MATHEMATICS
Despite developing quite independently of Chinese (and probably also of Babylonian mathematics), some very advanced mathematical discoveries were made at a very early time in India.
Mantras from the early Vedic period (before 1000 BC) invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of
arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. A 4th Century AD Sanskrit text reports Buddha enumerating numbers up to 1053, as well as describing six more numbering systems over and above these, leading to a number equivalent to 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. It also describes a series of iterations in decreasing size, in order to demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a metre).
As early as the 8th Century BC, long before Pythagoras, a text known as the “Sulba Sutras” (or "Sulva Sutras") listed several simple Pythagorean triples, as well as a statement of the simplified Pythagorean theorem for the sides of a square and for a rectangle (indeed, it seems quite likely that Pythagoras learned his basic geometry from the "Sulba Sutras"). The Sutras also contain geometric solutions of linear and quadratic equations in a single unknown, and give a remarkably accurate figure for the square root of 2, obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) + 1⁄(3 x 4 x 34), which yields a value of 1.4142156, correct to 5 decimal places.
As early as the 3rd or 2nd Century BC, Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Ancient Buddhist literature also demonstrates a prescient awareness of indeterminate and infinite numbers, with numbers deemed to be of three types: countable, uncountable and infinite.
Like the Chinese, the Indians early discovered the benefits of a decimal place value number system, and were certainly using it before about the 3rd Century AD. They refined and perfected the system, particularly the written representation of the numerals, creating the ancestors of the nine numerals that (thanks to its dissemination by medieval Arabic mathematicans) we use across the world today, sometimes considered one of the greatest intellectual innovations of all time.
The Indians were also responsible for another hugely important
development in mathematics. The earliest recorded usage of a circle
character for the number zero is usually attributed to a 9th Century
engraving in a temple in Gwalior in central India. But the brilliant
conceptual leap to include zero as a number in its own right (rather
than merely as a placeholder, a blank or empty space within a number, as
it had been treated until that time) is usually credited to the 7th
Century Indian mathematicians Brahmagupta
- or possibly another Indian, Bhaskara I - even though it may well have
been in practical use for centuries before that. The use of zero as a
number which could be used in calculations and mathematical
investigations, would revolutionize mathematics.
Brahmagupta established the basic mathematical rules for dealing
with zero: 1 + 0 = 1; 1 - 0 = 1; and 1 x 0 = 0 (the breakthrough which
would make sense of the apparently non-sensical operation 1 ÷ 0 would
also fall to an Indian, the 12th Century mathematician Bhaskara II). Brahmagupta also established rules for dealing
with negative numbers, and pointed out that quadratic equations could
in theory have two possible solutions, one of which could be negative.
He even attempted to write down these rather abstract concepts, using
the initials of the names of colours to represent unknowns in his
equations, one of the earliest intimations of what we now know as
algebra.
The so-called Golden Age of Indian mathematics can be said to extend from the 5th to 12th Centuries, and many of its mathematical discoveries predated similar discoveries in the West by several centuries, which has led to some claims of plagiarism by later European mathematicians, at least some of whom were probably aware of the earlier Indian work. Certainly, it seems that Indian contributions to mathematics have not been given due acknowledgement until very recently in modern history.
Golden Age Indian mathematicians made fundamental advances in the
theory of trigonometry, a method of linking geometry and numbers first
developed by the Greeks.
They used ideas like the sine, cosine and tangent functions (which
relate the angles of a triangle to the relative lengths of its sides) to
survey the land around them, navigate the seas and even chart the
heavens. For instance, Indian astronomers used trigonometry to
calculated the relative distances between the Earth and the Moon and the
Earth and the Sun. They realized that, when the Moon is half full and
directly opposite the Sun, then the Sun, Moon and Earth form a right
angled triangle, and were able to accurately measure the angle as 1⁄7°.
Their sine tables gave a ratio for the sides of such a triangle as
400:1, indicating that the Sun is 400 times further away from the Earth
than the Moon.
