SUMERIAN/BABYLONIAN

MATHEMATICS

Sumer (a region of Mesopotamia,

modern-day Iraq) was the

birthplace of writing, the wheel,

agriculture, the arch, the plow,

irrigation and many other

innovations, and is often referred

to as the Cradle of Civilization.

The Sumerians developed the

earliest known writing system - a

pictographic writing system

known as cuneiform script, using

wedge-shaped characters

inscribed on baked clay tablets -

and this has meant that we

actually have more knowledge of

ancient Sumerian and

Babylonian mathematics than of

early Egyptian mathematics.

Indeed, we even have what appear to school exercises in arithmetic and geometric problems.

As in Egypt, Sumerian mathematics initially developed largely as a response to bureaucratic

needs when their civilization settled and developed agriculture (possibly as early as the 6th

millennium BC) for the measurement of plots of land, the taxation of individuals, etc. In

addition, the Sumerians and Babylonians needed to describe quite large numbers as they

attempted to chart the course of the night sky and develop their sophisticated lunar calendar.

They were perhaps the first people to assign symbols to groups of objects in an attempt to make

the description of larger numbers easier. They moved from using separate tokens or symbols to

represent sheaves of wheat, jars of oil, etc, to the more abstract use of a symbol for specific

numbers of anything. Starting as early as the 4th millennium BC, they began using a small clay

cone to represent one, a clay ball for ten, and a large cone for sixty. Over the course of the third

millennium, these objects were replaced by cuneiform equivalents so that numbers could be

written with the same stylus that was being used for the words in the text. A rudimentary model

of the abacus was probably in use in Sumeria from as early as 2700 - 2300 BC.

Sumerian Clay Cones

Sumerian and Babylonian

mathematics was based on a

sexegesimal, or base 60, numeric

system, which could be counted

physically using the twelve

knuckles on one hand the five

fingers on the other hand. Unlike

those of

theEgyptians, Greeks and Roman

s, Babylonian numbers used a

true place-value system, where

digits written in the left column

represented larger values, much

as in the modern decimal system,

although of course using base 60

not base 10. Thus, in the

Babylonian system represented

3,600 plus 60 plus 1, or 3,661.

Also, to represent the numbers 1 - 59 within each place value, two distinct symbols were used, a

unit symbol ( ) and a ten symbol ( ) which were combined in a similar way to the familiar

system ofRoman numerals (e.g. 23 would be shown as ). Thus, represents 60 plus

23, or 83. However, the number 60 was represented by the same symbol as the number 1 and,

because they lacked an equivalent of the decimal point, the actual place value of a symbol often

had to be inferred from the context.

It has been conjectured that Babylonian advances in mathematics were probably facilitated by

the fact that 60 has many divisors (1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 - in fact, 60 is the

smallest integer divisible by all integers from 1 to 6), and the continued modern-day usage of of

60 seconds in a minute, 60 minutes in an hour, and 360 (60 x 6) degrees in a circle, are all

testaments to the ancient Babylonian system. It is for similar reasons that 12 (which has factors

of 1, 2, 3, 4 and 6) has been such a popular multiple historically (e.g. 12 months, 12 inches, 12

pence, 2 x 12 hours, etc).

The Babylonians also developed another revolutionary mathematical concept, something else

that theEgyptians, Greeks and Romans did not have, a circle character for zero, although its

symbol was really still more of a placeholder than a number in its own right.

We have evidence of the development of a complex system of metrology in Sumer from about

3000 BC, and multiplication and reciprocal (division) tables, tables of squares, square roots and

cube roots, geometrical exercises and division problems from around 2600 BC onwards. Later

Babylonian tablets dating from about 1800 to 1600 BC cover topics as varied as fractions,

algebra, methods for solving linear, quadratic and even some cubic equations, and the calculation

of regular reciprocal pairs (pairs of number which multiply together to give 60). One Babylonian

tablet gives an approximation to √2 accurate to an astonishing five decimal places. Others list the

squares of numbers up to 59, the cubes of numbers up to 32 as well as tables of compound

Babylonian Numerals

interest. Yet another gives an estimate for π of 3 1⁄8 (3.125, a reasonable approximation of the

real value of 3.1416).

The idea of square numbers and

quadratic equations (where the

unknown quantity is multiplied

by itself, e.g. x2) naturally arose

in the context of the meaurement

of land, and Babylonian

mathematical tablets give us the

first ever evidence of the solution

of quadratic equations. The

Babylonian approach to solving

them usually revolved around a

kind of geometric game of

slicing up and rearranging

shapes, although the use of

algebra and quadratic equations

also appears. At least some of the

examples we have appear to

indicate problem-solving for its own sake rather than in order to resolve a concrete practical

problem.

The Babylonians used geometric shapes in their buildings and design and in dice for the leisure

games which were so popular in their society, such as the ancient game of backgammon. Their

geometry extended to the calculation of the areas of rectangles, triangles and trapezoids, as well

as the volumes of simple shapes such as bricks and cylinders (although not pyramids).

