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HELLENISTIC MATHEMATICS

By the 3rd Century BC, in the wake of the conquests of Alexander the Great, mathematical breakthroughs were also beginning to be made on the edges of the Greek Hellenistic empire.

In particular, Alexandria in Egypt became a great centre of learning under the beneficent rule of the Ptolemies, and its famous Library soon gained a reputation to rival that of the Athenian Academy. The patrons of the Library were arguably the first professional scientists, paid for their devotion to research. Among the best known and most influential mathematicians who studied and taught at Alexandria were Euclid, Archimedes, Eratosthenes, Heron, Menelaus and Diophantus.

During the late 4th and early 3rd Century BC, Euclid was the great chronicler of the mathematics of the time, and one of the most influential teachers in history. He virtually invented classical (Euclidean) geometry as we know it. Archimedes spent most of his life in Syracuse, Sicily, but also studied for a while in Alexandria. He is perhaps best known as an engineer and inventor but, in the light of recent discoveries, he is now considered of one of the greatest pure mathematicians of all time. Eratosthenes of Alexandria was a near contemporary of Archimedes in the 3rd Century BC. A mathematician, astronomer and geographer, he devised the first system of latitude and longitude, and calculated the circumference of the earth to a remarkable degree of accuracy. As a mathematician, his greatest legacy is the “Sieve of Eratosthenes” algorithm for identifying prime numbers.

The Sieve of Eratosthenes

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It is not known exactly when the great Library of Alexandria burned down, but Alexandria remained an important intellectual centre for some centuries. In the 1st century BC, Heron (or Hero) was another great Alexandrian inventor, best known in mathematical circles for Heronian triangles (triangles with integer sides and integer area), Heron’s Formula for finding the area of a triangle from its side lengths, and Heron’s Method for iteratively computing a square root. He was also the first mathematician to confront at least the idea of √-1 (although he had no idea how to treat it, something which had to wait for Tartaglia and Cardano in the 16th Century).

Menelaus of Alexandria, who lived in the 1st - 2nd Century AD, was the first to recognize geodesics on a curved surface as the natural analogues of straight lines on a flat plane. His book “Sphaerica” dealt with the geometry of the sphere and its application in astronomical measurements and calculations, and introduced the concept of spherical triangle (a figure formed of three great circle arcs, which he named "trilaterals").

In the 3rd Century AD, Diophantus of Alexandria was the first to recognize fractions as numbers, and is considered an early innovator in the field of what would later become known as algebra. He applied himself to some quite complex algebraic problems, including what is now known as Diophantine Analysis, which deals with finding integer solutions to kinds of problems that lead to equations in several unknowns (Diophantine equations). Diophantus’ “Arithmetica”, a collection of problems giving numerical solutions of both determinate and indeterminate equations, was the most prominent work on algebra in all Greek mathematics, and his problems exercised the minds of many of the world's best mathematicians for much of the next two millennia.

Menelaus of Alexandria introduced the concept of spherical triangle

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But Alexandria was not the only centre of learning in the Hellenistic Greek empire. Mention should also be made of Apollonius of Perga (a city in modern-day southern Turkey) whose late 3rd Century BC work on geometry (and, in particular, on conics and conic sections) was very influential on later European mathematicians. It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them, and showed how they could be derived from different sections through a cone.

Hipparchus, who was also from Hellenistic Anatolia and who live in the 2nd Century BC, was perhaps the greatest of all ancient astronomers. He revived the use of arithmetic techniques first developed by the Chaldeans and Babylonians, and is usually credited with the beginnings of trigonometry. He calculated (with remarkable accuracy for the time) the distance of the moon from the earth by measuring the different parts of the moon visible at different locations and calculating the distance using the properties of triangles. He went on to create the first table of chords (side lengths corresponding to different angles of a triangle). By the time of the great Alexandrian astronomer Ptolemy in the 2nd Century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his “Almagest” a table of trigonometric chords in a circle for steps of ¼° which (although expressed sexagesimally in the Babylonian style) is accurate to about five decimal places.

By the middle of the 1st Century BC and thereafter, however, the Romans had tightened their grip on the old Greek empire. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it even less so. The final blow to the Hellenistic mathematical heritage at Alexandria might be seen in the figure of Hypatia, the first recorded female mathematician, and a renowned teacher who had written some respected commentaries on Diophantus and Apollonius. She was dragged to her death by a Christian mob in 415 AD.