Although the Greeks had been able to calculate the sine function of some angles, the Indian astronomers wanted to be able to calculate the sine function of any given angle. A text called the “Surya Siddhanta”, by unknown authors and dating from around 400 AD, contains the roots of modern trigonometry, including the first real use of sines, cosines, inverse sines, tangents and secants.
As early as the 6th Century AD, the great Indian mathematician and astronomer Aryabhata produced categorical definitions of sine, cosine, versine and inverse sine, and specified complete sine and versine tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. Aryabhata also demonstrated solutions to simultaneous quadratic equations, and produced an approximation for the value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the circumference of the Earth, arriving at a figure of 24,835 miles, only 70 miles off its true value. But, perhaps even more astonishing, he seems to have been aware that π is an irrational number, and that any calculation can only ever be an approximation, something not proved in Europe until 1761.
Bhaskara II, who lived in the 12th Century, was one of the most
accomplished of all India’s great mathematicians. He is credited with
explaining the previously misunderstood operation of division by zero.
He noticed that dividing one into two pieces yields a half, so 1 ÷ 1⁄2 = 2. Similarly, 1 ÷ 1⁄3
= 3. So, dividing 1 by smaller and smaller factions yields a larger and
larger number of pieces. Ultimately, therefore, dividing one into
pieces of zero size would yield infinitely many pieces, indicating that 1
÷ 0 = ∞ (the symbol for infinity).
However, Bhaskara II also made important contributions to many different areas of mathematics from solutions of quadratic, cubic and quartic equations (including negative and irrational solutions) to solutions of Diophantine equations of the second order to preliminary concepts of infinitesimal calculus and mathematical analysis to spherical trigonometry and other aspects of trigonometry. Some of his findings predate similar discoveries in Europe by several centuries, and he made important contributions in terms of the systemization of (then) current knowledge and improved methods for known solutions.
The Kerala School of Astronomy and Mathematics was founded in the late 14th Century by Madhava of Sangamagrama, sometimes called the greatest mathematician-astronomer of medieval India. He developed infinite series approximations for a range of trigonometric functions, including π, sine, etc. Some of his contributions to geometry and algebra and his early forms of differentiation and integration for simple functions may have been transmitted to Europe via Jesuit missionaries, and it is possible that the later European development of calculus was influenced by his work to some extent
BRAHMAGUPTA
The great 7th Century Indian mathematician and astronomer Brahmagupta
wrote some important works on both mathematics and astronomy. He was
from the state of Rajasthan of northwest India (he is often referred to
as Bhillamalacarya, the teacher from Bhillamala), and later became the
head of the astronomical observatory at Ujjain in central India. Most of
his works are composed in elliptic verse, a common practice in Indian
mathematics at the time, and consequently have something of a poetic
ring to them.
It seems likely that Brahmagupta's works, especially his most famous text, the “Brahmasphutasiddhanta”, were brought by the 8th Century Abbasid caliph Al-Mansur to his newly founded centre of learning at Baghdad on the banks of the Tigris, providing an important link between Indian mathematics and astronomy and the nascent upsurge in science and mathematics in the Islamic world.
In his work on arithmetic, Brahmagupta explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots. He also gave rules for dealing with five types of combinations of fractions. He gave 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)².
Brahmagupta’s genius, though, came in his treatment of the concept of
(then relatively new) the number zero. Although often also attributed
to the 7th Century Indian mathematician Bhaskara I, his
“Brahmasphutasiddhanta” is probably the earliest known text to treat
zero as a number in its own right, rather than as simply a placeholder
digit as was done by the Babylonians, or as a symbol for a lack of quantity as was done by the Greeks and Romans.
Brahmagupta established the basic mathematical rules for dealing
with zero (1 + 0 = 1; 1 - 0 = 1; and 1 x 0 = 0), although his
understanding of division by zero was incomplete (he thought that 1 ÷ 0 =
0). Almost 500 years later, in the 12th Century, another Indian
mathematician, Bhaskara II, showed that the answer should be infinity,
not zero (on the grounds that 1 can be divided into an infinite number
of pieces of size zero), an answer that was considered correct for
centuries. However, this logic does not explain why 2 ÷ 0, 7 ÷ 0, etc,
should also be zero - the modern view is that a number divided by zero
is actually "undefined" (i.e. it doesn't make sense).
