Project On History of Mathematics/MathsProject On History of Mathematics/Maths Project for School Students.
Some lines From The Project History of Mathematics:
History of Mathematics in India
In early civilizations, the first expression of maths understanding appears in the form of counting systems.Numbers in early societies were typically represented by groups of lines, though later different numbers came to be assigned specific numeral names and symbols (India) or were designated by alphabetic letters (Rome). Although today, we take our decimal system for granted, not all ancient civilizations based their numbers on a ten-base system. In ancient Babylon, a sexagesimal (base 60) system was in use.
Section 1: Zero and the Place-Value Notation
The number zero is the subtle gift of the Hindus of antiquity to mankind. The concept itself was one of the most significant inventions in the ascent of Man for the growth of culture and civilization. To it must be credited the enormous usefulness of its counterpart, the place value system of expressing all numbers with just ten symbols. And to these two concepts we owe all the arithmetic and mathematics based upon them, the great ease which it has lent to all computations for two millenia and the binary system which now lies at the foundation of communicating with computers. Already in the first three centuries A.D.. the Hindu ancients were using a decimal positional system, that is, a system in which numerals in different positions represent different numbers and in which one of the ten symbols used was a fully functional zero. They called it 'Sunya'. The word and its meaning ‘void’ were obviously borrowed from its use in philosophical literature. Though the Babylonians used a special symbol for zero as early as the 3rd century B.C. , they used it only as a place holder; they did not have the concept of zero as an actual value. It appears the Maya civilisation of South America had a zero in the first century A.D. . but they did not use it in a fixed base system. The Greeks were hampered by their use of letters for the numbers. Before zero was invented, the art of reckoning remained an exclusive and highly skilled profession. It was difficult to distinguish, say, 27, 207, 270, 2007, because the latter three were all written 2 7, with a ‘space’ in between. The positional system is not possible in the Roman numeral system which had no expression or symbol for zero. A number, say, 101,000, would have to be written only by 101 consecutive M’s. The Egyptians had no zero and never reached the idea of expressing all numbers with ten digits. The mathematical climate among the Hindus, however, was congenial for the invention of zero and for its use as the null-value in all facets of calculation, due to four factors:
1. A notation for powers of 10 upto the power 17 was already in use even from vedic times. Single words have been used to denote the powers of the number 10. The numbers one, ten, hundred, thousand, ten thousand, … are given by the sequence of words in the list: eka, dasa, Sata, sahasra, ayuta, laksha, prayuta, koTi, arbuda, abja, kharva, nikharva, mahA-padma, Sankha, jaladhi, antya, mahASankha, parArdha. Thus the decimal system was in the culture even in the early part of the first millenium B.C. . The Yajurveda, in its description of rituals and the mantras employed therein, the Mahabharata and the Ramayanaa in their descriptions of statistics and measurements, used all these words, with total abandon.
2. Counting boards with columns representing units and tens were in use from very ancient times. The numberless content of an empty column in course of time was symbolized to be ‘nothing’.
3. The thriving activity in astrology, astronomy, navigation and business during the first few centuries A.D. naturally looked forward to a superior numerical system that lent itself to complicated calculations.
4. Distinct symbols for the numbers 1 to 9 already existed and the counting system used the base 10 in all its secular, religious and ritual activities. Compare this with the Babylonian numeration which had only three figures, one for 1, one for 10, and one for 100, so that a number, say, 999, would require 27 symbols, namely, nine of each of the symbols.
Of these, the first and fourth factors are probably unique to Hindu culture and contributed most to the thought process that led to the decimal place value notation as well as zero having a value. When exactly the invention of this most modest of all numerals took place, we do not know. The first time it reached Europe was during the Moorish invasion of Spain around 700 A.D. . Later, when massive Latin translations of books from Baghdad took place around the close of the first millenium A.D. the concept was found in an arithmetic book dated 820 A.D. . by Muhammad Ibn Musa al-Khouarizmi who explained the whole system in great detail. The Arabs themselves had no number system of their own. It was the Hindu system he was explaining. That is why the numerals are called the Indo-Arabic numerals even today. It is however a misnomer to call them so; they are already found on the Rock Edits of Ashoka (256 B.C. .). In spite of its being so crucial to our living, it took centuries for the western world to appreciate and incorporate this most valuable numeral, zero, in their recording of accounts or in scholarly writings. For by the time ‘Zero’ reached the West, the Dark Ages of the western world had begun. There are traces, however, of its knowledge in Spain in the tenth century A.D. . But the final breakthrough of the introduction to the West was by Leonardo of Pisa, through his popular text Liber Abaci, 1202 A.D. ., The first European book (in French) that used the zero appeared in 1275.
