On December 9, 1766 Hershcel went to Bath, where he stayed for 15 years. On October 4, 1767, he became the organist at the Octagon Chapel in Bath. Situated in south-west England, on the bank of the river Avon. Bath was an early Roman centre known as Aquae Sulis because of its hot natural springs, at a temperature of around 49O C. The elaborate Roman baths have survived and are considered the best Roman remains in England. Bath became fashionable as an elegant town in the 18th century. In Bath he lived a hard and successful life. He taught many pupils and wrote many musical pieces.

After the death of his father in 1765, his brother Alexander joined him in Bath. His sister Caroline Lucretia Herschel (1750-1848), who herself devoted her entire life to astronomy, joined him in 1772 when she was twenty-two. Initially Caroline did not share her brother's passion for astronomy. However, at the age of 32 she become on apprentice to her brother. Caroline discovered eight comets and the companion of the Andromeda Nebula. She was the first women to play a key role in astronomy. In 1787 she was granted an annual salary of 50 pounds by the king as her brother's assistant at Slough. This was the first instance when a woman was recognised for her scientific position. Her Index to Flamsteed's Observations of the Fixed Stars alongwith a list of errata was published by the Royal Society in 1798. After William's death Caroline went back to Hanover where she worked on the reorganisation of his catalogue of nebulae.

While leading the musical life of Bath, Herschel became deeply involved in optics and astronomy. At night Herschel studied mathematics, optics, Italian or Greek. His position as organist gave him enough money to help finance his growing interest in astronomy. Herschel had become so fascinated with astronomy that he read hundreds of books on astronomy, calculus and optics. He also bought a small telescope and spent most of his nights by gazing at the sky. And as time passed Herschel's enthusiasm for watching the night sky increased. To Herschel the night sky was a vast, dark ocean mostly uncharted and filled with the perpetual promise of new discoveries. As Caroline would later recall: "If it had not been for the intervention of a cloudy or moonlit night, I knew not when he or I either would have got any sleep."

The first obstacle that Herschel encountered in exploring the poorly uncharted sky was the lack of good telescope. However, he was not daunted. He decided to build a telescope himself. In 1774 he was successful in making a 5.5 ft telescope. He was never tired of making better and better instruments. William Herschel and his sister Caroline were considered to be the world's best telescope-makers of their time. They built a large number of telescopes culminating in the enormous 48-feet (12 m) reflector.

Leisure was unknown to Herschel. He grinded mirrors in the day, performed in concerts in the evening and spent the night gazing at the sky. How his health permitted all this was a wonder. The way they worked could be guessed from the observations made by Caroline in her diary: "My time was taken up with copying music and practising, besides attendance on my brother when polishing, since by way of keeping him alive I was constantly obliged to feed him by putting the victuals by bits into his mouth. This was once the case when, in order to finish a 7-foot mirror, he had not taken his hands from it for sixteen hours together. In general he was never unemployed at meals, but was always at those times contriving or making drawings of whatever came in his mind. Generally I was obliged to read to him whilst he was at the turning-lathe, or polishing mirrors_Don Quixote, Arabian Nights' Entertainments, the novels of Sterne, Fielding, &c.; serving tea and supper without interrupting the work with which he was engaged,...and sometimes lending a hand. I became, in time, as useful a member of the workshop as a boy might be to his master in the first year of his apprenticeship... But as I was to take part the next year in the oratorios, I had, for a whole twelvemonth, two lessons per week from Miss Fleming, the celebrated dancing-mistress, to drill me for a gentlewoman. So we lived on without interruption. My brother Alex, was absent from Bath for some months every summer, but when at home he took much pleasure in executing some turning or clockmaker's work for his brother."

