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A Brief History of Ancient Indian Mathematics
|by Dr. Rajen Barua|
India’s Romance with Numbers
It is without doubt that mathematics, the science of numbers, today owes a huge debt to the outstanding contributions made by Indian mathematicians over many hundreds of years. From zero to geometry, Indian mathematicians made some great historical achievements. Frankly speaking, without the Indian numerals, mathematics, as we know it today, would simply not exist. This article is to focus on some of these achievements and to show that mathematics was not only very much rooted in Indian soil but also that ancient Indians had great romance with mathematics.
Indian mathematics may be said to have started with the Vedic rituals which required knowledge of geometry for accurate construction of Vedic altars. It developed further under the Jain and the Buddhist scholars who pioneered some phenomenal ground level achievements. It is now generally admitted that the Indian system of numbers has its roots firmly planted in India and that it is the Indians who first invented and used the decimal place value system including the use of the zero. The novel Indian numerals were subsequently adopted by the Arabs, and eventually became known to Europe as Arabic numerals. The ancient Indians provided a unique, useful, flexible and intuitive model for the world to use.
The great French Mathematician, Pascal was one who appreciated the contributions of the Indians and he put this with great clarity, when he commented, “The ingenious method of expressing every possible number using a set of ten symbols (each symbol having a place value and an absolute value) emerged in India. The idea seems so simple nowadays that its significance and profound importance is no longer appreciated. Its simplicity lies in the way it facilitated calculation and placed arithmetic foremost amongst useful inventions. The importance of this invention is more readily appreciated when one considers that it was beyond the two greatest men of Antiquity, Archimedes and Apollonius.”
India has a long tradition, both historical and mythical, of its fascination with numbers. In ancient India, mathematics was considered as one of the highest sciences. There is a statement in the Vedanga Jyotisa, which proclaims, “As are the crests of a peacock, as are the gem-stones of a snake, placed on the highest place of the body, the forehead, so is mathematics (Ganita) the head of all Vedah and shastras.” The quotation suggests a reverent, almost elitist concept of mathematics in ancient India. In another mythical statement we find, “What is the use of much speaking. Whatever object exists in this moving and nonmoving world, can not be understood without the base of Ganita (Mathematics)”. Three thousands years later, Gaileo would realise the same when he said, “It (the universe) is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures, without which it is humanly impossible to understand a single word; without these, one is wandering about in a dark labyrinth.”
The tradition of mathematics in India, in fact, started much earlier of which we have solid historical facts in the Indus valley which was associated with the Harappan civilization established around 2,500 B.C. We do know that the Harappans had adopted a uniform system of weights and measures. An analysis of the weights discovered suggests that they are decimal in nature, giving for the main series ratios of 0.05, 0.1, 0.2, 0.5, 1, 2, 5, 10, 20, 50, 100, 200, and 500. Several scales for the measurement of length were also discovered during excavations. One was a decimal scale based on a unit of measurement of 1.32 inches (3.35 centimeters) which has been called the "Indus inch". Of course ten units is then 13.2 inches which is quite believable as the measure of a "foot". Another scale was discovered when a bronze rod was found which was marked in lengths of 0.367 inches. It is surprising to see the accuracy with which these scales are marked. Now 100 units of this measure is 36.7 inches which is the measure of a stride. Measurements of the ruins of the buildings which have been excavated show that these units of length were accurately used by the Harappans in construction.
It is not known fully if the Harappan knowledge in mathematics was continued in later Vedic period. However we do know that from ancient time, Indian minds fascinated with higher numbers. While the Greeks had no terminology for denominations above a myriad (104) and the Romans above millie (103), ancient Indians dealt freely with no less than eighteen denominations. We know from a record of an early Buddhist work in 5th century BC, how the prince Gautama Buddha correctly recited the counting beyond the koti on the centesimal scale: “Hundred kotis are called ayuta, hundred ayutas niyuata, hundred niyutas kankara, hundred kankaras vivara, hundred vivaras ksobhya, hundred ksobhyas vivaha, hundred vivahas ustanga, hundred ustangas babula, hundred babulas nagabala, hundred nagabalas tithi lambha and so on upto another twelve terms ending with tallaksana. (Thus one Tallaksna is 1053).