How the Sunya of the Hindus became the Zero of the modern world is interesting. The 'Sunya' of Sanskrit became the Arabic ‘sifr’ which means empty space. In medieval Latin it manifested as ‘ciphra’ , then in middle English as ‘siphre’, in English as ‘cypher’ and in American as ‘cipher’. In the middle ages, the word ‘ciphra’ evolved to stand for the whole system. In the wake of this general meaning, the Latin ‘zephirum’ came to be used to denote the Sunya. And that entered English finally as ‘zero’. In medieval Europe some countries banned the positional number system, along with zero, brought by the Arabs whom they considered as heathens. So they took Sunya to be a creation of the devil! As a result ‘ciphra’ came to mean a secret code. From this came ‘deciphering’, the resolution of a code!
Section 2. Vedic Mathematics and arithmetical operations
Vedic Mathematics provides an original and refreshing approach to subjects which are usually dismissed as mechanical and tedious. Bharati Krishna Tirtha who published his reconstruction of Vedic Mathematics in 1965, maintains that there are 16 aphorisms and 13 secondary aphorisms which forms his base of the so-called Vedic Mathematics. Though the origins of Vedic Mathematics have not yet been historically established, if nothing else, it provides tremendous insights into the place-value system of numbers without which it would not work. It is amazing that Vedic Mathematics does not require of cramming of multiplication tables beyond 5 x 5. One can improvise all the necessary multiplication tables for oneself and with the aid of the relevant Vedic formulae get the required products very easily, speedily, and correctly, almost immediately. The formulae can be used to evaluate determinants, solve simultaneous linear equations, evaluate logarithms and exponentials. Vedic Mathematics recognises that any algebraic polynomial may be expressed in terms of a positional notation without specifying the base. The same algorithmic scheme as applied to arithmetical operations will easily apply to algebraic problems. And this brings it to the Modern Algebra of Polynomials. It is difficult, in a historical introduction like this to get into the details of Vedic Mathematics. Suffice it to say that with today's over-dependence on calculators for even simple arithmetical computations, the Vedic methods have great pedagogical value and, through their revival, the skills of mental arithmetic may not be lost for posterity.
Section 3. Geometry of the Sulba Sutras
Hailing from the times of the Vedas, the ritual literature which gave directions for constructing sacrificial fires at different times of the year dealt with the their measurement and construction in a systematic and logical way, thus giving rise to the Sulba Sutras. The construction of altars (vedi) and the location of sacrificial fires had to conform to clearly laid down instructions about their shapes and areas in order that they may be effective instruments of sacrifice. The Sulba Sutras provide such instructions for two types of ritual – one for worship at home and one for communal worship. The instructions were mainly for the benefit of craftsmen laying out and building the altars. Bodhayana, Apastamba and Katyayana who have recorded these Sulbasutras were not only priests in the conventional sense but must have been craftsmen themselves. The earliest of them, The Bodhayana Sutras , in three chapters, (800 – 600 B.C.) contains a general statement of the Pythagorean theorem, an approximation procedure for obtaining the square root of two correct to five decimal places and a number of geometric constructions. These latter include an approximate squaring the circle, and construction of rectilinear shapes whose area is equal to the sum or difference of areas of other shapes. The Bodhayana version of the Pythagorean theorem sates as follows:
The rope which is stretched across the diagonal of a square
produces an area double the size of the original square.
