Mathematicians of India

Aryabhata  (476–550 CE) was the first in the line of greatmathematician-astronomers from the classical age of Indian mathematics and Indian astronomy. His most famous works are the Aryabhatiya (499 CE, when he was 23 years old) and the Arya-siddhanta. Name While there is a tendency to misspell his name as “Aryabhatta” by analogy with other names having the “bhatta” suffix, his name is properly spelled Aryabhata: every astronomical text spells his name thus,[1] including Brahmagupta’s references to him “in more than a hundred places by name”. 2] Furthermore, in most instances “Aryabhatta” does not fit the metre either. [1] Birth Aryabhata mentions in the Aryabhatiya that it was composed 3,600 years into the Kali Yuga, when he was 23 years old. This corresponds to 499 CE, and implies that he was born in 476 CE. [1] Aryabhata provides no information about his place of birth. The only information comes from Bhaskara I, who describes Aryabhata as asmakiya, “one belonging to the asmaka country. While asmaka was originally situated in the northwest of India, it is widely attested that, during the Buddha’s time, a branch of the Asmaka people settled in the region between the Narmada and Godavari rivers, in the South Gujarat–North Maharashtra region of central India. Aryabhata is believed to have been born there. [1][3] However, early Buddhist texts describe Ashmaka as being further south, in dakshinapath or the Deccan, while other texts describe the Ashmakas as having fought Alexander, which would put them further north. 3] Mathematics Place value system and zero The place-value system, first seen in the 3rd century Bakhshali Manuscript, was clearly in place in his work. While he did not use a symbol for zero, the French mathematician Georges Ifrah argues that knowledge of zero was implicit in Aryabhata’s place-value system as a place holder for the powers of ten with null coefficients[10] However, Aryabhata did not use the brahmi numerals. Continuing the Sanskritic tradition from Vedic times, he used letters of the alphabet to denote numbers, expressing quantities, such as the table of sines in a mnemonic form. 11] ]Pi as irrational Aryabhata worked on the approximation for Pi (? ), and may have come to the conclusion that ? is irrational. In the second part of the Aryabhatiyam (ga? itapada10), he writes: caturadhikam satama?? agu? am dva? a?? istatha sahasra? am ayutadvayavi? kambhasyasanno v? ttapari? aha?. “Add four to 100, multiply by eight, and then add 62,000. By this rule the circumference of a circle with a diameter of 20,000 can be approached. “[12] This implies that the ratio of the circumference to the diameter is ((4+100)? +62000)/20000 = 62832/20000 = 3. 1416, which is accurate to five significant figures. It is speculated that Aryabhata used the word asanna (approaching), to mean that not only is this an approximation but that the value is incommensurable (orirrational). If this is correct, it is quite a sophisticated insight, because the irrationality of pi was proved in Europe only in 1761 by Lambert). [13] After Aryabhatiya was translated into Arabic (ca. 820 CE) this approximation was mentioned in Al-Khwarizmi’s book on algebra. 3] Mensuration and trigonometry In Ganitapada 6, Aryabhata gives the area of a triangle as tribhujasya phalashariram samadalakoti bhujardhasamvargah that translates to: “for a triangle, the result of a perpendicular with the half-side is the area Aryabhata discussed the concept of sine in his work by the name of ardha-jya. Literally, it means “half-chord”. For simplicity, people started calling it jya. When Arabic writers translated his works from Sanskrit into Arabic, they referred it as jiba. However, in Arabic writings, vowels are omitted, and it was abbreviated as jb.

Later writers substituted it with jiab, meaning “cove” or “bay. ” (In Arabic, jiba is a meaningless word. ) Later in the 12th century, when Gherardo of Cremonatranslated these writings from Arabic into Latin, he replaced the Arabic jiab with its Latin counterpart, sinus, which means “cove” or “bay”. And after that, the  sinus became sine in English. Indeterminate equations A problem of great interest to Indian mathematicians since ancient times has been to find integer solutions to equations that have the form ax + b = cy, a topic that has come to be known as diophantine equations.