Brahmagupta’s view of numbers as abstract entities, rather than just for counting and measuring, allowed him to make yet another huge conceptual leap which would have profound consequence for future mathematics. Previously, the sum 3 - 4, for example, was considered to be either meaningless or, at best, just zero. Brahmagupta, however, realized that there could be such a thing as a negative number, which he referred to as “debt” as a opposed to “property”. He expounded on the rules for dealing with negative numbers (e.g. a negative times a negative is a positive, a negative times a positive is a negative, etc).
Furthermore, he pointed out, quadratic equations (of the type x2 + 2 = 11, for example) could in theory have two possible solutions, one of which could be negative, because 32 = 9 and -32 = 9. In addition to his work on solutions to general linear equations and quadratic equations, Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables), and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657.
MADHAVA
Madhava sometimes called the greatest mathematician-astronomer of medieval India. He came from the town of Sangamagrama in Kerala, near the southern tip of India, and founded the Kerala School of Astronomy and Mathematics in the late 14th Century.
Although almost all of Madhava's original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine, cosine, tangent and arctangent functions and the value of π), representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.
Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancient Egyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions.
But Madhava went further and linked the idea of an infinite series
with geometry and trigonometry. He realized that, by successively adding
and subtracting different odd number fractions to infinity, he could
home in on an exact formula for π (this was two centuries before Leibniz was to come to the same conclusion in Europe). Through his application of this series, Madhava obtained a value for π correct to an astonishing 13 decimal places.
He went on to use the same mathematics to obtain infinite series expressions for the sine formula, which could then be used to calculate the sine of any angle to any degree of accuracy, as well as for other trigonometric functions like cosine, tangent and arctangent. Perhaps even more remarkable, though, is that he also gave estimates of the error term or correction term, implying that he quite understood the limit nature of the infinite series.
Madhava’s use of infinite series to approximate a range of trigonometric functions, which were further developed by his successors at the Kerala School, effectively laid the foundations for the later development of calculus and analysis, and either he or his disciples developed an early form of integration for simple functions. Some historians have suggested that Madhava's work, through the writings of the Kerala School, may have been transmitted to Europe via Jesuit missionaries and traders
who were active around the ancient port of Cochin (Kochi) at the time,
and may have had an influence on later European developments in
calculus.
Among his other contributions, Madhava discovered the solutions of some transcendental equations by a process of iteration, and found approximations for some transcendental numbers by continued fractions. In astronomy, he discovered a procedure to determine the positions of the Moon every 36 minutes, and methods to estimate the motions of the planets
 |
 The evolution of Hindu-Arabic numerals |
Mantras from the early Vedic period (before 1000 BC) invoke powers of ten from a hundred all the way up to a trillion, and provide evidence of the use of
arithmetic operations such as addition, subtraction, multiplication, fractions, squares, cubes and roots. A 4th Century AD Sanskrit text reports Buddha enumerating numbers up to 1053, as well as describing six more numbering systems over and above these, leading to a number equivalent to 10421. Given that there are an estimated 1080 atoms in the whole universe, this is as close to infinity as any in the ancient world came. It also describes a series of iterations in decreasing size, in order to demonstrate the size of an atom, which comes remarkably close to the actual size of a carbon atom (about 70 trillionths of a metre).
As early as the 8th Century BC, long before Pythagoras, a text known as the “Sulba Sutras” (or "Sulva Sutras") listed several simple Pythagorean triples, as well as a statement of the simplified Pythagorean theorem for the sides of a square and for a rectangle (indeed, it seems quite likely that Pythagoras learned his basic geometry from the "Sulba Sutras"). The Sutras also contain geometric solutions of linear and quadratic equations in a single unknown, and give a remarkably accurate figure for the square root of 2, obtained by adding 1 + 1⁄3 + 1⁄(3 x 4) + 1⁄(3 x 4 x 34), which yields a value of 1.4142156, correct to 5 decimal places.
