Srinivasa Ramanujan

The arithmetic class was in progress. The teacher was solving questions on division. On the backboard were drawn three bananas.

“We have three bananas,” the teacher said, “and we have three boys. Can you tell me how many each will get?”

A smart boy in the front row replied, “Each will get one.”

“Right,” the teacher said. “Now, similarly, if 1,000 bananas are distributed among 1,000 boys, each will get one, isn’t that so?”

While the teacher was explaining, a boy sitting in one corner raised his hand and stood up. The teacher stopped and waiting for the boy to speak.

“Sir,” the boy asked, “if no bananas is distributed among no one, will everyone still get one banana?” There was roar of laughter in the class. What a silly question to ask!

“Quite,” the teacher said loudly and thumped the desk. “There is nothing to laugh at. I will just explain what he means to say. For the division of bananas, we divided three by three, saying that each boy will get one banana. Similarly, we divided 1,000 by 1,000 to get one. What he is asking is that if zero banana is divided among zero, will each one get one? The answer is ‘no’. Mathematically, each will get an infinite number of bananas!”

Everyone laughed again. The boys understood the trick arithmetic had played upon them. What they could not understand was why the teacher later complimented the boy who had asked that absurd question.

The boy had asked a question that had taken mathematicians several centuries to answer. Some mathematicians claimed that zero divided by zero was zero. Others claimed it to be unity. It was the Indian mathematician Bhaskara who proved that it is infinity. The boy who asked the intriguing question was Srinivasa Ramanujan. Throughout his life, whether in his native Kumbakonam or Cambridge, he was always ahead of his mathematics teachers.

Ramanujan was born at Erode in Tamil Nadu on December 22, 1887. His father was a petty clerk in a cloth shop. From early childhood it was evident that he was a prodigy. Senior students used to go to his dingy house to get their difficulties in mathematics solved. At the age of 13 Ramanujan was able to get Loney’s Trigonometry from a college library. Not only did he master this rather difficult book but also began his own research. He came forth with many mathematical theorems and formulae not given in the book, though they had been discovered much earlier by great mathematicians.

The most significant turn came two years later when one of his senior friends showed him Synopsis of Elementary Results in Pure and Applied Mathematics by George Shoobridge Carr. For a boy of 15 the title itself must be frightening, but Ramanujan was delighted. He took the book home and began to work on the problems given in it. This book triggered the mathematical genius in him.

Mathematical ideas began to come in such a flood to his mind that he was not able to write all of them down. He used to do problems on loose sheets of paper or on a slate and to jot the results down in notebooks. Before he went abroad he had filled three notebooks, which later became famous as Ramanujan’s Frayed Notebooks. Even today mathematicians are studying them to prove or disprove the results given in them.

Although Ramanujan secured a first class in mathematics in the matriculation examination and was awarded the Subramanyan Scholarship, he failed twice in his first year arts examination in college, as he neglected other subjects such as history, English and physiology. This disappointed his father. When he found the boy always scribbling numbers and not doing much else, he thought Ramanujan had gone mad. “To set him right”, he forced his son to marry. The girl chosen was eight-year-old Janaki.

Ramanujan began to look for a job. He had to find money not only for bread but for paper as well to do his calculations. He needed about 2,000 sheets of paper every month. Ramanujan started using even scraps of paper he found lying in the streets. Sometimes he used a red pen to write over what was written in blue ink on the piece of paper he had picked up.

Unkempt and uncouth, he would visit offices, showing everyone his frayed notebooks and telling them that he knew mathematics and could do a clerical job. But no one could understand what was written in the notebooks and his applications for jobs were turned down.

Luckily for him, he at last found someone who was impressed by his notebooks. He was the Director of the Madras Port Trust, Francis Spring, and he gave Ramanujan a clerical job on a monthly salary of Rs. 25. Later some teachers and educationists interested in mathematics initiated a move to provide Ramanujan with a research fellowship. On May 1, 1913, the University of Madras granted him a fellowship of Rs. 75 a month, though he had no qualifying degree.