HELLENISTIC MATHEMATICS - EUCLID

Conic sections of Apollonius

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The Greek mathematician Euclid lived and flourished in Alexandria in Egypt around 300 BC, during the reign of Ptolemy I. Almost nothing is known of his life, and no likeness or first-hand description of his physical appearance has survived antiquity, and so depictions of him (with a long flowing beard and cloth cap) in works of art are necessarily the products of the artist's imagination.

He probably studied for a time at Plato's Academy in Athens but, by Euclid's time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy.

Euclid is often referred to as the “Father of Geometry”, and he wrote perhaps the most important and successful mathematical textbook of all time, the “Stoicheion” or “Elements”, which represents the culmination of the mathematical revolution which had taken place in Greece up to that time. He also wrote works on the division of geometrical figures into into parts in given ratios, on catoptrics (the mathematical theory of mirrors and reflection), and on spherical astronomy (the determination of the location of objects on the "celestial sphere"), as well as important texts on optics and music.

The "Elements” was a lucid and comprehensive compilation and explanation of all the known mathematics of his time, including the work of Pythagoras, Hippocrates, Theudius, Theaetetus and Eudoxus. In all, it contains 465 theorems and proofs, described in a clear, logical and elegant style, and using only a compass and a straight edge. Euclid reworked the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean geometry, which is still as valid today as it was 2,300 years ago, even in higher mathematics dealing with higher dimensional spaces. It was only with the work of Bolyai, Lobachevski and Riemann in the first half of the 19th Century that any kind of non-Euclidean geometry was even considered.

The "Elements” remained the definitive textbook on geometry and mathematics for well over two millennia, surviving the eclipse in classical learning in Europe during the Dark Ages through Arabic translations. It set, for all time, the model for mathematical argument, following logical deductions from inital assumptions (which Euclid called “axioms” and "postulates") in order to establish proven theorems.

Euclid (c.330-275 BC, fl. c.300 BC)

Euclid’s method for constructing of an equilateral triangle from a given straight line segment AB using only a compass and straight edge was Proposition 1 in Book 1 of the "Elements"

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Euclid’s five general axioms were:

1. Things which are equal to the same thing are equal to each other.

2. If equals are added to equals, the wholes (sums) are equal.

3. If equals are subtracted from equals, the remainders (differences) are equal.

4. Things that coincide with one another are equal to one another.

5. The whole is greater than the part.

His five geometrical postulates were:

1. It is possible to draw a straight line from any point to any point.2. It is possible to extend a finite straight line continuously in a straight line (i.e. a line

segment can be extended past either of its endpoints to form an arbitrarily large line segment).

3. It is possible to create a circle with any center and distance (radius).4. All right angles are equal to one another (i.e. "half" of a straight angle).5. If a straight line crossing two straight lines makes the interior angles on the same side

less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.

Euclid’s Postulates (1 - 5)

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Among many other mathematical gems, the thirteen volumes of the “Elements” contain formulas for calculating the volumes of solids such as cones, pyramids and cylinders; proofs about geometric series, perfect numbers and primes; algorithms for finding the greatest common divisor and least common multiple of two numbers; a proof and generalization of Pythagoras’ Theorem, and proof that there are an infinite number of Pythagorean Triples; and a final definitive proof that there can be only five possible regular Platonic Solids.

However, the “Elements” also includes a series of theorems on the properties of numbers and integers, marking the first real beginnings of number theory. For example, Euclid proved what has become known as the Fundamental Theorem of Arithmethic (or the Unique Factorization Theorem), that every positive integer greater than 1 can be written as a product of prime numbers (or is itself a prime number). Thus, for example: 21 = 3 x 7; 113 = 1 x 113; 1,200 = 2 x 2 x 2 x 2 x 3 x 5 x 5; 6,936 = 2 x 2 x 2 x 3 x 17 x 17; etc. His proof was the first known example of a proof by contradiction (where any counter-example, which would otherwise prove an idea false, is shown to makes no logical sense itself).

He was the first to realize - and prove - that there are infinitely many prime numbers. The basis of his proof, often known as Euclid’s Theorem, is that, for any given (finite) set of primes, if you multiply all of them together and then add one, then a new prime has been added to the set (for example, 2 x 3 x 5 = 30, and 30 + 1 = 31, a prime number) a process which can be repeated indefinitely.