Examples of geometric knowledge (rekha-ganit) are to be found in the Sulva-Sutras of Baudhayana (800 BC) and Apasthmaba (400 BC). The term Sulvasutra means “the rules of the chord”; it is the name given to the supplements of the Kalpasutras which explain the construction of sacrificial Vedic altars. A statement in Baudhayana's Sulvasutra runs,” In a Deerghchatursh (Rectangle) the Chetra (Square) of Rajju (hypotenuse) is equal to sum of squares of Parshvamani (base) and Triyangmani (perpendicular).” This is what is commonly known as the Pythagoras theorem. This shows that Vedic Indians had knowledge of the Pythagoras theorem as early as 8th century B.C. Apasthamba's sutra provides a value for the square root of 2 that is accurate to the fifth decimal place. Apasthamba also looked at the problems of squaring a circle, dividing a segment into seven equal parts, and a solution to the general linear equation.
However we donot find any proof of any of the theorems in Vedic mathematics, just the statements. This may be because, the involved mathematics was considered a sacred and secret knowledge just for the Vedic rituals reserved for the priests only. This has been one of the main problems for further development on Indian science during the Vedic Brahmanic age. As a result, none of these geometrical constructions appeared in any subsequent Indian literature, and later mathematicians did not carry these discussions to any higher level like what the Greeks did. As Gordon Childe, the famous historian rightly puts it: while the Greeks were free to speculate on ‘facts of common experience’ and ‘the practice of the craft’, the Vedic Indians were restricted by their ‘inheriting from the Bronze age the sacred hymns of Veda and ritual manuals verbally remembered’. In fact the Indians could never got rid of the Vedas completely which prevented them, throughout the ages, from exploring ‘secular’ scientific speculation to higher limits.
Thus it was mainly under the religious and philosophical impulses of the Jains and the Buddhists, that Indian mind learned to speculate outside the Vedas, and science and mathematics got some freedom. In ancient India, mathematics was a very lively passion. In Sanskrit, Ganita literally means the science of calculations which were generally done on a board (pati) with a piece of chalk or on sand (dhuli) spread on the ground. Thus the terms Pati-Ganita (science of calculations on the borad) and Dhuli-Karma (dust work) came to be used for higher mathematics. In ancient Buddhist literature we find mention of three classes of Ganita (1) Mudra – finger arithmetic (2) Ganana – mental arithmetic (3) Samkhyana – higher mathematics in general.
These Jains and Buddhist scholars worked on problems such as number theory, cubic equations, quadratic equations, and statistics. They also had an understanding of advanced ideas such as that of infinity. The Jains were the first to discard the idea that all infinites were the same or equal. They recognized four different types of infinities: infinite in length (one dimension), infinite in area (two dimension), infinite in volume (three dimensions) and infinite perpetually (infinite number of dimensions). Jain texts from the 6th C BC such as the Surya Pragyapti describe ellipses. Buddhist literature also demonstrates an awareness of indeterminate and infinite numbers. Numbers were deemed to be of three types: Sankheya (countable), Asankheya (uncountable) and Anant (infinite). The Buddhist philosophical formulations concerning Shunya - i.e. emptiness or the void may have facilitated in the introduction of the concept of zero. While the zero (bindu) as an empty place holder in the place-value numeral system appears much earlier, algebraic definitions of the zero and it's relationship to mathematical functions appear in the mathematical treatises of Brahmagupta in the 7th C AD.