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This is an example from Bhaskara’s commentary on Aryabhatiya:Find the number which gives 5 as the remainder when divided by 8, 4 as the remainder when divided by 9, and 1 as the remainder when divided by 7That is, find N = 8x+5 = 9y+4 = 7z+1. It turns out that the smallest value for N is 85. In general, diophantine equations, such as this, can be notoriously difficult. They were discussed extensively in ancient Vedic text Sulba Sutras, whose more ancient parts might date to 800 BCE. Aryabhata’s method of solving such problems is called the ku?? aka (?????? ) method.

Kuttaka means “pulverizing” or “breaking into small pieces”, and the method involves a recursive algorithm for writing the original factors in smaller numbers. Today this algorithm, elaborated by Bhaskara in 621 CE, is the standard method for solving first-order diophantine equations and is often referred to as the Aryabhata algorithm. [16] The diophantine equations are of interest in cryptology, and the RSA Conference, 2006, focused on the kuttaka method and earlier work in the Sulvasutras. AlgebraIn Aryabhatiya Aryabhata provided elegant results for the summation of series of squares and cubes:[17] and

Srinivasa Aiyangar Ramanujan FRS, better known as Srinivasa Iyengar Ramanujan  (22 December 1887 – 26 April 1920) was an Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made substantial contributions to mathematical analysis, number theory, infinite series and continued fractions. Born and raised in Erode, Tamil Nadu, India, Ramanujan first encountered formal mathematics at age 10. He demonstrated a natural ability, and was given books on advanced trigonometry written by S L Loney. [1] He had mastered them by age 12, and even discovered theorems of his own.

He demonstrated unusual mathematical skills at school, winning accolades and awards. By 17, Ramanujan conducted his own mathematical research on Bernoulli numbers and the Euler–Mascheroni constant. He received a scholarship to study at Government College in Kumbakonam, but lost it when he failed his non-mathematical coursework. He joined another college to pursue independent mathematical research, working as a clerk in the Accountant-General’s office at the Madras Port Trust Office to support himself. [2] In 1912–1913, he sent samples of his theorems to three academics at the University of Cambridge.

Only G. H. Hardy recognized the brilliance of his work, subsequently inviting Ramanujan to visit and work with him at Cambridge. He became a Fellow of the Royal Society and a Fellow of Trinity College, Cambridge, dying of illness, malnutrition and possibly liver infection in 1920 at the age of 32. Attention from mathematicians He met deputy collector V. Ramaswamy Aiyer, who had recently founded the Indian Mathematical Society. [33] Ramanujan, wishing for a job at the revenue department where Iyer worked, showed him his mathematics notebooks.

As Iyer later recalled:I was struck by the extraordinary mathematical results contained in it [the notebooks]. I had no mind to smother his genius by an appointment in the lowest rungs of the revenue department. [34] Iyer sent Ramanujan, with letters of introduction, to his mathematician friends in Madras. [33] Some of these friends looked at his work and gave him letters of introduction to R. Ramachandra Rao, the district collector for Nellore and the secretary of the Indian Mathematical Society. [35][36][37] Ramachandra Rao was impressed by Ramanujan’s research but doubted that it was actually his own work.

Ramanujan mentioned a correspondence he had with Professor Saldhana, a notable Bombay mathematician, in which Saldhana expressed a lack of understanding for his work but concluded that he was not a phony. [38] Ramanujan’s friend, C. V. Rajagopalachari, persisted with Ramachandra Rao and tried to quell any doubts over Ramanujan’s academic integrity. Rao agreed to give him another chance, and he listened as Ramanujan discussed elliptic integrals, hypergeometric series, and his theory of divergent series, which Rao said ultimately “converted” him to a belief in Ramanujan’s mathematical brilliance. 38] When Rao asked him what he wanted, Ramanujan replied that he needed some work and financial support. Rao consented and sent him to Madras. He continued his mathematical research with Rao’s financial aid taking care of his daily needs. Ramanujan, with the help of V. Ramaswamy Aiyer, had his work published in the Journal of Indian Mathematical Society. [39] One of the first problems he posed in the journal was: He waited for a solution to be offered in three issues, over six months, but failed to receive any. At the end, Ramanujan supplied the solution to the problem himself.