The famous and controversial Plimpton 322 clay tablet, believed to date from around 1800 BC,

suggests that the Babylonians may well have known the secret of right-angled triangles (that the

square of the hypotenuse equals the sum of the square of the other two sides) many centuries

before the Greek Pythagoras. The tablet appears to list 15 perfect Pythagorean triangles with

whole number sides, although some claim that they were merely academic exercises, and not

deliberate manifestations of Pythagorean triples.

Babylonian Clay tablets from c. 2100 BC showing a problem

concerning the area of an irregular shape

EGYPTIAN MATHEMATICS

The early Egyptians settled along

the fertile Nile valley as early as

about 6000 BC, and they began

to record the patterns of lunar

phases and the seasons, both for

agricultural and religious

reasons. The Pharaoh’s surveyors

used measurements based on

body parts (a palm was the width

of the hand, a cubit the

measurement from elbow to

fingertips) to measure land and

buildings very early in Egyptian

history, and a decimal numeric system was developed based on our ten fingers. The oldest

mathematical text from ancient Egypt discovered so far, though, is the Moscow Papyrus, which

dates from the Egyptian Middle Kingdom around 2000 - 1800 BC.

It is thought that the Egyptians introduced the earliest fully-developed base 10 numeration

system at least as early as 2700 BC (and probably much early). Written numbers used a stroke

for units, a heel-bone symbol for tens, a coil of rope for hundreds and a lotus plant for thousands,

as well as other hieroglyphic symbols for higher powers of ten up to a million. However, there

was no concept of place value, so larger numbers were rather unwieldy (although a million

required just one character, a million minus one required fifty-four characters).

Ancient Egyptian hieroglyphic numerals

The Rhind Papyrus, dating from

around 1650 BC, is a kind of

instruction manual in arithmetic

and geometry, and it gives us

explicit demonstrations of how

multiplication and division was

carried out at that time. It also

contains evidence of other

mathematical knowledge,

including unit fractions,

composite and prime numbers,

arithmetic, geometric and

harmonic means, and how to

solve first order linear equations

as well as arithmetic and

geometric series. The Berlin

Papyrus, which dates from

around 1300 BC, shows that

ancient Egyptians could solve

second-order algebraic

(quadratic) equations.

Multiplication, for example, was

achieved by a process of repeated

doubling of the number to be

multiplied on one side and of one

on the other, essentially a kind of

multiplication of binary factors similar to that used by modern computers (see the example at

right). These corresponding blocks of counters could then be used as a kind of multiplication

reference table: first, the combination of powers of two which add up to the number to be

multiplied by was isolated, and then the corresponding blocks of counters on the other side

yielded the answer. This effectively made use of the concept of binary numbers, over 3,000 years

before Leibniz introduced it into the west, and many more years before the development of the

computer was to fully explore its potential.

Practical problems of trade and the market led to the development of a notation for fractions. The

papyri which have come down to us demonstrate the use of unit fractions based on the symbol of

the Eye of Horus, where each part of the eye represented a different fraction, each half of the

previous one (i.e. half, quarter, eighth, sixteenth, thirty-second, sixty-fourth), so that the total was

one-sixty-fourth short of a whole, the first known example of a geometric series.

Ancient Egyptian method of multiplication

Unit fractions could also be used

for simple division sums. For

example, if they needed to divide

3 loaves among 5 people, they

would first divide two of the

loaves into thirds and the third

loaf into fifths, then they would

divide the left over third from the

second loaf into five pieces.

Thus, each person would receive

one-third plus one-fifth plus one-

fifteenth (which totals three-

fifths, as we would expect).

The Egyptians approximated the

area of a circle by using shapes

whose area they did know. They

observed that the area of a circle

of diameter 9 units, for example,

was very close to the area of a

square with sides of 8 units, so

that the area of circles of other

diameters could be obtained by

multiplying the diameter

by 8⁄9 and then squaring it. This

gives an effective approximation

of π accurate to within less than

one percent.

The pyramids themselves are

another indication of the

sophistication of Egyptian mathematics. Setting aside claims that the pyramids are first known

structures to observe the golden ratio of 1 : 1.618 (which may have occurred for purely aesthetic,

and not mathematical, reasons), there is certainly evidence that they knew the formula for the

volume of a pyramid -1⁄3 times the height times the length times the width - as well as of a

truncated or clipped pyramid. They were also aware, long before Pythagoras, of the rule that a

triangle with sides 3, 4 and 5 units yields a perfect right angle, and Egyptian builders used ropes

knotted at intervals of 3, 4 and 5 units in order to ensure exact right angles for their stonework (in

fact, the 3-4-5 right triangle is often called "Egyptian").

Ancient Egyptian method of division

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 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.

Mayan numerals

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 inMesopotamia, 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

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 untilCarl 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

Early Chinese method of solving equations

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.

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.

