CHINESE MATHEMATICS

Even as mathematical developments in the ancient Greek world were beginning to falter during the final centuries BCE, 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 BCE, 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 BCE). 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. 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 BCE (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 BCE). 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

BCE 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 CE,

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.

1
 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 CE 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.

EARLY COMPUTATION

From tally sticks, bones, and stones to Papyrus to abacus to the IBM personal computer, calculating devices have come a long way. Let's

take a look through the history of these machines and the remarkable progress that came with the 20th century.

Early Computing Devices:

✔ The Chinese Abacus 'Suan Pan' and the Roman Abacus;

✔ The Antikythera mechanism, designed to calculate astronomical positions (early 1st century BCE);

✔ The Pascaline or Pascal's Calculator, by Blaise Pascal. It could add, subtract, multiply and divide two numbers (1642);

✔ The Stepped Reckoner, invented by Gottfried Wilhelm Leibniz, completed in 1694. Two prototypes were built, only one survived;

✔ The Arithmométre, the first mass-produced mechanical calculator, by Charles Xavier Thomas de Colmar, based on Leibniz's work,

around 1820, but manufactured until 1915;

✔ The punched card system, developed by 1801 by Joseph-Marie Jacquard. Used in music machines, mechanical organs, calculators,

mechanical counters looms, and other automatons and early computers;

✔ The Difference engines, the first mechanical computers, by Charles Babbage in the early 1800s;

✔ The hand-cranked calculator Curta, invented by Curt Herzstark in 1948. Type II was introduced in 1954 and produced until 1972;

2
 ✔ The Water Integrator, that could solve (partial) differential equations, built by Vladimir Lukyanov, 1936;

✔ The water levels in the chambers represented stored numbers, and the rate of flow between them represented the mathematical

operations;

✔ The Mallock Machine, built by Rawlyn Richard Manconchy Mallock of Cambridge University to solve simultaneous linear differential

equations (1933);

✔ The MONIAC or the Philips Hydraulic Computer, developed by Bill Philips to model the economic processes of the UK (1949);

✔ Differential analyzers of the 1930s and 1940s;

✔ Model K, the first electric digital computer by George Robert Stibitz at Bell Laboratories (1937);

✔ The Z1, the first freely programmable computer, which used Boolean logic and binary floating point numbers, built by Konrad Use

between 1936 and 1938;

✔ Colossus, the British codebreaker computer of WWII, developed by Tommy Flowers (Type I: 1943, Type II: 1944);

✔ ENIAC – Electronic Numerical Integrator And Computer, the first electronic general-purpose computer, designed by John Mauchly

and J. Presper Eckert, announced in 1946;

✔ Modern Computer and Calculator.

HINDU-ARABIC NUMBER SYSTEM

Hindu-Arabic numerals, set of 10 symbols—1, 2, 3, 4, 5, 6, 7, 8, 9, 0—that represent numbers in the decimal number system. They

originated in India in the 6th or 7th century and were introduced to Europe through the writings of Middle Eastern mathematicians, especially

al-Khwarizmi and al-Kindi, about the 12th century. They represented a profound break with previous methods of counting, such as the abacus,

and paved the way for the development of algebra. Our own number system, composed of the ten symbols {0,1,2,3,4,5,6,7,8,9} is called the

Hindu-Arabic system. This is a base-ten (decimal) system since place values increase by powers of ten. Furthermore, this system is positional,

which means that the position of a symbol has bearing on the value of that symbol within the number. For example, the position of the symbol 3

in the number 435,681 gives it a value much greater than the value of the symbol 8 in that same number. We’ll explore base systems more

thoroughly later. The development of these ten symbols and their use in a positional system comes to us primarily from India. It was not until the

fifteenth century that the symbols that we are familiar with today first took form in Europe. However, the history of these numbers and their

development goes back hundreds of years. One important source of information on this topic is the writer al-Biruni. Al-Biruni, who was born in

modern day Uzbekistan, had visited India on several occasions and made comments on the Indian number system. It is then that the Brahmi

numerals were being used.

3
BIRTH OF DEMONSTRATIVE MATHEMATICS

The static outlook of the ancient orient became impossible and in developing atmosphere of rationalism men began to ask why as well as

how. The empirical processes of the ancient orient, quite sufficient for the question how, no longer sufficed to answer these more scientific

inquiries of why. Some attempt at demonstrative methods was bound to assert itself, and the deductive feature, which modern scholars regard

as a fundamental characteristic of mathematics, came into being.

Demonstrative geometry began with Thales of Miletus, one of the, "Seven Wise Men" of antiquity, during the first half of the sixth century

B.C. He is the first known individual with whom mathematical discoveries are associated.

In Geometry he is credited with the following elementary results:

1. A circle is bisected by any diameter

2. The base angles of an isosceles triangle are equal.

3. The vertical angles formed by two intersecting lines are equal.

The value of these results is not to be measured by the theories themselves, but rather by its belief that Thales supported them by some

logical reasoning, instead of intuition and experiment.

Pythagoras of Samos

He was born on Island of Samos in Greece, and did much traveling through Egypt, learning among other things mathematics. Not much

more is known of his early years. Pythagoras gained his famous status by founding a group of brotherhood, Pythagoreans, which was develop

to the study of mathematics.

The group was almost cult-like in that it had symbols, rituals and prayers, in addition, Pythagoras believed that, "number rules the universe",

and the Pythagoreans gave numerical values to many objects and ideas. These numerical values in turn, were endowed with mystical and

spiritual qualities.