On 13 March 1781 Herschel in one of his careful 'sweeps' discovered the planet Uranus. Before Herschel many astronomers had noticed it but they invariably took it as a star and so they did not further ponder over it. But Herschel had memorised the positions of thousands of stars and so when he found Uranus, he was sure that no star could be expected in that position. Describing his finding Herschel wrote to the Royal Society: "On this night in examining the small stars near _ Geminorum, I perceived one visibly larger than the rest. Struck with its uncommon appearance, I compared it to _ Geminorium and another star, and finding it so much larger than either I suspected it to be a comet."

Professional astronomers computed the orbit of Herschel's 'comet' and it was found to move in nearly a circle and not in elongated ellipse that a comet would be expected to move. So it was a new planet, more than 100 times bigger than the Earth and nearly twice as far way as Saturn. Herschel wished to name it after his patron as Georgium Sidus (George's Star) but finally the name 'Uranus' was universally adopted. This unique and utterly unexpected discovery, as only five planets were known since the prehistoric times and there was no increase in their number, made Herschel famous overnight. The Royal Society made him a Fellow the same year, the Oxford University awarded him a doctoral degree and what is more the king of United kingdom and Ireland, George III (1738-1820) appointed him his court astronomer. Thus the discovery of Uranus made Herschel a practical astronomer. So from Bath, Herschel moved to a small house at Datchet, near Windsor. Herschel was awarded an annual stipend by the king. This enabled Herschel to devote all his time to astronomy without resorting to earning a living as a professional musician. However, his stipend was not enough to take care of his experimental work and his frequent trips to London. So Herschel and his sister had to supplement their income by making and selling telescopes. Among Herschel's customers were the King of Spain, the Prince of Canino, the Russian court and the Austrian Emperor. One of his instruments was sold to China. Herschel also made telescopes for well-known astronomers like Johann Elert Bode (1742-1826) Johann Hieronymus Schroter (1745-1816), Giuseppe Piazzi (1746-1826) and John Pond (1767-1836).

With the king's patronage Herschel was able to build a telescope with a 48 in (1.22m) mirror and a focal length of 40 ft (12.2m) the largest in the world then. The telescope cost the king 400 pounds plus 200 pounds a year for its maintenance. The eyepiece of the telescope was attached to the open end. This arrangement eliminated the loss of light caused by the secondary mirror used in Newtonian and Gregorian reflectors. However, there was a serious disadvantage arising out of this arrangement. One was required to climb up to the open end in the dark. While doing this manouvering Giuseppe Piazzi fell and broke his arm. With this telescope Herschel in 1787 discovered the two satellites of Uranus and two more of Saturn. (It may be noted that after Herschel's discovery the two satellites of Uranus were not seen till some forty years later, when his son Sir John Frederick William Herschel (1792-1871) observed them again). The telescope was dismantled in 1839 while John conducted his family in a special requiem specially composed for the occasion.

On May 8, 1788, Herschel at the age of 50, married Mrs. Mary Pitt, a wealthy widow and his financial worries came to an end. They moved to a more spacious house at Slough, where he remained for the rest of his life. However, after the marriage Caroline lived in lodgings and went over at night time to help Herschel observe. Herschel and his sister for all practical purpose turned night into day. They slept only during daytime as they often watched the sky till daylight. At the time they moved from Datchet to Slough, Caroline wrote: "The last night at Datchet was spent in sweeping till daylight, and by the next evening the telescope stood ready for observation at Slough."

Before Herschel the stars had been mainly observed for nautical and other practical purposes. Astronomers noted their times of rising and setting but nobody bothered to observe them in detail. They were just considered as fixed points of reference. People treated them as clock or piece of dead mechanism. In the meantime all the attention of astronomers was concentrated on the Solar System, studying the planets and satellites. Tycho Brahe (1546-1601) patiently and meticulously observed the positions and movements of the planets over two decades . Based on Tycho's observations Johannes Kepler (1571-1630) formulated his celebrated laws of planetary motions. Galileo Galilie (1564-1642) discovered their peculiarities and satellites. Newton and Laplace elaborated every detail of their laws. But as far as the stars were concerned they remained the same as Ptolemaic system assumed them to be, some fixed points in the sky. Herschel found that the stars were not of the same kind. He found a variety in them. Every star was not at the same distance from the Earth. The stars were not at rest, Herschel found them moving and full of activity. It is worthwhile to note that in 1718 Sir Edmond Halley (1656-1742) pointed out that stars had proper motion. His conclusion was based on the fact the brightest stars had changed position since the time of Ptolemy's Almagest. The stars were moving in all directions and in all manners. He found stars revolving around stars at mind boggling distances but at the same time obeying the law of gravitation. It may be noted that the stars are at such great distances that by moving even at very high speeds their positions may appear to us unchanged for thousands of years. For Herschel every star was a Solar System.