On page 105 of his first notebook, he formulated an equation that could be used to solve the infinitely nested radicals problem. Using this equation, the answer to the question posed in the Journal was simply 3. [40] Ramanujan wrote his first formal paper for the Journal on the properties ofBernoulli numbers. One property he discovered was that the denominators (sequence A027642 in OEIS) of the fractions of Bernoulli numbers were always divisible by six. He also devised a method of calculating Bn based on previous Bernoulli numbers.

One of these methods went as follows It will be observed that if n is even but not equal to zero, (i) Bn is a fraction and the numerator of  in its lowest terms is a prime number, (ii) the denominator of Bn contains each of the factors 2 and 3 once and only once, (iii)  is an integer and  consequently is an odd integer. In his 17–page paper, “Some Properties of Bernoulli’s Numbers”, Ramanujan gave three proofs, two corollaries and three conjectures. [41] Ramanujan’s writing initially had many flaws. As Journal editor M. T. Narayana Iyengar noted: Mr.

Ramanujan’s methods were so terse and novel and his presentation so lacking in clearness and precision, that the ordinary [mathematical reader], unaccustomed to such intellectual gymnastics, could hardly follow him. [42] Ramanujan later wrote another paper and also continued to provide problems in the Journal. [43]  Contacting English mathematicians Spring, Narayana Iyeru, Ramachandra Rao and E. W. Middlemast tried to present Ramanujan’s work to British mathematicians. One mathematician, M. J. M. Hill of University College London, commented that Ramanujan’s papers were riddled with holes. 48] He said that although Ramanujan had “a taste for mathematics, and some ability”, he lacked the educational background and foundation needed to be accepted by mathematicians. [49] Although Hill did not offer to take Ramanujan on as a student, he did give thorough and serious professional advice on his work. With the help of friends, Ramanujan drafted letters to leading mathematicians at Cambridge University. [50] The first two professors, H. F. Baker and E. W. Hobson, returned Ramanujan’s papers without comment. [51] On 16 January 1913, Ramanujan wrote to G. H. Hardy.

Coming from an unknown mathematician, the nine pages of mathematical wonder made Hardy originally view Ramanujan’s manuscripts as a possible “fraud”. [52] Hardy recognized some of Ramanujan’s formulae but others “seemed scarcely possible to believe. “[53] One of the theorems Hardy found so incredible was found on the bottom of page three (valid for 0 < a < b + 1/2): Hardy was also impressed by some of Ramanujan’s other work relating to infinite series: The first result had already been determined by a mathematician named Bauer. The second one was new to Hardy.

It was derived from a class of functions called a hypergeometric series which had first been researched by Leonhard Euler and Carl Friedrich Gauss. Compared to Ramanujan’s work on integrals, Hardy found these results “much more intriguing”. [54] After he saw Ramanujan’s theorems on continued fractions on the last page of the manuscripts, Hardy commented that the “[theorems] defeated me completely; I had never seen anything in the least like them before. “[55] He figured that Ramanujan’s theorems “must be true, because, if they were not true, no one would have the imagination to invent them. [55] Hardy asked a colleague, J. E. Littlewood, to take a look at the papers. Littlewood was amazed by the mathematical genius of Ramanujan. After discussing the papers with Littlewood, Hardy concluded that the letters were “certainly the most remarkable I have received” and commented that Ramanujan was “a mathematician of the highest quality, a man of altogether exceptional originality and power In mathematics, there is a distinction between having an insight and having a proof. Ramanujan’s talent suggested a plethora of formulae that could then be investigated in depth later.

It is said that Ramanujan’s discoveries are unusually rich and that there is often more in it than what initially meets the eye. As a by-product, new directions of research were opened up. Examples of the most interesting of these formulae include the intriguing infinite series for ? , one of which is given below This result is based on the negative fundamental discriminant d = ? 4? 58 with class number h(d) = 2 (note that 5? 7? 13? 58 = 26390) and is related to the fact that Compare to Heegner numbers, which have class number 1 and yield similar formulae. Ramanujan’s series for ? onverges extraordinarily rapidly (exponentially) and forms the basis of some of the fastest algorithms currently used to calculate ?. Truncating the sum to the first term also gives the approximation  for ? , which is correct to six decimal places. One of his remarkable capabilities was the rapid solution for problems. He was sharing a room with P. C. Mahalanobis who had a problem, “Imagine that you are on a street with houses marked 1 through n. There is a house in between (x) such that the sum of the house numbers to left of it equals the sum of the house numbers to its right.