Legend has it upon completion of his famous theorem, Pythagoras sacrificed 100 oxen. Although he is credited with the discovery of famous

theorem, it is not possible to tell if Pythagoras is the actual author. The Pythagoreans wrote many geometric proofs, but it is difficult to ascertain

who proved what, as the group wanted to keep their findings secret. Unfortunately, this vow of secrecy prevented an important mathematical

idea from being made public. The Pythagoreans had discovered irrational numbers. If we take an isosceles right triangle with legs of measure 1,

the hypotenuse will measure square root of2. But this number cannot be expressed as a length that can measure with a ruler divided into

fractional parts, and that deeply disturbed by Pythagoreans, who believed that, "all is number". They called this numbers, "alogon", which

means, "unutterable". So shocked were the Pythagoreans by these numbers, they put to death a member who dared to mention their existence

to the public. It would be 200 years later that the Greek mathematician Eudoxus developed a way go deal with these unutterable numbers.

The Pythagoreans

4
 About 5 centuries B.C. the school founded by the philosopher, mathematician and astronomer Pythagoras flourished in Samos, Greece. The

Pythagoreans believed (but failed to prove) that the universe can be understood in terms of whole numbers. This belief stemmed from

observations in music, mathematics and astronomy. For example, they noticed that vibrating strings produce harmonious tones when the ratios

of their lengths are whole numbers. From this first attempt to express the universe in terms of numbers the idea was born that the world could

be understood through mathematics, a central concept to the development of mathematics and science.

Pythagoras also developed a rather sophisticated cosmology. He and his followers believed the earth to be perfectly spherical and that

heavenly bodies, likewise perfectly spheres, moved as the earth around a central fire invisible to human eyes (this was not the sun for it also

circled this central fire). There were 10 objects circling the central fire which included a counter-earth assumed to be there to account from

some eclipses but also because they believed the number 10 to be particularly sacred.

It was also stated that heavenly bodies give forth musical sounds, "the harmony of the spheres" as they moved to the cosmos, a music

which we cannot discern, being used to it from childhood ( a sort of background noise);through we could certainly noticed if anything went

wrong. The Pythagoreans did not believed that music, numbers and cosmos were just related, they believed that music was number, and

cosmos was music.

Pythagorean Theorem

A Brief History of Pythagorean Theorem

The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek

mathematician and philosopher Pythagoras. Pythagoras founded the Pythagorean School of mathematics in Crotona, a Greek seaport in

southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.

The Pythagorean Theorem is Pythagoras' most famous mathematical contribution.

According to legend, Pythagoras was so happy when he discovered the theorem, that he offered a sacrifice of oxen. The later discovery that

the square root of 2 is irrational and therefore, cannot be expressed as ratio of 2 integers, greatly troubled Pythagoras and his followers. They

were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the

5
knowledge that the square root of w is irrational. It is even said that the man who divulged the secret was drowned at sea. The Pythagorean

Theorem is a statement about triangles containing a right angle.

The Pythagorean Theorem states that:

“The area of a square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.”

According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares a and b is equal to the area of the blue

square, square c.

Area of Square A = a²

Area of Square B = b²

Area of Square C = c²

Thus, the Pythagorean Theorem stated algebraically is:

a² + b² = c²

For the right triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse.

The Pythagorean Problem

Find all right triangles whose sides are of integral length, thus finding all solutions in the positive integers of the Pythagorean equation:

X² + y² = z²

Pythagorean Triplets/Triples

Pythagorean triples are integer solutions to the Pythagorean Theorem, a²+b²=c². For a right triangle, the c side is the hypotenuse, side that is

opposite the right angle.

The a side is shorter of the two sides adjacent to the right angle. The first rule that became aware of for determining a subset of

Pythagorean triples are as follows:

6
ODD NUMBERS

a b c

3 4 5

5 12 13

7 24 25

9 40 41

11 60 61

13 84 85

15 112 113

17 144 145

19 180 181

Every odd number is the a side of a Pythagorean triple

The b side of a Pythagorean triple is simply:

(a² -1) / 2.

The c side is:

b+1

Here, a and c are always odd; b is always even. These relationships hold because the difference between successive square numbers is

successive odd numbers.

Every odd number that is itself a square (and the square of every odd number is an odd number) thus make for a Pythagorean triple.

EVEN NUMBERS

a b c

2 0 2

4 3 5

6 8 10

8 15 17

10 24 26

12 35 37

14 48 50

16 63 65

18 80 82

20 99 101

7
 It was originally pointed out by Matthew Q. Boeke that there are Pythagorean triple in which the a side is an even number. An infinite series

of Pythagorean solutions can be generated with the following rules:

Every even number is the a side of Pythagorean triple

The b side is simply:

(a/2)²-1

The c side is:

b+2

In the table, we see the interesting result that for triplets where a is not evenly divisible by 4, the numbers are all even. These are simply

multiples of 2 of triples in the table above where b is odd.

EUCLID AND THE ELEMENTS

The Greek Mathematician Euclid lived and flourished in Alexandria in Egypt around 300 BCE, during the reign Ptolemy I. Almost nothing is

known in Euclid’s Life, even His dates and Birthplace are not known. No likeness or first-hand description of his physical appearance as

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 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 parts in given ratios, on catoptrics (the mathematical theory of mirrors

and reflection), and 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, Theatetus, 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 Straightedge.