A few months earlier, Ramanujan had sent a letter to the great mathematician G.H. Hardy, of Cambridge University, in which he set out 120 theorems and formulae. Among them was what is known as the Reimann series, a topic in the definite integral of calculus. But Ramanujan was ignorant of the work of the German mathematician, George F. Reimann, who had earlier arrived at the series, a rare achievement. Also included was Ramanujan’s conjecture about the kind of questions called “modular”. Pierre Deligne subsequently proved this conjecture to be correct. He also gave a key formula in the hypergeometric series, which came to be named after him.

It did not take long for Hardy and his colleague, J.E. Littlewood, to realize that they had discovered a rare mathematical genius. They made arrangements for Ramanujan’s passage and stay at Cambridge University. On March 17, 1914, he sailed for Britain.

Ramanujan found himself a stranger at Cambridge. The cold was hard to bear and, being a Brahmin and a vegetarian, he had to cook his own food. However, he continued his research in mathematics with determination. In the company of Hardy and littlewood he could forget much of the hardship he had to endure.

In Ramanujan Hardy found an unsystematic mathematician, similar to one who knows the Pythagoras theorem but does not know that a congruent triangle means. Several discrepancies in his research could be attributed to his lack of formal education. Ramanujan played with numbers, as a child would with a toy. It was sheer genius that led him to mathematical “truths”. The task of proving them, so important in science, he left to lesser mortals.

Ramanujan was elected Fellow of the Royal Society on February 28, 1918. He was the second Indian to receive this distinguished fellowship. In October that year he became the first Indian to be elected Fellow of Trinity College, Cambridge. His achievements at Cambridge include the Hardy-Ramanujan-Littlewood circle method in number theory, Roger-Ramanujan’s identities in partition of integers, a long list of the highest composite numbers, besides work on the number theory and the algebra of inequalities. In algebra his work on continued fractions is considered to be equal in importance to that of great mathematicians like Leonard Eular and Jacobi.

While Ramanujan continued his research work, tuberculosis, then an incurable disease, was devouring him. Ramanujan was sent back to India and when he disembarked, his friends found him pale, exhausted and emaciated. To forget the agonizing pain, he continued to play with numbers even on his deathbed. On April 26, 1920, he died at Chetpet in Madras.

Besides being a mathematician, Ramanujan was an astrologer of repute and a good speaker. He used to give lectures on subjects like “God, Zero and Infinity”.

Bhaskara

Leelavati looked entranced at the water-clock her father had brought home. Its movements were fascinating. She had a slight feeling of guilt, for her father had told her never to enter that room. But because it was forbidden territory, its exploration gave her a sense of adventure. And she continued looking at the clock.

Then came disaster, though she was never to know about it. A tiny pearl slipped out of her nose-ring and fell into the clock. She was so alarmed that she fled. And in the excitement of the arrangements being made for her wedding the next day she forget all about the clock and the pearl. Which was not surprising for she was only six years old.

Leelavati was married, but a week later her husband fell off a cliff and died. This was what her father, Bhaskara, a great mathematician and astrologer, had feared. Astrological calculations had shown Bhaskara that, if the marriage of his daughter was not performed at a particular hour on that particular day, she would become a widow. And he had bought the water-clock to ensure that he would know the right time. He did not know that the pearl in it had made the clock inexact. And, going by that clock, he had made an error. Bhaskara thought that it was his astrological calculations that had gone wrong and blamed himself for the tragedy.

In those days, widowed girls were not allowed to marry again. Bhaskara, therefore, began to try to arouse her interest in mathematics, so that she would forget her grief. It is not known how good a mathematician she turned out to be, but he made her immortal in the history of mathematics in India by titling after his daughter a chapter of the book Siddhantasiromani that he wrote when he was only 30 years old. At one time there was even a popular saying: “Whosoever is well-versed with Leelavati can tell the exact number of leaves on a tree.”

The part of the book titled Leelavati dealt essentially with arithmetic. The other three parts were on different aspects of mathematics: Bijaganita dealt with algebra, Coladhyaya with spheres and Grahaganita with planetary mathematics. Basically the book was a text-book, a collection of the works of some eminent scholars like Brahmagupta, Mahavira and Sridhara, after they had been simplified to help students. The book contained problems presented in such a way as to stimulate the student’s interest. It was so popular and authoritative that four to five centuries later it was translated twice into Persian.