It is therefore in the fitness of things that the Pythagorean theorem of Mathematics may be renamed as the Bodhayana theorem.! The other sutras are two centuries later but all of them are prior to Panini of the fourth century B.C. The geometry arising from these sutras give several geometric constructions. Some of these are:
1. To merge two equal or unequal squares to obtain a third square.
2. To transform a rectangle into a square of equal area
3. Squaring a circle and circling a square (approximately)
A remarkable achievement was the discovery of a procedure for evaluating square roots to a high degree of approximation. The square root of two is obtained as
the true value being 1.414213… . The fact that such procedures were used successfully by the Sulbasutra geometers to operations with other irrational numbers, is clear proof for negating the western-held opinion that the Sulba sutra geometers borrowed their methods from the Babylonians. The latter's calculation of the square root of two is an isolated instance and further they used the sexagesimal notation for numbers. The achievement of geometrical constructs in Indian mathematics reached its peak later when they arrived at the construction of Sriyantra, which is a complicated diagram, consisting of nine interwoven isosceles triangles, four pointing upwards and four pointing downwards. The triangles are arranged in such a way that they produce 43 subsidiary triangles, at the centre of the smallest of which there is a big dot called the bindu. The difficult problem is to construct the diagram in such a way that all the intersections are correct and the vertices of the largest triangles fall on the circumference of the enclosing circle. In all cases the base angles of the largest triangles is about 51.5 degrees. This has connections with the two most famous irrational numbers of Mathematics, namely p and f. The quantity f, called the golden ratio, has remarkable mathematical properties and is almost a semi-mystical number.
Section 4. Jaina contribution to Fundamentals of Numbers
By the time of the Jains, the role of rituals in the development of mathematics declined and mathematics began to be pursued also for its own sake. The Jains had a fascination for large numbers. Their definitions of the various types of infinities they comprehended are sophisticated, though lacking in mathematical precision. But it must be said to their credit that they were the first, in the chronology of scientific thinking, to have recognised that all infinities were not the same or equal. In fact this idea was established in the mathematical world only in the latter half of the nineteenth century when Cantor initiated his theory of sets.
The Jains were also aware of the theory of indices, though they did not have the modern notation or any convenient notation for the same. Calling the successive squares and square roots as the first, the second, etc. they make the following statement: The first square root multiplied by the second square root is the cube of the second square root. In modern notation this is nothing but the identity in the theory of indices:
a1/2 x a1/4 = (a1/4 )3
They have several such rules for working with powers of a number. They also seem to have had an idea of the logarithm of a number though they don't seem to have put them to practical use in calculation. Another favourite topic with them was the study of permutations and combinations. They had also a great interest in sequences and progressions developed out of their philosophical theory of cosmological structures. A Jain canonical text entitled Triloka prajnApati has a very detailed treatment of arithmetic progressions.
Section 5: The Anonymous Bakshali Manuscript
This manuscript was discovered in 1881 A.D. near a village called Bakshali. It is written in an old form of Sanskrit on seventy leaves of birch bark. It is probably a copy of a manuscript composed in the early centuries A.D. It is a handbook of rules and illustrative examples together with solutions, all mainly on arithmetic and algebra. Fractions, Square roots, Profit and Loss, Interest, Rule of Three, Approximation to surds, Simple equations as well as Simultaneous equations, Quadratic equations, Arithmetic and Geometric Progressions — all these are covered. Very unusually in the entire history of Ancient Indian mathematics, the subject matter is organised in a sequence: first, a rule or a sutra; then a relevant example in word form; the same in notational form; then the solution and finally the demonstration or the proof. Here for the first time in the history of world mathematics, the Rule of Three is stated in its abstract form. It was from here that the rule was taken to Europe by the Arabs and it was then known as the Golden Rule. It became very popular in Europe after the Renaissance. The rule goes as follows:
If p yields f what will i yield?
Here p stands for pramAna, f for phala and i for icchA.
Here p and i are of the same denomination and
f is of a different denomination.
Write p, f, i in that order. Multiply the middle quantity by the last quantity and divide by the first. The result is fi / p.
The first appearance of indeterminate equations is in the Bakshali mss. This marks the beginning of the continued work on indeterminate equations in India. Such an interest, though originally generated from the demands made by astronomical calculations, was also pursued for its own sake, sometimes even for recreational purposes. The ease with which the place value system is worked in the Bakshali mss. suggests that the system predates the mss. by probably a few centuries.
History of Mathematics/History of Maths: Section 6. Astronomy
The contribution to Astronomy by ancient Indians is so great that it does not befit it to include it as one of the contributions of Indian Mathematics to the rest of the world. It needs a separate forum all for itself. We shall leave it right there except to add a note on the ancient contribution to the problem of telling time at night by a look at the stars on the meridian. This part is usually not emphasized.