Euclid reworked the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean Geometry,

which is still 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 Reimann in the first half of 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 the time, the model for mathematical argument, following deductions from

initial assumptions (which Euclid called axioms and postulates) in order to establish proven theorems.

8
Euclid of Alexandria

Euclid of Alexandria (lived c. 300 BCE) an ancient Greek known for his major work The Elements. Disappointingly little is known about the

Euclid life and personality of Euclid except that he was a professor of mathematics at the University of Alexandria and apparently the founder of

the illustrious and long-lived Alexandrian School of Mathematics. Even his dates and Birthplace are not known, but it seems probable that he

received his mathematical training in the Platonic school at Athens. Many years later, when comparing Euclid with Apollonius for the latter’s

discredit, Pappus praised Euclid for his modesty and considerations of others.

Proclus augmented his Eudemian Summary with the frequently old story of Euclid’s reply to Ptolemy’s request for a short out to geometric

knowledge that “there is no royal road in geometry”. But same has been told of Menaechmus when serving as instructor to Alexander the Great.

Another pretty story has been told by Stobeaus. Some student studying geometry under Euclid questioned what he would get from learning the

subject, whereupon Euclid ordered a slave give the fellow a penny, ‘since he must make gain from what he learns.’

3 STAFF AT THE UNIVERSITY

1. Demetrius Phaleneus - (from Athens) tale change of the library

2. Able and Talent men - were selected and develop various fields of study.

3. Euclid - (from Athens) chosen head the department of mathematics

Proclus (412-485)

A Greek mathematician who lived around seven centuries after Euclid, wrote on his commentary on the ELEMENTS Euclid, who put

together the elements collecting many of Eudoxus’ theorems, perfecting many of Theaetetus’,

ELEMENTS have been referred to as the most successful and influential textbook ever written.

It was one of the very earliest mathematical works after the invention of the printing press and has been estimated to be second only in the

number edition published since the printing in 1482. At the time of its introduction, Element was the most comprehensive and logically rigorous

examination of the basic principles of geometry. It survived the eclipse of classical learning, which occurred with the fall of roman Empire

through Arabic translation.

The translation of the Elements

The first complete translation of the elements was not made from the Greek but from Arabic.

8th Century - a number of Byzantine manuscripts of Greek works were translated by the Arabians.

1120 - An English scholar Adeland of Bath, made a Latin translation

From ancient times to the last 19th Century CE people considered the elements as perfect example of correct reasoning more than a

thousand editions have been published making it one of the most popular books after the Bible Euclid element contains Collection of definitions

9
of postulates, Propositions, theorem and constructions and the mathematical proofs of the propositions. He gave definitions, postulate and

axioms. He called axioms “common notions”. If the ideas seem obvious, that’s the point. Euclid wanted to base his geometry on ideas so

obvious that no one could reasonably doubt them. For his definitions, postulates and common notions, Euclid deduces the rest of geometry. His

geometry describes the normal space around we us. Modern ‘non-Euclidean’ geometries describe space over astronomical distances at

near-light speeds, or warped by gravity. Finally, a collection of theorem. Theorems are statements that are proved by the logical conclusion of a

combination of axioms, definitions and undefined terms

Euclid’s Postulates, Axioms, and Definitions

In the elements, Euclid attempted to bring together the various geometric facts known in his day (including some that he discovered himself)

in order to form an axiomatic system , in which theses “facts” could be subjected to rigorous proof. His undefined terms were point, line, straight

line, surface, and plane. (To Euclid, the word line meant any finite curve, and hence a “straight” line is what we would call a line segment.)

Euclid divided his axioms into two categories, calling the first five postulates and the next five “common notions.” The distinction between

postulates and common notions is that the postulates are geometric in character, whereas common notions were considered by Euclid to be

true in general.

Elements have thirteen (13) books.

BOOK 1:

✔ Commences of course, with the necessary preliminary definition, postulate and axiom

Postulates

(Geometrical Postulates)

A part from these common notations Euclid also made the following geometrical postulates deduce propositions:

Postulate 1: A Straight line segment can be drawn joining any two points.

Postulate 2: Any straight line segment can be extended in definitely in a straight line.

Postulate 3: Given any straight line segment a circle can be drawn having the segment as radius and one endpoint as centre.

Postulate 4: All line angles are congruent.

Postulate 5: If a straight meets to the other lines, so as to make the two interior on one side of it together less than two rights angles. The other

straight will meet if produce on the side on which the angles are less than two right angles.

10
 Axioms

Axiom 1: Thing which are equal to the same thing are equal to one another.

Axiom 2: If equals are added to equal the whole are equal.

Axiom 3: If equals are subtracted from equals, then the remainders are equal

Axiom 4: Thing which coincide with one another must be equal to one another

Axiom 5: The Whole is Greater than the part.

Definitions are also of axiomatic system as are undefined terms. Thus, an axiomatic system consists of the following: a collection of undefined

words; a collection of definitions; a collection of axioms (also called postulates).

Definitions

1. A point is that which has no part

2. A line is breadth less length.

3. The extremities of a line are points.

4. A straight line is a line which lies evenly with the points on itself.

Many of Euclid’s are constructive, demonstrating the existence of some figure by detailing the steps he used to construct the object using

Compass and Straightedge.

The Three Classical Problems

1. Doubling the cube

✔ Also known as the Delian Problem is ancient number geometric problem. Given the cube, the problem requires the construction of

the edge of a second cube whose volume in double that of first of it.