The early Jainas seem to have great liking for the subject of combinations and permutations. Mahabira, the founder of Jainism, was himself a great mathematician. In the Bhagawati sutra are set forth simple problems such as finding the number of combinations that can be obtained from a given number of fundamental philosophical categories taken one at a time, two at a time, three at a time or more at a time. The Jaina commentator Silanka has quoted three rules regarding permutations and combinations. The Jains were the first to conceive of transfinite numbers, a concept, which was brought to Europe by Cantor in the late 19th century. The two thousand year old Jaina literature may hold valuable clues to the very nature of mathematics. This is one area where further research could prove very fruitful.
The works of the early Jain and Buddhist scholars were later summarized and expanded by Aryabhatta (476-550), the most important ancient mathematician of
Bhaskar I continued where Aryabhatta left off, and discussed in further detail topics such as the longitudes of the planets; conjunctions of the planets with each other and with bright stars; risings and settings of the planets; and the lunar crescent. Again, these studies required still more advanced mathematics and Bhaskar I expanded on the trigonometric equations provided by Aryabhatta, and like Aryabhatta correctly assessed π (pi) to be an irrational number. Amongst his most important contributions was his formula for calculating the sine function which was 99% accurate. He also did pioneering work on indeterminate equations and considered for the first time quadrilaterals with all the four sides unequal and none of the opposite sides parallel.
Another important astronomer/mathematician was Varahamira (6th C, Ujjain) who compiled previously written texts on astronomy and made important additions to Aryabhatta's trigonometric formulas. His works on permutations and combinations complemented what had been previously achieved by Jain mathematicians and provided a method of calculation of n-C-r that closely resembles the much more recent Pascal's Triangle. In the 7th century, Brahmagupta (born 598) did important work in enumerating the basic principles of algebra. In 628 he wrote his Brahma-sphuta-siddhanta (“The Revised System of Brahma”). In addition to listing the algebraic properties of zero, he also listed the algebraic properties of negative numbers and used these signs for addition, subtraction and multiplication (+, -, x). He was the first to postulate that, “The multiplication of a positive number with a negative number comes out to be a negative number. Further when a positive number is divided by a negative number or a negative number is divided by a positive number the result is a negative number.”
His work on solutions to quadratic indeterminate equations anticipated the work of Euler and Lagrange. It is a sobering thought that even eight hundred years later European mathematics would be struggling to cope with the use of negative numbers and of zero. More than that, he made other major contributions to the understanding of integer solutions to indeterminate equations and to interpolation formulas invented to aid the computation of sine tables. Although almost all ancient countries used quantities of unknown values for solution of problems, the expansion of Algebra (Biz Ganit) became possible only when Indians realized that all the calculations of Numerical Mathematics could be done by notations, and for the first time used Sanskrit Alphabet to denote unknown quantities.
Brahmagupta is also credited with the following:
Developments also took place in applied mathematics such as in creation of trigonometric tables and measurement units. Yativrsabha's work Tiloyapannatti (6th C) gives various units for measuring distances and time and also describes the system of infinite time measures
Aryabhatta's equations were elaborated on by Manjula (10th C) and Bhaskaracharya (12th C) who derived the differential of the sine function. Later mathematicians used their intuitive understanding of integration in deriving the areas of curved surfaces and the volumes enclosed by them.
Between the 7th C and the 11th C, Indian numerals developed into their modern form, and along with the symbols denoting various mathematical functions eventually became the foundation stones of modern mathematical notation.
But this was also the time when Indian mathematical texts were increasingly being translated into Arabic and Persian. The last notable Indian mathematician may be said to be Bhaskaracharya who came from a long-line of mathematicians and was head of the astronomical observatory at Ujjain. He left several important mathematical texts including the Lilavati, Bijaganita and the Siddhanta Shiromani, an astronomical text. He is however more popularly known today for his book, Lilavati which is a unique book which shows how mathematics was brought to the reach of the common people. It is a collection of worked out examples of arithmetic, algebra, geometry, and mensuration. The language is written in Sanskrit verse, and the level of mathematics ranges between high school algebra to pre-Calculus. In its time, it represented the height of 12th-century mathematics. The problems are generally addressed to one Lilavati, traditionally taken to be either his wife or his daughter. Reading Lilavati, any reader will find that learning mathematics can be fun which also flourishes in wonder.