Bhaskara was an original thinker, too. He was the first mathematician to declare confidently that any term and infinity is infinity.

In algebra, Bhaskara considered Brahmagupta his guru and mostly extended Brahmagupta’s work. But his introduction of Chakrawal, or the cyclic method, to solve algebraic equation is a remarkable contribution. It was only after six centuries that European mathematicians like Galois, Euler and Lagrange rediscovered this method and called it “inverse cyclic”. Determination of the area and volume of a sphere in a rough integral calculus manner was also mentioned for the first time in his book. It contained some important formulas and theorems in trigonometry and permutation and combination.

Bhaskara can also be called the founder of differential calculus. He had conceived it several centuries before Isaac Newton and Gottfried Leibniz, who are considered in the West to be the founders of this subject. He had even given an example of what is now called “differential coefficient” and the basic idea of what is now known as “Rolle’s theorem”. Although Bhaskara attained such excellence in calculus, no one in the land took any notice of it.

As an astronomer Bhaskara is renowned for his concept of Tatkalikagati, which means instantaneous motion. This enables astronomers to determine the motion of the planets accurately.

Bhaskara was born in 1114 at Bijjada Bida (Bijapur, Karnataka) in the Sahyadri Hills. He learnt mathematics from his saintly father. Later, the works of Brahmagupta inspired him so much that he devoted himself entirely to mathematics. At the age of 69 he wrote his second book, Karanakutuhala, a manual of astronomical calculations. Though it is not as well known as his other book, it is still referred to in making calendars.

Brahmagupta

The mathematician who first framed the rules of operation for zero was Brahmagupta. He was also to give a solution to indeterminate equations of the type ax² + 1 = y² and the founder of a branch of higher mathematics called “Numerical analysis”. No wonder Bhaskara, the great mathematician, conferred on him the title of Ganakachakrachudamani, the gem of the circle of mathematicians.

Brahmagupta was born at Bhillamala (Bhinmal), in Gujarat, in 598 A.D. He became court astronomer to King Vyaghramukha of the Chapa dynasty. Of his two treatises, Brahmasphutasiddhanta and Karanakhandakhadyaka, the first is the more famous. It was a corrected version of the old astronomical text, Brahmasiddhanta. It was translated into Arabic, but erroneously titled Sind Hind. For several centuries the treatise remained a standard work of reference in India and the Arab countries.

Brahmasphutasiddhanta also contains chapters on arithmetic and algebra. Brahmagupta’s major contribution is the rules of operation for zero. He declared that addition or subtraction of zero to or from any quantity, negative or positive, does not affect it. He also added that the product of any quantity by zero is zero and division of any quantity by zero is infinity. He, however, wrongly claimed that division of zero by zero was zero.

He also framed rules to solve a simple equation of the type ax + b = 0 and a quadratic equation of the type ax² + bx + c = 0, as well as methods to sum up a geometric series. Besides, he noted the difference between algebra and arithmetic and so was the first mathematician to treat them as two separate branches of mathematics.

Brahmagupta’s Karanakhandakhadyaka is a hand-book on astronomical calculations. In this he effectively used algebra for the first time in calculations. But Brahmagupta always was careful not to anger the priests. His wives were orthodox, in keeping with the beliefs held during those times, and he criticized Aryabhata, who said the earth was not stationary. But he believed that the earth was round.

About gravity he said: “Bodies fall towards the Earth”.

Patanjali

Although the Upanishads and the Atharvaveda mention yoga, it was only in the second century B.C. that its fundamentals and techniques were adequately presented. The man who did this was Patanjali in his Yogasutras (Yoga aphorisms).

According to Patanjali, there are channels called nadi and centres called chakra in the human body. If these are tapped, the hidden energy in the body called Kundalini can be released, enabling the body to acquire “supernatural” powers. Patanjali gives eight stages. Yama (universal moral commandments), Niyama (self-purification through discipline), Asana (posture), Pranayama (breath-control), Pratyahara (withdrawal of mind from external objects), Dharana (concentration), Dhyana (meditation), and Samadhi (state of superconsciousness). It is the last stage which is the more difficult and which enables one to attain “Godhead”.