2. Trisecting the angle

✔ One of the most famous problems in the history of mathematics is how to divide an angle drawn on a piece of paper into three equal

parts using just a compass and a straightedge

3. Quadrature of a Circle

✔ The problem of constructing a square of equal area as the given circle; one of the classical Ancient problems on constructions with a

ruler and compass. The side of a square equal in area to a circle of radius r.

11
Euclidean Geometry

Is a great practical value it has been used by Ancient Greeks through modern society to design buildings.

Non-Euclidean Geometry

Is any geometry that is different from Euclidean Geometry. Each Non-Euclidean geometry is a consistent system of definition, assumption,

and proves that describe such as objects, as points, lines and planes.

Euclid’s Other Works

About half of Euclid’s works are lost. We only know about them because other ancient writers refer to them. Lost works include books on

Conic section, Logical fallacies and Prisms. Euclid’s works that still exist are Elements, Data, Division of Figures, Phenomena and Optics.

Data

✔ deals with the nature and implications of “given” information in geometrical problems

On Divisions of Figures

✔ which survives only partially in Arabic translation, concerns the division of geometrical figures into two or more equal parts in given

ratios.

Catoptrics

✔ which concerns the mathematical theory of mirrors , particularly the images formed in plane and spherical concave mirrors.

Phenomena

✔ a treatise on spherical astronomy survives in Greek; it is quite similar to On the Moving Sphere by Autolycus of Pitane, who

flourished around 310 BC.

12
 INDIAN AND ARABIAN MATHEMATICS

INDIAN 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 BCE) 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 CE 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 BCE, 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 , which yields a value of 1.4142156, correct to 5 decimal places. As early as the 3rd or 2nd Century BCE, Jain mathematicians
x 4 x 34)

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 CE. 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

13
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.

Surya Siddhanta

A text called the “Surya Siddhanta”, by unknown authors and dating from around 400 CE, 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 CE, 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

14
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.

The Rise of a Moslem Culture

Arabian Empire was the most spectacular and prominent episodes history in Asia. During that period, Arabian people were scattered in the

Arabia peninsula. Muslim Religion collaborated again to make their nation powerful.

Baghdad one of the productive places occupied under the Arabian Empire. Mongols, an East-Central Asian Ethnic Group, native to Mongolia

and China’s Inner Mongolia Autonomous Region. The Arabians were invited to have a scholar for them to translate those Greek and Hindu

erudition or learning books.

Erudition - the quality of having or showing great knowledge or learning.

When their education began to grow up, Brahmagupta’s work was brought to Bagdad in (ca. 766) During the reign of Caliph al-mansur and

the Hindu numerical were brought and translated in Arabian Mathematics. The next to Caliph was Harun al-rashld who reigned from

15
786-706.C.A. – chartered account or the preceding date or amount. Caliph - the Chief Muslim Civil and religious ruler, regarded as the

successor of Muhammad. He also ruled in Bagdad.

BRAHMAGUPTA

Brahmagupta (598–668 CE)

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 nnatural numbers

as n(n + 1)(2n + 1)⁄ 6 and the sum of the cubes of the first nnatural 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

16
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 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. Brahmagupta dedicated a substantial portion of his work to geometry and trigonometry. He established √10 (3.162277) as a good

practical approximation for π (3.141593), and gave a formula, now known as Brahmagupta's Formula, for the area of a cyclic quadrilateral, as

well as a celebrated theorem on the diagonals of a cyclic quadrilateral, usually referred to as Brahmagupta's Theorem. He was also an Indian

mathematician and astronomer who wrote two important works on mathematics: The Brahmasphutadssidhanta and Khandakhadyaka; written c.

628, in Sanskrit, it contains ideas including a good understanding of the mathematical role of zero, rules for manipulating both negative and

positive numbers. It also contains a method for computing square roots, methods of solving linear and some quadratic equations, and rules for

summing series, Brahmagupta's identity, and the Brahmagupta’s theorem. Khandakhadyaka meaning "edible bite; morsel of food") is an

astronomical treatise written by Indian mathematician and astronomer Brahmagupta in 665 AD. The treatise contains eight chapters covering

such topics as the longitudes of the planets, diurnal rotation, lunar and solar eclipses, risings and settings, the moon's crescent and

conjunctions of the planets. The treatise also includes an appendix which is some versions has only one chapter, and in other has three. The

treatise was written as a response to Aryabhata's Ardharatrikapaksa. Khandakhadyaka was known in Sanskrit to al-Biruni.

Arithmetic and Algebra

Islam gave birth to a new civilization that spread from China in the east, India in the south east, Russia in the north, and

Anatolia in the west of Asia, to East and North Africa up to the Mediterranean regions of Southern Europe. Mathematics flourished

during the golden age of Islamic science, which began around the seventh century AD and continued through to about the

17
fourteenth century. Both arithmetic and algebra were advanced dramatically by Muslim mathematicians, who adopted Indian innovations

such as decimal numbers and considerably extended them – they also developed earlier Greek concepts of geometry, trigonometry,

number theory and the resolution of equations. Islamic mathematicians did far more than just copy Greek and Indian techniques –

their additional researches developed and systematized several fields of mathematics.

Even modern mathematical language, including terms like ͞algebra͟, ͞root͟ and ͞zero͟, owes an important debt to Arabic scientists.