If n is between 50 and 500, what are n and x. ” This is a bivariate problem with multiple solutions. Ramanujan thought about it and gave the answer with a twist: He gave a continued fraction. The unusual part was that it was the solution to the whole class of problems. Mahalanobis was astounded and asked how he did it. “It is simple. The minute I heard the problem, I knew that the answer was a continued fraction. Which continued fraction, I asked myself. Then the answer came to my mind”, Ramanujan replied. [81][82] His intuition also led him to derive some previously unknown identities, such as or all ? , where ? (z) is the gamma function. Equating coefficients of ? 0, ? 4, and ? 8 gives some deep identities for the hyperbolic secant. In 1918, Hardy and Ramanujan studied the partition function P(n) extensively and gave a non-convergent asymptotic series that permits exact computation of the number of partitions of an integer. Hans Rademacher, in 1937, was able to refine their formula to find an exact convergent series solution to this problem. Ramanujan and Hardy’s work in this area gave rise to a powerful new method for finding asymptotic formulae, called the circle method. 83] He discovered mock theta functions in the last year of his life. For many years these functions were a mystery, but they are now known to be the holomorphic parts of harmonic weak Maass forms. Hardy–Ramanujan number 1729 A common anecdote about Ramanujan relates to the number 1729. Hardy arrived at Ramanujan’s residence in a cab numbered 1729. Hardy commented that the number 1729 seemed to be uninteresting. Ramanujan is said to have stated on the spot that it was actually a very interesting number mathematically, being the smallest number representable in two different ways as a sum of two cubes: ———————————————— Brahmagupta  (598–668) was an Indian mathematician and astronomer. Brahmagupta wrote important works on mathematics and astronomy. In particular hewrote Brahmasphutasiddhanta (Correctly Established Doctrine of Brahma), in 628. The work was written in 25 chapters and Brahmagupta tells us in the text that he wrote it at Bhillamal which today is the city of Bhinmal. Mathematics Brahmagupta’s most famous work is his Brahmasphutasiddhanta. It is composed in elliptic verse, as was common ractice in Indian mathematics, and consequently has a poetic ring to it. As no proofs are given, it is not known how Brahmagupta’s mathematics was derived. [3] ]Algebra Brahmagupta gave the solution of the general linear equation in chapter eighteen of Brahmasphutasiddhanta, 18. 43 The difference between rupas, when inverted and divided by the difference of the unknowns, is the unknown in the equation. The rupas are [subtracted on the side] below that from which the square and the unknown are to be subtracted. [4] Which is a solution equivalent to , where rupas represents constants.

He further gave two equivalent solutions to the general quadratic equation, 18. 44. Diminish by the middle [number] the square-root of the rupas multiplied by four times the square and increased by the square of the middle [number]; divide the remainder by twice the square. [The result is] the middle [number]. 18. 45. Whatever is the square-root of the rupas multiplied by the square [and] increased by the square of half the unknown, diminish that by half the unknown [and] divide [the remainder] by its square. [The result is] the unknown. 4] Which are, respectively, solutions equivalent to, And He went on to solve systems of simultaneous indeterminate equations stating that the desired variable must first be isolated, and then the equation must be divided by the desired variable’s coefficient. In particular, he recommended using “the pulverizer” to solve equations with multiple unknown 18. 51. Subtract the colors different from the first color. [The remainder] divided by the first [color’s coefficient] is the measure of the first. [Terms] two by two [are] considered [when reduced to] similar divisors, [and so on] repeatedly.