The ancients of India have passed on to us 27 mathematical formulae coded in the Sanskrit language, but not very difficult to remember. In fact, very possibly it has mostly come down to us by oral transmission from generation to generation. For instance, the formula
krittikA simhe kAyA
says that if you see the asterism krittikA (Pleides, in modern terminology) on the meridian, that is the time the Leo (= simha) constellation (of the zodiac) has risen above the horizon by an amount indicated by the word: kAyA. This latter word interpreted in katapayA sankhyA, which is the notation used by astronomers, astrologers and mathematicians to represent numbers, means in this context that the amount of Leo above the (Eastern) horizon is 27 minutes of time. From this and the known position of the Sun on the date in question, one mentally calculates the time of night. On November 7 for example, the Sun is in the middle of Scorpio. So if you see krittikA on the meridian it means Leo has risen 27 minutes before and this means the Sun is behind by
93m (remaining portion of simha)
+ 2h (full portion of kanyA)
+ 2h (full portion of tulA)
+ 60m (half portion of vrischika)
that is 6 hours 33 minutes. In other words it is 6h 33m before sunrise. So it is 11-27 P.M.
Suffice it to say these beautiful formulae constitute an intellectual marvel put to the most mundane use. Never perhaps was so much achieved with so little so early in the Ascent of Man. For details on this, see reference no. .
History of Mathematics/History of Maths: Section 7. Classical contribution to Indeterminate Equations and Algebra
The apex of Mathematical achievement of ancient India occurred during the so-called classical period of Indian Mathematics. The great names are: Aryabhata I (b.476 A.D.) ; Brahmagupta (b.598 A.D.); Bhaskara I (circa 620 A.D.) ; Mahavira (circa 850 A.D.); Sridhara (circa 900 A.D.) ; Bhaskara II (b.1114 A.D.); Nilakantha Somayaji (1445 – 1545 A.D.).
Aryabhata wrote the famous Aryabhatiyam which is an exhaustive exposition of Astronomy. In addition he gave a unique method of representing large numbers by word forms . He systematized all the knowledge of astronomy and mathematics prior to him. The first one in Indian mathematics to give the formula for the area of a triangle was Aryabhata. Several results on Triangles and circles and on Progressions, algorithm for finding cube roots, approximation of p, all these give him a unique position in the development of mathematics. Aryabhata ushered in a Renaissance in Indian Mathematics and Astronomy, that resulted in a remarkable flourishing of science and technology in India. Aryabhata, for the first time, secularised mathematics and astronomy in India and established these as intellectual disciplines in their own right. Excellent commentators followed Aryabhata and to them are due several modifications and applications, explanation of subtle points, and finally, the proofs of results embodied in the sutras of the Master.
Bhaskara I takes a large share of the credit of explaining the too brief and aphoristic statements of Aryabhata. On the important topic of indeterminate equations the Kuttaka method was introduced by Aryabhata and elucidated by Bhaskara I.
Brahmagupta is generally known as the Indian mathematician par excellence. His monumental work Brahma SiddhAnta has 24 chapters of which the latter 14 contain original results on arithmetic algebra and on astronomical instruments. The 12th chapter is on mensuration. The 18th chapter is on Kuttaka. Among his famous results are those on rational right-angled triangles, and cyclic quadrilaterals. He is the earliest one, in the history of world mathematics, to have discussed cyclic quadrilaterals. There is every reason for us to name cyclic quadrilaterals as Brahmagupta Quadrilaterals. It was partly through a translation of Brahma-siddhAnta that the Arabs became aware of Indian astronomy and mathematics.
Bhaskara II's famous work SiddhAnta Siromani has four parts of which the first two are Mathematics and the latter two are astronomy. The first part, LilAvati is an extremely popular text dealing with arithmetic, algebra, geometry and mensuration. The second part, BIjaganitam is a treatise on Advanced Algebra. It contains problems on determining unknown quantities, evaluating surds and solving simple and quadratic equations.
The sheer ingenuity and versatility of Brahmagupta's approach to indeterminate equations of the second degree of the form
N x2 + 1 = y2
is the climax of Indian work in this area. Bhaskara II's cakravAla method to solve such equations is world-famous. By using this powerful method he solved, as one example, the above equation with N = 61 and gave the least integral solution as
x = 226153980 and y = 1766319049.