Euclid also identified the first four “perfect numbers”, numbers that are the sum of all their divisors (excluding the number itself):    6 = 1 + 2 + 3;    28 = 1 + 2 + 4 + 7 + 14;    496 = 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248; and    8,128 = 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1,016 + 2,032 + 4,064.He noted that these numbers also have many other interesting properties. For example:

They are triangular numbers, and therefore the sum of all the consecutive numbers up to their largest prime factor: 6 = 1 + 2 + 3; 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7; 496 = 1 + 2 + 3 + 4 + 5 + .... + 30 + 31; 8,128 = 1 + 2 + 3 + 4 + 5 + ... + 126 + 127. Their largest prime factor is a power of 2 less one, and the number is always a product of this number and the previous power of two: 6 = 21(22 - 1); 28 = 22(23 - 1); 496 = 24(25 - 1); 8,128 =

Part of Euclid’s proof of Pythagoras’ Theorem

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26(27 - 1).

Although the Pythagoreans may have been aware of the Golden Ratio (φ, approximately equal to 1.618), Euclid was the first to define it in terms of ratios (AB:AC = AC:CB), and demonstrated its appearance within many geometric shapes.

HELLENISTIC MATHEMATICS - ARCHIMEDES

Another Greek mathematician who studied at Alexandria in the 3rd Century BC was Archimedes, although he was born, died and lived most of his life in Syracuse, Sicily (a Hellenic Greek

Archimedes (c.287-212 BC)

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colony in Magna Graecia). Little is known for sure of his life, and many of the stories and anecdotes about him were written long after his death by the historians of ancient Rome.

Also an engineer, inventor and astronomer, Archimedes was best known throughout most of history for his military innovations like his siege engines and mirrors to harness and focus the power of the sun, as well as levers, pulleys and pumps (including the famous screw pump known as Archimedes’ Screw, which is still used today in some parts of the world for irrigation).

But his true love was pure mathematics, and the discovery in 1906 of previously unknown works, referred to as the "Archimedes Palimpsest", has provided new insights into how he obtained his mathematical results. Today, Archimedes is widely considered to have been one of the greatest mathematicians of antiquity, if not of all time, in the august company of mathematicians such as Newton and Gauss.

Archimedes produced formulas to calculate the areas of regular shapes, using a revolutionary method of capturing new shapes by using shapes he already understood. For example, to estimate the area of a circle, he constructed a larger polygon outside the circle and a smaller one inside it. He first enclosed the circle in a triangle, then in a square, pentagon, hexagon, etc, etc, each time approximating the area of the circle more closely. By this so-called “method of exhaustion” (or simply “Archimedes’ Method”), he effectively homed in on a value for one of the most important numbers in all of mathematics, π. His estimate was between 31⁄7 (approximately 3.1429) and 310⁄71 (approximately 3.1408), which compares well with its actual value of approximately 3.1416.

Interestingly, Archimedes seemed quite aware that a range was all that could be established and that the actual value might never be known. His method for estimating π was taken to the extreme by Ludoph van Ceulen in the 16th Century, who used a polygon with an extraordinary 4,611,686,018,427,387,904 sides to arrive at a value of π correct to 35 digits. We now know that π is in fact an irrational number, whose value can never be known with complete accuracy.

Similarly, he calculated the approximate volume of a solid like a sphere by slicing it up into a series of cylinders, and adding up the volumes of the constituent cylinders. He saw that by making the slices ever thinner, his approximation became more and more exact, so that, in the limit, his approximation became an exact calculation. This use of infinitesimals, in a way similar to modern integral calculus, allowed him to give answers to problems to an arbitrary degree of accuracy, while specifying the limits within which the answer lay.

Approximation of the area of circle by Archimedes’ method of exhaustion

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Archimedes’ most sophisticated use of the method of exhaustion, which remained unsurpassed until the development of integral calculus in the 17th Century, was his proof - known as the Quadrature of the Parabola - that the area of a parabolic segment is 4⁄3 that of a certain inscribed triangle. He dissected the area of a parabolic segment (the region enclosed by a parabola and a line) into infinitely many triangles whose areas form a geometric progression. He then computed the sum of the resulting geometric series, and proved that this is the area of the parabolic segment.

In fact, Archimedes had perhaps the most prescient view of the concept of infinity of all the Greek mathematicians. Generally speaking, the Greeks’ preference for precise, rigorous proofs and their distrust of paradoxes meant that they completely avoided the concept of actual infinity. Even Euclid, in his proof of the infinitude of prime numbers, was careful to conclude that there are “more primes than any given finite number” i.e. a kind of “potential infinity” rather than the “actual infinity” of, for example, the number of points on a line. Archimedes, however, in the "Archimedes Palimpsest", went further than any other Greek mathematician when, on compared two infinitely large sets, he noted that they had an equal number of members, thus for the first time considering actual infinity, a concept not seriously considered again until Georg Cantor in the 19th Century.