If there are many [colors], the pulverizer [is to be used]. [4]Like the algebra of Diophantus, the algebra of Brahmagupta was syncopated. Addition was indicated by placing the numbers side by side, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, similar to our notation but without the bar. Multiplication, evolution, and unknown quantities were represented by abbreviations of appropriate terms. [5] The extent of Greek influence on this syncopation, if any, is not known and it is possible that both Greek and Indian syncopation may be derived from a common Babylonian source. 5] ]Arithmetic In the beginning of chapter twelve of his Brahmasphutasiddhanta, entitled Calculation, Brahmagupta details operations on fractions. The reader is expected to know the basic arithmetic operations as far as taking the square root, although he explains how to find the cube and cube-root of an integer and later gives rules facilitating the computation of squares and square roots. He then gives rules for dealing with five types of combinations of fractions, , , , , and . [6]]Series Brahmagupta then goes on to give the sum of the squares and cubes of the first n integers. 2. 20. The sum of the squares is that [sum] multiplied by twice the [number of] step[s] increased by one [and] divided by three. The sum of the cubes is the square of that [sum] Piles of these with identical balls [can also be computed]. [7] It is important to note here Brahmagupta found the result in terms of the sum of the first n integers, rather than in terms of n as is the modern practice. [8] He gives the sum of the squares of the first n natural numbers as n(n+1)(2n+1)/6 and the sum of the cubes of the first n natural numbers as (n(n+1)/2)?.

Zero Brahmaguptasiddhanta is the very first book that mentions zero as a number. hence Brahmagupta is considered as the man who found zero. He gave rules of using zero with other numbers. Zero plus a positive number is the positive number etc. Brahmagupta made use of an important concept in mathematics, thenumber zero. The Brahmasphutasiddhanta is the earliest known text to treat zero as a number in its own right, rather than as simply a placeholder digit in representing another number as was done by the Babylonians or as a symbol for a lack of quantity as was done by Ptolemy and the Romans.

In chapter eighteen of his Brahmasphutasiddhanta, Brahmagupta describes operations on negative numbers. He first describes addition and subtraction, 18. 30. [The sum] of two positives is positives, of two negatives negative; of a positive and a negative [the sum] is their difference; if they are equal it is zero. The sum of a negative and zero is negative, [that] of a positive and zero positive, [and that] of two zeros zero. 18. 32. A negative minus zero is negative, a positive [minus zero] positive; zero [minus zero] is zero. When a positive is to be subtracted from a negative or a negative from a positive, then it is to be added. 4] He goes on to describe multiplication, 18. 33. The product of a negative and a positive is negative, of two negatives positive, and of positives positive; the product of zero and a negative, of zero and a positive, or of two zeros is zero. [4]But his description of division by zero differs from our modern understanding, 18. 34. A positive divided by a positive or a negative divided by a negative is positive; a zero divided by a zero is zero; a positive divided by a negative is negative; a negative divided by a positive is [also] negative. 18. 35.

A negative or a positive divided by zero has that [zero] as its divisor, or zero divided by a negative or a positive [has that negative or positive as its divisor]. The square of a negative or of a positive is positive; [the square] of zero is zero. That of which [the square] is the square is [its] square-root Here Brahmagupta states that  and as for the question of  where  he did not commit himself. [9] His rules for arithmetic on negative numbers and zero are quite close to the modern understanding, except that in modern mathematics division by zero is left undefined.

Diophantine analysis Pythagorean triples In chapter twelve of his Brahmasphutasiddhanta, Brahmagupta finds Pythagorean triples, 12. 39. The height of a mountain multiplied by a given multiplier is the distance to a city; it is not erased. When it is divided by the multiplier increased by two it is the leap of one of the two who make the same journey. [7] or in other words, for a given length m and an arbitrary multiplier x, let a = mx and b = m + mx/(x + 2). Then m, a, and b form a Pythagorean triple. 7] Pell’s equation Brahmagupta went on to give a recurrence relation for generating solutions to certain instances of Diophantine equations of the second degree such asNx2 + 1 = y2 (called Pell’s equation) by using the Euclidean algorithm. The Euclidean algorithm was known to him as the “pulverizer” since it breaks numbers down into ever smaller pieces. The nature of squares: 18. 64. [Put down] twice the square-root of a given square by a multiplier and increased or diminished by an arbitrary [number]. The product product of the first [pair], multiplied by the multiplier, with the product of the last [pair], is the last computed. 8. 65. The sum of the thunderbolt products is the first. The additive is equal to the product of the additives. The two square-roots, divided by the additive or the subtractive, are the additive rupas. [4] The key to his solution was the identity,[11] which is a generalization of an identity that was discovered by Diophantus, Using his identity and the fact that if (x1, y1) and (x2, y2) are solutions to the equations x2 ? Ny2 = k1 and x2 ? Ny2 = k2, respectively, then (x1x2 + Ny1y2,x1y2 + x2y1) is a solution to x2 ?