Herschel was fascinated with the nebulae, the mysterious objects as initially they appeared to be. He freely speculated on the nature of these objects. Herschel considered them in various forms _ as cluster of stars, other universes at almost infinite distances or nascent stars. He visualised the universe as conglomeration of innumerable worlds _ some dead, some old, some at the prime stage of their life and some in their infancy or in the process of being born. He likened the universe to a garden with all manner of plants growing at different stages.

Herschel died on August 25, 1822.

In Herschel's time any fixed, extended and somewhat fuzzy white haze observed in the sky with a telescope was termed nebula. Many of these objects can now be resolved into clouds of individual faint stars and have been identified as galaxies. As early as in 1864 William Huggins (1824-1910) demonstrated that true 'nebulae' could be distinguished from those composed of stars on the basis of their spectra. Now-a-days the term nebula refers to a gaseous nebula, which cannot be resolved into individual stars and consists, for most part, of interstellar dust and gas. The gaseous nebulae have been classified into three broad groups viz., emission nebulae, reflection nebulae and dark nebulae.

 An emission nebula is a luminous cloud of gas and dust in space which shines with its own light. For example the Orion Nebula, H II regions, and planetary nebulae. Emission nebulae glow brightly because the gas in these nebulae is energised by stars that have formed within them. There are other ways by which the light can be generated. A gas cloud or nebula can glow because it has become ionised in a violent collision with another gas cloud. Herbig-Haro objects (small nebulae found in the regions of star formation) are examples of emission nebulae. In Crab Nebula, a supernova remanent, the light is produced by a process known as synchrotron radiation (electromagnetic radiation emitted by a charged particle moving in a magnetic field at a velocity very close to that of light).

Reflection nebulae appear bright because they reflect or scatter starlight. A reflection nebula surrounds the stars of the Pleiades cluster, a star cluster about 400 light years away in the Constellation Taurus, representing the seven Sisters of Greek mythology.

A dark nebula is a dense cloud, composed of interstellar gas and dust which partially or completely absorbs light behind it. Examples include the Coalsack Nebula in Crux, the smallest Constellation better known as the Southern Cross and the Horsehead Nebula in Orion, a magnificent Constellation on the celestial equator.

Tarantula Nebula, an emission nebula, is the largest and the brightest nebula. Its name comes from its spider-like shape (tarantula is the name of a tropical or subtropical American spider). The nebula is located in the Large Magellanic cloud. It has a diameter of over 800 light years with faint extensions to 6000 light years and contains half-a-million solar masses of ionised gases. The ionisation is caused by several clusters of O and B stars. O star is the brightest, hottest and most massive of all normal, hydrogen-burning stars. B-type stars are hot and appear blue in colour, emitting strongly in the ultraviolet. Normally they have masses upto 25 solar masses and luminosities as high as 260,000 times the Sun's.

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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I have enjoyed the article or Galileo (published in July issue) very much. It is interesting to learn from your article that Leonard Digges built a telescope more than 60 years before Lippershey. Can you please send me the paper in which this fact has been established ? In this connection I would like to mention that out of jealously, Galileo did not send a telescope to Kepler though kepler asked for it. I am quoting a few lines from the book `Coming of Age in the Milky Way' by Timothy Ferris.

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