Patanjali gives a beautiful analogy to explain how God is attained through yoga. Our mind, he says, is like the surface of a pond. Just as we cannot see the jewel lying at the bottom of the pond because of the breeze disturbing its surface, we do not see the divine in us because our mind is in constant agitation. If the surface of the pond is undisturbed, one can see the jewel at the bottom. So also, if the mind is kept calm, with the doors to the external world closed, one can see God within.

Only in the last few decades have scientists begun to recognize the powers of yoga. It has now been established by experiments that through the practice of yoga several ailments, mental and physical, can be cured. Tests conducted on yogis show that they do acquire extraordinary powers. For instance, they can live without oxygen or food for a long a time. Research is now in progress in laboratories all over the world to probe further the merits of yoga. What Patanjali advocated several centuries is now receiving the attention it deserves.

Aryabhata

Nearly five hundred years after the birth of Christ a ritual was held near Khagola, the famous astronomical observatory at the University of Nalanda near Kusumapura (Patna), to mark the “birth” of a treatise that was to lay the foundation of a new school of thought in astronomy.

When the bell at the university tolled at 12 noon on March 21,499 A.D., a chorus of vedic chants filled the air. And priests, after prayers before a havan, led a 23-year-old astronomer to a platform. Silence prevailed as the astronomer sprinkled holy water on the parchment and pen lying on a desk placed on the platform. Chanting holy verses, he gazed at the sun overhead and prostrated himself in obeisance before sitting at the desk. Taking the pen, he wrote the first letter of the treatise while the priests chanted slokas and the large crowd of learned men showered flowers on him.

The young astronomer was Aryabhata and the treatise was Aryabhatiya. Born in 476 in Kerala, Aryabhata had come to complete his studies at the University of Nalanda, which was then a great centre of learning. When his treatise was recognized as a masterpiece, the then Gupta ruler Buddhagupta, made him head of the university.

Aryabhata was the first to deduce that the earth is round and that it rotates on its own axis, creating day and night. He declared that the moon is dark and shines only because of sunlight. Solar and lunar eclipses, he believed, occurred not because Rahu gobbled the sun and the moon, as Hindu mythology claimed, but because of the shadows cast by the earth and the moon.

He, however, believed in the geocentric concept of the universe that the earth is the center of the universe. To explain the “erratic” movements of some planets, he like the Greek king Ptolemy, made use of “epicycle”. But his method was superior to Ptolemy’s.

In mathematics Aryabhata’s contributions are equally valuable. He gave the value of ? (pi) as 3.1416, claiming, for the first time, that it was an approximation. And he was the first mathematician to give what later came to be called the tables of Sines. His method to find a solution to indeterminate equations of the type ax – by = c is also recognized the world over. He also devised a novel method to express large numbers such as 100,000,000,000 in words. He developed this method to write unwieldy numbers in poetic form. The consise but somewhat difficult-to-grasp Aryabhatiya also dealt with other aspects of mathematics and astronomical calculations, namely, geometry, mensuration, square root, cube root, progression and celestial sphere.

In his old age Aryabhata also wrote another treatise, Aryabhatasiddhanta. It was a text-book for day-to-day astronomical calculations as well as a guide to determine auspicious times for various rituals. Even today Aryabhata’s astronomical data are used in preparing panchangs (Hindu calendars).

Kanada

What is the relationship between man, the universe and their creator? This question has always intrigued philosophers and thinkers. Between 600 B.C. and 200 A.D. there were several attempts by Indian philosophers to find an answer to the question.

One of these was Kanada, who, in about 600 B.C. at Prabhasa propounded the Vaisesikasutra (Peculiarity Aphorisms). Today, we realize that these sutras are a blend of science, philosophy and religion. Their essence is the atomic theory of matter. If Kanada’s sutras are analysed, one would find that his atomic theory was far more advanced than those forwarded later by the Greek philosophers, Leucippus and Democritus. In fact, he gave the name paramanu (atom) to an indivisible entity of matter.

According to Kanada, everything is made up of paramanu. When matter is divided, then further divided, till no further division is possible, the remaining indivisible entity is called paramanu. This entity does not exist in a free state, nor can it be sensed through any human organ. It is eternal and indestructible.