Ny2 = k1k2, he was able to find integral solutions to the Pell’s equation through a series of equations of the formx2 ? Ny2 = ki. Unfortunately, Brahmagupta was not able to apply his solution uniformly for all possible values of N, rather he was only able to show that ifx2 ? Ny2 = k has an integral solution for k = ±1, ±2, or ±4, then x2 ? Ny2 = 1 has a solution. The solution of the general Pell’s equation would have to wait forBhaskara II in c. 1150 CE. [11] Brahmagupta’s formula The approximate area is the product of the halves of the sums of the sides and opposite sides of a triangle and a quadrilateral.

The accurate [area] is the square root from the product of the halves of the sums of the sides diminished by [each] side of the quadrilateral. [7] So given the lengths p, q, r and s of a cyclic quadrilateral, the approximate area is while, letting the exact area is Although Brahmagupta does not explicitly state that these quadrilaterals are cyclic, it is apparent from his rules that this is the case. [12] Heron’s formula is a special case of this formula and it can be derived by setting one of the sides equal to zero. Triangles Brahmagupta dedicated a substantial portion of his work to geometry.

One theorem states that the two lengths of a triangle’s base when divided by its altitude then follows, 12. 22. The base decreased and increased by the difference between the squares of the sides divided by the base; when divided by two they are the true segments. The perpendicular [altitude] is the square-root from the square of a side diminished by the square of its segment. [7] Thus the lengths of the two segments are . He further gives a theorem on rational triangles. A triangle with rational sides a, b, c and rational area is of the form: for some rational numbers u, v, and w. 13] BHASKARACHARYA ————————————————- Mathematics Some of Bhaskara’s contributions to mathematics include the following: A proof of the Pythagorean theorem by calculating the same area in two different ways and then canceling out terms to get a? + b? = c?. In Lilavati, solutions of  quadratic, cubic and quartic indeterminate equations. Solutions of indeterminate quadratic equations (of the type ax? + b = y? ). Integer solutions of linear and quadratic indeterminate equations (Kuttaka). The rules he gives are (in effect) the same as those given by he RenaissanceEuropean mathematicians of the 17th century A cyclic Chakravala method for solving indeterminate equations of the form ax? + bx + c = y. The solution to this equation was traditionally attributed to William Brouncker in 1657, though his method was more difficult than the chakravala method The first general method for finding the solutions of the problem x? ? ny? = 1 (so-called “Pell’s equation”) was given by Bhaskara II. [10] Solutions of Diophantine equations of the second order, such as 61x? + 1 = y?.

This very equation was posed as a problem in 1657 by the Frenchmathematician Pierre de Fermat, but its solution was unknown in Europe until the time of Euler in the 18th century. Solved quadratic equations with more than one unknown, and found negative and irrational solutions. Preliminary concept of mathematical analysis. Preliminary concept of infinitesimal calculus, along with notable contributions towards integral calculus. Conceived differential calculus, after discovering the derivative and differential coefficient. Stated Rolle’s theorem, a special case of one of the most important theorems in analysis, the mean value theorem.

Traces of the general mean value theoremare also found in his works. Calculated the derivatives of trigonometric functions and formulae. (See Calculus section below. ) In Siddhanta Shiromani, Bhaskara developed spherical trigonometry along with a number of other trigonometric results. (See Trigonometry section below. ) Arithmetic Bhaskara’s arithmetic text Lilavati covers the topics of definitions, arithmetical terms, interest computation, arithmetical and geometrical progressions, plane geometry, solid geometry, the shadow of the gnomon, methods to solve indeterminate equations, and combinations.

Lilavati is divided into 13 chapters and covers many branches of mathematics, arithmetic, algebra, geometry, and a little trigonometry and mensuration. More specifically the contents include: Definitions. Properties of zero (including division, and rules of operations with zero) Further extensive numerical work, including use of negative numbers and surds. Estimation of ?. Arithmetical terms, methods of multiplication, and squaring. Inverse rule of three, and rules of 3, 5, 7, 9, and 11. Problems involving interest and interest computation. Arithmetical and geometrical progressions.