Another example of the meticulousness and precision of Archimedes’ work is his calculation of the value of the square root of 3 as lying between 265⁄153 (approximately 1.7320261) and 1351⁄780

(approximately 1.7320512) - the actual value is approximately 1.7320508. He even calculated the number of grains of sand required to fill the universe, using a system of counting based on the myriad (10,000) and myriad of myriads (100 million). His estimate was 8 vigintillion, or 8 x 1063.

Archimedes’ quadrature of the parabola using his method of exhaustion

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The discovery of which Archimedes claimed to be most proud was that of the relationship between a sphere and a circumscribing cylinder of the same height and diameter. He calculated the volume of a sphere as 4⁄3πr3, and that of a cylinder of the same height and diameter as 2πr3. The surface area was 4πr2 for the sphere, and 6πr2 for the cylinder (including its two bases). Therefore, it turns out that the sphere has a volume equal to two-thirds that of the cylinder, and a surface area also equal to two-thirds that of the cylinder. Archimedes was so pleased with this result that a sculpted sphere and cylinder were supposed to have been placed on his tomb of at his request.

Despite his important contributions to pure mathematics, though, Archimedes is probably best remembered for the anecdotal story of his discovery of a method for determining the volume of an object with an irregular shape. King Hieron of Syracuse had asked Archimedes to find out if the royal goldsmith had cheated him by putting silver in his new gold crown, but Archimedes clearly could not melt it down in order to measure it and establish its density, so he was forced to search for an alternative solution.

While taking his bath on day, he noticed that that the level of the water in the tub rose as he got in, and he had the sudden inspiration that he could use this effect to determine the volume (and therefore the density) of the crown. In his excitement, he

Archimedes showed that the volume and surface area of a sphere are two-thirds that of its circumscribing cylinder

An experiment to demonstrate Archimedes’ Principle

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apparently rushed out of the bath and ran naked through the streets shouting, "Eureka! Eureka!" (“I found it! I found it!”). This gave rise to what has become known as Archimedes’ Principle: an object is immersed in a fluid is buoyed up by a force equal to the weight of the fluid displaced by the object.

Another well-known quotation attributed to Archimedes is: “Give me a place to stand on and I will move the Earth”, meaning that, if he had a fulcrum and a lever long enough, he could move the Earth by his own effort, and his work on centres of gravity was very important for future developments in mechanics.

According to legend, Archimedes was killed by a Roman soldier after the capture of the city of Syracuse. He was contemplating a mathematical diagram in the sand and enraged the soldier by refusing to go to meet the Roman general until he had finished working on the problem. His last words are supposed to have been “Do not disturb my circles!”

HELLENISTIC MATHEMATICS - DIOPHANTUS

Diophantus was a Hellenistic Greek (or possibly Egyptian, Jewish or even Chaldean) mathematician who lived in Alexandria during the 3rd Century AD. He is sometimes called “the father of algebra”, and wrote an influential series of books called the “Arithmetica”, a collection of algebraic problems which greatly influenced the subsequent development of number theory.

He also made important advances in mathematical notation, and was one of the first mathematicians to introduce symbolism into algebra, using an abridged notation for frequently occurring operations, and an abbreviation for the unknown and for the powers of the unknown. He was perhaps the first to recognize fractions as numbers in their own right, allowing positive rational numbers for the coefficients and solutions of his equations.

Diophantus of Alexandria (c.200-284 AD)

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Diophantus applied himself to some quite complex algebraic problems, particularly what has since become known as Diophantine Analysis, which deals with finding integer solutions to kinds of problems that lead to equations in several unknowns. Diophantine equations can be defined as polynomial equations with integer coefficients to which only integer solutions are sought.

For example, he would explore problems such as: two integers such that the sum of their squares is a square (x2 + y2 = z2, examples being x = 3 and y = 4 giving z = 5, or x = 5 and y =12 giving z = 13); or two integers such that the sum of their cubes is a square (x3 + y3 = z2, a trivial example being x = 1 and y = 2, giving z = 3); or three integers such that their squares are in arithmetic progression (x2 + z2 = 2y2, an example being x = 1, z = 7 and y = 5). His general approach was to determine if a problem has infinitely many, or a finite number of solutions, or none at all.