Algebra, for example, comes from the ninth‐century Arabic Astronomer and mathematician al‐Khwārizmī. One of the

greatest minds of the early mathematical production in Arabic was Abu Abdullah Muhammad ibn Musa al-Khwarizmi who was a

mathematician and astronomer as well as a geographer and a historian. It is said that he is the author in Arabic of one of the

oldest astronomical tables, of one the oldest works on arithmetic and the oldest work on algebra; some of his scientific

contributions were translated into Latin and were used until the 16th century as the principal mathematical textbooks in European

universities. Originally he belonged to Khwârazm (modern Khiwa) situated in Turkistan but he carried on his scientific career in

Baghdad and all his works are in Arabic. He was summoned to Baghdad by Abbasid Caliph Al-Ma'mun, who was a patron of

knowledge and learning. Al-Ma'mun established the famous Bayt al-Hikma (House of Wisdom) which worked on the model of a

library and a research academy. It had a large and rich library and distinguished scholars of various faiths were assembled to

produce scientific masterpieces as well as to translate faithfully nearly all the great and important ancient works of Greek, Sanskrit,

Pahlavi and of other languages into Arabic.

Contrast between Greek and Hindu Mathematics

The contrast between Greek and Hindu mathematics are: first and foremost, Hindu mathematics was precisely cultivated when the first

important astronomical works by the Hindu astronomer used Hindu mathematics for its own computation and measurement generally the study

of heaven. Besides Hindu trigonometry was meritorious and arithmetical in nature very uneven quality regarding mathematics. Whereas, the

Greek mathematics was obtained an independent existence and studied for its own sake and due to their caste system. Moreover, their

mathematics was open to anyone who cared to study the subject. They are also particular to determine good from poor quality to preserve their

mathematics.

ISLAMIC MATHEMATICS

The Islamic Empire established across Persia, the Middle

East, Central Asia, North Africa, Iberia and parts of India from the

8th Century onwards made significant contributions towards

mathematics. They were able to draw on and fuse together the

mathematical developments of both Greece and India.

One consequence of the Islamic prohibition on depicting the

human form was the extensive use of complex geometric

patterns to decorate their buildings, raising mathematics to the

18
form of an art. In fact, over time, Muslim artists discovered all the different forms of symmetry that can be depicted on a 2-dimensional surface.

The Qu’ran itself encouraged the accumulation of knowledge, and a Golden Age of Islamic science and mathematics flourished throughout the

medieval period from the 9th to 15th Centuries. The House of Wisdom was set up in Baghdad around 810, and work started almost immediately

on translating the major Greek and Indian mathematical and astronomy works into Arabic. The outstanding Persian mathematician Muhammad

Al-Khwarizmi was an early Director of the House of Wisdom in the 9th Century, and one of the greatest of early Muslim mathematicians.

Perhaps Al-Khwarizmi’s most important contribution to mathematics was his strong advocacy of the Hindu numerical system (1 - 9 and 0),

which he recognized as having the power and efficiency needed to revolutionize Islamic (and, later, Western) mathematics, and which was soon

adopted by the entire Islamic world, and later by Europe as well.

Al-Khwarizmi's other important contribution was algebra, and he introduced the fundamental algebraic methods of “reduction” and

“balancing” and provided an exhaustive account of solving polynomial equations up to the second degree. In this way, he helped create the

powerful abstract mathematical language still used across the world today, and allowed a much more general way of analyzing problems other

than just the specific problems previously considered by the Indians and Chinese.

The 10th Century Persian mathematician Muhammad Al-Karaji worked to

extend algebra still further, freeing it from its geometrical heritage, and

introduced the theory of algebraic calculus. Al-Karaji was the first to use the

method of proof by mathematical induction to prove his results, by proving

that the first statement in an infinite sequence of statements is true, and then

proving that, if any one statement in the sequence is true, then so is the next

one.

Among other things, Al-Karaji used mathematical induction to prove the

binomial theorem. A binomial is a simple type of algebraic expression which

has just two terms which are operated on only by addition, subtraction,

multiplication and positive whole-number exponents, such as (x +y)2. The

co-efficients needed when a binomial is expanded form a symmetrical

triangle, usually referred to as Pascal’s Triangle after the 17th Century French mathematician Blaise Pascal, although many other

mathematicians had studied it centuries before him in India, Persia, China and Italy, including Al-Karaji. Some hundred years after Al-Karaji,

Omar Khayyam (perhaps better known as a poet and the writer of the “Rubaiyat”, but an important mathematician and astronomer in his own

right) generalized Indian methods for extracting square and cube roots to include fourth, fifth and higher roots in the early 12th Century. He

carried out a systematic analysis of cubic problems, revealing there were actually several different sorts of cubic equations. Although he did in

fact succeed in solving cubic equations, and although he is usually credited with identifying the foundations of algebraic geometry, he was held

back from further advances by his inability to separate the algebra from the geometry, and a purely algebraic method for the solution of cubic

equations had to wait another 500 years and the Italian mathematicians del Ferro and Tartaglia.

19
 The 13th Century Persian astronomer, scientist and mathematician

Nasir Al-Din Al-Tusi was perhaps the first to treat trigonometry as a

separate mathematical discipline, distinct from astronomy.

Building on earlier work by Greek mathematicians such as Menelaus

of Alexandria and Indian work on the sine function, he gave the first

extensive exposition of spherical trigonometry, including listing the six

distinct cases of a right triangle in spherical trigonometry.

One of his major mathematical contributions was the formulation of

the famous law of sines for plane triangles, a⁄(sin A) = b⁄(sin B) = c⁄(sin C),

although the sine law for spherical triangles had been discovered earlier

by the 10th Century Persians Abul Wafa Buzjani and Abu Nasr Mansur.