As early as the 3rd or 2nd Century BC, Jain mathematicians recognized five different types of infinities: infinite in one direction, in two directions, in area, infinite everywhere and perpetually infinite. Ancient Buddhist literature also demonstrates a prescient awareness of indeterminate and infinite numbers, with numbers deemed to be of three types: countable, uncountable and infinite.
Like the Chinese, the Indians early discovered the benefits of a decimal place value number system, and were certainly using it before about the 3rd Century AD. They refined and perfected the system, particularly the written representation of the numerals, creating the ancestors of the nine numerals that (thanks to its dissemination by medieval Arabic mathematicans) we use across the world today, sometimes considered one of the greatest intellectual innovations of all time.
 |
|
The earliest use of a circle character for the number zero was in India |
Brahmagupta established the basic mathematical rules for dealing
with zero: 1 + 0 = 1; 1 - 0 = 1; and 1 x 0 = 0 (the breakthrough which
would make sense of the apparently non-sensical operation 1 ÷ 0 would
also fall to an Indian, the 12th Century mathematician Bhaskara II). Brahmagupta also established rules for dealing
with negative numbers, and pointed out that quadratic equations could
in theory have two possible solutions, one of which could be negative.
He even attempted to write down these rather abstract concepts, using
the initials of the names of colours to represent unknowns in his
equations, one of the earliest intimations of what we now know as
algebra.The so-called Golden Age of Indian mathematics can be said to extend from the 5th to 12th Centuries, and many of its mathematical discoveries predated similar discoveries in the West by several centuries, which has led to some claims of plagiarism by later European mathematicians, at least some of whom were probably aware of the earlier Indian work. Certainly, it seems that Indian contributions to mathematics have not been given due acknowledgement until very recently in modern history.
 |
|
Indian astronomers used trigonometry tables to estimate the relative distance of the Earth to the Sun and Moon |
Although the Greeks had been able to calculate the sine function of some angles, the Indian astronomers wanted to be able to calculate the sine function of any given angle. A text called the “Surya Siddhanta”, by unknown authors and dating from around 400 AD, contains the roots of modern trigonometry, including the first real use of sines, cosines, inverse sines, tangents and secants.
As early as the 6th Century AD, the great Indian mathematician and astronomer Aryabhata produced categorical definitions of sine, cosine, versine and inverse sine, and specified complete sine and versine tables, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. Aryabhata also demonstrated solutions to simultaneous quadratic equations, and produced an approximation for the value of π equivalent to 3.1416, correct to four decimal places. He used this to estimate the circumference of the Earth, arriving at a figure of 24,835 miles, only 70 miles off its true value. But, perhaps even more astonishing, he seems to have been aware that π is an irrational number, and that any calculation can only ever be an approximation, something not proved in Europe until 1761.
 |
|
Illustration of infinity as the reciprocal of zero |
However, Bhaskara II also made important contributions to many different areas of mathematics from solutions of quadratic, cubic and quartic equations (including negative and irrational solutions) to solutions of Diophantine equations of the second order to preliminary concepts of infinitesimal calculus and mathematical analysis to spherical trigonometry and other aspects of trigonometry. Some of his findings predate similar discoveries in Europe by several centuries, and he made important contributions in terms of the systemization of (then) current knowledge and improved methods for known solutions.
The Kerala School of Astronomy and Mathematics was founded in the late 14th Century by Madhava of Sangamagrama, sometimes called the greatest mathematician-astronomer of medieval India. He developed infinite series approximations for a range of trigonometric functions, including π, sine, etc. Some of his contributions to geometry and algebra and his early forms of differentiation and integration for simple functions may have been transmitted to Europe via Jesuit missionaries, and it is possible that the later European development of calculus was influenced by his work to some extent
BRAHMAGUPTA
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|
Brahmagupta (598–668 AD) |
It seems likely that Brahmagupta's works, especially his most famous text, the “Brahmasphutasiddhanta”, were brought by the 8th Century Abbasid caliph Al-Mansur to his newly founded centre of learning at Baghdad on the banks of the Tigris, providing an important link between Indian mathematics and astronomy and the nascent upsurge in science and mathematics in the Islamic world.