The Chinese Remainder Theorem

MEDIEVAL MATHEMATICS - FIBONACCI

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 Romannumerals. However, common sense eventually prevailed and the new system

was adopted throughout Europe by the 15th century, making theRoman 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).

Leonardo of Pisa (Fibonacci)

(c.1170-1250)

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.

The discovery of the famous Fibonacci sequence

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 φ

2 = 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 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.

The Golden Ratio φ can be derived from the Fibonacci

Sequence

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 asBrahmagupta’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 + 4

2)(2

2 + 7

2) =

262 + 15

2 = 30

2 + 1

2.

Fibonacci introduced lattice multiplication to Europe

16TH CENTURY

MATHEMATICS

The cultural, intellectual and

artistic movement of the

Renaissance, which saw a

resurgence of learning based on

classical sources, began in Italy

around the 14th Century, and

gradually spread across most of

Europe over the next two

centuries. Science and art were

still very much interconnected

and intermingled at this time, as

exemplified by the work of

artist/scientists such as Leonardo

da Vinci, and it is no surprise

that, just as in art, revolutionary

work in the fields of philosophy

and science was soon taking

place.

It is a tribute to the respect in

which mathematics was held in

Renaissance Europe that the

famed German artist Albrecht

Dьrer included an order-4 magic

square in his engraving

"Melencolia I". In fact, it is a so-

called "supermagic square" with

many more lines of addition

symmetry than a regular 4 x 4

magic square (see image at

right). The year of the work,

1514, is shown in the two bottom

central squares.

An important figure in the

late15th and early 16th Centuries is an Italian Franciscan friar called Luca Pacioli, who

published a book on arithmetic, geometry and book-keeping at the end of the 15th Century which

became quite popular for the mathematical puzzles it contained. It also introduced symbols for

plus and minus for the first time in a printed book (although this is also sometimes attributed to

Giel Vander Hoecke, Johannes Widmann and others), symbols that were to become standard

notation. Pacioli also investigated the Golden Ratio of 1 : 1.618... (see the section on Fibonacci)

in his 1509 book "The Divine Proportion", concluding that the number was a message from God

and a source of secret knowledge about the inner beauty of things.

The supermagic square shown in Albrecht Dьrer's engraving

"Melencolia I"

During the 16th and early 17th

Century, the equals,

multiplication, division, radical

(root), decimal and inequality

symbols were gradually

introduced and standardized. The

use of decimal fractions and

decimal arithmetic is usually

attributed to the Flemish

mathematician Simon Stevin the

late 16th Century, although the

decimal point notation was not

popularized until early in

the 17th Century. Stevin was

ahead of his time in enjoining

that all types of numbers,

whether fractions, negatives, real

numbers or surds (such as √2)

should be treated equally as

numbers in their own right.

In the Renaissance Italy of the

early 16th Century, Bologna

University in particular was

famed for its intense public

mathematics competitions. It was

in just such a competion that the

unlikely figure of the young, self-

taughtNiccolт Fontana

Tartaglia revealed to the world

the formula for solving first one type, and later all types, of cubic equations (equations with

terms including x3), an achievement hitherto considered impossible and which had stumped the

best mathematicians ofChina, India and the Islamic world.

Building on Tartaglia’s work, another young Italian, Lodovico Ferrari, soon devised a similar

method to solve quartic equations (equations with terms including x4) and both solutions were

published by Gerolamo Cardano. Despite a decade-long fight over the publication, Tartaglia,

Cardano and Ferrari between them demonstrated the first uses of what are now known as

complex numbers, combinations of real and imaginary numbers (although it fell to another

Bologna resident, Rafael Bombelli, to explain what imaginary numbers really were and how they

could be used). Tartaglia went on to produce other important (although largely ignored) formulas

and methods, and Cardano published perhaps the first systematic treatment of probability.

With Hindu-Arabic numerals, standardized notation and the new language of algebra at their

disposal, the stage was set for the European mathematical revolution of the 17th Century.

Basic mathematical notation, with dates of first use

17TH CENTURY

MATHEMATICS

In the wake of the Renaissance,

the 17th Century saw an

unprecedented explosion of

mathematical and scientific ideas

across Europe, a period

sometimes called the Age of

Reason. Hard on the heels of the

“Copernican Revolution” of

Nicolaus Copernicus in the 16th

Century, scientists like Galileo

Galilei, Tycho Brahe and

Johannes Kepler were making

equally revolutionary discoveries

in the exploration of the Solar

system, leading to Kepler’s

formulation of mathematical

laws of planetary motion.

The invention of the logarithm in

the early 17th Century by John

Napier (and later improved by

Napier and Henry Briggs)

contributed to the advance of

science, astronomy and

mathematics by making some

difficult calculations relatively

easy. It was one of the most significant mathematical developments of the age, and 17th Century

physicists like Kepler and Newton could never have performed the complex calculatons needed

for their innovations without it. The French astronomer and mathematician Pierre Simon Laplace

remarked, almost two centuries later, that Napier, by halving the labours of astronomers, had

doubled their lifetimes.