Other medieval Muslim mathematicians worthy of note include:

❖ the 9th Century Arab Thabit ibn Qurra, who developed a

general formula by which amicable numbers could be derived, re-discovered much later by both Fermat and Descartes(amicable

numbers are pairs of numbers for which the sum of the divisors of one number equals the other number, e.g. the proper divisors of

220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, of which the sum is 284; and the proper divisors of 284 are 1, 2, 4, 71, and 142, of

which the sum is 220);

❖ the 10th Century Arab mathematician Abul Hasan al-Uqlidisi, who wrote the earliest surviving text showing the positional use of

Arabic numerals, and particularly the use of decimals instead of fractions (e.g. 7.375 insead of 73⁄8);

❖ the 10th Century Arab geometer Ibrahim ibn Sinan, who continued Archimedes' investigations of areas and volumes, as well as on

tangents of a circle;

❖ the 11th Century Persian Ibn al-Haytham (also known as Alhazen), who, in addition to his groundbreaking work on optics and

physics, established the beginnings of the link between algebra and geometry, and devised what is now known as "Alhazen's

problem" (he was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily

generalizable); and

❖ the 13th Century Persian Kamal al-Din al-Farisi, who applied the theory of conic sections to solve optical problems, as well as

pursuing work in number theory such as on amicable numbers, factorization and combinatorial methods;

20
 ❖ the 13th Century Moroccan Ibn al-Banna al-Marrakushi, whose works included topics such as computing square roots and the theory

of continued fractions, as well as the discovery of the first new pair of amicable numbers since ancient times (17,296 and 18,416,

later re-discovered by Fermat) and the the first use of algebraic notation since Brahmagupta.

With the stifling influence of the Turkish Ottoman Empire from the 14th or 15th Century onwards, Islamic mathematics stagnated, and further

developments moved to Europe.

AL-KHWARIZMI

Muhammad

Al-Khwarizmi (c.780-850

CE)

One of the first Directors of the House of Wisdom in Bagdad in the

early 9th Century was an outstanding Persian mathematician called

Muhammad Al-Khwarizmi. He oversaw the translation of the

major Greek and Indianmathematical and astronomy works (including

those of Brahmagupta) into Arabic, and produced original work which

had a lasting influence on the advance of Muslim and (after his works

spread to Europe through Latin translations in the 12th Century) later

European mathematics. The word “algorithm” is derived from the

Latinization of his name, and the word "algebra" is derived from the

Latinization of "al-jabr", part of the title of his most famous book, in

which he introduced the fundamental algebraic methods and

techniques for solving equations. Perhaps his most important

contribution to mathematics was his strong advocacy of the Hindu

21
numerical system, which Al-Khwarizmi recognized as having the power and efficiency needed to revolutionize Islamic and Western

mathematics. The Hindu numerals 1 - 9 and 0 - which have since become known as Hindu-Arabic numerals - were soon adopted by the entire

Islamic world. Later, with translations of Al-Khwarizmi’s work into Latin by Adelard of Bath and others in the 12th Century, and with the influence

of Fibonacci’s “Liber Abaci” they would be adopted throughout Europe as well.

Al-Khwarizmi’s other important contribution was algebra, a word derived from the title of a mathematical text he published in about 830

called “Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala” (“The Compendious Book on Calculation by Completion and Balancing”).

Al-Khwarizmi wanted to go from the specific problems considered by the Indians and Chinese to a more general way of analyzing problems,

and in doing so he created an abstract mathematical language which is used across the world today.

His book is considered the foundational text of modern algebra, although he did not employ the kind of algebraic notation used today (he

used words to explain the problem, and diagrams to solve it). But the book provided an exhaustive account of solving polynomial equations up

to the second degree, and introduced for the first time the fundamental algebraic methods of “reduction” (rewriting an expression in a simpler

form), “completion” (moving a negative quantity from one side of the equation to the other side and changing its sign) and “balancing”

(subtraction of the same quantity from both sides of an equation, and the cancellation of like terms on opposite sides). In particular,

Al-Khwarizmi developed a formula for systematically solving quadratic equations (equations involving unknown numbers to the power of 2,

or x2) by using the methods of completion and balancing to reduce any equation to one of six standard forms, which were then solvable. He

described the standard forms in terms of "squares" (what would today be "x2"), "roots" (what would today be "x") and "numbers" (regular

constants, like 42), and identified the six types as: squares equal roots (ax2 = bx), squares equal number (ax2= c), roots equal number (bx = c),

squares and roots equal number (ax2 + bx = c), squares and number equal roots (ax2 + c = bx), and roots and number equal squares

(bx + c = ax2).

Al-Khwarizmi is usually credited with the development of lattice (or sieve) multiplication method of multiplying large numbers, a method

algorithmically equivalent to long multiplication. His lattice method was later introduced into Europe by Fibonacci.

In addition to his work in mathematics, Al-Khwarizmi made important contributions to astronomy, also largely based on methods from India, and

he developed the first quadrant (an instrument used to determine time by observations of the Sun or stars), the second most widely used

astronomical instrument during the Middle Ages after the astrolabe. He also produced a revised and completed version of Ptolemy's

“Geography”, consisting of a list of 2,402 coordinates of cities throughout the known world.

Moreover, one of the greatest scientific minds of the medieval period and the most important Muslim mathematician, justly called

the 'father of algebra'. He wrote the Kitâb al-Jem wa'l Tafrîq bi Hisâb al-Hind also called ͞The Book of Chapters on Hindu

Arithmetic͟, on arithmetic in which he used Indian numerals including zero. It deals with the four basic operations of addition,

subtraction, multiplication and division as well as with both common and sexagesimal fractions and the extraction of the square root.