Diophantus’ major work (and the most prominent work on algebra in all Greek mathematics) was his “Arithmetica”, a collection of problems giving numerical solutions of both determinate and indeterminate equations. Of the original thirteen books of the “Arithmetica”, only six have survived, although some Diophantine problems from “Arithmetica” have also been found in later Arabic sources. His problems exercised the minds of many of the world's best mathematicians for much of the next two millennia, with some particularly celebrated solutions provided by Brahmagupta, Pierre de Fermat, Joseph Louis Lagrange and Leonhard Euler, among others. In recognition of their depth, David Hilbert proposed the solvability of all Diophantine problems as the tenth of his celebrated problems in 1900, a definitive solution to which only emerged with the work of Robinson and Matiyasevich in the mid-20th Century.

One of the problems in a later 5th Century Greek anthology of number games is sometimes considered to be Diophantus’ epitaph:

“Here lies Diophantus.God gave him his boyhood one-sixth of his life; One twelfth more as youth while whiskers grew rife; And then yet one-seventh ‘ere marriage begun.In five years there came a bouncing new son;Alas, the dear child of master and sage,After attaining half the measure of his father's life, chill fate took him.After consoling his fate by the science of numbers for four years, he ended his life.”

Diophantine equations

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The puzzle implies that Diophantus lived to be about 84 years old (although its biographical accuracy is uncertain).

ROMAN MATHEMATICS

By the middle of the 1st Century BC, the Roman had tightened their grip on the old Greek and Hellenistic empires, and the mathematical revolution of the Greeks ground to halt. Despite all their advances in other respects, no mathematical innovations occurred under the Roman Empire and Republic, and there were no mathematicians of note. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it (after Christianity became the official religion of the Roman empire) even less so.

Roman numerals are well known today, and were the dominant number system for trade and administration in most of Europe for the best part of a millennium. It was decimal (base 10) system but not directly positional, and did not include a zero, so that, for arithmetic and mathematical purposes, it was a clumsy and inefficient system. It was based on letters of the Roman alphabet - I, V, X, L, C, D and M - combines to signify the sum of their values (e.g. VII = V + I + I = 7).

Roman numerals

Roman arithmetic

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Later, a subtractive notation was also adopted, where VIIII, for example, was replaced by IX (10 - 1 = 9), which simplified the writing of numbers a little, but made calculation even more difficult, requiring conversion of the subtractive notation at the beginning of a sum and then its re-application at the end (see image at right). Due to the difficulty of written arithmetic using Roman numeral notation, calculations were usually performed with an abacus, based on earlier Babylonian and Greek abaci.

MAYAN MATHEMATICS

The Mayan civilisation had settled in the region of Central America from about 2000 BC, although the so-called Classic Period stretches from about 250 AD to 900 AD. At its peak, it was one of the most densely populated and culturally dynamic societies in the world.

The importance of astronomy and calendar calculations in Mayan society required mathematics, and the Maya constructed quite early a very sophisticated number system, possibly more advanced than any other in the world at the time (although the dating of developments is quite difficult).

The Mayan and other Mesoamerican cultures used a vigesimal number system based on base 20 (and, to some extent, base 5), probably originally developed from counting on fingers and toes. The numerals consisted of only three symbols: zero, represented as a shell shape; one, a dot; and five, a bar. Thus, addition and subtraction was a relatively simple matter of adding up dots and bars. After the number 19, larger numbers were written in a kind of vertical place value format using powers of 20: 1, 20, 400, 8000, 160000, etc (see image above), although in their calendar calculations they gave the third position a value of 360 instead of 400 (higher positions revert to multiples of 20).

The pre-classic Maya and their neighbours had independently developed the concept of zero by at least as early as 36 BC, and we have evidence of their working with sums up to the hundreds of millions, and with dates so large it took several lines just to represent them. Despite not possessing the concept of a fraction, they produced extremely accurate astronomical observations using no instruments other than sticks, and were able to measure the length of the solar year to a far higher degree of accuracy than that used in Europe (their calculations

Mayan numerals

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produced 365.242 days, compared to the modern value of 365.242198), as well as the length of the lunar month (their estimate was 29.5308 days, compared to the modern value of 29.53059).

However, due to the geographical disconnect, Mayan and Mesoamerican mathematics had absolutely no influence on Old World (European and Asian) numbering systems and mathematics.