Kanada added-and it is here that he took a lead over other philosophers-that there are a variety of paramanu as different as the different classes of substances then believed to exist, namely, earth, water, air and fire. Each paramanu has a peculiar property which is the same as the class of substance it belongs to. It is only because of this peculiarity of paramanu that the theory was called Vaisesikasutras.

Kanada also claimed that an inherent urge made one paramanu combine with another. If two paramanu belonging to one class of substance combined, a dwinuka (binary molecule) was produced, which had properties similar to those of the two original paramanu. Paramanu belonging to different classes of substance could also combine in large numbers.

The idea of chemical change was also put forward by Kanada. He claimed heat was responsible for any change. The properties of paramanu also changed when heated. He gave the examples of the blackening of a new earthen pot and the ripening of a mango to illustrate the action of heat. All things seen in the universe were, therefore, formed, Kanada said, because of the peculiarity of paramanu, their variety, the variety of ways in which they combined and the action of heat.

Susruta

It was midnight when Susruta was awakened by a frantic knocking at the door.

“Who’s out there?” asked the aged doctor, taking a lighted torch from its socket in the wall and approaching the door.

“I’m a traveler, by revered Susruta,” was the anguished reply. “A tragedy has befallen me. I need your help…”

Susruta opened the door. What he saw was a man kneeling before him, tears flowing from his eyes and blood from his disfigured nose.

“Get up, my son, and come in,” said Susruta. “Everything will be all right. But be quiet, now.”

He led the stranger to a neat and clean room, with surgical instruments on its walls. Unfolding a mattress he asked him to sit on it after taking off his robe and washing his face with water and the juice of a medicinal plant. Susruta then offered the traveler a mug of wine and began preparing for the operation.

With a large leaf of creeper brought from the garden, he measured the size of the stranger’s nose. Taking a knife and forceps from the wall, he held them over a flam and cut a strip of flesh from the stranger’s cheek. The man moaned, but the wine had numbed his senses.

After banding the cut in the cheek, Susruta cautiously inserted two pipes into the stranger’s nostrils and transplanted the flesh to the disfigured nose. Moulding the flesh into shape he dusted the nose with powdered liquorice, red sandalwood and extract of Indian barberry. He then enveloped the nose in cotton, sprinkled some refined oil of sesame on it and finally put a bandage. Before the traveler left, he was given instructions on what to do and what not to and a list of medicines and herbs he was to take regularly. He was also asked to come back after a few weeks to be examined.

In this manner did Susruta mend a nose some 26 centuries ago. And what he did is not greatly different from what a plastic surgeon would do today. In fact, Susruta is today recognized as the father of plastic surgery all over the world. His treatise, Susrutasamhita, has considerable medical knowledge of relevance even today. It indicates that India was far ahead of the rest of the world in medical knowledge. In the eighth century A.D. Susrutasamhita was translated into Arabic as Kitab-Shaw Shoon-a-Hindi and Kitab-i-Susrud.

Born in the sixth century B.C., Susruta was a descendant of the Vedic sage Visvamitra. He learnt surgery and medicine at the feet of Divodasa Dhanvantari in his hermitage at Varanasi. Later, he became an authority in not only surgery but other branches of medicine.

He was the first physician to advocate what is today known as the “caesarean” operation. He was also expert in removing urinary stones, locating and treating fractures and doing eye operations for cataract. Several centuries before Joseph Lister, he put forth the concept of asepsis. His suggestion to give wine to patients about to be operated upon makes him also the father of anaesthesia.

In his treatise, Susruta lists 101 types of instruments. His Samdamsa yantras are the first forms of the modern surgeon’s spring forceps and dissection and dressing forceps. In fact, his system of naming surgical tools after the animals or birds they resemble in shape, for example crocodile forceps, hawkbill forceps, is adopted even today.

Susruta was also an excellent teacher. He told his pupils that one could become a good physician only if one knew both theory and practice. He advised his pupils to use carcases and models for practice before surgery.

iaddition to classifying worms that infect the human body, leeches for bloodletting, medicinal herbs, alkalis and metals, Susruta gave a vague classification of animals.