Other mathematical writings of al-Khwarizmi are also known. His best known classical work on algebra is the Kitâb al-Mukhtasar fî

Hisâb al-Jabr wa'l-Muqâbala or ͞The Book of Integration and Equation͟. The original work in Arabic was written in 820 CE and was

translated into Latin in the 12th century. It is worth remarking that the term al-jabr, in the Latinized form of algebra, has found its

22
way into the modern languages. The meaning of the Arabic word Al-Jabr is restoration by transposing negative quantities to the

other side of the equation to make them positive; and the term Al-Muqâbalah refers to the process of eliminating identical quantities

from the two sides of the equation.

Al-Khwarizmi had given the rules for the solution of quadratic equations which are supported in a number of cases by

geometrical proofs. The common equation (linear or quadratic) was reduced in this book to one of six standard forms:

1. Squares equal to roots (ax^2 = bx)

2. Squares equal to numbers (ax^2 = c)

3. Roots equal to numbers (bx = c)

4. Squares and roots equal to numbers (ax^2 + bx = c)

5. Squares and numbers equal to roots (ax^2 + c = bx)

6. Roots and numbers equal to squares (bx + c = ax^2)

The mathematical works of al-Khwarizmi were used in European universities up to the 17th century. It was through his work on

mathematics that the Indian system of numeration became known to the Arabs and later through its Latin translation to the people

of Europe. He synchronized Greek and Indian mathematical knowledge but he was the first mathematician to distinguish clearly

between algebra and geometry and gave geometrical solutions of linear and quadratic equations.

Medieval Mathematics

During the centuries in which

the Chinese, Indian and Islamicmathematicians had been in the

ascendancy, Europe had fallen into the Dark Ages, in which

science, mathematics and almost all intellectual endeavour

stagnated. Scholastic scholars only valued studies in the

humanities, such as philosophy and literature, and spent much

of their energies quarrelling over subtle subjects in metaphysics

and theology, such as "How many angels can stand on the point

of a needle?"

From the 4th to 12th Centuries, European knowledge and

study of arithmetic, geometry, astronomy and music was limited

mainly to Boethius’ translations of some of the works of ancient

Greek masters such as Nicomachus and Euclid. All trade and

calculation was made using the clumsy and

23
inefficient Roman numeral system, and with an abacus based on Greekand Roman models. By the 12th Century, though, Europe, and

particularly Italy, was beginning to trade with the East, and Eastern knowledge gradually began to spread to the West. Robert of Chester

translated Al-Khwarizmi's important book on algebra into Latin in the 12th Century, and the complete text of Euclid's “Elements” was translated

in various versions by Adelard of Bath, Herman of Carinthia and Gerard of Cremona. The great expansion of trade and commerce in general

created a growing practical need for mathematics, and arithmetic entered much more into the lives of common people and was no longer limited

to the academic realm.

The advent of the printing press in the mid-15th Century also had a huge impact. Numerous books on arithmetic were published for the

purpose of teaching business people computational methods for their commercial needs and mathematics gradually began to acquire a more

important position in education. Europe’s first great medieval mathematician was the Italian Leonardo of Pisa, better known by his

nickname Fibonacci. Although best known for the so-called Fibonacci Sequence of numbers, perhaps his most important contribution to

European mathematics was his role in spreading the use of the Hindu-Arabic numeral system throughout Europe early in the 13th Century,

which soon made the Roman numeral system obsolete, and opened the way for great advances in European mathematics.

An important (but largely unknown and underrated) mathematician and

scholar of the 14th Century was the Frenchman Nicole Oresme. He used a

system of rectangular coordinates centuries before his countryman René

Descartespopularized the idea, as well as perhaps the first

time-speed-distance graph. Also, leading from his research into musicology, he

was the first to use fractional exponents, and also worked on infinite series,

being the first to prove that the harmonic series 1⁄1 + 1⁄2 + 1⁄3 + 1⁄4 + 1⁄5... is a

divergent infinite series (i.e. not tending to a limit, other than infinity).

The German scholar Regiomontatus was perhaps the most capable

mathematician of the 15th Century, his main contribution to mathematics being

in the area of trigonometry. He helped separate trigonometry from astronomy,

and it was largely through his efforts that trigonometry came to be considered

an independent branch of mathematics. His book "De Triangulis", in which he

described much of the basic trigonometric knowledge which is now taught in high school and college, was the first great book on trigonometry to

appear in print. Mention should also be made of Nicholas of Cusa (or Nicolaus Cusanus), a 15th Century German philosopher, mathematician

and astronomer, whose prescient ideas on the infinite and the infinitesimal directly influenced later mathematicians like Gottfried

Leibniz and Georg Cantor. He also held some distinctly non-standard intuitive ideas about the universe and the Earth's position in it, and about

the elliptical orbits of the planets and relative motion, which foreshadowed the later discoveries of Copernicus and Kepler.

24
Fibonacci

Leonardo of Pisa (Fibonacci)

(c.1170-1250)

The 13th Century Italian Leonardo of Pisa, better known

by his nickname Fibonacci, was perhaps the most talented

Western mathematician of the Middle Ages. Little is known of

his life except that he was the son of a customs offical and, as

a child, he travelled around North Africa with his father, where

he learned about Arabic mathematics. On his return to Italy,

he helped to disseminate this knowledge throughout Europe,

thus setting in motion a rejuvenation in European

mathematics, which had lain largely dormant for centuries

during the Dark Ages.