The logarithm of a number is the exponent when that number is expressed as a power of 10 (or

any other base). It is effectively the inverse of exponentiation. For example, the base 10

logarithm of 100 (usually written log10 100 or lg 100 or just log 100) is 2, because 102 = 100.

The value of logarithms arises from the fact that multiplication of two or more numbers is

equivalent to adding their logarithms, a much simpler operation. In the same way, division

involves the subtraction of logarithms, squaring is as simple as multiplying the logarithm by two

(or by three for cubing, etc), square roots requires dividing the logarithm by 2 (or by 3 for cube

roots, etc).

Although base 10 is the most popular base, another common base for logarithms is the

number e which has a value of 2.7182818... and which has special properties which make it very

Logarithms were invented by John Napier, early in the 17th

Century

useful for logarithmic calculations. These are known as natural logarithms, and are written

loge or ln. Briggs produced extensive lookup tables of common (base 10) logarithms, and by

1622 William Oughted had produced a logarithmic slide rule, an instrument which became

indispensible in technological innovation for the next 300 years.

Napier also improved Simon Stevin's decimal notation and popularized the use of the decimal

point, and made lattice multiplication (originally developed by the Persian mathematician Al-

Khwarizmi and introduced into Europe by Fibonacci) more convenient with the introduction of

“Napier's Bones”, a

multiplication tool using a set of

numbered rods.

Although not principally a

mathematician, the role of the

Frenchman Marin Mersenne as a

sort of clearing house and go-

between for mathematical

thought in France during this

period was crucial. Mersenne is

largely remembered in

mathematics today in the term

Mersenne primes - prime

numbers that are one less than a

power of 2, e.g. 3 (22-1), 7 (2

3-1),

31 (25-1), 127 (2

7-1), 8191 (2

13-

1), etc. In modern times, the

largest known prime number has

almost always been a Mersenne

prime, but in actual fact,

Mersenne’s real connection with

the numbers was only to compile

a none-too-accurate list of the

smaller ones (when Edouard

Lucas devised a method of checking them in the 19th Century, he pointed out that Mersenne had

incorrectly included 267

-1 and left out 261

-1, 289

-1 and 2107

-1 from his list).

The Frenchman Renй Descartes is sometimes considered the first of the modern school of

mathematics. His development of analytic geometry and Cartesian coordinates in the mid-17th

Century soon allowed the orbits of the planets to be plotted on a graph, as well as laying the

foundations for the later development of calculus (and much later multi-dimensional geometry).

Descartes is also credited with the first use of superscripts for powers or exponents.

Two other great French mathematicians were close contemporaries of Descartes: Pierre de

Fermat andBlaise Pascal. Fermat formulated several theorems which greatly extended our

knowlege of number theory, as well as contributing some early work on infinitesimal

Graph of the number of digits in the known Mersenne primes

calculus. Pascal is most famous for Pascal’s Triangle of binomial coefficients, although similar

figures had actually been produced by Chinese and Persian mathematicians long before him.

It was an ongoing exchange of letters between Fermat and Pascal that led to the development of

the concept of expected values and the field of probability theory. The first published work on

probability theory, however, and the first to outline the concept of mathematical expectation, was

by the Dutchman Christiaan Huygens in 1657, although it was largely based on the ideas in the

letters of the two Frenchmen.

The French mathematician and

engineer Girard Desargues is

considered one of the founders of

the field of projective geometry,

later developed further by Jean

Victor Poncelet and Gaspard

Monge. Projective geometry

considers what happens to shapes

when they are projected on to a

non-parallel plane. For example,

a circle may be projected into an

ellipse or a hyperbola, and so

these curves may all be regarded

as equivalent in projective

geometry. In particular,

Desargues developed the pivotal

concept of the “point at infinity”

where parallels actually meet.

His perspective theorem states

that, when two triangles are in

perspective, their corresponding

sides meet at points on the same

collinear line.

By “standing on the shoulders of giants”, the Englishman Sir Isaac Newton was able to pin down

the laws of physics in an unprecedented way, and he effectively laid the groundwork for all of

classical mechanics, almost single-handedly. But his contribution to mathematics should never

be underestimated, and nowadays he is often considered, along with Archimedes and Gauss, as

one of the greatest mathematicians of all time.

Newton and, independently, the German philosopher and mathematician Gottfried Leibniz,

completely revolutionized mathematics (not to mention physics, engineering, economics and

science in general) by the development of infinitesimal calculus, with its two main operations,

differentiation and integration. Newtonprobably developed his work before Leibniz,

but Leibniz published his first, leading to an extended and rancorous dispute. Whatever the truth

behind the various claims, though, it is Leibniz’s calculus notation that is the one still in use

Desargues’ perspective theorem

today, and calculus of some sort is used extensively in everything from engineering to economics

to medicine to astronomy.