A typical problem reads, “O deer eyed one! Tell me if one sixth of the number of bees in a colony entered a jasmine flower tree, one-third went to kadamba tree, one fourth flew to a mango tree, one-fifth went to a tree blooming with sampaka flowers, one-thirtieth went to a beautiful bed of lotuses bloomed by the Sun’s rays and if the remaining one bee was roving about, how many total bees were there in the colony?”
This is a problem of unknown quantity which is solved by now a days by assuming X for the unknown quantity. In this case, the problem resolves to
(X) – (X/6) – (X/3) – (X/4) – (X/5) – (X/30) = 1; Or, (X/60) =1; Or, X = 60
There were also problems that excite real passion for mathematics. Note the following verse:
This is another problem of finding the unknown number. The modern way of solving is to assume X for the unknown number. Then the problem can be written as:
(X) - (X/3) – (X/5) – (X/6) – (X/10) = 6;
Another problem describes as “A flock of swans contained total x2 members. As clouds gathered, (10x) of these went to Manasa lake, and (1/8 x2) flew away to a garden. The remaining three couples played about in the water. O young woman, how many swans were there in that lake full of beautiful lotuses?”
This is a problem of quadratic equation which can be written as:
Another problem that deals higher mathematics runs like this: “Dear Lilavaty, Suppose different kinds of Chatnies are made by mixing 1, 2, 3, 4, 5 or 6 at a time from six substances which are respectively sweet, bitter, astringent, sour, salty and hot. O, my pretty mathematician, tell me how many different Chatnies can be prepared from these all?”
This a problem of Combination which can be solved by the formula for Combination of n things taken 1, 2, 3 … r things at a time. The formula is:
The solution is: = n!/0!(n!), n!/1!(n-1)!, n!/2!(n-2)!, n!/3!(n-3)!,….n!/n!(n-n)!
= 6!/0!(6!), 6!/1!(5!), 6!/2!(4!), 6!/3!(3!), 6!/4!(2!), 6!/5!(1!), 6!/6!(0!)
= 1, 6, 15, 20, 15, 6, 1 = 64, the total number of different chatnies that can be made.
[In the above, [n! is called factorial n ; example, Factorial 3!=3.2.1=6] The answers also can be found from modern day Pascels Triangle which was used by ancient Indian mathematicians and was known as Khandmeru.
The lucid, scholarly and literary presentation in Lilavaty has attracted several cultural areas. The graphic descriptions of a drove of swans; a flock of elephants; a colony of bees; the attack of a snake by a domesticated peacock; sinking of a lotus in water owing to strong wind; and many others were to train the students not only in mathematics but also in appreciation of nature. No wonder that Lilavati has not only been used widely in India in the medieval times as a standard base text-book for about 800 years, but it was commented and translated into several languages of the world.
Besides that, like poetry, mathematics has a beauty and truth, and can be enjoyed as such. Plato once said, “Arithmetic has a very great elevating effect, compelling the soul to reason about abstract numbers refusing to be satisfied visible and tangible objects.” The joy of mathematics is similar to the experience of discovering something new for the first time. In Lilavaty, Bhaskara showed that he was not only a great mathematician but was also a great poet who may be compared to Omar Khayyam. His conclusion to Lilavati states: “Joy and happiness is indeed ever increasing in this world for those who have Lilavati clasped to their throats.” Overall, Bhaskaracharya’s Lilavati proves the point that Indians had great romance with mathematics and that it is very much ingrained in the Indian culture. The creation of a book like Lilavaty was possible in India because of its long tradition of culture with mathematics; one will not find such book in any other civilization.
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