CHINESE MATHEMATICS

Even as mathematical developments in the ancient Greek world were beginning to falter during the final centuries BC, the burgeoning trade empire of China was leading Chinese mathematics to ever greater heights.

The simple but efficient ancient Chinese numbering system, which dates back to at least the 2nd millennium BC, used small bamboo rods arranged to represent the numbers 1 to 9, which were then places in columns representing units, tens, hundreds, thousands, etc. It was therefore a decimal place value system, very similar to the one we use today - indeed it was the first such number system, adopted by the Chinese over a thousand years before it was adopted in the West - and it made even quite complex calculations very quick and easy.

Written numbers, however, employed the slightly less efficient system of using a different symbol for tens, hundreds, thousands, etc. This was largely because there was no concept or symbol of zero, and it had the effect of limiting the usefulness of the written number in Chinese.

The use of the abacus is often thought of as a Chinese idea, although some type of abacus was in use in Mesopotamia, Egypt and Greece, probably much earlier than in China (the first Chinese abacus, or “suanpan”, we know of dates to about the 2nd Century BC).

There was a pervasive fascination with numbers and mathematical patterns in ancient China, and different numbers were believed to have cosmic significance. In particular, magic squares - squares of numbers where each row, column and diagonal added up to the same total - were regarded as having great spiritual and religious significance.

Ancient Chinese number system

Lo Shu magic square, with its traditional graphical representation

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The Lo Shu Square, an order three square where each row, column and diagonal adds up to 15, is perhaps the earliest of these, dating back to around 650 BC (the legend of Emperor Yu’s discovery of the the square on the back of a turtle is set as taking place in about 2800 BC). But soon, bigger magic squares were being constructed, with even greater magical and mathematical powers, culminating in the elaborate magic squares, circles and triangles of Yang Hui in the 13th Century (Yang Hui also produced a trianglular representation of binomial coefficients identical to the later Pascals’ Triangle, and was perhaps the first to use decimal fractions in the modern form).

But the main thrust of Chinese mathematics developed in response to the empire’s growing need for mathematically competent administrators. A textbook called “Jiuzhang Suanshu” or “Nine Chapters on the Mathematical Art” (written over a period of time from about 200 BC onwards, probably by a variety of authors) became an important tool in the education of such a civil service, covering hundreds of problems in practical areas such as trade, taxation, engineering and the payment of wages.

It was particularly important as a guide to how to solve equations - the deduction of an unknown number from other known information - using a sophisticated matrix-based method which did not appear in the West until Carl Friedrich Gauss re-discovered it at the beginning of the 19th Century (and which is now known as Gaussian elimination).

Among the greatest mathematicians of ancient China was Liu Hui, who produced a detailed commentary on the “Nine Chapters” in 263 AD, was one of the first mathematicians known to leave roots unevaluated, giving more exact results instead of approximations. By an approximation using a regular polygon with 192 sides, he also formulated an algorithm which calculated the value of π as 3.14159 (correct to five decimal places), as well as developing a very early forms of both integral and differential calculus.

Early Chinese method of solving equations

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The Chinese went on to solve far more complex equations using far larger numbers than those outlined in the “Nine Chapters”, though. They also started to pursue more abstract mathematical problems (although usually couched in rather artificial practical terms), including what has become known as the Chinese Remainder Theorem. This uses the remainders after dividing an unknown number by a succession of smaller numbers, such as 3, 5 and 7, in order to calculate the smallest value of the unknown number. A technique for solving such problems, initially posed by Sun Tzu in the 3rd Century AD and considered one of the jewels of mathematics, was being used to measure planetary movements by Chinese astronomers in the 6th Century AD, and even today it has practical uses, such as in Internet cryptography.

By the 13th Century, the Golden Age of Chinese mathematics, there were over 30 prestigious mathematics schools scattered across China. Perhaps the most brilliant Chinese mathematician of this time was Qin Jiushao, a rather violent and corrupt imperial administrator and warrior, who explored solutions to quadratic and even cubic equations using a method of repeated approximations very similar to that later devised in the West by Sir Isaac Newton in the 17th Century. Qin even extended his technique to solve (albeit approximately) equations involving numbers up to the power of ten, extraordinarily complex mathematics for its time.

INDIAN MATHEMATICS

The Chinese Remainder Theorem

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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

The evolution of Hindu-Arabic numerals

The earliest use of a circle character for the number zero was in India

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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

Indian astronomers used trigonometry tables to estimate the relative distance of the Earth to the Sun and Moon

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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

Illustration of infinity as the reciprocal of zero

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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.


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