Both Newton and Leibniz also contributed greatly in other areas of mathematics,

including Newton’s contributions to a generalized binomial theorem, the theory of finite

differences and the use of infinite power series, and Leibniz’s development of a mechanical

forerunner to the computer and the use of matrices to solve linear equations.

However, credit should also be given to some earlier 17th Century mathematicians whose work

partially anticipated, and to some extent paved the way for, the development of infinitesimal

calculus. As early as the 1630s, the Italian mathematician Bonaventura Cavalieri developed a

geometrical approach to calculus known as Cavalieri's principle, or the “method of indivisibles”.

The Englishman John Wallis, who systematized and extended the methods of analysis

of Descartes and Cavalieri, also made significant contributions towards the development of

calculus, as well as originating the idea of the number line, introducing the symbol ∞ for infinity

and the term “continued fraction”, and extending the standard notation for powers to include

negative integers and rational numbers. Newton's teacher Isaac Barrow is usually credited with

the discovery (or at least the first rigorous statrement of) the fundamental theorem of calculus,

which essentially showed that integration and differentiation are inverse operations, and he also

made complete translations of Euclid into Latin and English.

18TH CENTURY

MATHEMATICS

Calculus of variations

Most of the late 17th Century and

a good part of the early 18th

were taken up by the work of disciples of Newtonand Leibniz, who applied their ideas on

calculus to solving a variety of problems in physics, astronomy and engineering.

The period was dominated, though, by one family, the Bernoulli’s of Basel in Switzerland, which

boasted two or three generations of exceptional mathematicians, particularly the brothers, Jacob

and Johann. They were largely responsible for further developing Leibniz’s infinitesimal

calculus - paricularly through the generalization and extension of calculus known as the

"calculus of variations" - as well as Pascal andFermat’s probability and number theory.

Basel was also the home town of the greatest of the 18th Century mathematicians, Leonhard

Euler, although, partly due to the difficulties in getting on in a city dominated by

theBernoulli family, Euler spent most of his time abroad, in Germany and St. Petersburg, Russia.

He excelled in all aspects of mathematics, from geometry to calculus to trigonometry to algebra

to number theory, and was able to find unexpected links between the different fields. He proved

numerous theorems, pioneered new methods, standardized mathematical notation and wrote

many influential textbooks throughout his long academic life.

In a letter to Euler in 1742, the German mathematician Christian Goldbach proposed the

Goldbach Conjecture, which states that every even integer greater than 2 can be expressed as the

sum of two primes (e.g. 4 = 2 + 2; 8 = 3 + 5; 14 = 3 + 11 = 7 + 7; etc) or, in another equivalent

version, every integer greater than 5 can be expressed as the sum of three primes. Yet another

version is the so-called “weak” Goldbach Conjecture, that all odd numbers greater than 7 are the

sum of three odd primes. They remain among the oldest unsolved problems in number theory

(and in all of mathematics), although the weak form of the conjecture appears to be closer to

resolution than the strong one. Goldbach also proved other theorems in number theory such as

the Goldbach-Euler Theorem on perfect powers.

Despite Euler’s and the Bernoullis’ dominance of 18th Century mathematics, many of the other

important mathematicians were from France. In the early part of the century, Abraham de

Moivre is perhaps best known for de Moivre's formula, (cosx + isinx)n = cos(nx) + isin(nx),

which links complex numbers and trigonometry. But he also generalized Newton’s famous

binomial theorem into the multinomial theorem, pioneered the development of analytic

geometry, and his work on the normal distribution (he gave the first statement of the formula for

the normal distribution curve) and probability theory were of great importance.

France became even more prominent towards the end of the century, and a handful of late 18th

Century French mathematicians in particular deserve mention at this point, beginning with “the

three L’s”.

Joseph Louis Lagrange collaborated with Euler in an important joint work on the calculus of

variation, but he also contributed to differential equations and number theory, and he is usually

credited with originating the theory of groups, which would become so important

in 19th and 20th Century mathematics. His name is given an early theorem in group theory,

which states that the number of elements of every sub-group of a finite group divides evenly into

the number of elements of the

original finite group.

Lagrange is also credited with

the four-square theorem, that any

natural number can be

represented as the sum of four

squares (e.g. 3 = 12 + 1

2 + 1

2 +

02; 31 = 5

2 + 2

2 + 1

2 + 1

2; 310 =

172 + 4

2 + 2

2 + 1

2; etc), as well as

another theorem, confusingly

also known as Lagrange’s

Theorem or Lagrange’s Mean

Value Theorem, which states

that, given a section of a smooth

continuous (differentiable) curve,

there is at least one point on that

section at which the derivative

(or slope) of the curve is equal

(or parallel) to the average (or

mean) derivative of the section.

Lagrange’s 1788 treatise on

analytical mechanics offered the most comprehensive treatment of classical mechanics

since Newton, and formed a basis for the development of mathematical physics in the 19th

Century.