Plane (geometry). Solid geometry. Permutations and combinations . Indeterminate equations (Kuttaka), integer solutions (first and second order). His contributions to this topic are particularly important, since the rules he gives are (in effect) the same as those given by the renaissance European mathematicians of the 17th century, yet his work was of the 12th century. Bhaskara’s method of solving was an improvement of the methods found in the work of Aryabhata and subsequent mathematicians. His work is outstanding for its systemisation, improved methods and the new topics that he has introduced.

Furthermore the Lilavati contained excellent recreative problems and it is thought that Bhaskara’s intention may have been that a student of ‘Lilavati’ should concern himself with the mechanical application of the method. Algebra His Bijaganita (“Algebra”) was a work in twelve chapters. It was the first text to recognize that a positive number has two square roots (a positive and negative square root). His work Bijaganita is effectively a treatise on algebra and contains the following topics:Positive and negative numbers. Zero. The ‘unknown’ (includes determining unknown quantities). Determining unknown quantities.

Surds (includes evaluating surds). Kuttaka (for solving indeterminate equations and Diophantine equations). Simple equations (indeterminate of second, third and fourth degree). Simple equations with more than one unknown. indeterminate quadratic equations (of the type ax? + b = y? ). Solutions of indeterminate equations of the second, third and fourth degree. Quadratic equations. Quadratic equations with more than one unknown. Operations with products of several unknowns. Bhaskara derived a cyclic, chakravala method for solving indeterminate quadratic equations of the form ax? + bx + c = y.

Bhaskara’s method for finding the solutions of the problem Nx? + 1 = y? (the so-called “Pell’s equation”) is of considerable importance. [10] He gave the general solutions of:Pell’s equation using the chakravala method. The indeterminate quadratic equation using the chakravala method. He also solved [Cubic equations Quartic equations. Indeterminate cubic equations. , Indeterminate quartic equations,. Indeterminate higher-order polynomial equations. Trigonometry The Siddhanta Shiromani (written in 1150) demonstrates Bhaskara’s knowledge of trigonometry, including the sine table and relationships between different trigonometric functions.

He also discovered spherical trigonometry, along with other interesting trigonometrical results. In particular Bhaskara seemed more interested in trigonometry for its own sake than his predecessors who saw it only as a tool for calculation. Among the many interesting results given by Bhaskara, discoveries first found in his works include the now well known results for  and : Calculus His work, the Siddhanta Shiromani, is an astronomical treatise and contains many theories not found in earlier works.

Preliminary concepts of infinitesimal calculus and mathematical analysis, along with a number of results in trigonometry, differential calculus and integral calculus that are found in the work are of particular interest. Evidence suggests Bhaskara was acquainted with some ideas of differential calculus. It seems, however, that he did not understand the utility of his researches, and thus historians of mathematics generally neglect this achievement.

Bhaskara also goes deeper into the ‘differential calculus’ and suggests the differential coefficient vanishes at an extremum value of the function, indicating knowledge of the concept of ‘infinitesimals’. [11] There is evidence of an early form of Rolle’s theorem in his work: If  then  for some  with  He gave the result that if  then , thereby finding the derivative of sine, although he never developed the notion of derivatives. [12]Bhaskara uses this result to work out the position angle of the ecliptic, a quantity required for accurately predicting the time of an eclipse.

In computing the instantaneous motion of a planet, the time interval between successive positions of the planets was no greater than a truti, or a 1? 33750 of a second, and his measure of velocity was expressed in this infinitesimal unit of time. He was aware that when a variable attains the maximum value, its differential vanishes. He also showed that when a planet is at its farthest from the earth, or at its closest, the equation of the centre (measure of how far a planet is from the position in which it is predicted to be, by assuming it is to move uniformly) vanishes.

He therefore concluded that for some intermediate position the differential of the equation of the centre is equal to zero. In this result, there are traces of the general mean value theorem, one of the most important theorems in analysis, which today is usually derived from Rolle’s theorem. The mean value theorem was later found by Parameshvara in the 15th century in the Lilavati Bhasya, a commentary on Bhaskara’s Lilavati. Madhava (1340–1425) and the Kerala School mathematicians (including Parameshvara) from the 14th century to the 16th century expanded on Bhaskara’s work and further advanced the development of calculus in India


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