In particular, in 1202, he wrote a hugely influential book

called “Liber Abaci” ("Book of Calculation"), in which he

promoted the use of the Hindu-Arabic numeral system,

describing its many benefits for merchants and

mathematicians alike over the clumsy system of Roman numerals then in use in Europe. Despite its obvious advantages, uptake of the system

in Europe was slow (this was after all during the time of the Crusades against Islam, a time in which anything Arabic was viewed with great

suspicion), and Arabic numerals were even banned in the city of Florence in 1299 on the pretext that they were easier to falsify

than Roman numerals. However, common sense eventually prevailed and the new system was adopted throughout Europe by the 15th century,

making the Roman system obsolete. The horizontal bar notation for fractions was also first used in this work (although following

the Arabic practice of placing the fraction to the left of the integer).

25
Fibonacci is best known, though, for his introduction into Europe of a particular number sequence, which has since become known as Fibonacci

Numbers or the Fibonacci Sequence. He discovered the sequence - the first recursive number sequence known in Europe - while considering a

practical problem in the “Liber Abaci” involving the growth of a hypothetical population of rabbits based on idealized assumptions. He noted that,

after each monthly generation, the number of pairs of rabbits increased from 1 to 2 to 3 to 5 to 8 to 13, etc, and identified how the sequence

progressed by adding the previous two terms (in mathematical terms, Fn = Fn-1 + Fn-2), a sequence which could in theory extend indefinitely.

The sequence, which had actually been known

to Indian mathematicians since the 6th Century, has many interesting

mathematical properties, and many of the implications and relationships

of the sequence were not discovered until several centuries after

Fibonacci's death. For instance, the sequence regenerates itself in

some surprising ways: every third F-number is divisible by 2 (F3 = 2),

every fourth F-number is divisible by 3 (F4 = 3), every fifth F-number is

divisible by 5 (F5 = 5), every sixth F-number is divisible by 8 (F6 = 8),

every seventh F-number is divisible by 13 (F7 = 13), etc. The numbers

of the sequence has also been found to be ubiquitous in nature: among

other things, many species of flowering plants have numbers of petals

in the Fibonacci Sequence; the spiral arrangements of pineapples occur

in 5s and 8s, those of pinecones in 8s and 13s, and the seeds of

sunflower heads in 21s, 34s, 55s or even higher terms in the sequence;

etc.

In the 1750s, Robert Simson noted that the ratio of each term in the Fibonacci Sequence to the previous term approaches, with ever greater

accuracy the higher the terms, a ratio of approximately 1 : 1.6180339887 (it is actually an irrational number equal to (1 + √5)⁄2 which has since

been calculated to thousands of decimal places). This value is referred to as the Golden Ratio, also known as the Golden Mean, Golden

Section, Divine Proportion, etc, and is usually denoted by the Greek letter phi φ (or sometimes the capital letter Phi Φ). Essentially, two

quantities are in the Golden Ratio if the ratio of the sum of the quantities to the larger quantity is equal to the ratio of the larger quantity to the

smaller one. The Golden Ratio itself has many unique properties, such as 1⁄φ = φ - 1 (0.618...) and φ2 = φ + 1 (2.618...), and there are countless

examples of it to be found both in nature and in the human world.

A rectangle with sides in the ratio of 1 : φ is known as a Golden Rectangle, and many artists and architects throughout history (dating back

to ancient Egypt and Greece, but particularly popular in the Renaissance art of Leonardo da Vinci and his contemporaries) have proportioned

their works approximately using the Golden Ratio and Golden Rectangles, which are widely considered to be innately aesthetically pleasing. An

arc connecting opposite points of ever smaller nested Golden Rectangles forms a logarithmic spiral, known as a Golden Spiral. The Golden

Ratio and Golden Spiral can also be found in a surprising number of instances in Nature, from shells to flowers to animal horns to human

bodies to storm systems to complete galaxies. It should be remembered, though, that the Fibonacci Sequence was actually only a very minor

element in “Liber Abaci” - indeed, the sequence only received Fibonacci's name in 1877 when Eduouard Lucas decided to pay tribute to him by

26
naming the series after him - and that Fibonacci himself was not responsible for identifying any of the interesting mathematical properties of the

sequence, its relationship to the Golden Mean and Golden Rectangles and Spirals, etc.

However, the book's influence on medieval mathematics is undeniable, and

it does also include discussions of a number of other mathematical

problems such as the Chinese Remainder Theorem, perfect numbers and

prime numbers, formulas for arithmetic series and for square pyramidal

numbers, Euclidean geometric proofs, and a study of simultaneous linear

equations along the lines of Diophantus and Al-Karaji. He also described

the lattice (or sieve) multiplication method of multiplying large numbers, a

method - originally pioneered by Islamic mathematicians

like Al-Khwarizmi - algorithmically equivalent to long multiplication. Neither

was “Liber Abaci” Fibonacci’s only book, although it was his most important

one.

His “Liber Quadratorum” (“The Book of Squares”), for example, is a

book on algebra, published in 1225 in which appears a statement of what is now called Fibonacci's identity - sometimes also known

as Brahmagupta’s identity after the much earlier Indian mathematician who also came to the same conclusions - that the product of two sums of

two squares is itself a sum of two squares e.g. (12 + 42)(22 + 72) = 262 + 152 = 302 + 12.

27