Pierre-Simon Laplace, sometimes referred to as “the French Newton”, was an important

mathematician and astronomer, whose monumental work “Celestial Mechanics” translated the

geometric study of classical mechanics to one based on calculus, opening up a much broader

range of problems. Although his early work was mainly on differential equations and finite

differences, he was already starting to think about the mathematical and philosophical concepts

of probability and statistics in the 1770s, and he developed his own version of the so-called

Bayesian interpretation of probability independently of Thomas Bayes. Laplace is well known

for his belief in complete scientific determinism, and he maintained that there should be a set of

scientific laws that would allow us - at least in principle - to predict everything about the

universe and how it works.

Lagrange’s Mean value Theorem

Adrien-Marie Legendre also

made important contributions to

statistics, number theory, abstract

algebra and mathematical

analysis in the late 18th and early

19th Centuries, athough much of

his work (such as the least

squares method for curve-fitting

and linear regression, the

quadratic reciprocity law, the

prime number theorem and his

work on elliptic functions) was

only brought to perfection - or at

least to general notice - by

others, particularlyGauss. His

“Elements of Geometry”, a re-

working of Euclid’s book,

became the leading geometry

textbook for almost 100 years,

and his extremely accurate

measurement of the terrestrial

meridian inspired the creation,

and almost universal adoption, of the metric system of measures and weights.

Yet another Frenchman, Gaspard Monge was the inventor of descriptive geometry, a clever

method of representing three-dimensional objects by projections on the two-dimensional plane

using a specific set of procedures, a technique which would later become important in the fields

of engineering, architecture and design. His orthographic projection became the graphical

method used in almost all modern mechanical drawing.

After many centuries of increasingly accurate approximations, Johann Lambert, a Swiss

mathematician and prominent astronomer, finally provided a rigorous proof in 1761 that π is

irrational, i.e. it can not be expressed as a simple fraction using integers only or as a terminating

or repeating decimal. This definitively proved that it would never be possible to calculate it

exactly, although the obsession with obtaining more and more accurate approximations continues

to this day. (Over a hundred years later, in 1882, Ferdinand von Lindemann would prove that π is

also transcendental, i.e. it cannot be the root of any polynomial equation with rational

coefficients). Lambert was also the first to introduce hyperbolic functions into trigonometry and

made some prescient conjectures regarding non-Euclidean space and the properties of hyperbolic

triangles.

The first six Legendre polynomials (solutions to Legendre’s

differential equation)

19TH CENTURY

MATHEMATICS

The 19th Century saw an

unprecedented increase in the

breadth and complexity of

mathematical concepts. Both

France and Germany were caught

up in the age of revolution which

swept Europe in the late 18th

Century, but the two countries

treated mathematics quite

differently.

After the French Revolution,

Napoleon emphasized the

practical usefulness of

mathematics and his reforms and

military ambitions gave French

mathematics a big boost, as

exemplified by “the three L’s”,

Lagrange, Laplace and Legendre

(see the section on 18th Century

Mathematics), Fourier and

Galois.

Joseph Fourier's study, at the

beginning of the 19th Century, of

infinite sums in which the terms are trigonometric functions were another important advance in

mathematical analysis. Periodic functions that can be expressed as the sum of an infinite series of

sines and cosines are known today as Fourier Series, and they are still powerful tools in pure and

applied mathematics. Fourier (following Leibniz,Euler, Lagrange and others) also contributed

towards defining exactly what is meant by a function, although the definition that is found in

texts today - defining it in terms of a correspondence between elements of the domain and the

range - is usually attributed to the 19th Century German mathematician Peter Dirichlet.

In 1806, Jean-Robert Argand published his paper on how complex numbers (of the form a + bi,

where i is √-1) could be represented on geometric diagrams and manipulated using trigonometry

and vectors. Even though the Dane Caspar Wessel had produced a very similar paper at the end

of the 18th Century, and even though it was Gauss who popularized the practice, they are still

known today as Argand Diagrams.

The Frenchman Йvariste Galois proved in the late 1820s that there is no general algebraic

method for solving polynomial equations of any degree greater than four, going further than the

Norwegian Niels Henrik Abel who had, just a few years earlier, shown the impossibility of

solving quintic equations, and breaching an impasse which had existed for centuries. Galois'

Approximation of a periodic function by the Fourier Series

work also laid the groundwork for further developments such as the beginnings of the field of

abstract algebra, including areas like algebraic geometry, group theory, rings, fields, modules,

vector spaces and non-commutative algebra.

Germany, on the other hand, under the influence of the great educationalist Wilhelm von

Humboldt, took a rather different approach, supporting pure mathematics for its own sake,

detached from the demands of the state and military. It was in this environment that the young

German prodigy Carl Friedrich Gauss, sometimes called the “Prince of Mathematics”, received

his education at the prestigious University of Gцttingen. Some ofGauss’ ideas were a hundred

years ahead of their time, and touched on many different parts of the mathematical world,

including geometry, number theory, calculus, algebra and probability. He is widely regarded as

one of the three greatest

mathematicians of all times,

along

with Archimedes and Newton.