In his work on arithmetic, Brahmagupta explained how to find the cube and cube-root of an integer and gave rules facilitating the computation of squares and square roots. He also gave rules for dealing
with five types of combinations of fractions. He gave 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)².
 |
|
Brahmagupta’s rules for dealing with zero and negative numbers |
Brahmagupta established the basic mathematical rules for dealing
with zero (1 + 0 = 1; 1 - 0 = 1; and 1 x 0 = 0), although his
understanding of division by zero was incomplete (he thought that 1 ÷ 0 =
0). Almost 500 years later, in the 12th Century, another Indian
mathematician, Bhaskara II, showed that the answer should be infinity,
not zero (on the grounds that 1 can be divided into an infinite number
of pieces of size zero), an answer that was considered correct for
centuries. However, this logic does not explain why 2 ÷ 0, 7 ÷ 0, etc,
should also be zero - the modern view is that a number divided by zero
is actually "undefined" (i.e. it doesn't make sense).Brahmagupta’s view of numbers as abstract entities, rather than just for counting and measuring, allowed him to make yet another huge conceptual leap which would have profound consequence for future mathematics. Previously, the sum 3 - 4, for example, was considered to be either meaningless or, at best, just zero. Brahmagupta, however, realized that there could be such a thing as a negative number, which he referred to as “debt” as a opposed to “property”. He expounded on the rules for dealing with negative numbers (e.g. a negative times a negative is a positive, a negative times a positive is a negative, etc).
Furthermore, he pointed out, quadratic equations (of the type x2 + 2 = 11, for example) could in theory have two possible solutions, one of which could be negative, because 32 = 9 and -32 = 9. In addition to his work on solutions to general linear equations and quadratic equations, Brahmagupta went yet further by considering systems of simultaneous equations (set of equations containing multiple variables), and solving quadratic equations with two unknowns, something which was not even considered in the West until a thousand years later, when Fermat was considering similar problems in 1657.
 |
|
Brahmagupta’s Theorem on cyclic quadrilateral |
MADHAVA
Madhava sometimes called the greatest mathematician-astronomer of medieval India. He came from the town of Sangamagrama in Kerala, near the southern tip of India, and founded the Kerala School of Astronomy and Mathematics in the late 14th Century.
Although almost all of Madhava's original work is lost, he is referred to in the work of later Kerala mathematicians as the source for several infinite series expansions (including the sine, cosine, tangent and arctangent functions and the value of π), representing the first steps from the traditional finite processes of algebra to considerations of the infinite, with its implications for the future development of calculus and mathematical analysis.
Unlike most previous cultures, which had been rather nervous about the concept of infinity, Madhava was more than happy to play around with infinity, particularly infinite series. He showed how, although one can be approximated by adding a half plus a quarter plus an eighth plus a sixteenth, etc, (as even the ancient Egyptians and Greeks had known), the exact total of one can only be achieved by adding up infinitely many fractions.
 |
|
Madhava’s method for approximating π by an infinite series of fractions |
He went on to use the same mathematics to obtain infinite series expressions for the sine formula, which could then be used to calculate the sine of any angle to any degree of accuracy, as well as for other trigonometric functions like cosine, tangent and arctangent. Perhaps even more remarkable, though, is that he also gave estimates of the error term or correction term, implying that he quite understood the limit nature of the infinite series.
Madhava’s use of infinite series to approximate a range of trigonometric functions, which were further developed by his successors at the Kerala School, effectively laid the foundations for the later development of calculus and analysis, and either he or his disciples developed an early form of integration for simple functions. Some historians have suggested that Madhava's work, through the writings of the Kerala School, may have been transmitted to Europe via Jesuit missionaries and traders
who were active around the ancient port of Cochin (Kochi) at the time,
and may have had an influence on later European developments in
calculus.Among his other contributions, Madhava discovered the solutions of some transcendental equations by a process of iteration, and found approximations for some transcendental numbers by continued fractions. In astronomy, he discovered a procedure to determine the positions of the Moon every 36 minutes, and methods to estimate the motions of the planets
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