The famous French mathematician, Fermat, in 1657 A.D. proposed this equation with N = 61 for solution as a challenge to his contemporaries. None of them succeeded in solving the equation in integers. It was not until 1767 A.D. that the western world through Euler, by Lagrange's method of continued fractions, had a complete solution to such types of equations, wrongly called Pell's equation by Euler. But the very same equation, though coincidentally, was completely solved by Bhaskara II five hundred years earlier.
The problem of determining integer solutions of such equations is called Diophantine Analysis after the Greek Mathematician Diophantus (3rd cen. A.D.). As soon as one finds a non-trivial solution (that is, other than the obvious solution x = 0, y = 1) an infinite number of new solutions can be found by repeated application of the Principle of Compositions, known as Brahmagupta's Bhavana Principle. It is Bhaskara's cakravAla method that makes the decisive step in determining a non-trivial solution. Under these circumstances it is appropriate to designate these equations as the Brahmagupta-Bhaskara equations.
Before we leave this topic it is important to mention Srinivasa Ramanujan, the 20th century genius, who revelled in such problems – namely, to determine the possible cases in which a number can be broken up into two or more equal sums of like or unlike powers or more generally to solve intermediate problems in rational numbers.
Bhaskara II introduces also the notion of instantaneous motion of planets. He clearly distinguishes between sthUla gati (average velocity) and sUkshma gati (accurate velocity) in terms of differentials. He also gave formulae for the surface area of a sphere and its volume, and volume of the frustum of a pyramid. Suffice it to say that his work on fundamental operations, his rules of three, five, seven, nine and eleven, his work on permutations and combinations and his handling of zero all speak of a maturity, a culmination of five hundred years of mathematical progress.
History of Mathematics/History of Maths: Section 8. Indian Trigonometry
Though Trigonometry goes back to the Greek period, the character of the subject started to resemble modern form only after the time of Aryabhata. From here it went to Europe through the Arabs and went into several modifications to reach its present form. In ancient times Trigonometry was considered a part of astronomy. Three functions were introduced: jya, kojya and ukramajya. The first one is r sina where r is the radius of the circle and a is the angle subtended at the centre. The second one is r cosa and the third one is r (1 – cosa). By taking the radius of the circle to be 1, we get the modern trigonometric functions. Various relationships between the sine of an arc and its integral and fractional multiples were used to construct sine tables for different arcs lying between 0 and 90°.
History of Mathematics/History of Maths: Section 9: Kerala contribution to Infinite Series and Calculus.
Kerala mathematicians produced rules for second order interpolation to calculate intermediate sine values. The Kerala mathematician Madhava may have discovered the sine and cosine series about three hundred years before Newton. In this sense we may consider Madhava to have been the founder of mathematical analysis. Madhava (circa 1340 – 1425 A.D. ) was the first to take the decisive step from the finite procedures of ancient Indian mathematics to treat their limit-passage to infinity. His contributions include infinite-series expansions of circular and trigonometric functions and finite-series approximations. His power series for p and for sine and cosine functions is referred to reverentially by later writers. Many later discoveries in European mathematics ( for example, the Gregory series for the inverse tangent) were anticipated by Kerala astronomer-mathematicians. Nilakantha was mainly an astronomer but his Aryabhatiya bhashya and tantra-sangraha contain work on infinite-series expansions, problems of algebra and spherical geometry.
History of Mathematics/History of Maths: Section 10. Modern Contribution:
Srinivasa Ramanujan onwards
The second decade of the 20th century compulsorily turned the attention of the mathematical world to India and the Number Theory genius, Srinivasa Ramanujan. The ideas and innovative genius of Ramanujan have not been surpassed ever before or even 100 years after him. His birth, his super-activity in Madras and Cambridge, his glorious rise to international fame and unfortunate death – all happened almost in a flash. But ever since, India has remained on the mathematical map of the world more and more prominently. The 20th century saw the growth of Mathematics in India in the same style, though not in the same speed, the western world has been taking the race for the quest of mathematical knowledge. May we hope that the new millenium will see not only further speed in the acquisition and spread of mathematical knowledge by Indian-born mathematicians, but also further depth in the intensity and relevance of that knowledge, true to its traditions which are at least three millenia old .
These were some of the lines from History of Mathematics/History of Maths.