Later in life, Gauss also claimed

to have investigated a kind of

non-Euclidean geometry using

curved space but, unwilling to

court controversy, he decided not

to pursue or publish any of these

avant-garde ideas. This left the

field open for Jбnos

Bolyai and Nikolai

Lobachevsky(respectively, a Hungarian and a Russian) who both independently explored the

potential of hyperbolic geometry and curved spaces.

The German Bernhard Riemann worked on a different kind of non-Euclidean geometry called

elliptic geometry, as well as on a generalized theory of all the different types of

geometry. Riemann, however, soon took this even further, breaking away completely from all

the limitations of 2 and 3 dimensional geometry, whether flat or curved, and began to think in

higher dimensions. His exploration of the zeta function in multi-dimensional complex numbers

revealed an unexpected link with the distribution of prime numbers, and his famous Riemann

Hypothesis, still unproven after 150 years, remains one of the world’s great unsolved

mathematical mysteries and the testing ground for new generations of mathematicians.

British mathematics also saw something of a resurgence in the early and mid-19th century.

Although the roots of the computer go back to the geared calculators of Pascal and Leibniz in the

17th Century, it was Charles Babbage in 19th Century England who designed a machine that

could automatically perform computations based on a program of instructions stored on cards or

tape. His large "difference engine" of 1823 was able to calculate logarithms and trigonometric

functions, and was the true forerunner of the modern electronic computer. Although never

actually built in his lifetime, a machine was built almost 200 years later to his specifications and

worked perfectly. He also designed a much more sophisticated machine he called the "analytic

Euclidean, hyperbolic and elliptic geometry

engine", complete with punched cards, printer and computational abilities commensurate with

modern computers.

Another 19th Century Englishman, George Peacock, is usually credited with the invention of

symbolic algebra, and the extension of the scope of algebra beyond the ordinary systems of

numbers. This recognition of the possible existence of non-arithmetical algebras was an

important stepping stone toward future developments in abstract algebra.

In the mid-19th Century, the British mathematician George Boole devised an algebra (now called

Boolean algebra or Boolean logic), in which the only operators were AND, OR and NOT, and

which could be applied to the solution of logical problems and mathematical functions. He also

described a kind of binary system which used just two objects, "on" and "off" (or "true" and

"false", 0 and 1, etc), in which, famously, 1 + 1 = 1. Boolean algebra was the starting point of

modern mathematical logic and

ultimately led to the development

of computer science.

The concept of number and

algebra was further extended by

the Irish mathematician William

Hamilton, whose 1843 theory of

quaternions (a 4-dimensional

number system, where a quantity

representing a 3-dimensional

rotation can be described by just

an angle and a vector).

Quaternions, and its later

generalization by Hermann

Grassmann, provided the first

example of a non-commutative

algebra (i.e. one in

which a x b does not always

equal bx a), and showed that

several different consistent

algebras may be derived by

choosing different sets of

axioms.

The Englishman Arthur Cayley extended Hamilton's quaternions and developed the octonions.

But Cayley was one of the most prolific mathematicians in history, and was a pioneer of modern

group theory, matrix algebra, the theory of higher singularities, and higher dimensional geometry

(anticipating the later ideas of Klein), as well as the theory of invariants.

Throughout the 19th Century, mathematics in general became ever more complex and abstract.

But it also saw a re-visiting of some older methods and an emphasis on mathematical rigour. In

the first decades of the century, the Bohemian priest Bernhard Bolzano was one of the earliest

Hamilton’s quaternion

mathematicians to begin instilling rigour into mathematical analysis, as well as giving the first

purely analytic proof of both the fundamental theorem of algebra and the intermediate value

theorem, and early consideration of sets (collections of objects defined by a common property,

such as "all the numbers greater than 7" or "all right triangles", etc). When the German

mathematician Karl Weierstrass discovered the theoretical existence of a continuous function

having no derivative (in other words, a continuous curve possessing no tangent at any of its

points), he saw the need for a rigorous “arithmetization” of calculus, from which all the basic

concepts of analysis could be derived.

Along with Riemann and, particularly, the Frenchman Augustin-Louis Cauchy, Weierstrass

completely reformulated calculus in an even more rigorous fashion, leading to the development

of mathematical analysis, a branch of pure mathematics largely concerned with the notion of

limits (whether it be the limit of a sequence or the limit of a function) and with the theories of

differentiation, integration, infinite series and analytic functions. In 1845, Cauchy also proved

Cauchy's theorem, a fundamental theorem of group theory, which he discovered while

examining permutation groups. Carl Jacobi also made important contributions to analysis,

determinants and matrices, and especially his theory of periodic functions and elliptic functions

and their relation to the elliptic

theta function.

August Ferdinand Mцbius is best

known for his 1858 discovery of

the Mцbius strip, a non-

orientable two-dimensional

surface which has only one side

when embedded in three-

dimensional Euclidean space

(actually a German, Johann

Benedict Listing, devised the

same object just a couple of

months before Mцbius, but it has

come to hold Mцbius' name).

Many other concepts are also

named after him, including the

Mцbius configuration, Mцbius

transformations, the Mцbius

transform of number theory, the

Mцbius function and the Mцbius

inversion formula. He also

introduced homogeneous

coordinates and discussed

geometric and projective

transformations.

Felix Klein also pursued more

developments in non-Euclidean

Non-orientable surfaces with no identifiable "inner" and

"outer" sides

geometry, include the Klein bottle, a one-sided closed surface which cannot be embedded in

three-dimensional Euclidean space, only in four or more dimensions. It can be best visualized as

a cylinder looped back through itself to join with its other end from the "inside". Klein’s 1872

Erlangen Program, which classified geometries by their underlying symmetry groups (or their

groups of transformations), was a hugely influential synthesis of much of the mathematics of the

day, and his work was very important in the later development of group theory and function

theory.

The Norwegian mathematician Marius Sophus Lie also applied algebra to the study of geometry.

He largely created the theory of continuous symmetry, and applied it to the geometric theory of

differential equations by means of continuous groups of transformations known as Lie groups.

In an unusual occurrence in 1866, an unknown 16-year old Italian, Niccolт Paganini, discovered

the second smallest pair of amicable numbers (1,184 and 1210), which had been completely

overlooked by some of the greatest mathematicians in history (including Euler, who had

identified over 60 such numbers in the 18th Century, some of them huge).

In the later 19th Century, Georg Cantor established the first foundations of set theory, which

enabled the rigorous treatment of the notion of infinity, and which has since become the common

language of nearly all mathematics. In the face of fierce resistance from most of his

contemporaries and his own battle against mental illness, Cantor explored new mathematical

worlds where there were many

different infinities, some of

which were larger than others.

Cantor’s work on set theory was

extended by another German,

Richard Dedekind, who defined

concepts such as similar sets and

infinite sets. Dedekind also came

up with the notion, now called a

Dedekind cut which is now a

standard definition of the real

numbers. He showed that any

irrational number divides the

rational numbers into two classes

or sets, the upper class being

strictly greater than all the

members of the other lower class. Thus, every location on the number line continuum contains

either a rational or an irrational number, with no empty locations, gaps or discontinuities. In

1881, the Englishman John Venn introduced his “Venn diagrams” which become useful and

ubiquitous tools in set theory.

Building on Riemann’s deep ideas on the distribution of prime numbers, the year 1896 saw two

independent proofs of the asymptotic law of the distribution of prime numbers (known as the

Prime Number Theorem), one by Jacques Hadamard and one by Charles de la Vallйe Poussin,

Venn diagram

which showed that the number of primes occurring up to any number x is asymptotic to (or tends

towards) x⁄log x.

Hermann Minkowski, a great

friend ofDavid Hilbert and

teacher of the young Albert

Einstein, developed a branch of

number theory called the

"geometry of numbers" late in

the 19th Century as a geometrical

method in multi-dimensional

space for solving number theory

problems, involving complex

concepts such as convex sets,

lattice points and vector space.

Later, in 1907, it was Minkowski

who realized that the Einstein’s

1905 special theory of relativity

could be best understood in a

four-dimensional space, often

referred to as Minkowski space-

time.

Gottlob Frege’s 1879

“Begriffsschrift” (roughly

translated as “Concept-Script”)

broke new ground in the field of logic, including a rigorous treatment of the ideas of functions

and variables. In his attempt to show that mathematics grows out of logic, he devised techniques

that took him far beyond the logical traditions of Aristotle (and even of George Boole). He was

the first to explicitly introduce the notion of variables in logical statements, as well as the notions

of quantifiers, universals and existentials. He extended Boole's "propositional logic" into a new

"predicate logic" and, in so doing, set the stage for the radical advances of Giuseppe

Peano, Bertrand Russell and David Hilbert in the early 20th Century.Henri Poincarй came to

prominence in the latter part of the 19th Century with at least a partial solution to the “three body

problem”, a deceptively simple problem which had stubbornly resisted resolution since the time

ofNewton, over two hundred years earlier. Although his solution actually proved to be erroneous,

its implications led to the early intimations of what would later become known as chaos theory.

In between his important work in theoretical physics, he also greatly extended the theory of

mathematical topology, leaving behind a knotty problem known as the Poincarй conjecture

which remined unsolved until 2002.Poincarй was also an engineer and a polymath, and perhaps

the last of the great mathematicians to adhere to an older conception of mathematics, which

championed a faith in human intuition over rigour and formalism. He is sometimes referred to as

the “Last Univeralist” as he was perhaps the last mathematician able to shine in almost all of the

various aspects of what had become by now a huge, encyclopedic and incredibly complex

subject. The 20th Century would belong to the specialists.

Minkowski space-time